Content-Type: multipart/mixed; boundary="-------------0204031251900" This is a multi-part message in MIME format. ---------------0204031251900 Content-Type: text/plain; name="02-166.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-166.keywords" Splitting of separatrices, Hamiltonian systems ---------------0204031251900 Content-Type: application/postscript; name="balfon.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="balfon.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: balfon.dvi %%Pages: 88 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips balfon %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.04.03:1916 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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5 4 bop 1682 251 a Fo(INTR)n(ODUCTION)1194 b(v)605 450 y Fu(P)n(oincar)n(\023)-39 b(e)21 b(found)k(this)f(exp)r(onen)n(tially) g(small)g(e\013ects)g(in)h([)p Fk(P)m(o)p Fu(].)36 b(In)25 b(his)f(study)g(of)h(p)r(erio)r(dic)456 550 y(orbits)h(in)i(t)n(w)n(o)f (degrees)f(of)h(freedom)g(Hamiltonian)h(systems)f(he)g(prop)r(osed)g(a) g(mo)r(del,)h(whic)n(h)456 649 y(after)f(reduction)g(b)r(ecame)h(the)g (follo)n(wing)e(p)r(erturb)r(ed)i(p)r(endulum)1455 790 y(\177)-48 b Fq(y)26 b Fu(=)c(2)p Fq(\026)14 b Fu(sin)f Fq(y)22 b Fu(+)c(2)p Fq(\026")c Fu(cos)e Fq(y)k Fu(cos)d Fq(t)456 935 y Fu(\(using)28 b(the)g(same)g(notation)f(as)h(P)n(oincar) n(\023)-39 b(e\).)35 b(He)29 b(deduced)f(that)g(the)h(splitting)f(of)h (separatri-)456 1035 y(ces)j(is)h(exp)r(onen)n(tial)f(small)h(in)g Fq(\026)p Fu(,)i(pro)n(vided)d(that)h Fq(")g Fu(is)f(less)h(than)g(an)g (exp)r(onen)n(tially)f(small)456 1135 y(quan)n(tit)n(y)-7 b(.)605 1234 y(Arnold)25 b([)p Fk(Ar1)p Fu(])h(found)f(the)h(exp)r 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(get)f(exp)r(onen)n(tially)h(smallness.)34 b(The)24 b(Hamiltonian)g(c)n (haracter)d(of)j(the)456 2031 y(equation)j(is)g(translated)g(to)g(the)h (symplectic)g(c)n(haracter)d(of)j(the)g(maps.)605 2131 y(In)20 b([)p Fk(La1)q Fu(])g(Lazutkin)g(studied)g(the)g(standard)f (map)h Fq(F)12 b Fu(\()p Fq(x;)i(y)s Fu(\))24 b(=)f(\()p Fq(x)s Fu(+)s Fq(y)6 b Fu(+)s Fq(")14 b Fu(sin)f Fq(x;)h(y)6 b Fu(+)s Fq(")14 b Fu(sin)g Fq(x)p Fu(\))456 2230 y(and)30 b(pro)n(vided)f(the)h(follo)n(wing)f(form)n(ula)h(for)f(the)i(angle)e (b)r(et)n(w)n(een)h(the)h(stable)f(and)g(unstable)456 2330 y(manifolds)d(at)g(a)h(homo)r(clinic)f(p)r(oin)n(t)1419 2487 y Fq(')c Fu(=)1594 2431 y Fq(\031)p 1594 2468 51 4 v 1600 2544 a(")1654 2487 y Fp(j)p Fu(\002)1742 2499 y Fo(1)1779 2487 y Fp(j)p Fq(e)1841 2453 y Fj(\000)p Fi(\031)1934 2428 y Fh(2)1966 2453 y Fi(=)2000 2411 y Fj(p)p 2054 2411 32 3 v 2054 2453 a Fi(")2090 2487 y Fu([1)18 b(+)g Fq(O)r Fu(\()p Fq(")2392 2453 y Fi(b)2426 2487 y Fu(\)])-2025 b(\(2\))456 2656 y(with)29 b(0)24 b Fq(<)h(b)g(<)f Fu(1)p Fq(=)p Fu(8.)40 b(This)28 b(w)n(as)g(the)h (\014rst)g(exp)r(onen)n(tially)f(small)g(asymptotic)h(form)n(ula)f(for) g(a)456 2755 y(non)n(trivial)c(problem)h(with)h(only)f(one)g (parameter.)35 b(Although)25 b(the)h(pro)r(of)f(w)n(as)f(not)i (complete)456 2855 y(Lazutkin)k(in)n(tro)r(duced)h(pioneering)f(new)g (analytic)h(to)r(ols)f(for)g(the)h(study)g(of)g(the)g(separatrix)456 2955 y(splitting)d(whic)n(h)f(ha)n(v)n(e)f(decisiv)n(ely)h (in\015uenced)h(the)g(dev)n(elopmen)n(t)f(of)h(this)g(area.)605 3054 y(Sev)n(eral)d(pap)r(ers)h(deal)g(with)h(the)g(computation)f(of)h (the)g(constan)n(t)f(\002)2777 3066 y Fo(1)2840 3054 y Fu([)p Fk(LST)q Fu(])h([)p Fk(Su)p Fu(].)36 b(The)456 3154 y(complete)27 b(pro)r(of)g(of)h(\(2\))f(is)h(in)g([)p Fk(Ge4)o Fu(].)605 3253 y(F)-7 b(on)n(tic)n(h)24 b(and)g(Sim\023)-42 b(o)25 b([)p Fk(FS1)o Fu(])g([)p Fk(FS2)p Fu(])f(study)h(the)f (splitting)h(of)f(separatrices)e(for)i(families)h(of)456 3353 y(di\013eomorphisms)g(in)h(a)f(neigh)n(b)r(orho)r(o)r(d)g(of)g (the)h(iden)n(tit)n(y)g(of)g(class)f Fq(C)2642 3323 y Fi(!)2716 3353 y Fu(and)g Fq(C)2940 3323 y Fi(r)3003 3353 y Fu(resp)r(ectiv)n(ely)-7 b(.)456 3453 y(Under)25 b(fairly)f(general)g(h)n(yp)r(otheses)h(exp)r(onen)n(tially)f(small)h (upp)r(er)h(b)r(ounds)f(are)f(obtained)h(for)456 3552 y(the)c(distance)g(b)r(et)n(w)n(een)f(in)n(v)-5 b(arian)n(t)20 b(manifolds)h(in)g(the)g(analytic)g(case)f(with)h(generally)f(optimal) 456 3652 y(v)-5 b(alues)27 b(of)g(the)h(constan)n(t)f(in)h(the)g(exp)r (onen)n(t.)605 3752 y(Other)f(w)n(orks)f(referring)g(to)i(maps)f(are)f ([)p Fk(Ch)p Fu(])i([)p Fk(DR2)p Fu(])g([)p Fk(DR1)o Fu(])g([)p Fk(Ge5)p Fu(])f([)p Fk(GS)p Fu(].)605 3851 y(Man)n(y)33 b(authors)g(ha)n(v)n(e)f(studied)j(the)f(phenomenon)f(of)h (separatrix)e(splitting)i(with)g(fast)456 3951 y(frequency)c(p)r(erio)r (dic)h(p)r(erturbations,)g(in)h(order)e(to)h(pro)n(v)n(e)e(that)i(in)h (certain)e(cases)g(the)i(Mel-)456 4050 y(nik)n(o)n(v)26 b(function)i(yields)g(the)g(righ)n(t)e(asymptotics)h(of)h(the)g (measure)e(of)i(separatrix)e(splitting.)605 4150 y(They)i(consider)1172 4291 y(_)-37 b Fq(z)26 b Fu(=)d Fq(f)9 b Fu(\()p Fq(z)t Fu(\))18 b(+)g Fq(\026")1658 4256 y Fi(p)1696 4291 y Fq(g)s Fu(\()p Fq(x;)c(t=";)g(\026)p Fu(\))p Fq(;)180 b(z)26 b Fp(2)d Fq(U)32 b Fp(\032)23 b Fl(R)2662 4256 y Fo(2)2705 4291 y Fq(;)456 4431 y Fu(where)f Fq(\026)g Fu(and)h Fq(")f(>)h Fu(0)f(are)g(parameters)f(a)h(priori)f(indep)r (enden)n(t)j(and)e(suc)n(h)g(that)h(the)g(origin)e(is)i(a)456 4531 y(saddle-t)n(yp)r(e)h(\014xed)h(p)r(oin)n(t.)36 b(There)24 b(has)g(b)r(een)h(a)g(lot)f(of)h(discussion)f(ab)r(out)h (the)g(optimal)g(v)-5 b(alue)456 4631 y(of)26 b Fq(p)h Fu(for)f(whic)n(h)h(one)f(gets)h(exp)r(onen)n(tially)f(small)g(upp)r (er)h(b)r(ounds)g(or)f(asymptotics.)36 b(In)27 b([)p Fk(F)-8 b(o2)o Fu(])456 4730 y(upp)r(er)30 b(b)r(ounds)h(for)f(the)h (splitting)g(are)e(giv)n(en)h(ev)n(en)g(for)g(negativ)n(e)f(v)-5 b(alues)30 b(of)h Fq(p)p Fu(,)g(sp)r(eci\014cally)456 4830 y Fq(p)22 b(>)h Fp(\000)p Fu(1)p Fq(=)p Fu(2.)35 b(If)28 b(the)g(mo)r(del)g(is)f(simpli\014ed,)i(considering)d (equations)h(of)g(the)h(form)1136 4970 y(\177)-47 b Fq(x)18 b Fu(+)g Fq(f)9 b Fu(\()p Fq(x)p Fu(\))24 b(=)f Fq(\026")1641 4936 y Fi(p)1679 4970 y Fq(g)s Fu(\()p Fq(x;)14 b(t=";)g(";)g(\026)p Fu(\))p Fq(;)179 b(x)24 b Fp(2)f Fq(V)42 b Fp(\032)23 b Fl(R)p Fq(;)-2294 b Fu(\(3\))456 5116 y(then)36 b(in)g([)p Fk(F)-8 b(o1)p Fu(])36 b(upp)r(er)g(b)r(ounds)g(are)e(giv)n(en)h(for)h (the)g(splitting)g(of)g(separatrices)e(for)h(v)-5 b(alues)456 5216 y(of)31 b Fq(p)f(>)f Fp(\000)p Fu(2.)49 b(In)31 b([)p Fk(HMS)p Fu(],)i(Holmes)e(et)h(al)f(are)g(able)g(to)h(giv)n(e)f (upp)r(er)h(and)f(lo)n(w)n(er)f(b)r(ounds)i(for)p eop %%Page: 6 6 6 5 bop 456 251 a Fo(vi)941 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(the)28 b(splitting)h(of)f(separatrices)e(for)i(quite)g(general)f (systems)h(and)g(for)f(v)-5 b(alues)28 b(of)h Fq(p)23 b(>)h Fu(8.)38 b(The)456 550 y(situation)28 b(impro)n(v)n(es)f(when)h (dealing)g(with)h(sp)r(eci\014c)f(systems.)39 b(The)29 b(most)f(studied)h(example)456 649 y(is)f(the)g(p)r(endulum.)40 b(In)29 b([)p Fk(Ge1)o Fu(])f(and)h(in)f([)p Fk(DS1)p Fu(],)h(asymptotic)e(expressions)g(are)g(giv)n(en)g(for)h(the)456 749 y(separatrix)d(splitting)j(of)g(the)g(equation)1560 898 y(\177)-47 b Fq(x)19 b Fu(+)f(sin)c Fq(x)23 b Fu(=)g Fq(\026")2067 864 y Fi(p)2119 898 y Fu(sin)14 b Fq(t=")456 1052 y Fu(for)30 b Fq(p)f(>)g Fu(5)i(and)g Fq(p)e(>)f Fu(0)j(resp)r(ectiv)n(ely)-7 b(.)47 b(See)31 b(also)f(T)-7 b(reshev)31 b([)p Fk(T)-8 b(r)p Fu(].)48 b(Later)31 b(on,)h(Delshams)f (and)456 1152 y(Seara)e(in)i([)p Fk(DS2)p Fu(],)g(could)g(get)f(an)h (asymptotic)f(expression)f(of)h(the)h(separatrix)e(splitting)i(for)456 1252 y(more)25 b(general)g(systems)h(giv)n(en)g(that)h Fq(p)f Fu(is)g(bigger)f(than)i(a)f(certain)g(quan)n(tit)n(y)g(whic)n(h) g(dep)r(ends)456 1351 y(on)h(the)h(p)r(erturbation)f(and)h(of)g(the)g (singularit)n(y)e(order)g(of)i(the)g(homo)r(clinic)g(orbit.)37 b(Gelfreic)n(h)456 1451 y(in)30 b([)p Fk(Ge2)p Fu(])g(also)g(giv)n(es)f (an)h(asymptotic)g(expression)e(for)i(the)h(separatrix)d(splitting,)k (but)f(it)g(is)456 1551 y(di\016cult)20 b(to)g(\014nd)g(out)g(whic)n(h) g Fq(p)g Fu(is)g(needed)g(in)g(order)e(to)i(apply)g(it.)34 b(Finally)-7 b(,)22 b(in)e([)p Fk(Ge3)p Fu(])g(Gelfreic)n(h)456 1650 y(studies)28 b(in)h(some)f(sp)r(eci\014c)h(examples)f(the)h Fq(p)24 b(<)h Fu(0)j(case.)39 b(The)28 b(prop)r(osed)g(metho)r(d)h(is)g (the)g(use)456 1750 y(of)k(an)g(auxiliary)g(system)g(whose)g(in)n(v)-5 b(arian)n(t)32 b(manifolds)i(are)e(a)i(go)r(o)r(d)f(appro)n(ximation)e (near)456 1849 y(the)26 b(singularities)g(of)g(the)h(in)n(v)-5 b(arian)n(t)25 b(manifolds)i(of)f(the)h(initial)g(system.)36 b(In)27 b([)p Fk(An)p Fu(])f(Angenen)n(t)456 1949 y(studies)32 b(the)h(splitting)g(using)f(v)-5 b(ariational)32 b(metho)r(ds.)52 b(It)33 b(is)f(w)n(orth)g(noting)g(that)h(there)f(are)456 2049 y(examples)21 b(for)g(whic)n(h)h(the)h(asymptotic)e(expressions)f (are)h(not)h(of)g(the)g(form)g Fq(")2878 2019 y Fi(r)2915 2049 y Fq(e)2954 2019 y Fj(\000)p Fi(a=")3111 2049 y Fu(,)h([)p Fk(SMH)o Fu(].)605 2148 y(In)g(all)g(these)g(cases,)g(one)f (deals)h(with)g(Hamiltonian)g(systems)g(of)g(one)f(and)h(a)g(half)g (degrees)456 2248 y(of)j(freedom)g(or)g(area-preserving)c(maps)27 b(suc)n(h)f(that)g(the)h(origin)e(is)i(a)f(h)n(yp)r(erb)r(olic)g (\014xed)g(p)r(oin)n(t)456 2348 y(of)21 b(the)g(non-p)r(erturb)r(ed)g (Hamiltonian.)35 b(Another)21 b(situation)g(where)f(the)i(separatrix)d (splitting)456 2447 y(phenomenon)26 b(app)r(ears)g(is)h(when)g(one)g (considers)f(quasi-p)r(erio)r(dic)g(p)r(erturbations.)36 b(W)-7 b(e)27 b(refer)456 2547 y(to)g([)p Fk(DG)p Fu(],)h([)p Fk(DGJS1)q Fu(],)f([)p Fk(DGJS2)q Fu(])h(and)f([)p Fk(GGM)p Fu(])h(for)f(suc)n(h)g(case.)605 2646 y(Exp)r(onen)n(tially)d(small)h (phenomena)f(are)g(also)g(found)h(b)n(y)g(Fiedler)g(and)g(Sc)n(heurle)f ([)p Fk(FS)p Fu(])h(in)456 2746 y(one)i(step)h(discretizations)e(of)i (autonomous)e(equations.)605 2846 y(This)36 b(memoir)g(is)h(dev)n(oted) f(to)g(study)h(the)g(splitting)f(for)g(one)g(and)h(a)f(half)g(degrees)g (of)456 2945 y(freedom)d(Hamiltonian)h(systems)g(of)g(the)h(form)f (\(3\))g(suc)n(h)g(that)h(the)g(origin)e(is)h(a)g(parab)r(olic)456 3045 y(\014xed)28 b(p)r(oin)n(t.)39 b(Sp)r(eci\014cally)28 b(w)n(e)g(assume)f(that)i(the)f(linear)g(part)g(of)g(the)g(v)n(ector)f (\014eld)i(at)f(\(0)p Fq(;)14 b Fu(0\))456 3145 y(is)1746 3206 y Fg(\022)1849 3272 y Fu(0)82 b(1)1849 3372 y(0)g(0)2056 3206 y Fg(\023)2131 3323 y Fu(.)456 3522 y(W)-7 b(e)28 b(consider)f(the)h(case)f(of)h(fast)f(frequency)h(p)r(erturbation.)37 b(The)28 b(pap)r(er)f([)p Fk(CFN)p Fu(])h(deals)f(with)456 3622 y(the)j(case)f(of)h(constan)n(t)g(frequency)-7 b(.)44 b(The)30 b(\014rst)g(p)r(oin)n(t)g(is)g(to)g(put)h(su\016cien)n(t)f (conditions)g(suc)n(h)456 3721 y(that)d(the)h(p)r(erturb)r(ed)g(system) g(also)e(has)h(in)n(v)-5 b(arian)n(t)27 b(manifolds.)605 3821 y(W)-7 b(e)33 b(ha)n(v)n(e)f(follo)n(w)n(ed)g(basically)g(the)h (structure)g(of)g([)p Fk(DS2)p Fu(].)53 b(Ho)n(w)n(ev)n(er,)32 b(due)h(to)g(the)g(fact)456 3920 y(that)25 b(man)n(y)f(of)g(their)h (argumen)n(ts)f(strongly)f(rely)h(on)h(the)g(h)n(yp)r(erb)r(olic)f(c)n (haracter)f(of)h(the)h(\014xed)456 4020 y(p)r(oin)n(t,)g(w)n(e)e(ha)n (v)n(e)g(had)h(to)g(in)n(tro)r(duce)f(new)h(tec)n(hniques)g(to)g(deal)f (with)i(the)f(parab)r(olic)f(case.)35 b(T)-7 b(o)456 4120 y(this)31 b(end)g(w)n(e)f(ha)n(v)n(e)g(also)g(used)h(to)r(ols)f (in)n(tro)r(duced)h(b)n(y)f(Lazutkin)h([)p Fk(La2)q Fu(])g([)p Fk(La1)p Fu(].)47 b(It)32 b(is)e(w)n(orth)456 4219 y(remarking)25 b(that)i(most)g(of)h(our)e(argumen)n(ts)g(are)g(can)h(b)r(e)g(adapted)g (for)g(the)h(h)n(yp)r(erb)r(olic)e(case.)605 4319 y(The)41 b(memoir)f(is)g(organized)f(as)h(follo)n(ws.)75 b(In)41 b(the)g(\014rst)f(c)n(hapter)g(w)n(e)g(in)n(tro)r(duce)g(the)456 4419 y(notation,)27 b(the)h(h)n(yp)r(otheses)f(and)g(the)h(main)g (theorem.)605 4518 y(In)k(the)g(second)f(c)n(hapter,)h(w)n(e)f(study)h (some)f(analytical)g(prop)r(erties)g(of)h(the)g(homo)r(clinic)456 4618 y(orbit)20 b(of)h(the)h(unp)r(erturb)r(ed)g(system.)34 b(In)22 b(particular)d(w)n(e)i(get)g(the)h(asymptotic)e(b)r(eha)n(vior) g(of)h(its)456 4717 y(parameterization.)34 b(W)-7 b(e)25 b(pro)n(v)n(e)e(that,)j(as)e(w)n(as)f(to)i(b)r(e)g(exp)r(ected,)h(this) f(b)r(eha)n(vior)f(is)g(algebraic,)456 4817 y(that)h(is,)g(there)f (exists)g Fq(T)35 b(>)22 b Fu(0)i(suc)n(h)h(that)g(if)g Fq(t)e Fp(2)g Fl(C)15 b Fu(,)32 b(Re)13 b Fq(t)23 b Fp(\025)g Fq(T)12 b Fu(,)25 b(the)g(stable)f(manifold)h(b)r(eha)n(v)n(es)456 4917 y(lik)n(e)g(1)p Fq(=t)720 4887 y Fi(p)783 4917 y Fu(with)h Fq(p)g Fu(a)f(certain)g(p)r(ositiv)n(e)g(n)n(um)n(b)r(er,)h (and,)g(analogously)-7 b(,)24 b(the)i(unstable)f(manifold)456 5016 y(has)i(the)h(form)f(1)p Fq(=)p Fu(\()p Fp(\000)p Fq(t)p Fu(\))1186 4986 y Fi(p)1251 5016 y Fu(for)g Fq(t)c Fp(2)h Fl(C)15 b Fu(,)33 b(Re)14 b Fq(t)23 b(<)g Fp(\000)p Fq(T)12 b Fu(.)605 5116 y(In)33 b(the)g(third)g(c)n(hapter,)g(w)n(e)f (establish,)i(under)e(the)i(stated)e(conditions,)i(the)f(existence)456 5216 y(of)28 b(stable)g(and)g(unstable)h(in)n(v)-5 b(arian)n(t)27 b(manifolds)h(for)g(the)h(p)r(erturb)r(ed)f(system.)39 b(Moreo)n(v)n(er)26 b(w)n(e)p eop %%Page: 7 7 7 6 bop 1682 251 a Fo(INTR)n(ODUCTION)1156 b(vii)456 450 y Fu(\014nd)28 b(useful)h(parameterizations)d Fq(\015)1584 420 y Fj(\003)1622 450 y Fu(\()p Fq(t;)14 b(s)p Fu(\))29 b(\()p Fp(\003)24 b Fu(=)g Fq(s;)14 b(u)p Fu(\),)28 b(of)h(the)f(lo)r (cal)g(in)n(v)-5 b(arian)n(t)27 b(manifolds)i(of)456 550 y(the)34 b(p)r(erturb)r(ed)h(Hamiltonian)f(system.)57 b(These)34 b(parameterizations)e(satisfy)i(that)h Fq(\015)3241 520 y Fj(\003)3313 550 y Fu(is)f(a)456 649 y(solution)27 b(with)h(resp)r(ect)f(to)h(the)g(v)-5 b(ariable)26 b Fq(t)d Fp(2)h Fl(R)p Fu(,)33 b(it)28 b(is)g(analytic)f(with)h(resp)r (ect)f(to)h Fq(s)f Fu(and)1395 784 y Fq(\015)1443 750 y Fj(\003)1481 784 y Fu(\()p Fq(t)18 b Fu(+)g(2)p Fq(\031)s(";)c(s)p Fu(\))23 b(=)g Fq(\015)2042 750 y Fj(\003)2080 784 y Fu(\()p Fq(t;)14 b(s)k Fu(+)g(2)p Fq(\031)s(")p Fu(\).)456 927 y(In)30 b(this)h(w)n(a)n(y)e(w)n(e)h(endo)n(w)g(the)g(v)-5 b(ariable)30 b Fq(s)g Fu(with)h(a)f(dynamic)g(c)n(haracter)e(since,)j (if)g Fq(P)3151 897 y Fi(t)3176 905 y Fh(0)3243 927 y Fu(is)f(the)456 1027 y(P)n(oincar)n(\023)-39 b(e)26 b(map)j(from)g Fq(t)d Fu(=)f Fq(t)1358 1039 y Fo(0)1425 1027 y Fu(to)k Fq(t)d Fu(=)f Fq(t)1704 1039 y Fo(0)1761 1027 y Fu(+)19 b(2)p Fq(\031)s(")p Fu(,)30 b Fq(\015)2077 996 y Fi(s)2112 1027 y Fu(\()p Fq(t)2174 1039 y Fo(0)2211 1027 y Fq(;)14 b(s)p Fu(\))30 b(represen)n(ts)e(the)h(stable)g(manifold)456 1126 y(of)e Fq(P)615 1096 y Fi(t)640 1104 y Fh(0)704 1126 y Fu(and)h(the)g(dynamics)f(of)g Fq(P)1535 1096 y Fi(t)1560 1104 y Fh(0)1625 1126 y Fu(on)g(it)h(is)f(simply)1381 1261 y Fq(P)1446 1226 y Fi(t)1471 1234 y Fh(0)1507 1261 y Fu(\()p Fq(\015)1587 1226 y Fi(s)1623 1261 y Fu(\()p Fq(t)1685 1273 y Fo(0)1722 1261 y Fq(;)14 b(s)p Fu(\)\))24 b(=)e Fq(\015)2021 1226 y Fi(s)2056 1261 y Fu(\()p Fq(t)2118 1273 y Fo(0)2156 1261 y Fq(;)14 b(s)k Fu(+)g(2)p Fq(\031)s(")p Fu(\).)456 1396 y(Moreo)n(v)n(er)24 b Fq(\015)868 1365 y Fj(\003)934 1396 y Fu(is)j(of)h(the)g(form)1312 1530 y Fq(\015)1360 1496 y Fj(\003)1397 1530 y Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b(=)e Fq(\015)1721 1542 y Fo(0)1759 1530 y Fu(\()p Fq(t)c Fu(+)g Fq(s)p Fu(\))h(+)f Fq(\026")2184 1496 y Fi(p)p Fo(+1)2306 1530 y Fq(\033)2356 1496 y Fj(\003)2395 1530 y Fu(\()p Fq(t;)c(s)p Fu(\),)456 1665 y(where)27 b Fq(\015)739 1677 y Fo(0)804 1665 y Fu(is)g(the)h(homo)r(clinic)f (orbit)h(of)f(the)h(unp)r(erturb)r(ed)g(system.)605 1765 y(In)35 b(the)g(fourth)g(c)n(hapter,)h(w)n(e)f(built)g(the)h(\015o)n(w) e(b)r(o)n(x)g(co)r(ordinates,)i(i.e.,)h(co)r(ordinates)c(in)456 1864 y(whic)n(h)25 b(the)h(\015o)n(w)f(straigh)n(tens.)35 b(These)25 b(co)r(ordinates)f(are)g(de\014ned)i(in)g(a)f(neigh)n(b)r (orho)r(o)r(d)g(of)g(the)456 1964 y(stable)33 b(manifold)g(not)g(con)n (taining)f(the)i(origin,)g(but)f(close)g(to)g(it)h(and)f(indep)r(enden) n(t)h(of)f(the)456 2063 y(parameters.)k(W)-7 b(e)29 b(built)g(them)f (follo)n(wing)g(sev)n(eral)e(steps.)39 b(W)-7 b(e)29 b(parameterize)e(the)h(solutions)456 2163 y(of)33 b(the)h(p)r(erturb)r (ed)f(system)g(near)g(a)g(piece)g(of)h(the)f(stable)g(manifolds)g(b)n (y)h(t)n(w)n(o)e(parameters.)456 2263 y(One)22 b(of)g(these)h(is)f (time,)i(and)e(the)h(other)f(is)g(a)g(complex)g(parameter)f Fq(s)i Fu(suc)n(h)f(that)g(the)h(solutions)456 2362 y(are)i(analytic)g (with)i(resp)r(ect)f(to)g Fq(s)g Fu(and)g(the)h(dynamics)f(of)g(the)g (P)n(oincar)n(\023)-39 b(e)23 b(mapping)j(is)g(simply)456 2462 y Fq(s)h Fp(7!)h Fq(s)21 b Fu(+)f(2)p Fq(\031)s(")p Fu(.)45 b(W)-7 b(e)31 b(can)f(write)g(them)h(in)g(the)g(form)f Fq(w)r Fu(\()p Fq(t)21 b Fu(+)f Fq(s;)14 b(t=")p Fu(\).)45 b(W)-7 b(e)31 b(pro)n(v)n(e)e(afterw)n(ards,)456 2562 y(thanks)h(to)h(this)g(go)r(o)r(d)g(parameterization,)e(that)j(the)f (solutions)f(in)n(tersect)h(a)f(\(real\))h(section)456 2661 y(transv)n(erse)21 b(to)i(the)g(\015o)n(w)g(for)f(some)h(v)-5 b(alue)23 b(\()p Fq(t)1847 2673 y Fo(0)1884 2661 y Fq(;)14 b(s)1960 2673 y Fo(0)1997 2661 y Fu(\),)25 b(th)n(us)e(w)n(e)g(are)f (able)h(to)g(straigh)n(ten)f(the)h(\015o)n(w)456 2761 y(in)28 b(a)h(neigh)n(b)r(orho)r(o)r(d)e(of)h(the)h(stable)g(manifold.) 39 b(Finally)29 b(w)n(e)f(sligh)n(tly)g(mo)r(dify)h(the)g(v)-5 b(ariables)456 2860 y(to)27 b(mak)n(e)g(them)h(canonical.)605 2960 y(In)d(the)f(\014fth)i(c)n(hapter,)e(w)n(e)g(presen)n(t)f(a)h (result)g(of)h(Delshams-Seara)d([)p Fk(DS2)p Fu(])j(whic)n(h)f(asserts) 456 3060 y(that)35 b(if)h Fq(p)f Fu(is)h(bigger)e(than)h(some)g(v)-5 b(alue,)37 b(whic)n(h)e(dep)r(ends)h(on)f(the)h(p)r(erturbation)f(and)g (the)456 3159 y(unp)r(erturb)r(ed)27 b(homo)r(clinic)g(orbit,)g(w)n(e)f (can)h(extend)g(the)g(parameterization)e(of)i(the)h(unstable)456 3259 y(manifold)f(un)n(til)h(it)g(reac)n(hes)e(the)i(domain)f(where)g (the)h(\015o)n(w)f(b)r(o)n(x)h(v)-5 b(ariables)26 b(are)g(de\014ned.) 605 3359 y(Finally)-7 b(,)26 b(in)g(the)g(last)f(c)n(hapter)g(w)n(e)g (in)n(tro)r(duce)g(the)h(splitting)g(function.)37 b(F)-7 b(rom)25 b(it)h(and)f(its)456 3458 y(prop)r(erties)e(w)n(e)h(deriv)n(e) g(the)h(asymptotic)f(form)n(ulas)f(for)h(the)h(area)e(of)h(the)h(lob)r (es)f(generated)g(b)n(y)456 3558 y(the)29 b(in)n(v)-5 b(arian)n(t)29 b(manifolds)g(b)r(et)n(w)n(een)g(t)n(w)n(o)g(homo)r (clinic)g(p)r(oin)n(ts)g(and)g(the)h(angle)f(b)r(et)n(w)n(een)g(the)456 3657 y(in)n(v)-5 b(arian)n(t)26 b(manifolds)i(at)g(a)g(homo)r(clinic)f (p)r(oin)n(t.)39 b(They)27 b(are)g(exp)r(onen)n(tially)h(small)f(in)i Fq(")p Fu(.)37 b(The)456 3757 y(main)20 b(di\013erence)h(with)g([)p Fk(DS2)p Fu(])g(is)g(that)g(here)f(w)n(e)h(consider)e(homo)r(clinic)i (orbits)f(with)h(algebraic)456 3857 y(branc)n(h)j(t)n(yp)r(e)i (singularities)e(and)h(consequen)n(tly)g(some)f(computations)h(are)g (somewhat)f(more)456 3956 y(in)n(v)n(olv)n(ed.)605 4106 y Fk(Akno)m(wledgemen)m(ts.)39 b Fu(The)g(authors)f(thank)i(the)f(supp) r(ort)g(of)g(the)h(Catalan)e(Gran)n(t)456 4205 y(CIRIT)f(2001SGR{70.)63 b(The)37 b(second)g(author)f(also)h(thanks)g(the)g(partial)g(supp)r (ort)g(of)g(the)456 4305 y(Spanish)27 b(Gran)n(t)g(DGICYT)h (BFM2000{0805)c(and)j(the)h(INT)-7 b(AS)29 b(pro)5 b(ject)26 b(00-221.)p eop %%Page: 8 8 8 7 bop eop %%Page: 1 9 1 8 bop 1094 566 a Fw(1.)61 b(Notation)46 b(and)f(main)g(results)605 849 y Fu(In)31 b(this)g(Chapter)f(w)n(e)g(presen)n(t)g(the)h(main)f (problem)g(w)n(e)h(consider,)f(the)h(h)n(yp)r(otheses)f(w)n(e)456 948 y(assume,)c(and)g(the)h(rigorous)e(statemen)n(t)h(of)h(the)g(main)g (results.)36 b(F)-7 b(or)26 b(that)h(w)n(e)f(ha)n(v)n(e)g(to)g(b)r (egin)456 1048 y(b)n(y)h(in)n(tro)r(ducing)g(some)g(notation.)605 1147 y(A)n(t)h(the)g(end)g(w)n(e)f(presen)n(t)g(an)g(example)g(where)h (the)f(ab)r(o)n(v)n(e)g(men)n(tioned)g(results)g(apply)-7 b(.)1330 1364 y Fk(1.1.)46 b(Notation)31 b(and)i(h)m(yp)s(otheses)605 1513 y Fu(W)-7 b(e)31 b(study)h(the)f(splitting)g(of)g(separatrices)e (in)i(the)g(case)f(whic)n(h)h(w)n(e)g(call)f(the)i(parab)r(olic)456 1613 y(case.)k(Next)28 b(w)n(e)f(describ)r(e)g(the)h(settings)f(of)h (this)g(case)f(and)g(the)h(h)n(yp)r(otheses)f(w)n(e)g(will)h(need.)605 1712 y(W)-7 b(e)35 b(consider)e(Hamiltonian)i(systems)f(of)g(one)g(and) h(a)f(half)g(degrees)f(of)i(freedom)f(with)456 1812 y(Hamiltonian)1059 1963 y Fq(H)7 b Fu(\()p Fq(x;)14 b(y)s(;)g(t=";)g(\026;)g(")p Fu(\))22 b(=)h Fq(h)1796 1975 y Fo(0)1833 1963 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(\026")2216 1928 y Fi(p)2254 1963 y Fq(h)2302 1975 y Fo(1)2339 1963 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g (\026;)g(")p Fu(\))456 2114 y(where)1554 2297 y Fq(h)1602 2309 y Fo(0)1639 2297 y Fu(\()p Fq(x;)g(y)s Fu(\))23 b(=)1952 2241 y Fq(y)1996 2211 y Fo(2)p 1952 2278 81 4 v 1972 2354 a Fu(2)2061 2297 y(+)18 b Fq(V)h Fu(\()p Fq(x)p Fu(\),)456 2476 y Fq(V)f Fu(\()p Fq(x)p Fu(\))29 b(is)f(an)f(analytic)g(function)h(of)g(order)e Fq(n)p Fu(,)h(that)h(is)1608 2627 y Fq(V)19 b Fu(\()p Fq(x)p Fu(\))24 b(=)f Fq(a)1942 2639 y Fi(n)1987 2627 y Fq(x)2034 2592 y Fi(n)2098 2627 y Fu(+)18 b Fp(\001)c(\001)g(\001)456 2782 y Fu(with)35 b Fq(n)g Fp(\025)g Fu(3.)58 b(With)36 b(these)f(assumptions,)h(for)e(the)h(unp)r(erturb)r(ed)h(system)e (\(i.e.)59 b(the)35 b(sys-)456 2882 y(tem)h(when)f Fq(\026)i Fu(=)f(0\))f(the)h(origin)e(is)i(a)f(parab)r(olic)f(\014xed)i(p)r(oin)n (t)g(and)f(the)h(deriv)-5 b(ativ)n(e)35 b(of)g(the)456 2982 y(Hamiltonian)27 b(v)n(ector)f(\014eld)i(at)g(\(0)p Fq(;)14 b Fu(0\))27 b(is)1746 3065 y Fg(\022)1849 3132 y Fu(0)82 b(1)1849 3231 y(0)g(0)2056 3065 y Fg(\023)2131 3182 y Fu(.)605 3378 y(The)28 b(di\013eren)n(tial)f(equations)g(asso)r (ciated)f(to)i(the)g(Hamiltonian)f(are)1230 3629 y(_)-38 b Fq(x)84 b Fu(=)e Fq(y)21 b Fu(+)d Fq(\026")1727 3595 y Fi(p)1765 3629 y Fq(@)1809 3641 y Fi(y)1849 3629 y Fq(h)1897 3641 y Fo(1)1935 3629 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g(\026;) g(")p Fu(\))-1982 b(\(1.1\))1234 3753 y(_)-38 b Fq(y)86 b Fu(=)c Fp(\000)p Fq(V)1625 3719 y Fj(0)1648 3753 y Fu(\()p Fq(x)p Fu(\))19 b Fp(\000)f Fq(\026")1950 3719 y Fi(p)1989 3753 y Fq(@)2033 3765 y Fi(x)2074 3753 y Fq(h)2122 3765 y Fo(1)2160 3753 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g(\026;) g(")p Fu(\).)605 3904 y(W)-7 b(e)36 b(will)f(assume)g(the)h(follo)n (wing)e(h)n(yp)r(othesis)h(related)g(to)g(the)h(unp)r(erturb)r(ed)g (system.)456 4004 y(Note)27 b(that)h(the)g(unp)r(erturb)r(ed)g(system)g (is)f(autonomous)f(and)i(indep)r(enden)n(t)g(on)g Fq(")p Fu(.)605 4186 y Fk(1.1.1.)46 b(Hyp)s(othesis)30 b(for)i(the)g(unp)s (erturb)s(ed)f(system.)565 4311 y(HP1)41 b Fu(W)-7 b(e)22 b(assume)f(that,)i Fq(h)1458 4323 y Fo(0)1496 4311 y Fu(\()p Fq(x;)14 b(y)s Fu(\))23 b(=)g Fq(y)1843 4281 y Fo(2)1880 4311 y Fq(=)p Fu(2)7 b(+)g Fq(V)17 b Fu(\()p Fq(x)p Fu(\))22 b(is)g(analytic)f(and)h Fq(V)c Fu(\()p Fq(x)p Fu(\))24 b(=)f Fq(a)3118 4323 y Fi(n)3163 4311 y Fq(x)3210 4281 y Fi(n)3262 4311 y Fu(+)7 b Fp(\001)14 b(\001)g(\001)794 4411 y Fu(with)33 b Fq(a)1032 4423 y Fi(n)1108 4411 y Fq(<)e Fu(0)h(and)g Fq(n)f Fp(\025)g Fu(3.)51 b(Moreo)n(v)n(er)30 b(w)n(e)i(assume)f(that)i Fq(h)2758 4423 y Fo(0)2828 4411 y Fu(has)f(a)g(homo)r(clinic)794 4510 y(orbit,)27 b(asso)r(ciated)g(to)g(the)h(equilibrium)g(p)r(oin)n (t)g(\(0)p Fq(;)14 b Fu(0\))944 4610 y(W)-7 b(e)28 b(denote)f(the)h (time)g(parameterization)e(of)h(the)h(homo)r(clinic)g(orbit)f(b)n(y) 1548 4761 y Fq(\015)1591 4773 y Fo(0)1629 4761 y Fu(\()p Fq(u)p Fu(\))c(=)f(\()p Fq(\013)1936 4773 y Fo(0)1974 4761 y Fu(\()p Fq(u)p Fu(\))p Fq(;)14 b(\014)2170 4773 y Fo(0)2207 4761 y Fu(\()p Fq(u)p Fu(\)\))794 4917 y(with)32 b(some)f(c)n(hosen)f(\(\014xed\))i(initial)f(condition)g Fq(\015)2391 4929 y Fo(0)2428 4917 y Fu(\(0\))f(=)e(\()p Fq(x)2736 4929 y Fo(0)2774 4917 y Fq(;)14 b(y)2852 4929 y Fo(0)2889 4917 y Fu(\))32 b(on)f(the)g(homo-)794 5016 y(clinic)d(orbit.)944 5116 y(W)-7 b(e)29 b(assume)g(that)g Fq(\015)1601 5128 y Fo(0)1638 5116 y Fu(\()p Fq(u)p Fu(\))g(is)g (analytic)g(in)g(a)g(complex)g(strip)g Fp(j)14 b Fu(Im)g Fq(u)p Fp(j)25 b Fq(<)g(a)k Fu(with)794 5216 y(branc)n(hing)g(p)r(oin)n (ts)g(at)h Fq(u)c Fu(=)g Fp(\006)p Fq(ia)p Fu(,)j(i.e.,)h(there)g (exists)f Fq(\032)d(>)g Fu(0)j(suc)n(h)h(that)g(for)f Fq(u)d Fp(2)g Fl(C)1933 5365 y Fo(1)p eop %%Page: 2 10 2 9 bop 456 251 a Fo(2)962 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)794 450 y Fu(satisfying)h Fp(j)p Fq(u)11 b Fp(\000)g Fq(ia)p Fp(j)22 b Fq(<)h(\032)p Fu(,)h(arg\()p Fq(u)11 b Fp(\000)g Fq(ia)p Fu(\))22 b Fp(2)h Fu(\()p Fp(\000)p Fu(3)p Fq(\031)s(=)p Fu(2)p Fq(;)14 b(\031)s(=)p Fu(2\),)23 b Fq(\015)2668 462 y Fo(0)2706 450 y Fu(\()p Fq(u)p Fu(\))h(can)f(b)r(e)i(expressed) 794 550 y(as)473 822 y Fq(\013)526 834 y Fo(0)564 822 y Fu(\()p Fq(u)p Fu(\))e(=)946 766 y Fq(c)982 778 y Fj(\000)p 796 803 391 4 v 796 880 a Fu(\()p Fq(u)c Fp(\000)f Fq(ia)p Fu(\))1083 856 y Fi(p=q)1197 822 y Fu(\(1)h(+)f Fq(O)r Fu(\()p Fq(u)g Fp(\000)g Fq(ia)p Fu(\))1724 787 y Fo(1)p Fi(=q)1828 822 y Fu(\),)56 b Fq(\014)1986 834 y Fo(0)2023 822 y Fu(\()p Fq(u)p Fu(\))23 b(=)2444 766 y Fq(d)2487 778 y Fj(\000)p 2256 803 476 4 v 2256 880 a Fu(\()p Fq(u)18 b Fp(\000)g Fq(ia)p Fu(\))2542 856 y Fo(1+)p Fi(p=q)2741 822 y Fu(\(1)g(+)g Fq(O)r Fu(\()p Fq(u)h Fp(\000)f Fq(ia)p Fu(\))3268 787 y Fo(1)p Fi(=q)3371 822 y Fu(\).)794 1022 y(and)31 b(for)f Fq(u)e Fp(2)h Fl(C)52 b Fu(suc)n(h)30 b(that)h Fp(j)p Fq(u)21 b Fu(+)f Fq(ia)p Fp(j)28 b Fq(<)g(\032)p Fu(,)j(arg)o(\()p Fq(u)21 b Fu(+)f Fq(ia)p Fu(\))28 b Fp(2)h Fu(\()p Fp(\000)p Fq(\031)s(=)p Fu(2)p Fq(;)14 b Fu(3)p Fq(\031)s(=)p Fu(2\),)30 b Fq(\015)3295 1034 y Fo(0)3332 1022 y Fu(\()p Fq(u)p Fu(\))794 1121 y(can)d(b)r(e)h (expressed)f(as)473 1409 y Fq(\013)526 1421 y Fo(0)564 1409 y Fu(\()p Fq(u)p Fu(\))c(=)946 1353 y Fq(c)982 1365 y Fo(+)p 796 1390 391 4 v 796 1468 a Fu(\()p Fq(u)c Fu(+)f Fq(ia)p Fu(\))1083 1444 y Fi(p=q)1197 1409 y Fu(\(1)h(+)f Fq(O)r Fu(\()p Fq(u)g Fu(+)g Fq(ia)p Fu(\))1724 1375 y Fo(1)p Fi(=q)1828 1409 y Fu(\),)56 b Fq(\014)1986 1421 y Fo(0)2023 1409 y Fu(\()p Fq(u)p Fu(\))23 b(=)2444 1353 y Fq(d)2487 1365 y Fo(+)p 2256 1390 476 4 v 2256 1468 a Fu(\()p Fq(u)18 b Fu(+)g Fq(ia)p Fu(\))2542 1444 y Fo(1+)p Fi(p=q)2741 1409 y Fu(\(1)g(+)g Fq(O)r Fu(\()p Fq(u)h Fu(+)f Fq(ia)p Fu(\))3268 1375 y Fo(1)p Fi(=q)3371 1409 y Fu(\).)794 1605 y(Moreo)n(v)n(er)25 b(on)i Fq(u)c Fu(=)k Fp(\006)p Fq(ia)g Fu(there)g(are)g(not)g(other)g(singularities)g (of)g Fq(\015)2917 1617 y Fo(0)2955 1605 y Fu(.)37 b(W)-7 b(e)28 b(de\014ne)1685 1771 y Fq(r)e Fu(=)c(1)c(+)1988 1715 y Fq(p)p 1988 1752 42 4 v 1989 1828 a(q)2063 1771 y(>)23 b Fu(1.)605 1961 y(Of)28 b(course,)e(p)r(oles)h(are)g(included)h (in)g(this)g(de\014nition)g(of)f(branc)n(hing)g(p)r(oin)n(ts.)605 2119 y Ft(Remark)33 b Fu(1.1)p Ft(.)41 b Fs(As)31 b(a)g(c)l(onse)l (quenc)l(e)g(of)h(Pr)l(op)l(osition)g(2.3)g(in)g(Chapter)g(2)f(always)i (exists)456 2218 y Fq(a)23 b(>)f Fu(0)29 b Fs(such)h(that)g Fq(\015)1083 2230 y Fo(0)1120 2218 y Fu(\()p Fq(u)p Fu(\))g Fs(is)g(analytic)h(on)f(the)g(strip)g Fp(f)p Fq(u)22 b Fp(2)h Fl(C)44 b Fu(:)24 b Fp(j)14 b Fu(Im)g Fq(u)p Fp(j)22 b Fq(<)h(a)p Fp(g)p Fs(.)605 2377 y Ft(Remark)40 b Fu(1.2)p Ft(.)45 b Fs(As)37 b(a)h(c)l(onse)l(quenc)l(e)f(of)h(Pr)l (op)l(osition)h(2.4)g(in)e(Chapter)i(2,)h(if)e Fq(V)19 b Fu(\()p Fq(x)p Fu(\))38 b(=)456 2476 y Fq(a)500 2488 y Fi(n)545 2476 y Fq(x)592 2446 y Fi(n)656 2476 y Fu(+)18 b Fp(\001)c(\001)g(\001)19 b Fu(+)f Fq(a)982 2488 y Fi(m)1045 2476 y Fq(x)1092 2446 y Fi(m)1185 2476 y Fs(is)30 b(a)h(p)l(olynomial)h (and)e(we)g(assume)g(that)g Fq(\013)2563 2488 y Fo(0)2601 2476 y Fu(\()p Fq(u)p Fu(\))g Fs(has)g(a)h(singularity)f(at)456 2576 y Fq(u)22 b Fu(=)h Fq(u)662 2546 y Fj(\003)723 2576 y Fp(2)g Fl(C)15 b Fs(,)36 b(then)30 b(for)g Fq(u)g Fs(in)f(a)i(neighb) l(orho)l(o)l(d)h(of)e Fq(u)2123 2546 y Fj(\003)2191 2576 y Fs(we)g(have)h(that)963 2763 y Fq(\013)1016 2775 y Fo(0)1053 2763 y Fu(\()p Fq(u)p Fu(\))83 b(=)1656 2707 y Fq(C)p 1406 2744 567 4 v 1406 2822 a Fu(\()p Fq(u)18 b Fp(\000)g Fq(u)1635 2798 y Fj(\003)1673 2822 y Fu(\))1705 2798 y Fo(2)p Fi(=)p Fo(\()p Fi(m)p Fj(\000)p Fo(2\))1982 2763 y Fu(\(1)g(+)g Fq(O)r Fu(\()p Fq(u)h Fp(\000)f Fq(u)2452 2729 y Fj(\003)2490 2763 y Fu(\))2522 2729 y Fo(2)p Fi(=)p Fo(\()p Fi(m)p Fj(\000)p Fo(2\))2789 2763 y Fu(\))969 2994 y Fq(\014)1016 3006 y Fo(0)1053 2994 y Fu(\()p Fq(u)p Fu(\))83 b(=)g Fp(\000)1722 2938 y Fq(C)1787 2908 y Fj(0)p 1470 2975 593 4 v 1470 3053 a Fu(\()p Fq(u)19 b Fp(\000)f Fq(u)1700 3029 y Fj(\003)1737 3053 y Fu(\))1769 3029 y Fi(m=)p Fo(\()p Fi(m)p Fj(\000)p Fo(2\))2072 2994 y Fu(\(1)h(+)f Fq(O)r Fu(\()p Fq(u)h Fp(\000)f Fq(u)2543 2960 y Fj(\003)2580 2994 y Fu(\))2612 2960 y Fo(2)p Fi(=)p Fo(\()p Fi(m)p Fj(\000)p Fo(2\))2880 2994 y Fu(\))p Fs(.)456 3190 y(As)38 b(a)i(c)l(onse)l(quenc)l(e,)h(the)f(exp)l(onents)e(of)i Fq(u)25 b Fp(\000)g Fq(u)2024 3160 y Fj(\003)2101 3190 y Fs(in)39 b(the)g(expr)l(essions)h(of)g Fq(\013)2963 3202 y Fo(0)3039 3190 y Fs(and)g Fq(\014)3257 3202 y Fo(0)3333 3190 y Fs(ar)l(e)456 3289 y(r)l(ational)30 b(numb)l(ers.)605 3447 y Fk(1.1.2.)46 b(Hyp)s(otheses)30 b(o)m(v)m(er)j(the)e(p)s(erturbation.)565 3572 y(HP2)41 b Fu(The)33 b(function)g Fq(h)1348 3584 y Fo(1)1385 3572 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))33 b(is)g(de\014ned)g(for)f(\()p Fq(x;)14 b(y)s Fu(\))32 b Fp(\032)f Fq(U)40 b Fp(\032)32 b Fl(C)2930 3542 y Fo(2)2974 3572 y Fu(,)i Fq(\022)g Fp(2)d Fl(R)p Fu(,)41 b Fq(\026)31 b Fp(2)794 3672 y Fl(D)14 b Fu(\(0)p Fq(;)g(\026)1009 3684 y Fo(0)1052 3672 y Fu(\),)28 b Fq(")23 b Fp(2)g Fu(\(0)p Fq(;)14 b(")1425 3684 y Fo(0)1462 3672 y Fu(\),)28 b(it)g(is)f Fq(C)1776 3642 y Fo(0)1814 3672 y Fu(,)h(2)p Fq(\031)s Fu(-p)r(erio)r(dic)f(in)h Fq(\022)r Fu(,)g(has)f(zero)f (mean:)1478 3768 y Fg(Z)1561 3789 y Fo(2)p Fi(\031)1524 3957 y Fo(0)1653 3881 y Fq(h)1701 3893 y Fo(1)1738 3881 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))g Fq(d\022)26 b Fu(=)c(0)794 4077 y(and)28 b(it)g(is)f(real)g(analytic)g (with)h(resp)r(ect)f(to)h(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\).)565 4177 y Fk(HP3)41 b Fu(The)29 b(function)h Fq(h)1341 4189 y Fo(1)1378 4177 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g (\026;)g(")p Fu(\))29 b(is)g(a)g(p)r(olynomial)f(of)h(order)f Fq(k)k Fu(in)d(the)h(\()p Fq(x;)14 b(y)s Fu(\))29 b(v)-5 b(ari-)794 4277 y(ables.)36 b(That)28 b(is)1248 4472 y Fq(h)1296 4484 y Fo(1)1333 4472 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))24 b(=)1955 4369 y Fi(\024)1914 4393 y Fg(X)1878 4572 y Fi(i)p Fo(+)p Fi(j)s Fo(=)p Fi(k)2084 4472 y Fq(a)2128 4484 y Fi(i;j)2206 4472 y Fu(\()p Fq(\022)r(;)14 b(\026;)g(")p Fu(\))p Fq(x)2521 4438 y Fi(i)2550 4472 y Fq(y)2594 4438 y Fi(j)2629 4472 y Fu(.)565 4708 y Fk(HP4)41 b Fu(The)28 b(order)e Fq(k)31 b Fu(of)c(the)h(p)r(erturbation)f(satis\014es)1743 4858 y(2)p Fq(k)21 b Fp(\000)d Fu(2)k Fp(\025)h Fq(n)p Fu(.)605 5016 y Ft(Remark)31 b Fu(1.3)p Ft(.)40 b Fs(We)29 b(observe)h(that)f Ff(HP4)h Fs(implies)h(that)e(the)g(origin)h(is)g(also)g(a)f(p)l(ar)l (ab)l(olic)456 5116 y(\014xe)l(d)k(p)l(oint)h(of)h(the)e(p)l(erturb)l (e)l(d)h(system)g(and)g(the)g(derivative)i(of)e(the)g(ve)l(ctor)g (\014eld)g(evaluate)l(d)456 5216 y(at)29 b(this)h(p)l(oint)g(is)h(the)f (same)g(as)g(the)g(one)g(of)g(the)g(unp)l(erturb)l(e)l(d)f(system.)p eop %%Page: 3 11 3 10 bop 1629 251 a Fo(1.2.)29 b(MAIN)g(RESUL)-5 b(TS)1139 b(3)605 450 y Fu(Consider)23 b(the)i(terms)f Fq(a)1360 462 y Fi(ij)1419 450 y Fu(\()p Fq(\022)r(;)14 b(\026;)g(")p Fu(\))p Fq(x)1734 420 y Fi(i)1762 450 y Fq(y)1806 420 y Fi(j)1865 450 y Fu(of)25 b Fq(h)2005 462 y Fo(1)2066 450 y Fu(ev)-5 b(aluated)24 b(on)g Fq(\015)2589 462 y Fo(0)2627 450 y Fu(.)36 b(W)-7 b(e)24 b(de\014ne)h Fq(`)f Fu(to)g(b)r(e)h(the)456 561 y(greatest)g(order)g(of)h(the)h(branc)n (hing)f(p)r(oin)n(ts)g Fp(\006)p Fq(ia)g Fu(corresp)r(onding)e(to)j Fq(a)2694 573 y Fi(ij)2752 561 y Fu(\()p Fq(\022)r(;)14 b(\026;)g(")p Fu(\))p Fq(\013)3073 530 y Fi(i)3073 581 y Fo(0)3111 561 y Fu(\()p Fq(u)p Fu(\))p Fq(\014)3274 521 y Fi(j)3270 583 y Fo(0)3309 561 y Fu(\()p Fq(u)p Fu(\).)456 660 y(That)27 b(is:)1213 801 y Fq(`)c Fu(=)f(max)p Fp(f)p Fq(i)p Fu(\()p Fq(r)e Fp(\000)f Fu(1\))f(+)g Fq(j)5 b(r)26 b Fu(:)d Fq(a)2124 813 y Fi(i;j)2202 801 y Fu(\()p Fq(\022)r(;)14 b(\026;)g(")p Fu(\))23 b Fp(6)p Fu(=)g(0)p Fp(g)p Fu(.)-2232 b(\(1.2\))605 941 y(Also)27 b(w)n(e)h(de\014ne)1771 1082 y Fq(\027)g Fu(=)23 b Fq(p)18 b Fp(\000)g Fq(`)p Fu(.)565 1223 y Fk(HP5)41 b Fu(The)28 b(constan)n(t)f Fq(\027)33 b Fu(is)27 b(greater)f(or)h(equal)g(than)g(0.)605 1375 y Ft(Remark)32 b Fu(1.4)p Ft(.)40 b Fs(Hyp)l(othesis)31 b Ff(HP5)f Fs(c)l(ontr)l(ols)g(the)g(gr)l(owth)g(of)h(the)f(p)l(erturb) l(ation)g(term)1593 1516 y Fq(\026")1682 1482 y Fi(p)1720 1516 y Fq(h)1768 1528 y Fo(1)1805 1516 y Fu(\()p Fq(x;)14 b(y)s(;)g(t=";)g(\026;)g(")p Fu(\))456 1657 y Fs(evaluate)l(d)27 b(at)f(the)h(homo)l(clinic)h(orbit,)g(ne)l(ar)e(the)h(singularities.)38 b(In)26 b(fact,)i(if)f(hyp)l(othesis)i Ff(HP5)456 1756 y Fs(is)h(assume)l(d:)1326 1897 y Fq(\026")1415 1862 y Fi(p)1453 1897 y Fp(k)p Fq(h)1543 1909 y Fo(1)1579 1897 y Fu(\()p Fq(\015)1654 1909 y Fo(0)1692 1897 y Fu(\()p Fq(u)p Fu(\))p Fq(;)14 b(t=";)g(\026;)g(")p Fu(\))p Fp(k)2189 1909 y Fj(1)2281 1897 y Fu(=)23 b Fq(O)r Fu(\()p Fq(\026)p Fu(\))p Fs(,)456 2042 y(for)30 b Fp(j)14 b Fu(Im)g Fq(u)p Fp(j)23 b(\024)f Fq(a)d Fp(\000)f Fq(")p Fs(.)605 2195 y Ft(Remark)35 b Fu(1.5)p Ft(.)42 b Fs(A)32 b(c)l(onse)l(quenc)l(e)h (of)g(hyp)l(otheses)h Ff(HP1)p Fs(-)p Ff(HP5)p Fs(,)h(is)e(that)g(if)g Fq(p)28 b(<)g Fu(1)p Fs(,)34 b(then)456 2294 y Fq(@)500 2306 y Fi(y)540 2294 y Fq(h)588 2306 y Fo(1)648 2294 y Fu(=)22 b(0)p Fs(.)38 b(Inde)l(e)l(d,)29 b(if)g Fq(`)22 b Fp(\025)h Fu(1)p Fs(,)28 b(then)g(by)g(hyp)l(othesis)i Ff(HP5)p Fs(,)g Fq(p)22 b Fp(\025)h Fu(1)p Fs(.)38 b(Ther)l(efor)l(e,) 30 b(we)e(c)l(onsider)456 2394 y(the)j(c)l(ase)h Fq(`)26 b(<)g Fu(1)p Fs(.)44 b(By)33 b(de\014nition)f(of)g Fq(`)f Fs(and)h(using)g(that)f Fq(r)f Fp(\025)c Fu(1)p Fs(,)32 b(we)f(have)i(that)f(for)g(any)g(p)l(air)456 2494 y(of)e(p)l(ositive)h (inte)l(gers,)g Fq(i)p Fs(,)e Fq(j)35 b Fs(such)30 b(that)f Fq(a)1741 2506 y Fi(ij)1800 2494 y Fu(\()p Fq(\022)r(;)14 b(\026;)g(")p Fu(\))23 b Fp(6)p Fu(=)g(0)p Fs(,)1404 2634 y Fu(1)f Fq(>)h(`)g Fp(\025)f Fq(i)p Fu(\()p Fq(r)f Fp(\000)d Fu(1\))h(+)f Fq(j)5 b(r)25 b Fp(\025)e Fq(j)5 b(r)26 b Fp(\025)c Fq(j)456 2775 y Fs(Ther)l(efor)l(e,)36 b Fq(j)d Fu(=)c(0)j Fs(and)i(this)f(implies)i(that)e Fq(h)1932 2787 y Fo(1)2002 2775 y Fs(has)g(no)g(terms)g(dep)l(ending)h (on)f(the)g(variable)456 2875 y Fq(y)s Fs(.)38 b(Ther)l(efor)l(e)31 b Fq(@)982 2887 y Fi(y)1022 2875 y Fq(h)1070 2887 y Fo(1)1131 2875 y Fu(=)22 b(0)p Fs(.)1590 3061 y Fk(1.2.)46 b(Main)32 b(results)605 3210 y Fu(It)c(is)f(a)g(w)n(ell)h(kno)n(wn)f(fact)g(that) h(P)n(oincar)n(\023)-39 b(e)24 b(maps)k(asso)r(ciated)e(to)h(p)r(erio)r (dic)h(Hamiltonian)456 3310 y(p)r(erturbations)j(of)g(one)g(degree)g (of)h(freedom)f(Hamiltonian)h(systems)f(ha)n(ving)f(a)i(homo)r(clinic) 456 3410 y(connexion,)g(ha)n(v)n(e)f(either)i(primary)e(homo)r(clinic)h (p)r(oin)n(ts)g(or)f(a)h(homo)r(clinic)g(connexion,)h(the)456 3509 y(latter)h(p)r(ossibilit)n(y)g(b)r(eing)h(non-generic.)57 b(These)35 b(p)r(oin)n(ts)f(are)g(related)g(to)h(the)g(zeros)e(of)i (the)456 3609 y(Melnik)n(o)n(v)26 b(function)i Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))28 b(de\014ned)g(b)n(y)1213 3796 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))22 b(=)1592 3683 y Fg(Z)1675 3704 y Fj(1)1638 3872 y(\0001)1760 3796 y Fp(f)p Fq(h)1850 3808 y Fo(0)1887 3796 y Fq(;)14 b(h)1972 3808 y Fo(1)2009 3796 y Fp(g)p Fu(\()p Fq(\015)2126 3808 y Fo(0)2163 3796 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))g Fq(dt)p Fu(.)456 3994 y(Let)35 b Fq(P)677 3964 y Fi(t)702 3972 y Fh(0)774 3994 y Fu(b)r(e)g(the)h(P)n (oincar)n(\023)-39 b(e)32 b(map)j(from)g Fq(t)1819 4006 y Fo(0)1892 3994 y Fu(to)g Fq(t)2031 4006 y Fo(0)2091 3994 y Fu(+)24 b(2)p Fq(\031)s(")p Fu(.)59 b(W)-7 b(e)36 b(denote)f(b)n(y)g Fq(A)g Fu(the)h(area)e(of)456 4093 y(the)h(lob)r(e)h(generated)e(b)n(y)h(the)h(stable)f(and)h(unstable)f (manifold)h(b)r(et)n(w)n(een)f(t)n(w)n(o)g(homo)r(clinic)456 4193 y(p)r(oin)n(ts)24 b(and)h(b)n(y)f Fq(#)h Fu(the)g(angle)f(b)r(et)n (w)n(een)h(the)g(stable)g(and)f(unstable)h(in)n(v)-5 b(arian)n(t)24 b(manifolds)g(at)h(a)456 4293 y(homo)r(clinic)j(p)r(oin) n(t.)38 b(W)-7 b(e)29 b(observ)n(e)d(that,)j(since)e(the)i(P)n(oincar)n (\023)-39 b(e)25 b(map)j(is)g(area)f(preserving,)f(the)456 4392 y(area)g Fq(A)i Fu(will)f(not)h(dep)r(end)g(on)g(the)g(homo)r (clinic)f(p)r(oin)n(ts.)605 4492 y(The)h(main)f(results)g(are:)605 4645 y Ft(Theorem)j Fu(1.1)p Ft(.)38 b Fs(Under)27 b(hyp)l(otheses)j Ff(HP1-HP5)p Fs(,)g(for)f Fq(")23 b Fp(!)g Fu(0)2643 4614 y Fo(+)2697 4645 y Fs(,)29 b Fq(\026)23 b Fp(!)g Fu(0)p Fs(,)28 b(and)g(for)h(any)456 4744 y Fq(t)486 4756 y Fo(0)546 4744 y Fp(2)23 b Fl(R)p Fs(,)36 b(the)30 b(fol)t(lowing)j(formulae)d(hold:)831 4935 y Fq(A)84 b Fu(=)e Fq(\026")1213 4901 y Fi(p)1265 4822 y Fg(Z)1351 4843 y Fo(\026)-36 b Fi(s)1379 4851 y Fh(0)1311 5011 y Fi(s)1342 5019 y Fh(0)1430 4935 y Fq(M)9 b Fu(\()p Fq(\035)s(;)14 b(")p Fu(\))g Fq(d\035)21 b Fu(+)d Fq(O)r Fu(\()p Fq(\026)2061 4901 y Fo(2)2099 4935 y Fq(")2138 4901 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)2296 4935 y Fq(;)c(\026)2383 4901 y Fo(2)2420 4935 y Fq(")2459 4901 y Fi(\027)t Fo(+)p Fi(p)p Fo(+1)2669 4935 y Fq(;)g(\026")2795 4901 y Fi(p)p Fo(+2)2917 4935 y Fu(\))p Fq(e)2988 4901 y Fj(\000)p Fi(a=")3146 4935 y Fs(,)729 5171 y Fu(sin)g Fq(#)83 b Fu(=)f Fq(\026")1213 5137 y Fi(p)1352 5115 y Fq(M)1442 5085 y Fj(0)1465 5115 y Fu(\()p Fq(s)1536 5127 y Fo(0)1573 5115 y Fq(;)14 b(")p Fu(\))p 1261 5152 510 4 v 1261 5228 a Fp(k)e Fu(_)-35 b Fq(\015)1346 5240 y Fo(0)1383 5228 y Fu(\()p Fq(t)1445 5240 y Fo(0)1501 5228 y Fu(+)18 b Fq(s)1623 5240 y Fo(0)1660 5228 y Fu(\))p Fp(k)1734 5204 y Fo(2)1800 5171 y Fu(+)g Fq(O)r Fu(\()p Fq(\026)2030 5137 y Fo(2)2068 5171 y Fq(")2107 5137 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(2)2350 5171 y Fq(;)c(\026)2437 5137 y Fo(2)2474 5171 y Fq(")2513 5137 y Fi(\027)t Fo(+)p Fi(p)p Fj(\000)p Fo(1)2724 5171 y Fq(;)g(\026")2850 5137 y Fi(p)2888 5171 y Fu(\))p Fq(e)2959 5137 y Fj(\000)p Fi(a=")3116 5171 y Fs(,)p eop %%Page: 4 12 4 11 bop 456 251 a Fo(4)962 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fs(wher)l(e)27 b Fq(s)726 462 y Fo(0)787 450 y Fq(<)f Fu(\026)-46 b Fq(s)913 462 y Fo(0)978 450 y Fs(ar)l(e)27 b(the)g(two)h(zer)l(os)f(of)h(the)f(Melnikov)i(function)f(\(asso)l (ciate)l(d)g(to)f(two)h(c)l(onse)l(c-)456 550 y(utive)h(homo)l(clinic)j (p)l(oints\),)f(closest)f(to)f(zer)l(o.)605 725 y Fu(W)-7 b(e)28 b(de\014ne)g(the)g(function)1440 905 y Fq(J)8 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fp(f)p Fq(h)1966 917 y Fo(0)2002 905 y Fq(;)14 b(h)2087 917 y Fo(1)2124 905 y Fp(g)p Fu(\()p Fq(x;)g(y)s(;)g(\022)r Fu(\).)456 1085 y(By)35 b(the)i(h)n(yp)r(othesis)e Fk(HP2)h Fu(on)f Fq(h)1558 1097 y Fo(1)1595 1085 y Fu(,)j Fq(J)44 b Fu(is)36 b(2)p Fq(\031)s Fu(-p)r(erio)r(dic)g(in)g Fq(\022)i Fu(and)e(has)f(zero)g(a)n(v)n(erage)e(with)456 1185 y(resp)r(ect)27 b(to)g Fq(\022)r Fu(.)38 b(Then)27 b(w)n(e)h(can)f(consider)f(its)i(F)-7 b(ourier)27 b(expansion)1445 1377 y Fq(J)8 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b Fp(\030)1881 1298 y Fg(X)1881 1477 y Fi(k)q Fj(6)p Fo(=0)2016 1377 y Fq(J)2062 1389 y Fi(k)2102 1377 y Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fq(e)2333 1343 y Fi(ik)q(\022)2431 1377 y Fu(.)456 1647 y(Moreo)n(v)n(er,)26 b(for)i(all)g Fq(k)g Fp(2)d Fl(Z)o Fu(,)e Fq(J)1393 1659 y Fi(k)1434 1647 y Fu(\()p Fq(\015)1509 1659 y Fo(0)1546 1647 y Fu(\()p Fq(u)p Fu(\)\))29 b(has)f(a)g(branc)n(hing)g(p)r(oin)n(t)h(of)f(order)f (at)i(most)f Fq(`)19 b Fu(+)f(1)29 b(at)456 1747 y Fq(u)22 b Fu(=)h Fp(\006)p Fq(ia)p Fu(.)36 b(Therefore,)26 b(near)h(the)h (singularit)n(y)e Fq(u)d Fu(=)f Fq(ia)p Fu(,)28 b Fq(J)2277 1759 y Fi(k)2318 1747 y Fu(\()p Fq(\015)2393 1759 y Fo(0)2430 1747 y Fu(\()p Fq(u)p Fu(\)\))g(has)f(the)h(form)935 2026 y Fq(J)981 2038 y Fi(k)1022 2026 y Fu(\()p Fq(\015)1097 2038 y Fo(0)1135 2026 y Fu(\()p Fq(u)p Fu(\)\))23 b(=)1580 1970 y(1)p 1400 2007 403 4 v 1400 2083 a(\()p Fq(u)18 b Fp(\000)g Fq(ia)p Fu(\))1686 2059 y Fi(`)p Fo(+1)1826 1859 y Fg(0)1826 2009 y(@)1898 2026 y Fq(J)1952 1991 y Fj(\000)1944 2051 y Fi(k)q(;)p Fo(0)2056 2026 y Fu(+)2151 1947 y Fg(X)2139 2126 y Fi(m)p Fj(\025)p Fo(1)2297 2026 y Fq(J)2351 1991 y Fj(\000)2343 2051 y Fi(k)q(;m)2463 2026 y Fu(\()p Fq(u)g Fp(\000)g Fq(ia)p Fu(\))2749 1992 y Fi(m=q)2878 1859 y Fg(1)2878 2009 y(A)456 2306 y Fu(and,)27 b(near)g(the)h(singularit)n(y)e Fq(u)d Fu(=)f Fp(\000)p Fq(ia)p Fu(,)27 b Fq(J)1771 2318 y Fi(k)1812 2306 y Fu(\()p Fq(\015)1887 2318 y Fo(0)1924 2306 y Fu(\()p Fq(u)p Fu(\)\),)i(has)e (the)h(form)924 2585 y Fq(J)970 2597 y Fi(k)1011 2585 y Fu(\()p Fq(\015)1086 2597 y Fo(0)1123 2585 y Fu(\()p Fq(u)p Fu(\)\))c(=)1568 2529 y(1)p 1388 2566 V 1388 2642 a(\()p Fq(u)18 b Fu(+)g Fq(ia)p Fu(\))1674 2618 y Fi(`)p Fo(+1)1814 2418 y Fg(0)1814 2568 y(@)1887 2585 y Fq(J)1941 2550 y Fo(+)1933 2610 y Fi(k)q(;)p Fo(0)2045 2585 y Fu(+)2140 2506 y Fg(X)2128 2685 y Fi(m)p Fj(\025)p Fo(1)2286 2585 y Fq(J)2340 2550 y Fo(+)2332 2610 y Fi(k)q(;m)2451 2585 y Fu(\()p Fq(u)g Fu(+)g Fq(ia)p Fu(\))2737 2551 y Fi(m=q)2867 2418 y Fg(1)2867 2568 y(A)2953 2585 y Fu(.)456 2891 y(W)-7 b(e)28 b(note)f(that)h Fq(J)1017 2855 y Fo(+)1009 2916 y Fi(k)q(;)p Fo(0)1126 2891 y Fu(=)p 1213 2809 192 4 v 22 w Fq(J)1267 2855 y Fj(\000)1259 2916 y(\000)p Fi(k)q(;)p Fo(0)1405 2891 y Fu(.)605 2993 y(W)-7 b(e)28 b(consider)f(the)h(follo)n (wing)e(condition:)565 3133 y Fk(HP6)41 b Fu(The)29 b(F)-7 b(ourier)27 b(co)r(e\016cien)n(ts)h Fq(J)1721 3145 y Fj(\006)p Fo(1)1839 3133 y Fu(ev)-5 b(aluated)28 b(on)g Fq(\015)2370 3145 y Fo(0)2408 3133 y Fu(\()p Fq(u)p Fu(\),)h(that)f(is) h Fq(J)2883 3145 y Fj(\006)p Fo(1)2972 3133 y Fu(\()p Fq(\015)3047 3145 y Fo(0)3084 3133 y Fu(\()p Fq(u)p Fu(\)\),)h(ha)n(v)n (e)794 3232 y(singularities)d(of)g(order)f(exactly)h Fq(`)18 b Fu(+)g(1)28 b(at)f(the)h(p)r(oin)n(ts)g Fq(u)22 b Fu(=)h Fp(\006)p Fq(ai)p Fu(.)605 3408 y Ft(Remark)h Fu(1.6)p Ft(.)34 b Fs(The)25 b(hyp)l(othesis)g Ff(HP6)f Fs(is)f(generic)i(b)l(e)l(c)l(ause)e(it)g(is)h(e)l(quivalent)f(to)h (supp)l(ose)456 3507 y(that)i(the)g(c)l(o)l(e\016cients)h Fq(J)1230 3472 y Fo(+)1222 3529 y Fj(\006)p Fo(1)p Fi(;)p Fo(0)1390 3507 y Fs(of)g(the)g(L)l(aur)l(ent)e(exp)l(ansion)i(of)g Fq(J)2442 3519 y Fj(\006)p Fo(1)2531 3507 y Fu(\()p Fq(\015)2606 3519 y Fo(0)2643 3507 y Fu(\()p Fq(u)p Fu(\)\))g Fs(ar)l(e)f(di\013er)l (ent)h(fr)l(om)456 3607 y(zer)l(o.)605 3782 y Fu(W)-7 b(e)32 b(can)e(obtain)h(an)g(asymptotic)g(expression)f(of)h(the)h (Melnik)n(o)n(v)e(function)h(and)h(conse-)456 3882 y(quen)n(tly)27 b(of)h(the)g(area)e(of)h(the)h(lob)r(e)g(and)f(of)h(the)g(angle.)605 4057 y Ft(Cor)n(ollar)-6 b(y)41 b Fu(1.1)p Ft(.)k Fs(If)37 b Ff(HP1-HP6)45 b Fs(hold,)40 b(then)d(for)h Fq(")d Fp(!)i Fu(0)2605 4027 y Fo(+)2659 4057 y Fs(,)i Fq(\026)d Fp(!)g Fu(0)h Fs(and)g(for)h(any)456 4157 y Fq(t)486 4169 y Fo(0)546 4157 y Fp(2)23 b Fl(R)p Fs(,)517 4371 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))82 b(=)h Fq(")1055 4337 y Fj(\000)p Fi(`)1250 4315 y Fu(4)p Fq(\031)p 1149 4352 295 4 v 1149 4428 a Fu(\000\()p Fq(`)18 b Fu(+)g(1\))1453 4371 y Fp(j)p Fq(J)1530 4336 y Fj(\000)1522 4394 y Fo(1)p Fi(;)p Fo(0)1612 4371 y Fp(j)c Fu(Re\()p Fq(e)1818 4337 y Fi(i)p Fo(\()p Fi(\022)r Fj(\000)p Fo(\()p Fi(`)p Fo(+1\))p Fi(\031)r(=)p Fo(2\))2254 4371 y Fq(e)2293 4337 y Fj(\000)p Fi(is=")p Fo(\))2495 4371 y Fu(\))p Fq(e)2566 4337 y Fj(\000)p Fi(a=")2742 4371 y Fu(+)k Fq(O)r Fu(\()p Fq(")2961 4337 y Fj(\000)p Fi(`)p Fo(+1)3129 4371 y Fq(e)3168 4337 y Fj(\000)p Fi(a=")3325 4371 y Fu(\))p Fs(,)723 4592 y Fq(A)83 b Fu(=)g Fq(\026")1105 4558 y Fi(\027)t Fo(+1)1341 4536 y Fu(8)p Fq(\031)p 1240 4573 V 1240 4649 a Fu(\000\()p Fq(`)18 b Fu(+)g(1\))1544 4592 y Fp(j)p Fq(J)1621 4556 y Fj(\000)1613 4614 y Fo(1)p Fi(;)p Fo(0)1703 4592 y Fp(j)p Fq(e)1765 4558 y Fj(\000)p Fi(a=")1941 4592 y Fu(+)g Fq(O)r Fu(\()p Fq(\026)2171 4558 y Fo(2)2209 4592 y Fq(")2248 4558 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)2406 4592 y Fq(;)c(\026)2493 4558 y Fo(2)2530 4592 y Fq(")2569 4558 y Fi(\027)t Fo(+)p Fi(p)p Fo(+1)2779 4592 y Fq(;)g(\026")2905 4558 y Fi(\027)t Fo(+2)3030 4592 y Fu(\))p Fq(e)3101 4558 y Fj(\000)p Fi(a=")3259 4592 y Fs(,)561 4813 y Fp(j)g Fu(sin)f Fq(#)p Fp(j)83 b Fu(=)g Fq(\026")1105 4778 y Fi(\027)t Fj(\000)p Fo(1)1342 4756 y Fu(4)p Fq(\031)p 1241 4793 V 1241 4869 a Fu(\000\()p Fq(`)18 b Fu(+)g(1\))1545 4813 y Fp(j)p Fq(J)1622 4777 y Fj(\000)1614 4835 y Fo(1)p Fi(;)p Fo(0)1704 4813 y Fp(j)1971 4756 y Fu(1)p 1737 4793 510 4 v 1737 4869 a Fp(k)12 b Fu(_)-35 b Fq(\015)1822 4881 y Fo(0)1859 4869 y Fu(\()p Fq(t)1921 4881 y Fo(0)1977 4869 y Fu(+)18 b Fq(s)2099 4881 y Fo(0)2136 4869 y Fu(\))p Fp(k)2210 4845 y Fo(2)2257 4813 y Fq(e)2296 4778 y Fj(\000)p Fi(a=")1016 5001 y Fu(+)p Fq(O)r Fu(\()p Fq(\026)1228 4967 y Fo(2)1266 5001 y Fq(")1305 4967 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(2)1548 5001 y Fq(;)c(\026)1635 4967 y Fo(2)1672 5001 y Fq(")1711 4967 y Fi(\027)t Fo(+)p Fi(p)p Fj(\000)p Fo(1)1922 5001 y Fq(;)g(\026")2048 4967 y Fi(\027)2089 5001 y Fu(\))p Fq(e)2160 4967 y Fj(\000)p Fi(a=")2317 5001 y Fs(,)456 5182 y(wher)l(e)30 b Fq(J)744 5147 y Fj(\000)736 5204 y Fo(1)p Fi(;)p Fo(0)849 5182 y Fu(=)23 b Fp(j)p Fq(J)1014 5147 y Fj(\000)1006 5204 y Fo(1)p Fi(;)p Fo(0)1096 5182 y Fp(j)p Fq(e)1158 5152 y Fi(i\022)1248 5182 y Fs(and)31 b Fu(\000)e Fs(is)h(the)g(Gamma)h (function.)p eop %%Page: 5 13 5 12 bop 1717 251 a Fo(1.3.)29 b(EXAMPLE)1227 b(5)1671 450 y Fk(1.3.)46 b(Example)605 600 y Fu(An)28 b(example)f(of)h(unp)r (erturb)r(ed)g(Hamiltonian)f(system)h(satisfying)g Fk(HP1)f Fu(is)h(giv)n(en)e(b)n(y)1565 783 y Fq(h)1613 795 y Fo(0)1650 783 y Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)1964 726 y Fq(y)2008 696 y Fo(2)p 1964 764 81 4 v 1983 840 a Fu(2)2073 783 y(+)18 b Fq(V)h Fu(\()p Fq(x)p Fu(\))-1878 b(\(1.3\))456 945 y(where)27 b Fq(V)19 b Fu(\()p Fq(x)p Fu(\))28 b(is)g(a)f(p)r (olynomial)g(of)g(the)h(form)1351 1087 y Fq(V)19 b Fu(\()p Fq(x)p Fu(\))24 b(=)f Fp(\000)p Fq(x)1753 1053 y Fi(n)1817 1087 y Fu(+)18 b Fq(x)1947 1053 y Fo(2\()p Fi(n)p Fj(\000)p Fo(1\))2162 1087 y Fu(,)139 b Fq(n)23 b Fp(\025)f Fu(3.)456 1226 y(Indeed,)32 b(the)f(Hamiltonian)g(system)g(has)f(a)h(homo)r (clinic)f(orbit)h(con)n(tained)f(in)h Fq(H)7 b Fu(\()p Fq(x;)14 b(y)s Fu(\))30 b(=)e(0.)456 1326 y(Let)f Fq(\015)5 b Fu(\()p Fq(t)p Fu(\))28 b(b)r(e)g(the)g(parameterization)e(of)h(the)h (homo)r(clinic)g(orbit)f(suc)n(h)g(that)h Fq(\015)5 b Fu(\(0\))23 b(=)f(\(1)p Fq(;)14 b Fu(0\).)605 1426 y(W)-7 b(e)19 b(can)f(giv)n(e)f(an)h(explicit)h(expression)e(for)h(the)g(homo) r(clinic)h(orbit)f Fq(\015)2721 1438 y Fo(0)2758 1426 y Fu(\()p Fq(t)p Fu(\))23 b(=)g(\()p Fq(\013)3048 1438 y Fo(0)3086 1426 y Fu(\()p Fq(t)p Fu(\))p Fq(;)14 b(\014)3264 1438 y Fo(0)3301 1426 y Fu(\()p Fq(t)p Fu(\)\):)841 1633 y Fq(\013)894 1645 y Fo(0)932 1633 y Fu(\()p Fq(t)p Fu(\))23 b(=)1137 1516 y Fg(\022)1440 1577 y Fu(2)p 1208 1614 505 4 v 1208 1690 a(2)18 b(+)g(\()p Fq(n)h Fp(\000)f Fu(2\))1609 1666 y Fo(2)1646 1690 y Fq(t)1676 1666 y Fo(2)1723 1516 y Fg(\023)1784 1533 y Fo(1)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2047 1633 y Fu(,)115 b Fq(\014)2232 1645 y Fo(0)2269 1633 y Fu(\()p Fq(t)p Fu(\))24 b(=)f Fp(\000)p Fu(\()p Fq(n)18 b Fp(\000)g Fu(2\))p Fq(t)p Fu(\()p Fq(\013)2912 1645 y Fo(0)2949 1633 y Fu(\()p Fq(t)p Fu(\)\))3075 1599 y Fi(n)p Fj(\000)p Fo(1)3206 1633 y Fu(.)456 1830 y(Therefore,)i(the)f(homo)r(clinic)h(orbit)f(has)g (singularities)f(at)h(the)h(p)r(oin)n(ts)f Fp(\006)p Fq(ia)g Fu(with)h Fq(a)j Fu(=)3143 1762 y Fp(p)p 3213 1762 42 4 v 3213 1830 a Fu(2)o Fq(=)p Fu(\()p Fq(n)r Fp(\000)456 1930 y Fu(2\))30 b(whic)n(h)h(are)f(branc)n(hing)f(p)r(oin) n(ts)i(\(if)h Fq(n)c Fu(=)g(3,)j(are)f(p)r(oles\).)46 b(It)31 b(is)g(not)g(di\016cult)g(to)g(see)f(that,)456 2029 y(near)c(the)i(singularities)f Fp(\006)p Fq(ia)p Fu(,)f(the)i(\014rst)g(comp)r(onen)n(t)f(of)h Fq(\015)2310 2041 y Fo(0)2375 2029 y Fu(reads)e(as)1490 2146 y Fq(C)1549 2158 y Fj(\006)p 1289 2183 518 4 v 1289 2260 a Fu(\()p Fq(t)19 b Fp(\006)f Fq(ia)p Fu(\))1558 2236 y Fo(1)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))1830 2202 y Fu(\()q(1)g(+)g Fq(O)r Fu(\()p Fq(t)h Fp(\006)f Fq(ia)p Fu(\))2340 2160 y Fo(1)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2589 2202 y Fu(\))456 2398 y(with)28 b Fq(C)704 2410 y Fj(\000)783 2398 y Fu(=)23 b(\()p Fq(a=)p Fu(2\))1063 2368 y Fo(1)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))1311 2398 y Fq(e)1350 2368 y Fj(\000)p Fi(i\031)r(=)p Fo(2\()p Fi(n)p Fj(\000)p Fo(2\))1743 2398 y Fu(and)28 b Fq(C)1964 2410 y Fo(+)2042 2398 y Fu(=)p 2130 2331 116 4 v 23 w Fq(C)2189 2410 y Fj(\000)2245 2398 y Fu(.)605 2497 y(W)-7 b(e)28 b(consider)f(a)g (family)g(of)h(p)r(erturbations)f(giv)n(en)g(b)n(y)1020 2632 y Fq(\026")1109 2598 y Fi(p)1147 2632 y Fq(h)1195 2644 y Fo(1)1232 2632 y Fu(\()p Fq(x;)14 b(y)s(;)g(t=")p Fu(\))23 b(=)g Fq(\026")1772 2598 y Fi(p)1810 2632 y Fq(x)1857 2598 y Fi(k)1912 2632 y Fu(cos)o(\()p Fq(t=")p Fu(\),)138 b Fq(p)23 b Fp(\025)g Fq(k)s(=)p Fu(\()p Fq(n)18 b Fp(\000)g Fu(2\).)456 2771 y(In)33 b(this)h(case,)g Fq(`)e Fu(=)h Fq(k)s(=)p Fu(\()p Fq(n)21 b Fp(\000)h Fu(2\).)54 b(Of)34 b(course)e(w)n(e)h(assume)g(that)g Fq(k)k Fu(satis\014es)32 b(the)i(h)n(yp)r(othesis)456 2871 y Fk(HP4)p Fu(,)27 b(that)h(is,)f(2)p Fq(k)21 b Fp(\000)d Fu(2)23 b Fp(\025)g Fq(n)p Fu(.)36 b(Hence)28 b(in)g(this)g(case)f Fq(`)22 b Fp(\025)h Fu(\()p Fq(n)18 b Fu(+)g(2\))p Fq(=)p Fu(\(2)p Fq(n)g Fp(\000)g Fu(4\).)605 2971 y(Then,)26 b(b)n(y)g(Corollary)d(1.1)i(the)h(area)f(of)g(the)h (lob)r(e)g(generated)f(b)r(et)n(w)n(een)g(t)n(w)n(o)g(consecutiv)n(e) 456 3070 y(homo)r(clinic)i(p)r(oin)n(ts)h(satis\014es)e(the)i (asymptotic)f(expression)1392 3239 y Fq(A)c Fp(\030)g Fq(\026")1654 3205 y Fi(\027)t Fo(+1)1898 3183 y Fu(4)p Fq(\031)p 1789 3220 311 4 v 1789 3326 a Fu(\000)1855 3233 y Fg(\020)1959 3293 y Fi(k)p 1914 3307 127 4 v 1914 3354 a(n)p Fj(\000)p Fo(2)2050 3233 y Fg(\021)2110 3239 y Fp(j)p Fq(C)2192 3251 y Fj(\000)2248 3239 y Fp(j)2271 3205 y Fi(k)2312 3239 y Fq(e)2351 3205 y Fj(\000)p Fi(a=")456 3485 y Fu(where)k Fq(\027)h Fu(=)23 b Fq(p)18 b Fp(\000)g Fq(k)s(=)p Fu(\()p Fq(n)g Fp(\000)g Fu(2\).)p eop %%Page: 6 14 6 13 bop 524 566 a Fw(2.)61 b(Analytic)45 b(prop)t(erties)g(of)h(the)f (homo)t(clinic)g(orbit)g(of)1217 716 y(the)g(unp)t(erturb)t(ed)f (system)1224 998 y Fk(2.1.)j(In)m(tro)s(duction)32 b(and)g(main)f (results)605 1147 y Fu(The)h(purp)r(ose)f(of)g(this)h(c)n(hapter)e(is)i (to)f(obtain)h(the)f(asymptotic)g(b)r(eha)n(viour)g(of)g(the)h(ho-)456 1247 y(mo)r(clinic)27 b(orbits)g(to)h(parab)r(olic)e(p)r(oin)n(ts)i(of) f(Hamiltonians)g(systems)g(of)h(the)g(form)1538 1396 y Fq(H)7 b Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e Fq(y)1961 1362 y Fo(2)1998 1396 y Fq(=)p Fu(2)c(+)g Fq(V)h Fu(\()p Fq(x)p Fu(\))456 1551 y(with)g Fq(V)f Fu(\()p Fq(x)p Fu(\))i(b)r(eing)e(an)h(analytic)e(function,)k(for)d(complex)g(v)-5 b(alues)18 b(of)g(time)h(in)g(a)f(certain)g(domain.)605 1650 y(W)-7 b(e)33 b(assume)f(that)h(the)g(origin)e(is)i(a)f(\014xed)h (p)r(oin)n(t)f(of)h(the)g(corresp)r(onding)e(Hamiltonian)456 1750 y(equation)1681 1899 y(_)-38 b Fq(x)84 b Fu(=)e Fq(y)1685 2024 y Fu(_)-38 b Fq(y)86 b Fu(=)c Fp(\000)p Fq(V)2076 1990 y Fj(0)2099 2024 y Fu(\()p Fq(x)p Fu(\).)456 2173 y(It)28 b(is)f(not)h(restrictiv)n(e)e(to)h(assume)g(that)h Fq(V)19 b Fu(\(0\))k(=)g(0.)36 b(W)-7 b(e)28 b(supp)r(ose)g(that)f Fq(V)47 b Fu(is)28 b(of)f(the)h(form)1608 2323 y Fq(V)19 b Fu(\()p Fq(x)p Fu(\))24 b(=)f Fq(a)1942 2335 y Fi(n)1987 2323 y Fq(x)2034 2288 y Fi(n)2098 2323 y Fu(+)18 b Fq(:)c(:)g(:)456 2472 y Fu(with)28 b Fq(n)23 b Fp(\025)f Fu(3)27 b(and)h Fq(a)1080 2484 y Fi(n)1148 2472 y Fp(6)p Fu(=)23 b(0.)605 2572 y(In)40 b(suc)n(h)f(case)g(the)i(origin)d(is)i(a)f(parab)r(olic)g (p)r(oin)n(t,)k(that)d(is,)j(the)d(linear)f(part)g(of)h(the)456 2672 y(equation)31 b(at)i(\(0)p Fq(;)14 b Fu(0\))32 b(has)f(a)h(double) h(zero)e(eigen)n(v)-5 b(alue.)50 b(Assuming)32 b(that)h(the)g(origin)e (has)h(an)456 2771 y(in)n(v)-5 b(arian)n(t)27 b(curv)n(e)g(passing)g (through)g(the)i(origin,)e(the)i(solution)e(on)h(this)h(curv)n(e)e(has) g(to)h(lie)h(on)456 2871 y(the)f(energy)e(lev)n(el)h Fq(H)7 b Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e(0.)37 b(Then)1580 3031 y(_)-37 b Fq(x)23 b Fu(=)g Fq(y)j Fu(=)c Fp(\006)1943 2956 y Fg(p)p 2026 2956 285 4 v 75 x Fp(\000)p Fu(2)p Fq(V)c Fu(\()p Fq(x)p Fu(\))q(.)456 3197 y(Hence)26 b(w)n(e)g(will)h(ha)n(v)n(e)e(that)42 b(_)-38 b Fq(x)24 b Fu(=)e Fq(ax)1596 3167 y Fi(k)1654 3197 y Fu(+)16 b Fq(:)e(:)g(:)40 b Fu(or)g(_)-37 b Fq(x)23 b Fu(=)g Fq(ax)2222 3167 y Fi(k)q Fo(+1)p Fi(=)p Fo(2)2430 3197 y Fu(+)16 b Fq(:)e(:)g(:)41 b Fu(according)24 b(to)j(the)g(cases)456 3297 y Fq(n)22 b Fu(=)h(2)p Fq(k)30 b Fu(or)d Fq(n)c Fu(=)f(2)p Fq(k)f Fu(+)d(1,)27 b Fq(k)f Fp(2)e Fl(N)t Fu(.)605 3396 y(F)-7 b(or)27 b(the)h(sak)n(e)e(of)i(generalit)n(y)e(w)n (e)h(consider)g(the)h(case)e Fq(k)g Fp(2)e Fl(R)p Fu(.)43 b(W)-7 b(e)28 b(de\014ne)g(the)f(set)1090 3546 y Fq(U)k Fu(=)23 b Fl(D)14 b Fu(\(0)p Fq(;)g(r)r Fu(\))25 b Fp(\000)18 b(f)p Fq(z)26 b Fp(2)d Fl(C)44 b Fu(:)24 b(Im)14 b Fq(z)26 b Fu(=)c(0)p Fq(;)14 b Fu(Re)g Fq(z)26 b Fp(\024)d Fu(0)p Fp(g)f(\032)g Fl(C)15 b Fu(.)456 3695 y(The)30 b(main)h(result)f(of)h (this)f(c)n(hapter)g(is)h(the)f(follo)n(wing)g(prop)r(osition)f(from)i (whic)n(h)f(w)n(e)g(deriv)n(e)456 3795 y(the)e(asymptotic)f(represen)n (tation)f(of)h Fq(x)p Fu(\()p Fq(t)p Fu(\),)i(and)e(then)h Fq(y)s Fu(\()p Fq(t)p Fu(\))g(follo)n(ws)f(from)g Fq(y)s Fu(\()p Fq(t)p Fu(\))c(=)38 b(_)-38 b Fq(x)q Fu(\()p Fq(t)p Fu(\).)605 3953 y Ft(Pr)n(oposition)31 b Fu(2.1)p Ft(.)40 b Fs(L)l(et)29 b Fq(f)39 b Fs(b)l(e)29 b(an)h(analytic)h (function)f(on)g Fq(U)9 b Fs(.)38 b(Supp)l(ose)30 b(that)1607 4102 y Fq(f)9 b Fu(\()p Fq(x)p Fu(\))24 b(=)e Fq(ax)1970 4068 y Fi(k)2030 4102 y Fu(+)c Fq(g)s Fu(\()p Fq(x)p Fu(\))p Fs(,)456 4265 y(with)30 b Fp(j)p Fq(g)s Fu(\()p Fq(x)p Fu(\))p Fp(j)24 b(\024)e Fq(B)t Fp(j)p Fq(x)p Fp(j)1107 4235 y Fi(`)1140 4265 y Fs(,)30 b Fq(k)s Fs(,)g Fq(`)23 b Fp(2)g Fl(R)p Fs(,)36 b Fu(1)23 b Fq(<)g(k)i(<)e(`)29 b Fs(and)i Fq(a)23 b(<)f Fu(0)p Fs(.)38 b(Consider)32 b(the)e(e)l(quation)1792 4414 y Fu(_)-38 b Fq(x)24 b Fu(=)f Fq(f)9 b Fu(\()p Fq(x)p Fu(\))p Fs(.)456 4569 y(Then,)30 b(ther)l(e)g(is)g(an)g(analytic)h(solution)f Fq(')p Fu(\()p Fq(t)p Fu(\))h Fs(de\014ne)l(d)f(on)1256 4718 y Fu(\012\()p Fq(T)7 b(;)14 b(\013)p Fu(\))23 b(=)g Fp(f)p Fq(t)f Fp(2)i Fl(C)44 b Fu(:)23 b Fp(j)p Fq(t)p Fp(j)g Fq(>)g(T)7 b(;)14 b Fp(j)g Fu(arg)e Fq(t)p Fp(j)24 b Fq(<)e(\013)p Fp(g)456 4872 y Fs(with)30 b Fq(\013)23 b(<)g Fu(min)p Fp(f)p Fq(\031)s(;)1077 4840 y Fi(\031)p 1077 4854 41 4 v 1080 4901 a(p)1128 4872 y Fp(g)29 b Fs(and)h Fq(T)41 b Fs(big)31 b(enough,)f(such)g(that)1565 5040 y Fq(')p Fu(\()p Fq(t)p Fu(\))24 b(=)e Fq(ct)1890 5006 y Fj(\000)p Fi(p)1999 5040 y Fu(+)c Fq(O)r Fu(\()p Fq(t)2209 5006 y Fj(\000)p Fi(\027)2303 5040 y Fu(\))456 5195 y Fs(with)30 b Fq(p)23 b Fu(=)f(1)p Fq(=)p Fu(\()p Fq(k)f Fp(\000)d Fu(1\))p Fs(,)30 b Fq(p)23 b(<)f(\027)29 b(<)22 b Fu(min)q Fp(f)p Fq(q)s(;)14 b(p)j Fu(+)i(1)p Fp(g)p Fs(,)29 b Fq(q)d Fu(=)c Fq(p)p Fu(\(1)c(+)g Fq(`)h Fp(\000)f Fq(k)s Fu(\))29 b Fs(and)i Fq(c)23 b Fu(=)f(\()p Fp(\000)p Fq(p=a)p Fu(\))3204 5165 y Fi(p)3242 5195 y Fs(.)1933 5365 y Fo(6)p eop %%Page: 7 15 7 14 bop 1258 251 a Fo(2.1.)29 b(INTR)n(ODUCTION)i(AND)e(MAIN)g(RESUL) -5 b(TS)768 b(7)605 450 y Ft(Remark)28 b Fu(2.1)p Ft(.)37 b Fs(Sinc)l(e)26 b(the)h(e)l(quation)f(is)h(one-dimensional,)i(every)e (solution)g(which)h(go)l(es)456 550 y(to)g(zer)l(o)h(as)g Fu(Re)14 b Fq(t)29 b Fs(go)l(es)g(to)g(+)p Fp(1)p Fs(,)g(with)g(the)g (r)l(e)l(al)g(p)l(art)g(of)h(the)f(initial)h(c)l(ondition)g(p)l (ositive,)g(is)g(of)456 649 y(the)g(form)g Fq(')p Fu(\()p Fq(t)19 b Fu(+)f Fq(\034)9 b Fu(\))p Fs(.)605 807 y Fu(Note)23 b(that)h(if)g(w)n(e)f(assume)f(that)i(the)g(leading)e(term)i(of)f Fq(')h Fu(is)f Fq(ct)2553 777 y Fj(\000)p Fi(p)2666 807 y Fu(a)g(formal)g(computation)456 907 y(already)j(sho)n(ws)g(that)959 1207 y Fq(p)d Fu(=)1195 1151 y(1)p 1122 1188 189 4 v 1122 1264 a Fq(k)e Fp(\000)d Fu(1)1320 1207 y(,)333 b Fq(c)22 b Fu(=)1822 1090 y Fg(\022)1893 1151 y Fp(\000)p Fq(p)p 1893 1188 107 4 v 1924 1264 a(a)2010 1090 y Fg(\023)2071 1108 y Fi(p)2132 1207 y Fu(=)2220 1090 y Fg(\022)2419 1151 y Fu(1)p 2291 1188 298 4 v 2291 1264 a(\(1)c Fp(\000)g Fq(k)s Fu(\))p Fq(a)2598 1090 y Fg(\023)2659 1108 y Fo(1)p Fi(=)p Fo(\()p Fi(k)q Fj(\000)p Fo(1\))2918 1207 y Fu(.)456 1402 y(The)27 b(pro)r(of)g(of)h(the)g(Prop)r(osition)e(2.1)h(is)g(giv)n (en)g(in)h(the)g(next)f(section.)605 1501 y(If)32 b(w)n(e)g(restrict)f (us)h(to)g(consider)f(functions)64 b Fq(f)41 b Fu(of)31 b(the)i(form)e Fq(f)9 b Fu(\()p Fq(x)p Fu(\))31 b(=)f Fq(x)2911 1471 y Fi(n=)p Fo(2)3024 1501 y Fq(g)s Fu(\()p Fq(x)p Fu(\),)k(whic)n(h)456 1601 y(are)27 b(the)h(ones)f(w)n(e)h(will) g(deal)f(with)i(in)f(the)g(next)g(c)n(hapters,)f(w)n(e)h(obtain)f(a)h (bigger)e(domain)i(for)456 1700 y(the)g(solution.)605 1858 y Ft(Pr)n(oposition)38 b Fu(2.2)p Ft(.)45 b Fs(L)l(et)35 b Fq(f)9 b Fu(\()p Fq(x)p Fu(\))36 b(=)e Fq(x)1817 1828 y Fi(n=)p Fo(2)1930 1858 y Fq(g)s Fu(\()p Fq(x)p Fu(\))p Fs(,)k Fq(n)d Fp(2)g Fl(N)t Fs(,)44 b Fq(g)s Fu(\(0\))35 b Fq(<)f Fu(0)h Fs(and)i Fq(g)i Fs(analytic)e(in)456 1958 y Fl(D)14 b Fu(\(0)p Fq(;)g(r)r Fu(\))p Fs(.)66 b(Then)38 b(the)f(e)l(quation)52 b Fu(_)-38 b Fq(x)36 b Fu(=)g Fq(f)9 b Fu(\()p Fq(x)p Fu(\))37 b Fs(has)h(an)f(analytic)h (solution)f Fq(')p Fs(,)j Fq(')p Fu(\()p Fq(t)p Fu(\))c(=)g Fq(ct)3266 1928 y Fj(\000)p Fi(p)3380 1958 y Fu(+)456 2057 y Fq(O)r Fu(\()p Fq(t)583 2027 y Fj(\000)p Fi(\027)677 2057 y Fu(\))p Fs(,)30 b(de\014ne)l(d)g(on)g Fu(\012\()p Fq(T)7 b(;)14 b(\031)s Fu(\))p Fs(.)605 2215 y Fu(The)31 b(pro)r(of)f(is)h(almost)f(the)h(same)f(as)g(the)i(one)e(of)h(Prop)r (osition)e(2.1.)45 b(W)-7 b(e)32 b(only)e(ha)n(v)n(e)g(to)456 2315 y(tak)n(e)f(in)n(to)g(accoun)n(t)g(that)h Fq(x)1346 2285 y Fi(n=)p Fo(2)1488 2315 y Fu(is)f(con)n(tin)n(ued)h(analytically) e(to)i(its)f(Riemman)h(surface)f(with)456 2414 y Fp(j)p Fq(x)526 2384 y Fi(n=)p Fo(2)638 2414 y Fp(j)24 b(\024)e(j)p Fq(x)p Fp(j)865 2384 y Fi(n=)p Fo(2)978 2414 y Fu(.)605 2572 y Ft(Remark)42 b Fu(2.2)p Ft(.)47 b Fs(As)38 b(a)i(c)l(onse)l (quenc)l(e)f(of)h(Pr)l(op)l(osition)g(2.2,)k(if)c(the)f(system)g(2.1)h (has)g(a)456 2672 y(homo)l(clinic)30 b(orbit,)g Fq(\033)s Fs(,)g(it)e(has)i(not)e(p)l(erio)l(ds.)40 b(This)29 b(is)g(b)l(e)l(c)l (ause)g Fu(lim)2569 2687 y Fj(j)p Fi(t)p Fj(j!)p Fo(+)p Fj(1)2835 2672 y Fq(\033)s Fu(\()p Fq(t)p Fu(\))24 b(=)e(0)p Fs(.)38 b(This)30 b(is)456 2771 y(in)g(c)l(ontr)l(ast)f(with)h(the)g (hyp)l(erb)l(olic)i(c)l(ase)f(wher)l(e)f(always)h(ther)l(e)f(exists)g (an)g(imaginary)h(p)l(erio)l(d.)456 2871 y(\(Se)l(e)e Fu([)p Fk(F)-8 b(o2)p Fu(])p Fs(\))605 3029 y Fu(F)h(rom)19 b(the)i(previous)d(results)i(w)n(e)f(easily)g(obtain)h(that)g(a)f(homo) r(clinic)h(orbit)g(to)f(a)h(parab)r(olic)456 3128 y(p)r(oin)n(t)27 b(has)g(singularities)g(and)g(they)h(do)f(not)h(accum)n(ulate)f(to)g (the)h(real)f(axis:)605 3286 y Ft(Pr)n(oposition)d Fu(2.3)p Ft(.)33 b Fs(L)l(et)23 b Fq(H)7 b Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e Fq(y)1859 3256 y Fo(2)1896 3286 y Fq(=)p Fu(2)5 b(+)g Fq(V)17 b Fu(\()p Fq(x)p Fu(\))24 b Fs(b)l(e)g(a)f (Hamiltonian)i(with)f(a)f(p)l(ar)l(ab)l(olic)456 3386 y(e)l(quilibrium)30 b(p)l(oint)g(and)h Fq(\033)s Fu(\()p Fq(t)p Fu(\))f Fs(a)g(homo)l(clinic)i(orbit)f(asso)l(ciate)l(d)g(to)f (it.)38 b(Then)679 3510 y Fu(1\))j Fq(\033)33 b Fs(has)e (singularities,)679 3610 y Fu(2\))41 b Fs(ther)l(e)29 b(exists)f Fq(\021)e(>)d Fu(0)28 b Fs(such)h(that)f Fq(\033)k Fs(is)d(analytic)h(in)f(the)f(strip)h Fp(f)p Fq(t)23 b Fp(2)g Fl(C)44 b Fu(:)24 b Fp(j)14 b Fu(Im)f Fq(t)p Fp(j)24 b Fq(<)e(\021)s Fp(g)p Fs(.)605 3767 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)28 b(ma)n(y)f(assume)g(that)g(the)h(equilibrium)g(p)r (oin)n(t)g(is)f(the)h(origin.)36 b(Therefore)1725 3916 y(lim)1678 3966 y Fi(t)p Fj(!\0061)1901 3916 y Fq(\033)s Fu(\()p Fq(t)p Fu(\))25 b(=)d(0.)456 4097 y(Applying)28 b(Prop)r(osition)e(2.2)i(to)g(the)g(equation)g(reduced)g(to)g(the)h (stable)e(in)n(v)-5 b(arian)n(t)27 b(manifold)456 4196 y(when)35 b Fq(t)h Fp(!)g Fu(+)p Fp(1)g Fu(and)f(to)g(the)h(unstable)f (one)g(when)h Fq(t)g Fp(!)g(\0001)f Fu(\(c)n(hanging)g Fq(t)g Fu(b)n(y)g Fp(\000)p Fq(t)p Fu(\))h(and)456 4296 y(taking)27 b(in)n(to)g(accoun)n(t)g(Remark)f(2.1,)h(w)n(e)g(obtain)h (that)g(there)f(exist)g Fq(T)12 b Fu(,)27 b Fq(t)2766 4308 y Fo(1)2831 4296 y Fu(and)h Fq(t)3023 4308 y Fo(2)3087 4296 y Fu(suc)n(h)g(that)766 4490 y Fq(\033)s Fu(\()p Fq(t)p Fu(\))84 b(=)1266 4434 y Fq(c)1302 4446 y Fo(1)p 1152 4471 302 4 v 1152 4547 a Fu(\()p Fq(t)18 b Fp(\000)g Fq(t)1345 4559 y Fo(1)1383 4547 y Fu(\))1415 4523 y Fi(p)1482 4490 y Fu(+)g Fq(O)1644 4373 y Fg(\022)1847 4434 y Fu(1)p 1715 4471 305 4 v 1715 4547 a(\()p Fq(t)h Fp(\000)f Fq(t)1909 4559 y Fo(1)1946 4547 y Fu(\))1978 4523 y Fi(\027)2030 4373 y Fg(\023)2105 4490 y Fu(,)166 b Fp(j)p Fq(t)p Fp(j)23 b(\025)g Fq(T)12 b Fu(,)26 b Fp(j)14 b Fu(arg)g Fq(t)p Fp(j)23 b Fq(<)g(\031)s Fu(,)766 4723 y Fq(\033)s Fu(\()p Fq(t)p Fu(\))84 b(=)1266 4666 y Fq(c)1302 4678 y Fo(2)p 1152 4703 302 4 v 1152 4779 a Fu(\()p Fq(t)18 b Fp(\000)g Fq(t)1345 4791 y Fo(2)1383 4779 y Fu(\))1415 4755 y Fi(p)1482 4723 y Fu(+)g Fq(O)1644 4605 y Fg(\022)1847 4666 y Fu(1)p 1715 4703 305 4 v 1715 4779 a(\()p Fq(t)h Fp(\000)f Fq(t)1909 4791 y Fo(2)1946 4779 y Fu(\))1978 4755 y Fi(\027)2030 4605 y Fg(\023)2105 4723 y Fu(,)166 b Fp(j)p Fq(t)p Fp(j)23 b(\025)g Fq(T)12 b Fu(,)26 b Fp(j)p Fq(\031)c Fp(\000)c Fu(arg)c Fq(t)p Fp(j)23 b Fq(<)g(\031)s Fu(.)456 4917 y(If)32 b(w)n(e)g(supp)r(ose)f(that)i Fq(\033)i Fu(is)d(an)g(en)n(tire) g(function,)i(the)e(previous)f(expressions)f(imply)j(that)f Fq(\033)456 5016 y Fu(is)d(b)r(ounded)h(outside)g(the)g(disc)g(of)f (radius)g Fq(T)12 b Fu(.)42 b(Then,)31 b(b)n(y)e(Liouville's)g (theorem,)h(it)g(m)n(ust)g(b)r(e)456 5116 y(constan)n(t.)40 b(This)30 b(con)n(tradiction)d(implies)j(that)f Fq(\033)k Fu(has)c(at)g(least)f(one)h(singularit)n(y)f(whic)n(h)h(has)456 5216 y(to)e(b)r(e)h(in)g Fl(D)14 b Fu(\(0)p Fq(;)g(T)e Fu(\).)p eop %%Page: 8 16 8 15 bop 456 251 a Fo(8)962 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(T)-7 b(o)37 b(pro)n(v)n(e)e(the)i(existence)f(of)h Fq(\021)42 b(>)c Fu(0)e(w)n(e)h(just)g(ha)n(v)n(e)f(to)h(note)f(that)i Fq(\033)i Fu(is)d(analytic)f(on)456 550 y(Re)13 b Fq(t)31 b(>)g(T)43 b Fu(and)32 b(on)g(Re)14 b Fq(t)31 b(<)f Fp(\000)p Fq(T)12 b Fu(.)50 b(Since)33 b Fq(\033)i Fu(is)e(analytic)e(on)h Fl(R)p Fu(,)40 b(and)32 b(therefore)f(on)h([)p Fp(\000)p 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Fu(\()p Fq(u)18 b Fp(\000)g Fq(u)1635 1301 y Fj(\003)1673 1325 y Fu(\))1705 1301 y Fo(2)p Fi(=)p Fo(\()p Fi(m)p Fj(\000)p Fo(2\))1982 1267 y Fu(\(1)g(+)g Fq(O)r Fu(\()p Fq(u)h Fp(\000)f Fq(u)2452 1232 y Fj(\003)2490 1267 y Fu(\))2522 1232 y Fo(2)p Fi(=)p Fo(\()p Fi(m)p Fj(\000)p Fo(2\))2789 1267 y Fu(\))-2365 b(\(2.1\))969 1498 y Fq(\014)1016 1510 y Fo(0)1053 1498 y Fu(\()p Fq(u)p Fu(\))83 b(=)g Fp(\000)1722 1442 y Fq(C)1787 1411 y Fj(0)p 1470 1479 593 4 v 1470 1556 a Fu(\()p Fq(u)19 b Fp(\000)f Fq(u)1700 1532 y Fj(\003)1737 1556 y Fu(\))1769 1532 y Fi(m=)p Fo(\()p Fi(m)p Fj(\000)p Fo(2\))2072 1498 y Fu(\(1)h(+)f Fq(O)r Fu(\()p Fq(u)h Fp(\000)f Fq(u)2543 1463 y Fj(\003)2580 1498 y Fu(\))2612 1463 y Fo(2)p Fi(=)p Fo(\()p Fi(m)p Fj(\000)p Fo(2\))2880 1498 y Fu(\))p Fs(.)-2481 b Fu(\(2.2\))605 1722 y Ft(Pr)n(oof.)41 b Fu(It)g(is)f(clear)f(that)i Fq(\013)1589 1734 y Fo(0)1667 1722 y Fu(is)f(a)h(solution)e(of)i(the)g (equation)54 b(_)-37 b Fq(x)45 b Fu(=)f Fp(\006)3054 1651 y Fg(p)p 3136 1651 285 4 v 3136 1722 a Fp(\000)p Fu(2)p Fq(V)18 b Fu(\()p Fq(x)p Fu(\))q(.)456 1821 y(P)n(erforming)25 b(the)j(c)n(hange)f Fq(w)e Fu(=)e(1)p Fq(=x)k Fu(w)n(e)g(obtain)h(the)g (equation)1178 1974 y Fq(du)p 1171 2011 105 4 v 1171 2087 a(dw)1309 2030 y Fu(=)1897 1974 y Fp(\007)p Fq(w)2023 1944 y Fi(m=)p Fo(2)p Fj(\000)p Fo(2)p 1407 2011 1323 4 v 1407 2028 a Fg(p)p 1490 2028 1240 4 v 71 x Fp(\000)p Fu(2\()p Fq(a)1673 2111 y Fi(m)1753 2099 y Fu(+)18 b Fq(w)r(a)1941 2111 y Fi(m)p Fj(\000)p Fo(1)2108 2099 y Fu(+)h Fq(:)14 b(:)g(:)k Fu(+)g Fq(w)2451 2075 y Fi(m)p Fj(\000)p Fi(n)2608 2099 y Fq(a)2652 2111 y Fi(n)2697 2099 y Fu(\))456 2241 y(whic)n(h,)27 b(in)h(a)f(neigh)n(b)r(orho)r(o)r (d)g(of)g Fq(w)f Fu(=)d(0,)k(can)g(b)r(e)h(written)g(as)1484 2373 y Fq(du)p 1477 2410 105 4 v 1477 2486 a(dw)1615 2429 y Fu(=)22 b Fq(w)1763 2395 y Fi(m=)p Fo(2)p Fj(\000)p Fo(2)1979 2429 y Fu(\()p Fq(c)2047 2441 y Fo(0)2103 2429 y Fu(+)c Fq(O)r Fu(\()p Fq(w)r Fu(\)\).)456 2603 y(In)n(tegrating)26 b(this)i(relation)e(w)n(e)i(obtain)1386 2755 y Fq(u)18 b Fp(\000)g Fq(u)1583 2721 y Fj(\003)1644 2755 y Fu(=)k Fq(w)1792 2721 y Fo(\()p Fi(m)p Fj(\000)p Fo(2\))p Fi(=)p Fo(2)2060 2755 y Fu(\()p Fq(c)2128 2767 y Fo(1)2184 2755 y Fu(+)c Fq(O)r Fu(\()p Fq(w)r Fu(\)\).)456 2906 y(In)n(v)n(erting)25 b(the)i(last)f(equation)g(and)h(going)e(bac)n(k)h(to)g(the)h(v)-5 b(ariable)26 b Fq(x)h Fu(w)n(e)f(obtain)h(the)g(claimed)456 3005 y(expressions)e(\(2.1\))j(and)f(\(2.2\).)1938 b Fe(\003)605 3177 y Fu(In)39 b(the)h(follo)n(wing)e(prop)r(osition)h(w)n (e)f(will)i(giv)n(e)e(a)h(qualitativ)n(e)g(study)g(of)g(the)h(domain) 456 3277 y Fq(\013)509 3289 y Fo(0)546 3277 y Fu(\()p Fp(f)p Fq(u)22 b Fp(2)i Fl(C)44 b Fu(:)23 b(Re)14 b Fq(u)22 b Fp(\025)h Fq(\024)p Fp(g)p Fu(.)605 3435 y Ft(Pr)n(oposition)44 b Fu(2.5)p Ft(.)j Fs(L)l(et)41 b Fq(V)19 b Fu(\()p Fq(x)p Fu(\))44 b(=)f Fq(a)1862 3447 y Fi(n)1907 3435 y Fq(x)1954 3405 y Fi(n)2027 3435 y Fu(+)26 b Fp(\001)14 b(\001)g(\001)54 b Fs(b)l(e)41 b(an)h(analytic)g(function.)72 b(Then,)456 3535 y(ther)l(e)32 b(exists)h Fq(\024)944 3547 y Fo(0)1014 3535 y Fs(such)f(that,)i(if)g Fq(\024)28 b Fp(\025)g Fq(\024)1705 3547 y Fo(0)1742 3535 y Fs(,)34 b Fq(\013)1854 3547 y Fo(0)1924 3535 y Fs(applies)h(the)e(semiplane)h Fp(f)p Fq(u)27 b Fp(2)i Fl(C)50 b Fu(:)28 b(Re)14 b Fq(u)28 b Fp(\025)g Fq(\024)p Fp(g)456 3634 y Fs(bije)l(ctively)d(to)f(a)g (domain)h Fq(D)1353 3604 y Fj(\003)1391 3634 y Fu(\()p Fq(\024)p Fu(\))f Fs(of)h Fl(C)15 b Fs(.)42 b(The)25 b(image)g(by)f Fq(\013)2295 3646 y Fo(0)2356 3634 y Fs(of)g(the)g(line) g Fp(f)p Fu(Re)14 b Fq(u)22 b Fu(=)h Fq(\024)p Fp(g)g Fs(to)l(gether)456 3734 y(with)35 b Fp(f)p Fu(0)p Fp(g)e Fs(is)i(a)g(single)h(close)l(d)g(curve)f(which)h(is)f(the)g(b)l (oundary)g(of)h Fq(D)2697 3704 y Fj(\003)2735 3734 y Fu(\()p Fq(\024)p Fu(\))p Fs(.)54 b(It)35 b(is)g(a)g(smo)l(oth)456 3834 y(curve)29 b(exc)l(ept)h(at)g(zer)l(o.)39 b(The)30 b(tangent)f(lines)i(ne)l(ar)e(zer)l(o)h(have)h(limit)g(slop)l(es)f Fp(\006)p Fq(\031)s(=)p Fu(\()p Fq(n)18 b Fp(\000)g Fu(2\))p Fs(.)605 3992 y Fu(In)32 b(Figures)f(1)g(and)g(2)g(w)n(e)h(displa)n(y)e (the)i(domain)g Fq(D)2253 3962 y Fj(\003)2291 3992 y Fu(\()p Fq(\024)p Fu(\),)h(with)f Fq(\024)g Fu(big,)g(and)g Fq(n)d Fu(=)g(2)j(and)456 4092 y Fq(n)22 b Fu(=)h(7)k(resp)r(ectiv)n (ely)538 4141 y 10815148 7458719 3289088 3289088 36443095 50651955 startTexFig 538 4141 a %%BeginDocument: domain1.ps %!PS-Adobe-2.0 %%Title: domain1.ps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Wed Mar 27 12:17:26 2002 %%DocumentFonts: (atend) %%BoundingBox: 50 50 554 770 %%Orientation: Portrait %%Pages: (atend) %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def 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4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V 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copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog %%Page: 1 1 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1 1 V 1 1 V 2 1 V 1 1 V 1 1 V 1 1 V 2 1 V 1 1 V 1 0 V 1 1 V 1 1 V 1 1 V 2 1 V 1 0 V 1 1 V 1 1 V 1 0 V 1 1 V 1 1 V 1 0 V 1 1 V 1 1 V 1 0 V 1 1 V 1 0 V 0 1 V 1 0 V 1 1 V 1 0 V 1 1 V 1 0 V 1 1 V 1 1 V 1 0 V 1 1 V 1 0 V 0 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 1 V 1 0 V 1 1 V 1 0 V 1 1 V 1 0 V 1 1 V 1 0 V 1 1 V 1 0 V 1 1 V 1 0 V 1 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 1 V 1 0 V 1 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 1 V 1 0 V 1 0 V 1 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 1 V -1 0 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V currentpoint stroke M -1 0 V 0 1 V -1 0 V -1 0 V -1 1 V -1 0 V -1 0 V -1 1 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 1 V -1 0 V -1 1 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 1 V -1 0 V -1 1 V -1 0 V -1 1 V -1 0 V -1 1 V -1 0 V -1 1 V -1 0 V -1 1 V -1 0 V -1 1 V -1 0 V 0 1 V -1 0 V -1 0 V 0 1 V -1 0 V -1 1 V -1 0 V -1 1 V -1 1 V -1 0 V -1 1 V -1 0 V -1 1 V -1 0 V 0 1 V -1 0 V -1 1 V -1 0 V -1 1 V -1 1 V -1 0 V -1 1 V -1 1 V -1 0 V -1 1 V -1 1 V -1 0 V -2 1 V -1 1 V -1 1 V -1 1 V -1 0 V -1 1 V -2 1 V -1 1 V -1 1 V -1 1 V -2 1 V -1 1 V -1 1 V -2 1 V -1 1 V -2 1 V -1 1 V -1 1 V -2 1 V -1 2 V -2 1 V -1 1 V -2 2 V -2 1 V -1 1 V -2 2 V -2 1 V -1 2 V -2 1 V -2 2 V -2 1 V -1 2 V -2 2 V -2 1 V -2 2 V -2 2 V -2 2 V -2 2 V -2 2 V -2 2 V -2 2 V -3 2 V -2 2 V -2 3 V -2 2 V -3 2 V -2 3 V -2 2 V -3 3 V -2 3 V -3 2 V -2 3 V -3 3 V -2 3 V -3 3 V -3 3 V -2 4 V -3 3 V -3 4 V -3 3 V -3 4 V -3 4 V -3 3 V -3 4 V -3 4 V -3 5 V -3 4 V -3 5 V -4 4 V -3 5 V -3 5 V -4 5 V -3 5 V -4 5 V -3 6 V -4 6 V -4 5 V -3 6 V -4 7 V -4 6 V -4 7 V -4 6 V -4 7 V -4 8 V -4 7 V -4 8 V -4 8 V -4 8 V -4 8 V -4 9 V -5 8 V -4 10 V -4 9 V -5 10 V -4 10 V -4 10 V -5 11 V -4 11 V -5 11 V -4 12 V -5 12 V -4 12 V -5 13 V -4 13 V -5 13 V -4 14 V -5 15 V -4 14 V -5 16 V -4 15 V -4 16 V -4 17 V -5 17 V -4 18 V -4 18 V -4 19 V -3 19 V -4 20 V -4 20 V -3 21 V -4 22 V -3 22 V -3 23 V -2 23 V -3 24 V -2 25 V -2 26 V -2 26 V -1 27 V -1 27 V -1 28 V -1 29 V 0 30 V 1 31 V 0 31 V 2 32 V 1 33 V 3 33 V 3 35 V 3 35 V 4 36 V 5 36 V 5 38 V 7 38 V 7 39 V 8 40 V 8 40 V 10 41 V 10 42 V 12 42 V 12 43 V 14 44 V 15 44 V 16 45 V 17 45 V 19 46 V 19 46 V 22 46 V 22 47 V 24 47 V 26 47 V 27 46 V 29 47 V 30 47 V 32 46 V 34 46 V 35 46 V 37 44 V 39 45 V 41 43 V 42 42 V 44 41 V 46 40 V 48 38 V 50 37 V 51 35 V 53 33 V 55 31 V 56 29 V 57 27 V 60 24 V 60 21 V 62 18 V 63 16 V 64 12 V 65 10 V 66 6 V 66 2 V 67 -1 V 67 -4 V 67 -8 V 68 -12 V 67 -15 V 67 -19 V 67 -23 V 66 -26 V 65 -29 V 64 -33 V 63 -36 V 61 -40 V 60 -42 V 59 -46 V 57 -48 V 55 -50 V 53 -53 V 51 -55 V 49 -57 V 47 -59 V 45 -60 V 42 -62 V 41 -62 V 38 -64 V 35 -64 V 34 -64 V 31 -65 V 29 -65 V 27 -64 V 25 -64 V 23 -64 V 22 -63 V 19 -62 V 18 -61 V 16 -61 V 14 -59 V 13 -58 V 12 -56 V 10 -56 V 9 -54 V 9 -52 V 7 -51 V 6 -50 V 6 -48 V 5 -47 V 4 -45 V 3 -44 V 3 -43 V 3 -41 V 2 -39 V 2 -38 V 2 -37 V 1 -36 V 1 -34 V 1 -34 V 0 -31 V 1 -31 V 0 -30 V 1 -28 V 0 -28 V 0 -26 V 0 -25 V 0 -25 V 0 -23 V 0 -22 V 0 -22 V 0 -20 V 0 -20 V 0 -18 V 0 -18 V 0 -17 V 0 -16 V 0 -15 V 0 -14 V 0 -14 V 0 -12 V 0 -12 V 0 -11 V 0 -10 V 0 -9 V 0 -8 V 0 -8 V 0 -7 V 0 -5 V 0 -5 V 0 -5 V 0 -3 V 0 -3 V 0 -1 V 0 -1 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%Pages: 1 %%EndDocument endTexFig 1992 4141 a 10815148 7458719 3289088 3289088 36443095 50651955 startTexFig 1992 4141 a %%BeginDocument: domain2.ps %!PS-Adobe-2.0 %%Title: domain2.ps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Wed Mar 27 12:23:57 2002 %%DocumentFonts: (atend) %%BoundingBox: 50 50 554 770 %%Orientation: Portrait %%Pages: (atend) %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog %%Page: 1 1 gnudict begin gsave 50 50 translate 0.100 0.100 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 658 280 M 63 0 V 4081 0 R -63 0 V 574 280 M (-0.04) Rshow 658 1124 M 63 0 V 4081 0 R -63 0 V -4165 0 R (-0.03) Rshow 658 1968 M 63 0 V 4081 0 R -63 0 V -4165 0 R (-0.02) Rshow 658 2812 M 63 0 V 4081 0 R -63 0 V -4165 0 R (-0.01) Rshow 658 3656 M 63 0 V 4081 0 R -63 0 V -4165 0 R (0) Rshow 658 4500 M 63 0 V 4081 0 R -63 0 V -4165 0 R (0.01) Rshow 658 5344 M 63 0 V 4081 0 R -63 0 V -4165 0 R (0.02) Rshow 658 6188 M 63 0 V 4081 0 R -63 0 V -4165 0 R (0.03) Rshow 658 7032 M 63 0 V 4081 0 R -63 0 V -4165 0 R (0.04) Rshow 658 280 M 0 63 V 0 6689 R 0 -63 V 658 140 M (0) Cshow 1118 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.02) Cshow 1579 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.04) Cshow 2039 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.06) Cshow 2500 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.08) Cshow 2960 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.1) Cshow 3421 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.12) Cshow 3881 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.14) Cshow 4342 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.16) Cshow 4802 280 M 0 63 V 0 6689 R 0 -63 V 0 -6829 R (0.18) Cshow 1.000 UL LTb 658 280 M 4144 0 V 0 6752 V -4144 0 V 658 280 L 1.000 UL LT0 4496 3656 M 0 -1 V 0 -2 V 0 -2 V 0 -3 V 0 -5 V 0 -6 V 0 -7 V 0 -10 V 0 -11 V 0 -13 V 0 -16 V 0 -18 V 0 -20 V 0 -24 V -1 -26 V 0 -30 V -1 -33 V -1 -37 V -1 -40 V -1 -44 V -2 -49 V -2 -52 V -3 -56 V -3 -61 V -3 -64 V -5 -69 V -5 -74 V -6 -77 V -7 -81 V -8 -86 V -10 -88 V -10 -92 V -13 -95 V -13 -98 V -15 -99 V -17 -101 V -19 -102 V -20 -101 V -21 -102 V -24 -100 V -25 -99 V -26 -97 V -28 -94 V -30 -91 V -30 -87 V -32 -83 V -32 -78 V -34 -75 V -34 -69 V -34 -66 V -35 -60 V -35 -56 V -35 -51 V -36 -47 V -35 -42 V -35 -39 V -35 -34 V -35 -31 V -34 -27 V -34 -24 V -34 -20 V -33 -18 V -33 -15 V -32 -12 V -31 -10 V -32 -8 V -30 -6 V -30 -3 V -30 -2 V -29 -1 V -28 1 V -28 3 V -28 3 V -26 5 V -27 6 V -26 7 V -25 8 V -25 9 V -24 9 V -24 10 V -23 11 V -23 11 V -23 11 V -22 13 V -21 12 V -22 13 V -21 13 V -20 14 V -20 13 V -20 14 V -19 14 V -19 15 V -19 14 V -18 15 V -18 15 V -17 14 V -18 15 V -17 15 V -16 15 V -17 15 V -16 15 V -16 15 V -15 15 V -16 15 V -15 15 V -15 15 V -14 15 V -14 15 V -15 15 V -13 14 V -14 15 V -14 15 V -13 14 V -13 15 V -13 14 V -12 14 V -13 14 V -12 15 V -12 14 V -12 13 V -12 14 V -11 14 V -11 14 V -12 13 V -11 14 V -11 13 V -10 13 V -11 13 V -10 13 V -11 13 V -10 13 V -10 13 V -10 12 V -10 13 V -9 12 V -10 13 V -9 12 V -9 12 V -9 12 V -9 12 V -9 12 V -9 11 V -9 12 V -8 12 V -9 11 V -8 11 V -8 12 V -9 11 V -8 11 V -8 11 V -7 11 V -8 10 V -8 11 V -7 10 V -8 11 V -7 10 V -8 11 V -7 10 V -7 10 V -7 10 V -7 10 V -7 10 V -7 10 V -7 9 V -6 10 V -7 9 V -7 10 V -6 9 V -7 10 V -6 9 V -6 9 V -6 9 V -7 9 V -6 9 V -6 9 V -6 9 V -5 8 V -6 9 V -6 8 V -6 9 V -6 8 V -5 9 V -6 8 V -5 8 V -6 8 V -5 8 V -5 8 V -6 8 V -5 8 V -5 8 V -5 8 V -5 7 V -5 8 V -5 8 V -5 7 V -5 8 V -5 7 V -5 7 V -5 8 V -4 7 V -5 7 V -5 7 V -4 7 V -5 7 V -4 7 V -5 7 V -4 7 V -5 7 V -4 6 V -4 7 V -5 7 V -4 6 V -4 7 V -4 6 V -5 7 V -4 6 V -4 6 V -4 7 V -4 6 V -4 6 V -4 6 V -4 7 V -4 6 V -3 6 V -4 6 V -4 6 V -4 5 V -4 6 V -3 6 V -4 6 V -4 6 V -3 5 V -4 6 V -3 6 V -4 5 V -3 6 V -4 5 V -3 6 V -4 5 V -3 6 V -3 5 V -4 5 V -3 6 V -3 5 V -4 5 V -3 5 V -3 5 V -3 5 V -4 5 V -3 5 V -3 6 V -3 5 V -3 4 V -3 5 V -3 5 V -3 5 V -3 5 V -3 5 V -3 4 V -3 5 V -3 5 V -3 4 V -3 5 V -3 5 V -2 4 V -3 5 V -3 4 V -3 5 V -3 4 V -2 5 V -3 4 V -3 4 V -2 5 V -3 4 V -3 4 V -2 5 V -3 4 V -3 4 V -2 4 V -3 5 V -2 4 V -3 4 V -2 4 V -3 4 V -2 4 V -3 4 V -2 4 V -3 4 V -2 4 V -3 4 V -2 4 V -2 4 V -3 3 V -2 4 V -2 4 V -3 4 V -2 4 V -2 3 V -3 4 V -2 4 V -2 3 V -2 4 V -3 4 V -2 3 V -2 4 V -2 4 V -2 3 V -3 4 V -2 3 V -2 4 V -2 3 V -2 4 V -2 3 V -2 4 V -2 3 V -2 3 V -2 4 V -2 3 V -2 3 V -2 4 V -2 3 V -2 3 V -2 4 V -2 3 V -2 3 V -2 3 V -2 3 V -2 4 V -2 3 V -2 3 V -2 3 V -2 3 V -2 3 V -2 3 V -1 3 V -2 3 V -2 3 V -2 3 V -2 3 V -1 3 V -2 3 V -2 3 V -2 3 V -2 3 V -1 3 V -2 3 V -2 3 V -2 3 V -1 3 V -2 2 V -2 3 V -1 3 V -2 3 V -2 3 V -1 2 V -2 3 V -2 3 V -1 3 V -2 2 V -2 3 V -1 3 V -2 2 V -1 3 V -2 3 V -2 2 V -1 3 V -2 3 V -1 2 V -2 3 V -1 2 V -2 3 V -1 2 V -2 3 V -1 3 V -2 2 V -2 3 V -1 2 V -1 2 V -2 3 V -1 2 V -2 3 V -1 2 V -2 3 V -1 2 V -2 3 V -1 2 V -2 2 V -1 3 V -1 2 V -2 2 V -1 3 V -2 2 V -1 2 V -1 3 V -2 2 V -1 2 V -1 3 V -2 2 V -1 2 V currentpoint stroke M -1 2 V -2 3 V -1 2 V -1 2 V -2 2 V -1 2 V -1 3 V -2 2 V -1 2 V -1 2 V -1 2 V -2 2 V -1 2 V -1 3 V -2 2 V -1 2 V -1 2 V -1 2 V -2 2 V -1 2 V -1 2 V -1 2 V -1 2 V -2 2 V -1 2 V -1 2 V -1 2 V -1 2 V -2 2 V -1 2 V -1 2 V -1 2 V -1 2 V -1 2 V -2 2 V -1 2 V -1 2 V -1 2 V -1 2 V -1 2 V -1 2 V -2 2 V -1 1 V -1 2 V -1 2 V -1 2 V -1 2 V -1 2 V -1 2 V -1 1 V -1 2 V -1 2 V -2 2 V -1 2 V -1 1 V -1 2 V -1 2 V -1 2 V -1 2 V -1 1 V -1 2 V -1 2 V -1 2 V -1 1 V -1 2 V -1 2 V -1 1 V -1 2 V -1 2 V -1 2 V -1 1 V -1 2 V -1 2 V -1 1 V -1 2 V -1 2 V -1 1 V -1 2 V -1 1 V -1 2 V -1 2 V -1 1 V -1 2 V -1 2 V -1 1 V -1 2 V -1 1 V -1 2 V 0 1 V -1 2 V -1 2 V -1 1 V -1 2 V -1 1 V -1 2 V -1 1 V -1 2 V -1 2 V -1 1 V 0 2 V -1 1 V -1 1 V -1 2 V -1 2 V -1 1 V -1 1 V -1 2 V 0 1 V -1 2 V -1 1 V -1 2 V -1 1 V -1 2 V -1 1 V 0 2 V -1 1 V -1 1 V -1 2 V -1 1 V -1 2 V 0 1 V -1 2 V -1 1 V -1 1 V -1 2 V -1 1 V 0 1 V -1 2 V -1 1 V -1 2 V -1 1 V 0 1 V -1 2 V -1 1 V -1 1 V -1 2 V 0 1 V -1 1 V -1 2 V -1 1 V 0 1 V -1 2 V -1 1 V -1 1 V -1 2 V 0 1 V -1 1 V -1 1 V -1 2 V 0 1 V -1 1 V -1 2 V -1 1 V 0 1 V -1 1 V -1 2 V 0 1 V -1 1 V -1 1 V -1 1 V 0 2 V -1 1 V -1 1 V -1 1 V 0 2 V -1 1 V -1 1 V 0 1 V -1 1 V -1 2 V 0 1 V -1 1 V -1 1 V -1 1 V 0 2 V -1 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 2 V -1 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 2 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 2 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 2 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 0 V 0 2 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V currentpoint stroke M 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V currentpoint stroke M 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 658 3656 L 115 202 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V currentpoint stroke M 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V currentpoint stroke M 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 2 V 1 0 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 2 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 2 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 2 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 2 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 2 V 1 1 V 1 1 V 1 1 V 0 1 V 1 2 V 1 1 V 0 1 V 1 1 V 1 1 V 0 2 V 1 1 V 1 1 V 1 1 V 0 2 V 1 1 V 1 1 V 1 1 V 0 1 V 1 2 V 1 1 V 0 1 V 1 1 V 1 2 V 1 1 V 0 1 V 1 2 V 1 1 V 1 1 V 0 1 V 1 2 V 1 1 V 1 1 V 1 2 V 0 1 V 1 1 V 1 2 V 1 1 V 0 1 V 1 2 V 1 1 V 1 1 V 1 2 V 0 1 V 1 1 V 1 2 V 1 1 V 1 2 V 0 1 V 1 1 V 1 2 V 1 1 V 1 1 V 1 2 V 0 1 V 1 2 V 1 1 V 1 2 V 1 1 V 1 1 V 0 2 V 1 1 V 1 2 V 1 1 V 1 2 V 1 1 V 1 2 V 0 1 V 1 2 V 1 1 V 1 1 V 1 2 V 1 2 V 1 1 V 1 1 V 0 2 V 1 1 V 1 2 V 1 2 V 1 1 V 1 2 V 1 1 V 1 2 V 1 1 V 1 2 V 1 2 V 0 1 V 1 2 V 1 1 V 1 2 V 1 1 V 1 2 V 1 2 V 1 1 V 1 2 V 1 2 V 1 1 V 1 2 V 1 1 V 1 2 V 1 2 V 1 1 V 1 2 V 1 2 V 1 1 V 1 2 V 1 2 V 1 2 V 1 1 V 1 2 V 1 2 V 1 1 V 1 2 V 1 2 V 1 2 V 1 1 V 1 2 V 1 2 V 1 2 V 1 2 V 1 1 V 1 2 V 2 2 V 1 2 V 1 2 V 1 1 V 1 2 V 1 2 V 1 2 V 1 2 V 1 2 V 1 2 V 1 1 V 2 2 V 1 2 V 1 2 V 1 2 V 1 2 V 1 2 V 1 2 V 2 2 V 1 2 V 1 2 V 1 2 V 1 2 V 1 2 V 2 2 V 1 2 V 1 2 V 1 2 V 1 2 V 2 2 V 1 2 V 1 2 V 1 2 V 1 2 V 2 2 V 1 2 V 1 2 V 1 2 V 2 2 V 1 3 V 1 2 V 2 2 V 1 2 V 1 2 V 1 2 V 2 2 V 1 3 V 1 2 V 2 2 V 1 2 V 1 2 V 2 3 V 1 2 V 1 2 V 2 2 V 1 3 V 1 2 V 2 2 V 1 3 V 1 2 V 2 2 V 1 3 V 2 2 V 1 2 V 1 3 V 2 2 V 1 2 V 2 3 V 1 2 V 2 3 V 1 2 V 2 3 V 1 2 V 2 3 V 1 2 V 1 2 V 2 3 V 2 2 V 1 3 V 2 3 V 1 2 V 2 3 V 1 2 V 2 3 V 1 2 V 2 3 V 1 3 V 2 2 V 2 3 V 1 3 V 2 2 V 1 3 V 2 3 V 2 2 V 1 3 V 2 3 V 2 3 V 1 2 V 2 3 V 2 3 V 1 3 V 2 3 V 2 2 V 1 3 V 2 3 V 2 3 V 2 3 V 1 3 V 2 3 V 2 3 V 2 3 V 2 3 V 1 3 V 2 3 V 2 3 V 2 3 V 2 3 V 1 3 V 2 3 V 2 3 V 2 3 V 2 3 V 2 3 V 2 3 V 2 4 V 2 3 V 2 3 V 2 3 V 2 3 V 2 4 V 2 3 V 2 3 V 2 4 V 2 3 V 2 3 V 2 4 V 2 3 V 2 3 V 2 4 V 2 3 V 2 4 V 2 3 V 2 4 V 2 3 V currentpoint stroke M 3 4 V 2 3 V 2 4 V 2 4 V 2 3 V 3 4 V 2 4 V 2 3 V 2 4 V 3 4 V 2 3 V 2 4 V 3 4 V 2 4 V 2 4 V 3 3 V 2 4 V 2 4 V 3 4 V 2 4 V 3 4 V 2 4 V 3 4 V 2 4 V 3 4 V 2 4 V 3 4 V 2 4 V 3 5 V 2 4 V 3 4 V 3 4 V 2 5 V 3 4 V 3 4 V 2 5 V 3 4 V 3 4 V 2 5 V 3 4 V 3 5 V 3 4 V 3 5 V 2 4 V 3 5 V 3 5 V 3 4 V 3 5 V 3 5 V 3 4 V 3 5 V 3 5 V 3 5 V 3 5 V 3 5 V 3 4 V 3 5 V 3 6 V 3 5 V 4 5 V 3 5 V 3 5 V 3 5 V 4 5 V 3 5 V 3 6 V 4 5 V 3 5 V 3 6 V 4 5 V 3 6 V 4 5 V 3 6 V 4 5 V 3 6 V 4 6 V 3 5 V 4 6 V 4 6 V 3 6 V 4 6 V 4 5 V 4 6 V 4 6 V 3 6 V 4 6 V 4 7 V 4 6 V 4 6 V 4 6 V 4 7 V 4 6 V 4 6 V 5 7 V 4 6 V 4 7 V 4 6 V 5 7 V 4 7 V 4 6 V 5 7 V 4 7 V 5 7 V 4 7 V 5 7 V 4 7 V 5 7 V 5 7 V 4 7 V 5 8 V 5 7 V 5 7 V 5 8 V 5 7 V 5 8 V 5 8 V 5 7 V 5 8 V 5 8 V 5 8 V 6 8 V 5 8 V 5 8 V 6 8 V 5 8 V 6 8 V 5 9 V 6 8 V 6 9 V 6 8 V 6 9 V 5 8 V 6 9 V 6 9 V 6 9 V 7 9 V 6 9 V 6 9 V 6 9 V 7 10 V 6 9 V 7 10 V 7 9 V 6 10 V 7 9 V 7 10 V 7 10 V 7 10 V 7 10 V 7 10 V 7 10 V 8 11 V 7 10 V 8 11 V 7 10 V 8 11 V 8 10 V 7 11 V 8 11 V 8 11 V 9 11 V 8 12 V 8 11 V 9 11 V 8 12 V 9 12 V 9 11 V 9 12 V 9 12 V 9 12 V 9 12 V 9 12 V 10 13 V 9 12 V 10 13 V 10 12 V 10 13 V 10 13 V 11 13 V 10 13 V 11 13 V 10 13 V 11 13 V 11 14 V 12 13 V 11 14 V 11 14 V 12 14 V 12 13 V 12 14 V 12 15 V 13 14 V 12 14 V 13 14 V 13 15 V 13 14 V 14 15 V 14 15 V 13 14 V 15 15 V 14 15 V 14 15 V 15 15 V 15 15 V 16 15 V 15 15 V 16 15 V 16 15 V 17 15 V 16 15 V 17 15 V 18 15 V 17 14 V 18 15 V 18 15 V 19 14 V 19 15 V 19 14 V 20 14 V 20 13 V 20 14 V 21 13 V 22 13 V 21 12 V 22 13 V 23 11 V 23 11 V 23 11 V 24 10 V 24 9 V 25 9 V 25 8 V 26 7 V 27 6 V 26 5 V 28 3 V 28 3 V 28 1 V 29 -1 V 30 -2 V 30 -3 V 30 -6 V 32 -8 V 31 -10 V 32 -12 V 33 -15 V 33 -18 V 34 -20 V 34 -24 V 34 -27 V 35 -31 V 35 -34 V 35 -39 V 35 -42 V 36 -47 V 35 -51 V 35 -56 V 35 -60 V 34 -66 V 34 -69 V 34 -75 V 32 -78 V 32 -83 V 30 -87 V 30 -91 V 28 -94 V 26 -97 V 25 -99 V 24 -100 V 21 -102 V 20 -101 V 19 -102 V 17 -101 V 15 -99 V 13 -98 V 13 -95 V 10 -92 V 10 -88 V 8 -86 V 7 -81 V 6 -77 V 5 -74 V 5 -69 V 3 -64 V 3 -61 V 3 -56 V 2 -52 V 2 -49 V 1 -44 V 1 -40 V 1 -37 V 1 -33 V 0 -30 V 1 -26 V 0 -24 V 0 -20 V 0 -18 V 0 -16 V 0 -13 V 0 -11 V 0 -10 V 0 -7 V 0 -6 V 0 -5 V 0 -3 V 0 -2 V 0 -2 V 0 -1 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%Pages: 1 %%EndDocument endTexFig 1072 5186 a Fu(Figure)g(1)1149 b(Figure)27 b(2)p eop %%Page: 9 17 9 16 bop 1313 251 a Fo(2.2.)29 b(PR)n(OOF)g(OF)g(THE)g(PR)n(OPOSITION)f (2.1)824 b(9)605 450 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)30 b(assume)e(that)i(there)f(exists)g Fq(u)2041 462 y Fo(1)2078 450 y Fq(;)14 b(u)2163 462 y Fo(2)2226 450 y Fp(2)26 b(f)p Fu(Re)13 b Fq(u)26 b Fp(\025)g Fq(\024)p Fp(g)i Fu(suc)n(h)h(that)h Fq(\013)3167 462 y Fo(0)3204 450 y Fu(\()p Fq(u)3284 462 y Fo(1)3321 450 y Fu(\))d(=)456 550 y Fq(\013)509 562 y Fo(0)546 550 y Fu(\()p Fq(u)626 562 y Fo(2)663 550 y Fu(\).)59 b(Then)35 b(the)g(p)r(oin)n(ts)g(\()p Fq(\013)1493 562 y Fo(0)1531 550 y Fu(\()p Fq(u)1611 562 y Fi(j)1646 550 y Fu(\))p Fq(;)14 b(\014)1762 562 y Fo(0)1799 550 y Fu(\()p Fq(u)1879 562 y Fi(j)1914 550 y Fu(\)\),)37 b Fq(j)j Fu(=)35 b(1)p Fq(;)14 b Fu(2)34 b(sta)n(y)g(in)h(the)g(zero)f(energy)g(lev)n(el,)456 649 y(and)c(hence)g Fq(\014)900 661 y Fo(0)937 649 y Fu(\()p Fq(u)1017 661 y Fo(1)1054 649 y Fu(\))e(=)e Fp(\006)p Fq(\014)1317 661 y Fo(0)1354 649 y Fu(\()p Fq(u)1434 661 y Fo(2)1471 649 y Fu(\).)45 b(In)31 b(the)f(case)f(that)i(the)f (sign)g(is)g(+,)h(w)n(e)e(should)h(ha)n(v)n(e)f(that)456 749 y Fq(\013)509 761 y Fo(0)546 749 y Fu(\()p Fq(t)16 b Fu(+)f Fq(u)752 761 y Fo(2)805 749 y Fp(\000)g Fq(u)933 761 y Fo(1)970 749 y Fu(\))23 b(=)g Fq(\013)1166 761 y Fo(0)1203 749 y Fu(\()p Fq(t)p Fu(\))k(for)f(all)g Fq(t)g Fu(in)h(the)g(domains)e(of)h Fq(\013)2330 761 y Fo(0)2368 749 y Fu(,)h(that)f(is)g Fq(u)2726 761 y Fo(2)2779 749 y Fp(\000)15 b Fq(u)2907 761 y Fo(1)2970 749 y Fu(is)26 b(a)g(p)r(erio)r(d)h(of)456 849 y Fq(\013)509 861 y Fo(0)546 849 y Fu(,)h(whic)n(h)f(is)h(a)f(con)n(tradiction)f (with)i(Remark)f(2.2)605 948 y(In)21 b(the)h(case)e(that)i(the)g(sign)e (is)h Fp(\000)p Fu(,)i(w)n(e)d(use)h(that,)i(due)f(to)f(the)g(symmetry) g(of)g(the)h(equation,)456 1048 y Fq(\013)509 1060 y Fo(0)546 1048 y Fu(\()p Fq(t)p Fu(\))27 b(=)f Fq(\013)811 1060 y Fo(0)848 1048 y Fu(\()p Fq(u)928 1060 y Fo(2)985 1048 y Fu(+)19 b Fq(u)1117 1060 y Fo(1)1174 1048 y Fp(\000)g Fq(t)p Fu(\))30 b(and)g Fq(\014)1561 1060 y Fo(0)1598 1048 y Fu(\()p Fq(t)p Fu(\))d(=)f Fp(\000)p Fq(\014)1922 1060 y Fo(0)1958 1048 y Fu(\()p Fq(u)2038 1060 y Fo(2)2095 1048 y Fu(+)20 b Fq(u)2228 1060 y Fo(1)2284 1048 y Fp(\000)g Fq(t)p Fu(\).)43 b(Then)30 b Fq(\014)2763 1060 y Fo(0)2800 1048 y Fu(\(\()p Fq(u)2912 1060 y Fo(1)2969 1048 y Fu(+)19 b Fq(u)3101 1060 y Fo(2)3138 1048 y Fu(\))p Fq(=)p Fu(2\))26 b(=)g(0)456 1147 y(and)g(therefore,)g Fq(V)19 b Fu(\()p Fq(\013)1139 1159 y Fo(0)1176 1147 y Fu(\(\()p Fq(u)1288 1159 y Fo(1)1342 1147 y Fu(+)d Fq(u)1471 1159 y Fo(2)1508 1147 y Fu(\))p Fq(=)p Fu(2\)\))23 b(=)g(0.)36 b(Since)26 b(0)h(is)f(an)g(isolated)g(zero)g(for)g Fq(V)19 b Fu(,)27 b(if)g Fq(\024)f Fu(is)h(big)456 1247 y(enough,)g Fq(\013)818 1259 y Fo(0)855 1247 y Fu(\()p Fp(f)p Fu(Re)14 b Fq(u)22 b Fp(\025)h Fq(\024)p Fp(g)p Fu(\))k(do)r(es)g(not)h(con)n(tain)f(an)n (y)g(zero)f(of)i Fq(V)19 b Fu(,)27 b(whic)n(h)h(is)f(a)h(con)n (tradiction.)605 1347 y(Let)1123 1495 y Fq(g)s Fu(\()p Fq(t)p Fu(\))23 b(=)g Fq(\013)1424 1507 y Fo(0)1461 1495 y Fu(\()p Fq(it)18 b Fu(+)g Fq(\024)p Fu(\))24 b(=)1992 1439 y Fq(c)p 1854 1476 311 4 v 1854 1552 a Fu(\()p Fq(it)19 b Fu(+)f Fq(\024)p Fu(\))2127 1528 y Fi(p)2194 1495 y Fu(+)g Fq(O)r Fu(\()p Fp(j)p Fq(it)h Fu(+)f Fq(\024)p Fp(j)2629 1461 y Fj(\000)p Fi(\027)2722 1495 y Fu(\),)456 1692 y(with)28 b Fq(p)23 b Fu(=)f(2)p Fq(=)p Fu(\()p Fq(n)c Fp(\000)g Fu(2\).)37 b(The)27 b(claim)h(on)f(the)h(tangen)n(t)f (lines)h(comes)e(from)1596 1861 y(lim)1549 1911 y Fi(t)p Fj(!\0061)1773 1861 y Fu(arg)13 b Fq(g)s Fu(\()p Fq(t)p Fu(\))23 b(=)g Fp(\007)p Fq(p)2268 1805 y(\031)p 2268 1842 51 4 v 2272 1918 a Fu(2)2328 1861 y(.)3380 2038 y Fe(\003)1264 2246 y Fk(2.2.)46 b(Pro)s(of)32 b(of)f(the)h(Prop)s (osition)e(2.1)605 2395 y Fu(Let)g Fq(U)36 b Fu(=)27 b Fl(D)14 b Fu(\()q(0)p Fq(;)g(r)r Fu(\))26 b Fp(\000)20 b(f)p Fq(z)30 b Fp(2)e Fl(C)48 b Fu(:)27 b(Im)14 b Fq(z)31 b Fu(=)c(0,)j(Re)14 b Fq(z)31 b Fp(\024)26 b Fu(0)p Fp(g)h(\032)g Fl(C)51 b Fu(and)30 b Fq(k)s(;)14 b(`)26 b Fp(2)i Fl(R)p Fu(,)37 b(1)27 b Fq(<)g(k)j(<)d(`)p Fu(.)456 2495 y(W)-7 b(e)28 b(consider)1591 2643 y Fq(f)9 b Fu(\()p Fq(x)p Fu(\))24 b(=)e Fq(f)1904 2655 y Fo(0)1941 2643 y Fu(\()p Fq(x)p Fu(\))e(+)e Fq(g)s Fu(\()p Fq(x)p Fu(\))456 2804 y(with)28 b Fq(f)686 2816 y Fo(0)723 2804 y Fu(\()p Fq(x)p Fu(\))c(=)e Fq(ax)1036 2774 y Fi(k)1078 2804 y Fu(,)27 b Fq(a)c(<)g Fu(0,)k(and)g Fq(g)f Fu(:)d Fq(U)32 b Fp(!)23 b Fl(C)49 b Fu(analytic)27 b(suc)n(h)g(that)h Fp(j)p Fq(g)s Fu(\()p Fq(x)p Fu(\))p Fp(j)c(\024)e Fq(B)t Fp(j)p Fq(x)p Fp(j)3085 2774 y Fi(`)3118 2804 y Fu(.)605 2904 y(A)28 b(solution)f(of)1819 3051 y(_)-37 b Fq(x)23 b Fu(=)g Fq(ax)2054 3017 y Fi(k)456 3208 y Fu(is)35 b Fq(x)p Fu(\()p Fq(t)p Fu(\))j(=)e Fq(ct)893 3178 y Fj(\000)p Fi(p)1020 3208 y Fu(with)g Fq(c)h Fu(=)f(\()p Fp(\000)p Fq(p=a)p Fu(\))1648 3178 y Fi(p)1722 3208 y Fu(and)f Fq(p)i Fu(=)f(1)p Fq(=)p Fu(\()p Fq(k)26 b Fp(\000)e Fu(1\).)61 b(W)-7 b(e)36 b(lo)r(ok)f(for)h(a)f(solution)h(of)470 3308 y(_)-37 b Fq(x)23 b Fu(=)g Fq(f)9 b Fu(\()p Fq(x)p Fu(\))28 b(of)g(the)g(form)1104 3456 y Fq(')p Fu(\()p Fq(t)p Fu(\))c(=)f Fq(')1418 3468 y Fo(0)1455 3456 y Fu(\()p Fq(t)p Fu(\))c(+)f Fq( )s Fu(\()p Fq(t)p Fu(\),)167 b(with)g Fq(')2374 3468 y Fo(0)2411 3456 y Fu(\()p Fq(t)p Fu(\))24 b(=)e Fq(ct)2682 3421 y Fj(\000)p Fi(p)2773 3456 y Fu(.)456 3608 y(W)-7 b(e)28 b(write)f(the)h(equation)41 b(_)-37 b Fq(x)23 b Fu(=)g Fq(f)9 b Fu(\()p Fq(x)p Fu(\))28 b(in)g(the)g(form)753 3763 y(_)-46 b Fq(')784 3775 y Fo(0)822 3763 y Fu(\()p Fq(t)p Fu(\))19 b(+)1044 3741 y(_)1018 3763 y Fq( )s Fu(\()p Fq(t)p Fu(\))83 b(=)g Fq(f)1441 3775 y Fo(0)1478 3763 y Fu(\()p Fq(')1564 3775 y Fo(0)1602 3763 y Fu(\()p Fq(t)p Fu(\)\))19 b(+)f Fq(D)r(f)1942 3775 y Fo(0)1979 3763 y Fu(\()p Fq(')2065 3775 y Fo(0)2103 3763 y Fu(\()p Fq(t)p Fu(\)\))p Fq( )s Fu(\()p Fq(t)p Fu(\))-1924 b(\(2.3\))1400 3888 y(+[)p Fq(f)9 b Fu(\()p Fq(')1624 3900 y Fo(0)1661 3888 y Fu(\()p Fq(t)p Fu(\))19 b(+)f Fq( )s Fu(\()p Fq(t)p Fu(\)\))h Fp(\000)f Fq(f)2183 3900 y Fo(0)2220 3888 y Fu(\()p Fq(')2306 3900 y Fo(0)2344 3888 y Fu(\()p Fq(t)p Fu(\)\))h Fp(\000)f Fq(D)r(f)2684 3900 y Fo(0)2721 3888 y Fu(\()p Fq(')2807 3900 y Fo(0)2845 3888 y Fu(\()p Fq(t)p Fu(\)\))p Fq( )s Fu(\()p Fq(t)p Fu(\)].)456 4035 y(First)27 b(w)n(e)g(consider)g(the)h(auxiliary)e (linear)h(equation)960 4223 y(_)-42 b Fq(\037)p Fu(\()p Fq(t)p Fu(\))23 b(=)g Fq(D)r(f)1310 4235 y Fo(0)1347 4223 y Fu(\()p Fq(')1433 4235 y Fo(0)1471 4223 y Fu(\()p Fq(t)p Fu(\)\))p Fq(\037)p Fu(\()p Fq(t)p Fu(\))h(=)f Fq(k)s(a)1959 4131 y Fg(\020)2034 4167 y Fq(c)p 2018 4204 69 4 v 2018 4280 a(t)2048 4256 y Fi(p)2096 4131 y Fg(\021)2146 4149 y Fi(k)q Fj(\000)p Fo(1)2285 4223 y Fq(\037)p Fu(\()p Fq(t)p Fu(\))h(=)2592 4167 y Fp(\000)p Fq(k)p 2553 4204 189 4 v 2553 4280 a(k)d Fp(\000)d Fu(1)2761 4167 y(1)p 2761 4204 42 4 v 2767 4280 a Fq(t)2813 4223 y(\037)p Fu(\()p Fq(t)p Fu(\))456 4402 y(whic)n(h)27 b(has)g(a)g(solution)1724 4568 y Fq(\037)p Fu(\()p Fq(t)p Fu(\))c(=)2046 4512 y(1)p 1991 4549 153 4 v 1991 4625 a Fq(t)2021 4601 y Fi(p)p Fo(+1)2153 4568 y Fu(.)456 4743 y(F)-7 b(rom)30 b(\(2.3\),)i(using)f(the)h(v)-5 b(ariation)30 b(of)h(constan)n(ts)g(form)n(ula)f(w)n(e)h(get)g(the)h (follo)n(wing)e(in)n(tegral)456 4843 y(equation)d(for)g Fq( )599 4991 y( )s Fu(\()p Fq(t)p Fu(\))84 b(=)e(\000)p Fq( )s Fu(\()p Fq(t)p Fu(\))834 5173 y Fp(\021)1047 5117 y Fu(1)p 991 5154 V 991 5230 a Fq(t)1021 5206 y Fi(p)p Fo(+1)1168 5060 y Fg(Z)1251 5081 y Fi(t)1214 5249 y(T)1294 5173 y Fq(s)1333 5139 y Fi(p)p Fo(+1)1455 5173 y Fu([)p Fq(f)9 b Fu(\()p Fq(')1614 5185 y Fo(0)1651 5173 y Fu(\()p Fq(s)p Fu(\))19 b(+)f Fq( )s Fu(\()p Fq(s)p Fu(\)\))i Fp(\000)e Fq(f)2192 5185 y Fo(0)2229 5173 y Fu(\()p Fq(')2315 5185 y Fo(0)2352 5173 y Fu(\()p Fq(s)p Fu(\)\))i Fp(\000)e Fq(D)r(f)2702 5185 y Fo(0)2739 5173 y Fu(\()p Fq(')2825 5185 y Fo(0)2862 5173 y Fu(\()p Fq(s)p Fu(\)\))p Fq( )s Fu(\()p Fq(s)p Fu(\)])c Fq(ds)p Fu(.)-2843 b(\(2.4\))p eop %%Page: 10 18 10 17 bop 456 251 a Fo(10)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(Here,)27 b(w)n(e)g(are)g(implicitly)h(assuming)f(that)h Fq( )s Fu(\()p Fq(T)12 b Fu(\))22 b(=)h(0.)36 b(W)-7 b(e)28 b(\014x)g Fq(\027)33 b Fu(suc)n(h)27 b(that)1542 591 y Fq(p)c(<)g(\027)28 b(<)23 b Fu(min)p Fp(f)p Fq(q)s(;)14 b(p)k Fu(+)g(1)p Fp(g)p Fu(.)456 732 y(W)-7 b(e)28 b(in)n(tro)r(duce)f (the)h(space)1053 872 y Fq(X)h Fu(=)23 b Fp(f)p Fq( )i Fu(:)e(\012\()p Fq(T)7 b(;)14 b(\013)p Fu(\))24 b Fp(!)f Fl(C)44 b Fu(:)23 b(analytic,)k Fp(j)p Fq(t)p Fp(j)2350 838 y Fi(\027)2391 872 y Fp(j)p Fq( )s Fu(\()p Fq(t)p Fu(\))p Fp(j)d Fq(<)f Fp(1g)p Fu(.)456 1013 y(W)-7 b(e)28 b(endo)n(w)f Fq(X)34 b Fu(with)28 b(the)g(norm)1510 1154 y Fp(k)p Fq( )s Fp(k)22 b Fu(=)99 b(sup)1761 1228 y Fi(t)p Fj(2)p Fo(\012\()p Fi(T)5 b(;\013)p Fo(\))2051 1154 y Fp(j)p Fq(t)p Fp(j)2127 1120 y Fi(\027)2169 1154 y Fp(j)p Fq( )s Fu(\()p Fq(t)p Fu(\))p Fp(j)p Fu(.)456 1362 y(W)-7 b(e)29 b(denote)f(b)n(y)g Fq(B)t Fu(\()p Fq(\032)p Fu(\))i(the)f (closed)f(ball)g(of)h(radius)f Fq(\032)g Fu(in)h Fq(X)35 b Fu(cen)n(tered)28 b(at)h(zero.)39 b(W)-7 b(e)29 b(consider)456 1462 y(the)f(op)r(erator)d(\000)e(:)h Fq(B)t Fu(\()p Fq(\032)p Fu(\))f Fp(!)g Fq(B)t Fu(\()p Fq(\032)p Fu(\))29 b(de\014ned)f(b)n(y)f(\(2.4\).)605 1561 y(The)33 b(rest)f(of)g(this)h (section)f(is)g(dev)n(oted)g(to)g(pro)n(v)n(e)f(that,)j(taking)e (suitable)g(v)-5 b(alues)33 b(for)e Fq(\032)456 1661 y Fu(and)c Fq(T)12 b Fu(,)27 b(w)n(e)g(ha)n(v)n(e)f(that)i(\000)23 b(:)g Fq(B)t Fu(\()p Fq(\032)p Fu(\))h Fp(!)f Fq(B)t Fu(\()p Fq(\032)p Fu(\))28 b(is)g(a)f(con)n(traction.)605 1760 y(W)-7 b(e)28 b(tak)n(e)f Fq(T)39 b Fu(satisfying)27 b(the)h(conditions)997 1901 y Fq(T)34 b Fp(\025)22 b Fu(1,)415 b Fq(T)1708 1867 y Fi(p)1769 1901 y Fq(>)22 b Fu(2)p Fq(c=r)r Fu(,)443 b Fq(T)2542 1867 y Fi(\027)t Fj(\000)p Fi(p)2691 1901 y Fq(>)23 b Fu(1)p Fq(=)p Fu(2)1070 1991 y Fg(\022)1132 2109 y Fu(1)18 b(+)1378 2052 y(1)p 1285 2089 229 4 v 1285 2166 a(2)p Fq(T)1388 2142 y Fi(\027)t Fj(\000)p Fi(p)1523 1991 y Fg(\023)1585 2009 y Fi(k)q Fj(\000)p Fo(2)1733 2109 y Fp(\024)23 b Fu(2,)2038 1991 y Fg(\022)2127 2109 y Fu(1)18 b(+)2373 2052 y(1)p 2280 2089 V 2280 2166 a(2)p Fq(T)2383 2142 y Fi(\027)t Fj(\000)p Fi(p)2519 1991 y Fg(\023)2580 2009 y Fi(`)2635 2109 y Fp(\024)k Fu(2)456 2294 y(and)27 b Fq(\032)h Fu(suc)n(h)f(that)1245 2430 y Fq(\032)c(<)g Fu(\()p Fq(c=)p Fu(2\))14 b(sin)o(\()p Fq(\031)22 b Fp(\000)c Fq(\013)p Fu(\))84 b(if)f Fq(\031)s(=)p Fu(2)23 b Fp(\024)f Fq(\013)i(<)e(\031)1245 2529 y(\032)h(<)g(c=)p Fu(2)532 b(if)83 b(0)23 b Fp(\024)f Fq(\013)i(<)f(\031)s(=)p Fu(2.)456 2671 y(First)k(w)n(e)g(c)n(hec)n(k)g(that)h(\000)f(is)h(w)n (ell)f(de\014ned.)38 b(If)28 b Fq( )e Fp(2)d Fq(B)t Fu(\()p Fq(\032)p Fu(\),)1207 2755 y Fg(\014)1207 2805 y(\014)1207 2855 y(\014)1265 2794 y Fq(c)p 1244 2831 78 4 v 1244 2907 a(s)1283 2883 y Fi(p)1350 2850 y Fu(+)18 b Fq( )s Fu(\()p Fq(s)p Fu(\))1593 2755 y Fg(\014)1593 2805 y(\014)1593 2855 y(\014)1644 2850 y Fp(\024)1773 2794 y Fq(c)p 1742 2831 99 4 v 1742 2907 a(T)1803 2883 y Fi(p)1869 2850 y Fu(+)1992 2794 y(1)p 1962 2831 102 4 v 1962 2907 a Fq(T)2023 2883 y Fi(\027)2073 2850 y Fp(k)p Fq( )s Fp(k)k(\024)2362 2794 y Fu(1)p 2334 2831 99 4 v 2334 2907 a Fq(T)2395 2883 y Fi(p)2442 2850 y Fu(2)p Fq(c)h(<)f(r)r Fu(.)456 3019 y(Moreo)n(v)n(er,)j(for)i Fq(s)c Fp(2)g Fu(\012\()p Fq(T)7 b(;)14 b(\013)p Fu(\),)1385 3160 y Fp(j)g Fu(arg)f Fq(')1606 3172 y Fo(0)1644 3160 y Fu(\()p Fq(s)p Fu(\))p Fp(j)23 b Fu(=)g Fp(j)14 b Fu(arg)f Fq(s)2087 3125 y Fi(p)2126 3160 y Fp(j)23 b Fq(<)f(p\013)i(<)e(\031)456 3301 y Fu(and)1271 3350 y Fg(\014)1271 3400 y(\014)1271 3449 y(\014)1271 3499 y(\014)1326 3414 y Fq( )s Fu(\()p Fq(s)p Fu(\))p 1309 3451 196 4 v 1309 3527 a Fq(')1363 3539 y Fo(0)1400 3527 y Fu(\()p Fq(s)p Fu(\))1514 3350 y Fg(\014)1514 3400 y(\014)1514 3449 y(\014)1514 3499 y(\014)1564 3470 y Fu(=)1652 3350 y Fg(\014)1652 3400 y(\014)1652 3449 y(\014)1652 3499 y(\014)1690 3414 y Fq( )s Fu(\()p Fq(s)p Fu(\))p 1690 3451 161 4 v 1693 3527 a Fq(c=s)1810 3503 y Fi(p)1860 3350 y Fg(\014)1860 3400 y(\014)1860 3449 y(\014)1860 3499 y(\014)1911 3470 y Fp(\024)2008 3414 y(k)p Fq( )s Fp(k)p 2008 3451 141 4 v 2061 3527 a Fq(c)2254 3414 y Fu(1)p 2168 3451 213 4 v 2168 3527 a Fp(j)p Fq(s)p Fp(j)2253 3503 y Fi(\027)t Fj(\000)p Fi(p)2414 3470 y Fp(\024)2512 3414 y Fu(2)p Fq(\032)p 2512 3451 85 4 v 2536 3527 a(c)2606 3470 y Fu(.)456 3661 y(This)36 b(implies)i(that)f Fp(j)14 b Fu(arg)o(\()p Fq(')1374 3673 y Fo(0)1412 3661 y Fu(\()p Fq(s)p Fu(\))25 b(+)f Fq( )s Fu(\()p Fq(s)p Fu(\)\))p Fp(j)40 b Fq(<)e(\031)i Fu(and)d(hence,)i(if)f Fq( )j Fp(2)e Fq(B)t Fu(\()p Fq(\032)p Fu(\),)h(then)d(for)g(all)456 3760 y Fq(s)23 b Fp(2)g Fu(\012\()p Fq(T)7 b(;)14 b(\013)p Fu(\),)28 b Fq(')971 3772 y Fo(0)1009 3760 y Fu(\()p Fq(s)p Fu(\))19 b(+)f Fq( )s Fu(\()p Fq(s)p Fu(\))28 b(b)r(elongs)f(to)g(the)h(domain)f(of)h Fq(f)9 b Fu(.)605 3860 y(W)-7 b(e)28 b(will)g(use)g(the)g(follo)n(wing)e(b)r(ounds.)37 b(F)-7 b(or)28 b Fq(k)e Fp(2)d Fl(R)p Fu(,)34 b Fq(k)26 b(>)d Fu(1)k(and)g Fq(z)t(;)14 b(w)25 b Fp(2)f Fl(C)15 b Fu(,)34 b(suc)n(h)27 b(that)h Fq(z)t Fu(,)456 3960 y Fq(z)21 b Fu(+)d Fq(w)26 b Fp(2)d Fq(U)37 b Fu(and)27 b Fp(j)p Fq(w)r(=z)t Fp(j)c Fq(<)g Fu(1,)k(w)n(e)g(ha)n(v)n(e)f(that) 782 4150 y Fp(j)p Fu(\()p Fq(z)c Fu(+)c Fq(w)r Fu(\))1074 4116 y Fi(k)1135 4150 y Fp(\000)g Fq(z)1261 4116 y Fi(k)1301 4150 y Fp(j)83 b Fu(=)f Fp(j)p Fq(z)t Fp(j)1643 4116 y Fi(k)1684 4150 y Fp(j)p Fu(\(1)18 b(+)g Fq(w)r(=z)t Fu(\))2060 4116 y Fi(k)2119 4150 y Fp(\000)g Fu(1)p Fp(j)23 b(\024)f(j)p Fq(z)t Fp(j)2466 4116 y Fi(k)2520 4071 y Fg(X)2522 4249 y Fi(j)s Fj(\025)p Fo(1)2654 4030 y Fg(\014)2654 4079 y(\014)2654 4129 y(\014)2654 4179 y(\014)2682 4033 y(\022)2743 4094 y Fq(k)2746 4207 y(j)2789 4033 y Fg(\023)2850 4030 y(\014)2850 4079 y(\014)2850 4129 y(\014)2850 4179 y(\014)2891 4150 y Fp(j)p Fq(w)r(=z)t Fp(j)3083 4116 y Fi(j)1407 4423 y Fp(\024)82 b(j)p Fq(z)t Fp(j)1643 4388 y Fi(k)q Fj(\000)p Fo(1)1769 4423 y Fp(j)p Fq(w)r Fp(j)1890 4344 y Fg(X)1892 4522 y Fi(j)s Fj(\025)p Fo(1)2024 4302 y Fg(\014)2024 4352 y(\014)2024 4402 y(\014)2024 4452 y(\014)2052 4305 y(\022)2113 4366 y Fq(k)2116 4479 y(j)2159 4305 y Fg(\023)2220 4302 y(\014)2220 4352 y(\014)2220 4402 y(\014)2220 4452 y(\014)2270 4423 y Fu(=)23 b Fq(C)2417 4435 y Fi(k)q(;)p Fo(1)2511 4423 y Fp(j)p Fq(z)t Fp(j)2600 4388 y Fi(k)q Fj(\000)p Fo(1)2725 4423 y Fp(j)p Fq(w)r Fp(j)-2376 b Fu(\(2.5\))456 4648 y(and)682 4789 y Fp(j)p Fu(\()p Fq(z)22 b Fu(+)c Fq(w)r Fu(\))974 4755 y Fi(k)1034 4789 y Fp(\000)g Fq(z)1160 4755 y Fi(k)1219 4789 y Fp(\000)g Fq(k)s(z)1391 4755 y Fi(k)q Fj(\000)p Fo(1)1516 4789 y Fq(w)r Fp(j)84 b Fu(=)e Fp(j)p Fq(z)t Fp(j)1920 4755 y Fi(k)1961 4789 y Fp(j)p Fu(\(1)18 b(+)g Fq(w)r(=z)t Fu(\))2337 4755 y Fi(k)2396 4789 y Fp(\000)g Fu(1)g Fp(\000)g Fq(k)s Fu(\()p Fq(w)r(=z)t Fu(\))p Fp(j)1684 4963 y(\024)82 b(j)p Fq(z)t Fp(j)1920 4929 y Fi(k)1974 4885 y Fg(X)1976 5063 y Fi(j)s Fj(\025)p Fo(2)2108 4843 y Fg(\014)2108 4893 y(\014)2108 4943 y(\014)2108 4992 y(\014)2136 4846 y(\022)2197 4907 y Fq(k)2200 5020 y(j)2243 4846 y Fg(\023)2304 4843 y(\014)2304 4893 y(\014)2304 4943 y(\014)2304 4992 y(\014)2345 4963 y Fp(j)p Fq(w)r(=z)t Fp(j)2537 4929 y Fi(j)2595 4963 y Fu(=)23 b Fq(C)2742 4975 y Fi(k)q(;)p Fo(2)2836 4963 y Fp(j)p Fq(z)t Fp(j)2925 4929 y Fi(k)q Fj(\000)p Fo(2)3050 4963 y Fp(j)p Fq(w)r Fp(j)3157 4929 y Fo(2)3195 4963 y Fu(,)456 5209 y(where)k Fq(C)755 5221 y Fi(k)q(;k)846 5229 y Fh(0)906 5209 y Fu(=)994 5147 y Fg(P)1082 5234 y Fi(j)s Fj(\025)p Fi(k)1199 5242 y Fh(0)1250 5138 y Fg(\014)1250 5188 y(\014)1278 5142 y(\000)1316 5172 y Fi(k)1319 5237 y(j)1352 5142 y Fg(\001)1390 5138 y(\014)1390 5188 y(\014)1418 5209 y Fu(.)p eop %%Page: 11 19 11 18 bop 1313 251 a Fo(2.2.)29 b(PR)n(OOF)g(OF)g(THE)g(PR)n(OPOSITION) f(2.1)791 b(11)605 450 y Fu(Giv)n(en)29 b Fq(t)d Fp(2)g Fu(\012\()p Fq(T)7 b(;)14 b(\013)p Fu(\),)30 b Fq(t)25 b Fu(=)g Fp(j)p Fq(t)p Fp(j)p Fq(e)1567 420 y Fi(i\022)1628 450 y Fu(,)30 b(to)f(ev)-5 b(aluate)29 b(the)g(in)n(tegral)f(in)i(the)f (de\014nition)h(of)f(\000)g(w)n(e)456 550 y(will)34 b(tak)n(e)g(the)h (path)g(of)f(in)n(tegration)g Fq(\015)39 b Fu(=)34 b Fq(\015)1912 562 y Fo(1)1972 550 y Fp(_)24 b Fq(\015)2094 562 y Fo(2)2166 550 y Fu(with)35 b Fq(\015)2405 562 y Fo(1)2442 550 y Fu(\()p Fq(u)p Fu(\))g(=)f Fq(T)12 b(e)2788 520 y Fi(iu)2854 550 y Fu(,)36 b Fq(u)e Fp(2)h Fu([0)p Fq(;)14 b(\022)r Fu(],)37 b(and)456 649 y Fq(\015)499 661 y Fo(2)536 649 y Fu(\()p Fq(u)p Fu(\))23 b(=)g Fq(T)12 b(e)859 619 y Fi(i\022)937 649 y Fu(+)18 b(\()p Fq(t)h Fp(\000)f Fq(T)12 b(e)1284 619 y Fi(i\022)1343 649 y Fu(\))p Fq(u)p Fu(,)28 b Fq(u)22 b Fp(2)i Fu([0)p Fq(;)14 b Fu(1].)36 b(W)-7 b(e)28 b(call)1029 854 y Fq(I)1065 866 y Fi(\016)1125 854 y Fu(=)1213 741 y Fg(Z)1259 930 y Fi(\015)1294 938 y Fh(2)1394 798 y Fu(1)p 1354 835 122 4 v 1354 911 a Fp(j)p Fq(s)p Fp(j)1439 887 y Fi(\016)1500 854 y Fq(ds)23 b Fu(=)1816 798 y(1)p 1702 835 269 4 v 1702 912 a Fq(e)1741 888 y Fi(i\022)r Fo(\()p Fi(\016)r Fj(\000)p Fo(1\))1995 741 y Fg(Z)2078 761 y Fo(1)2041 930 y(0)2317 798 y Fp(j)p Fq(t)p Fp(j)c(\000)f Fq(T)p 2139 835 594 4 v 2139 911 a Fu([)p Fq(T)30 b Fu(+)18 b(\()p Fp(j)p Fq(t)p Fp(j)h(\000)f Fq(T)12 b Fu(\))p Fq(u)p Fu(])2698 887 y Fi(\016)2757 854 y Fq(du)p Fu(.)456 1079 y(W)-7 b(e)28 b(ha)n(v)n(e)e Fp(j)p Fq(I)849 1091 y Fi(\016)886 1079 y Fp(j)d Fu(=)1072 1046 y Fo(1)p 1030 1060 118 4 v 1030 1107 a(1)p Fj(\000)p Fi(\016)1157 1079 y Fu(\()p Fp(j)p Fq(t)p Fp(j)1265 1048 y Fo(1)p Fj(\000)p Fi(\016)1405 1079 y Fp(\000)18 b Fq(T)1549 1048 y Fo(1)p Fj(\000)p Fi(\016)1670 1079 y Fu(\))28 b(if)g Fq(\016)e Fp(6)p Fu(=)c(1)28 b(and)f Fp(j)p Fq(I)2246 1091 y Fi(\016)2283 1079 y Fp(j)c Fu(=)g(log)2548 1038 y Fj(j)p Fi(t)p Fj(j)p 2548 1059 65 4 v 2556 1107 a Fi(T)2650 1079 y Fu(if)28 b Fq(\016)e Fu(=)d(1.)605 1180 y(W)-7 b(e)28 b(in)n(tro)r(duce)832 1325 y Fq(\037)884 1337 y Fo(1)921 1325 y Fu(\()p Fq(s)p Fu(\))c(=)f Fq(s)1175 1291 y Fi(p)p Fo(+1)1297 1325 y Fq(a)1341 1258 y Fg(\002)1375 1325 y Fu(\()p Fq(')1461 1337 y Fo(0)1499 1325 y Fu(\()p Fq(s)p Fu(\))c(+)f Fq( )s Fu(\()p Fq(s)p Fu(\)\))1896 1291 y Fi(k)1956 1325 y Fp(\000)g Fu(\()p Fq(')2125 1337 y Fo(0)2163 1325 y Fu(\()p Fq(s)p Fu(\)\))2298 1291 y Fi(k)2358 1325 y Fp(\000)g Fq(k)s Fu(\()p Fq(')2573 1337 y Fo(0)2611 1325 y Fu(\()p Fq(s)p Fu(\)\))2746 1291 y Fi(k)q Fj(\000)p Fo(1)2873 1325 y Fq( )s Fu(\()p Fq(s)p Fu(\))3033 1258 y Fg(\003)456 1481 y Fu(and)27 b Fq(\037)669 1493 y Fo(2)706 1481 y Fu(\()p Fq(s)p Fu(\))d(=)e Fq(s)959 1451 y Fi(p)p Fo(+1)1082 1481 y Fq(g)s Fu(\()p Fq(')1211 1493 y Fo(0)1248 1481 y Fu(\()p Fq(s)p Fu(\))d(+)f Fq( )s Fu(\()p Fq(s)p Fu(\)\).)38 b(W)-7 b(e)28 b(ha)n(v)n(e)1122 1555 y Fg(\014)1122 1605 y(\014)1122 1655 y(\014)1122 1705 y(\014)1149 1563 y(Z)1195 1751 y Fi(\015)1230 1759 y Fh(1)1281 1676 y Fq(\037)1333 1688 y Fo(1)1370 1676 y Fu(\()p Fq(s)p Fu(\))14 b Fq(ds)1569 1555 y Fg(\014)1569 1605 y(\014)1569 1655 y(\014)1569 1705 y(\014)1680 1676 y Fp(\024)83 b Fq(C)1887 1688 y Fi(k)q(;)p Fo(2)1981 1676 y Fq(\013)p Fp(j)p Fq(a)p Fp(j)p Fq(c)2160 1641 y Fi(k)q Fj(\000)p Fo(2)2286 1676 y Fp(k)p Fq( )s Fp(k)2427 1641 y Fo(2)2579 1620 y Fq(T)p 2473 1657 273 4 v 2473 1734 a(T)2534 1710 y Fo(2\()p Fi(\027)t Fj(\000)p Fi(p)p Fo(\))2755 1676 y Fu(,)1122 1796 y Fg(\014)1122 1846 y(\014)1122 1895 y(\014)1122 1945 y(\014)1149 1803 y(Z)1195 1992 y Fi(\015)1230 2000 y Fh(1)1281 1916 y Fq(\037)1333 1928 y Fo(2)1370 1916 y Fu(\()p Fq(s)p Fu(\))14 b Fq(ds)1569 1796 y Fg(\014)1569 1846 y(\014)1569 1895 y(\014)1569 1945 y(\014)1680 1916 y Fp(\024)83 b Fu(2)p Fq(B)t(c)1973 1882 y Fi(`)2004 1916 y Fq(\013)2129 1860 y(T)p 2067 1897 183 4 v 2067 1973 a(T)2128 1949 y Fi(q)r Fj(\000)p Fi(p)2260 1916 y Fu(,)1122 2036 y Fg(\014)1122 2086 y(\014)1122 2136 y(\014)1122 2186 y(\014)1149 2044 y(Z)1195 2232 y Fi(\015)1230 2240 y Fh(2)1281 2157 y Fq(\037)1333 2169 y Fo(1)1370 2157 y Fu(\()p Fq(s)p Fu(\))14 b Fq(ds)1569 2036 y Fg(\014)1569 2086 y(\014)1569 2136 y(\014)1569 2186 y(\014)1680 2157 y Fp(\024)83 b Fq(C)1887 2169 y Fi(k)q(;)p Fo(2)1981 2157 y Fp(j)p Fq(a)p Fp(j)p Fq(c)2107 2123 y Fi(k)q Fj(\000)p Fo(2)2232 2157 y Fp(k)p Fq( )s Fp(k)2373 2123 y Fo(2)2410 2157 y Fp(j)p Fq(I)2469 2172 y Fo(2\()p Fi(\027)t Fj(\000)p Fi(p)p Fo(\))2682 2157 y Fp(j)456 2355 y Fu(and)1454 2409 y Fg(\014)1454 2459 y(\014)1454 2509 y(\014)1454 2559 y(\014)1482 2416 y(Z)1528 2605 y Fi(\015)1563 2613 y Fh(2)1614 2529 y Fq(\037)1666 2541 y Fo(2)1703 2529 y Fu(\()p Fq(s)p Fu(\))14 b Fq(ds)1902 2409 y Fg(\014)1902 2459 y(\014)1902 2509 y(\014)1902 2559 y(\014)1953 2529 y Fp(\024)23 b Fu(2)p Fq(B)t(c)2186 2495 y Fi(`)2217 2529 y Fp(j)p Fq(I)2276 2541 y Fi(q)r Fj(\000)p Fi(p)2399 2529 y Fp(j)p Fu(.)456 2733 y(W)-7 b(e)35 b(ha)n(v)n(e)f(to)h(distinguish)g(the)g(cases)f(2\() p Fq(\027)29 b Fp(\000)23 b Fq(p)p Fu(\))35 b Fq(<)g Fu(1,)i(2\()p Fq(\027)28 b Fp(\000)23 b Fq(p)p Fu(\))36 b(=)f(1,)h(2\()p Fq(\027)29 b Fp(\000)23 b Fq(p)p Fu(\))35 b Fq(>)g Fu(1)g(and)456 2832 y Fq(q)17 b Fp(\000)e Fq(p)23 b(<)f Fu(1,)k Fq(q)18 b Fp(\000)c Fq(p)23 b Fu(=)g(1)i(and)h Fq(q)18 b Fp(\000)c Fq(p)23 b(>)f Fu(1.)36 b(Using)26 b(the)g(previous)f(estimates)g(w)n(e)g(\014nd)i(that)f(in)g(all)456 2932 y(cases)1669 3078 y Fp(j)p Fu(\000\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))p Fp(j)e(\024)2141 3021 y Fq(\032)p 2104 3058 118 4 v 2104 3135 a Fp(j)p Fq(t)p Fp(j)2180 3111 y Fi(\027)456 3271 y Fu(if)k Fq(T)38 b Fu(is)28 b(big)f(enough,)g(and)h (hence)f(\000)p Fq( )f Fp(2)e Fq(B)t Fu(\()p Fq(\032)p Fu(\).)605 3378 y(Next)f(w)n(e)e(see)h(that)h(\000)f(is)g(a)g(con)n (traction.)33 b(Indeed,)24 b(let)e Fq( )k Fu(and)2561 3356 y(~)2544 3378 y Fq( )f Fu(b)r(e)e(t)n(w)n(o)e(functions)i(whic)n (h)456 3477 y(b)r(elong)k(to)g Fq(B)t Fu(\()p Fq(\032)p Fu(\),)464 3672 y Fp(j)p Fu(\(\000)p Fq( )22 b Fp(\000)c Fu(\000)799 3651 y(~)782 3672 y Fq( )s Fu(\)\()p Fq(t)p Fu(\))p Fp(j)84 b(\024)1308 3616 y Fu(1)p 1229 3653 199 4 v 1229 3729 a Fp(j)p Fq(t)p Fp(j)1305 3705 y Fi(p)p Fo(+1)1452 3552 y Fg(\014)1452 3602 y(\014)1452 3652 y(\014)1452 3701 y(\014)1479 3555 y(\022)1540 3559 y(Z)1586 3748 y Fi(\015)1621 3756 y Fh(1)1672 3672 y Fu(+)1751 3559 y Fg(Z)1796 3748 y Fi(\015)1831 3756 y Fh(2)1868 3555 y Fg(\023)1943 3672 y Fq(s)1982 3638 y Fi(p)p Fo(+1)2104 3605 y Fg(\000)2142 3672 y Fq(f)2183 3684 y Fo(0)2220 3672 y Fu(\()p Fq(')2306 3684 y Fo(0)2344 3672 y Fu(\()p Fq(s)p Fu(\))19 b(+)f Fq( )s Fu(\()p Fq(s)p Fu(\)\))h Fp(\000)f Fq(D)r(f)2955 3684 y Fo(0)2992 3672 y Fu(\()p Fq(')3078 3684 y Fo(0)3116 3672 y Fu(\()p Fq(s)p Fu(\)\))p Fq( )s Fu(\()p Fq(s)p Fu(\))1219 3870 y Fp(\000)p Fu([)p Fq(f)1348 3882 y Fo(0)1385 3870 y Fu(\()p Fq(')1471 3882 y Fo(0)1509 3870 y Fu(\()p Fq(s)p Fu(\))h(+)1731 3848 y(~)1714 3870 y Fq( )s Fu(\()p Fq(s)p Fu(\)\))g Fp(\000)f Fq(D)r(f)2120 3882 y Fo(0)2157 3870 y Fu(\()p Fq(')2243 3882 y Fo(0)2281 3870 y Fu(\()p Fq(s)p Fu(\)\))2433 3848 y(~)2416 3870 y Fq( )t Fu(\()p Fq(s)p Fu(\)])2600 3803 y Fg(\001)2652 3870 y Fq(ds)-2278 b Fu(\(2.6\))1229 4053 y(+)1308 3936 y Fg(\022)1369 3940 y(Z)1415 4128 y Fi(\015)1450 4136 y Fh(1)1500 4053 y Fu(+)1579 3940 y Fg(Z)1625 4128 y Fi(\015)1660 4136 y Fh(2)1696 3936 y Fg(\023)1771 4053 y Fq(s)1810 4018 y Fi(p)p Fo(+1)1932 4053 y Fu([)p Fq(g)s Fu(\()p Fq(')2084 4065 y Fo(0)2122 4053 y Fu(\()p Fq(s)p Fu(\))19 b(+)f Fq( )s Fu(\()p Fq(s)p Fu(\)\))h Fp(\000)f Fq(g)s Fu(\()p Fq(')2750 4065 y Fo(0)2788 4053 y Fu(\()p Fq(s)p Fu(\))h(+)3010 4031 y(~)2993 4053 y Fq( )s Fu(\()p Fq(s)p Fu(\)\)])14 b Fq(ds)3304 3932 y Fg(\014)3304 3982 y(\014)3304 4032 y(\014)3304 4082 y(\014)3346 4053 y Fu(.)456 4251 y(T)-7 b(o)27 b(ev)-5 b(aluate)27 b(the)h(\014rst)f (di\013erence)h(w)n(e)f(consider)g(the)h(function)1249 4396 y Fq(\037)p Fu(\()p Fq(z)t Fu(\))23 b(=)g Fq(a)p Fu([\()p Fq(')1672 4408 y Fo(0)1709 4396 y Fu(\()p Fq(s)p Fu(\))c(+)f Fq(z)t Fu(\))1989 4362 y Fi(k)2048 4396 y Fp(\000)g Fq(k)s Fu(\()p Fq(')2263 4408 y Fo(0)2301 4396 y Fu(\()p Fq(s)p Fu(\)\))2436 4362 y Fi(k)q Fj(\000)p Fo(1)2562 4396 y Fq(z)t Fu(].)456 4542 y(By)27 b(the)h(mean)f(v)-5 b(alue)28 b(theorem,)f(w)n(e)g(ha)n(v)n(e)g(that)1164 4742 y Fq(\037)p Fu(\()5 b(~)-47 b Fq(z)s Fu(\))19 b Fp(\000)f Fq(\037)p Fu(\()p Fq(z)t Fu(\))23 b(=)1694 4629 y Fg(Z)1777 4649 y Fo(1)1740 4817 y(0)1828 4742 y Fq(\037)1880 4707 y Fj(0)1903 4742 y Fu(\()p Fq(z)f Fu(+)c Fq(\020)6 b Fu(\()f(~)-47 b Fq(z)23 b Fp(\000)18 b Fq(z)t Fu(\)\)[)5 b(~)-47 b Fq(z)21 b Fp(\000)d Fq(z)t Fu(])c Fq(d\020)456 4942 y Fu(and,)27 b(since)h Fq(\037)896 4912 y Fj(0)919 4942 y Fu(\()p Fq(w)r Fu(\))c(=)f Fq(ak)s Fu([\()p Fq(')1355 4954 y Fo(0)1392 4942 y Fu(\()p Fq(s)p Fu(\))c(+)f Fq(w)r Fu(\))1690 4912 y Fi(k)q Fj(\000)p Fo(1)1835 4942 y Fp(\000)h Fu(\()p Fq(')2005 4954 y Fo(0)2042 4942 y Fu(\()p Fq(s)p Fu(\)\))2177 4912 y Fi(k)q Fj(\000)p Fo(1)2304 4942 y Fu(])28 b(then,)g(using)f(\(2.5\),)907 5137 y Fp(j)p Fq(\037)p Fu(\()1031 5115 y(~)1014 5137 y Fq( )s Fu(\()p Fq(s)p Fu(\)\))20 b Fp(\000)e Fq(\037)p Fu(\()p Fq( )s Fu(\()p Fq(s)p Fu(\)\))p Fp(j)24 b(\024)e(j)p Fq(a)p Fp(j)p Fq(k)1869 5020 y Fg(\024)1913 5137 y Fq(C)1972 5149 y Fi(k)q(;)p Fo(1)2066 5045 y Fg(\020)2169 5081 y Fq(c)p 2125 5118 124 4 v 2125 5194 a Fp(j)p Fq(s)p Fp(j)2210 5170 y Fi(p)2259 5045 y Fg(\021)2308 5062 y Fi(k)q Fj(\000)p Fo(2)2503 5081 y Fq(\032)p 2444 5118 160 4 v 2444 5194 a Fp(j)p Fq(s)p Fp(j)2529 5170 y Fo(2)p Fi(\027)2614 5137 y Fp(k)2672 5115 y Fu(~)2656 5137 y Fq( )f Fp(\000)d Fq( )s Fp(k)2913 5020 y Fg(\025)2970 5137 y Fu(.)p eop %%Page: 12 20 12 19 bop 456 251 a Fo(12)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(W)-7 b(e)28 b(b)r(ound)g(the)g(\014rst)f(in)n(tegrals)f(in)i (\(2.6\),)670 514 y Fg(\014)670 564 y(\014)670 614 y(\014)670 663 y(\014)697 521 y(Z)744 710 y Fi(\015)779 718 y Fh(1)829 634 y Fq(s)868 600 y Fi(p)p Fo(+1)990 567 y Fg(\002)1025 634 y Fq(\037)p Fu(\()1126 613 y(~)1109 634 y Fq( )s Fu(\()p Fq(s)p Fu(\)\))19 b Fp(\000)f Fq(\037)p Fu(\()p Fq( )s Fu(\()p Fq(s)p Fu(\)\))1679 567 y Fg(\003)1729 634 y Fq(ds)1811 514 y Fg(\014)1811 564 y(\014)1811 614 y(\014)1811 663 y(\014)1922 634 y Fp(\024)82 b Fq(C)2128 646 y Fi(k)q(;)p Fo(1)2222 634 y Fq(k)s(\013)p Fp(j)p Fq(a)p Fp(j)p Fq(c)2447 600 y Fi(k)q Fj(\000)p Fo(2)2573 634 y Fq(\032)p Fp(k)2675 613 y Fu(~)2658 634 y Fq( )21 b Fp(\000)d Fq( )s Fp(k)3031 578 y Fq(T)p 2925 615 273 4 v 2925 693 a(T)2986 669 y Fo(2\()p Fi(\027)t Fj(\000)p Fi(p)p Fo(\))3207 634 y Fu(,)670 755 y Fg(\014)670 804 y(\014)670 854 y(\014)670 904 y(\014)697 762 y(Z)744 951 y Fi(\015)779 959 y Fh(2)829 875 y Fq(s)868 841 y Fi(p)p Fo(+1)990 808 y Fg(\002)1025 875 y Fq(\037)p Fu(\()1126 853 y(~)1109 875 y Fq( )s Fu(\()p Fq(s)p Fu(\)\))h Fp(\000)f Fq(\037)p Fu(\()p Fq( )s Fu(\()p Fq(s)p Fu(\)\))1679 808 y Fg(\003)1729 875 y Fq(ds)1811 755 y Fg(\014)1811 804 y(\014)1811 854 y(\014)1811 904 y(\014)1922 875 y Fp(\024)82 b Fq(C)2128 887 y Fi(k)q(;)p Fo(1)2222 875 y Fq(k)s Fp(j)p Fq(a)p Fp(j)p Fq(c)2394 841 y Fi(k)q Fj(\000)p Fo(2)2520 875 y Fq(\032)p Fp(k)2621 853 y Fu(~)2605 875 y Fq( )21 b Fp(\000)d Fq( )s Fp(k)p Fq(I)2898 890 y Fo(2\()p Fi(\027)t Fj(\000)p Fi(p)p Fo(\))3110 875 y Fu(.)456 1067 y(T)-7 b(o)32 b(deal)h(with)h(the)f(second)g(in)n (tegrals)e(w)n(e)i(use)g(that,)i(since)e Fq(g)j Fu(is)d(analytic)f(in)h Fq(U)9 b Fu(,)35 b Fp(j)p Fq(g)3195 1037 y Fj(0)3218 1067 y Fu(\()p Fq(z)t Fu(\))p Fp(j)d(\024)456 1167 y Fq(B)519 1179 y Fo(2)556 1167 y Fp(j)p Fq(z)t Fp(j)645 1137 y Fi(`)p Fj(\000)p Fo(1)789 1167 y Fu(in)c(a)f(domain)893 1302 y Fp(f)p Fq(z)f Fp(2)d Fl(C)44 b Fu(:)23 b Fp(j)p Fq(z)t Fp(j)g Fq(<)f(r)1443 1314 y Fo(1)1481 1302 y Fu(,)28 b Fp(j)14 b Fu(arg)f Fq(z)t Fp(j)22 b Fq(<)h(\013)1928 1314 y Fo(1)1965 1302 y Fp(g)p Fu(,)83 b(0)23 b Fq(<)f(r)2302 1314 y Fo(1)2363 1302 y Fq(<)h(r)r Fu(,)56 b Fq(\013)23 b(<)g(\013)2786 1314 y Fo(1)2846 1302 y Fq(<)g(\031)s Fu(.)456 1436 y(Then)k(the)h(in)n(tegrals)e(that)i(w)n(e)g(are)e (analyzing)g(are)h(b)r(ounded)h(b)n(y)1158 1611 y Fq(\013B)1274 1623 y Fo(2)1311 1611 y Fq(c)1347 1577 y Fi(`)p Fj(\000)p Fo(1)1464 1519 y Fg(\020)1514 1611 y Fu(1)18 b(+)1757 1555 y Fq(\032)p 1667 1592 224 4 v 1667 1668 a(cT)1764 1644 y Fi(\027)t Fj(\000)p Fi(p)1900 1519 y Fg(\021)1950 1536 y Fi(`)p Fj(\000)p Fo(1)2067 1611 y Fp(k)2125 1589 y Fu(~)2109 1611 y Fq( )j Fp(\000)d Fq( )s Fp(k)2524 1555 y Fq(T)p 2376 1592 358 4 v 2376 1668 a(T)2437 1644 y Fi(q)r Fj(\000)p Fi(p)p Fo(+)p Fi(\027)t Fj(\000)p Fi(p)456 1769 y Fu(and)1206 1927 y Fq(B)1269 1939 y Fo(2)1306 1927 y Fq(c)1342 1893 y Fi(`)p Fj(\000)p Fo(1)1459 1835 y Fg(\020)1509 1927 y Fu(1)g(+)1752 1871 y Fq(\032)p 1662 1908 224 4 v 1662 1984 a(cT)1759 1960 y Fi(\027)t Fj(\000)p Fi(p)1895 1835 y Fg(\021)1945 1852 y Fi(`)p Fj(\000)p Fo(1)2062 1927 y Fp(k)2120 1905 y Fu(~)2104 1927 y Fq( )j Fp(\000)d Fq( )s Fp(k)p Fq(I)2397 1939 y Fi(q)r Fj(\000)p Fi(p)p Fo(+)p Fi(\027)t Fj(\000)p Fi(p)456 2085 y Fu(resp)r(ectiv)n(ely)-7 b(,)24 b(where)h(w)n(e)g(ha)n (v)n(e)e(used)i(that)h Fq(p`)c Fu(=)h Fq(q)16 b Fu(+)d(1.)36 b(Distinguishing)25 b(cases)f(as)g(b)r(efore)h(w)n(e)456 2185 y(get)i(that)1359 2354 y Fp(j)p Fu(\(\000)p Fq( )21 b Fp(\000)d Fu(\000)1693 2332 y(~)1676 2354 y Fq( )s Fu(\)\()p Fq(t)p Fu(\))p Fp(j)24 b(\024)f Fq(C)2053 2366 y Fi(T)2106 2354 y Fp(k)2164 2332 y Fu(~)2148 2354 y Fq( )e Fp(\000)d Fq( )s Fp(k)2452 2298 y Fu(1)p 2415 2335 118 4 v 2415 2411 a Fp(j)p Fq(t)p Fp(j)2491 2387 y Fi(\027)456 2532 y Fu(and)27 b Fq(C)676 2544 y Fi(T)756 2532 y Fu(is)h(smaller)e(than)i(1)f(if)h Fq(T)39 b Fu(is)28 b(big)f(enough.)605 2632 y(Hence,)33 b(if)g Fq(T)43 b Fu(is)32 b(big)g(enough,)h(\000)f(is)g(a)f(con)n(traction)g(in)h Fq(B)t Fu(\()p Fq(\032)p Fu(\))h(and,)g(b)n(y)f(the)g(\014xed)g(p)r (oin)n(t)456 2732 y(theorem,)d(there)h(exists)f(a)g(unique)h(solution)g (of)f(\(2.4\))h(whic)n(h)f(b)r(elongs)g(to)h Fq(B)t Fu(\()p Fq(\032)p Fu(\).)44 b(This)30 b(ends)456 2831 y(the)e(pro)r(of)f(of)g (Prop)r(osition)f(2.1.)p eop %%Page: 13 21 13 20 bop 543 566 a Fw(3.)60 b(P)l(arameterization)48 b(of)d(lo)t(cal)h(in)l(v)-7 b(arian)l(t)46 b(manifolds)1589 849 y Fk(3.1.)g(In)m(tro)s(duction)605 998 y Fu(In)25 b(this)f(c)n(hapter)g(w)n(e)g(pro)n(v)n(e)e(the)j(existence)f(of)g(the) h(lo)r(cal)f(stable)g(and)g(the)h(lo)r(cal)f(unstable)456 1098 y(manifolds)33 b(asso)r(ciated)f(to)i(the)g(origin)f(for)g(the)h (p)r(erturb)r(ed)g(system.)55 b(These)33 b(are)g(obtained)456 1197 y(through)21 b(sp)r(ecial)i(parameterizations)d(with)j(prop)r (erties)f(whic)n(h)g(will)h(b)r(e)g(useful)g(later)f(on.)35 b(W)-7 b(e)456 1297 y(only)23 b(deal)g(with)i(the)f(lo)r(cal)f(stable)g (manifold,)i(but)f(it)g(is)g(clear)f(that)h(the)g(results)f(are)g(also) g(true)456 1397 y(for)k(the)h(unstable)f(one,)g(w)n(orking)f(with)i (the)g(system)g(obtained)f(rev)n(ersing)f(time.)605 1496 y(In)g(order)d(to)j(pro)n(v)n(e)d(this)j(result)f(w)n(e)g(need)g(a)g (go)r(o)r(d)g(initial)g(appro)n(ximation)f(of)h(the)h(stable)456 1596 y(\(and)h(unstable\))h(manifold)g(and)f(suitable)h(co)r(ordinates) e(to)h(w)n(ork)g(with.)605 1695 y(In)33 b(Section)f(3.3,)h(w)n(e)g (obtain)f(these)g(co)r(ordinates)f(b)n(y)i(canonical)e(c)n(hanges)g(of) i(v)-5 b(ariables)456 1795 y(using)20 b(the)h(a)n(v)n(eraging)d(metho)r (d.)35 b(This)21 b(metho)r(d)g(allo)n(ws)f(us)h(to)f(obtain)h(t)n(w)n (o)f(imp)r(ortan)n(t)g(things:)456 1895 y(remo)n(v)n(e)25 b(the)i(terms)f(of)h(order)e Fq(\026")1510 1865 y Fi(p)1575 1895 y Fu(and)i(remo)n(v)n(e)e(the)i(smallest)f(degree)g(terms)g (\(with)i(resp)r(ect)456 1994 y(to)j(\()p Fq(x;)14 b(y)s Fu(\)\))33 b(of)e Fq(h)964 2006 y Fo(1)1001 1994 y Fu(.)49 b(W)-7 b(e)32 b(m)n(ust)g(a)n(v)n(erage)d(sev)n(eral)h(times)h(in)h (order)f(to)g(obtain)g(a)h(high)f(enough)456 2094 y(degree.)605 2194 y(The)j(initial)g(appro)n(ximation)e(of)i(the)h(stable)e(manifold) h(is)g(ac)n(hiev)n(ed)f(as)g(the)i(in)n(v)-5 b(arian)n(t)456 2293 y(manifold)26 b(of)g(an)g(appropriate)f(in)n(termediate)h(system)g (whic)n(h)g(is)g(obtained)h(b)n(y)f(dropping)f(the)456 2393 y(non-autonomous)g(part)i(in)g(the)h(a)n(v)n(eraged)d(system.)36 b(W)-7 b(e)28 b(remark)e(that)h(the)h(construction)f(of)456 2492 y(the)j(initial)h(appro)n(ximation)e(is)h(only)g(necessary)e(when) j Fq(k)g(<)c(n)p Fu(.)45 b(F)-7 b(or)30 b Fq(k)g Fp(\025)e Fq(n)i Fu(w)n(e)g(can)g(use)g(as)456 2592 y(initial)d(appro)n(ximation) f(the)i(unp)r(erturb)r(ed)g(homo)r(clinic)g(orbit.)605 2692 y(Finally)h(w)n(e)g(obtain)g(a)f(functional)i(equation)e(for)h (the)g(parameterization)e(of)i(the)h(stable)456 2791 y(manifold)39 b(and)g(w)n(e)g(pro)n(v)n(e)e(it)j(has)e(a)h(solution)g (applying)g(the)g(\014xed)g(p)r(oin)n(t)h(theorem)e(in)i(a)456 2891 y(suitable)27 b(Banac)n(h)f(space.)605 2991 y(It)41 b(is)f(imp)r(ortan)n(t)f(to)h(sa)n(y)f(that)i(although)e(the)i(system)f (is)g Fq(C)2634 2960 y Fo(0)2711 2991 y Fu(in)h Fq(t=")p Fu(,)h(w)n(e)e(obtain)g(a)456 3090 y(parameterization)26 b(with)i(t)n(w)n(o)f(parameters)g(sa)n(y)f(\()p Fq(t;)14 b(s)p Fu(\),)29 b(whic)n(h)f(is)g(analytic)f(in)h Fq(s)p Fu(,)g(considered)456 3190 y(as)f(a)g(complex)g(v)-5 b(ariable,)27 b(and)g(w)n(e)g(pro)n(vide)g(a)g(dynamic)g(sense)g(for)g Fq(s)p Fu(.)1281 3444 y Fk(3.2.)46 b(De\014nitions)30 b(and)j(main)d(result)605 3593 y Fu(W)-7 b(e)28 b(b)r(egin)f(b)n(y)g (in)n(tro)r(ducing)g(some)g(notation.)36 b(Giv)n(en)27 b Fq(T)34 b(>)23 b Fu(0)k(and)g Fq(\034)33 b(>)23 b Fu(0)k(w)n(e)f (de\014ne)i(the)456 3693 y(follo)n(wing)e(sets:)857 3857 y Fq(D)928 3822 y Fi(s)986 3857 y Fu(=)d Fq(D)1145 3822 y Fi(s)1180 3857 y Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))24 b(=)f Fp(f)o Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b Fp(2)f Fl(R)h Fp(\002)18 b Fl(C)44 b Fu(:)j Fq(t)18 b Fu(+)g(Re)c Fq(s)23 b Fp(\025)g Fq(T)7 b(;)27 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)f Fq(\034)9 b Fp(g)456 4021 y Fu(and)805 4185 y Fq(D)876 4151 y Fi(u)942 4185 y Fu(=)23 b Fq(D)1101 4151 y Fi(u)1144 4185 y Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))24 b(=)f Fp(f)o Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b Fp(2)f Fl(R)i Fp(\002)18 b Fl(C)44 b Fu(:)i Fq(t)18 b Fu(+)g(Re)c Fq(s)23 b Fp(\024)g(\000)p Fq(T)7 b(;)26 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)g Fq(\034)9 b Fp(g)14 b Fu(.)605 4354 y(Note)28 b(that)g(if)g(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)h Fq(D)1405 4324 y Fi(s)1440 4354 y Fu(,)k(\()p Fq(t)18 b Fu(+)h(2)p Fq(\031)s(";)14 b(s)p Fu(\))27 b(and)g(\()p Fq(t;)14 b(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\))27 b(also)g(b)r(elong)g(to)g Fq(D)3116 4324 y Fi(s)3152 4354 y Fu(.)605 4454 y(F)-7 b(or)27 b Fq(k)f Fp(2)d Fl(R)p Fu(,)34 b Fq(k)26 b Fp(\025)d Fu(0,)k(w)n(e)g(de\014ne)h(the)g(space)f Fp(X)2059 4424 y Fi(s)2047 4477 y(k)2122 4454 y Fu(of)h(functions)g Fq(h)23 b Fu(:)g Fq(D)2763 4424 y Fi(s)2821 4454 y Fp(!)g Fl(C)49 b Fu(suc)n(h)27 b(that)647 4586 y(\(a\))41 b Fq(h)28 b Fu(is)f(con)n(tin)n(uous.)642 4685 y(\(b\))42 b(F)-7 b(or)27 b Fq(t)h Fu(\014xed,)f Fq(s)c Fp(7!)h Fq(h)p Fu(\()p Fq(t;)14 b(s)p Fu(\))27 b(is)h(analytic)f(in)h Fp(j)14 b Fu(Re)f Fq(s)p Fp(j)23 b(\024)g Fq(T)30 b Fp(\000)18 b Fq(t)p Fu(,)27 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)g Fq(\034)9 b Fu(.)651 4785 y(\(c\))42 b Fq(h)p Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\))22 b(=)h Fq(h)p Fu(\()p Fq(t;)14 b(s)k Fu(+)g(2)p Fq(\031)s(")p Fu(\))28 b(for)f(all)g(\()p Fq(t;)14 b(s)p Fu(\))24 b Fp(2)f Fq(D)2418 4755 y Fi(s)2454 4785 y Fu(.)642 4884 y(\(d\))42 b Fp(k)p Fq(h)p Fp(k)926 4896 y Fi(k)989 4884 y Fp(\021)23 b Fu(sup)p Fp(f)p Fu(\()p Fq(t)18 b Fu(+)g(Re)c Fq(s)p Fu(\))1590 4854 y Fi(k)1631 4884 y Fp(j)p Fq(h)p Fu(\()p Fq(t;)g(s)p Fu(\))p Fp(j)p Fq(;)28 b Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)h Fq(D)2289 4854 y Fi(s)2324 4884 y Fp(g)f Fq(<)f Fp(1)p Fu(.)605 5016 y(In)28 b(the)g(ob)n(vious)e (analogous)f(w)n(a)n(y)i(w)n(e)g(de\014ne)h Fp(X)2137 4986 y Fi(u)2125 5040 y(k)2181 5016 y Fu(.)605 5116 y(W)-7 b(e)27 b(endo)n(w)f Fp(X)1073 5086 y Fi(s)1061 5139 y(k)1136 5116 y Fu(with)h(the)g(norm)f Fp(k)16 b(\001)g(k)1821 5128 y Fi(k)1888 5116 y Fu(in)n(tro)r(duced)27 b(in)f(\(d\))i(and)e(it) h(b)r(ecomes)g(a)f(Banac)n(h)456 5216 y(space.)36 b(Clearly)26 b(w)n(e)h(ha)n(v)n(e)g Fp(X)1383 5185 y Fi(s)1371 5239 y(k)q Fo(+1)1519 5216 y Fp(\032)c(X)1678 5185 y Fi(s)1666 5239 y(k)1714 5216 y Fu(.)1917 5365 y Fo(13)p eop %%Page: 14 22 14 21 bop 456 251 a Fo(14)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(The)33 b(main)g(theorem)g(of)g(this)g(c)n(hapter)f(states)h(the)g (existence)g(of)g(in)n(v)-5 b(arian)n(t)32 b(manifolds)456 550 y(for)27 b(the)h(p)r(erturb)r(ed)g(Hamiltonian)f(equation)1241 713 y(_)-37 b Fq(x)83 b Fu(=)g Fq(y)21 b Fu(+)d Fq(\026")1739 679 y Fi(p)1777 713 y Fq(@)1821 725 y Fi(y)1861 713 y Fq(h)1909 725 y Fo(1)1946 713 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g(\026;)g (")p Fu(\))-1993 b(\(3.1\))1245 838 y(_)-38 b Fq(y)86 b Fu(=)d Fp(\000)p Fq(V)1636 804 y Fj(0)1660 838 y Fu(\()p Fq(x)p Fu(\))19 b Fp(\000)f Fq(\026")1962 804 y Fi(p)2000 838 y Fq(@)2044 850 y Fi(x)2086 838 y Fq(h)2134 850 y Fo(1)2171 838 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g(\026;)g(")p Fu(\))456 1001 y(assuming)31 b(conditions)i(on)f(the)h(orders)e(of)i Fq(h)1895 1013 y Fo(0)1964 1001 y Fu(and)g Fq(h)2179 1013 y Fo(1)2216 1001 y Fu(.)52 b(It)33 b(is)g(w)n(orth)e(noting)i (that)g(here,)g(in)456 1101 y(con)n(trast)26 b(with)i(h)n(yp)r(othesis) f Fk(HP3)p Fu(,)h(w)n(e)f(do)g(not)h(assume)f(that)g Fq(h)2513 1113 y Fo(1)2578 1101 y Fu(is)h(a)f(p)r(olynomial.)605 1267 y Ft(Theorem)32 b Fu(3.1)p Ft(.)40 b Fs(L)l(et)1657 1431 y Fq(H)30 b Fu(=)23 b Fq(h)1892 1443 y Fo(0)1947 1431 y Fu(+)18 b Fq(\026")2119 1396 y Fi(p)2158 1431 y Fq(h)2206 1443 y Fo(1)456 1605 y Fs(wher)l(e)30 b Fq(h)738 1617 y Fo(0)775 1605 y Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e Fq(y)1122 1574 y Fo(2)1159 1605 y Fq(=)p Fu(2)17 b(+)h Fq(V)h Fu(\()p Fq(x)p Fu(\))31 b Fs(and)f Fq(V)48 b Fs(is)30 b(analytic,)h Fq(V)19 b Fu(\()p Fq(x)p Fu(\))24 b(=)f Fq(a)2571 1617 y Fi(n)2616 1605 y Fq(x)2663 1574 y Fi(n)2727 1605 y Fu(+)18 b Fq(:)c(:)g(:)f Fs(,)30 b Fq(a)3019 1617 y Fi(n)3087 1605 y Fq(<)23 b Fu(0)p Fs(,)30 b Fq(h)3320 1617 y Fo(1)3380 1605 y Fu(=)456 1704 y Fq(h)504 1716 y Fo(1)541 1704 y Fu(\()p Fq(x;)14 b(y)s(;)g(t=";)g(\026;)g(")p Fu(\))25 b Fs(is)i(c)l(ontinuous,)f(analytic)i(in)e Fu(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\))27 b Fs(in)g(a)f(neighb)l(orho)l(o)l(d) j(of)e Fu(\(0)p Fq(;)14 b Fu(0)p Fq(;)g Fu(0\))p Fs(,)27 b(of)456 1804 y(or)l(der)g Fq(k)j Fs(in)e Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fs(,)28 b Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)h(in)e Fq(t=")f Fs(with)i(zer)l(o)f(aver)l(age.)39 b(We)28 b(assume)f(that)g Fu(2)p Fq(k)15 b Fp(\000)e Fu(2)21 b Fp(\025)i Fq(n)p Fs(,)456 1903 y(and)30 b(in)g(c)l(ase)g(that)g Fq(p)22 b(<)h Fu(1)p Fs(,)30 b Fq(@)1360 1915 y Fi(x)1402 1903 y Fq(h)1450 1915 y Fo(1)1487 1903 y Fq(@)1531 1915 y Fi(y)1571 1903 y Fq(h)1619 1915 y Fo(1)1679 1903 y Fu(=)22 b(0)p Fs(.)605 2003 y(Then,)29 b Fq(H)34 b Fs(has)28 b(stable)f(and)h(unstable)f(invariants)h(manifolds)h(asso)l(ciate)l(d)g (to)e Fu(\(0)p Fq(;)14 b Fu(0\))p Fs(,)27 b(and,)456 2103 y(given)f Fq(\034)33 b(>)22 b Fu(0)p Fs(,)27 b(ther)l(e)e(exist)h Fq(T)34 b(>)23 b Fu(0)i Fs(big)h(enough)g(and)g(p)l(ar)l (ameterizations)i Fq(\015)2823 2073 y Fi(s)2818 2123 y(\026;")2913 2103 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fs(,)27 b Fq(\015)3183 2073 y Fi(u)3178 2123 y(\026;")3274 2103 y Fu(\()p Fq(t;)14 b(s)p Fu(\))456 2205 y Fs(of)30 b(the)g(lo)l(c)l(al) h(stable)f(and)g(unstable)g(invariant)h(manifolds,)h(de\014ne)l(d)e(in) g Fq(D)2809 2175 y Fi(s)2844 2205 y Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))p Fs(,)31 b Fq(D)3173 2175 y Fi(u)3216 2205 y Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))p Fs(,)456 2305 y(r)l(esp)l(e)l(ctively,)31 b(such)f(that)g(\()p Fp(\003)f Fs(stands)g(for)i Fq(s)f Fs(or)g Fq(u)p Fs(\):)679 2436 y Fu(1\))41 b Fq(t)46 b Fp(7!)f Fq(\015)1046 2406 y Fj(\003)1041 2457 y Fi(\026;")1137 2436 y Fu(\()p Fq(t;)14 b(s)p Fu(\))42 b Fs(is)h(a)g(solution)f(of)h(system)f(\(3.1\))i(and)e Fq(s)k Fp(7!)f Fq(\015)2911 2406 y Fj(\003)2906 2457 y Fi(\026;")3002 2436 y Fu(\()p Fq(t;)14 b(s)p Fu(\))42 b Fs(is)h(r)l(e)l(al)794 2544 y(analytic.)d(Mor)l(e)l(over)31 b(the)f(map)h Fu(\()p Fq(t;)14 b(s;)g(\026;)g(")p Fu(\))23 b Fp(7!)g Fq(\015)2338 2514 y Fj(\003)2333 2564 y Fi(\026;")2429 2544 y Fu(\()p Fq(t;)14 b(s)p Fu(\))30 b Fs(is)g(c)l(ontinuous,)f Fq(C)3226 2514 y Fo(1)3294 2544 y Fs(with)794 2646 y(r)l(esp)l(e)l(ct)h (to)f Fq(t)h Fs(and)g(analytic)h(with)g(r)l(esp)l(e)l(ct)e(to)h Fu(\()p Fq(s;)14 b(\026)p Fu(\))p Fs(.)679 2746 y Fu(2\))41 b Fs(F)-6 b(or)30 b(al)t(l)h Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)h Fq(D)1411 2716 y Fj(\003)1449 2746 y Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))1354 2910 y Fq(\015)1402 2875 y Fj(\003)1397 2930 y Fi(\026;")1492 2910 y Fu(\()p Fq(t)19 b Fp(\006)f Fu(2)p Fq(\031)s(";)c(s)p Fu(\))23 b(=)f Fq(\015)2053 2875 y Fj(\003)2048 2930 y Fi(\026;")2144 2910 y Fu(\()p Fq(t;)14 b(s)k Fp(\006)g Fu(2)p Fq(\031)s(")p Fu(\))794 3074 y Fs(wher)l(e)31 b(we)f(take)g Fu(+)f Fs(for)i Fp(\003)22 b Fu(=)h Fq(s)29 b Fs(and)i Fp(\000)e Fs(for)i Fp(\003)22 b Fu(=)h Fq(u)p Fs(.)679 3173 y Fu(3\))41 b Fs(F)-6 b(or)30 b Fq(\026)23 b Fu(=)g(0)p Fs(,)30 b Fq(\015)1255 3143 y Fj(\003)1250 3194 y Fi(\026;")1345 3173 y Fu(\()p Fq(t;)14 b(s)p Fu(\))30 b Fs(c)l(oincides)i(with)e(the)g (r)l(estriction)g(of)h(the)f(homo)l(clinic)i(solu-)794 3276 y(tion)e Fq(\015)1009 3288 y Fo(0)1046 3276 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))30 b Fs(to)g Fq(D)1482 3245 y Fj(\003)1520 3276 y Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))p Fs(,)31 b(and)f(for)g Fq(\026)23 b Fp(6)p Fu(=)g(0)29 b Fs(the)h(fol)t(lowing)i(estimate)e(holds:)1270 3440 y Fq(\015)1318 3406 y Fj(\003)1313 3461 y Fi(\026;")1408 3440 y Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b(=)e Fq(\015)1732 3452 y Fo(0)1769 3440 y Fu(\()p Fq(t)d Fu(+)f Fq(s)p Fu(\))h(+)f Fq(\026")2195 3406 y Fi(p)p Fo(+1)2317 3440 y Fq(\033)2367 3406 y Fj(\003)2364 3461 y Fi(\026;")2460 3440 y Fu(\()p Fq(t;)c(s)p Fu(\))794 3620 y Fs(wher)l(e)31 b Fq(\033)1079 3589 y Fj(\003)1076 3640 y Fi(\026;")1172 3620 y Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)g(X)1514 3589 y Fj(\003)1502 3643 y Fi(\025)1571 3620 y Fp(\002)18 b(X)1725 3589 y Fj(\003)1713 3643 y Fi(\025)1794 3620 y Fs(with)30 b Fq(\025)24 b Fu(=)2143 3587 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)p 2143 3601 155 4 v 2158 3648 a Fi(n)p Fj(\000)p Fo(2)2338 3620 y Fs(and)1259 3790 y Fq(\033)1309 3755 y Fj(\003)1306 3810 y Fi(\026;")1402 3790 y Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b(=)g Fq(G)1748 3802 y Fi(\026;")1844 3790 y Fu(\()p Fq(\015)1919 3802 y Fo(0)1956 3790 y Fu(\()p Fq(t)c Fu(+)f Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(")p Fu(\))794 3958 y Fs(wher)l(e)31 b Fq(G)1094 3970 y Fi(\026;")1213 3958 y Fu(=)22 b(\()p Fq(G)1397 3970 y Fo(1)1435 3958 y Fq(;)14 b(G)1537 3970 y Fo(2)1574 3958 y Fu(\))30 b Fs(is)g(determine)l(d)h(by)f(the)g(c)l (onditions)939 4122 y Fq(@)983 4134 y Fi(\022)1021 4122 y Fq(G)1086 4134 y Fi(\026;")1181 4122 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f(\()p Fq(@)1639 4134 y Fi(y)1679 4122 y Fq(h)1727 4134 y Fo(1)1764 4122 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))p Fq(;)g Fp(\000)p Fq(@)2343 4134 y Fi(x)2385 4122 y Fq(h)2433 4134 y Fo(1)2470 4122 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\)\))p Fs(,)794 4286 y(and)30 b(having)h(zer)l(o)g(me)l(an.)605 4452 y Ft(Remark)39 b Fu(3.1)p Ft(.)44 b Fs(In)36 b(this)g(the)l(or)l (em)g(we)g(have)i(intr)l(o)l(duc)l(e)l(d)e(a)g(new)g(c)l(ondition:)52 b(if)37 b Fq(p)d(<)g Fu(1)456 4551 y Fs(then)g Fq(@)689 4563 y Fi(x)730 4551 y Fq(h)778 4563 y Fo(1)816 4551 y Fq(@)860 4563 y Fi(y)899 4551 y Fq(h)947 4563 y Fo(1)1016 4551 y Fu(=)c(0)p Fs(.)51 b(By)35 b(R)l(emark)f(1.5,)j(hyp)l(othesis)f Ff(HP5)f Fs(implies)h(this)e(c)l(ondition,)j(and)456 4651 y(ther)l(efor)l(e,)31 b(under)e(hyp)l(otheses)j Ff(HP1)p Fs(-)p Ff(HP5,)j Fs(The)l(or)l(em)c(3.1)g(applies.)605 4817 y Fu(The)f(pro)r(of)f(of)h(this)h(theorem)e(is)h(done)g(in)g(sev)n (eral)e(steps)i(in)g(the)h(presen)n(t)e(c)n(hapter.)43 b(W)-7 b(e)456 4917 y(pro)n(v)n(e)26 b(the)i(statemen)n(ts)f(for)g Fq(\015)1416 4887 y Fi(s)1411 4937 y(\026;")1507 4917 y Fu(,)g(the)h(ones)f(for)g Fq(\015)2060 4887 y Fi(u)2055 4937 y(\026;")2179 4917 y Fu(are)f(obtained)h(c)n(hanging)g Fq(t)g Fu(b)n(y)h Fp(\000)p Fq(t)p Fu(.)605 5016 y(In)e(the)h(follo)n (wing)e(sections)g(w)n(e)h(assume)f(the)h(h)n(yp)r(otheses)g(of)g (Theorem)f(3.1.)36 b(F)-7 b(rom)25 b(no)n(w)456 5116 y(on,)k(to)f(simplify)i(the)f(notation,)g(w)n(e)f(omit)h(the)h(dep)r (endence)f(on)g Fq(")f Fu(and)h Fq(\026)g Fu(at)g(sev)n(eral)e(places) 456 5216 y(where)g(do)r(es)g(not)h(pla)n(y)e(an)i(essen)n(tial)e(role.) p eop %%Page: 15 23 15 22 bop 1339 251 a Fo(3.3.)30 b(A)-7 b(VERA)n(GING)29 b(OF)g(THE)g(EQUA)-5 b(TION)817 b(15)1307 450 y Fk(3.3.)47 b(Av)m(eraging)32 b(of)g(the)f(equation)605 600 y Fu(Some)i(steps)g(of) f(a)n(v)n(eraging)e(are)i(necessary)f(to)i(transform)f(the)h(equation)g (\(3.1\))f(in)n(to)h(a)456 699 y(suitable)27 b(form.)37 b(W)-7 b(e)28 b(b)r(egin)f(b)n(y)h(in)n(tro)r(ducing)f(notation.)605 861 y Ft(Definition)g Fu(3.1)p Ft(.)36 b Fs(Given)26 b Fq(l)f Fp(2)e Fl(Z)1668 831 y Fo(+)1717 861 y Fs(and)j Fq(U)32 b Fp(\032)22 b Fl(C)2104 831 y Fo(2)2173 861 y Fs(we)k(denote)g(by)g Fq(P)2710 873 y Fi(l)2761 861 y Fs(the)g(set)f(of)h(functions)456 961 y Fq(p)h Fu(:)i Fq(U)g Fp(\002)20 b Fl(R)26 b Fp(\002)20 b Fq(B)t Fu(\(0)p Fq(;)14 b(\026)1141 973 y Fo(0)1178 961 y Fu(\))21 b Fp(\002)f Fu([0)p Fq(;)14 b(")1457 973 y Fo(0)1493 961 y Fu(\))29 b Fp(!)f Fl(C)53 b Fs(that)33 b(ar)l(e)g(c)l(ontinuous,)g Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)h(in)e Fq(\022)r Fs(,)i(analytic)456 1060 y(in)29 b Fu(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\))p Fs(,)31 b(and)f(have)h(or)l(der)g Fq(l)r Fs(.)38 b(Ther)l(efor)l(e)31 b(they)g(c)l(an)e(b)l(e)h(r)l(epr)l(esente)l(d)g (in)g(the)g(form)1040 1284 y Fq(p)1082 1296 y Fi(\026;")1177 1284 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fq(p)p Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))23 b(=)2200 1180 y Fj(1)2173 1205 y Fg(X)2144 1383 y Fi(i)p Fo(+)p Fi(j)s Fo(=)p Fi(l)2336 1284 y Fq(a)2380 1296 y Fi(ij)2438 1284 y Fu(\()p Fq(\022)r(;)14 b(\026;)g(")p Fu(\))p Fq(x)2753 1249 y Fi(i)2782 1284 y Fq(y)2826 1249 y Fi(j)456 1531 y Fs(with)30 b(the)g(c)l(o)l(e\016cients)g Fq(a)1241 1543 y Fi(ij)1299 1531 y Fu(\()p Fq(\022)r(;)14 b(\026;)g(")p Fu(\))31 b Fs(c)l(ontinuous,)e Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)i(in)f Fq(\022)r Fs(,)h(and)f(analytic)h(in)f Fq(\026)p Fs(.)605 1630 y(We)h(wil)t(l)i(write)e Fq(p)26 b Fu(=)f(\()p Fq(p)1360 1642 y Fi(l)1381 1650 y Fh(1)1418 1630 y Fq(;)14 b(p)1497 1642 y Fi(l)1518 1650 y Fh(2)1554 1630 y Fu(\))26 b Fp(2)g Fq(P)1746 1642 y Fi(l)1767 1650 y Fh(1)1823 1630 y Fp(\002)20 b Fq(P)1961 1642 y Fi(l)1982 1650 y Fh(2)2050 1630 y Fs(if)32 b Fq(p)2174 1642 y Fi(l)2195 1650 y Fh(1)2257 1630 y Fp(2)26 b Fq(P)2391 1642 y Fi(l)2412 1650 y Fh(1)2480 1630 y Fs(and)32 b Fq(p)2685 1642 y Fi(l)2706 1650 y Fh(2)2768 1630 y Fp(2)26 b Fq(P)2902 1642 y Fi(l)2923 1650 y Fh(2)2960 1630 y Fs(.)43 b(However,)32 b(if)456 1730 y Fq(l)481 1742 y Fo(1)541 1730 y Fu(=)22 b Fq(l)653 1742 y Fo(2)720 1730 y Fs(we)30 b(wil)t(l)h(simply)g(write)f (that)g Fq(p)23 b Fp(2)g Fq(P)1838 1742 y Fi(l)1859 1750 y Fh(1)1896 1730 y Fs(.)605 1829 y(F)-6 b(or)30 b(notational)h(c)l (onvenienc)l(e)f(we)g(also)h(de\014ne)1434 1986 y Fq(P)1487 1998 y Fi(l)1536 1986 y Fu(=)22 b Fq(P)1676 1998 y Fo(0)2011 1986 y Fs(for)149 b Fq(l)25 b(<)d Fu(0)p Fs(.)605 2148 y Ft(Pr)n(oposition)32 b Fu(3.1)p Ft(.)40 b Fs(Ther)l(e)31 b(exists)f(a)h(c)l(anonic)l(al)h(change)f(of)g(variables)i Fp(C)i Fs(de\014ne)l(d)30 b(in)h(a)456 2248 y(neighb)l(orho)l(o)l(d)h (of)e(the)g(origin,)i(that)e(tr)l(ansforms)f(the)h(Hamiltonian)h Fq(H)36 b Fs(to)893 2405 y Fp(H)q Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(t=";)g(\026;)g(")p Fu(\))82 b(=)g Fq(h)1745 2417 y Fo(0)1782 2405 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s Fu(\))18 b(+)g Fq(\026")2165 2370 y Fi(p)p Fo(+3)2288 2405 y Fq(F)2341 2417 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2504 2405 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(t=";)g(\026;)g(")p Fu(\))1697 2530 y(+)p Fq(\026)1812 2496 y Fo(2)1849 2530 y Fq(")1888 2496 y Fi(p)p Fo(+2)2010 2530 y Fq(R)2073 2542 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2232 2530 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\026;)g(")p Fu(\))456 2687 y Fs(in)28 b(a)h(neighb)l(orho)l(o)l(d)j(of)d(the)g(origin,)i(wher)l(e)e Fq(F)1907 2699 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2094 2687 y Fp(2)23 b Fq(P)2225 2699 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2417 2687 y Fs(and)29 b(has)h(zer)l(o)f(me)l(an,)g Fq(R)3207 2699 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)3389 2687 y Fp(2)456 2786 y Fq(P)509 2798 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)698 2786 y Fs(do)l(es)h(not)f(dep)l(end)i(on)f Fq(t)g Fs(and)1460 2945 y Fq(R)1523 2957 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1705 2945 y Fu(=)p 1793 2878 311 4 v 23 w Fq(@)1837 2957 y Fi(y)1877 2945 y Fq(h)1925 2957 y Fo(1)1962 2945 y Fq(@)2006 2957 y Fi(x)2048 2945 y Fq(S)23 b Fu(+)18 b Fq("r)2281 2957 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)456 3107 y Fs(with)31 b Fq(S)e Fp(2)c Fq(P)850 3119 y Fi(k)921 3107 y Fs(such)31 b(that)f Fq(@)1325 3119 y Fi(\022)1363 3107 y Fq(S)5 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))24 b(=)g Fp(\000)p Fq(h)2078 3119 y Fo(1)2115 3107 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))31 b Fs(and)g(has)g(zer)l(o)g(me)l(an,)g(and)456 3206 y Fq(r)493 3218 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)675 3206 y Fp(2)23 b Fq(P)806 3218 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)966 3206 y Fs(.)38 b(The)31 b(change)g Fp(C)j Fs(has)d(the)f(form)861 3363 y Fp(C)5 b Fu(\()g(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(t=";)g(\026;)g(")p Fu(\))22 b(=)h(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r Fu(\))19 b(+)f Fq(\026")1906 3329 y Fi(p)p Fo(+1)2028 3363 y Fq(G)p Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(t=";)g(\026;)g(")p Fu(\))j(+)h Fq(O)r Fu(\()p Fq(\026")2883 3329 y Fi(p)p Fo(+2)3006 3363 y Fu(\))456 3524 y Fs(wher)l(e)30 b Fq(G)g Fs(is)g(determine)l(d)h(by)f Fq(@)1454 3536 y Fi(\022)1491 3524 y Fq(G)24 b Fu(=)e(\()p Fq(@)1743 3536 y Fi(y)1784 3524 y Fq(h)1832 3536 y Fo(1)1869 3524 y Fq(;)14 b Fp(\000)p Fq(@)2015 3536 y Fi(x)2056 3524 y Fq(h)2104 3536 y Fo(1)2141 3524 y Fu(\))30 b Fs(and)g(having)i(zer)l (o)e(me)l(an.)605 3624 y(Mor)l(e)l(over)h Fp(C)k Fs(and)30 b Fp(H)h Fs(ar)l(e)f(c)l(ontinuous)f(in)h Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)26 b Fu(=)d Fq(t=";)14 b(\026;)g(")p Fu(\))p Fs(,)29 b Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)i(in)f Fq(\022)r Fs(,)h(and)456 3724 y(analytic)g(in)e Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\026)p Fu(\))30 b Fs(and)g Fp(C)k Fs(is)c Fq(C)1574 3694 y Fo(1)1642 3724 y Fs(in)f Fq(\022)r Fs(.)605 3886 y Fu(T)-7 b(o)30 b(a)n(v)n(erage)d(\014rst)j(w) n(e)g(scale)g(time)g(b)n(y)g Fq(\022)g Fu(=)d Fq(t=")p Fu(.)45 b(The)30 b(Hamiltonian)g(b)r(ecomes)g Fq("H)37 b Fu(and)456 3985 y(the)28 b(corresp)r(onding)d(equation)i(is)1214 4142 y(_)-37 b Fq(x)83 b Fu(=)g Fq("y)21 b Fu(+)d Fq(\026")1751 4108 y Fi(p)p Fo(+1)1873 4142 y Fq(@)1917 4154 y Fi(y)1957 4142 y Fq(h)2005 4154 y Fo(1)2042 4142 y Fu(\()p Fq(x;)c(y)s(;)g(\022)r (;)g(\026;)g(")p Fu(\))1218 4271 y(_)-37 b Fq(y)85 b Fu(=)e Fp(\000)p Fq("V)1648 4236 y Fj(0)1671 4271 y Fu(\()p Fq(x)p Fu(\))20 b Fp(\000)e Fq(\026")1974 4236 y Fi(p)p Fo(+1)2096 4271 y Fq(@)2140 4283 y Fi(x)2182 4271 y Fq(h)2230 4283 y Fo(1)2267 4271 y Fu(\()p Fq(x;)c(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))456 4427 y(where)50 b(_)-37 b Fq(x)37 b Fu(and)52 b(_)-38 b Fq(y)39 b Fu(no)n(w)e(mean)f(deriv)-5 b(ativ)n(es)36 b(with)h(resp)r(ect)g(to)f(the)i(new)f(time)g Fq(\022)r Fu(.)65 b(Next)37 b(w)n(e)456 4527 y(a)n(v)n(erage)31 b(sev)n(eral)i(times)i(with)g(resp)r(ect)f(to)g Fq(\022)j Fu(in)e(order)e(to)h(mo)n(v)n(e)g(the)h(con)n(tribution)f(of)g(the)456 4626 y(p)r(erturbation)f(to)h(terms)g(of)g(order)f Fq(\026")1709 4596 y Fi(p)p Fo(+3)1865 4626 y Fu(and)h Fq(\026)2083 4596 y Fo(2)2120 4626 y Fq(")2159 4596 y Fi(p)p Fo(+3)2316 4626 y Fu(in)g(the)h(parameters,)f(and)g(also)f(to)456 4726 y(increase)24 b(the)i(orders)d(with)j(resp)r(ect)f(to)h Fq(x)p Fu(,)g Fq(y)i Fu(of)e(some)f(terms)g(of)g Fq(H)7 b Fu(.)36 b(W)-7 b(e)26 b(will)f(mak)n(e)g(t)n(w)n(o)g(sets)456 4826 y(of)i(a)n(v)n(eraging)e(steps.)36 b(W)-7 b(e)28 b(b)r(egin)g(with)g(the)g(\014rst)g(set.)605 4925 y(Let)i Fq(\027)i Fp(2)27 b Fl(Z)972 4895 y Fo(+)1022 4925 y Fu(.)43 b(F)-7 b(or)29 b(the)h(inductiv)n(e)g(step)g(of)g(a)n(v)n (eraging)d(w)n(e)i(assume)g(w)n(e)h(ha)n(v)n(e)e(a)i(Hamil-)456 5025 y(tonian)1201 5182 y Fq(")p Fp(H)1311 5147 y Fi(\027)1375 5182 y Fu(=)23 b Fq("h)1550 5194 y Fo(0)1605 5182 y Fu(+)18 b Fq(\026")1777 5147 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)1988 5182 y Fq(F)2053 5147 y Fi(\027)2112 5182 y Fu(+)g Fq(\026)2245 5147 y Fo(2)2282 5182 y Fq(")2321 5147 y Fo(2)p Fi(p)p Fo(+2)2477 5182 y Fq(R)2541 5147 y Fi(\027)2540 5202 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)p eop %%Page: 16 24 16 23 bop 456 251 a Fo(16)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(with)28 b Fq(R)709 420 y Fi(\027)708 474 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)890 450 y Fp(2)23 b Fq(P)1021 462 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1208 450 y Fu(and)k Fq(F)1434 420 y Fi(\027)1503 450 y Fu(ha)n(ving)g(the)h(form)1163 616 y Fq(F)1228 582 y Fi(\027)1292 616 y Fu(=)1434 537 y Fg(X)1418 729 y Fi(l)p Fj(2)p Fd(Z)1529 703 y Fh(+)1379 782 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\025)p Fo(0)1622 616 y Fq(y)1666 582 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)1813 616 y Fq(p)1855 628 y Fi(i)1878 609 y Fc(\027)1878 649 y(l)1938 616 y Fu(+)2117 537 y Fg(X)2101 729 y Fi(l)p Fj(2)p Fd(Z)2212 703 y Fh(+)2021 782 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)p Fj(\025)p Fo(0)2348 616 y Fq(y)2392 582 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)2624 616 y Fq(p)2666 628 y Fi(j)2696 609 y Fc(\027)2693 649 y(l)456 616 y Fu(\(3.2\))456 912 y(with)h Fq(i)675 882 y Fi(\027)675 935 y(l)742 912 y Fu(=)c(max)o Fp(f)p Fu(0)p Fq(;)14 b(k)21 b Fu(+)e Fq(nl)i Fp(\000)e Fq(\027)5 b Fp(g)p Fu(,)29 b Fq(j)1614 882 y Fi(\027)1609 935 y(l)1681 912 y Fu(=)c(max)o Fp(f)p Fu(0)p Fq(;)14 b(k)22 b Fu(+)c Fq(n)p Fu(\()p Fq(l)j Fu(+)e(1\))h Fp(\000)f Fq(\027)24 b Fp(\000)19 b Fu(1)p Fp(g)28 b Fu(and)h Fq(p)3050 924 y Fi(i)3103 912 y Fp(2)d Fq(P)3237 924 y Fi(i)3265 912 y Fu(.)41 b(W)-7 b(e)456 1011 y(assume)29 b(that)i Fq(F)993 981 y Fi(\027)1064 1011 y Fu(has)f(zero)f(mean.)44 b(F)-7 b(rom)30 b(\(3.2\),)h(it)f(is)g(clear)f(that)i Fq(F)2720 981 y Fi(\027)2788 1011 y Fp(2)d Fq(P)2924 1023 y Fi(k)2965 1011 y Fu(.)45 b(Indeed,)31 b(the)456 1111 y(terms)c(are)g(of)g(order) 1008 1251 y Fq(\027)c Fp(\000)18 b Fu(2)p Fq(l)i Fu(+)e Fq(i)1354 1217 y Fi(\027)1354 1271 y(l)1418 1251 y Fp(\025)k Fq(\027)i Fp(\000)18 b Fu(2)p Fq(l)i Fu(+)e Fq(k)j Fu(+)d Fq(nl)i Fp(\000)e Fq(\027)28 b Fu(=)23 b Fq(k)e Fu(+)d Fq(l)r Fu(\()p Fq(n)g Fp(\000)g Fu(2\))23 b Fp(\025)f Fq(k)456 1391 y Fu(or)k(of)i(order)526 1602 y Fq(\027)23 b Fp(\000)18 b Fu(2)p Fq(l)i Fp(\000)e Fu(1)g(+)g Fq(j)1025 1567 y Fi(\027)1020 1622 y(l)1089 1602 y Fp(\025)23 b Fq(\027)g Fp(\000)18 b Fu(2)p Fq(l)i Fp(\000)e Fu(1)g(+)g Fq(k)j Fu(+)d Fq(n)p Fu(\()p Fq(l)i Fu(+)e(1\))g Fp(\000)g Fq(\027)24 b Fp(\000)18 b Fu(1)23 b(=)f Fq(k)g Fu(+)c Fq(n)g Fp(\000)g Fu(2)g(+)g Fq(l)r Fu(\()p Fq(n)g Fp(\000)g Fu(2\))23 b Fp(\025)f Fq(k)s Fu(.)605 1742 y(W)-7 b(e)37 b(observ)n(e)e(that)i(the)g(original)e(Hamiltonian)h Fq("H)43 b Fu(has)36 b(this)h(form)f(for)g Fq(\027)43 b Fu(=)38 b(0)e(with)456 1842 y Fq(F)521 1811 y Fo(0)581 1842 y Fu(=)22 b Fq(h)716 1854 y Fo(1)781 1842 y Fu(and)28 b Fq(R)1007 1811 y Fo(0)1006 1865 y(2)p Fi(k)q Fj(\000)p Fo(2)1188 1842 y Fu(=)22 b(0.)605 1994 y Ft(Lemma)i Fu(3.1)p Ft(.)34 b Fs(Under)24 b(the)g(pr)l(evious)g(c)l(onditions)h(and)f (assuming)g(that)f Fq(n)g Fp(\025)g Fu(3)g Fs(and)h Fq(k)i Fp(\025)c Fu(2)p Fs(,)456 2094 y(ther)l(e)32 b(exists)g(a)g(c)l(anonic) l(al)i(change)f(of)g(variables)i Fu(\()p Fq(x;)14 b(y)s Fu(\))28 b(=)f Fp(C)2424 2064 y Fi(\027)t Fo(+1)2549 2094 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))32 b Fs(which)i(is)f Fq(C)3407 2064 y Fo(0)456 2193 y Fs(in)25 b Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))p Fs(,)27 b Fq(C)1103 2163 y Fo(1)1141 2193 y Fs(and)f Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)h(in)e Fq(\022)j Fs(and)e(analytic)h(in)e Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\026)p Fu(\))25 b Fs(that)h(tr)l (ansforms)f(the)456 2293 y(Hamiltonian)30 b Fq(")p Fp(H)1045 2263 y Fi(\027)1116 2293 y Fs(to)1117 2433 y Fq(")p Fp(H)1227 2399 y Fi(\027)t Fo(+1)1375 2433 y Fu(=)23 b Fq("h)1550 2445 y Fo(0)1605 2433 y Fu(+)18 b Fq(\026")1777 2399 y Fi(p)p Fo(+)p Fi(\027)t Fo(+2)1988 2433 y Fq(F)2053 2399 y Fi(\027)t Fo(+1)2196 2433 y Fu(+)g Fq(\026)2329 2399 y Fo(2)2367 2433 y Fq(")2406 2399 y Fo(2)p Fi(p)p Fo(+2)2561 2433 y Fq(R)2625 2397 y Fi(\027)t Fo(+1)2624 2458 y(2)p Fi(k)q Fj(\000)p Fo(2)456 2573 y Fs(in)29 b(a)i(neighb)l(orho)l(o)l(d)h(of)e(the)g(origin,)i(wher)l(e)1360 2713 y Fq(F)1425 2679 y Fi(\027)t Fo(+1)1573 2713 y Fu(=)d(\026)-48 b Fq(y)s(@)1749 2725 y Fi(x)1790 2713 y Fq(S)1846 2678 y Fi(\027)t Fo(+1)1841 2735 y(1)1990 2713 y Fp(\000)18 b Fq(V)2140 2679 y Fj(0)2163 2713 y Fu(\()5 b(\026)-47 b Fq(x)q Fu(\))p Fq(@)2324 2725 y Fo(\026)-38 b Fi(y)2359 2713 y Fq(S)2415 2678 y Fi(\027)t Fo(+1)2410 2735 y(1)456 2869 y Fs(and)30 b Fq(S)673 2833 y Fi(\027)t Fo(+1)668 2891 y(1)821 2869 y Fu(=)23 b Fq(S)965 2833 y Fi(\027)t Fo(+1)960 2891 y(1)1090 2869 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))30 b Fs(satis\014es)1665 3012 y Fq(@)1709 3024 y Fi(\022)1747 3012 y Fq(S)1803 2977 y Fi(\027)t Fo(+1)1798 3035 y(1)1951 3012 y Fu(=)23 b Fp(\000)p Fq(F)2169 2978 y Fi(\027)2209 3012 y Fs(,)-1778 b Fu(\(3.3\))456 3153 y Fs(and)30 b(has)g(zer)l(o)g(me)l(an.)605 3252 y(Mor)l(e)l(over)g Fq(F)1034 3222 y Fi(\027)t Fo(+1)1189 3252 y Fs(has)g(the)f(form)h (\(3.2\),)g Fq(F)1971 3222 y Fi(\027)t Fo(+1)2120 3252 y Fp(2)23 b Fq(P)2251 3264 y Fi(k)2321 3252 y Fs(and)30 b(has)g(zer)l(o)f(me)l(an)g(with)h(r)l(esp)l(e)l(ct)456 3354 y(to)f Fq(\022)r Fs(,)i Fq(R)716 3319 y Fi(\027)t Fo(+1)715 3379 y(2)p Fi(k)q Fj(\000)p Fo(2)897 3354 y Fp(2)23 b Fq(P)1028 3366 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1217 3354 y Fs(and)1089 3508 y Fq(R)1153 3472 y Fi(\027)t Fo(+1)1152 3533 y(2)p Fi(k)q Fj(\000)p Fo(2)1334 3508 y Fu(=)g Fq(")1461 3474 y Fo(2)p Fi(\027)1535 3508 y Fq(@)1579 3520 y Fi(y)1619 3508 y Fq(F)1684 3474 y Fi(\027)1725 3508 y Fq(@)1769 3520 y Fi(x)1811 3508 y Fq(S)1867 3472 y Fi(\027)t Fo(+1)1862 3530 y(1)2011 3508 y Fu(+)18 b Fq(R)2158 3474 y Fi(\027)2157 3528 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2334 3508 y Fu(+)g Fq(")2456 3474 y Fo(2)p Fi(\027)t Fo(+1)2614 3508 y Fq(r)2651 3520 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)456 3508 y Fu(\(3.4\))456 3648 y Fs(with)30 b Fq(r)673 3660 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)855 3648 y Fp(2)24 b Fq(P)987 3660 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1146 3648 y Fs(.)605 3801 y Ft(Remark)k Fu(3.2)p Ft(.)36 b Fs(T)-6 b(o)27 b(pr)l(ove)g(this)f(lemma)h(we)g(do)g(not)e(assume)h(the)h(c)l (ondition)g Fu(2)p Fq(k)13 b Fp(\000)e Fu(2)21 b Fp(\025)i Fq(n)p Fs(.)605 3953 y Ft(Pr)n(oof.)41 b Fu(T)-7 b(o)19 b(simplify)h(the)f(notation,)i(in)e(this)h(pro)r(of)f(w)n(e)f(will)i (not)f(write)g(the)h(dep)r(endence)456 4053 y(of)31 b(the)i(functions)f (on)f Fq(\026;)14 b(")p Fu(.)50 b(Also,)32 b(along)f(the)h(pro)r(of)g Fq(r)2229 4065 y Fi(j)2296 4053 y Fu(and)g Fq(g)2502 4065 y Fi(j)2568 4053 y Fu(will)g(mean)g(generic)f(terms)456 4152 y(of)25 b Fq(P)601 4164 y Fi(j)637 4152 y Fu(;)h(therefore)f(they) h(ma)n(y)f(b)r(e)h(di\013eren)n(t)g(at)g(di\013eren)n(t)g(places.)36 b(W)-7 b(e)26 b(consider)f(a)g(generating)456 4252 y(function)32 b Fq(S)841 4222 y Fi(\027)t Fo(+1)966 4252 y Fu(\()p Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))33 b(whic)n(h)e(will)h(pro)n(vide)f(a)g(canonical)g(c)n(hange)g(of)g(v)-5 b(ariables)31 b(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s Fu(\))30 b Fp(7!)456 4352 y Fu(\()p Fq(x;)14 b(y)s Fu(\))28 b(implicitly)g(through)1547 4493 y(\026)-47 b Fq(x)84 b Fu(=)e Fq(@)1869 4505 y Fo(\026)-38 b Fi(y)1904 4493 y Fq(S)1960 4459 y Fi(\027)t Fo(+1)2085 4493 y Fu(\()p Fq(x;)21 b Fu(\026)-49 b Fq(y)t(;)14 b(\022)r Fu(\))1546 4621 y Fq(y)86 b Fu(=)c Fq(@)1864 4633 y Fi(x)1906 4621 y Fq(S)1962 4587 y Fi(\027)t Fo(+1)2087 4621 y Fu(\()p Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))-1901 b(\(3.5\))456 4761 y(and)27 b(then)h(the)g(new)g(Hamiltonian)f(will)h (b)r(e)1102 4901 y Fq(")p Fp(H)1212 4867 y Fi(\027)t Fo(+1)1337 4901 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r Fu(\))24 b(=)e Fq(")p Fp(H)1828 4867 y Fi(\027)1869 4901 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))20 b(+)e Fq(@)2286 4913 y Fi(\022)2323 4901 y Fq(S)2379 4867 y Fi(\027)t Fo(+1)2504 4901 y Fu(\()p Fq(x;)j Fu(\026)-49 b Fq(y)t(;)14 b(\022)r Fu(\))p Fq(:)456 5042 y Fu(W)-7 b(e)28 b(tak)n(e)1197 5182 y Fq(S)1253 5147 y Fi(\027)t Fo(+1)1378 5182 y Fu(\()p Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))24 b(=)e Fq(x)6 b Fu(\026)-48 b Fq(y)22 b Fu(+)c Fq(\026")2041 5147 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)2251 5182 y Fq(S)2307 5146 y Fi(\027)t Fo(+1)2302 5204 y(1)2432 5182 y Fu(\()p Fq(x;)j Fu(\026)-49 b Fq(y)t(;)14 b(\022)r Fu(\))p eop %%Page: 17 25 17 24 bop 1339 251 a Fo(3.3.)30 b(A)-7 b(VERA)n(GING)29 b(OF)g(THE)g(EQUA)-5 b(TION)817 b(17)456 450 y Fu(with)30 b Fq(S)703 415 y Fi(\027)t Fo(+1)698 472 y(1)858 450 y Fu(satisfying)f(\(3.3\).)43 b(This)30 b(c)n(hoice)f(is)h(motiv)-5 b(ated)30 b(b)n(y)f(the)i(next)f(calculations.)42 b(W)-7 b(e)456 550 y(observ)n(e)22 b(that)i Fq(S)980 520 y Fi(\027)t Fo(+1)1129 550 y Fu(is)g Fq(C)1274 520 y Fo(0)1312 550 y Fu(,)h(2)p Fq(\031)s Fu(-p)r(erio)r(dic)e(in)h Fq(\022)r Fu(,)i Fq(C)2042 520 y Fo(1)2103 550 y Fu(in)e(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026)p Fu(\))25 b(and)f(analytic)g(in)g(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\).)456 649 y(With)33 b(it)f(w)n(e)g(will) h(cancel)f(the)g(terms)g(of)h(orders)d Fq(\026")2131 619 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)2374 649 y Fu(in)i(the)h(a)n(v)n (eraged)d(Hamiltonian.)456 749 y(Since)e Fq(F)738 719 y Fi(\027)807 749 y Fu(has)g(zero)f(mean)g(with)i(resp)r(ect)f(to)g Fq(\022)i Fu(w)n(e)e(can)f(c)n(ho)r(ose)g Fq(S)2594 713 y Fi(\027)t Fo(+1)2589 771 y(1)2747 749 y Fu(with)i(the)f(additional) 456 849 y(condition)f(of)h(ha)n(ving)e(zero)h(mean.)605 948 y(F)-7 b(rom)31 b(\(3.3\),)g Fq(S)1108 913 y Fi(\027)t Fo(+1)1103 970 y(1)1265 948 y Fu(is)g(of)g(order)e Fq(k)s Fu(.)48 b(T)-7 b(o)30 b(get)h(the)h(c)n(hange)e(of)h(v)-5 b(ariables,)30 b(from)h(\(3.5\))g(w)n(e)456 1048 y(ha)n(v)n(e)26 b(to)h(apply)h(the)g(implicit)g(function)g(theorem)f(\(I.F.T.\))i(to) 763 1193 y(\()p Fq(x;)14 b(y)s(;)20 b Fu(\026)-48 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))23 b Fp(7!)g Fu(\()5 b(\026)-47 b Fq(x)20 b Fp(\000)e Fq(@)1721 1205 y Fo(\026)-38 b Fi(y)1756 1193 y Fq(S)1812 1159 y Fi(\027)t Fo(+1)1937 1193 y Fu(\()p Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))p Fq(;)g(y)21 b Fp(\000)d Fq(@)2596 1205 y Fi(x)2638 1193 y Fq(S)2694 1159 y Fi(\027)t Fo(+1)2819 1193 y Fu(\()p Fq(x;)i Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\)\).)456 1348 y(This)41 b(map)g(is)g Fq(C)1019 1318 y Fo(1)1097 1348 y Fu(with)h(resp)r(ect)f(to)g(\()p Fq(x;)14 b(y)s(;)19 b Fu(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026)p Fu(\))41 b(and)g(con)n(tin)n(uous.)76 b(A)42 b(generalized)456 1447 y(v)n(ersion)26 b(of)h(the)h(I.F.T.)g (giv)n(es)f(that)g(w)n(e)h(can)f(obtain)1495 1592 y(\()p Fq(x;)14 b(y)s Fu(\))23 b(=)g Fp(C)1847 1557 y Fi(\027)t Fo(+1)1972 1592 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))456 1746 y(with)28 b Fp(C)694 1716 y Fi(\027)t Fo(+1)846 1746 y Fu(b)r(eing)g Fq(C)1135 1716 y Fo(1)1200 1746 y Fu(with)g(resp)r(ect)g(to)f Fq(x)p Fu(,)h Fq(y)s Fu(,)g Fq(\022)r Fu(,)g Fq(\026)f Fu(and)h(con)n(tin)n(uous.)605 1846 y(A)23 b(new)h(application)e(of)h(the)h(I.F.T.)f(for)g(analytic)f (functions,)j(with)e Fq(\022)j Fu(and)d Fq(")f Fu(\014xed,)j(giv)n(es,) 456 1945 y(b)n(y)i(uniqueness,)g(that)h Fq(C)1257 1915 y Fi(\027)t Fo(+1)1411 1945 y Fu(also)e(is)i(analytic)e(with)j(resp)r (ect)e(to)g(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\).)605 2045 y(Moreo)n(v)n(er)25 b(it)j(is)f(clear)g(that)1104 2189 y Fq(x)84 b Fu(=)j(\026)-47 b Fq(x)19 b Fp(\000)f Fq(\026")1620 2155 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)1830 2189 y Fq(@)1879 2201 y Fo(\026)-38 b Fi(y)1914 2189 y Fq(S)1970 2154 y Fi(\027)t Fo(+1)1965 2212 y(1)2114 2189 y Fu(+)18 b Fq(\026)2247 2155 y Fo(2)2284 2189 y Fq(")2323 2155 y Fo(2)p Fi(p)p Fo(+2)p Fi(\027)t Fo(+2)2600 2189 y Fq(r)2637 2201 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1108 2319 y Fq(y)86 b Fu(=)i(\026)-48 b Fq(y)21 b Fu(+)d Fq(\026")1616 2285 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)1827 2319 y Fq(@)1871 2331 y Fi(x)1912 2319 y Fq(S)1968 2284 y Fi(\027)t Fo(+1)1963 2341 y(1)2112 2319 y Fu(+)g Fq(\026)2245 2285 y Fo(2)2282 2319 y Fq(")2321 2285 y Fo(2)p Fi(p)p Fo(+2)p Fi(\027)t Fo(+2)2598 2319 y Fq(r)2635 2331 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)456 2467 y Fu(where)32 b(the)h(deriv)-5 b(ativ)n(es)32 b(of)h Fq(S)1424 2432 y Fi(\027)t Fo(+1)1419 2489 y(1)1582 2467 y Fu(and)g Fq(r)1786 2479 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1978 2467 y Fu(are)f(ev)-5 b(aluated)33 b(at)g(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\).)53 b(The)33 b(a)n(v)n(eraged)456 2567 y(Hamiltonian)27 b(is)456 2711 y Fq(")p Fp(H)566 2677 y Fi(\027)t Fo(+1)691 2711 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))83 b(=)g Fq(")p Fp(H)1302 2677 y Fi(\027)1343 2711 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))19 b Fp(\000)f Fq(\026")1804 2677 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)2014 2711 y Fq(F)2079 2677 y Fi(\027)2121 2711 y Fu(\()p Fq(x;)i Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))1044 2857 y(=)1203 2801 y Fq(")p 1202 2838 42 4 v 1202 2914 a Fu(2)1253 2857 y([)6 b(\026)-48 b Fq(y)22 b Fu(+)c Fq(\026")1511 2823 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)1721 2857 y Fq(@)1765 2869 y Fi(x)1807 2857 y Fq(S)1863 2822 y Fi(\027)t Fo(+1)1858 2879 y(1)2006 2857 y Fu(+)g Fq(\026)2139 2823 y Fo(2)2177 2857 y Fq(")2216 2823 y Fo(2)p Fi(p)p Fo(+2)p Fi(\027)t Fo(+2)2492 2857 y Fq(r)2529 2869 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2688 2857 y Fu(])2711 2823 y Fo(2)1192 3020 y Fu(+)p Fq("V)g Fu(\()5 b(\026)-47 b Fq(x)19 b Fp(\000)f Fq(\026")1632 2986 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)1843 3020 y Fq(@)1892 3032 y Fo(\026)-38 b Fi(y)1927 3020 y Fq(S)1983 2985 y Fi(\027)t Fo(+1)1978 3042 y(1)2126 3020 y Fu(+)18 b Fq(\026)2259 2986 y Fo(2)2296 3020 y Fq(")2335 2986 y Fo(2)p Fi(p)p Fo(+2)p Fi(\027)t Fo(+2)2612 3020 y Fq(r)2649 3032 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2808 3020 y Fu(\))1192 3149 y(+)p Fq(\026")1346 3115 y Fi(p)p Fo(+)p Fi(\027)t Fo(+1)1556 3149 y Fu([)p Fq(F)1644 3115 y Fi(\027)1685 3149 y Fu(\()p Fq(x;)c(y)s(;)g(\022)r Fu(\))19 b Fp(\000)f Fq(F)2122 3115 y Fi(\027)2164 3149 y Fu(\()p Fq(x;)i Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\)])19 b(+)f Fq(\026)2609 3115 y Fo(2)2646 3149 y Fq(")2685 3115 y Fo(2)p Fi(p)p Fo(+2)2840 3149 y Fq(R)2904 3115 y Fi(\027)2903 3169 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1044 3301 y Fu(=)1203 3245 y Fq(")p 1202 3282 V 1202 3358 a Fu(2)1259 3301 y(\026)-48 b Fq(y)1297 3267 y Fo(2)1353 3301 y Fu(+)18 b Fq("V)g Fu(\()5 b(\026)-47 b Fq(x)q Fu(\))19 b(+)f Fq(\026")1844 3267 y Fi(p)p Fo(+)p Fi(\027)t Fo(+2)2054 3301 y Fu([)6 b(\026)-48 b Fq(y)s(@)2165 3313 y Fi(x)2207 3301 y Fq(S)2263 3266 y Fi(\027)t Fo(+1)2258 3324 y(1)2406 3301 y Fp(\000)18 b Fq(V)2556 3267 y Fj(0)2580 3301 y Fu(\()5 b(\026)-47 b Fq(x)p Fu(\))p Fq(@)2740 3313 y Fo(\026)-38 b Fi(y)2776 3301 y Fq(S)2832 3266 y Fi(\027)t Fo(+1)2827 3324 y(1)2957 3301 y Fu(])1192 3465 y(+)p Fq(\026)1307 3430 y Fo(2)1344 3465 y Fq(")1383 3430 y Fo(2)p Fi(p)p Fo(+2)p Fi(\027)t Fo(+2)1659 3465 y Fq(@)1703 3477 y Fi(y)1743 3465 y Fq(F)1808 3430 y Fi(\027)1850 3465 y Fq(@)1894 3477 y Fi(x)1935 3465 y Fq(S)1991 3429 y Fi(\027)t Fo(+1)1986 3487 y(1)2135 3465 y Fu(+)18 b Fq(\026)2268 3430 y Fo(2)2305 3465 y Fq(")2344 3430 y Fo(2)p Fi(p)p Fo(+2)2499 3465 y Fq(R)2563 3430 y Fi(\027)2562 3485 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2740 3465 y Fu(+)g Fq(\026)2873 3430 y Fo(2)2910 3465 y Fq(")2949 3430 y Fo(2)p Fi(p)p Fo(+2)p Fi(\027)t Fo(+3)3226 3465 y Fq(r)3263 3477 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)3422 3465 y Fu(.)456 3609 y(Therefore)26 b(w)n(e)h(can)g(tak)n(e)1078 3753 y Fq(R)1142 3718 y Fi(\027)t Fo(+1)1141 3778 y(2)p Fi(k)q Fj(\000)p Fo(2)1323 3753 y Fu(=)22 b Fq(")1449 3719 y Fo(2)p Fi(\027)1524 3753 y Fq(@)1568 3765 y Fi(y)1608 3753 y Fq(F)1673 3719 y Fi(\027)1714 3753 y Fq(@)1758 3765 y Fi(x)1800 3753 y Fq(S)1856 3718 y Fi(\027)t Fo(+1)1851 3775 y(1)1999 3753 y Fu(+)c Fq(R)2146 3719 y Fi(\027)2145 3774 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2323 3753 y Fu(+)g Fq(")2445 3719 y Fo(2)p Fi(\027)t Fo(+1)2603 3753 y Fq(r)2640 3765 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2799 3753 y Fu(.)-2366 b(\(3.6\))456 3898 y(W)-7 b(e)33 b(will)g(need)g(information)f(on)h (the)g(orders)e(of)i(the)h(terms)e(in)h(the)h(Hamiltonian)e(and)h(the) 456 3997 y(factors)26 b Fq(y)k Fu(they)e(ha)n(v)n(e.)36 b(F)-7 b(rom)27 b(\(3.2\))g(and)h(\(3.3\))f(w)n(e)g(write)1127 4158 y Fq(S)1183 4123 y Fi(\027)t Fo(+1)1178 4180 y(1)1331 4158 y Fu(=)1473 4079 y Fg(X)1419 4258 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\025)p Fo(0)1661 4158 y Fq(y)1705 4124 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)1853 4158 y Fq(g)1893 4170 y Fi(i)1916 4151 y Fc(\027)1916 4191 y(l)1975 4158 y Fu(+)2155 4079 y Fg(X)2058 4258 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)p Fj(\025)p Fo(0)2385 4158 y Fq(y)2429 4124 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)2662 4158 y Fq(g)2702 4170 y Fi(j)2732 4151 y Fc(\027)2729 4191 y(l)456 4388 y Fu(when)g(here)g(and)h(in)g (what)f(follo)n(ws)g Fq(l)i Fu(is)e(a)h(non-negativ)n(e)e(in)n(teger.) 605 4487 y(Therefore,)g Fq(F)1069 4457 y Fi(\027)t Fo(+1)1218 4487 y Fu(=)c Fq(y)s(@)1393 4499 y Fi(x)1435 4487 y Fq(S)1491 4452 y Fi(\027)t Fo(+1)1486 4509 y(1)1635 4487 y Fp(\000)c Fq(V)1784 4457 y Fj(0)1808 4487 y Fu(\()p Fq(x)p Fu(\))p Fq(@)1963 4499 y Fi(y)2004 4487 y Fq(S)2060 4452 y Fi(\027)t Fo(+1)2055 4509 y(1)2212 4487 y Fu(is)28 b(of)f(the)h(form)569 4651 y Fq(F)634 4617 y Fi(\027)t Fo(+1)842 4651 y Fu(=)1044 4572 y Fg(X)990 4751 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\025)p Fo(0)1232 4651 y Fq(y)1276 4617 y Fi(\027)t Fo(+1)p Fj(\000)p Fo(2)p Fi(l)1507 4651 y Fq(g)1547 4666 y Fo(max)o Fj(f)p Fo(0)p Fi(;i)1779 4646 y Fc(\027)1779 4686 y(l)1816 4666 y Fj(\000)p Fo(1)p Fj(g)1958 4651 y Fu(+)2138 4572 y Fg(X)2041 4751 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)p Fj(\025)p Fo(0)2368 4651 y Fq(y)2412 4617 y Fi(\027)t Fo(+1)p Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)2729 4651 y Fq(g)2769 4666 y Fo(max)o Fj(f)p Fo(0)p Fi(;j)3008 4646 y Fc(\027)3005 4686 y(l)3045 4666 y Fj(\000)p Fo(1)p Fj(g)990 4891 y Fu(+)1122 4812 y Fg(X)1069 4990 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\025)p Fo(0)1310 4891 y Fq(y)1354 4856 y Fi(\027)t Fo(+1)p Fj(\000)p Fo(2\()p Fi(l)p Fo(+1\))1722 4891 y Fq(g)1762 4903 y Fi(i)1785 4883 y Fc(\027)1785 4923 y(l)1821 4903 y Fo(+)p Fi(n)p Fj(\000)p Fo(1)2021 4891 y Fu(+)2158 4812 y Fg(X)2104 4990 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\025)p Fo(0)2346 4891 y Fq(y)2390 4856 y Fi(\027)t Fo(+1)p Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)2707 4891 y Fq(g)2747 4903 y Fi(i)2770 4883 y Fc(\027)2770 4923 y(l)2806 4903 y Fo(+)p Fi(n)p Fj(\000)p Fo(2)990 5130 y Fu(+)1165 5051 y Fg(X)1069 5230 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)p Fj(\025)p Fo(0)1395 5130 y Fq(y)1439 5096 y Fi(\027)t Fo(+1)p Fj(\000)p Fo(2\()p Fi(l)p Fo(+1\))p Fj(\000)p Fo(1)1892 5130 y Fq(g)1932 5142 y Fi(j)1962 5122 y Fc(\027)1959 5162 y(l)1999 5142 y Fo(+)p Fi(n)p Fj(\000)p Fo(1)2198 5130 y Fu(+)2378 5051 y Fg(X)2281 5230 y Fi(\027)t Fj(\000)p Fo(2)p Fi(l)p Fj(\000)p Fo(1)p Fj(\025)p Fo(0)2609 5130 y Fq(y)2653 5096 y Fi(\027)t Fo(+1)p Fj(\000)p Fo(2\()p Fi(l)p Fo(+1\))3020 5130 y Fq(g)3060 5142 y Fi(j)3090 5122 y Fc(\027)3087 5162 y(l)3127 5142 y Fo(+)p Fi(n)p Fj(\000)p Fo(2)3308 5130 y Fu(.)p eop %%Page: 18 26 18 25 bop 456 251 a Fo(18)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(The)27 b(follo)n(wing)g(relations)f(hold)973 592 y(max)p Fp(f)p Fu(0)p Fq(;)14 b(i)1278 558 y Fi(\027)1278 613 y(l)1336 592 y Fp(\000)k Fu(1)p Fp(g)82 b(\025)h Fq(i)1762 556 y Fi(\027)t Fo(+1)1762 617 y Fi(l)1887 592 y Fu(,)180 b(max)o Fp(f)p Fu(0)p Fq(;)14 b(j)2404 558 y Fi(\027)2399 613 y(l)2463 592 y Fp(\000)k Fu(1)p Fp(g)k(\025)g Fq(j)2778 556 y Fi(\027)t Fo(+1)2773 617 y Fi(l)2904 592 y Fu(,)1138 723 y Fq(i)1167 689 y Fi(\027)1167 744 y(l)1226 723 y Fu(+)c Fq(n)g Fp(\000)h Fu(1)82 b Fp(\025)h Fq(i)1762 688 y Fi(\027)t Fo(+1)1762 748 y Fi(l)p Fo(+1)1887 723 y Fu(,)166 b Fq(j)2115 689 y Fi(\027)2110 744 y(l)2174 723 y Fu(+)18 b Fq(n)h Fp(\000)f Fu(1)k Fp(\025)h Fq(j)2600 688 y Fi(\027)t Fo(+1)2595 748 y Fi(l)p Fo(+1)2725 723 y Fu(,)1138 861 y Fq(i)1167 827 y Fi(\027)1167 882 y(l)1226 861 y Fu(+)18 b Fq(n)g Fp(\000)h Fu(1)82 b Fp(\025)h Fq(i)1762 826 y Fi(\027)t Fo(+1)1762 886 y Fi(l)p Fo(+1)1887 861 y Fu(,)166 b Fq(j)2115 827 y Fi(\027)2110 882 y(l)2174 861 y Fu(+)18 b Fq(n)h Fp(\000)f Fu(1)k Fp(\025)h Fq(j)2600 826 y Fi(\027)t Fo(+1)2595 886 y Fi(l)p Fo(+1)2725 861 y Fu(,)456 1012 y(and)k(then)h(w)n(e)f(deduce)h(that)g Fq(F)1450 981 y Fi(\027)t Fo(+1)1603 1012 y Fu(has)f(the)h(form)f (\(3.2\).)605 1114 y(Since)h Fq(S)878 1078 y Fi(\027)t Fo(+1)873 1136 y(1)1031 1114 y Fu(has)f(zero)f(mean,)i(w)n(e)f(also)f (ha)n(v)n(e)h(that)h Fq(F)2324 1083 y Fi(\027)t Fo(+1)2477 1114 y Fu(has)f(zero)f(mean.)605 1213 y(Finally)f(the)g(regularit)n(y)d (on)j Fp(H)1584 1183 y Fi(\027)t Fo(+1)1734 1213 y Fu(follo)n(ws)f (from)g(the)h(regularit)n(y)e(of)h Fp(H)2875 1183 y Fi(\027)2941 1213 y Fu(and)h Fp(C)3149 1183 y Fi(\027)t Fo(+1)3274 1213 y Fu(.)83 b Fe(\003)605 1373 y Fu(No)n(w)27 b(w)n(e)g(use)h(the)g (previous)e(lemma)i(to)f(p)r(erform)g(2)p Fq(n)18 b Fu(+)g(2)27 b(steps)h(of)f(a)n(v)n(eraging.)605 1527 y Ft(Lemma)22 b Fu(3.2)p Ft(.)31 b Fs(Ther)l(e)23 b(exists)e(a)g(c)l(anonic)l(al)i (change)f(of)g(variables)h Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)f Fp(C)2980 1539 y Fo(0)3017 1527 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))456 1626 y Fs(which)32 b(is)g Fq(C)846 1596 y Fo(0)914 1626 y Fs(in)f Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))p Fs(,)32 b Fq(C)1573 1596 y Fo(1)1641 1626 y Fs(and)g Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)g(in)f Fq(\022)j Fs(and)d(analytic)i(in)e Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\026)p Fu(\))31 b Fs(that)456 1726 y(tr)l(ansforms)e(the)h (Hamiltonian)h Fq("H)36 b Fs(to)1206 1868 y Fq(")p Fp(H)1315 1880 y Fo(0)1375 1868 y Fu(=)23 b Fq("h)1550 1880 y Fo(0)1605 1868 y Fu(+)18 b Fq(\026")1777 1834 y Fi(p)p Fo(+2)p Fi(n)p Fo(+3)2024 1868 y Fq(F)31 b Fu(+)18 b Fq(\026)2241 1834 y Fo(2)2278 1868 y Fq(")2317 1834 y Fo(2)p Fi(p)p Fo(+2)2472 1868 y Fq(R)2535 1880 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)456 2010 y Fs(in)32 b(a)h(neighb)l(orho)l(o)l(d)i(of)e(the)f (origin,)j(wher)l(e)e Fq(F)40 b Fp(2)28 b Fq(P)2109 2022 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2305 2010 y Fs(and)33 b(has)g(zer)l(o)f(me)l(an)h(with)g(r)l(esp)l(e)l(ct)456 2110 y(to)c Fq(\022)r Fs(,)i Fq(R)715 2122 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)897 2110 y Fp(2)23 b Fq(P)1028 2122 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1217 2110 y Fs(and)1441 2252 y Fq(R)1504 2264 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1686 2252 y Fu(=)g Fq(@)1818 2264 y Fi(y)1858 2252 y Fq(h)1906 2264 y Fo(1)1943 2252 y Fq(@)1987 2264 y Fi(x)2029 2252 y Fq(S)2085 2218 y Fo(1)2080 2272 y(1)2140 2252 y Fu(+)18 b Fq("r)2299 2264 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)456 2404 y Fs(with)33 b Fq(S)695 2374 y Fo(1)690 2425 y(1)765 2404 y Fs(such)f(that)h Fq(@)1173 2416 y Fi(\022)1211 2404 y Fq(S)1267 2374 y Fo(1)1262 2425 y(1)1332 2404 y Fu(=)28 b Fp(\000)p Fq(h)1538 2416 y Fo(1)1608 2404 y Fs(and)33 b(has)g(zer)l(o)g(me)l(an,)h(and)f Fq(r)2552 2416 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2740 2404 y Fp(2)c Fq(P)2877 2416 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)3036 2404 y Fs(.)48 b(Mor)l(e)l(over)456 2504 y Fp(H)526 2516 y Fo(0)593 2504 y Fs(is)30 b(c)l(ontinuous)f(in)h Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g (")p Fu(\))30 b Fs(and)g(analytic)h(in)f Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\026)p Fu(\))p Fs(.)605 2657 y Ft(Pr)n(oof.)41 b Fu(Since)d Fq(h)1208 2669 y Fo(1)1286 2657 y Fp(2)j Fq(P)1435 2669 y Fi(k)1476 2657 y Fu(,)g(w)n(e)d(note)g(that)h Fq(H)45 b Fu(has)37 b(the)i(form)f(\(3.2\))g(for)f Fq(\027)47 b Fu(=)40 b(0)e(and)456 2757 y Fq(R)520 2727 y Fo(0)519 2780 y(2)p Fi(k)q Fj(\000)p Fo(2)703 2757 y Fu(=)25 b(0.)41 b(Then)30 b(w)n(e)e(b)r(egin)i(with)f Fq(F)1722 2727 y Fo(0)1785 2757 y Fu(=)c Fq(h)1923 2769 y Fo(1)1990 2757 y Fu(and)j Fq(R)2216 2727 y Fo(0)2215 2780 y(2)p Fi(k)q Fj(\000)p Fo(2)2400 2757 y Fu(=)d(0)k(and)g(w)n(e)g (apply)f(iterativ)n(ely)456 2863 y(Lemma)f(3.1)g(2)p Fq(n)17 b Fu(+)i(2)27 b(times)h(.)37 b(In)27 b(this)h(w)n(a)n(y)f(w)n (e)g(obtain)g(that)h Fq(F)35 b Fp(\021)23 b Fq(F)2665 2833 y Fo(2)p Fi(n)p Fo(+2)2854 2863 y Fu(has)28 b(the)g(form)992 3017 y Fq(F)1057 2983 y Fo(2)p Fi(n)p Fo(+2)1243 3017 y Fu(=)1412 2938 y Fg(X)1330 3117 y Fo(0)p Fj(\024)p Fi(l)p Fj(\024)p Fi(n)p Fo(+1)1633 3017 y Fu(\026)-48 b Fq(y)1671 2983 y Fo(2)p Fi(n)p Fo(+2)p Fj(\000)p Fo(2)p Fi(l)1940 3017 y Fq(p)1982 3029 y Fi(i)2005 3038 y Fc(l)2052 3017 y Fu(+)2249 2938 y Fg(X)2135 3117 y Fo(0)p Fj(\024)p Fo(2)p Fi(l)p Fj(\024)p Fo(2)p Fi(n)p Fo(+1)2504 3017 y Fu(\026)g Fq(y)2542 2983 y Fo(2)p Fi(n)p Fo(+1)p Fj(\000)p Fo(2)p Fi(l)2810 3017 y Fq(p)2852 3029 y Fi(j)2879 3038 y Fc(l)456 3017 y Fu(\(3.7\))456 3249 y(with)25 b Fq(i)671 3261 y Fi(l)719 3249 y Fu(=)e(max)p Fp(f)p Fu(0)p Fq(;)14 b(k)h Fu(+)e Fq(nl)h Fp(\000)f Fu(2)p Fq(n)g Fp(\000)g Fu(2)p Fp(g)p Fu(,)25 b Fq(j)1735 3261 y Fi(l)1784 3249 y Fu(=)e(max)o Fp(f)p Fu(0)p Fq(;)14 b(k)h Fu(+)e Fq(n)p Fu(\()p Fq(l)i Fu(+)e(1\))g Fp(\000)g Fu(2)p Fq(n)g Fp(\000)g Fu(3)p Fp(g)25 b Fu(and)g Fq(p)3141 3261 y Fi(m)3227 3249 y Fp(2)e Fq(P)3358 3261 y Fi(m)3421 3249 y Fu(.)456 3351 y(Also)k(from)g(\(3.6\),)h(w)n(e)f(can)g(write)g Fq(R)1611 3316 y Fo(2)p Fi(n)p Fo(+2)1610 3376 y(2)p Fi(k)q Fj(\000)p Fo(2)1801 3351 y Fu(as)1439 3507 y Fq(R)1503 3471 y Fo(2)p Fi(n)p Fo(+2)1502 3532 y(2)p Fi(k)q Fj(\000)p Fo(2)1689 3507 y Fu(=)22 b Fq(@)1820 3519 y Fi(y)1860 3507 y Fq(h)1908 3519 y Fo(1)1945 3507 y Fq(@)1989 3519 y Fi(x)2031 3507 y Fq(S)2087 3472 y Fo(1)2082 3527 y(1)2142 3507 y Fu(+)c Fq("r)2301 3519 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)456 3659 y Fu(where)29 b Fq(S)754 3629 y Fo(1)749 3679 y(1)820 3659 y Fu(is)g(the)h(one)f(whic)n(h)h(corresp)r(onds)d(to) j(the)g(\014rst)f(c)n(hange)f Fp(C)2648 3629 y Fo(1)2685 3659 y Fu(.)43 b(Moreo)n(v)n(er)26 b(the)k(func-)456 3758 y(tion)k Fq(F)698 3728 y Fo(2)p Fi(n)p Fo(+2)894 3758 y Fu(has)g(zero)f(mean)h(with)g(resp)r(ect)g(to)g Fq(\022)r Fu(.)57 b(W)-7 b(e)35 b(observ)n(e)d(that)i(the)h (Hamiltonians)456 3858 y Fp(H)527 3828 y Fo(1)564 3858 y Fq(;)14 b(:)g(:)g(:)27 b(;)14 b Fp(H)833 3828 y Fo(2)p Fi(n)p Fo(+2)1029 3858 y Fu(are)33 b Fq(C)1239 3828 y Fo(0)1277 3858 y Fu(,)i(2)p Fq(\031)s Fu(-p)r(erio)r(dic)f(in)g Fq(\022)i Fu(and)e(analytic)f(with)h(resp)r(ect)g(to)g Fq(x;)14 b(y)37 b Fu(and)c Fq(\026)p Fu(.)456 3958 y(The)25 b(c)n(hanges)f Fp(C)978 3927 y Fo(1)1014 3958 y Fq(;)14 b(:)g(:)g(:)28 b(;)14 b Fp(C)1262 3927 y Fo(2)p Fi(n)p Fo(+2)1449 3958 y Fu(are)24 b Fq(C)1650 3927 y Fo(1)1713 3958 y Fu(with)h(resp)r(ect)g(to)g Fq(\022)r Fu(.)37 b(W)-7 b(e)25 b(tak)n(e)g Fp(C)2743 3970 y Fo(0)2803 3958 y Fu(=)d Fp(C)2939 3927 y Fo(2)p Fi(n)p Fo(+2)3115 3958 y Fp(\016)14 b(\001)g(\001)g(\001)e(\016)i(C)3385 3927 y Fo(1)3421 3958 y Fu(.)605 4057 y(W)-7 b(e)25 b(pro)n(v)n(e)f(no) n(w)g(that,)i(if)g(a)e(function)i(has)e(the)i(form)e(giv)n(en)h(in)g (\(3.7\),)g(then)h(it)f(b)r(elongs)g(to)456 4157 y Fq(P)509 4169 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)672 4157 y Fu(.)37 b(F)-7 b(or)27 b(that)h(w)n(e)f(c)n(hec)n(k)g(the)h(order)f(of)g(ev)n (ery)f(term)i(in)g(\(3.7\).)37 b(In)2712 4095 y Fg(P)2799 4182 y Fi(l)p Fj(\024)p Fi(n)p Fo(+1)3021 4157 y Fu(\026)-48 b Fq(y)3059 4127 y Fo(2)p Fi(n)p Fo(+2)p Fj(\000)p Fo(2)p Fi(l)3328 4157 y Fq(p)3370 4169 y Fi(i)3393 4178 y Fc(l)3421 4157 y Fu(,)456 4259 y(the)28 b(term)f(indexed)h(b)n(y)f Fq(l)i Fu(has)e(order)g(2)p Fq(n)17 b Fu(+)h(2)g Fp(\000)g Fu(2)p Fq(l)i Fu(+)e Fq(i)2173 4271 y Fi(l)2221 4259 y Fp(\025)23 b Fu(2)p Fq(n)k Fu(if)h Fq(l)c Fu(=)f(0)p Fq(;)14 b Fu(1;)26 b(and)1038 4400 y(2)p Fq(n)18 b Fu(+)g(2)g Fp(\000)g Fu(2)p Fq(l)h Fu(+)f Fq(i)1572 4412 y Fi(l)1620 4400 y Fp(\025)23 b Fu(\()p Fq(n)c Fp(\000)f Fu(2\))p Fq(l)h Fu(+)f Fq(k)26 b Fp(\025)d Fu(2\()p Fq(n)18 b Fp(\000)g Fu(1\))g(+)g Fq(k)j Fp(\000)d Fu(2,)456 4556 y(if)28 b Fq(l)c Fp(\025)f Fu(2.)36 b(In)874 4493 y Fg(P)962 4580 y Fo(2)p Fi(l)p Fj(\024)p Fo(2)p Fi(n)p Fo(+1)1250 4556 y Fu(\026)-48 b Fq(y)1288 4525 y Fo(2)p Fi(n)p Fo(+1)p Fj(\000)p Fo(2)p Fi(l)1557 4556 y Fq(p)1599 4568 y Fi(j)1626 4577 y Fc(l)1654 4556 y Fu(,)28 b(it)g(has)f(order)f(bigger)g(than)i(2) p Fq(n)18 b Fu(+)g(1)27 b(if)h Fq(l)d Fu(=)d(0)27 b(and)943 4704 y(2)p Fq(n)18 b Fu(+)g(1)g Fp(\000)g Fu(2)p Fq(l)h Fu(+)f Fq(j)1482 4716 y Fi(l)1531 4704 y Fp(\025)23 b Fq(k)e Fu(+)d(\()p Fq(n)g Fp(\000)g Fu(2\)\()p Fq(l)i Fu(+)e(1\))23 b Fp(\025)g Fu(2\()p Fq(n)18 b Fp(\000)g Fu(1\))g(+)g Fq(k)k Fp(\000)c Fu(2)456 4856 y(if)28 b Fq(l)c Fp(\025)f Fu(1.)36 b(Hence)28 b Fq(F)35 b Fu(=)23 b Fq(F)1258 4826 y Fo(2)p Fi(n)p Fo(+2)1443 4856 y Fp(2)g Fq(P)1574 4868 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1738 4856 y Fu(.)1619 b Fe(\003)605 5016 y Ft(Remark)34 b Fu(3.3)p Ft(.)41 b Fs(We)31 b(observe)i(that)e Fq(\026)1811 4986 y Fo(2)1848 5016 y Fq(")1887 4986 y Fo(2)p Fi(p)p Fo(+2)2043 5016 y Fq(R)2106 5028 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2296 5016 y Fs(c)l(an)g(b)l(e)h(written)f(as)h Fq(\026)2999 4986 y Fo(2)3036 5016 y Fq(")3075 4986 y Fi(p)p Fo(+3)3197 5016 y Fq(R)3260 5028 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)3419 5016 y Fs(.)456 5116 y(Inde)l(e)l(d,)i(if)g Fq(p)29 b Fp(\025)g Fu(1)k Fs(it)g(fol)t(lows)i(fr)l(om)f(the)f(c)l(omp)l(arison) i(of)f(p)l(owers)g(of)g Fq(")p Fs(.)48 b(A)n(nd,)34 b(if)g Fq(p)29 b(<)g Fu(1)p Fs(,)34 b(by)456 5216 y(hyp)l(othesis)d Fq(@)894 5228 y Fi(y)934 5216 y Fq(h)982 5228 y Fo(1)1019 5216 y Fq(@)1063 5228 y Fi(x)1105 5216 y Fq(h)1153 5228 y Fo(1)1213 5216 y Fu(=)23 b(0)29 b Fs(which)j(implies)f(that)f Fq(@)2104 5228 y Fi(y)2144 5216 y Fq(h)2192 5228 y Fo(1)2229 5216 y Fq(@)2273 5228 y Fi(x)2315 5216 y Fq(S)2371 5185 y Fo(1)2366 5236 y(1)2431 5216 y Fu(=)22 b(0)p Fs(.)p eop %%Page: 19 27 19 26 bop 1339 251 a Fo(3.3.)30 b(A)-7 b(VERA)n(GING)29 b(OF)g(THE)g(EQUA)-5 b(TION)817 b(19)605 450 y Fu(W)-7 b(e)37 b(rename)f(the)i(v)-5 b(ariables)35 b(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r Fu(\))38 b(b)n(y)e(\()p Fq(x;)14 b(y)s Fu(\).)65 b(As)37 b(a)g(consequence)f(of)g(the)i (previous)456 550 y(remark,)26 b(the)i(transformed)e(Hamiltonian)i(is) 1075 698 y Fq(")p Fp(H)1184 710 y Fo(0)1304 698 y Fu(=)83 b Fq("h)1539 710 y Fo(0)1594 698 y Fu(+)18 b Fq(\026")1766 663 y Fi(p)p Fo(+2)p Fi(n)p Fo(+3)2013 698 y Fq(F)2066 710 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2248 698 y Fu(+)g Fq(\026)2381 663 y Fo(2)2419 698 y Fq(")2458 663 y Fi(p)p Fo(+3)2580 698 y Fq(R)2643 710 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2802 698 y Fu(.)605 846 y(Next)40 b(w)n(e)g(p)r(erform)f(a)g(new)h (set)g(of)g(steps)f(of)h(a)n(v)n(eraging)d(in)j(order)e(to)i(eliminate) g(the)456 945 y(dep)r(endence)35 b(on)g Fq(\022)i Fu(in)f(the)f(terms)g (of)g Fq(R)1766 957 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1960 945 y Fu(of)g(order)f(\(in)h(the)h(\()p Fq(x;)14 b(y)s Fu(\))35 b(v)-5 b(ariables\))34 b(smaller)456 1045 y(than)27 b(2)p Fq(n)18 b Fp(\000)g Fu(2.)605 1144 y(As)23 b(usual,)g(giv)n(en)f Fq(f)32 b Fp(2)23 b Fq(P)1375 1156 y Fi(j)1410 1144 y Fu(,)h(w)n(e)e(will)h(denote)g(its)g(mean)f(with)h(resp)r(ect)g(to)f Fq(\022)j Fu(b)n(y)3069 1122 y(\026)3051 1144 y Fq(f)8 b Fu(.)36 b(W)-7 b(e)23 b(pro-)456 1244 y(ceed)k(in)h(a)f(similar)f(w)n (a)n(y)h(as)g(b)r(efore.)36 b(F)-7 b(or)27 b(the)h(inductiv)n(e)f (step,)h(w)n(e)f(consider)g(a)g(Hamiltonian)456 1344 y(system)g(of)h(the)f(form)637 1493 y Fq(")700 1472 y Fu(~)676 1493 y Fp(H)747 1459 y Fi(\027)788 1493 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))84 b(=)e Fq("h)1539 1505 y Fo(0)1576 1493 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(\026)1920 1459 y Fo(2)1957 1493 y Fq(")1996 1459 y Fi(p)p Fo(+3)2119 1493 y Fq(R)2183 1459 y Fi(\027)2182 1513 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2341 1493 y Fu(\()p Fq(x;)c(y)s(;)g(\026;)g(")p Fu(\))1452 1625 y(+)p Fq(\026")1606 1591 y Fi(p)p Fo(+4)1728 1625 y Fq(F)1793 1591 y Fi(\027)1781 1646 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1945 1625 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))k(+)g Fq(\026)2529 1591 y Fo(2)2567 1625 y Fq(")2606 1591 y Fi(p)p Fo(+3)2728 1625 y Fq(T)2789 1591 y Fi(\027)2829 1625 y Fu(\()p Fq(x;)c(y)s(;)g (\022)r(;)g(\026;)g(")p Fu(\))456 1773 y(where)27 b Fq(F)761 1743 y Fi(\027)749 1794 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)935 1773 y Fp(2)d Fq(P)1067 1785 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1230 1773 y Fu(,)k Fq(R)1345 1743 y Fi(\027)1344 1797 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1526 1773 y Fp(2)c Fq(P)1658 1785 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1817 1773 y Fu(,)k Fq(T)1929 1743 y Fi(\027)1997 1773 y Fu(has)f(zero)f(mean)h (and)h(has)f(the)h(form)1751 1922 y Fq(T)1812 1888 y Fi(\027)1875 1922 y Fu(=)23 b Fq(y)2007 1888 y Fi(\027)2048 1922 y Fq(r)2085 1934 y Fi(i)2108 1942 y Fc(\027)456 1922 y Fu(\(3.8\))456 2075 y(with)28 b Fq(i)674 2087 y Fi(\027)738 2075 y Fu(=)22 b(max)p Fp(f)p Fu(0)p Fq(;)14 b Fu(2)p Fq(k)20 b Fp(\000)e Fu(2)g Fp(\000)g Fq(\027)5 b Fp(g)27 b Fu(and)g Fq(r)1745 2087 y Fi(i)1768 2095 y Fc(\027)1833 2075 y Fp(2)d Fq(P)1965 2087 y Fi(i)1988 2095 y Fc(\027)2029 2075 y Fu(.)605 2175 y(W)-7 b(e)24 b(observ)n(e)e(that)h Fq(")p Fp(H)1320 2187 y Fo(0)1381 2175 y Fu(has)g(this)h(form)f(for)g Fq(\027)28 b Fu(=)23 b(0,)h(with)g Fq(F)2494 2145 y Fo(0)2482 2195 y(2)p Fi(n)p Fj(\000)p Fo(2)2669 2175 y Fu(=)e Fq(")2795 2145 y Fo(2)p Fi(n)p Fj(\000)p Fo(1)2958 2175 y Fq(F)3011 2187 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)3175 2175 y Fu(,)i Fq(R)3286 2145 y Fo(0)3285 2198 y(2)p Fi(k)q Fj(\000)p Fo(2)456 2281 y Fu(the)k(mean)f(of)h Fq(R)978 2293 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1164 2281 y Fu(and)g Fq(T)1387 2251 y Fo(0)1446 2281 y Fu(=)23 b Fq(R)1597 2293 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1774 2281 y Fp(\000)18 b Fq(R)1921 2251 y Fo(0)1920 2305 y(2)p Fi(k)q Fj(\000)p Fo(2)2079 2281 y Fu(.)605 2438 y Ft(Lemma)47 b Fu(3.3)p Ft(.)i Fs(Under)44 b(the)f(pr)l(evious)i(c)l(onditions)f(and)g(assuming)g(that)f Fq(n)48 b Fp(\025)g Fu(3)43 b Fs(and)456 2539 y Fu(2)p Fq(k)22 b Fp(\000)e Fu(2)27 b Fp(\025)g Fq(n)p Fs(,)34 b(ther)l(e)e(exists)g(a)g(c)l(anonic)l(al)i(change)f(of)g(variables)i Fu(\()p Fq(x;)14 b(y)s Fu(\))28 b(=)2852 2518 y(~)2837 2539 y Fp(C)2886 2509 y Fi(\027)t Fo(+1)3011 2539 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))456 2639 y Fs(which)35 b(is)g Fq(C)852 2609 y Fo(0)890 2639 y Fs(,)g Fq(C)1015 2609 y Fo(1)1053 2639 y Fs(and)g Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)g (in)g Fq(\022)h Fs(and)f(analytic)h(in)e Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\026)p Fu(\))35 b Fs(that)f(tr)l(ansforms)g(the)456 2744 y(Hamiltonian)c Fq(")998 2723 y Fu(~)974 2744 y Fp(H)1045 2714 y Fi(\027)1116 2744 y Fs(to)553 2892 y Fq(")616 2871 y Fu(~)592 2892 y Fp(H)663 2858 y Fi(\027)t Fo(+1)788 2892 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))83 b(=)f Fq("h)1539 2904 y Fo(0)1576 2892 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r Fu(\))19 b(+)f Fq(\026)1920 2858 y Fo(2)1957 2892 y Fq(")1996 2858 y Fi(p)p Fo(+3)2119 2892 y Fq(R)2183 2857 y Fi(\027)t Fo(+1)2182 2917 y(2)p Fi(k)q Fj(\000)p Fo(2)2341 2892 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\026;)g(")p Fu(\))1452 3030 y(+)p Fq(\026")1606 2996 y Fi(p)p Fo(+4)1728 3030 y Fq(F)1793 2995 y Fi(\027)t Fo(+1)1781 3052 y(2)p Fi(n)p Fj(\000)p Fo(2)1945 3030 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))k(+)g Fq(\026)2529 2996 y Fo(2)2567 3030 y Fq(")2606 2996 y Fi(p)p Fo(+3)2728 3030 y Fq(T)2789 2996 y Fi(\027)t Fo(+1)2913 3030 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))456 3178 y Fs(in)29 b(a)i(neighb)l(orho)l(o)l(d)h(of)e(the)g(origin,)i (wher)l(e)1280 3326 y Fq(T)1341 3292 y Fi(\027)t Fo(+1)1548 3326 y Fu(=)83 b Fq(")6 b Fu(\026)-48 b Fq(y)r(@)1822 3338 y Fi(x)1864 3326 y Fq(S)1920 3291 y Fi(\027)t Fo(+1)1915 3348 y(1)1243 3454 y Fq(R)1307 3419 y Fi(\027)t Fo(+1)1306 3479 y(2)p Fi(k)q Fj(\000)p Fo(2)1548 3454 y Fu(=)83 b Fq(R)1760 3420 y Fi(\027)1759 3475 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1936 3454 y Fu(+)18 b Fq("r)2095 3466 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1248 3593 y Fq(F)1313 3557 y Fi(\027)t Fo(+1)1301 3615 y(2)p Fi(n)p Fj(\000)p Fo(2)1548 3593 y Fu(=)83 b Fq(F)1761 3558 y Fi(\027)1749 3613 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1931 3593 y Fu(+)18 b Fq(\026)p Fu(\()p Fq(r)2133 3605 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2315 3593 y Fp(\000)k Fu(\026)-46 b Fq(r)2435 3605 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2599 3593 y Fu(\))p Fs(.)456 3751 y(The)30 b(function)g Fq(S)1008 3716 y Fi(\027)t Fo(+1)1003 3773 y(1)1133 3751 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))30 b Fs(is)g(determine)l(d)h(by)f(the)g(c)l(onditions)1403 3919 y Fq(@)1447 3931 y Fi(\022)1485 3919 y Fq(S)1541 3884 y Fi(\027)t Fo(+1)1536 3942 y(1)1689 3919 y Fu(=)23 b Fp(\000)p Fq(T)1903 3885 y Fi(\027)1943 3919 y Fs(,)p 2138 3836 182 4 v 170 w Fq(S)2194 3884 y Fi(\027)t Fo(+1)2189 3942 y(1)2342 3919 y Fu(=)g(0)p Fs(.)-2041 b Fu(\(3.9\))456 4067 y Fs(Mor)l(e)l(over)1616 4215 y Fq(T)1677 4181 y Fi(\027)t Fo(+1)1825 4215 y Fu(=)22 b Fq(y)1956 4181 y Fi(\027)t Fo(+1)2081 4215 y Fq(p)2123 4227 y Fi(i)2146 4235 y Fc(\027)s Fh(+1)2258 4215 y Fs(,)456 4382 y(with)33 b Fq(p)681 4394 y Fi(j)745 4382 y Fp(2)d Fq(P)883 4394 y Fi(j)918 4382 y Fs(,)35 b(and)f Fq(T)1204 4352 y Fi(\027)t Fo(+1)1361 4382 y Fs(has)g(zer)l(o)f(me)l(an.)49 b(A)n(lso)2171 4361 y Fu(~)2147 4382 y Fp(H)2218 4352 y Fi(\027)t Fo(+1)2376 4382 y Fs(is)34 b(c)l(ontinuous)e(and)i(analytic)g(in)456 4482 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\026)p Fu(\))p Fs(.)605 4639 y Ft(Pr)n(oof.)41 b Fu(In)30 b(the)h(pro)r(of)f(w)n(e)g(will)h(not)f(write)g(the)h(dep)r (endence)g(of)g(the)f(functions)h(on)f Fq(\026)p Fu(,)456 4739 y Fq(")p Fu(.)56 b(Along)34 b(the)h(pro)r(of)e Fq(r)1233 4751 y Fi(j)1303 4739 y Fu(will)i(mean)f(a)g(generic)f(term)h(of)g Fq(P)2418 4751 y Fi(j)2454 4739 y Fu(.)57 b(As)34 b(b)r(efore,)i(w)n(e) e(consider)f(a)456 4838 y(generating)23 b(function)i Fq(S)1237 4808 y Fi(\027)t Fo(+1)1362 4838 y Fu(\()p Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))25 b(whic)n(h)g(will)f(pro)n(vide)g(a)g(canonical)f(c)n(hange)g(of)i(v)-5 b(ariables)456 4938 y(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s Fu(\))23 b Fp(7!)g Fu(\()p Fq(x;)14 b(y)s Fu(\))29 b(implicitly)f(through)1536 5087 y(\026)-47 b Fq(x)83 b Fu(=)g Fq(@)1858 5099 y Fo(\026)-38 b Fi(y)1893 5087 y Fq(S)1949 5053 y Fi(\027)t Fo(+1)2074 5087 y Fu(\()p Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))1534 5216 y Fq(y)86 b Fu(=)d Fq(@)1853 5228 y Fi(x)1895 5216 y Fq(S)1951 5181 y Fi(\027)t Fo(+1)2076 5216 y Fu(\()p Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\).)-1913 b(\(3.10\))p eop %%Page: 20 28 20 27 bop 456 251 a Fo(20)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(W)-7 b(e)28 b(tak)n(e)1222 611 y Fq(S)1278 577 y Fi(\027)t Fo(+1)1403 611 y Fu(\()p Fq(x;)21 b Fu(\026)-49 b Fq(y)t(;)14 b(\022)r Fu(\))23 b(=)g Fq(x)6 b Fu(\026)-48 b Fq(y)21 b Fu(+)d Fq(\026)2027 577 y Fo(2)2065 611 y Fq(")2104 577 y Fi(p)p Fo(+3)2226 611 y Fq(S)2282 576 y Fi(\027)t Fo(+1)2277 634 y(1)2407 611 y Fu(\()p Fq(x;)i Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))456 788 y(with)33 b Fq(S)706 753 y Fi(\027)t Fo(+1)701 811 y(1)865 788 y Fu(satisfying)f(\(3.9\).)54 b(W)-7 b(e)33 b(observ)n(e)f(that)i Fq(S)2180 758 y Fi(\027)t Fo(+1)2338 788 y Fu(is)f Fq(C)2492 758 y Fo(0)2530 788 y Fu(,)h Fq(C)2652 758 y Fo(1)2690 788 y Fu(and)f(2)p Fq(\031)s Fu(-p)r(erio)r(dic)g(in)g Fq(\022)456 888 y Fu(and)27 b(analytic)g(in)h(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\).)605 990 y(F)-7 b(rom)27 b(\(3.8\),)h(it)g(is)f(clear)g(that)h Fq(T)1651 960 y Fi(\027)1714 990 y Fp(2)23 b Fq(P)1845 1002 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2005 990 y Fu(.)37 b(Hence)27 b Fq(S)2367 955 y Fi(\027)t Fo(+1)2362 1012 y(1)2515 990 y Fp(2)d Fq(P)2647 1002 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2806 990 y Fu(.)605 1090 y(Analogously)h(as)h(in)g(the)h (pro)r(of)f(of)g(Lemma)h(3.1)e(w)n(e)h(get)g(the)h(regularit)n(y)e (statemen)n(ts)h(for)471 1170 y(~)456 1191 y Fp(C)505 1160 y Fi(\027)t Fo(+1)657 1191 y Fu(and)i(w)n(e)f(ha)n(v)n(e)f(that) 1190 1352 y Fq(x)84 b Fu(=)j(\026)-47 b Fq(x)19 b Fp(\000)f Fq(\026)1667 1318 y Fo(2)1704 1352 y Fq(")1743 1318 y Fi(p)p Fo(+3)1865 1352 y Fq(@)1914 1364 y Fo(\026)-38 b Fi(y)1949 1352 y Fq(S)2005 1316 y Fi(\027)t Fo(+1)2000 1374 y(1)2149 1352 y Fu(+)18 b Fq(\026)2282 1318 y Fo(4)2319 1352 y Fq(")2358 1318 y Fo(2)p Fi(p)p Fo(+6)2514 1352 y Fq(r)2551 1364 y Fo(4)p Fi(k)q Fj(\000)p Fo(7)1194 1482 y Fq(y)86 b Fu(=)i(\026)-48 b Fq(y)21 b Fu(+)d Fq(\026)1663 1447 y Fo(2)1701 1482 y Fq(")1740 1447 y Fi(p)p Fo(+3)1862 1482 y Fq(@)1906 1494 y Fi(x)1948 1482 y Fq(S)2004 1446 y Fi(\027)t Fo(+1)1999 1504 y(1)2147 1482 y Fu(+)g Fq(\026)2280 1447 y Fo(4)2317 1482 y Fq(")2356 1447 y Fo(2)p Fi(p)p Fo(+6)2512 1482 y Fq(r)2549 1494 y Fo(4)p Fi(k)q Fj(\000)p Fo(7)456 1647 y Fu(where)32 b(the)h(deriv)-5 b(ativ)n(es)32 b(of)h Fq(S)1424 1611 y Fi(\027)t Fo(+1)1419 1669 y(1)1582 1647 y Fu(and)g Fq(r)1786 1659 y Fo(4)p Fi(k)q Fj(\000)p Fo(7)1978 1647 y Fu(are)f(ev)-5 b(aluated)33 b(at)g(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\).)53 b(The)33 b(a)n(v)n(eraged)456 1746 y(Hamiltonian)27 b(is)g(therefore)735 1908 y Fq(")798 1887 y Fu(~)774 1908 y Fp(H)845 1873 y Fi(\027)t Fo(+1)970 1908 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r Fu(\))84 b(=)e Fq(")1534 1887 y Fu(~)1510 1908 y Fp(H)1581 1873 y Fi(\027)1622 1908 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))19 b Fp(\000)f Fq(\026)2044 1873 y Fo(2)2082 1908 y Fq(")2121 1873 y Fi(p)p Fo(+3)2243 1908 y Fq(T)2304 1873 y Fi(\027)2344 1908 y Fu(\()p Fq(x;)j Fu(\026)-49 b Fq(y)t(;)14 b(\022)r Fu(\))1324 2054 y(=)1483 1997 y Fq(")p 1481 2035 42 4 v 1481 2111 a Fu(2)1533 2054 y([)6 b(\026)-48 b Fq(y)21 b Fu(+)d Fq(\026)1751 2019 y Fo(2)1788 2054 y Fq(")1827 2019 y Fi(p)p Fo(+3)1949 2054 y Fq(@)1993 2066 y Fi(x)2035 2054 y Fq(S)2091 2018 y Fi(\027)t Fo(+1)2086 2076 y(1)2235 2054 y Fu(+)g Fq(\026)2368 2019 y Fo(4)2405 2054 y Fq(")2444 2019 y Fo(2)p Fi(p)p Fo(+6)2599 2054 y Fq(r)2636 2066 y Fo(4)p Fi(k)q Fj(\000)p Fo(7)2796 2054 y Fu(])2819 2019 y Fo(2)1471 2217 y Fu(+)p Fq("V)g Fu(\()5 b(\026)-47 b Fq(x)20 b Fp(\000)e Fq(\026)1873 2182 y Fo(2)1910 2217 y Fq(")1949 2182 y Fi(p)p Fo(+3)2071 2217 y Fq(@)2120 2229 y Fo(\026)-38 b Fi(y)2155 2217 y Fq(S)2211 2181 y Fi(\027)t Fo(+1)2206 2239 y(1)2355 2217 y Fu(+)18 b Fq(\026)2488 2182 y Fo(4)2525 2217 y Fq(")2564 2182 y Fo(2)p Fi(p)p Fo(+6)2719 2217 y Fq(r)2756 2229 y Fo(4)p Fi(k)q Fj(\000)p Fo(7)2915 2217 y Fu(\))1471 2345 y(+)p Fq(\026)1586 2311 y Fo(2)1623 2345 y Fq(")1662 2311 y Fi(p)p Fo(+3)1784 2345 y Fu([)p Fq(T)1868 2311 y Fi(\027)1909 2345 y Fu(\()p Fq(x;)c(y)s(;)g(\022)r Fu(\))19 b Fp(\000)f Fq(T)2342 2311 y Fi(\027)2382 2345 y Fu(\()p Fq(x;)j Fu(\026)-49 b Fq(y)t(;)14 b(\022)r Fu(\)])1471 2471 y(+)p Fq(\026)1586 2437 y Fo(2)1623 2471 y Fq(")1662 2437 y Fi(p)p Fo(+3)1784 2471 y Fq(R)1848 2437 y Fi(\027)1847 2492 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2025 2471 y Fu(+)k Fq(\026")2197 2437 y Fi(p)p Fo(+4)2319 2471 y Fq(F)2384 2437 y Fi(\027)2372 2492 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2554 2471 y Fu(+)g Fq(\026)2687 2437 y Fo(4)2724 2471 y Fq(")2763 2437 y Fo(2)p Fi(p)p Fo(+6)2919 2471 y Fq(r)2956 2483 y Fo(4)p Fi(k)q Fj(\000)p Fo(6)1324 2624 y Fu(=)1483 2568 y Fq(")p 1481 2605 V 1481 2681 a Fu(2)1538 2624 y(\026)-47 b Fq(y)1577 2589 y Fo(2)1632 2624 y Fu(+)18 b Fq("V)h Fu(\()5 b(\026)-47 b Fq(x)p Fu(\))19 b(+)f Fq(\026)2084 2589 y Fo(2)2121 2624 y Fq(")2160 2589 y Fi(p)p Fo(+3)2283 2624 y Fq(R)2347 2589 y Fi(\027)2346 2644 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2523 2624 y Fu(+)g Fq(\026)2656 2589 y Fo(2)2693 2624 y Fq(")2732 2589 y Fi(p)p Fo(+4)2860 2624 y Fu(\026)-48 b Fq(y)s(@)2942 2636 y Fi(x)2984 2624 y Fq(S)3040 2588 y Fi(\027)t Fo(+1)3035 2646 y(1)1471 2786 y Fu(+)p Fq(\026")1625 2751 y Fi(p)p Fo(+4)1747 2786 y Fq(F)1812 2751 y Fi(\027)1800 2806 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1982 2786 y Fu(+)18 b Fq(\026)2115 2751 y Fo(2)2152 2786 y Fq(")2191 2751 y Fi(p)p Fo(+4)2314 2786 y Fq(r)2351 2798 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)456 2947 y Fu(where)39 b(in)h(the)g(last)f(equalit)n(y)g (w)n(e)g(ha)n(v)n(e)g(used)g(that,)k(since)d(2)p Fq(k)28 b Fp(\000)f Fu(2)42 b Fp(\025)h Fq(n)p Fu(,)f(w)n(e)e(ha)n(v)n(e)e (that)456 3047 y(4)p Fq(k)20 b Fp(\000)e Fu(6)23 b Fp(\025)g Fu(2)p Fq(n)17 b Fp(\000)h Fu(2.)605 3146 y(Therefore)26 b(w)n(e)i(can)f(tak)n(e)1261 3307 y Fq(F)1326 3272 y Fi(\027)t Fo(+1)1314 3330 y(2)p Fi(n)p Fj(\000)p Fo(2)1561 3307 y Fu(=)82 b Fq(F)1773 3273 y Fi(\027)1761 3328 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1944 3307 y Fu(+)18 b Fq(\026)p Fu(\()p Fq(r)2146 3319 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2328 3307 y Fp(\000)k Fu(\026)-46 b Fq(r)2448 3319 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2612 3307 y Fu(\))1256 3443 y Fq(R)1320 3407 y Fi(\027)t Fo(+1)1319 3468 y(2)p Fi(k)q Fj(\000)p Fo(2)1561 3443 y Fu(=)82 b Fq(R)1772 3408 y Fi(\027)1771 3463 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1949 3443 y Fu(+)18 b Fq(")t Fu(\026)-46 b Fq(r)2108 3455 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1292 3581 y Fq(T)1353 3547 y Fi(\027)t Fo(+1)1561 3581 y Fu(=)82 b Fq(")6 b Fu(\026)-48 b Fq(y)s(@)1835 3593 y Fi(x)1877 3581 y Fq(S)1933 3545 y Fi(\027)t Fo(+1)1928 3603 y(1)2058 3581 y Fu(.)605 3742 y(No)n(w)21 b(w)n(e)h(c)n(hec)n(k)f (that)h Fq(T)1358 3712 y Fi(\027)t Fo(+1)1504 3742 y Fu(has)g(the)g(claimed)g(form.)34 b(It)22 b(is)g(clear)f(that)h(it)h (has)e(zero)g(mean,)456 3842 y(and)27 b(that,)h(from)f(\(3.8\))h(and)f (\(3.9\),)g Fq(T)1658 3812 y Fi(\027)t Fo(+1)1810 3842 y Fu(has)h(the)g(form)1500 4004 y Fq(T)1561 3970 y Fi(\027)t Fo(+1)1709 4004 y Fu(=)22 b Fq(y)1840 3970 y Fi(\027)t Fo(+1)1965 4004 y Fq(p)2007 4019 y Fo(max)p Fj(f)p Fo(0)p Fi(;i)2240 4027 y Fc(\027)2277 4019 y Fj(\000)p Fo(1)p Fj(g)456 4170 y Fu(and)27 b Fq(i)646 4182 y Fi(\027)t Fo(+1)794 4170 y Fu(=)c(max)o Fp(f)p Fu(0)p Fq(;)14 b(i)1186 4182 y Fi(\027)1244 4170 y Fp(\000)19 b Fu(1)p Fp(g)p Fu(.)605 4276 y(Finally)-7 b(,)23 b(the)g(regularit)n(y)d(statemen)n(t) i(on)1924 4255 y(~)1900 4276 y Fp(H)1971 4246 y Fi(\027)t Fo(+1)2118 4276 y Fu(follo)n(ws)g(from)f(the)i(regularit)n(y)d(of)3200 4255 y(~)3176 4276 y Fp(H)3247 4246 y Fi(\027)3311 4276 y Fu(and)471 4356 y(~)456 4377 y Fp(C)505 4347 y Fi(\027)t Fo(+1)630 4377 y Fu(,)27 b(in)h(the)g(same)f(w)n(a)n(y)f(as)h(in)h (Lemma)g(3.1.)1458 b Fe(\003)605 4565 y Fu(No)n(w)27 b(w)n(e)g(use)h(the)g(previous)e(lemma)i(to)f(p)r(erform)g(2)p Fq(n)18 b Fp(\000)g Fu(2)27 b(steps)h(of)f(a)n(v)n(eraging.)605 4729 y Ft(Lemma)22 b Fu(3.4)p Ft(.)31 b Fs(Ther)l(e)23 b(exists)e(a)g(c)l(anonic)l(al)i(change)f(of)g(variables)h Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)f Fp(C)2980 4741 y Fo(1)3017 4729 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))456 4829 y Fs(which)33 b(is)f Fq(C)847 4799 y Fo(0)885 4829 y Fs(,)h Fq(C)1008 4799 y Fo(1)1077 4829 y Fs(and)f Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)i(in)e Fq(\022)i Fs(and)e(analytic)h(in)f Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\026)p Fu(\))32 b Fs(that)g(tr)l(ansforms)g(the)456 4929 y(Hamiltonian)e Fq(")p Fp(H)1044 4941 y Fo(0)1111 4929 y Fs(to)905 5090 y Fq(")p Fp(H)1014 5102 y Fo(1)1052 5090 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))83 b(=)f Fq("h)1802 5102 y Fo(0)1839 5090 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s Fu(\))18 b(+)g Fq(\026")2222 5056 y Fi(p)p Fo(+4)2345 5090 y Fq(F)2398 5102 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2561 5090 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r(;)g(\026;)g(")p Fu(\))1715 5216 y(+)p Fq(\026)1830 5181 y Fo(2)1867 5216 y Fq(")1906 5181 y Fi(p)p Fo(+3)2028 5216 y Fq(R)2091 5228 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2250 5216 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\026;)g(")p Fu(\))p eop %%Page: 21 29 21 28 bop 1222 251 a Fo(3.4.)29 b(ESTIMA)-5 b(TES)28 b(F)n(OR)h(THE)h(POINCAR)2428 236 y(\023)2422 251 y(E)e(MAP)729 b(21)456 450 y Fs(in)28 b(a)h(neighb)l(orho)l(o)l(d)j(of)d(the)g (origin,)i(wher)l(e)e Fq(F)1907 462 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2094 450 y Fp(2)23 b Fq(P)2225 462 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2417 450 y Fs(and)29 b(has)h(zer)l(o)f(me)l(an,)g Fq(R)3207 462 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)3389 450 y Fp(2)456 550 y Fq(P)509 562 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)698 550 y Fs(do)l(es)h(not)f(dep)l(end)i(on)f Fq(\022)i Fs(and)1441 706 y Fq(R)1504 718 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1686 706 y Fu(=)p 1774 630 348 4 v 23 w Fq(@)1818 718 y Fi(y)1858 706 y Fq(h)1906 718 y Fo(1)1943 706 y Fq(@)1987 718 y Fi(x)2029 706 y Fq(S)2085 678 y Fo(1)2080 728 y(1)2140 706 y Fu(+)18 b Fq("r)2299 718 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)456 862 y Fs(with)33 b Fq(S)695 832 y Fo(1)690 883 y(1)765 862 y Fs(such)f(that)h Fq(@)1173 874 y Fi(\022)1211 862 y Fq(S)1267 832 y Fo(1)1262 883 y(1)1332 862 y Fu(=)28 b Fp(\000)p Fq(h)1538 874 y Fo(1)1608 862 y Fs(and)33 b(has)g(zer)l(o)g(me)l(an,)h(and)f Fq(r)2552 874 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2740 862 y Fp(2)c Fq(P)2877 874 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)3036 862 y Fs(.)48 b(Mor)l(e)l(over)456 962 y Fp(H)526 974 y Fo(1)593 962 y Fs(is)30 b(c)l(ontinuous,)f(and)h(analytic)h(in)f Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\026)p Fu(\))p Fs(.)605 1118 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)24 b(apply)f(the)h(Lemma)f(3.3)g(iterativ)n(ely)f(2)p Fq(n)10 b Fp(\000)g Fu(2)23 b(times.)35 b(W)-7 b(e)24 b(omit)g(the)g(details)456 1217 y(since)j(they)h(are)e(the)i(same)f(as) g(the)h(ones)f(in)h(Lemma)f(3.2.)1092 b Fe(\003)605 1383 y Ft(Pr)n(oof)31 b(of)h(pr)n(oposition)f(3.1.)41 b Fu(First)34 b(w)n(e)h(scale)e(time)j(to)e(obtain)h(the)g(Hamiltonian)456 1483 y Fq("H)7 b Fu(.)34 b(Then)21 b(w)n(e)f(apply)h(the)g(c)n(hanges)e Fp(C)1655 1495 y Fo(0)1713 1483 y Fu(and)i Fp(C)1912 1495 y Fo(1)1970 1483 y Fu(giv)n(en)f(b)n(y)g(Lemmas)h(3.2)f(and)g(3.4) g(resp)r(ectiv)n(ely)456 1582 y(to)27 b(obtain)h(the)g(Hamiltonian)f Fq(")p Fp(H)1545 1594 y Fo(1)1583 1582 y Fu(,)h(and)f(\014nally)h(w)n (e)f(scale)g(bac)n(k)g(to)h(the)g(original)e(time.)38 b(W)-7 b(e)456 1682 y(tak)n(e)31 b Fp(H)g Fu(=)e Fp(H)905 1694 y Fo(1)943 1682 y Fu(\()p Fq(x;)14 b(y)s(;)g(t=";)g(\026;)g(")p Fu(\).)48 b(T)-7 b(o)32 b(obtain)f(the)h(form)g(of)g Fp(C)k Fu(just)c(recall)f(that)h Fp(C)c Fu(=)23 b Fp(C)3219 1694 y Fo(1)3278 1682 y Fp(\016)d(C)3384 1694 y Fo(0)3421 1682 y Fu(,)456 1781 y Fp(C)500 1793 y Fo(0)560 1781 y Fu(=)i Fp(C)696 1751 y Fo(2)p Fi(n)p Fo(+2)866 1781 y Fp(\016)8 b(\001)14 b(\001)g(\001)8 b(\016)g(C)1120 1751 y Fo(1)1179 1781 y Fu(with)22 b Fp(C)1411 1751 y Fi(j)1446 1781 y Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e(\()p Fq(x;)14 b(y)s Fu(\))8 b(+)g Fq(O)r Fu(\()p Fq(\026")2208 1751 y Fi(p)p Fo(+)p Fi(j)2329 1781 y Fu(\),)24 b Fp(C)2452 1793 y Fo(1)2489 1781 y Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e(\()p Fq(x;)14 b(y)s Fu(\))8 b(+)g Fq(O)r Fu(\()p Fq(\026)3212 1751 y Fo(2)3251 1781 y Fq(")3290 1751 y Fi(p)p Fo(+3)3412 1781 y Fu(\))456 1881 y(and)28 b(that)h Fp(C)848 1851 y Fo(1)885 1881 y Fu(\()p Fq(x;)14 b(y)s Fu(\))26 b(=)e(\()p Fq(x;)14 b(y)s Fu(\))20 b(+)f Fq(\026")1577 1851 y Fi(p)p Fo(+1)1699 1881 y Fu(\()p Fp(\000)p Fq(@)1840 1893 y Fi(y)1880 1881 y Fq(S)1936 1851 y Fo(1)1931 1902 y(1)1973 1881 y Fq(;)14 b(@)2054 1893 y Fi(x)2096 1881 y Fq(S)2152 1851 y Fo(1)2147 1902 y(1)2189 1881 y Fu(\))19 b(+)g Fq(O)r Fu(\()p Fq(\026)2471 1851 y Fo(2)2509 1881 y Fq(")2548 1851 y Fo(2)p Fi(p)p Fo(+2)2703 1881 y Fu(\))29 b(where)g Fq(@)3050 1893 y Fi(\022)3087 1881 y Fq(S)3143 1851 y Fo(1)3138 1902 y(1)3205 1881 y Fu(=)c Fp(\000)p Fq(h)3408 1893 y Fo(1)456 1981 y Fu(and)35 b Fq(S)681 1951 y Fo(1)676 2001 y(1)753 1981 y Fu(has)g(zero)g(mean.)60 b(The)35 b(form)g(of)h Fq(R)1918 1993 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2112 1981 y Fu(comes)f(from)g(\(3.4\),)i(and)f(the)g (fact)f(that)456 2083 y(from)29 b(the)i(second)e(set)h(of)g(steps)g(of) g(a)n(v)n(eraging)d(w)n(e)j(b)r(egin)g(with)2541 2062 y(\026)2523 2083 y Fq(R)2586 2095 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2775 2083 y Fu(and)g(then)h Fq(R)3195 2047 y Fi(\027)t Fo(+1)3194 2108 y(2)p Fi(k)q Fj(\000)p Fo(2)3380 2083 y Fu(=)456 2182 y Fq(R)520 2152 y Fi(\027)519 2206 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)696 2182 y Fu(+)18 b Fq(")t Fu(\026)-46 b Fq(r)855 2194 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1018 2182 y Fu(.)2339 b Fe(\003)1188 2384 y Fk(3.4.)47 b(Estimates)29 b(for)j(the)g(P)m(oincar)n(\023)-46 b(e)33 b(map)605 2533 y(3.4.1.)46 b(Notation.)40 b Fu(In)26 b(this)g(section)f(w)n(e)g(calculate)g(the)h(P)n(oincar)n(\023)-39 b(e)23 b(map)i(asso)r(ciated)g(to)456 2633 y Fp(H)j Fu(obtained)g(in)f (Prop)r(osition)f(3.1.)36 b(Let)1018 2779 y Fq(x)1065 2744 y Fj(0)1172 2779 y Fu(=)82 b Fq(y)21 b Fu(+)d Fq(\026")1553 2744 y Fi(p)p Fo(+3)1676 2779 y Fq(@)1720 2791 y Fi(y)1760 2779 y Fq(F)1813 2791 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1995 2779 y Fu(+)g Fq(\026)2128 2744 y Fo(2)2165 2779 y Fq(")2204 2744 y Fi(p)p Fo(+2)2326 2779 y Fq(@)2370 2791 y Fi(y)2410 2779 y Fq(R)2473 2791 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)456 2779 y Fu(\(3.11\))1022 2907 y Fq(y)1066 2873 y Fj(0)1172 2907 y Fu(=)82 b Fp(\000)p Fq(V)1451 2873 y Fj(0)1474 2907 y Fu(\()p Fq(x)p Fu(\))20 b Fp(\000)e Fq(\026")1777 2873 y Fi(p)p Fo(+3)1899 2907 y Fq(@)1943 2919 y Fi(x)1985 2907 y Fq(F)2038 2919 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2220 2907 y Fp(\000)g Fq(\026)2353 2873 y Fo(2)2390 2907 y Fq(")2429 2873 y Fi(p)p Fo(+2)2551 2907 y Fq(@)2595 2919 y Fi(x)2637 2907 y Fq(R)2700 2919 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2859 2907 y Fu(.)456 3053 y(b)r(e)33 b(the)g(corresp)r(onding)e (equation.)52 b(T)-7 b(o)32 b(simplify)h(the)g(notation)g(w)n(e)f(in)n (tro)r(duce)h Fq(z)i Fu(=)c(\()p Fq(x;)14 b(y)s Fu(\))456 3153 y(and)839 3299 y Fq(F)892 3311 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1078 3299 y Fu(=)23 b(\()p Fq(@)1242 3311 y Fi(y)1282 3299 y Fq(F)1335 3311 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1499 3299 y Fq(;)14 b Fp(\000)p Fq(@)1645 3311 y Fi(x)1686 3299 y Fq(F)1739 3311 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1903 3299 y Fu(\))p Fq(;)180 b(R)2201 3311 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2383 3299 y Fu(=)23 b(\()p Fq(@)2547 3311 y Fi(y)2587 3299 y Fq(R)2650 3311 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2809 3299 y Fq(;)14 b Fp(\000)p Fq(@)2955 3311 y Fi(x)2996 3299 y Fq(R)3059 3311 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)3218 3299 y Fu(\))p Fq(:)605 3445 y Fu(W)-7 b(e)28 b(de\014ne)1346 3620 y Fq(X)1415 3632 y Fo(0)1452 3620 y Fu(\()p Fq(x;)14 b(y)s Fu(\))84 b(=)1876 3502 y Fg(\022)2090 3569 y Fq(y)1978 3668 y Fp(\000)p Fq(V)2110 3638 y Fj(0)2133 3668 y Fu(\()p Fq(x)p Fu(\))2287 3502 y Fg(\023)2362 3620 y Fu(,)1501 3803 y Fq(Y)1549 3815 y Fi(\026;")1728 3803 y Fu(=)f Fq(X)1945 3815 y Fo(0)2000 3803 y Fu(+)18 b Fq(\026)2133 3769 y Fo(2)2170 3803 y Fq(")2209 3769 y Fi(p)p Fo(+2)2332 3803 y Fq(R)2395 3815 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)456 3949 y Fu(and)1464 4095 y Fq(X)1533 4107 y Fi(\026;")1652 4095 y Fu(=)k Fq(Y)1787 4107 y Fi(\026;")1902 4095 y Fu(+)c Fq(\026")2074 4061 y Fi(p)p Fo(+3)2196 4095 y Fq(F)2249 4107 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2413 4095 y Fu(.)456 4246 y(Hence,)27 b(equation)g(\(3.11\))g(b)r (ecomes)h Fq(z)1678 4216 y Fj(0)1723 4246 y Fu(=)23 b Fq(X)1880 4258 y Fi(\026;")1975 4246 y Fu(\()p Fq(z)t(;)14 b(t=")p Fu(\).)605 4345 y(Let)37 b Fq(')817 4357 y Fi(\026;")913 4345 y Fu(\()p Fq(t;)14 b(t)1042 4357 y Fo(0)1080 4345 y Fq(;)g(z)t Fu(\))36 b(b)r(e)i(the)f(solution)g(of)g(the)g(equation)g Fq(z)2475 4315 y Fj(0)2536 4345 y Fu(=)i Fq(X)2709 4357 y Fi(\026;")2804 4345 y Fu(\()p Fq(z)t(;)14 b(t=")p Fu(\))36 b(suc)n(h)h(that)456 4445 y Fq(')510 4457 y Fi(\026;")606 4445 y Fu(\()p Fq(t)668 4457 y Fo(0)705 4445 y Fq(;)14 b(t)772 4457 y Fo(0)809 4445 y Fq(;)g(z)t Fu(\))46 b(=)g Fq(z)f Fu(and)d Fq(\036)1387 4457 y Fi(\026;")1483 4445 y Fu(\()p Fq(t;)14 b(t)1612 4457 y Fo(0)1649 4445 y Fq(;)g(z)t Fu(\))42 b(b)r(e)g(the)g(solution)f(of)h(the)g(system)f Fq(z)3013 4415 y Fj(0)3083 4445 y Fu(=)46 b Fq(Y)3242 4457 y Fi(\026;")3338 4445 y Fu(\()p Fq(z)t Fu(\))456 4545 y(suc)n(h)35 b(that)h Fq(\036)888 4557 y Fi(\026;")984 4545 y Fu(\()p Fq(t)1046 4557 y Fo(0)1084 4545 y Fq(;)14 b(t)1151 4557 y Fo(0)1188 4545 y Fq(;)g(z)t Fu(\))36 b(=)h Fq(z)t Fu(.)61 b(If)36 b(there)g(is)g(not)g(danger)e(of)i (confusion,)i(w)n(e)d(will)i(denote)456 4644 y Fq(')510 4656 y Fi(\026;")606 4644 y Fu(\()p Fq(t;)14 b(t)735 4656 y Fo(0)772 4644 y Fq(;)g(z)t Fu(\))27 b(b)n(y)g Fq(')1080 4656 y Fi(\026;")1176 4644 y Fu(\()p Fq(t)p Fu(\))h(and)g Fq(\036)1509 4656 y Fi(\026;")1605 4644 y Fu(\()p Fq(t;)14 b(t)1734 4656 y Fo(0)1771 4644 y Fq(;)g(z)t Fu(\))27 b(b)n(y)h Fq(\036)2075 4656 y Fi(\026)2120 4644 y Fu(\()p Fq(t;)14 b(")p Fu(\).)605 4744 y(W)-7 b(e)28 b(consider)f(the)h(P)n(oincar)n(\023)-39 b(e)24 b(maps)1418 4890 y Fq(P)1483 4855 y Fi(t)1508 4863 y Fh(0)1471 4910 y Fi(\026;")1567 4890 y Fu(\()p Fq(z)t Fu(\))f(=)g Fq(')1839 4902 y Fi(\026;")1935 4890 y Fu(\()p Fq(t)1997 4902 y Fo(0)2053 4890 y Fu(+)18 b(2)p Fq(\031)s(";)c(t)2334 4902 y Fo(0)2370 4890 y Fq(;)g(z)t Fu(\))-2026 b(\(3.12\))456 5036 y(and)1124 5161 y(^)1105 5182 y Fq(P)1158 5194 y Fi(\026;")1254 5182 y Fu(\()p Fq(z)t Fu(\))23 b(=)f Fq(\036)1520 5194 y Fi(\026;")1616 5182 y Fu(\()p Fq(t)1678 5194 y Fo(0)1734 5182 y Fu(+)c(2)p Fq(\031)s(";)c(t)2015 5194 y Fo(0)2052 5182 y Fq(;)g(z)t Fu(\))23 b(=)f Fq(\036)2323 5194 y Fi(\026;")2419 5182 y Fu(\(2)p Fq(\031)s(";)14 b Fu(0)p Fq(;)g(z)t Fu(\))p Fq(:)-2340 b Fu(\(3.13\))p eop %%Page: 22 30 22 29 bop 456 251 a Fo(22)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(Let)28 b Fq(U)k Fp(\032)22 b Fl(C)984 420 y Fo(2)1055 450 y Fu(b)r(e)28 b(a)f(neigh)n(b)r(orho)r(o)r(d)g(of)g(the)h(origin)f (and)g(let)759 608 y Fq(V)18 b Fu(\()p Fq(\022)896 620 y Fo(0)934 608 y Fu(\))23 b(=)1077 541 y Fg(\000)1115 608 y Fp([)1170 623 y Fi(s)p Fj(2)p Fo([0)p Fi(;)p Fo(1])1374 608 y Fq(')1428 620 y Fi(\026;")1524 608 y Fu(\()p Fq(\022)1595 620 y Fo(0)1651 608 y Fu(+)18 b Fq(s)p Fu(2)p Fq(\031)s(;)c(\022)1941 620 y Fo(0)1978 608 y Fq(;)g(U)9 b Fu(\))2113 541 y Fg(\001)2165 529 y([)2271 541 y(\000)2309 608 y Fp([)2364 623 y Fi(s)p Fj(2)p Fo([0)p Fi(;)p Fo(1])2568 608 y Fq(\036)2617 620 y Fi(\026;")2713 608 y Fu(\()p Fq(\022)2784 620 y Fo(0)2840 608 y Fu(+)18 b Fq(s)p Fu(2)p Fq(\031)s(;)c(\022)3130 620 y Fo(0)3167 608 y Fq(;)g(U)9 b Fu(\))3302 541 y Fg(\001)456 766 y Fu(and)1664 912 y Fq(V)42 b Fu(=)1870 833 y Fg([)1841 1012 y Fi(\022)1873 1020 y Fh(0)1905 1012 y Fj(2)p Fd(R)2006 912 y Fq(V)19 b Fu(\()p Fq(\022)2144 924 y Fo(0)2181 912 y Fu(\).)456 1146 y(Therefore,)31 b(since)h Fq(')1122 1158 y Fi(\026;")1218 1146 y Fu(\()p Fq(\022)1289 1158 y Fo(0)1347 1146 y Fu(+)21 b Fq(s)p Fu(2)p Fq(\031)s(;)14 b(\022)1640 1158 y Fo(0)1677 1146 y Fq(;)g(z)t Fu(\))31 b(dep)r(ends)h(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dically)f(on)g Fq(\022)2916 1158 y Fo(0)2953 1146 y Fu(,)i(the)f(set)g Fq(V)51 b Fu(is)456 1245 y(b)r(ounded.)605 1401 y Ft(Pr)n(oposition)31 b Fu(3.2)p Ft(.)40 b Fs(We)30 b(have)h(the)f(fol)t(lowing)i(expr)l (essions)e(for)h(the)f(Poinc)l(ar)n(\023)-40 b(e)31 b(maps)851 1572 y Fu(^)832 1593 y Fq(P)885 1605 y Fi(\026;")981 1593 y Fu(\()p Fq(z)t Fu(\))83 b(=)1319 1476 y Fg(\022)1421 1542 y Fq(x)19 b Fu(+)f(2)p Fq(\031)s("y)1561 1642 y(y)1786 1476 y Fg(\023)1866 1593 y Fu(+)g(2)p Fq(\031)s(")2094 1476 y Fg(\022)2380 1542 y Fu(2)p Fq(\031)s("q)2548 1554 y Fo(1)2585 1542 y Fu(\()p Fq(z)t(;)c(")p Fu(\))2196 1642 y Fp(\000)p Fq(V)2327 1612 y Fj(0)2350 1642 y Fu(\()p Fq(x)p Fu(\))20 b(+)e(2)p Fq(\031)s("q)2732 1654 y Fo(2)2769 1642 y Fu(\()p Fq(z)t(;)c(")p Fu(\))2993 1476 y Fg(\023)1319 1777 y Fu(+)p Fq(\026)1434 1742 y Fo(2)1471 1777 y Fq(")1510 1742 y Fi(p)p Fo(+3)1632 1777 y Fq(T)1681 1789 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1839 1777 y Fu(\()p Fq(z)t Fu(\))456 1923 y Fs(and)1217 2069 y Fq(P)1282 2034 y Fi(t)1307 2042 y Fh(0)1270 2089 y Fi(\026;")1366 2069 y Fu(\()p Fq(z)t Fu(\))23 b(=)1602 2048 y(^)1584 2069 y Fq(P)1637 2081 y Fi(\026;")1733 2069 y Fu(\()p Fq(z)t Fu(\))18 b(+)g Fq(\026")2030 2034 y Fi(p)p Fo(+4)2152 2069 y Fq(S)2203 2081 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2366 2069 y Fu(\()p Fq(z)t(;)c(t)2508 2081 y Fo(0)2545 2069 y Fq(=")p Fu(\))p Fs(.)456 2220 y(wher)l(e)30 b Fq(q)727 2232 y Fo(1)764 2220 y Fq(;)g(q)854 2232 y Fo(2)914 2220 y Fp(2)24 b Fq(P)1046 2232 y Fi(n)p Fj(\000)p Fo(1)1176 2220 y Fs(\(indep)l(endent) 31 b(of)f Fq(\026)p Fu(\))p Fs(,)h Fq(S)1957 2232 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2143 2220 y Fp(2)23 b Fq(P)2274 2232 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2468 2220 y Fs(and)30 b Fq(T)2678 2232 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2859 2220 y Fp(2)24 b Fq(P)2991 2232 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)3150 2220 y Fs(.)605 2376 y Fk(3.4.2.)46 b(Some)32 b(preliminary)h(b)s(ounds.)40 b Fu(In)30 b(order)e(to)i(determine)g (the)g(prop)r(erties)f(of)456 2475 y(the)36 b(P)n(oincar)n(\023)-39 b(e)32 b(map)k(de\014ned)g(in)g(\(3.12\))f(w)n(e)g(need)h(a)g(precise)f (kno)n(wledge)f(of)i(the)g(distance)456 2575 y(b)r(et)n(w)n(een)30 b(a)g(solution)g(and)h(its)g(initial)g(condition,)g(the)g(distance)f(b) r(et)n(w)n(een)h(the)g(solutions)f(of)456 2675 y(the)e(unp)r(erturb)r (ed)h(system,)f Fq(')1433 2687 y Fo(0)1471 2675 y Fu(\()p Fq(t)p Fu(\),)h(and)f(the)h(solutions)e(of)i(the)f(p)r(erturb)r(ed)h (one,)f Fq(')3128 2687 y Fi(\026;")3224 2675 y Fu(\()p Fq(t)p Fu(\),)h(as)456 2774 y(w)n(ell)c(as)g(the)i(distance)e(b)r(et)n (w)n(een)h Fq(\036)1553 2786 y Fi(\026;")1649 2774 y Fu(\()p Fq(t)p Fu(\))g(and)g Fq(')1983 2786 y Fi(\026;")2079 2774 y Fu(\()p Fq(t)p Fu(\).)37 b(This)26 b(is)f(studied)i(in)f(this)g (subsection.)605 2877 y(W)-7 b(e)28 b(mak)n(e)f(the)h(con)n(v)n(en)n (tion)e(that)i(if)g Fq(l)d(<)e Fu(0)k(in)h Fp(k)p Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fp(k)2359 2847 y Fi(l)2412 2877 y Fu(w)n(e)27 b(understand)h(that)g(it)g(repre-)456 2977 y(sen)n(ts)f(a)g(constan)n(t)g(term.)605 3076 y(W)-7 b(e)28 b(need)g(a)f(simple)h(lemma:)605 3232 y Ft(Lemma)h Fu(3.5)p Ft(.)37 b Fs(L)l(et)27 b Fu(\012)c Fp(\032)g Fl(C)1449 3202 y Fo(2)1492 3232 y Fp(\002)p Fl(R)h Fp(\002)18 b Fl(C)40 b Fp(\002)18 b Fl(R)33 b Fs(b)l(e)27 b(a)h(neighb)l(orho)l(o) l(d)i(of)e Fp(f)p Fu(\(0)p Fq(;)14 b Fu(0\))p Fp(g)f(\002)g Fl(R)23 b Fp(\002)18 b(f)p Fu(0)p Fp(g)13 b(\002)456 3332 y(f)p Fu(0\))p Fp(g)30 b Fs(and)i(let)f Fq(f)k Fu(:)26 b(\012)g Fp(!)g Fl(R)38 b Fs(b)l(e)31 b(a)h(function)g(that)f(is)h(c)l (ontinuous,)g Fq(C)2616 3302 y Fo(1)2684 3332 y Fs(with)h(r)l(esp)l(e)l (ct)e(to)g Fq(\022)j Fs(and)456 3431 y(analytic)d(with)f(r)l(esp)l(e)l (ct)f(to)h Fu(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\))30 b Fs(such)g(that)g(ther)l(e)g(exists)f(a)h(c)l(onstant)f Fq(c)23 b(>)g Fu(0)29 b Fs(verifying)1384 3582 y Fp(k)p Fq(f)9 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))p Fp(k)22 b(\024)h Fq(c)p Fp(j)p Fq(y)s Fp(j)2187 3547 y Fi(i)2214 3582 y Fp(k)p Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fp(k)2490 3547 y Fi(l)456 3738 y Fs(for)30 b(al)t(l)h Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))24 b Fp(2)g Fu(\012)p Fs(.)39 b(Then)30 b(ther)l(e)h(exists)e(a)h (function)g Fq(f)2458 3750 y Fi(l)2507 3738 y Fp(2)23 b Fq(P)2638 3750 y Fi(l)2664 3738 y Fs(,)31 b Fq(C)2785 3708 y Fo(1)2852 3738 y Fs(with)g(r)l(esp)l(e)l(ct)e(to)h Fq(\022)456 3838 y Fs(such)f(that)1355 3984 y Fq(f)9 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))23 b(=)g Fq(y)1993 3949 y Fi(i)2020 3984 y Fq(f)2061 3996 y Fi(l)2086 3984 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))p Fs(.)605 4153 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)41 b(tak)n(e)f Fq(f)1323 4165 y Fi(l)1348 4153 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))45 b(=)f Fq(f)9 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))p Fq(=y)2504 4123 y Fi(l)2529 4153 y Fu(.)76 b(Ob)n(viously)39 b(w)n(e)h(ha)n(v)n(e)g(to)456 4253 y(pro)n(v)n(e)e(that)i Fq(f)925 4265 y Fi(l)991 4253 y Fu(is)f(analytic)h(at)g(p)r(oin)n(ts)g (of)g(the)g(form)g(\()p Fq(x;)14 b Fu(0)p Fq(;)g(\022)r(;)g(\026;)g(")p Fu(\))44 b Fp(2)g Fu(\012.)74 b(W)-7 b(e)41 b(consider)456 4352 y(\()p Fq(x;)14 b Fu(0)p Fq(;)g(\022)r(;)g(\026;)g(")p Fu(\))23 b Fp(2)h Fu(\012)k(and)g Fq(y)j Fu(small)c(enough)g(so)h(that) g(the)g(T)-7 b(a)n(ylor)26 b(series)h(of)h Fq(f)37 b Fu(with)28 b(resp)r(ect)g(to)456 4452 y Fq(y)i Fu(at)d(\()p Fq(x;)14 b Fu(0)p Fq(;)g(\022)r(;)g(\026;)g(")p Fu(\))28 b(con)n(v)n(erges)d(at)j Fq(y)s Fu(.)36 b(Then)28 b(b)n(y)f(T)-7 b(a)n(ylor's)26 b(theorem)h(the)h(result)f(follo)n(ws.)101 b Fe(\003)605 4618 y Fu(It)35 b(is)f(clear)g(that)h Fq(X)1254 4630 y Fi(\026;")1384 4618 y Fu(is)f(b)r(ounded)h(in)g Fq(V)54 b Fu(and)34 b(it)h(is)g(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)e (on)i Fq(t)p Fu(,)h(th)n(us)f(there)456 4717 y(exists)29 b(some)g(constan)n(t)g Fq(M)38 b Fu(\(indep)r(enden)n(t)31 b(on)e Fq(\022)r Fu(\))h(suc)n(h)f(that,)i Fp(k)p Fq(X)2584 4729 y Fi(\026;")2679 4717 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))p Fp(k)26 b(\024)g Fq(M)39 b Fu(for)29 b(all)456 4817 y(\()p Fq(x;)14 b(y)s Fu(\))23 b Fp(2)h Fq(V)46 b Fu(and)28 b Fq(\022)d Fp(2)e Fl(R)p Fu(.)605 4917 y(Moreo)n(v)n(er)17 b Fq(X)1031 4929 y Fi(\026;")1146 4917 y Fu(and)j Fq(Y)1348 4929 y Fi(\026;")1463 4917 y Fu(are)f(Lipsc)n(hitz)h(in)g Fq(V)e Fu(.)35 b(W)-7 b(e)20 b(denote)f(b)n(y)h Fq(L)f Fu(a)g(common)h(Lipsc)n(hitz)456 5016 y(constan)n(t)26 b(for)i Fq(X)987 5028 y Fi(\026;")1110 5016 y Fu(and)f Fq(Y)1319 5028 y Fi(\026;")1415 5016 y Fu(.)605 5116 y(T)-7 b(o)24 b(simplify)h(the)g(argumen)n(ts)e(related)h(to)g(the)h (dep)r(endence)g(with)f(resp)r(ect)h(to)f Fq(")p Fu(,)h(\014rst)f(w)n (e)456 5216 y(will)j(obtain)h(estimates)f(for)g(the)h(solutions)f(of)g (the)h(scaled)f(equations.)p eop %%Page: 23 31 23 30 bop 1222 251 a Fo(3.4.)29 b(ESTIMA)-5 b(TES)28 b(F)n(OR)h(THE)h(POINCAR)2428 236 y(\023)2422 251 y(E)e(MAP)729 b(23)605 450 y Ft(Lemma)31 b Fu(3.6)p Ft(.)40 b Fs(L)l(et)j Fu(~)-55 b Fq(')1286 462 y Fi(\026;")1382 450 y Fu(\()p Fq(\022)r Fu(\))24 b(=)36 b(~)-56 b Fq(')1652 462 y Fi(\026;")1748 450 y Fu(\()p Fq(\022)r(;)14 b(\022)1897 462 y Fo(0)1935 450 y Fq(;)g(z)t Fu(\))29 b Fs(b)l(e)h(the)g(solution)g(of)1374 613 y Fu(_)-37 b Fq(z)26 b Fu(=)d Fq("X)1621 625 y Fi(\026;")1716 613 y Fu(\()p Fq(z)t(;)14 b(\022)r Fu(\))p Fs(,)184 b Fu(~)-56 b Fq(')2150 625 y Fi(\026;")2246 613 y Fu(\()p Fq(\022)2317 625 y Fo(0)2355 613 y Fu(\))23 b(=)g Fq(z)-2081 b Fu(\(3.14\))456 796 y Fs(and)628 774 y Fu(~)617 796 y Fq(\036)666 808 y Fi(\026;")762 796 y Fu(\()p Fq(\022)r Fu(\))24 b(=)989 774 y(~)978 796 y Fq(\036)1027 808 y Fi(\026;")1124 796 y Fu(\()p Fq(\022)r(;)14 b(\022)1273 808 y Fo(0)1310 796 y Fq(;)g(z)t Fu(\))29 b Fs(b)l(e)h(the)g(solution)g (of)1426 968 y Fu(_)-37 b Fq(z)26 b Fu(=)d Fq("Y)1652 980 y Fi(\026;")1747 968 y Fu(\()p Fq(z)t Fu(\))p Fs(,)2060 946 y Fu(~)2049 968 y Fq(\036)2098 980 y Fi(\026;")2194 968 y Fu(\()p Fq(\022)2265 980 y Fo(0)2303 968 y Fu(\))g(=)g Fq(z)-2029 b Fu(\(3.15\))456 1136 y Fs(Then,)29 b(if)h Fq(\022)25 b Fp(2)f Fu([)p Fq(\022)981 1148 y Fo(0)1018 1136 y Fq(;)14 b(\022)1094 1148 y Fo(0)1147 1136 y Fu(+)h(2)p Fq(\031)s Fu(])28 b Fs(and)h Fq(z)d Fu(=)d(\()p Fq(x;)14 b(y)s Fu(\))24 b Fp(2)f Fq(U)9 b Fs(,)29 b(ther)l(e)f(exist)h(some)f(c) l(onstants)g Fq(C)6 b Fs(,)29 b Fq(C)3247 1148 y Fi(F)3303 1136 y Fs(,)g Fq(\026)3407 1148 y Fo(0)456 1236 y Fs(and)h Fq(")656 1248 y Fo(0)723 1236 y Fs(such)f(that)h(for)h(al)t(l)f Fp(j)p Fq(\026)p Fp(j)24 b(\024)e Fq(\026)1589 1248 y Fo(0)1656 1236 y Fs(and)31 b Fp(j)p Fq(")p Fp(j)22 b(\024)h Fq(")2052 1248 y Fo(0)2119 1236 y Fs(the)30 b(fol)t(lowing)i(b)l(ounds) e(hold:)679 1374 y Fu(1\))41 b Fp(k)13 b Fu(~)-55 b Fq(')890 1386 y Fi(\026;")986 1374 y Fu(\()p Fq(\022)r Fu(\))p Fp(k)23 b(\024)g Fq(C)6 b Fp(k)p Fq(z)t Fp(k)28 b Fs(and)i Fp(k)1678 1352 y Fu(~)1667 1374 y Fq(\036)1716 1386 y Fi(\026;")1812 1374 y Fu(\()p Fq(\022)r Fu(\))p Fp(k)23 b(\024)g Fq(C)6 b Fp(k)p Fq(z)t Fp(k)p Fs(.)679 1476 y Fu(2\))41 b Fp(k)13 b Fu(~)-55 b Fq(')890 1488 y Fi(\026;")986 1476 y Fu(\()p Fq(\022)r Fu(\))19 b Fp(\000)f Fq(z)t Fp(k)k(\024)g Fq("C)6 b Fu(\()p Fp(j)p Fq(y)s Fp(j)19 b Fu(+)f Fp(k)p Fq(z)t Fp(k)1842 1446 y Fi(n)p Fj(\000)p Fo(1)1989 1476 y Fu(+)g Fq(\026)2122 1446 y Fo(2)2159 1476 y Fq(")2198 1446 y Fi(p)p Fo(+2)2320 1476 y Fp(k)p Fq(z)t Fp(k)2447 1446 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2604 1476 y Fu(\))p Fs(.)679 1586 y Fu(3\))41 b Fs(The)31 b(solutions)1323 1564 y Fu(~)1312 1586 y Fq(\036)1361 1598 y Fi(\026;")1487 1586 y Fs(and)43 b Fu(~)-55 b Fq(')1702 1598 y Fi(\026;")1827 1586 y Fs(c)l(an)30 b(b)l(e)g(expr)l(esse)l(d)g (as)1280 1736 y Fu(~)1269 1758 y Fq(\036)1318 1770 y Fi(\026;")1414 1758 y Fu(\()p Fq(\022)r Fu(\))24 b(=)e Fq(')1684 1770 y Fo(0)1722 1758 y Fu(\()p Fq(\022)r Fu(\))d(+)f Fq(\026)1979 1724 y Fo(2)2016 1758 y Fq(")2055 1724 y Fi(p)p Fo(+3)2178 1758 y Fu(\010)2238 1770 y Fi(\026;")2333 1758 y Fu(\()p Fq(\022)r(;)c(\022)2482 1770 y Fo(0)2520 1758 y Fq(;)g(z)t Fu(\))794 1921 y Fs(with)1439 2084 y Fp(k)p Fu(\010)1541 2096 y Fi(\026;")1636 2084 y Fu(\()p Fq(\022)r(;)g(\022)1785 2096 y Fo(0)1823 2084 y Fq(;)g(z)t Fu(\))p Fp(k)22 b(\024)g Fq(C)6 b Fp(k)p Fq(z)t Fp(k)2278 2050 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2436 2084 y Fs(,)794 2247 y(and)1269 2410 y Fu(~)-55 b Fq(')1310 2422 y Fi(\026;")1406 2410 y Fu(\()p Fq(\022)r Fu(\))24 b(=)1633 2388 y(~)1622 2410 y Fq(\036)1671 2422 y Fi(\026;")1767 2410 y Fu(\()p Fq(\022)r Fu(\))c(+)e Fq(\026")2064 2376 y Fi(p)p Fo(+4)2186 2410 y Fu(\011)2251 2422 y Fi(\026;")2346 2410 y Fu(\()p Fq(\022)r(;)c(\022)2495 2422 y Fo(0)2533 2410 y Fq(;)g(z)t Fu(\))794 2573 y Fs(with)1411 2736 y Fp(k)p Fu(\011)1518 2748 y Fi(\026;")1613 2736 y Fu(\()p Fq(\022)r(;)g(\022)1762 2748 y Fo(0)1799 2736 y Fq(;)g(z)t Fu(\))p Fp(k)22 b(\024)h Fq(C)2122 2748 y Fi(F)2177 2736 y Fp(k)p Fq(z)t Fp(k)2304 2702 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2466 2736 y Fq(:)794 2909 y Fs(F)-6 b(urthermor)l(e,)35 b(if)f Fq(F)1441 2921 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1634 2909 y Fu(=)29 b(0)p Fs(,)35 b Fu(\011)1895 2921 y Fi(\026;")2019 2909 y Fu(=)29 b(0)p Fs(.)49 b(Mor)l(e)l(over,)37 b Fu(\011)2690 2921 y Fi(\026;")2818 2909 y Fs(and)d Fu(\010)3043 2921 y Fi(\026;")3172 2909 y Fs(ar)l(e)f Fq(C)3381 2879 y Fo(0)3419 2909 y Fs(,)794 3009 y Fq(C)859 2979 y Fo(1)927 3009 y Fs(with)e(r)l(esp)l(e)l(ct)f(to)h Fq(\022)h Fs(and)f Fq(\022)1752 3021 y Fo(0)1820 3009 y Fs(and)g(analytic)h(with)f(r)l (esp)l(e)l(ct)f(to)g Fq(\026)h Fs(and)g(the)f(initial)794 3108 y(c)l(ondition)h Fq(z)t Fs(.)679 3208 y Fu(4\))41 b Fs(The)31 b(functions)1274 3371 y Fq(S)1325 3383 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1489 3371 y Fu(\()p Fq(z)t(;)14 b(\022)1640 3383 y Fo(0)1676 3371 y Fu(\))84 b Fp(\021)e Fu(\011)2004 3383 y Fi(\026;")2099 3371 y Fu(\()p Fq(\022)2170 3383 y Fo(0)2226 3371 y Fu(+)18 b(2)p Fq(\031)s(;)c(\022)2477 3383 y Fo(0)2514 3371 y Fq(;)g(z)t Fu(\))1394 3495 y Fq(T)1443 3507 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1602 3495 y Fu(\()p Fq(z)t Fu(\))83 b Fp(\021)f Fu(\010)1999 3507 y Fi(\026;")2095 3495 y Fu(\()p Fq(\022)2166 3507 y Fo(0)2222 3495 y Fu(+)18 b(2)p Fq(\031)s(;)c(\022)2473 3507 y Fo(0)2510 3495 y Fq(;)g(z)t Fu(\))794 3658 y Fs(satisfy)33 b(that,)h Fq(S)1305 3670 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1495 3658 y Fp(2)28 b Fq(P)1631 3670 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1827 3658 y Fs(and)k Fq(T)2039 3670 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2225 3658 y Fp(2)c Fq(P)2361 3670 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2520 3658 y Fs(.)46 b(Mor)l(e)l(over)33 b(if)g Fq(F)3094 3670 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)3285 3658 y Fu(=)27 b(0)p Fs(,)794 3758 y Fq(S)845 3770 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1031 3758 y Fp(\021)c Fu(0)p Fs(.)605 3923 y Ft(Remark)33 b Fu(3.4)p Ft(.)41 b Fs(We)31 b(observe)g(that,)h(sinc)l(e)f(the)g(system)f(\(3.15\))j(is)e (autonomous,)g Fq(T)3286 3935 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)456 4023 y Fs(actual)t(ly)f(do)l(es)h(not)e(dep)l(end)i(on)f Fq(\022)1530 4035 y Fo(0)1567 4023 y Fs(.)605 4189 y Ft(Pr)n(oof.)41 b Fu(The)29 b(pro)r(of)g(of)h(estimates)f(1\))g(and)h (2\))f(follo)n(ws)g(from)g(Gron)n(w)n(all's)e(lemma.)43 b(T)-7 b(o)456 4288 y(deal)27 b(with)h(the)g(third)g(prop)r(ert)n(y)-7 b(,)26 b(w)n(e)i(lo)r(ok)e(for)i(the)g(solutions)e(of)i(\(3.15\))f(in)h (the)g(form)1268 4436 y(~)1257 4458 y Fq(\036)1306 4470 y Fi(\026;")1402 4458 y Fu(\()p Fq(\022)r Fu(\))c(=)f Fq(')1673 4470 y Fo(0)1710 4458 y Fu(\()p Fq(\022)r Fu(\))d(+)e Fq(\026)1968 4424 y Fo(2)2005 4458 y Fq(")2044 4424 y Fi(p)p Fo(+3)2166 4458 y Fu(\010)2226 4470 y Fi(\026;")2322 4458 y Fu(\()p Fq(\022)r(;)c(\022)2471 4470 y Fo(0)2508 4458 y Fq(;)g(z)t Fu(\).)456 4626 y(W)-7 b(e)28 b(denote)f(\010)926 4638 y Fi(\026;")1022 4626 y Fu(\()p Fq(\022)r(;)14 b(\022)1171 4638 y Fo(0)1208 4626 y Fq(;)g(z)t Fu(\))27 b(b)n(y)h(\010)1523 4638 y Fi(\026;")1618 4626 y Fu(\()p Fq(\022)r Fu(\).)38 b(F)-7 b(rom)27 b(the)h(iden)n(tit)n(y)819 4798 y Fq(X)888 4810 y Fo(0)925 4798 y Fu(\()968 4776 y(~)957 4798 y Fq(\036)1006 4810 y Fi(\026;")1103 4798 y Fu(\()p Fq(\022)r Fu(\)\))c(=)f Fq(X)1421 4810 y Fo(0)1458 4798 y Fu(\()p Fq(')1544 4810 y Fo(0)1581 4798 y Fu(\()p Fq(\022)r Fu(\)\))d(+)e Fq(X)1890 4810 y Fo(0)1927 4798 y Fu(\()p Fq(')2013 4810 y Fo(0)2051 4798 y Fu(\()p Fq(\022)r Fu(\))h(+)f Fq(\026)2308 4764 y Fo(2)2345 4798 y Fq(")2384 4764 y Fi(p)p Fo(+3)2507 4798 y Fu(\010)2567 4810 y Fi(\026;")2662 4798 y Fu(\()p Fq(\022)r Fu(\)\))i Fp(\000)e Fq(X)2971 4810 y Fo(0)3008 4798 y Fu(\()p Fq(')3094 4810 y Fo(0)3132 4798 y Fu(\()p Fq(\022)r Fu(\)\),)456 4961 y(w)n(e)27 b(deduce)h(that)846 5121 y(_)827 5142 y(\010)887 5154 y Fi(\026;")1006 5142 y Fu(=)1207 5086 y(1)p 1104 5123 249 4 v 1104 5199 a Fq(\026)1154 5175 y Fo(2)1191 5199 y Fq(")1230 5175 y Fi(p)p Fo(+2)1362 5142 y Fu([)p Fq(X)1454 5154 y Fo(0)1491 5142 y Fu(\()p Fq(')1577 5154 y Fo(0)1633 5142 y Fu(+)18 b Fq(\026)1766 5108 y Fo(2)1804 5142 y Fq(")1843 5108 y Fi(p)p Fo(+3)1965 5142 y Fu(\010)2025 5154 y Fi(\026;")2120 5142 y Fu(\))h Fp(\000)f Fq(X)2323 5154 y Fo(0)2360 5142 y Fu(\()p Fq(')2446 5154 y Fo(0)2484 5142 y Fu(\)])h(+)f Fq(R)2704 5154 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2863 5142 y Fu(\()2906 5121 y(~)2895 5142 y Fq(\036)2944 5154 y Fi(\026;")3040 5142 y Fu(\))p eop %%Page: 24 32 24 31 bop 456 251 a Fo(24)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(with)30 b(the)g(initial)g(condition)g(\010)1461 462 y Fi(\026)1505 450 y Fu(\()p Fq(\022)1576 462 y Fo(0)1613 450 y Fu(\))d(=)g(\(0)p Fq(;)14 b Fu(0\).)43 b(W)-7 b(e)30 b(observ)n(e)e(that)i(since)f Fq(')2899 462 y Fo(0)2963 450 y Fu(=)e Fq(')3109 462 y Fo(0)3146 450 y Fu(\()p Fq(\022)r(;)14 b(\022)3295 462 y Fo(0)3333 450 y Fq(;)g(z)t Fu(\))456 550 y(is)26 b(analytic,)f(the)i(function)g(\010)1401 562 y Fi(\026;")1522 550 y Fu(is)f Fq(C)1669 520 y Fo(0)1707 550 y Fu(,)g Fq(C)1821 520 y Fo(1)1885 550 y Fu(with)h(resp)r(ect)e(to) h Fq(\022)j Fu(and)d Fq(\022)2722 562 y Fo(0)2759 550 y Fu(,)h(and)f(analytic)f(with)456 649 y(resp)r(ect)i(to)g Fq(\026)p Fu(.)37 b(F)-7 b(rom)28 b(the)g(di\013eren)n(tial)f(equation) g(for)g(\010)2254 661 y Fi(\026)711 855 y Fu(\010)771 867 y Fi(\026;")866 855 y Fu(\()p Fq(\022)r Fu(\))84 b(=)1316 799 y(1)p 1213 836 249 4 v 1213 912 a Fq(\026)1263 888 y Fo(2)1300 912 y Fq(")1339 888 y Fi(p)p Fo(+2)1485 742 y Fg(Z)1568 763 y Fi(\022)1531 931 y(\022)1563 939 y Fh(0)1606 855 y Fu([)p Fq(X)1698 867 y Fo(0)1735 855 y Fu(\()p Fq(')1821 867 y Fo(0)1858 855 y Fu(\()p Fq(s)p Fu(\))19 b(+)f Fq(\026)2113 821 y Fo(2)2151 855 y Fq(")2190 821 y Fi(p)p Fo(+3)2312 855 y Fu(\010)2372 867 y Fi(\026;")2467 855 y Fu(\()p Fq(s)p Fu(\)\))i Fp(\000)e Fq(X)2774 867 y Fo(0)2811 855 y Fu(\()p Fq(')2897 867 y Fo(0)2934 855 y Fu(\()p Fq(s)p Fu(\)\)])c Fq(ds)1203 1105 y Fu(+)1282 992 y Fg(Z)1364 1013 y Fi(\022)1327 1181 y(\022)1359 1189 y Fh(0)1416 1105 y Fq(R)1479 1117 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1638 1105 y Fu(\()1681 1084 y(~)1670 1105 y Fq(\036)1719 1117 y Fi(\026;")1815 1105 y Fu(\()p Fq(s)p Fu(\)\))g Fq(ds)456 1316 y Fu(and,)27 b(since)h Fq(X)913 1328 y Fo(0)977 1316 y Fu(is)g(Lipsc)n(hitz)f(and)h Fp(k)1630 1294 y Fu(~)1619 1316 y Fq(\036)1668 1328 y Fi(\026;")1763 1316 y Fu(\()p Fq(s)p Fu(\))p Fp(k)23 b(\024)g Fq(C)6 b Fp(k)p Fq(z)t Fp(k)26 b Fu(w)n(e)h(ha)n(v)n(e)g(that)847 1522 y Fp(k)p Fu(\010)949 1534 y Fi(\026;")1044 1522 y Fu(\()p Fq(\022)r Fu(\))p Fp(k)83 b(\024)g Fq(L")1532 1409 y Fg(Z)1614 1430 y Fi(\022)1577 1598 y(\022)1609 1606 y Fh(0)1665 1522 y Fp(k)p Fu(\010)1767 1534 y Fi(\026;")1862 1522 y Fu(\()p Fq(s)p Fu(\))p Fp(k)14 b Fq(ds)19 b Fu(+)2205 1409 y Fg(Z)2288 1430 y Fi(\022)2251 1598 y(\022)2283 1606 y Fh(0)2339 1522 y Fp(k)p Fq(R)2444 1534 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2603 1522 y Fu(\()2646 1500 y(~)2635 1522 y Fq(\036)2684 1534 y Fi(\026;")2780 1522 y Fu(\()p Fq(s)p Fu(\)\))p Fp(k)14 b Fq(ds)1274 1772 y Fp(\024)83 b Fq(L")1532 1659 y Fg(Z)1614 1680 y Fi(\022)1577 1848 y(\022)1609 1856 y Fh(0)1665 1772 y Fp(k)p Fu(\010)1767 1784 y Fi(\026;")1862 1772 y Fu(\()p Fq(s)p Fu(\))p Fp(k)14 b Fq(ds)19 b Fu(+)f Fq(C)6 b Fp(k)p Fq(z)t Fp(k)2397 1738 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2554 1772 y Fu(.)456 1963 y(An)28 b(application)f(of)g(Gron)n(w)n(all's)f(lemma)h(giv)n(es)g (the)h(b)r(ound)934 2104 y Fp(k)p Fu(\010)1036 2116 y Fi(\026;")1132 2104 y Fu(\()p Fq(\022)r Fu(\))p Fp(k)23 b(\024)g Fq(C)6 b(e)1494 2070 y Fi(L")p Fo(2)p Fi(\031)1649 2104 y Fp(k)p Fq(z)t Fp(k)1776 2070 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1933 2104 y Fu(,)415 b Fq(\022)25 b Fp(2)f Fu([)p Fq(\022)2576 2116 y Fo(0)2613 2104 y Fq(;)14 b(\022)2689 2116 y Fo(0)2745 2104 y Fu(+)k(2)p Fq(\031)s Fu(].)-2510 b(\(3.16\))456 2245 y(It)28 b(is)f(clear)g(that)1397 2386 y Fq(T)1446 2398 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1604 2386 y Fu(\()p Fq(z)t Fu(\))c Fp(\021)g Fu(\010)1882 2398 y Fi(\026;")1977 2386 y Fu(\()p Fq(\022)2048 2398 y Fo(0)2104 2386 y Fu(+)18 b(2)p Fq(\031)s(;)c(\022)2355 2398 y Fo(0)2392 2386 y Fq(;)g(z)t Fu(\))456 2527 y(do)r(es)k(not)h (dep)r(end)h(on)f Fq(\022)1199 2539 y Fo(0)1255 2527 y Fu(and)g(it)h(is)f(analytic)f(with)i(resp)r(ect)f(to)g(initial)g (conditions.)34 b(Therefore,)456 2627 y(b)n(y)27 b(Lemma)g(3.5,)g Fq(T)1073 2639 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1255 2627 y Fp(2)c Fq(P)1386 2639 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1545 2627 y Fu(.)605 2726 y(Analogously)-7 b(,)26 b(w)n(e)h(lo)r(ok)g(for)g (solutions)g(of)h(\(3.14\))f(of)g(the)h(form)1258 2874 y(~)-56 b Fq(')1298 2886 y Fi(\026;")1394 2874 y Fu(\()p Fq(\022)r Fu(\))24 b(=)1622 2852 y(~)1611 2874 y Fq(\036)1660 2886 y Fi(\026;")1756 2874 y Fu(\()p Fq(\022)r Fu(\))19 b(+)f Fq(\026")2052 2840 y Fi(p)p Fo(+4)2174 2874 y Fu(\011)2239 2886 y Fi(\026;")2335 2874 y Fu(\()p Fq(\022)r(;)c(\022)2484 2886 y Fo(0)2521 2874 y Fq(;)g(z)t Fu(\).)456 3020 y(As)31 b(b)r(efore)g(w)n(e)f(denote)i(\011)1297 3032 y Fi(\026;")1392 3020 y Fu(\()p Fq(\022)r(;)14 b(\022)1541 3032 y Fo(0)1578 3020 y Fq(;)g(z)t Fu(\))31 b(b)n(y)g(\011)1905 3032 y Fi(\026;")2000 3020 y Fu(\()p Fq(\022)r Fu(\).)49 b(The)31 b(function)g(\011)2744 3032 y Fi(\026;")2871 3020 y Fu(satis\014es)f (the)i(fol-)456 3119 y(lo)n(wing)26 b(equation:)884 3274 y(_)864 3295 y(\011)929 3307 y Fi(\026;")1047 3295 y Fu(=)1229 3239 y(1)p 1144 3276 212 4 v 1144 3352 a Fq(\026")1233 3328 y Fi(p)p Fo(+3)1366 3295 y Fu([)p Fq(Y)1437 3307 y Fi(\026;")1532 3295 y Fu(\()1575 3273 y(~)1564 3295 y Fq(\036)1613 3307 y Fi(\026;")1728 3295 y Fu(+)18 b Fq(\026")1900 3260 y Fi(p)p Fo(+4)2022 3295 y Fu(\011)2087 3307 y Fi(\026;")2183 3295 y Fu(\))g Fp(\000)g Fq(Y)2364 3307 y Fi(\026;")2460 3295 y Fu(\()2503 3273 y(~)2492 3295 y Fq(\036)2541 3307 y Fi(\026;")2638 3295 y Fu(\)])g(+)g Fq(F)2847 3307 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)3011 3295 y Fu(\()13 b(~)-55 b Fq(')3097 3307 y Fi(\026;")3193 3295 y Fu(\).)456 3475 y(Therefore,)26 b(w)n(e)h(obtain)h(the)g (estimate)803 3673 y Fp(k)p Fu(\011)910 3685 y Fi(\026;")1005 3673 y Fu(\()p Fq(\022)r Fu(\))p Fp(k)83 b(\024)g Fq(L")1493 3560 y Fg(Z)1575 3581 y Fi(\022)1538 3749 y(\022)1570 3757 y Fh(0)1626 3673 y Fp(k)p Fu(\011)1733 3685 y Fi(\026;")1828 3673 y Fu(\()p Fq(\022)r Fu(\))p Fp(k)14 b Fq(ds)19 b Fu(+)2173 3560 y Fg(Z)2256 3581 y Fi(\022)2219 3749 y(\022)2251 3757 y Fh(0)2307 3673 y Fp(k)p Fq(F)2402 3685 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2565 3673 y Fu(\()13 b(~)-55 b Fq(')2651 3685 y Fi(\026;")2748 3673 y Fu(\()p Fq(s)p Fu(\))p Fq(;)14 b(s)p Fu(\))p Fp(k)g Fq(ds)1235 3923 y Fp(\024)83 b Fq(L")1493 3810 y Fg(Z)1575 3831 y Fi(\022)1538 3999 y(\022)1570 4007 y Fh(0)1626 3923 y Fp(k)p Fu(\011)1733 3935 y Fi(\026;")1828 3923 y Fu(\()p Fq(\022)r Fu(\))p Fp(k)14 b Fq(ds)19 b Fu(+)2192 3902 y(~)2173 3923 y Fq(C)2232 3935 y Fi(F)2287 3923 y Fp(k)p Fq(z)t Fp(k)2414 3889 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2576 3923 y Fu(.)456 4114 y(As)27 b(b)r(efore,)h(Gron)n(w)n(all's)d(lemma)j(giv)n(es)e(the)i(b)r (ound)1030 4255 y Fp(k)p Fu(\011)1137 4267 y Fi(\026;")1232 4255 y Fu(\()p Fq(\022)r Fu(\))p Fp(k)23 b(\024)g Fq(C)1549 4267 y Fi(F)1604 4255 y Fp(k)p Fq(z)t Fp(k)1731 4221 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1893 4255 y Fu(,)360 b Fq(\022)25 b Fp(2)e Fu([)p Fq(\022)2480 4267 y Fo(0)2518 4255 y Fq(;)14 b(\022)2594 4267 y Fo(0)2649 4255 y Fu(+)k(2)p Fq(\031)s Fu(].)-2414 b(\(3.17\))456 4396 y(It)28 b(is)f(clear)g(that)g (\011)1072 4408 y Fi(\026;")1195 4396 y Fu(is)h(zero)e(if)i Fq(F)1583 4408 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1775 4396 y Fu(is)f(zero)g(and)g(that)h(the)g(function)1334 4537 y Fq(S)1385 4549 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1549 4537 y Fu(\()p Fq(z)t(;)14 b(\022)1700 4549 y Fo(0)1736 4537 y Fu(\))24 b(=)e(\011)1944 4549 y Fi(\026;")2039 4537 y Fu(\()p Fq(\022)2110 4549 y Fo(0)2166 4537 y Fu(+)c(2)p Fq(\031)s(;)c(\022)2417 4549 y Fo(0)2454 4537 y Fq(;)g(z)t Fu(\))456 4683 y(is)33 b(2)p Fq(\031)s Fu(-p)r(erio)r(dic)g(in)g Fq(\022)1130 4695 y Fo(0)1201 4683 y Fu(and)g(analytic)g(with)h(resp)r (ect)f(to)g Fq(z)t Fu(.)54 b(Moreo)n(v)n(er)31 b(b)n(y)i(estimate)g (\(3.17\))456 4783 y(and)27 b(Lemma)g(3.5,)g Fq(S)1121 4795 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1307 4783 y Fp(2)d Fq(P)1439 4795 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1602 4783 y Fu(.)1755 b Fe(\003)605 4941 y Ft(Pr)n(oof)31 b(of)h(the)f(pr)n (oposition)h(3.2.)40 b Fu(W)-7 b(e)42 b(ha)n(v)n(e)e(that)h(for)g(all)f Fq(t)p Fu(,)45 b Fq(t)2859 4953 y Fo(0)2937 4941 y Fu(for)c(whic)n(h)g (the)456 5041 y(solutions)26 b(are)h(de\014ned)1387 5182 y Fq(')1441 5194 y Fi(\026;")1537 5182 y Fu(\()p Fq(t;)14 b(t)1666 5194 y Fo(0)1703 5182 y Fq(;)g(z)t Fu(\))22 b(=)36 b(~)-55 b Fq(')1979 5194 y Fi(\026;")2075 5182 y Fu(\()p Fq(t=";)14 b(t)2285 5194 y Fo(0)2322 5182 y Fq(=";)g(z)t Fu(\))p eop %%Page: 25 33 25 32 bop 1222 251 a Fo(3.4.)29 b(ESTIMA)-5 b(TES)28 b(F)n(OR)h(THE)h(POINCAR)2428 236 y(\023)2422 251 y(E)e(MAP)729 b(25)456 450 y Fu(and)27 b(the)h(analogous)d(relation)i(for)g Fq(\036)1624 462 y Fi(\026;")1748 450 y Fu(and)1920 428 y(~)1909 450 y Fq(\036)1958 462 y Fi(\026;")2054 450 y Fu(.)37 b(Then)28 b(w)n(e)f(can)g(write)1357 591 y Fq(P)1422 557 y Fi(t)1447 565 y Fh(0)1410 612 y Fi(\026;")1506 591 y Fu(\()p Fq(z)t Fu(\))c(=)36 b(~)-55 b Fq(')1778 603 y Fi(\026;")1874 591 y Fu(\()p Fq(t)1936 603 y Fo(0)1973 591 y Fq(=")18 b Fu(+)g(2)p Fq(\031)s(;)c(t)2314 603 y Fo(0)2351 591 y Fq(=";)g(z)t Fu(\))456 730 y(and)1544 848 y(^)1526 869 y Fq(P)1579 881 y Fi(\026;")1675 869 y Fu(\()p Fq(z)t Fu(\))23 b(=)1903 847 y(~)1892 869 y Fq(\036)1941 881 y Fi(\026;")2037 869 y Fu(\(2)p Fq(\031)s(;)14 b Fu(0)p Fq(;)g(z)t Fu(\))p Fq(:)605 1012 y Fu(By)33 b(prop)r(erties)g(3\))g(and)h(4\))f(of)h(Lemma)f(3.6)f(it)i(is)g (enough)f(to)g(compute)h(the)g(P)n(oincar)n(\023)-39 b(e)456 1112 y(map)28 b(of)g(the)h(unp)r(erturb)r(ed)g(system,)f(whic)n (h)g(is)g(indep)r(enden)n(t)h(of)g Fq(t)2579 1124 y Fo(0)2644 1112 y Fu(since)f(the)h(unp)r(erturb)r(ed)456 1211 y(system)e(is)g (autonomous:)774 1350 y Fq(P)839 1313 y Fi(t)864 1321 y Fh(0)827 1372 y Fo(0)p Fi(;")915 1350 y Fu(\()p Fq(z)t Fu(\))83 b(=)g Fq(')1307 1362 y Fo(0)1344 1350 y Fu(\(2)p Fq(\031)s(";)14 b Fu(0)p Fq(;)g(z)t Fu(\))1105 1547 y(=)83 b Fq(z)22 b Fu(+)c(2)p Fq(\031)s("')1582 1512 y Fj(0)1582 1567 y Fo(0)1619 1547 y Fu(\(0\))g(+)g(4)p Fq(\031)1918 1512 y Fo(2)1955 1547 y Fq(")1994 1512 y Fo(2)2045 1434 y Fg(Z)2128 1454 y Fo(1)2091 1622 y(0)2165 1547 y Fu(\(1)h Fp(\000)f Fq(s)p Fu(\))p Fq(')2466 1512 y Fj(00)2466 1567 y Fo(0)2509 1547 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\))c Fq(ds)1105 1776 y Fu(=)83 b Fq(z)22 b Fu(+)c(2)p Fq(\031)s(")1542 1659 y Fg(\022)1755 1725 y Fq(y)1644 1825 y Fp(\000)p Fq(V)1775 1795 y Fj(0)1798 1825 y Fu(\()p Fq(x)p Fu(\))1952 1659 y Fg(\023)1253 2033 y Fu(+4)p Fq(\031)1410 1999 y Fo(2)1447 2033 y Fq(")1486 1999 y Fo(2)1536 1891 y Fg( )1855 1916 y(R)1910 1936 y Fo(1)1894 2012 y(0)1948 1983 y Fu(\(1)g Fp(\000)g Fq(s)p Fu(\))p Fq(V)2261 1952 y Fj(0)2284 1983 y Fu(\()p Fq(')2370 1952 y Fo(1)2370 2003 y(0)2408 1983 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\)\))c Fq(ds)1644 2096 y Fp(\000)1723 2029 y Fg(R)1777 2050 y Fo(1)1761 2126 y(0)1815 2096 y Fu(\(1)k Fp(\000)g Fq(s)p Fu(\))p Fq(V)2128 2066 y Fj(00)2170 2096 y Fu(\()p Fq(')2256 2066 y Fo(1)2256 2117 y(0)2294 2096 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\)\))p Fq(')2614 2066 y Fo(2)2614 2117 y(0)2652 2096 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\))c Fq(ds)3024 1891 y Fg(!)3103 2033 y Fu(.)456 2246 y(It)28 b(is)f(clear)g(that)g (for)h Fq(s)23 b Fp(2)g Fu([0)p Fq(;)14 b Fu(1])1572 2386 y Fp(j)p Fq(V)1662 2352 y Fj(0)1685 2386 y Fu(\()p Fq(')1771 2352 y Fo(1)1771 2407 y(0)1809 2386 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\)\))p Fp(j)83 b(\024)g Fq(C)6 b Fp(k)p Fq(z)t Fp(k)2521 2352 y Fi(n)p Fj(\000)p Fo(1)1227 2512 y Fp(j)p Fq(V)1317 2478 y Fj(00)1360 2512 y Fu(\()p Fq(')1446 2478 y Fo(1)1446 2532 y(0)1483 2512 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\)\))p Fq(')1803 2478 y Fo(2)1803 2532 y(0)1841 2512 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\))p Fp(j)83 b(\024)g Fq(C)6 b Fp(k)p Fq(z)t Fp(k)2521 2478 y Fi(n)p Fj(\000)p Fo(1)2650 2512 y Fu(.)456 2650 y(Hence,)27 b(b)n(y)h(Lemma)f(3.5,)1668 2789 y Fq(V)1735 2755 y Fj(0)1758 2789 y Fu(\()p Fq(')1844 2755 y Fo(1)1844 2810 y(0)1882 2789 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\)\))84 b Fp(2)f Fq(P)2423 2801 y Fi(n)p Fj(\000)p Fo(1)1324 2915 y Fq(V)1390 2880 y Fj(0)q(0)1433 2915 y Fu(\()p Fq(')1519 2880 y Fo(1)1519 2935 y(0)1557 2915 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\)\))p Fq(')1877 2880 y Fo(2)1877 2935 y(0)1915 2915 y Fu(\()p Fq(s)p Fu(2)p Fq(\031)s(")p Fu(\))g Fp(2)g Fq(P)2423 2927 y Fi(n)p Fj(\000)p Fo(1)2553 2915 y Fu(.)456 3053 y(Therefore,)921 3192 y Fq(P)986 3155 y Fi(t)1011 3163 y Fh(0)974 3214 y Fo(0)p Fi(;")1062 3192 y Fu(\()p Fq(z)t Fu(\))g(=)g Fq(')1454 3204 y Fo(0)1491 3192 y Fu(\(2)p Fq(\031)s(";)14 b Fu(0)p Fq(;)g(z)t Fu(\))1252 3379 y(=)83 b Fq(z)22 b Fu(+)c(2)p Fq(\031)s(")1689 3262 y Fg(\022)1902 3328 y Fq(y)1791 3428 y Fp(\000)p Fq(V)1922 3398 y Fj(0)1945 3428 y Fu(\()p Fq(x)p Fu(\))2099 3262 y Fg(\023)2178 3379 y Fu(+)g(4)p Fq(\031)2353 3345 y Fo(2)2391 3379 y Fq(")2430 3345 y Fo(2)2480 3262 y Fg(\022)2583 3328 y Fq(q)2620 3340 y Fo(1)2657 3328 y Fu(\()p Fq(z)t(;)c(")p Fu(\))2583 3428 y Fq(q)2620 3440 y Fo(2)2657 3428 y Fu(\()p Fq(z)t(;)g(")p Fu(\))2881 3262 y Fg(\023)2956 3379 y Fq(:)3380 3561 y Fe(\003)605 3716 y Fk(3.4.3.)46 b(The)29 b(stable)g(manifold)e(of)j(the)f(auxiliary)h(system)e Fq(z)2757 3686 y Fj(0)2802 3716 y Fu(=)23 b Fq(Y)2938 3728 y Fi(\026;")3034 3716 y Fu(\()p Fq(z)t Fu(\))p Fk(.)41 b Fu(In)25 b(this)456 3816 y(subsection)i(w)n(e)g(compute)h(the)g (parameterization)d(of)j(the)g(stable)f(manifold)h(of)f Fq(z)3060 3786 y Fj(0)3106 3816 y Fu(=)c Fq(Y)3242 3828 y Fi(\026;")3338 3816 y Fu(\()p Fq(z)t Fu(\))456 3916 y(and)32 b(w)n(e)g(compare)f(it)h(with)h Fq(\015)1410 3928 y Fo(0)1447 3916 y Fu(.)51 b(F)-7 b(or)32 b(that)h(w)n(e)e(will)i (tak)n(e)e(adv)-5 b(an)n(tage)31 b(of)h(the)h(fact)g(that)f(the)456 4015 y(v)n(ector)26 b(\014eld)i Fq(Y)933 4027 y Fi(\026;")1056 4015 y Fu(is)g(Hamiltonian)f(and)h(autonomous.)605 4167 y Ft(Pr)n(oposition)23 b Fu(3.3)p Ft(.)33 b Fs(Consider)25 b(the)e(system)g Fq(z)2084 4137 y Fj(0)2129 4167 y Fu(=)g Fq(Y)2265 4179 y Fi(\026;")2361 4167 y Fu(\()p Fq(z)t Fu(\))f Fs(with)i Fq(\026)f Fs(and)g Fq(")g Fs(smal)t(l)g(enough.)456 4267 y(Then)679 4386 y Fu(1\))41 b Fs(The)31 b(system)f Fq(z)1279 4356 y Fj(0)1325 4386 y Fu(=)23 b Fq(Y)1461 4398 y Fi(\026;")1557 4386 y Fu(\()p Fq(z)t Fu(\))29 b Fs(has)i(lo)l(c)l(al)g(stable)f(manifold.)41 b(Mor)l(e)l(over,)32 b(given)f Fq(\034)i(>)23 b Fu(0)794 4497 y Fs(ther)l(e)31 b(exists)f Fq(T)35 b(>)24 b Fu(0)29 b Fs(such)i(that)f(it)h(c)l(an)f(b) l(e)h(r)l(epr)l(esente)l(d)f(by)k Fu(^)-45 b Fq(\015)t Fu(\()p Fq(u)p Fu(\))25 b(=)f(\()8 b(^)-50 b Fq(\013)p Fu(\()p Fq(u)p Fu(\))p Fq(;)3235 4475 y Fu(^)3223 4497 y Fq(\014)5 b Fu(\()p Fq(u)p Fu(\)\))p Fs(,)794 4597 y Fq(u)23 b Fp(2)g Fq(D)r Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))p Fs(,)31 b(with)814 4785 y Fu(^)-50 b Fq(\013)p Fu(\()p Fq(u)p Fu(\))23 b(=)1204 4729 y Fq(c)1240 4741 y Fo(1)p 1092 4766 297 4 v 1092 4844 a Fq(u)1140 4820 y Fo(2)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))1412 4668 y Fg(\022)1473 4785 y Fu(1)18 b(+)g Fq(O)1696 4668 y Fg(\022)1791 4729 y Fu(1)p 1767 4766 89 4 v 1767 4842 a Fq(u)1815 4818 y Fi(\027)1866 4668 y Fg(\023\023)2002 4785 y Fs(,)119 b Fq(c)2182 4750 y Fi(n)p Fj(\000)p Fo(2)2182 4807 y(1)2335 4785 y Fu(=)2572 4729 y Fp(\000)p Fu(2)p 2433 4766 384 4 v 2433 4842 a Fq(a)2477 4854 y Fi(n)2522 4842 y Fu(\()p Fq(n)18 b Fp(\000)g Fu(2\))2779 4818 y Fo(2)2845 4785 y Fu(+)g Fq(O)r Fu(\()p Fq(\026)3075 4751 y Fo(2)3113 4785 y Fq(")3152 4751 y Fi(p)p Fo(+2)3274 4785 y Fu(\))794 4969 y Fs(and)992 5115 y Fu(^)980 5137 y Fq(\014)t Fu(\()p Fq(u)p Fu(\))23 b(=)1380 5081 y Fq(c)1416 5093 y Fo(2)p 1264 5118 305 4 v 1264 5195 a Fq(u)1312 5171 y Fi(n=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))1593 5020 y Fg(\022)1654 5137 y Fu(1)18 b(+)g Fq(O)1876 5020 y Fg(\022)1971 5081 y Fu(1)p 1947 5118 89 4 v 1947 5194 a Fq(u)1995 5170 y Fi(\027)2046 5020 y Fg(\023\023)2182 5137 y Fs(,)179 b Fq(c)2422 5149 y Fo(2)2482 5137 y Fu(=)23 b Fq(c)2606 5149 y Fo(1)2728 5081 y Fp(\000)p Fu(2)p 2653 5118 258 4 v 2653 5194 a(\()p Fq(n)18 b Fp(\000)g Fu(2\))p eop %%Page: 26 34 26 33 bop 456 251 a Fo(26)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)794 450 y Fs(wher)l(e)31 b Fq(\027)d(>)23 b Fu(0)p Fs(.)679 550 y Fu(2\))41 b Fs(Ther)l(e)31 b(exists)e Fq(M)j(>)22 b Fu(0)30 b Fs(such)g(that)f(for)i Fq(u)22 b Fp(2)i Fq(D)r Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))818 749 y Fp(j)f Fu(^)-50 b Fq(\013)q Fu(\()p Fq(u)p Fu(\))19 b Fp(\000)f Fq(\013)1162 761 y Fo(0)1199 749 y Fu(\()p Fq(u)p Fu(\))p Fp(j)23 b(\024)g Fq(M)1592 693 y(\026)1642 663 y Fo(2)1679 693 y Fq(")1718 663 y Fi(p)p Fo(+2)p 1544 730 343 4 v 1544 807 a Fp(j)p Fq(u)p Fp(j)1638 784 y Fo(2)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))1897 749 y Fs(,)268 b Fp(j)2225 727 y Fu(^)2213 749 y Fq(\014)t Fu(\()p Fq(u)p Fu(\))19 b Fp(\000)f Fq(\014)2525 761 y Fo(0)2562 749 y Fu(\()p Fq(u)p Fu(\))p Fp(j)23 b(\024)g Fq(M)2959 693 y(\026)3009 663 y Fo(2)3046 693 y Fq(")3085 663 y Fi(p)p Fo(+2)p 2907 730 351 4 v 2907 807 a Fp(j)p Fq(u)p Fp(j)3001 784 y Fi(n=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))3268 749 y Fs(.)605 961 y Ft(Pr)n(oof.)41 b Fu(1\))24 b(Since)h Fq(Y)1293 973 y Fi(\026;")1389 961 y Fu(\()p Fq(z)t Fu(\))e(=)g Fq(J)8 b Fu([)p Fq(D)r(h)1803 973 y Fo(0)1853 961 y Fu(+)13 b Fq(\026)1981 930 y Fo(2)2017 961 y Fq(")2056 930 y Fi(p)p Fo(+2)2179 961 y Fq(D)r(R)2313 973 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2472 961 y Fu(])25 b(where)f Fq(J)33 b Fu(is)24 b(the)i(usual)e(sym-)456 1060 y(plectic)k(matrix,)f(w)n(e)g (kno)n(w)g(that,)h(if)g(the)g(stable)f(manifold)h(exists,)f(it)h(is)f (con)n(tained)g(in)1364 1206 y Fq(h)1412 1218 y Fo(0)1449 1206 y Fu(\()p Fq(z)t Fu(\))19 b(+)f Fq(\026")1747 1172 y Fi(p)p Fo(+2)1869 1206 y Fq(R)1932 1218 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2091 1206 y Fu(\()p Fq(z)t(;)c(\026;)g(")p Fu(\))22 b(=)h(0.)-2080 b(\(3.18\))456 1357 y(W)-7 b(e)39 b(kno)n(w)g(that)g(when)h Fq(\026)i Fu(=)g(0)d(the)h(lo)r(cal)e(stable) h(manifold)h(is)f(giv)n(en)f(b)n(y)h Fq(y)46 b Fu(=)c Fq(f)3194 1369 y Fo(0)3230 1357 y Fu(\()p Fq(z)t Fu(\))h(=)456 1464 y Fp(\000)521 1393 y Fg(p)p 603 1393 285 4 v 603 1464 a Fp(\000)p Fu(2)p Fq(V)18 b Fu(\()p Fq(x)p Fu(\))q(,)37 b(whic)n(h)d(is)h(de\014ned)g(in)g(\012\()p Fq(\031)s(;)14 b(r)r Fu(\).)59 b(F)-7 b(or)34 b Fq(\026)h Fp(6)p Fu(=)f(0)g(w)n(e)h (lo)r(ok)f(for)g(the)h(lo)r(cal)f(stable)456 1564 y(manifold)27 b(in)h(the)g(form)1433 1710 y Fq(y)e Fu(=)c Fq(f)1628 1722 y Fo(0)1665 1710 y Fu(\()p Fq(x)p Fu(\))e(+)e Fq(\026)1929 1676 y Fo(2)1966 1710 y Fq(")2005 1676 y Fi(p)p Fo(+2)2127 1710 y Fq(g)s Fu(\()p Fq(x;)c(\026;)g(")p Fu(\).)456 1861 y(Putting)27 b(this)h(expression)e(in)n(to)i(\(3.18\))e(w)n(e)i (get)f(the)h(follo)n(wing)e(equation)i(for)f Fq(g)584 2165 y(g)s Fu(\()p Fq(x;)14 b(\026;)g(")p Fu(\))456 1999 y(\(3.19\))729 2356 y(=)1420 2300 y(2)p Fq(R)1525 2312 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1684 2300 y Fu(\()p Fq(x;)g(f)1841 2312 y Fo(0)1878 2300 y Fu(\()p Fq(x)p Fu(\))20 b(+)e Fq(\026)2142 2270 y Fo(2)2179 2300 y Fq(")2218 2270 y Fi(p)p Fo(+2)2340 2300 y Fq(g)s Fu(\()p Fq(x;)c(\026;)g(")p Fu(\)\))p 827 2337 2457 4 v 827 2353 a Fg(p)p 910 2353 285 4 v 71 x Fp(\000)p Fu(2)p Fq(V)k Fu(\()p Fq(x)p Fu(\))h(+)1296 2353 y Fg(p)p 1379 2353 1905 4 v 71 x Fp(\000)p Fu(2)p Fq(V)f Fu(\()p Fq(x)p Fu(\))i Fp(\000)e Fq(\026)1816 2400 y Fo(2)1853 2424 y Fq(")1892 2400 y Fi(p)p Fo(+2)2014 2424 y Fu(2)p Fq(R)2119 2436 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2277 2424 y Fu(\()p Fq(x;)c(f)2434 2436 y Fo(0)2472 2424 y Fu(\()p Fq(x)p Fu(\))19 b(+)f Fq(\026)2735 2400 y Fo(2)2773 2424 y Fq(")2812 2400 y Fi(p)p Fo(+2)2934 2424 y Fq(g)s Fu(\()p Fq(x;)c(\026;)g(")p Fu(\)\))3293 2356 y(.)456 2567 y(T)-7 b(o)23 b(solv)n(e)f(this)h(equation,)h(w)n(e)f (in)n(tro)r(duce)g(the)h(domain)f(\012)2262 2579 y Fo(1)2322 2567 y Fu(=)g(\012)2470 2579 y Fo(1)2507 2567 y Fu(\()p Fq(\031)s(;)14 b(r)n(;)g(\026)2748 2579 y Fo(0)2786 2567 y Fq(;)g(")2862 2579 y Fo(0)2899 2567 y Fu(\))23 b(and)g(the)h(space) 576 2862 y(\006)f(=)746 2745 y Fg(\032)809 2862 y Fq(g)i Fu(:)e(\012)980 2874 y Fo(1)1041 2862 y Fp(!)g Fl(C)44 b Fu(:)51 b(con)n(tin)n(uous,)26 b(analytic)h(in)h(\()p Fq(x;)14 b(\026)p Fu(\),)43 b(sup)2441 2932 y Fo(\012)2488 2940 y Fh(1)2592 2806 y Fp(j)p Fq(g)s Fu(\()p Fq(x;)14 b(\026;)g(")p Fu(\))p Fp(j)p 2567 2843 413 4 v 2567 2921 a(j)p Fq(x)p Fp(j)2660 2897 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)p Fj(\000)p Fi(n=)p Fo(2)3013 2862 y Fq(<)23 b Fu(+)p Fp(1)3249 2745 y Fg(\033)456 3077 y Fu(with)28 b(the)g(norm)f Fp(k)p Fq(g)s Fp(k)1132 3089 y Fo(\006)1205 3077 y Fu(=)c(sup)1418 3098 y Fo(\012)1465 3106 y Fh(1)1515 3077 y Fp(j)p Fq(g)s Fu(\()p Fq(x;)14 b(\026;)g(")p Fu(\))p Fp(j)p Fq(=)p Fp(j)p Fq(x)p Fp(j)2013 3047 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)p Fj(\000)p Fi(n=)p Fo(2)2333 3077 y Fu(.)605 3177 y(Let)28 b(\000)f(b)r(e)h(the)g(op)r(erator)e(suc)n(h)h(that)h(\000)p Fq(g)i Fu(is)d(de\014ned)h(b)n(y)f(the)h(righ)n(t)f(hand)g(side)h(of)f (\(3.19\).)456 3277 y(It)i(is)g(easy)f(to)h(see)f(that)i(there)e (exists)h(a)g(closed)f(ball)h(\006)2220 3289 y Fi(M)2294 3277 y Fu(,)g(of)g(radius)f Fq(M)9 b Fu(,)29 b(in)h(\006)f(suc)n(h)f (that)i(\000)456 3376 y(has)d(a)g(\014xed)g(p)r(oin)n(t)h(in)g(\006) 1247 3388 y Fi(M)1321 3376 y Fu(,)g(whic)n(h)f(pro)n(vides)f(us)i(with) g(the)g(solution)f(w)n(e)g(w)n(ere)g(lo)r(oking)f(for.)605 3476 y(Restricting)g(the)h(di\013eren)n(tial)g(equation)f(to)g(the)h (lo)r(cal)f(manifold)h(w)n(e)f(\014nd)h(the)g(equation)1413 3622 y Fq(x)1460 3588 y Fj(0)1507 3622 y Fu(=)c Fq(f)1636 3634 y Fo(0)1672 3622 y Fu(\()p Fq(x)p Fu(\))d(+)e Fq(\026)1936 3588 y Fo(2)1973 3622 y Fq(")2012 3588 y Fi(p)p Fo(+2)2134 3622 y Fq(f)2175 3634 y Fo(1)2212 3622 y Fu(\()p Fq(x;)c(\026;)g(")p Fu(\))-2030 b(\(3.20\))456 3768 y(where)948 3915 y Fq(f)989 3927 y Fo(1)1026 3915 y Fu(\()p Fq(x)p Fu(\))24 b(=)e Fq(g)s Fu(\()p Fq(x;)14 b(\026;)g(")p Fu(\))19 b(+)f Fq(@)1711 3927 y Fi(y)1751 3915 y Fq(R)1814 3927 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1973 3915 y Fu(\()p Fq(x;)c(f)2130 3927 y Fo(0)2167 3915 y Fu(\()p Fq(x)p Fu(\))20 b(+)e Fq(\026)2431 3880 y Fo(2)2468 3915 y Fq(")2507 3880 y Fi(p)p Fo(+2)2629 3915 y Fq(g)s Fu(\()p Fq(x;)c(\026;)g(")p Fu(\))p Fq(;)g(\026;)g(")p Fu(\).)456 4066 y(No)n(w)24 b(applying)h(Prop)r(osition)e(2.1)h(of)h(Chapter)g(2)g(w)n(e)f(get)h (that)h(a)e(solution)h(of)g(\(3.20\))f(is)h(giv)n(en)456 4165 y(b)n(y)1013 4357 y(^)-50 b Fq(\013)p Fu(\()p Fq(u)p Fu(\))23 b(=)1402 4301 y Fq(c)1438 4313 y Fo(1)p 1291 4338 297 4 v 1291 4415 a Fq(u)1339 4391 y Fo(2)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))1611 4240 y Fg(\022)1672 4357 y Fu(1)18 b(+)g Fq(O)1895 4240 y Fg(\022)1990 4301 y Fu(1)p 1966 4338 89 4 v 1966 4414 a Fq(u)2014 4390 y Fi(\027)2065 4240 y Fg(\023\023)2201 4357 y Fu(,)249 b Fq(u)22 b Fp(2)i Fq(D)r Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))456 4548 y(with)28 b Fq(\027)g(>)23 b Fu(0)k(and)g Fq(c)1068 4560 y Fo(1)1133 4548 y Fu(dep)r(ending)h(on)f Fq(\026)p Fu(,)h Fq(")g Fu(and)f(satisfying)1358 4729 y Fq(c)1394 4693 y Fi(n)p Fj(\000)p Fo(2)1394 4751 y(1)1547 4729 y Fu(=)1784 4672 y Fp(\000)p Fu(2)p 1645 4710 384 4 v 1645 4786 a Fq(a)1689 4798 y Fi(n)1734 4786 y Fu(\()p Fq(n)19 b Fp(\000)f Fu(2\))1992 4762 y Fo(2)2057 4729 y Fu(+)g Fq(O)r Fu(\()p Fq(\026)2287 4694 y Fo(2)2325 4729 y Fq(")2364 4694 y Fi(p)p Fo(+2)2486 4729 y Fu(\).)467 4919 y(^)456 4941 y Fq(\014)t Fu(\()p Fq(u)p Fu(\))28 b(is)f(obtained)h(as)1185 4919 y(^)1173 4941 y Fq(\014)5 b Fu(\()p Fq(u)p Fu(\))23 b(=)g Fq(f)1489 4953 y Fo(0)1525 4941 y Fu(\()8 b(~)-50 b Fq(\013)q Fu(\()p Fq(u)p Fu(\)\))19 b(+)f Fq(\026)1907 4911 y Fo(2)1944 4941 y Fq(")1983 4911 y Fi(p)p Fo(+2)2105 4941 y Fq(g)s Fu(\()8 b(~)-50 b Fq(\013)q Fu(\()p Fq(u)p Fu(\))p Fq(;)14 b(\026;)g(")p Fu(\))27 b(and)h(hence)1349 5115 y(^)1337 5137 y Fq(\014)5 b Fu(\()p Fq(u)p Fu(\))23 b(=)1737 5081 y Fq(c)1773 5093 y Fo(2)p 1622 5118 305 4 v 1622 5195 a Fq(u)1670 5171 y Fi(n=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))1950 5020 y Fg(\022)2011 5137 y Fu(1)18 b(+)g Fq(O)2234 5020 y Fg(\022)2328 5081 y Fu(1)p 2305 5118 89 4 v 2305 5194 a Fq(u)2353 5170 y Fi(\027)2403 5020 y Fg(\023)q(\023)2539 5137 y Fu(,)p eop %%Page: 27 35 27 34 bop 1222 251 a Fo(3.4.)29 b(ESTIMA)-5 b(TES)28 b(F)n(OR)h(THE)h(POINCAR)2428 236 y(\023)2422 251 y(E)e(MAP)729 b(27)456 450 y Fu(where)27 b Fq(\027)h(>)23 b Fu(0)k(and)1671 628 y Fq(c)1707 640 y Fo(2)1768 628 y Fu(=)22 b Fq(c)1891 640 y Fo(1)2014 571 y Fp(\000)p Fu(2)p 1938 609 258 4 v 1938 685 a(\()p Fq(n)d Fp(\000)f Fu(2\))2206 628 y(.)456 832 y(All)31 b(solutions)f(ha)n(v)n(e)g(the)i(form)h(^)-45 b Fq(\015)5 b Fu(\()p Fq(u)21 b Fp(\000)f Fq(u)1768 844 y Fo(0)1805 832 y Fu(\))31 b(and)g(ha)n(v)n(e)f(the)h(same)g (asymptotic)f(expressions.)456 932 y(W)-7 b(e)28 b(c)n(ho)r(ose)e Fq(u)910 944 y Fo(0)974 932 y Fu(suc)n(h)i(that)36 b(^)-50 b Fq(\013)p Fu(\()p Fq(T)12 b Fu(\))23 b(=)f Fq(\013)1683 944 y Fo(0)1721 932 y Fu(\()p Fq(T)12 b Fu(\).)605 1031 y(2\))28 b(W)-7 b(e)27 b(de\014ne)h Fq(\030)t Fu(\()p Fq(u)p Fu(\))c(=)31 b(^)-51 b Fq(\013)q Fu(\()p Fq(u)p Fu(\))18 b Fp(\000)g Fq(\013)1672 1043 y Fo(0)1710 1031 y Fu(\()p Fq(t)p Fu(\).)37 b(Since)45 b(_)-40 b Fq(\013)2134 1043 y Fo(0)2171 1031 y Fu(\()p Fq(u)p Fu(\))23 b(=)g Fq(f)2435 1043 y Fo(0)2472 1031 y Fu(\()p Fq(\013)2557 1043 y Fo(0)2595 1031 y Fu(\()p Fq(u)p Fu(\)\))28 b(w)n(e)f(ha)n(v)n(e) f(that)733 1172 y(_)715 1194 y Fq(\030)87 b Fu(=)c Fq(f)1027 1206 y Fo(0)1064 1194 y Fu(\()8 b(^)-50 b Fq(\013)q Fu(\))18 b Fp(\000)g Fq(f)1324 1206 y Fo(0)1361 1194 y Fu(\()p Fq(\013)1446 1206 y Fo(0)1484 1194 y Fu(\))h(+)f Fq(\026)1668 1160 y Fo(2)1705 1194 y Fq(")1744 1160 y Fi(p)p Fo(+2)1866 1194 y Fq(f)1907 1206 y Fo(1)1944 1194 y Fu(\()8 b(^)-50 b Fq(\013)q(;)14 b(\026;)g(")p Fu(\))838 1320 y(=)83 b Fq(D)r(f)1098 1332 y Fo(0)1135 1320 y Fu(\()p Fq(\013)1220 1332 y Fo(0)1258 1320 y Fu(\))p Fq(\030)23 b Fu(+)18 b([)p Fq(f)1496 1332 y Fo(0)1533 1320 y Fu(\()8 b(^)-50 b Fq(\013)p Fu(\))19 b Fp(\000)f Fq(f)1793 1332 y Fo(0)1830 1320 y Fu(\()p Fq(\013)1915 1332 y Fo(0)1952 1320 y Fu(\))h Fp(\000)f Fq(D)r(f)2198 1332 y Fo(0)2235 1320 y Fu(\()p Fq(\013)2320 1332 y Fo(0)2358 1320 y Fu(\))p Fq(\030)t Fu(])h(+)f Fq(\026)2605 1286 y Fo(2)2642 1320 y Fq(")2681 1286 y Fi(p)p Fo(+2)2803 1320 y Fq(f)2844 1332 y Fo(1)2881 1320 y Fu(\()8 b(^)-50 b Fq(\013)q(;)14 b(\026;)g(")p Fu(\).)456 1483 y(Since)23 b Fq(\030)t Fu(\()p Fq(u)p Fu(\))g(=)40 b(_)-41 b Fq(\013)983 1495 y Fo(0)1021 1483 y Fu(\()p Fq(u)p Fu(\))23 b(=)g Fq(f)1285 1495 y Fo(0)1322 1483 y Fu(\()p Fq(\013)1407 1495 y Fo(0)1444 1483 y Fu(\()p Fq(u)p Fu(\)\))g(is)g(a)f(solution)h(of)2171 1461 y(_)2153 1483 y Fq(\030)28 b Fu(=)22 b Fq(D)r(f)2416 1495 y Fo(0)2453 1483 y Fu(\()p Fq(\013)2538 1495 y Fo(0)2576 1483 y Fu(\))p Fq(\030)t Fu(,)i(applying)e(the)h(form)n(ula)456 1582 y(of)k(v)-5 b(ariation)27 b(of)g(parameters)f(and)h(using)h(that)g Fq(\030)t Fu(\()p Fq(T)12 b Fu(\))22 b(=)h(0)k(w)n(e)g(can)h(write,)531 1788 y Fq(\030)t Fu(\()p Fq(u)p Fu(\))84 b(=)e Fq(f)955 1800 y Fo(0)992 1788 y Fu(\()p Fq(\013)1077 1800 y Fo(0)1115 1788 y Fu(\()p Fq(u)p Fu(\)\))1273 1671 y Fg(\024)1317 1675 y(Z)1400 1695 y Fi(u)1363 1864 y(T)1614 1732 y Fu(1)p 1467 1769 337 4 v 1467 1845 a Fq(f)1508 1857 y Fo(0)1545 1845 y Fu(\()p Fq(\013)1630 1857 y Fo(0)1667 1845 y Fu(\()p Fq(s)p Fu(\)\))1813 1788 y([)p Fq(f)1877 1800 y Fo(0)1914 1788 y Fu(\()8 b(^)-50 b Fq(\013)q Fu(\()p Fq(s)p Fu(\)\))19 b Fp(\000)f Fq(f)2278 1800 y Fo(0)2315 1788 y Fu(\()p Fq(\013)2400 1800 y Fo(0)2438 1788 y Fu(\()p Fq(s)p Fu(\)\))h Fp(\000)f Fq(D)r(f)2787 1800 y Fo(0)2824 1788 y Fu(\()p Fq(\013)2909 1800 y Fo(0)2946 1788 y Fu(\()p Fq(s)p Fu(\)\))p Fq(\030)t Fu(\()p Fq(s)p Fu(\)])c Fq(ds)924 2020 y Fu(+)p Fq(\026)1039 1986 y Fo(2)1076 2020 y Fq(")1115 1986 y Fi(p)p Fo(+2)1251 1907 y Fg(Z)1334 1928 y Fi(u)1297 2096 y(T)1549 1964 y Fu(1)p 1401 2001 V 1401 2077 a Fq(f)1442 2089 y Fo(0)1479 2077 y Fu(\()p Fq(\013)1564 2089 y Fo(0)1602 2077 y Fu(\()p Fq(s)p Fu(\)\))1747 2020 y Fq(f)1788 2032 y Fo(1)1825 2020 y Fu(\()8 b(^)-50 b Fq(\013)q Fu(\()p Fq(s)p Fu(\))p Fq(;)14 b(\026;)g(")p Fu(\))g Fq(ds)2305 1903 y Fg(\025)2363 2020 y Fu(.)-1930 b(\(3.21\))456 2238 y(W)-7 b(e)36 b(ha)n(v)n(e)e(the)i(a)f(priori)f(estimate)i Fp(j)p Fq(\030)t Fu(\()p Fq(u)p Fu(\))p Fp(j)g(\024)g(j)8 b Fu(^)-50 b Fq(\013)q Fu(\()p Fq(u)p Fu(\))p Fp(j)24 b Fu(+)f Fp(j)p Fq(\013)2344 2250 y Fo(0)2381 2238 y Fu(\()p Fq(u)p Fu(\))p Fp(j)37 b(\024)f Fq(C)2713 2250 y Fo(1)2750 2238 y Fq(=)p Fp(j)p Fq(u)p Fp(j)2886 2207 y Fo(2)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))3170 2238 y Fu(for)f Fq(u)h Fp(2)456 2337 y Fq(D)r Fu(\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\).)37 b(Then)28 b(w)n(e)f(also)g(ha)n(v)n (e)777 2535 y Fp(j)p Fq(D)871 2501 y Fo(2)908 2535 y Fq(f)949 2547 y Fo(0)986 2535 y Fu(\()p Fq(\013)1071 2547 y Fo(0)1109 2535 y Fu(\()p Fq(s)p Fu(\))19 b(+)f Fq(\020)6 b(\030)t Fu(\()p Fq(s)p Fu(\)\))p Fp(j)24 b(\024)1867 2479 y Fq(C)1926 2491 y Fo(2)p 1676 2516 480 4 v 1676 2594 a Fp(j)p Fq(s)p Fp(j)1761 2570 y Fo(\()p Fi(n)p Fj(\000)p Fo(4\))p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2303 2535 y Fu(and)138 b Fp(j)p Fq(f)2639 2547 y Fo(1)2676 2535 y Fu(\()8 b(^)-50 b Fq(\013)q Fu(\()p Fq(s)p Fu(\))p Fq(;)14 b(\026;)g(")p Fu(\))p Fp(j)23 b(\024)3216 2479 y Fq(C)3275 2491 y Fo(3)p 3204 2516 122 4 v 3204 2592 a Fp(j)p Fq(s)p Fp(j)3289 2568 y Fi(q)456 2775 y Fu(with)28 b Fq(q)e Fu(=)c(min)948 2683 y Fg(n)1013 2735 y Fo(2\(2)p Fi(k)q Fj(\000)p Fo(2)p Fj(\000)p Fi(n=)p Fo(2\))p 1013 2756 400 4 v 1150 2803 a Fi(n)p Fj(\000)p Fo(2)1423 2775 y Fq(;)1470 2735 y Fo(2\(2)p Fi(n)p Fj(\000)p Fo(3\))p 1470 2756 245 4 v 1529 2803 a Fi(n)p 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Fo(1)1766 3918 y(0)1840 3842 y Fu(\(1)g Fp(\000)g Fq(\020)6 b Fu(\))p Fq(D)2160 3808 y Fo(2)2198 3842 y Fq(f)2239 3854 y Fo(0)2276 3842 y Fu(\()p Fq(\013)2361 3854 y Fo(0)2399 3842 y Fu(\()p Fq(s)p Fu(\))18 b(+)g Fq(\020)6 b(\030)t Fu(\()p Fq(s)p Fu(\)\))p Fq(\030)2860 3808 y Fo(2)2899 3842 y Fu(\()p Fq(s)p Fu(\))14 b Fq(d\020)3101 3722 y Fg(\014)3101 3772 y(\014)3101 3822 y(\014)3101 3872 y(\014)1604 4067 y Fp(\024)1893 4011 y Fq(C)1952 4023 y Fo(5)p 1702 4048 480 4 v 1702 4126 a Fp(j)p Fq(s)p Fp(j)1787 4102 y Fo(\()p Fi(n)p Fj(\000)p Fo(4\))p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2191 4067 y Fq(\026)2241 4033 y Fo(4)2278 4067 y Fq(")2317 4033 y Fo(2)p Fi(p)p Fo(+4)2472 4067 y Fq(M)2562 4033 y Fo(2)2755 4011 y Fu(1)p 2609 4048 335 4 v 2609 4126 a Fp(j)p Fq(s)p Fp(j)2694 4102 y Fo(4)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2953 4067 y Fu(.)456 4274 y(Putting)27 b(the)h(previous)f(b)r(ounds)g(in)h (\(3.21\))f(w)n(e)g(obtain)728 4558 y Fp(j)p Fq(\030)t Fu(\()p Fq(u)p Fu(\))p Fp(j)84 b(\024)1295 4502 y Fq(C)1354 4514 y Fo(4)p 1167 4539 351 4 v 1167 4617 a Fp(j)p Fq(u)p Fp(j)1261 4593 y Fi(n=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))1542 4441 y Fg(\024)1586 4445 y(Z)1669 4466 y Fi(u)1632 4634 y(T)1764 4502 y Fu(1)p 1736 4539 97 4 v 1755 4604 a(~)1736 4625 y Fq(C)1795 4637 y Fo(4)1843 4558 y Fp(j)p Fq(s)p Fp(j)1928 4524 y Fi(n=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2185 4558 y Fq(C)2244 4570 y Fo(5)2281 4558 y Fq(\026)2331 4524 y Fo(4)2369 4558 y Fq(")2408 4524 y Fo(2)p Fi(p)p Fo(+4)2563 4558 y Fq(M)2653 4524 y Fo(2)2850 4502 y Fu(1)p 2700 4539 343 4 v 2700 4617 a Fp(j)p Fq(s)p Fp(j)2785 4593 y Fi(n=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))3066 4558 y Fq(ds)1167 4791 y Fu(+)p Fq(\026)1282 4757 y Fo(2)1319 4791 y Fq(")1358 4757 y Fi(p)p Fo(+2)1494 4678 y Fg(Z)1577 4699 y Fi(u)1540 4867 y(T)1672 4735 y Fu(1)p 1644 4772 97 4 v 1663 4838 a(~)1644 4859 y Fq(C)1703 4871 y Fo(4)1751 4791 y Fp(j)p Fq(s)p Fp(j)1836 4757 y Fi(n=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2093 4791 y Fq(C)2152 4803 y Fo(3)2240 4735 y Fu(1)p 2200 4772 122 4 v 2200 4848 a Fp(j)p Fq(s)p Fp(j)2285 4824 y Fi(q)2345 4791 y Fq(ds)2427 4674 y Fg(\025)1010 5014 y Fp(\024)e Fq(\026)1207 4980 y Fo(2)1245 5014 y Fq(")1284 4980 y Fi(p)p Fo(+2)1406 5014 y Fu([)p Fq(\026)1479 4980 y Fo(2)1516 5014 y Fq(")1555 4980 y Fi(p)p Fo(+2)1677 5014 y Fq(C)1736 5026 y Fo(6)1774 5014 y Fq(M)1864 4980 y Fo(2)1919 5014 y Fu(+)18 b Fq(C)2061 5026 y Fo(7)2099 5014 y Fu(])2283 4958 y(1)p 2132 4995 343 4 v 2132 5073 a Fp(j)p Fq(u)p Fp(j)2226 5049 y Fo(2)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))605 5216 y Fu(with)28 b Fq(C)853 5228 y Fo(7)918 5216 y Fu(indep)r(enden)n(t)h(of)e Fq(M)9 b Fu(.)p eop %%Page: 28 36 28 35 bop 456 251 a Fo(28)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(W)-7 b(e)28 b(c)n(ho)r(ose)f Fq(M)32 b Fu(=)23 b(2)p Fq(C)1314 462 y Fo(7)1351 450 y Fu(,)28 b(then,)g(since)g Fq(M)k(>)23 b(C)2078 462 y Fo(7)2143 450 y Fu(and)28 b Fq(\026)p Fu(,)g Fq(")f Fu(are)g(small)h(w)n(e)f(get)h(that)g(for)f Fq(u)456 550 y Fu(suc)n(h)e(that)g(Re)14 b Fq(u)22 b Fu(=)h Fq(t)1118 562 y Fo(1)1155 550 y Fu(,)j Fp(j)p Fq(\030)t Fu(\()p Fq(u)p Fu(\))p Fp(j)e Fq(<)e(\026)1563 520 y Fo(2)1601 550 y Fq(")1640 520 y Fi(p)p Fo(+2)1762 550 y Fq(M)t(=)p Fp(j)p Fq(u)p Fp(j)1983 520 y Fo(2)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2256 550 y Fu(whic)n(h)j(is)h(a) e(con)n(tradiction)g(with)i(the)456 649 y(de\014nition)i(of)f Fq(t)949 661 y Fo(1)986 649 y Fu(.)37 b(Therefore,)27 b(for)g Fq(u)22 b Fp(2)i Fu(\012\()p Fq(T)7 b(;)14 b(\034)9 b Fu(\))1434 842 y Fp(j)p Fq(\030)t Fu(\()p Fq(u)p Fu(\))p Fp(j)23 b(\024)g Fq(\026)1793 807 y Fo(2)1830 842 y Fq(")1869 807 y Fi(p)p Fo(+2)1991 842 y Fq(M)2241 785 y Fu(1)p 2091 823 343 4 v 2091 900 a Fp(j)p Fq(u)p Fp(j)2185 876 y Fo(2)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2443 842 y Fu(.)456 1062 y(The)k(estimate)h(for)1098 1040 y(^)1086 1062 y Fq(\014)23 b Fp(\000)18 b Fq(\014)1286 1074 y Fo(0)1351 1062 y Fu(follo)n(ws)26 b(from)1175 1198 y(^)1163 1220 y Fq(\014)d Fp(\000)18 b Fq(\014)1363 1232 y Fo(0)1423 1220 y Fu(=)23 b Fq(f)1552 1232 y Fo(0)1589 1220 y Fu(\()8 b(~)-50 b Fq(\013)p Fu(\))19 b(+)f Fq(\026)1858 1186 y Fo(2)1895 1220 y Fq(")1934 1186 y Fi(p)p Fo(+2)2057 1220 y Fq(g)s Fu(\()8 b(^)-50 b Fq(\013;)14 b(\026;)g(")p Fu(\))k Fp(\000)g Fq(f)2522 1232 y Fo(0)2559 1220 y Fu(\()p Fq(\013)2644 1232 y Fo(0)2682 1220 y Fu(\).)3380 1373 y Fe(\003)1371 1596 y Fk(3.5.)46 b(The)32 b(op)s(erators)g Fq(B)k Fk(and)c Fp(B)605 1745 y Fu(The)d(Banac)n(h)f(spaces)g(w)n(e)g (use)h(in)g(this)g(section)g(w)n(ere)f(in)n(tro)r(duced)h(at)f(the)i(b) r(eginning)f(of)456 1845 y(Section)e(3.2.)605 1945 y(W)-7 b(e)28 b(will)g(need)g(the)g(op)r(erator)d Fq(B)1639 1957 y Fi(k)1703 1945 y Fu(:)e Fp(X)1820 1914 y Fi(s)1808 1968 y(k)1879 1945 y Fp(\000)-14 b(!)23 b(X)2107 1914 y Fi(s)2095 1968 y(k)2171 1945 y Fu(de\014ned)28 b(b)n(y)1313 2098 y(\()p Fq(B)1408 2110 y Fi(k)1449 2098 y Fq(\033)s Fu(\)\()p Fq(t;)14 b(s)p Fu(\))24 b(=)e Fq(\033)s Fu(\()p Fq(t)e Fu(+)e(2)p Fq(\031)s(";)c(s)p Fu(\))k Fp(\000)g Fq(\033)s Fu(\()p Fq(t;)c(s)p Fu(\))-2131 b(\(3.22\))456 2256 y(with)27 b Fq(")c(>)f Fu(0.)37 b(It)27 b(is)g(a)f(w)n(ell)h (de\014ned)g(linear)f(op)r(erator)g(with)h Fp(k)p Fq(B)2446 2268 y Fi(k)2486 2256 y Fp(k)c(\024)g Fu(2.)36 b(Indeed,)27 b(it)h(is)e(readily)456 2356 y(seen)h(that)h(if)g Fq(\033)e Fp(2)e(X)1115 2326 y Fi(s)1103 2379 y(k)1178 2356 y Fu(then)k Fq(B)1430 2368 y Fi(k)1471 2356 y Fq(\033)f Fp(2)c(X)1694 2326 y Fi(s)1682 2379 y(k)1758 2356 y Fu(and)k(that)502 2516 y(\()p Fq(t)18 b Fu(+)g(Re)c Fq(s)p Fu(\))848 2481 y Fi(k)889 2516 y Fp(j)p Fu(\()p Fq(B)1007 2528 y Fi(k)1048 2516 y Fq(\033)s Fu(\)\()p Fq(t;)g(s)p Fu(\))p Fp(j)84 b(\024)f Fu(\()p Fq(t)19 b Fu(+)f(Re)c Fq(s)p Fu(\))1902 2481 y Fi(k)1943 2516 y Fp(j)p Fq(\033)s Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\))p Fp(j)k Fu(+)g(\()p Fq(t)h Fu(+)f(Re)c Fq(s)p Fu(\))2890 2481 y Fi(k)2931 2516 y Fp(j)p Fq(\033)s Fu(\()p Fq(t;)g(s)p Fu(\))p Fp(j)1407 2710 y(\024)83 b Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(")g Fu(+)g(Re)c Fq(s)p Fu(\))2134 2676 y Fi(k)2175 2710 y Fp(j)p Fq(\033)s Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\))p Fp(j)2688 2593 y Fg(\022)2875 2654 y Fq(t)k Fu(+)g(Re)c Fq(s)p 2759 2691 515 4 v 2759 2767 a(t)k Fu(+)g(2)p Fq(\031)s(")g Fu(+)g(Re)c Fq(s)3283 2593 y Fg(\023)3344 2610 y Fi(k)1555 2897 y Fu(+\()p Fq(t)k Fu(+)g(Re)c Fq(s)p Fu(\))1966 2862 y Fi(k)2007 2897 y Fp(j)p Fq(\033)s Fu(\()p Fq(t;)g(s)p Fu(\))p Fp(j)1407 3021 y(\024)83 b Fu(2)p Fp(k)p Fq(\033)s Fp(k)1731 3033 y Fi(k)1771 3021 y Fu(.)456 3174 y(Of)27 b(course,)g(w)n(e)g(ha)n(v)n(e)f(used)i(that)g Fq(t)18 b Fu(+)g(Re)c Fq(s)23 b Fp(\025)g Fq(T)34 b(>)22 b Fu(0.)605 3334 y Ft(Remark)32 b Fu(3.5)p Ft(.)40 b Fs(In)29 b(fact)i Fp(k)p Fq(B)1516 3346 y Fi(k)1556 3334 y Fp(k)23 b Fu(=)f(2)p Fs(.)38 b(F)-6 b(or)30 b(the)g(function)g Fq(\033)c Fp(2)e(X)2656 3304 y Fi(s)2644 3358 y(k)2721 3334 y Fs(de\014ne)l(d)31 b(by)1240 3530 y Fq(\033)s Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b(=)1787 3473 y(1)p 1582 3510 453 4 v 1582 3586 a(cosh)o(\()p Fq(a=)p Fu(\(2)p Fq(")p Fu(\)\))2171 3473 y(1)p 2054 3510 276 4 v 2054 3586 a(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))2289 3562 y Fi(k)2353 3530 y Fu(sin)2479 3473 y Fq(t)h Fu(+)f Fq(s)p 2479 3510 171 4 v 2524 3586 a Fu(2)p Fq(")456 3731 y Fs(we)30 b(have)h Fp(k)p Fq(\033)s Fp(k)904 3743 y Fi(k)967 3731 y Fu(=)23 b(1)29 b Fs(and)h Fp(k)p Fq(B)1392 3743 y Fi(k)1432 3731 y Fq(\033)s Fp(k)1524 3743 y Fi(k)1588 3731 y Fu(=)23 b(2)p Fs(.)605 3891 y Fu(W)-7 b(e)33 b(are)e(in)n(terested)h(in)g(\014nding)h(a)e(righ)n(t)h (in)n(v)n(erse)f(of)h(the)g(op)r(erator)f Fq(B)2873 3903 y Fi(k)2914 3891 y Fu(.)51 b(F)-7 b(or)31 b(that,)j(w)n(e)456 3991 y(write)27 b Fq(B)731 4003 y Fi(k)772 3991 y Fq(\033)f Fu(=)d Fq( )31 b Fu(from)c(whic)n(h)g(w)n(e)h(can)f(obtain)1350 4144 y Fq(\033)s Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b(=)g Fp(\000)p Fq( )s Fu(\()p Fq(t;)14 b(s)p Fu(\))19 b(+)f Fq(\033)s Fu(\()p Fq(t)h Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\))p Fq(:)-2095 b Fu(\(3.23\))456 4303 y(Applying)27 b(\(3.23\))g(iterativ)n(ely)1001 4537 y Fq(\033)s Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b(=)e Fp(\000)1441 4433 y Fi(N)1411 4458 y Fg(X)1413 4635 y Fi(j)s Fo(=0)1545 4537 y Fq( )s Fu(\()p Fq(t)c Fu(+)g(2)p Fq(\031)s("j;)c(s)p Fu(\))19 b(+)f Fq(\033)s Fu(\()p Fq(t)h Fu(+)f(2)p Fq(\031)s(")p Fu(\()p Fq(N)27 b Fu(+)18 b(1\))p Fq(;)c(s)p Fu(\))p Fq(:)-2443 b Fu(\(3.24\))456 4779 y(If)32 b Fq(\033)i Fp(2)c(X)780 4749 y Fi(s)768 4802 y(k)816 4779 y Fu(,)j(lim)988 4791 y Fi(t)p Fj(!1)1163 4779 y Fq(\033)s Fu(\()p Fq(t;)14 b(s)p Fu(\))31 b(=)f(0)h(so)h(that)g(w)n(e)f(are)g(allo)n(w)n(ed)g(to)h (tak)n(e)f(limit)i(as)e Fq(N)39 b Fp(!)30 b(1)i Fu(in)456 4878 y(\(3.24\))26 b(and)i(w)n(e)f(obtain)g(the)h(formal)f(expression) 1306 5098 y(\()p Fq(B)1405 5062 y Fj(\000)p Fo(1)1401 5123 y Fi(k)1495 5098 y Fq( )s Fu(\)\()p Fq(t;)14 b(s)p Fu(\))23 b(=)g Fp(\000)1970 4994 y Fj(1)1943 5019 y Fg(X)1946 5196 y Fi(j)s Fo(=0)2077 5098 y Fq( )s Fu(\()p Fq(t)c Fu(+)f(2)p Fq(\031)s("j;)c(s)p Fu(\).)-2138 b(\(3.25\))p eop %%Page: 29 37 29 36 bop 1424 251 a Fo(3.5.)29 b(THE)g(OPERA)-5 b(TORS)29 b Fi(B)j Fo(AND)d Fj(B)903 b Fo(29)605 450 y Ft(Lemma)26 b Fu(3.7)p Ft(.)36 b Fs(The)26 b(op)l(er)l(ator)g Fq(B)1629 462 y Fi(k)1693 450 y Fu(:)d Fp(X)1810 420 y Fi(s)1798 474 y(k)1869 450 y Fp(\000)-14 b(!)23 b(X)2097 420 y Fi(s)2085 474 y(k)2158 450 y Fs(has)i(right)h(inverses)g Fq(B)2877 415 y Fj(\000)p Fo(1)2873 475 y Fi(k)2989 450 y Fu(:)d Fp(X)3106 420 y Fi(s)3094 471 y(m)3181 450 y Fp(\000)-15 b(!)24 b(X)3409 420 y Fi(s)3397 474 y(k)456 550 y Fs(with)30 b Fq(m)23 b Fp(\025)g Fq(k)e Fu(+)d(1)29 b Fs(and)1052 725 y Fp(k)p Fq(B)1161 689 y Fj(\000)p Fo(1)1157 750 y Fi(k)1249 725 y Fq( )s Fp(k)1348 737 y Fi(k)1412 725 y Fp(\024)1637 668 y Fu(1)p 1509 705 297 4 v 1509 782 a Fq(T)1570 758 y Fi(m)p Fj(\000)p Fi(k)q Fj(\000)p Fo(1)1830 608 y Fg(\022)1931 668 y Fu(1)p 1901 705 102 4 v 1901 782 a(2)p Fq(T)2031 725 y Fu(+)2309 668 y(1)p 2124 705 411 4 v 2124 782 a(2)p Fq(\031)s(")p Fu(\()p Fq(m)18 b Fp(\000)g Fu(1\))2545 608 y Fg(\023)2620 725 y Fp(k)p Fq( )s Fp(k)2761 737 y Fi(m)2823 725 y Fs(.)456 909 y(In)29 b(p)l(articular,)i(if)g Fq(T)j Fp(\025)23 b Fu(\()p Fq(m)18 b Fp(\000)g Fu(1\))p Fq(\031)s(=)p Fu(4)p Fs(,)1490 1082 y Fp(k)p Fq(B)1599 1047 y Fj(\000)p Fo(1)1595 1108 y Fi(k)1688 1082 y Fq( )s Fp(k)1787 1094 y Fi(k)1850 1082 y Fp(\024)1948 1026 y Fu(1)g(+)g(4)p Fq(")p 1948 1063 224 4 v 1971 1139 a Fu(2)p Fq(\031)s("k)2181 1082 y Fp(k)p Fq( )s Fp(k)2322 1094 y Fi(m)2384 1082 y Fs(.)605 1275 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)28 b(de\014ne)g Fq( )1370 1287 y Fi(N)1433 1275 y Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b(=)1714 1212 y Fg(P)1801 1233 y Fi(N)1801 1299 y(j)s Fo(=0)1934 1275 y Fq( )s Fu(\()p Fq(t)c Fu(+)f(2)p Fq(\031)s("j;)c(s)p Fu(\))28 b(and)1374 1427 y(\()p Fq(B)1473 1391 y Fj(\000)p Fo(1)1469 1452 y Fi(k)1562 1427 y Fq( )s Fu(\)\()p Fq(t;)14 b(s)p Fu(\))24 b(=)f Fp(\000)51 b Fu(lim)2012 1480 y Fi(N)6 b Fj(!1)2216 1427 y Fq( )2270 1439 y Fi(N)2333 1427 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(:)456 1590 y Fu(Let)31 b Fq(m)e Fp(\025)g Fq(k)24 b Fu(+)c(1.)48 b(First)31 b(w)n(e)g(c)n(hec)n(k)f(that)i(if)g Fq( )g Fp(2)e(X)2134 1559 y Fi(s)2122 1610 y(m)2185 1590 y Fu(,)i Fq( )2294 1602 y Fi(N)2389 1590 y Fu(con)n(v)n(erges)c (uniformly)-7 b(.)48 b(Indeed,)456 1689 y(from)828 1857 y Fp(j)p Fq( )s Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s("j;)c(s)p Fu(\))p Fp(j)23 b(\024)1809 1801 y Fu(1)p 1489 1838 681 4 v 1489 1914 a(\()p Fq(t)c Fu(+)f(2)p Fq(\031)s("j)23 b Fu(+)18 b(Re)c Fq(s)p Fu(\))2107 1890 y Fi(m)2180 1857 y Fp(k)p Fq( )s Fp(k)2321 1869 y Fi(m)2406 1857 y Fp(\024)2494 1740 y Fg(\022)2709 1801 y Fu(1)p 2565 1838 331 4 v 2565 1914 a Fq(T)29 b Fu(+)18 b(2)p Fq(\031)s("j)2906 1740 y Fg(\023)2967 1757 y Fi(m)3044 1857 y Fp(k)p Fq( )s Fp(k)3185 1869 y Fi(m)3247 1857 y Fq(;)456 2037 y Fu(the)28 b(claim)f(follo)n(ws)g(form)g(the)h(M-test)f(of)h(W)-7 b(eierstrass.)605 2137 y(One)37 b(immediately)g(sho)n(ws)e(that)i Fq(B)1775 2101 y Fj(\000)p Fo(1)1771 2162 y Fi(k)1865 2137 y Fq( )j Fu(satis\014es)c(the)h(\014rst)g(three)f(conditions)h (whic)n(h)456 2236 y(de\014ne)27 b Fp(X)766 2206 y Fi(s)754 2260 y(k)802 2236 y Fu(.)605 2336 y(Moreo)n(v)n(er)e(giv)n(en)i Fq( )f Fp(2)d(X)1416 2306 y Fi(s)1404 2356 y(m)998 2537 y Fp(k)p Fq(B)1107 2501 y Fj(\000)p Fo(1)1103 2562 y Fi(k)1196 2537 y Fq( )s Fp(k)1295 2549 y Fi(k)1419 2537 y Fu(=)150 b(sup)1566 2610 y Fo(\()p Fi(t;s)p Fo(\))p Fj(2)p Fi(D)1795 2594 y Fc(s)1868 2433 y Fj(1)1841 2458 y Fg(X)1843 2635 y Fi(j)s Fo(=0)1961 2537 y Fu(\()p Fq(t)19 b Fu(+)f(Re)13 b Fq(s)p Fu(\))2307 2502 y Fi(k)2348 2537 y Fp(j)p Fq( )s Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s("j;)c(s)p Fu(\))p Fp(j)1419 2824 y(\024)150 b Fu(sup)1566 2898 y Fo(\()p Fi(t;s)p Fo(\))p Fj(2)p Fi(D)1795 2881 y Fc(s)1868 2721 y Fj(1)1841 2746 y Fg(X)1843 2922 y Fi(j)s Fo(=0)2131 2768 y Fu(\()p Fq(t)19 b Fu(+)f(Re)c Fq(s)p Fu(\))2478 2738 y Fi(k)p 1985 2805 681 4 v 1985 2881 a Fu(\()p Fq(t)k Fu(+)g(2)p Fq(\031)s("j)23 b Fu(+)18 b(Re)c Fq(s)p Fu(\))2602 2857 y Fi(m)2675 2824 y Fp(k)p Fq( )s Fp(k)2816 2836 y Fi(m)2879 2824 y Fq(:)456 3043 y Fu(T)-7 b(o)27 b(b)r(ound)h(the)g (sum)g(w)n(e)f(in)n(tro)r(duce)g Fq(u)c Fu(=)f Fq(t)d Fu(+)f(Re)13 b Fq(s)28 b Fu(and)g(w)n(e)f(b)r(ound)1005 3126 y Fj(1)978 3151 y Fg(X)981 3328 y Fi(j)s Fo(=0)1269 3174 y Fu(\()p Fq(t)18 b Fu(+)g(Re)c Fq(s)p Fu(\))1615 3144 y Fi(k)p 1122 3211 V 1122 3287 a Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s("j)23 b Fu(+)18 b(Re)13 b Fq(s)p Fu(\))1739 3263 y Fi(m)1836 3230 y Fu(=)1978 3174 y(1)p 1933 3211 131 4 v 1933 3287 a(2)p Fq(\031)s(")2118 3174 y Fu(2)p Fq(\031)s(")p 2084 3211 200 4 v 2084 3287 a(u)2132 3263 y Fi(m)p Fj(\000)p Fi(k)2334 3126 y Fj(1)2307 3151 y Fg(X)2309 3328 y Fi(j)s Fo(=0)2649 3174 y Fu(1)p 2450 3211 439 4 v 2450 3234 a Fg(\000)2489 3301 y Fu(1)18 b(+)2641 3264 y Fo(2)p Fi(\031)r("j)p 2641 3282 137 4 v 2690 3330 a(u)2788 3234 y Fg(\001)2826 3246 y Fi(m)2899 3230 y Fq(:)456 3448 y Fu(Then)27 b(the)h(sum)g(can)f(b)r(e)h(b)r (ounded)g(b)n(y)630 3572 y(1)p 586 3609 131 4 v 586 3685 a(2)p Fq(\031)s(")858 3572 y Fu(1)p 736 3609 285 4 v 736 3685 a Fq(u)784 3661 y Fi(m)p Fj(\000)p Fi(k)q Fj(\000)p Fo(1)1044 3511 y Fg(\024)1098 3572 y Fu(2)p Fq(\031)s(")p 1098 3609 131 4 v 1119 3685 a Fu(2)p Fq(u)1257 3628 y Fu(+)1340 3515 y Fg(Z)1423 3536 y Fj(1)1386 3704 y Fo(0)1655 3572 y Fu(1)p 1517 3609 319 4 v 1517 3685 a(\(1)18 b(+)g Fq(x)p Fu(\))1771 3661 y Fi(m)1859 3628 y Fq(dx)1949 3511 y Fg(\025)2076 3628 y Fu(=)2278 3572 y(1)p 2234 3609 131 4 v 2234 3685 a(2)p Fq(\031)s(")2506 3572 y Fu(1)p 2384 3609 285 4 v 2384 3685 a Fq(u)2432 3661 y Fi(m)p Fj(\000)p Fi(k)q Fj(\000)p Fo(1)2692 3511 y Fg(\024)2746 3572 y Fu(2)p Fq(\031)s(")p 2746 3609 131 4 v 2767 3685 a Fu(2)p Fq(u)2905 3628 y Fu(+)3085 3572 y(1)p 2998 3609 216 4 v 2998 3685 a Fq(m)g Fp(\000)g Fu(1)3224 3511 y Fg(\025)2076 3850 y Fu(=)2334 3794 y(1)p 2234 3831 241 4 v 2234 3907 a(2)p Fq(u)2324 3883 y Fi(m)p Fj(\000)p Fi(k)2503 3850 y Fu(+)2923 3794 y(1)p 2596 3831 696 4 v 2596 3907 a(2)p Fq(\031)s(")p Fu(\()p Fq(m)g Fp(\000)g Fu(1\))p Fq(u)3055 3883 y Fi(m)p Fj(\000)p Fi(k)q Fj(\000)p Fo(1)3301 3850 y Fq(:)456 4028 y Fu(F)-7 b(rom)27 b(the)h (de\014nitions)f(of)h(b)r(oth)g(op)r(erators)e(w)n(e)h(easily)g(see)g (that)1606 4163 y Fq(B)1669 4175 y Fi(k)1728 4163 y Fp(\016)18 b Fq(B)1855 4127 y Fj(\000)p Fo(1)1851 4188 y Fi(k)1967 4163 y Fu(=)23 b(Id)2131 4178 y Fj(jX)2208 4161 y Fc(s)2199 4194 y(m)2271 4163 y Fu(.)3380 4297 y Fe(\003)605 4447 y Fu(W)-7 b(e)28 b(de\014ne)g Fp(B)d Fu(:)e Fp(X)1185 4417 y Fi(s)1173 4470 y(k)1239 4447 y Fp(\002)18 b(X)1393 4417 y Fi(s)1381 4470 y(k)q Fo(+1)1530 4447 y Fp(\000)-15 b(!)23 b(X)1757 4417 y Fi(s)1745 4470 y(k)1812 4447 y Fp(\002)18 b(X)1966 4417 y Fi(s)1954 4470 y(k)q Fo(+1)2107 4447 y Fu(b)n(y)1449 4583 y Fp(B)s Fu(\()p Fq(\033)1586 4595 y Fo(1)1623 4583 y Fq(;)c(\033)1707 4595 y Fo(2)1745 4583 y Fu(\))23 b(=)g(\()p Fq(B)1983 4595 y Fi(k)2024 4583 y Fq(\033)2071 4595 y Fo(1)2109 4583 y Fq(;)14 b(B)2209 4595 y Fi(k)q Fo(+1)2334 4583 y Fq(\033)2381 4595 y Fo(2)2418 4583 y Fu(\))456 4727 y(where)27 b Fq(B)759 4739 y Fi(k)827 4727 y Fu(is)h(de\014ned)g(in)f(\(3.22\))g(and)h Fp(B)1753 4697 y Fj(\000)p Fo(1)1864 4727 y Fu(:)23 b Fp(X)1981 4697 y Fi(s)1969 4751 y(k)q Fo(+1)2113 4727 y Fp(\002)18 b(X)2267 4697 y Fi(s)2255 4751 y(k)q Fo(+2)2403 4727 y Fp(\000)-14 b(!)23 b(X)2631 4697 y Fi(s)2619 4751 y(k)2685 4727 y Fp(\002)18 b(X)2839 4697 y Fi(s)2827 4751 y(k)q Fo(+1)2980 4727 y Fu(b)n(y)1350 4874 y Fp(B)1408 4840 y Fj(\000)p Fo(1)1496 4874 y Fu(\()p Fq( )1582 4886 y Fo(1)1619 4874 y Fq(;)c( )1710 4886 y Fo(2)1748 4874 y Fu(\))23 b(=)g(\()p Fq(B)1990 4839 y Fj(\000)p Fo(1)1986 4899 y Fi(k)q Fo(+1)2111 4874 y Fq( )2165 4886 y Fo(1)2202 4874 y Fq(;)14 b(B)2306 4839 y Fj(\000)p Fo(1)2302 4899 y Fi(k)q Fo(+2)2427 4874 y Fq( )2481 4886 y Fo(2)2518 4874 y Fu(\))456 5024 y(where)27 b Fq(B)763 4989 y Fj(\000)p Fo(1)759 5047 y Fi(j)880 5024 y Fu(is)g(de\014ned)h(in)g(\(3.25\).)36 b(Clearly)1544 5174 y Fp(B)s(B)1660 5139 y Fj(\000)p Fo(1)1770 5174 y Fu(=)23 b(Id)1934 5189 y Fj(jX)2011 5169 y Fc(s)2002 5209 y(k)q Fh(+1)2108 5189 y Fj(\002X)2217 5169 y Fc(s)2208 5209 y(k)q Fh(+2)2333 5174 y Fq(:)p eop %%Page: 30 38 30 37 bop 456 251 a Fo(30)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)1319 450 y Fk(3.6.)46 b(Pro)s(of)32 b(of)g(the)g(Theorem)e(3.1)605 600 y Fu(W)-7 b(e)28 b(lo)r(ok)e(for)h(the)g(parameterization)e(of)i (the)h(stable)f(manifold)g(of)g(the)h(equation)e(\(3.11\))456 699 y(in)h(the)h(form)1335 844 y(~)-45 b Fq(\015)1380 810 y Fi(s)1375 865 y(\026;")1471 844 y Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b(=)j(^)-45 b Fq(\015)t Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))h(+)f Fq(\026")2225 810 y Fi(p)p Fo(+2)2347 844 y Fq(\033)s Fu(\()p Fq(t;)c(s)p Fu(\))456 995 y(with)31 b Fq(\033)i Fu(=)c(\()p Fq(\033)901 1007 y Fo(1)939 995 y Fq(;)14 b(\033)1023 1007 y Fo(2)1060 995 y Fu(\))32 b(b)r(elonging)f(to)g(a)g(suitable)g(space)g(of)g (functions)h(decreasing)d(to)i(zero)g(at)456 1094 y(some)c(prescrib)r (ed)g(rate.)36 b(\(See)28 b(b)r(elo)n(w)f(the)h(c)n(hoice)f(of)g(the)h (spaces.\))605 1194 y(W)-7 b(e)28 b(in)n(tro)r(duce)f(the)h(notation) 1083 1368 y Fq(Q)1149 1380 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1307 1368 y Fu(\()p Fq(z)t Fu(\))23 b(=)g(4)p Fq(\031)1617 1334 y Fo(2)1668 1251 y Fg(\022)1771 1317 y Fq(q)1808 1329 y Fo(1)1845 1317 y Fu(\()p Fq(z)t(;)14 b(")p Fu(\))1771 1417 y Fq(q)1808 1429 y Fo(2)1845 1417 y Fu(\()p Fq(z)t(;)g(")p Fu(\))2069 1251 y Fg(\023)2148 1368 y Fu(+)k Fq(\026)2281 1334 y Fo(2)2319 1368 y Fq(")2358 1334 y Fi(p)p Fo(+1)2480 1368 y Fq(T)2529 1380 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2687 1368 y Fu(\()p Fq(z)t Fu(\))p Fq(:)456 1559 y Fu(Therefore)26 b(w)n(e)h(can)g(write)1151 1712 y(^)1132 1733 y Fq(P)1185 1745 y Fi(\026;")1281 1733 y Fu(\()p Fq(z)t Fu(\))c(=)g Fq(z)e Fu(+)d(2)p Fq(\031)s(")1787 1616 y Fg(\022)2001 1682 y Fq(y)1889 1782 y Fp(\000)p Fq(V)2021 1752 y Fj(0)2044 1782 y Fu(\()p Fq(x)p Fu(\))2198 1616 y Fg(\023)2277 1733 y Fu(+)g Fq(")2399 1698 y Fo(2)2436 1733 y Fq(Q)2502 1745 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2661 1733 y Fu(\()p Fq(z)t Fu(\))456 1923 y(and)1237 2068 y Fq(P)1302 2034 y Fi(t)1290 2089 y(\026;")1386 2068 y Fu(\()p Fq(z)t Fu(\))23 b(=)1622 2047 y(^)1603 2068 y Fq(P)1656 2080 y Fi(\026;")1752 2068 y Fu(\()p Fq(z)t Fu(\))c(+)f Fq(\026")2050 2034 y Fi(p)p Fo(+4)2172 2068 y Fq(S)2223 2080 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2386 2068 y Fu(\()p Fq(z)t(;)c(t=")p Fu(\))p Fq(:)456 2214 y Fu(W)-7 b(e)28 b(imp)r(ose)f(the)h(conditions) 1363 2359 y Fq(P)1428 2325 y Fi(t)1416 2380 y(\026;")1512 2359 y Fu(\()s(~)-45 b Fq(\015)1592 2325 y Fi(s)1587 2380 y(\026;")1683 2359 y Fu(\()p Fq(t;)14 b(s)p Fu(\)\))23 b(=)j(~)-45 b Fq(\015)2044 2325 y Fi(s)2039 2380 y(\026;")2135 2359 y Fu(\()p Fq(t)18 b Fu(+)g(2)p Fq(\031)s(";)c(s)p Fu(\))-2081 b(\(3.26\))456 2505 y(and)1345 2650 y(~)-45 b Fq(\015)1390 2616 y Fi(s)1385 2671 y(\026;")1481 2650 y Fu(\()p Fq(t)18 b Fu(+)g(2)p Fq(\031)s(";)c(s)p Fu(\))23 b(=)j(~)-45 b Fq(\015)2042 2616 y Fi(s)2037 2671 y(\026;")2132 2650 y Fu(\()p Fq(t;)14 b(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\))p Fq(:)605 2800 y Fu(Expanding)27 b(b)n(y)g(T)-7 b(a)n(ylor)26 b(around)j(^)-45 b Fq(\015)1724 2812 y Fi(\026;")1820 2800 y Fu(,)28 b(the)g(condition)f(\(3.26\))g(b)r (ecomes)553 3045 y Fq(P)618 3011 y Fi(t)618 3066 y(\026;")714 3045 y Fu(\()s(~)-45 b Fq(\015)794 3011 y Fi(s)789 3066 y(\026;")885 3045 y Fu(\()p Fq(t;)14 b(s)p Fu(\)\))641 3191 y(=)748 3170 y(^)729 3191 y Fq(P)782 3203 y Fi(\026;")878 3191 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)h Fq(s)p Fu(\)\))f(+)g Fq(\026")1415 3157 y Fi(p)p Fo(+4)1538 3191 y Fq(S)1589 3203 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1752 3191 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))k(+)h Fq(\026")2437 3157 y Fi(p)p Fo(+2)2559 3191 y Fq(D)2648 3170 y Fu(^)2630 3191 y Fq(P)2683 3203 y Fi(\026;")2779 3191 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\)\))p Fq(\033)s Fu(\()p Fq(t;)c(s)p Fu(\))692 3330 y(+)k Fq(\026)825 3296 y Fo(2)862 3330 y Fq(")901 3296 y Fo(2)p Fi(p)p Fo(+6)1057 3330 y Fq(D)r(S)1179 3342 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1342 3330 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))p Fq(\033)s Fu(\()p Fq(t;)g(s)p Fu(\))19 b(+)f Fq(\026)2208 3296 y Fo(2)2246 3330 y Fq(")2285 3296 y Fo(2)p Fi(p)p Fo(+4)2440 3330 y Fq(O)r Fu(\()p Fp(j)p Fq(\033)s Fu(\()p Fq(t;)c(s)p Fu(\))p Fp(j)2803 3296 y Fo(2)2842 3330 y Fu(\))p Fq(:)456 3480 y Fu(Th)n(us,)27 b(\(3.26\))g(is)g(equiv)-5 b(alen)n(t)28 b(to)585 3754 y Fq(\033)s Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\))83 b(=)g Fq(\033)s Fu(\()p Fq(t;)14 b(s)p Fu(\))19 b(+)f(2)p Fq(\031)s(")1736 3637 y Fg(\022)1838 3703 y Fq(\033)1885 3715 y Fo(2)1923 3703 y Fu(\()p Fq(t;)c(s)p Fu(\))1838 3803 y Fp(\000)p Fq(V)1969 3773 y Fj(0)q(0)2012 3803 y Fu(\()8 b(^)-50 b Fq(\013)q Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\)\))p Fq(\033)2411 3815 y Fo(1)2449 3803 y Fu(\()p Fq(t;)c(s)p Fu(\))2661 3637 y Fg(\023)1269 3938 y Fu(+)p Fq(")1373 3904 y Fo(2)1409 3938 y Fq(D)r(Q)1546 3950 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1705 3938 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\)\))p Fq(\033)s Fu(\()p Fq(t;)c(s)p Fu(\))20 b(+)e Fq(")2413 3904 y Fo(2)2450 3938 y Fq(S)2501 3950 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2664 3938 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))1269 4063 y(+)p Fq(\026")1423 4029 y Fi(p)p Fo(+2)1544 4063 y Fq(O)r Fu(\()p Fp(j)p Fq(\033)s Fu(\()p Fq(t;)g(s)p Fu(\))p Fp(j)1907 4029 y Fo(2)1947 4063 y Fu(\))k(+)g Fq(\026")2169 4029 y Fi(p)p Fo(+4)2291 4063 y Fq(D)r(S)2413 4075 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2577 4063 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))p Fq(\033)s Fu(\()p Fq(t;)g(s)p Fu(\))p Fq(:)456 4209 y Fu(T)-7 b(o)27 b(simplify)h(the)g(notation,)f(w)n(e)g (further)h(in)n(tro)r(duce)1240 4399 y Fq(A)p Fu(\()8 b(^)-50 b Fq(\013)q Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\)\))24 b(=)1766 4282 y Fg(\022)2111 4348 y Fu(0)326 b(1)1869 4448 y Fp(\000)p Fq(V)2000 4418 y Fj(00)2043 4448 y Fu(\()8 b(^)-50 b Fq(\013)p Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\)\))84 b(0)2562 4282 y Fg(\023)2637 4399 y Fu(,)936 4640 y Fq(H)7 b Fu(\()p Fq(\033)s Fu(\)\()p Fq(t;)14 b(s)p Fu(\))84 b(=)f Fq(D)r(Q)1665 4652 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1824 4640 y Fu(\()s(^)-45 b Fq(\015)t Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\)\))p Fq(\033)s Fu(\()p Fq(t;)c(s)p Fu(\))1528 4765 y(+)p Fq(S)1644 4777 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1807 4765 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))k(+)g Fq(\026")2491 4731 y Fi(p)2529 4765 y Fq(O)r Fu(\()p Fp(j)p Fq(\033)s Fu(\()p Fq(t;)c(s)p Fu(\))p Fp(j)2892 4731 y Fo(2)2932 4765 y Fu(\))-2508 b(\(3.27\))1528 4891 y(+)p Fq(\026")1682 4857 y Fi(p)p Fo(+2)1804 4891 y Fq(D)r(S)1926 4903 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2089 4891 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))p Fq(\033)s Fu(\()p Fq(t;)g(s)p Fu(\))456 5036 y(and)1054 5182 y Fp(F)8 b Fu(\()p Fq(\033)s Fu(\)\()p Fq(t;)14 b(s)p Fu(\))24 b(=)f(2)p Fq(\031)s("A)p Fu(\()8 b(^)-50 b Fq(\013)p Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\)\))p Fq(\033)s Fu(\()p Fq(t;)c(s)p Fu(\))20 b(+)e Fq(")2425 5147 y Fo(2)2462 5182 y Fq(H)7 b Fu(\()p Fq(\033)s Fu(\)\()p Fq(t;)14 b(s)p Fu(\).)-2389 b(\(3.28\))p eop %%Page: 31 39 31 38 bop 1375 251 a Fo(3.6.)29 b(PR)n(OOF)g(OF)g(THE)g(THEOREM)g(3.1) 853 b(31)456 450 y Fu(Then)27 b(the)h(problem)f(is)h(reduced)f(to)h (\014nd)g Fq(\033)e Fu(=)d(\()p Fq(\033)2039 462 y Fo(1)2077 450 y Fq(;)14 b(\033)2161 462 y Fo(2)2198 450 y Fu(\))28 b(suc)n(h)g(that)1738 586 y Fp(B)s Fq(\033)d Fu(=)e Fp(F)8 b Fu(\()p Fq(\033)s Fu(\).)456 721 y(In)23 b(the)h(next)g(lemma)g(w)n (e)f(will)h(see)f(that)h Fp(F)31 b Fu(sends)23 b Fp(X)2098 691 y Fi(s)2086 742 y(m)2160 721 y Fp(\002)10 b(X)2306 691 y Fi(s)2294 742 y(m)p Fo(+1)2465 721 y Fu(to)23 b Fp(X)2633 691 y Fi(s)2621 742 y(m)p Fo(+1)2779 721 y Fp(\002)10 b(X)2925 691 y Fi(s)2913 742 y(m)p Fo(+2)3084 721 y Fu(and)23 b(hence)456 821 y(w)n(e)31 b(can)g(use)g(a)g(righ)n(t)g (in)n(v)n(erse)f(of)h Fp(B)s Fu(,)h(as)f(giv)n(en)g(in)g(Section)h (3.5,)f(to)h(obtain)f(the)h(\014xed)f(p)r(oin)n(t)456 921 y(equation)1693 1056 y Fq(\033)c Fu(=)22 b Fp(B)1912 1022 y Fj(\000)p Fo(1)2001 1056 y Fp(F)8 b Fu(\()p Fq(\033)s Fu(\).)605 1192 y(W)-7 b(e)23 b(lo)r(ok)f(for)h Fq(\033)j Fp(2)d(X)1263 1162 y Fi(s)1251 1212 y(m)1324 1192 y Fp(\002)9 b(X)1469 1162 y Fi(s)1457 1212 y(m)p Fo(+1)1626 1192 y Fu(with)23 b(a)g(suitable)g(v)-5 b(alue)22 b(of)h Fq(m)p Fu(.)35 b(W)-7 b(e)23 b(de\014ne)g(the)h(follo)n(wing)456 1291 y(w)n(eigh)n(ted)j(norm)g(in)g(the)h(pro)r(duct)g(space)f Fp(X)1863 1261 y Fi(s)1851 1312 y(m)1933 1291 y Fp(\002)18 b(X)2087 1261 y Fi(s)2075 1312 y(m)p Fo(+1)2222 1291 y Fu(:)1198 1427 y Fp(k)p Fu(\()p Fq(h)1320 1439 y Fi(m)1383 1427 y Fq(;)c(h)1468 1439 y Fi(m)p Fo(+1)1614 1427 y Fu(\))p Fp(k)1688 1439 y Fi(m)1774 1427 y Fu(=)23 b Fq(L)p Fp(k)p Fq(h)2009 1439 y Fi(m)2071 1427 y Fp(k)2113 1439 y Fi(m)2194 1427 y Fu(+)18 b Fp(k)p Fq(h)2367 1439 y Fi(m)p Fo(+1)2513 1427 y Fp(k)2555 1439 y Fi(m)p Fo(+1)456 1563 y Fu(with)1421 1730 y Fq(L)23 b Fu(=)1598 1674 y Fq(n)c Fp(\000)f Fu(1)p 1598 1711 193 4 v 1598 1787 a Fq(n)h Fp(\000)f Fu(2)1820 1730 y(+)2147 1674 y Fq(n)2197 1644 y Fo(2)p 1913 1711 557 4 v 1913 1787 a Fu(\(3)p Fq(n)g Fp(\000)g Fu(4\)\()p Fq(n)g Fp(\000)g Fu(2\))456 1914 y(and)30 b(w)n(e)h(denote)g Fq(B)t Fu(\()p Fq(r)n(;)14 b(m;)g(m)20 b Fu(+)h(1\))28 b Fp(\032)g(X)1743 1884 y Fi(s)1731 1935 y(m)1815 1914 y Fp(\002)20 b(X)1971 1884 y Fi(s)1959 1935 y(m)p Fo(+1)2137 1914 y Fu(the)31 b(closed)g(ball)f (of)h(radius)f Fq(r)k Fu(with)d(this)456 2014 y(norm.)605 2164 y Ft(Lemma)j Fu(3.8)p Ft(.)41 b Fs(L)l(et)32 b Fq(m)27 b Fu(=)1439 2131 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)p 1439 2145 160 4 v 1456 2193 a Fi(n)p Fj(\000)p Fo(2)1608 2164 y Fs(.)46 b(Ther)l(e)33 b(exists)e Fq(r)f(>)d Fu(0)k Fs(indep)l(endent)i(of)f Fq(\026)g Fs(and)h Fq(")p Fs(,)f(such)456 2265 y(that)d(the)h(op)l(er)l(ator)1150 2401 y Fp(B)1208 2367 y Fj(\000)p Fo(1)1314 2401 y Fp(\016)18 b(F)31 b Fu(:)23 b Fq(B)t Fu(\()p Fq(r)n(;)14 b(m;)g(m)19 b Fu(+)f(1\))23 b Fp(\000)-14 b(!)23 b Fq(B)t Fu(\()p Fq(r)n(;)14 b(m;)g(m)k Fu(+)g(1\))456 2537 y Fs(is)30 b(wel)t(l)g(de\014ne)l(d)h(and)f(it)f (is)i(a)f(c)l(ontr)l(action.)605 2687 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)45 b(recall)f(that,)50 b(b)n(y)44 b(Prop)r(osition)f (3.3,)49 b(the)c(function)k(^)-45 b Fq(\015)56 b Fp(2)c(X)3112 2656 y Fi(s)3100 2713 y Fo(2)p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))3380 2687 y Fp(\002)456 2794 y(X)527 2764 y Fi(s)515 2821 y(n=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))800 2794 y Fu(and)27 b(hence,)h(if)g Fq(f)1332 2806 y Fi(l)1380 2794 y Fp(2)23 b Fq(P)1511 2806 y Fi(l)1537 2794 y Fu(,)1627 2938 y Fq(f)1668 2950 y Fi(l)1711 2938 y Fp(\016)e Fu(^)-45 b Fq(\015)28 b Fp(2)23 b(X)1991 2904 y Fi(s)1979 2961 y Fo(2)p Fi(l=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))2250 2938 y Fu(.)456 3082 y(Let)k Fq(\033)g Fp(2)c Fq(B)t Fu(\()p Fq(r)n(;)14 b(m;)g(m)19 b Fu(+)f(1\).)37 b(Since)27 b Fq(V)1629 3052 y Fj(00)1690 3082 y Fp(\016)f Fu(^)-50 b Fq(\013)24 b Fp(2)f(X)1976 3052 y Fi(s)1964 3103 y Fo(2)2012 3082 y Fu(,)28 b(it)g(is)f(clear)g(that)1515 3218 y Fq(A)p Fu(\()8 b(^)-50 b Fq(\013)q Fu(\))p Fq(\033)27 b Fp(2)c(X)1918 3183 y Fi(s)1906 3238 y(m)p Fo(+1)2072 3218 y Fp(\002)18 b(X)2226 3183 y Fi(s)2214 3238 y(m)p Fo(+2)2362 3218 y Fu(.)-1929 b(\(3.29\))456 3353 y(W)-7 b(e)28 b(c)n(hec)n(k)e(that)1369 3489 y Fq(S)1420 3501 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1584 3489 y Fu(\()s(^)-45 b Fq(\015)t Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))83 b Fp(2)g(X)2370 3454 y Fi(s)2358 3509 y(m)p Fo(+2)2524 3489 y Fp(\002)18 b(X)2678 3454 y Fi(s)2666 3509 y(m)p Fo(+2)1527 3613 y Fu(\()p Fq(D)r(Q)1696 3625 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1874 3613 y Fp(\016)j Fu(^)-45 b Fq(\015)t Fu(\))14 b Fq(\033)87 b Fp(2)c(X)2370 3579 y Fi(s)2358 3634 y(m)p Fo(+1)2524 3613 y Fp(\002)18 b(X)2678 3579 y Fi(s)2666 3634 y(m)p Fo(+2)456 3613 y Fu(\(3.30\))1064 3738 y Fq(D)r(S)1186 3750 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1349 3738 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)18 b Fu(+)h Fq(s)p Fu(\))p Fq(;)14 b(t=")p Fu(\))g Fq(\033)s Fu(\()p Fq(t;)g(s)p Fu(\))83 b Fp(2)g(X)2370 3703 y Fi(s)2358 3758 y(m)p Fo(+4)2524 3738 y Fp(\002)18 b(X)2678 3703 y Fi(s)2666 3758 y(m)p Fo(+4)2813 3738 y Fu(.)456 3873 y(Indeed,)26 b(w)n(e)g(recall)f(that)h(if)h Fq(S)1392 3885 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1578 3873 y Fp(2)c Fq(P)1709 3885 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1873 3873 y Fu(,)j(then)h Fq(S)2161 3885 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2324 3873 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t)16 b Fu(+)e Fq(s)p Fu(\))p Fq(;)g(t=")p Fu(\))23 b Fp(2)g(X)2984 3843 y Fi(s)2972 3894 y(m)p Fo(+2)3135 3873 y Fp(\002)15 b(X)3286 3843 y Fi(s)3274 3894 y(m)p Fo(+2)3421 3873 y Fu(.)605 3973 y(No)n(w)27 b(w)n(e)g(deal)h(with)g(\() p Fq(D)r(Q)1450 3985 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1627 3973 y Fp(\016)21 b Fu(^)-45 b Fq(\015)5 b Fu(\))14 b Fq(\033)s Fu(.)37 b(W)-7 b(e)28 b(denote)1385 4167 y Fq(D)r(Q)1522 4179 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1699 4167 y Fp(\016)21 b Fu(^)-45 b Fq(\015)28 b Fu(=)1918 4050 y Fg(\022)2039 4095 y Fu(^)2020 4116 y Fq(Q)2086 4128 y Fo(11)2258 4095 y Fu(^)2239 4116 y Fq(Q)2305 4128 y Fo(12)2039 4204 y Fu(^)2020 4225 y Fq(Q)2086 4237 y Fo(21)2258 4204 y Fu(^)2239 4225 y Fq(Q)2305 4237 y Fo(22)2417 4050 y Fg(\023)2492 4167 y Fu(.)456 4377 y(It)i(is)g(clear)f(that,)i (since)e(all)h(elemen)n(ts)g(of)g Fq(D)r(Q)1940 4389 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)2129 4377 y Fu(b)r(elong)f(to)h Fq(P)2553 4389 y Fi(n)p Fj(\000)p Fo(2)2684 4377 y Fu(,)g(w)n(e)g(ha)n (v)n(e)f(that)3257 4356 y(^)3238 4377 y Fq(Q)3304 4389 y Fi(ij)3389 4377 y Fp(2)456 4477 y(X)527 4447 y Fi(s)515 4504 y Fo(2\()p Fi(n)p Fj(\000)p Fo(2\))p Fi(=)p Fo(\()p Fi(n)p Fj(\000)p Fo(2\))965 4477 y Fu(=)23 b Fp(X)1124 4447 y Fi(s)1112 4498 y Fo(2)1187 4477 y Fu(and)28 b(therefore)1362 4625 y(\()p Fq(D)r(Q)1531 4637 y Fo(2)p Fi(k)q Fj(\000)p Fo(3)1709 4625 y Fp(\016)21 b Fu(^)-45 b Fq(\015)t Fu(\))p Fq(\033)27 b Fp(2)c(X)2071 4590 y Fi(s)2059 4645 y(m)p Fo(+2)2225 4625 y Fp(\002)18 b(X)2379 4590 y Fi(s)2367 4645 y(m)p Fo(+2)2515 4625 y Fu(.)605 4765 y(T)-7 b(o)27 b(deal)h(with)g(\()p Fq(D)r(S)1246 4777 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1428 4765 y Fp(\016)21 b Fu(^)-45 b Fq(\015)t Fu(\))p Fq(\033)s Fu(,)29 b(w)n(e)e(observ)n(e)f(that)i Fq(D)r(S)2389 4777 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)2570 4765 y Fp(\016)21 b Fu(^)-45 b Fq(\015)28 b Fp(2)23 b(X)2850 4735 y Fi(s)2838 4786 y Fo(4)2914 4765 y Fu(and)k(therefore)1163 4900 y Fq(D)r(S)1285 4912 y Fo(2)p Fi(n)p Fj(\000)p Fo(3)1448 4900 y Fu(\()s(^)-45 b Fq(\015)5 b Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))p Fq(\033)s Fu(\()p Fq(t;)g(s)p Fu(\))24 b Fp(2)f(X)2271 4866 y Fi(s)2259 4921 y(m)p Fo(+4)2425 4900 y Fp(\002)18 b(X)2579 4866 y Fi(s)2567 4921 y(m)p Fo(+4)2714 4900 y Fu(.)456 5046 y(Th)n(us,)27 b(b)n(y)g(\(3.30\))g(and)h(since)f Fq(O)r Fu(\()p Fp(j)p Fq(\033)s Fp(j)1602 5016 y Fo(2)1641 5046 y Fu(\))c Fp(2)g(X)1845 5016 y Fi(s)1833 5067 y Fo(2)p Fi(m)p Fo(+2)2014 5046 y Fu(,)28 b(w)n(e)f(ha)n(v)n(e)f(that)1535 5182 y Fq(H)7 b Fu(\()p Fq(\033)s Fu(\))24 b Fp(2)f(X)1898 5147 y Fi(s)1886 5202 y(m)p Fo(+1)2052 5182 y Fp(\002)18 b(X)2206 5147 y Fi(s)2194 5202 y(m)p Fo(+2)2342 5182 y Fu(.)-1909 b(\(3.31\))p eop %%Page: 32 40 32 39 bop 456 251 a Fo(32)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(Therefore,)k(b)n(y)h(\(3.29\),)g(\(3.31\))g(and)h(b)n(y)f (de\014nition)h(\(3.28\))f(of)g Fp(F)36 b Fu(w)n(e)27 b(ha)n(v)n(e)f(that)1119 614 y Fp(F)8 b Fu(\()p Fq(\033)s Fu(\))24 b(=)f(2)p Fq(\031)s("A)p Fu(\()8 b(^)-50 b Fq(\013)p Fu(\))p Fq(\033)22 b Fu(+)c Fq(")1914 580 y Fo(2)1951 614 y Fq(H)7 b Fu(\()p Fq(\033)s Fu(\))24 b Fp(2)g(X)2315 580 y Fi(s)2303 635 y(m)p Fo(+1)2468 614 y Fp(\002)18 b(X)2622 580 y Fi(s)2610 635 y(m)p Fo(+2)2758 614 y Fu(.)456 778 y(Finally)-7 b(,)27 b(b)n(y)h(Lemma)f(3.7,)1476 941 y Fp(B)1534 906 y Fj(\000)p Fo(1)1622 941 y Fu(\()p Fp(F)8 b Fu(\()p Fq(\033)s Fu(\)\))25 b Fp(2)e(X)2042 906 y Fi(s)2030 961 y(m)2112 941 y Fp(\002)18 b(X)2266 906 y Fi(s)2254 961 y(m)p Fo(+1)2401 941 y Fu(.)456 1114 y(Next)32 b(w)n(e)f(will)h(pro)n(v)n(e)e(that)i Fp(kB)1462 1084 y Fj(\000)p Fo(1)1550 1114 y Fu(\()p Fp(F)8 b Fu(\()p Fq(\033)s Fu(\)\))p Fp(k)1838 1126 y Fi(m)1932 1114 y Fp(\024)29 b Fq(r)35 b Fu(if)d Fp(k)p Fq(\033)s Fp(k)2312 1126 y Fi(m)2404 1114 y Fp(\024)e Fq(r)r Fu(.)50 b(W)-7 b(e)32 b(b)r(egin)g(b)n(y)f(b)r(ounding)456 1214 y(the)g(norms)g Fp(kF)958 1226 y Fo(1)994 1214 y Fp(k)1036 1226 y Fi(m)p Fo(+1)1214 1214 y Fu(and)g Fp(kF)1481 1226 y Fo(2)1518 1214 y Fp(k)1560 1226 y Fi(m)p Fo(+2)1707 1214 y Fu(.)48 b(F)-7 b(rom)31 b(\(3.27\),)g(whic)n(h)g(de\014nes)h Fq(H)7 b Fu(,)32 b(it)g(is)f(clear)f(that)456 1313 y(there)d(exist)g (constan)n(ts)g Fq(M)1313 1325 y Fo(1)1378 1313 y Fu(and)g Fq(M)1620 1325 y Fo(2)1684 1313 y Fu(suc)n(h)h(that)1164 1476 y Fp(k)p Fq(H)1275 1488 y Fo(1)1312 1476 y Fu(\()p Fq(\033)s Fu(\))p Fp(k)1468 1488 y Fi(m)p Fo(+1)1638 1476 y Fp(\024)23 b Fq(M)1807 1488 y Fo(1)1844 1476 y Fu(,)166 b Fp(k)p Fq(H)2144 1488 y Fo(2)2181 1476 y Fu(\()p Fq(\033)s Fu(\))p Fp(k)2337 1488 y Fi(m)p Fo(+2)2507 1476 y Fp(\024)23 b Fq(M)2676 1488 y Fo(2)2713 1476 y Fu(.)456 1640 y(Then)933 1803 y Fp(kF)1035 1815 y Fo(1)1072 1803 y Fu(\()p Fq(\033)s Fu(\))p Fp(k)1228 1815 y Fi(m)p Fo(+1)1458 1803 y Fp(\024)83 b Fu(2)p Fq(\031)s(")65 b Fu(sup)1751 1876 y Fo(\()p Fi(t;s)p Fo(\))p Fj(2)p Fi(D)1993 1803 y Fp(j)p Fq(\033)2063 1815 y Fo(2)2101 1803 y Fu(\()p Fq(t;)14 b(s)p Fu(\)\()p Fq(t)19 b Fu(+)f(Re)c Fq(s)p Fu(\))2618 1768 y Fi(m)p Fo(+1)2765 1803 y Fp(j)1606 2004 y Fu(+)p Fq(")1710 1970 y Fo(2)1812 2004 y Fu(sup)1760 2078 y Fo(\()p Fi(t;s)p Fo(\))p Fj(2)p Fi(D)2003 2004 y Fp(j)p Fq(H)2095 2016 y Fo(1)2132 2004 y Fu(\()p Fq(\033)s Fu(\)\()p Fq(t;)g(s)p Fu(\)\)\()p Fq(t)21 b Fu(+)d(Re)13 b Fq(s)p Fu(\))2796 1970 y Fi(m)p Fo(+1)2944 2004 y Fp(j)1458 2205 y(\024)83 b Fu(2)p Fq(\031)s(")p Fp(k)p Fq(\033)1826 2217 y Fo(2)1863 2205 y Fp(k)1905 2217 y Fi(m)p Fo(+1)2070 2205 y Fu(+)18 b Fq(")2192 2171 y Fo(2)2229 2205 y Fq(M)2310 2217 y Fo(1)456 2368 y Fu(and,)27 b(using)g(the)h(expression)e(of)36 b(^)-50 b Fq(\013)28 b Fu(obtained)f(in)h(Prop)r(osition)e(3.3)791 2531 y Fp(kF)893 2543 y Fo(2)930 2531 y Fu(\()p Fq(\033)s Fu(\))p Fp(k)1086 2543 y Fi(m)p Fo(+2)1317 2531 y Fp(\024)82 b Fu(2)p Fq(\031)s(")65 b Fu(sup)1609 2605 y Fo(\()p Fi(t;s)p Fo(\))p Fj(2)p Fi(D)1851 2531 y Fp(j)p Fq(V)1941 2497 y Fj(00)1984 2531 y Fu(\()8 b(^)-50 b Fq(\013)p Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\)\))p Fq(\033)2383 2543 y Fo(1)2421 2531 y Fu(\()p Fq(t;)c(s)p Fu(\)\()p Fq(t)19 b Fu(+)f(Re)c Fq(s)p Fu(\))2938 2497 y Fi(m)p Fo(+2)3085 2531 y Fp(j)1464 2733 y Fu(+)p Fq(")1568 2698 y Fo(2)1670 2733 y Fu(sup)1619 2806 y Fo(\()p Fi(t;s)p Fo(\))p Fj(2)p Fi(D)1861 2733 y Fp(j)p Fq(H)1953 2745 y Fo(2)1991 2733 y Fu(\()p Fq(\033)s Fu(\)\()p Fq(t;)g(s)p Fu(\)\)\()p Fq(t)20 b Fu(+)e(Re)c Fq(s)p Fu(\))2655 2698 y Fi(m)p Fo(+2)2802 2733 y Fp(j)1317 2983 y(\024)82 b Fu(2)p Fq(\031)s(")1609 2866 y Fg(\022)1680 2927 y Fu(2)p Fq(n)p Fu(\()p Fq(n)18 b Fp(\000)g Fu(1\))p 1680 2964 349 4 v 1707 3040 a(\()p Fq(n)g Fp(\000)g Fu(2\))1964 3016 y Fo(2)2057 2983 y Fu(+)g Fq(O)r Fu(\()p Fq(T)2298 2948 y Fj(\000)p Fi(\027)2391 2983 y Fu(\))2423 2866 y Fg(\023)2498 2983 y Fp(k)p Fq(\033)2587 2995 y Fo(1)2624 2983 y Fp(k)2666 2995 y Fi(m)2747 2983 y Fu(+)g Fq(")2869 2948 y Fo(2)2906 2983 y Fq(M)2987 2995 y Fo(2)3024 2983 y Fu(,)456 3191 y(with)28 b Fq(\027)g(>)23 b Fu(0.)36 b(Therefore,)27 b(using)g(Lemma)g(3.7)g(w)n(e)g(obtain)530 3354 y Fp(kB)630 3320 y Fj(\000)p Fo(1)737 3354 y Fp(\016)17 b(F)8 b Fu(\()p Fq(\033)s Fu(\))p Fp(k)1020 3366 y Fi(m)1167 3354 y Fu(=)83 b Fp(k)p Fu(\()p Fp(B)1447 3318 y Fj(\000)p Fo(1)1444 3376 y(1)1535 3354 y Fp(F)1595 3366 y Fo(1)1631 3354 y Fu(\()p Fq(\033)s Fu(\))p Fq(;)14 b Fp(B)1840 3318 y Fj(\000)p Fo(1)1837 3376 y(2)1930 3354 y Fp(F)1990 3366 y Fo(2)2026 3354 y Fu(\()p Fq(\033)s Fu(\)\))p Fp(k)2214 3366 y Fi(m)1167 3482 y Fu(=)83 b Fq(L)p Fp(kB)1472 3447 y Fj(\000)p Fo(1)1469 3504 y(1)1559 3482 y Fp(F)1619 3494 y Fo(1)1656 3482 y Fu(\()p Fq(\033)1735 3494 y Fo(1)1773 3482 y Fq(;)14 b(\033)1857 3494 y Fo(2)1894 3482 y Fu(\))p Fp(k)1968 3494 y Fi(m)2050 3482 y Fu(+)k Fp(kB)2233 3447 y Fj(\000)p Fo(1)2230 3504 y(2)2320 3482 y Fp(F)2380 3494 y Fo(2)2417 3482 y Fu(\()p Fq(\033)2496 3494 y Fo(1)2534 3482 y Fq(;)c(\033)2618 3494 y Fo(2)2656 3482 y Fu(\)\))p Fp(k)2762 3494 y Fi(m)p Fo(+1)1167 3647 y Fp(\024)83 b Fq(L)p Fp(kF)1474 3659 y Fo(1)1510 3647 y Fp(k)1552 3659 y Fi(m)p Fo(+1)1708 3591 y Fu(1)18 b(+)g(4)p Fq(")p 1708 3628 224 4 v 1718 3704 a Fu(2)p Fq(m\031)s(")1960 3647 y Fu(+)g Fp(kF)2145 3659 y Fo(2)2181 3647 y Fp(k)2223 3659 y Fi(m)p Fo(+2)2474 3591 y Fu(1)g(+)g(4)p Fq(")p 2380 3628 411 4 v 2380 3704 a Fu(2\()p Fq(m)g Fu(+)g(1\))p Fq(\031)s(")1167 3879 y Fp(\024)83 b Fq(L)p Fp(k)p Fq(\033)1461 3891 y Fo(2)1497 3879 y Fp(k)1539 3891 y Fi(m)p Fo(+1)1717 3822 y Fq(n)18 b Fp(\000)g Fu(2)p 1696 3859 235 4 v 1696 3936 a(2)p Fq(n)g Fp(\000)g Fu(2)1959 3879 y(+)2042 3762 y Fg(\022)2217 3822 y Fu(2)p Fq(n)p Fu(\()p Fq(n)f Fp(\000)h Fu(1\))p 2113 3859 557 4 v 2113 3936 a(\(3)p Fq(n)g Fp(\000)g Fu(4\)\()p Fq(n)g Fp(\000)g Fu(2\))2697 3879 y(+)g Fq(O)r Fu(\()p Fq(T)2938 3844 y Fj(\000)p Fi(\027)3032 3879 y Fu(\))3064 3762 y Fg(\023)3139 3879 y Fp(k)p Fq(\033)3228 3891 y Fo(1)3265 3879 y Fp(k)3307 3891 y Fi(m)1315 4053 y Fu(+)p Fq(O)r Fu(\()p Fq(")p Fu(\))1167 4231 y Fp(\024)83 b Fq(L)1402 4174 y(n)18 b Fp(\000)g Fu(2)p 1382 4212 235 4 v 1382 4288 a(2)p Fq(n)f Fp(\000)h Fu(2)1625 4231 y Fp(k)p Fq(\033)1714 4243 y Fo(2)1752 4231 y Fp(k)1794 4243 y Fi(m)p Fo(+1)1959 4231 y Fu(+)2265 4174 y(2)p Fq(n)2357 4144 y Fo(2)p 2052 4212 557 4 v 2052 4288 a Fu(\(3)p Fq(n)g Fp(\000)g Fu(4\)\()p Fq(n)g Fp(\000)g Fu(2\))2618 4231 y Fp(k)p Fq(\033)2707 4243 y Fo(1)2744 4231 y Fp(k)2786 4243 y Fi(m)2867 4231 y Fu(+)g Fq(O)r Fu(\()p Fq(")p Fu(\))-2662 b(\(3.32\))456 4450 y(if)28 b Fq(T)38 b Fu(is)28 b(big,)f(in)h(particular)e Fq(O)r Fu(\()p Fq(T)1502 4420 y Fj(\000)p Fi(\027)1596 4450 y Fu(\))d Fp(\024)1906 4417 y Fo(2)p Fi(n)p 1749 4431 389 4 v 1749 4479 a Fo(\(3)p Fi(n)p Fj(\000)p Fo(4\)\()p Fi(n)p Fj(\000)p Fo(2\))2147 4450 y Fu(.)37 b(W)-7 b(e)28 b(in)n(tro)r(duce)1513 4683 y Fq(a)83 b Fu(=)1798 4627 y(2\()p Fq(n)18 b Fp(\000)g Fu(1\))p 1798 4664 299 4 v 1851 4740 a Fq(n)g Fp(\000)g Fu(2)1521 4904 y Fq(b)83 b Fu(=)2011 4848 y(2)p Fq(n)2103 4818 y Fo(2)p 1798 4885 557 4 v 1798 4961 a Fu(\(3)p Fq(n)18 b Fp(\000)g Fu(4\)\()p Fq(n)g Fp(\000)g Fu(2\))2364 4904 y(.)456 5116 y(W)-7 b(e)40 b(observ)n(e)e(that)i(since)g Fq(a)j(>)g(b)d Fu(and)g Fq(L)j Fu(=)g(\()p Fq(a)26 b Fu(+)h Fq(b)p Fu(\))p Fq(=)p Fu(2)39 b(w)n(e)g(ha)n(v)n(e)g Fq(b)k(<)g(L)g(<)g(a)p Fu(.)74 b(Th)n(us,)456 5216 y(1)21 b Fp(\000)h Fq(La)707 5185 y Fj(\000)p Fo(1)795 5216 y Fu(and)33 b Fq(L)21 b Fp(\000)h Fq(b)33 b Fu(are)f(p)r(ositiv)n(e)g(n)n(um)n(b)r(ers.)52 b(Moreo)n(v)n(er)30 b(if)k(w)n(e)e(in)n(tro)r(duce)h(the)g(constan)n(t) p eop %%Page: 33 41 33 40 bop 1375 251 a Fo(3.6.)29 b(PR)n(OOF)g(OF)g(THE)g(THEOREM)g(3.1) 853 b(33)456 450 y Fq(K)28 b Fu(=)677 417 y Fo(2)p Fi(a)p 653 431 118 4 v 653 479 a(a)p Fj(\000)p Fi(b)803 450 y Fq(>)22 b Fu(2)28 b(w)n(e)f(ha)n(v)n(e)f(that)1592 672 y Fq(L)18 b Fp(\000)g Fq(b)82 b Fu(=)2026 616 y Fq(a)18 b Fp(\000)g Fq(b)p 2026 653 181 4 v 2096 729 a Fu(2)2240 672 y Fq(>)2348 616 y(L)p 2338 653 77 4 v 2338 729 a(K)1453 876 y Fu(1)g Fp(\000)g Fq(La)1697 842 y Fj(\000)p Fo(1)1868 876 y Fu(=)2026 820 y Fq(a)g Fp(\000)g Fq(b)p 2026 857 181 4 v 2074 933 a Fu(2)p Fq(a)2240 876 y Fu(=)2355 820 y(1)p 2338 857 77 4 v 2338 933 a Fq(K)2424 876 y Fu(.)456 1067 y(Therefore,)26 b(w)n(e)h(can)g(b)r(ound)h(\(3.32\))f(as)g(follo)n (ws)592 1270 y Fq(La)693 1236 y Fj(\000)p Fo(1)782 1270 y Fp(k)p Fq(\033)871 1282 y Fo(2)908 1270 y Fp(k)950 1282 y Fi(m)p Fo(+1)1115 1270 y Fu(+)18 b Fq(b)p Fp(k)p Fq(\033)1323 1282 y Fo(1)1360 1270 y Fp(k)1402 1282 y Fi(m)1483 1270 y Fu(+)g Fq(O)r Fu(\()p Fq(")p Fu(\))84 b Fp(\024)e Fu(\()p Fp(k)p Fq(\033)2086 1282 y Fo(2)2124 1270 y Fp(k)2166 1282 y Fi(m)p Fo(+1)2331 1270 y Fu(+)18 b Fq(L)p Fp(k)p Fq(\033)2560 1282 y Fo(1)2597 1270 y Fp(k)2639 1282 y Fi(m)2701 1270 y Fu(\)\(1)h Fp(\000)2936 1214 y Fu(1)p 2919 1251 V 2919 1327 a Fq(K)3005 1270 y Fu(\))g(+)f Fq(O)r Fu(\()p Fq(")p Fu(\))1818 1481 y Fp(\024)1965 1364 y Fg(\022)2026 1481 y Fu(1)g Fp(\000)2197 1425 y Fu(1)p 2179 1462 V 2179 1538 a Fq(K)2266 1364 y Fg(\023)2341 1481 y Fq(r)j Fu(+)d Fq(O)r Fu(\()p Fq(")p Fu(\))1818 1638 y Fq(<)82 b(r)456 1801 y Fu(if)28 b Fq(")f Fu(is)h(small)f(enough.)605 1901 y(No)n(w,)e(w)n(e)f(pro)n(v)n(e)e (that)j Fp(B)1389 1871 y Fj(\000)p Fo(1)1489 1901 y Fp(\016)12 b(F)32 b Fu(is)25 b(a)e(con)n(traction.)35 b(Let)24 b Fq(\033)j Fu(=)22 b(\()p Fq(\033)2634 1913 y Fo(1)2672 1901 y Fq(;)14 b(\033)2756 1913 y Fo(2)2794 1901 y Fu(\))23 b Fp(2)h Fq(B)t Fu(\()p Fq(r)n(;)14 b(m;)g(m)e Fu(+)g(1\))456 2001 y(and)31 b(\026)-46 b Fq(\033)27 b Fu(=)22 b(\()t(\026)-46 b Fq(\033)857 2013 y Fo(1)895 2001 y Fq(;)18 b Fu(\026)-46 b Fq(\033)979 2013 y Fo(2)1017 2001 y Fu(\))23 b Fp(2)g Fq(B)t Fu(\()p Fq(r)n(;)14 b(m;)g(m)19 b Fu(+)f(1\).)37 b(F)-7 b(rom)27 b(\(3.27\))g(it)h(is)f(easy)g(to)g(see)h(that)1373 2164 y Fp(k)p Fq(H)7 b Fu(\()p Fq(\033)s Fu(\))19 b Fp(\000)f Fq(H)7 b Fu(\()t(\026)-46 b Fq(\033)s Fu(\))p Fp(k)1939 2176 y Fi(m)2025 2164 y Fp(\024)23 b Fq(C)6 b Fp(k)p Fq(\033)22 b Fp(\000)g Fu(\026)-46 b Fq(\033)s Fp(k)2464 2176 y Fi(m)456 2333 y Fu(for)24 b(some)h(constan)n(t)f Fq(C)29 b(>)23 b Fu(0)i(.)36 b(Th)n(us,)25 b(b)n(y)g(de\014nition)h (\(3.28\))e(of)h Fp(F)8 b Fu(,)26 b(w)n(e)e(obtain,)i(doing)e(similar) 456 2433 y(estimations)j(as)g(ab)r(o)n(v)n(e)498 2596 y Fp(kB)598 2562 y Fj(\000)p Fo(1)704 2596 y Fp(\016)18 b(F)8 b Fu(\()p Fq(\033)911 2608 y Fo(1)949 2596 y Fq(;)14 b(\033)1033 2608 y Fo(2)1070 2596 y Fu(\))19 b Fp(\000)f(B)1262 2562 y Fj(\000)p Fo(1)1369 2596 y Fp(\016)f(F)8 b Fu(\()t(\026)-46 b Fq(\033)1575 2608 y Fo(1)1613 2596 y Fq(;)19 b Fu(\026)-47 b Fq(\033)1697 2608 y Fo(2)1735 2596 y Fp(k)1777 2608 y Fi(m)855 2787 y Fp(\024)22 b Fq(L)1029 2731 y(n)d Fp(\000)f Fu(2)p 1009 2768 235 4 v 1009 2844 a(2)p Fq(n)g Fp(\000)g Fu(2)1253 2787 y Fp(k)p Fq(\033)1342 2799 y Fo(2)1398 2787 y Fp(\000)k Fu(\026)-46 b Fq(\033)1528 2799 y Fo(2)1565 2787 y Fp(k)1607 2799 y Fi(m)p Fo(+1)1772 2787 y Fu(+)2079 2731 y(2)p Fq(n)2171 2701 y Fo(2)p 1865 2768 557 4 v 1865 2844 a Fu(\(3)p Fq(n)19 b Fp(\000)f Fu(4\)\()p Fq(n)g Fp(\000)g Fu(2\))2432 2787 y Fp(k)p Fq(\033)2521 2799 y Fo(1)2576 2787 y Fp(\000)23 b Fu(\026)-47 b Fq(\033)2706 2799 y Fo(1)2744 2787 y Fp(k)2786 2799 y Fi(m)2867 2787 y Fu(+)18 b Fq("C)6 b Fp(k)p Fq(\033)21 b Fp(\000)i Fu(\026)-47 b Fq(\033)t Fp(k)3340 2799 y Fi(m)855 3019 y Fp(\024)942 2901 y Fg(\022)1003 3019 y Fu(1)18 b Fp(\000)1195 2962 y Fu(1)p 1156 2999 118 4 v 1156 3075 a(2)p Fq(K)1284 2901 y Fg(\023)1359 3019 y Fp(k)p Fq(\033)k Fp(\000)g Fu(\026)-46 b Fq(\033)s Fp(k)1645 3031 y Fi(m)456 3227 y Fu(if)28 b Fq(")f Fu(is)h(small)f(enough.)605 3327 y(Then,)d(applying)e(the)h(\014xed)g(p)r(oin)n(t)g(theorem)g(w)n(e)f (obtain)g(the)i(existence)e(and)h(uniqueness)456 3427 y(of)k(a)g(\014xed)h(p)r(oin)n(t)g(\()p Fq(\033)1116 3439 y Fo(1)1154 3427 y Fq(;)14 b(\033)1238 3439 y Fo(2)1275 3427 y Fu(\))24 b Fp(2)f Fq(B)t Fu(\()p Fq(r)n(;)14 b(m;)g(m)19 b Fu(+)f(1\).)36 b(This)28 b(ends)g(the)g(pro)r(of)f(of)g(the)h(lemma.) 142 b Fe(\003)605 3618 y Fu(No)n(w)27 b(w)n(e)g(can)h(\014nish)f(the)h (pro)r(of)f(of)h(Theorem)f(3.1.)605 3809 y Ft(End)k(of)g(the)h(pr)n (oof)g(of)f(Theorem)h(3.1.)41 b Fu(Going)35 b(bac)n(k)g(to)h(the)g (original)f(v)-5 b(ariables)456 3909 y(w)n(e)23 b(obtain)h(the)g (result)g(w)n(e)g(ha)n(v)n(e)f(stated)h(in)g(Theorem)f(3.1.)35 b(Indeed,)25 b(b)n(y)f(Prop)r(osition)e(3.1)h(the)456 4008 y(c)n(hange)j Fp(C)32 b Fu(has)27 b(the)h(form)978 4172 y(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)f Fp(C)5 b Fu(\()g(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(t=")p Fu(\))22 b(=)h(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s Fu(\))19 b(+)f Fq(\026")2164 4138 y Fi(p)p Fo(+1)2286 4172 y Fq(G)p Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(t=")p Fu(\))j(+)h Fq(O)r Fu(\()p Fq(\026")2978 4138 y Fi(p)p Fo(+2)3101 4172 y Fu(\))456 4341 y(where)23 b(\()p Fq(x;)14 b(y)s Fu(\))24 b(are)f(the)h(original)e(v)-5 b(ariables)22 b(and)h(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s Fu(\))24 b(are)e(the)i(v)-5 b(ariables)23 b(for)g(whic)n(h)g(w)n(e)h(ha)n(v)n(e)456 4440 y(obtained)33 b(the)h(parameterization)d(of)j(the)f(in)n(v)-5 b(arian)n(t)33 b(lo)r(cal)g(stable)g(curv)n(e.)53 b(Therefore,)34 b(the)456 4540 y(lo)r(cal)27 b(stable)g(manifold,)h Fq(\037)1311 4510 y Fi(s)1311 4560 y(\026;")1406 4540 y Fu(,)g(of)f(the)h(equation)f (\(3.1\))h(has)f(the)h(form)833 4716 y Fq(\037)885 4682 y Fi(s)885 4736 y(\026;")980 4716 y Fu(\()p Fq(t;)14 b(s)p Fu(\))84 b(=)h(~)-45 b Fq(\015)1429 4682 y Fi(s)1424 4736 y(\026;")1520 4716 y Fu(\()p Fq(t;)14 b(s)p Fu(\))19 b(+)f Fq(\026")1881 4682 y Fi(p)p Fo(+1)2003 4716 y Fq(G)p Fu(\()s(~)-45 b Fq(\015)2148 4682 y Fi(s)2143 4736 y(\026;")2239 4716 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(\026")2876 4682 y Fi(p)p Fo(+2)2999 4716 y Fu(\))1234 4853 y(=)82 b Fq(\015)1424 4865 y Fo(0)1462 4853 y Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\))h(+)f Fq(\026")1887 4818 y Fi(p)p Fo(+1)2009 4853 y Fq(G)p Fu(\()p Fq(\015)2149 4865 y Fo(0)2187 4853 y Fu(\()p Fq(t)h Fu(+)f Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(\026")2889 4818 y Fi(p)p Fo(+2)3012 4853 y Fu(\).)456 5016 y(and)h(it)h (satis\014es)f(all)g(the)h(conditions)f(of)g(Theorem)g(3.1)g(except)g (that)h(it)g(need)f(not)h(b)r(e)g(a)f(solution)456 5116 y(of)25 b(the)i(equation)e(\(3.1\))h(with)g(resp)r(ect)g(to)f Fq(t)p Fu(.)37 b(Conclusion)25 b(2\))h(follo)n(ws)e(b)r(ecause)i(the)g (c)n(hange)f Fp(C)456 5216 y Fu(is)i(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)g(in)h Fq(t)p Fu(.)p eop %%Page: 34 42 34 41 bop 456 251 a Fo(34)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(In)40 b(order)f(to)g(\014nd)i(a)e(parameterization)f(of)i(the)g(lo)r (cal)g(stable)f(manifold)h(whic)n(h)g(is)g(a)456 550 y(solution)25 b(with)i(resp)r(ect)f(to)g Fq(t)g Fu(for)f(an)n(y)h Fq(s)d Fp(2)g Fl(C)47 b Fu(suc)n(h)26 b(that)h Fp(j)14 b Fu(Im)f Fq(s)p Fp(j)23 b(\024)g Fq(\034)36 b Fu(w)n(e)26 b(tak)n(e)f Fq(t)2990 562 y Fo(0)3050 550 y Fu(=)e Fq(T)k Fp(\000)15 b Fu(Re)e Fq(s)456 649 y Fu(and)1418 784 y Fq(\015)1466 750 y Fi(s)1461 804 y(\026;")1556 784 y Fu(\()p Fq(t;)h(s)p Fu(\))24 b(=)e Fq(')p Fu(\()p Fq(t;)14 b(t)2020 796 y Fo(0)2058 784 y Fq(;)g(\037)2147 750 y Fi(s)2147 804 y(\026;")2242 784 y Fu(\()p Fq(t)2304 796 y Fo(0)2342 784 y Fq(;)g(s)p Fu(\)\))456 923 y(where)28 b(here)g Fq(')p Fu(\()p Fq(t;)14 b(t)1061 935 y Fo(0)1099 923 y Fq(;)g(x;)g(y)s Fu(\))29 b(is)g(the)g(general)f(solution)g(of)h (equation)f(\(3.1\).)41 b(W)-7 b(e)29 b(observ)n(e)e(that,)456 1023 y(for)33 b(all)h(\()p Fq(t;)14 b(s)p Fu(\))34 b Fp(2)g Fq(D)1075 993 y Fi(s)1111 1023 y Fu(,)i Fq(\037)1222 993 y Fi(s)1222 1043 y(\026;")1317 1023 y Fu(\()p Fq(t;)14 b(s)p Fu(\))35 b(b)r(elongs)e(to)h(the)g(lo)r(cal)g(stable)g(manifold)g (of)g(the)g(equation)456 1125 y(\(3.1\),)29 b(hence)h Fq(\015)960 1095 y Fi(s)955 1146 y(\026;")1050 1125 y Fu(\()p Fq(t;)14 b(s)p Fu(\))30 b(is)g(a)f(parameterization)e(of)i(the) h(stable)f(manifold.)43 b(It)29 b(is)h(clear)e(that)456 1228 y(the)g(prop)r(erties)e(1\),)i(3\))f(and)h(4\))f(of)h(Theorem)e (3.1)h(are)g(satis\014ed)g(b)n(y)g Fq(\015)2678 1198 y Fi(s)2673 1248 y(\026;")2769 1228 y Fu(.)37 b(Moreo)n(v)n(er)632 1365 y Fq(\015)680 1331 y Fi(s)675 1386 y(\026;")770 1365 y Fu(\()p Fq(t;)14 b(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\))83 b(=)f Fq(')p Fu(\()p Fq(t;)14 b(t)1586 1377 y Fo(0)1624 1365 y Fq(;)g(\037)1713 1331 y Fi(s)1713 1386 y(\026;")1809 1365 y Fu(\()p Fq(t)1871 1377 y Fo(0)1908 1365 y Fq(;)g(s)k Fu(+)g(2)p Fq(\031)s(")p Fu(\)\))23 b(=)g Fq(')p Fu(\()p Fq(t;)14 b(t)2574 1377 y Fo(0)2612 1365 y Fq(;)g(\037)2701 1331 y Fi(s)2701 1386 y(\026;")2796 1365 y Fu(\()p Fq(t)2858 1377 y Fo(0)2914 1365 y Fu(+)k(2)p Fq(\031)s(";)c(s)p Fu(\)\))1256 1492 y(=)82 b Fq(')p Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(t)1819 1504 y Fo(0)1875 1492 y Fu(+)k(2)p Fq(\031)s(";)c(')p Fu(\()p Fq(t)2242 1504 y Fo(0)2297 1492 y Fu(+)k(2)p Fq(\031)s(";)c(t)2578 1504 y Fo(0)2615 1492 y Fq(;)g(\037)2704 1458 y Fi(s)2704 1513 y(\026;")2799 1492 y Fu(\()p Fq(t)2861 1504 y Fo(0)2899 1492 y Fq(;)g(s)p Fu(\)\)\))1256 1620 y(=)82 b Fq(')p Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(t)1819 1632 y Fo(0)1856 1620 y Fq(;)g(\037)1945 1585 y Fi(s)1945 1640 y(\026;")2041 1620 y Fu(\()p Fq(t)2103 1632 y Fo(0)2140 1620 y Fq(;)g(s)p Fu(\)\))1256 1747 y(=)82 b Fq(\015)1451 1713 y Fi(s)1446 1767 y(\026;")1542 1747 y Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\).)456 1881 y(Therefore,)32 b Fq(\015)909 1851 y Fi(s)904 1902 y(\026;")1032 1881 y Fu(is)g(the)h(parameterization)d(that)j(w)n(e)f(lo)r(ok)g(for.)50 b(W)-7 b(e)33 b(observ)n(e)e(that)i Fq(\015)3266 1851 y Fi(s)3261 1902 y(\026;")3389 1881 y Fu(is)456 1984 y(de\014ned)28 b(for)f(all)g(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)h Fq(D)1327 1954 y Fi(s)1362 1984 y Fu(.)1995 b Fe(\003)p eop %%Page: 35 43 35 42 bop 1239 566 a Fw(4.)61 b(Flo)l(w)45 b(b)t(o)l(x)g(co)t (ordinates)1589 849 y Fk(4.1.)h(In)m(tro)s(duction)605 998 y Fu(In)29 b(this)h(c)n(hapter)e(w)n(e)h(pro)n(v)n(e)f(the)i (existence)f(of)g(\015o)n(w)f(b)r(o)n(x)h(co)r(ordinates)f(of)h(a)g (quasi)g(in)n(te-)456 1098 y(grable)e(system)i(under)g(general)e(h)n (yp)r(otheses,)i(in)g(a)g(neigh)n(b)r(orho)r(o)r(d)f(of)g(the)i(stable) e(manifold)456 1197 y(of)34 b(the)h(unp)r(erturb)r(ed)g(system,)g(whic) n(h)g(do)r(es)f(not)g(con)n(tain)g(the)h(origin)e(and)h(it)h(is)f (indep)r(en-)456 1297 y(den)n(t)29 b(of)h(the)g(parameters)e Fq(\026)h Fu(and)h Fq(")p Fu(.)42 b(A)30 b(similar)f(result)g(is)g(in)h ([)p Fk(Ge2)p Fu(].)43 b(There,)29 b(the)h(\015o)n(w)f(b)r(o)n(x)456 1397 y(co)r(ordinates)d(are)g(found)i(implicitly)h(using)e(the)h(v)-5 b(ariational)26 b(equations)h(in)g(a)h(neigh)n(b)r(orho)r(o)r(d)456 1496 y(of)33 b(the)g(stable)g(manifold.)53 b(Our)33 b(pro)r(of)f(giv)n (es)g(these)h(co)r(ordinates)f(in)i(an)e(explicit)i(w)n(a)n(y)e(and)456 1596 y(giv)n(es)f(a)h(careful)h(estimate)f(of)h(the)g(distance)g(b)r (et)n(w)n(een)f(the)h(c)n(hange)f(of)h(co)r(ordinates)e(in)i(the)456 1695 y(unp)r(erturb)r(ed)23 b(case)g(and)g(the)h(c)n(hange)e(in)h(the)h (p)r(erturb)r(ed)f(case,)h(using)f(v)-5 b(ariational)21 b(equations)456 1795 y(with)28 b(resp)r(ect)f(to)h(the)g(parameter)e Fq(\026)p Fu(.)605 1895 y(In)e([)p Fk(DS2)p Fu(])h([)p Fk(DS1)p Fu(])f(the)g(authors)f(use)h(\015o)n(w-b)r(o)n(x)f(co)r (ordinates)g(de\014ned)h(near)f(a)h(h)n(yp)r(erb)r(olic)456 1994 y(\014xed)30 b(p)r(oin)n(t.)47 b(T)-7 b(o)30 b(construct)g(suc)n (h)g(co)r(ordinates)g(they)h(use)f(the)h(Birkho\013)f(Normal)g(F)-7 b(orm)30 b(in)456 2094 y(an)c(essen)n(tial)g(w)n(a)n(y)-7 b(.)35 b(Also,)27 b(in)g([)p Fk(DG)p Fu(],)g(the)g(authors)f(construct) g(\015o)n(w)g(b)r(o)n(x)g(co)r(ordinates)f(near)h(a)456 2194 y Fq(n)p Fu(-dimensional)g(h)n(yp)r(erb)r(olic)h(in)n(v)-5 b(arian)n(t)27 b(tori,)g(but)h(only)f(in)h(real)f(domains.)605 2293 y(W)-7 b(e)29 b(b)r(egin)g(b)n(y)f(in)n(tro)r(ducing)g(notation)g (and)g(the)h(h)n(yp)r(otheses)f Fk(H1)g Fu(and)g Fk(H2)h Fu(w)n(e)f(will)g(as-)456 2393 y(sume)23 b(in)i(this)f(c)n(hapter.)34 b(With)25 b(these)f(h)n(yp)r(otheses)f(w)n(e)h(will)g(pro)n(v)n(e)e(a)h (result)h(on)g(the)g(existence)456 2492 y(of)34 b(\015o)n(w)g(b)r(o)n (x)g(co)r(ordinates:)50 b(Theorem)33 b(4.1.)58 b(Then,)36 b(the)f(application)f(of)h(this)g(theorem)f(to)456 2592 y(equation)k(\(1.1\))g(of)h(Chapter)f(1)h(giv)n(es)e(Theorem)h(4.2,)j (in)e(whic)n(h)g(the)g(result)f(is)h(obtained)456 2692 y(applying)27 b(Theorem)f(4.1,)h(not)h(directly)f(to)h(\(1.1\))f(but)h (to)g(the)g(a)n(v)n(eraged)c(equation.)605 2791 y(T)-7 b(o)27 b(pro)n(v)n(e)f(Theorem)h(4.1)g(\014rst,)g(in)h(Section)g(4.3,)f (w)n(e)g(translate)g(the)h(stable)f(manifold)h(to)456 2891 y(the)h(\014rst)f(axis)g(of)h(co)r(ordinates)e(and)i(in)g(these)f (co)r(ordinates,)g(for)g(the)h(unp)r(erturb)r(ed)h(system,)456 2991 y(w)n(e)d(construct)g(explicitly)h(the)g(\015o)n(w)f(b)r(o)n(x)g (co)r(ordinates,)f(just)i(in)n(tegrating)f(the)h(equation)f(and)456 3090 y(using)g(that)h(the)g(system)f(is)h(Hamiltonian.)605 3190 y(T)-7 b(o)23 b(construct)f(the)h(\015o)n(w)g(b)r(o)n(x)f(co)r (ordinates)g(for)g(the)i(general)d(system)i(in)g(a)g(neigh)n(b)r(orho)r (o)r(d)456 3289 y(of)f(\(a)g(part)g(of)6 b(\))23 b(the)f(stable)g (manifold,)i(w)n(e)d(use)h(a)g(sp)r(ecial)g(parameterization)e(of)j (the)f(solutions)456 3389 y(of)38 b(the)i(equation.)70 b(W)-7 b(e)39 b(parameterize)e(the)j(solutions,)h Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)j Fu(\))40 b(with)f(t)n(w)n(o)f(parameters) 456 3489 y(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)g Fl(R)i Fp(\002)18 b Fl(C)49 b Fu(in)27 b(suc)n(h)h(a)f(w)n(a)n(y)f(that)i Fq(t)23 b Fp(2)g Fl(R)34 b Fu(is)28 b(a)f(time)h(parameter,)e Fq(Y)42 b Fp(2)23 b Fl(C)49 b Fu(and)1346 3643 y Fq(z)t Fu(\()p Fq(t)18 b Fu(+)g(2)p Fq(\031)s(";)c(s;)g(Y)k Fu(\))24 b(=)e Fq(z)t Fu(\()p Fq(t;)14 b(s)k Fu(+)g(2)p Fq(\031)s(";)c(Y)k Fu(\))456 3797 y(in)31 b(a)g(suitable)g(domain.)48 b(T)-7 b(o)31 b(obtain)g(this)g(w)n(e)g(use)g(a)g(tec)n(hnique)h (designed)e(b)n(y)i(Lazutkin)f(to)456 3897 y(do)d(a)h(con)n(trolled)f (analytic)g(con)n(tin)n(uation.)41 b(This)29 b(is)g(done)g(in)g (Subsections)g(4.5.2)f(and)h(4.5.3.)456 3996 y(F)-7 b(rom)27 b(this)h(parameterization)d(w)n(e)i(easily)g(obtain)h(another)e (parameterization)g(of)h(the)h(form)1665 4150 y Fq(w)r Fu(\()p Fq(t)19 b Fu(+)f Fq(s;)c(t=";)g(Y)k Fu(\),)456 4309 y(that)27 b(is,)h(w)n(e)f(separate)f(in)i(some)f(w)n(a)n(y)g(the)h (slo)n(w)e(time)i Fq(t)19 b Fu(+)f Fq(s)27 b Fu(and)h(the)g(fast)f (time)i Fq(t=")p Fu(.)605 4409 y(Next)f(w)n(e)f(\014nd)h(a)f(\014rst)h (\015o)n(w)f(b)r(o)n(x)g(co)r(ordinates)f(\()p Fp(T)c Fq(;)14 b Fp(Y)7 b Fu(\))28 b(from)f(the)h(condition)1343 4563 y Fq(w)r Fu(\()p Fp(T)23 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g(\022)r(;)g Fp(Y)7 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\)\))26 b(=)d(\()p Fq(x;)14 b(v)s Fu(\))456 4717 y(using)32 b(the)i(sc)n(heme)e(of)h(the)h(pro)r(of)e(of)h(the)h(implicit)f (function)h(theorem.)53 b(W)-7 b(e)33 b(obtain)g(them)456 4817 y(close)f(to)h(the)h(analogous)d(ones)i(w)n(e)f(ha)n(v)n(e)h (calculated)f(in)i(the)g(non)f(p)r(erturb)r(ed)g(case.)53 b(Then)456 4917 y(easily)18 b(w)n(e)g(pass)h(to)f(new)i(\015o)n(w)e(b)r (o)n(x)g(co)r(ordinates)g(\()p Fp(T)k Fq(;)14 b Fp(F)8 b Fu(\))19 b(with)g Fp(F)27 b Fu(close)18 b(to)h(the)h(energy)e(v)-5 b(ariable)456 5016 y(\(the)28 b(Hamiltonian\).)605 5116 y(Finally)-7 b(,)33 b(using)e(the)h(Hamiltonian)g(c)n(haracter)e(of)h (the)h(equations,)g(w)n(e)g(sligh)n(tly)f(mo)r(dify)456 5216 y(these)c(co)r(ordinates)f(to)i(mak)n(e)f(them)h(canonical.)1917 5365 y Fo(35)p eop %%Page: 36 44 36 43 bop 456 251 a Fo(36)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)1281 450 y Fk(4.2.)46 b(De\014nitions)30 b(and)j(main)d(result)605 600 y Fu(W)-7 b(e)28 b(consider)f(Hamiltonian)g(systems)g(of)h(the)g (form)1120 743 y Fq(H)7 b Fu(\()p Fq(x;)14 b(y)s(;)g(t=")p Fu(\))22 b(=)h Fq(H)1715 755 y Fo(0)1752 743 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(\026")2135 708 y Fi(q)2172 743 y Fq(H)2241 755 y Fo(1)2278 743 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g (\026;)g(")p Fu(\))456 886 y(where)1543 1061 y Fq(H)1612 1073 y Fo(0)1649 1061 y Fu(\()p Fq(x;)g(y)s Fu(\))24 b(=)1963 1005 y Fq(y)2007 975 y Fo(2)p 1963 1042 81 4 v 1982 1118 a Fu(2)2072 1061 y(+)18 b Fq(V)h Fu(\()p Fq(x)p Fu(\))p Fq(:)605 1239 y Ft(Remark)28 b Fu(4.1)p Ft(.)36 b Fs(Sinc)l(e)27 b(we)f(wil)t(l)i(apply)g(the)e(r)l(esult)g(of) h(existenc)l(e)f(of)h(\015ow)f(b)l(ox)g(c)l(o)l(or)l(dinates)456 1339 y(to)41 b(an)h(aver)l(age)l(d)h(system,)i(her)l(e)d Fq(q)j Fs(and)d Fq(H)1873 1351 y Fo(1)1951 1339 y Fs(me)l(an)g(a)g (generic)g(exp)l(onent)f(and)i(a)e(generic)456 1438 y(Hamiltonian)23 b(r)l(esp)l(e)l(ctively)g(which)h(\(in)e(gener)l(al\))h(do)g(not)f(c)l (oincide)i(with)f Fq(p)e Fs(and)i Fq(h)3019 1450 y Fo(1)3078 1438 y Fs(intr)l(o)l(duc)l(e)l(d)456 1538 y(in)29 b(Chapter)j(1.)605 1692 y Fu(The)c(asso)r(ciated)e(equations)h(of)g(the)h(Hamiltonian)g Fq(H)34 b Fu(are)1220 1835 y(_)-38 b Fq(x)84 b Fu(=)e Fq(y)22 b Fu(+)c Fq(\026")1718 1801 y Fi(q)1754 1835 y Fq(@)1798 1847 y Fi(y)1838 1835 y Fq(H)1907 1847 y Fo(1)1944 1835 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g(\026;)g(")p Fu(\))-1991 b(\(4.1\))1224 1960 y(_)-38 b Fq(y)86 b Fu(=)c Fp(\000)p Fq(V)1615 1926 y Fj(0)1638 1960 y Fu(\()p Fq(x)p Fu(\))20 b Fp(\000)e Fq(\026")1941 1926 y Fi(q)1977 1960 y Fq(@)2021 1972 y Fi(x)2063 1960 y Fq(H)2132 1972 y Fo(1)2169 1960 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g(\026;)g(")p Fu(\).)456 2103 y(F)-7 b(or)27 b Fq(w)e Fu(=)e(\()p Fq(w)868 2115 y Fo(1)906 2103 y Fq(;)14 b(w)1002 2115 y Fo(2)1039 2103 y Fu(\))24 b Fp(2)f Fl(C)1227 2073 y Fo(2)1270 2103 y Fu(,)28 b(w)n(e)f(de\014ne)1531 2246 y Fp(k)p Fq(w)r Fp(k)c Fu(=)f(max)p Fp(fj)p Fq(w)2065 2258 y Fo(1)2102 2246 y Fp(j)p Fq(;)14 b Fp(j)p Fq(w)2244 2258 y Fo(2)2282 2246 y Fp(jg)p Fu(.)456 2389 y(W)-7 b(e)28 b(will)f(assume)g(the)h (follo)n(wing)f(h)n(yp)r(otheses)630 2511 y Fk(H1)41 b Fu(The)34 b(p)r(oten)n(tial)g Fq(V)53 b Fu(is)34 b(an)g(analytic)f (function)i(in)f Fp(f)p Fq(x)g Fp(2)g Fl(C)55 b Fu(:)34 b Fp(j)p Fq(x)p Fp(j)g Fq(<)g(\032)3030 2523 y Fo(0)3067 2511 y Fp(g)p Fu(,)h Fq(V)19 b Fu(\()p Fq(x)p Fu(\))35 b(=)794 2610 y Fq(a)838 2622 y Fi(n)883 2610 y Fq(x)930 2580 y Fi(n)994 2610 y Fu(+)18 b Fq(:)c(:)g(:)42 b Fu(with)28 b Fq(a)1449 2622 y Fi(n)1517 2610 y Fq(<)23 b Fu(0,)k Fq(n)c Fp(2)g Fl(N)38 b Fu(and)27 b Fq(n)c Fp(\025)g Fu(3.)630 2710 y Fk(H2)41 b Fq(H)863 2722 y Fo(1)900 2710 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(";)g(\026)p Fu(\))23 b(is)f Fq(C)1499 2680 y Fo(0)1537 2710 y Fu(,)h(2)p Fq(\031)s Fu(-p)r(erio)r(dic)f(in)h Fq(\022)h Fu(and)e(analytic)g(in)h (the)f Fq(x;)14 b(y)s(;)g(\026)23 b Fu(v)-5 b(ariables.)794 2810 y(The)28 b(v)-5 b(ariables)26 b(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\))28 b(b)r(elong)g(to)1054 2954 y Fp(f)p Fu(\()p Fq(x;)14 b(y)s Fu(\))23 b Fp(2)g Fl(C)1443 2919 y Fo(2)1510 2954 y Fu(:)g Fp(k)p Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fp(k)23 b Fq(<)f(\032)1985 2966 y Fo(0)2022 2954 y Fp(g)c(\002)g(f)p Fq(\026)23 b Fp(2)g Fl(C)44 b Fu(:)23 b Fp(j)p Fq(\026)p Fp(j)h Fq(<)e(\026)2744 2966 y Fo(0)2782 2954 y Fp(g)p Fu(,)794 3110 y Fq(\022)39 b Fp(2)f Fl(R)k Fu(and)36 b(0)g Fq(<)h(")f(<)h(")1627 3122 y Fo(0)1664 3110 y Fu(.)62 b(Moreo)n(v)n(er)33 b Fq(H)2191 3122 y Fo(1)2228 3110 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(";)g(\026)p Fu(\))38 b(=)e Fq(O)r Fu(\()p Fp(k)p Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fp(k)3173 3080 y Fi(k)3215 3110 y Fu(\))36 b(with)794 3210 y Fq(k)26 b Fp(\025)d Fu(2.)605 3331 y(In)34 b(the)h(applications) e(of)i(the)f(results)g(of)g(this)h(c)n(hapter,)g Fq(k)i Fu(will)d(b)r(e)h(alw)n(a)n(ys)d(greater)h(of)456 3431 y(equal)27 b(than)g Fq(n)19 b Fp(\000)f Fu(1.)605 3530 y(F)-7 b(or)36 b(the)i(unp)r(erturb)r(ed)f(system,)i(the)e(origin)f (has)h(a)f(stable)h(manifold)g(whic)n(h)g(can)f(b)r(e)456 3630 y(represen)n(ted)26 b(as)h(the)h(graph)e(of)i(the)g(function)1586 3779 y Fq(f)9 b Fu(\()p Fq(x)p Fu(\))24 b(=)e Fp(\000)1923 3704 y Fg(p)p 2006 3704 285 4 v 75 x Fp(\000)p Fu(2)p Fq(V)c Fu(\()p Fq(x)p Fu(\))q(.)456 3923 y(W)-7 b(e)28 b(observ)n(e)d(that)j Fq(f)37 b Fu(is)27 b(analytic)g(in)989 4066 y(\012\()p Fq(\032)1124 4078 y Fo(0)1162 4066 y Fq(;)14 b(\031)s Fu(\))23 b(=)g Fp(f)p Fq(x)g Fp(2)g Fl(C)44 b Fu(:)23 b Fp(j)p Fq(x)p Fp(j)h Fq(<)f(\032)1959 4078 y Fo(0)1996 4066 y Fp(g)18 b(n)g(f)p Fu(Re)13 b Fq(x)24 b Fp(\024)e Fu(0,)41 b(Im)14 b Fq(x)24 b Fu(=)f(0)p Fp(g)p Fq(:)605 4225 y Ft(Remark)k Fu(4.2)p Ft(.)36 b Fs(By)26 b(Pr)l(op)l(osition)h(2.2)g(of)g(Chapter)f(2,)i(the)d(unp)l (erturb)l(e)l(d)g(system)g(\()p Fq(\026)e Fu(=)g(0)p Fs(\))456 4325 y(has)30 b(a)g(p)l(ar)l(ameterization)i(of)e(the)g (stable)g(manifold)i(de\014ne)l(d)e(in)1395 4468 y Fp(f)p Fq(u)22 b Fp(2)i Fl(C)44 b Fu(:)23 b Fp(j)p Fq(u)p Fp(j)g(\025)f Fq(T)12 b Fs(,)59 b Fp(j)14 b Fu(arg)g Fq(u)p Fp(j)22 b Fq(<)h(\031)s Fp(g)456 4616 y Fs(which)31 b(we)f(denote)g(by)g Fq(\015)1228 4628 y Fo(0)1266 4616 y Fu(\()p Fq(u)p Fu(\))23 b(=)f(\()p Fq(\013)1573 4628 y Fo(0)1611 4616 y Fu(\()p Fq(u)p Fu(\))p Fq(;)14 b(\014)1807 4628 y Fo(0)1844 4616 y Fu(\()p Fq(u)p Fu(\)\))p Fs(.)605 4770 y Fu(In)28 b(the)g(next)g (de\014nition)g(w)n(e)f(\014x)g(some)g(parameters.)605 4924 y Ft(Definition)32 b Fu(4.1)p Ft(.)40 b Fs(L)l(et)29 b Fq(C)1451 4936 y Fo(0)1519 4924 y Fs(and)h Fq(C)1739 4936 y Fo(1)1806 4924 y Fs(b)l(e)g(such)g(that)1501 5068 y Fq(C)1560 5080 y Fo(0)1597 5068 y Fp(j)p Fq(x)p Fp(j)1690 5033 y Fi(\014)1759 5068 y Fp(\024)22 b(j)p Fq(f)9 b Fu(\()p Fq(x)p Fu(\))p Fp(j)24 b(\024)f Fq(C)2224 5080 y Fo(1)2261 5068 y Fp(j)p Fq(x)p Fp(j)2354 5033 y Fi(\014)456 5216 y Fs(for)30 b(al)t(l)h Fq(x)24 b Fp(2)f Fu(\012\()p Fq(\032)991 5228 y Fo(0)1028 5216 y Fq(;)14 b(\031)s Fu(\))31 b Fs(wher)l(e)f Fq(\014)d Fu(=)c Fq(n=)p Fu(2)p Fs(.)p eop %%Page: 37 45 37 44 bop 1323 251 a Fo(4.2.)29 b(DEFINITIONS)g(AND)g(MAIN)g(RESUL)-5 b(T)800 b(37)605 450 y Fs(L)l(et)32 b Fq(\024)799 462 y Fo(0)865 450 y Fp(2)c Fl(R)39 b Fs(b)l(e)33 b(as)g(in)g(Pr)l(op)l (osition)h(2.5.)49 b(We)33 b(take)g Fq(\016)2357 462 y Fo(0)2422 450 y Fu(=)28 b Fq(\013)2568 462 y Fo(0)2605 450 y Fu(\()p Fq(\024)2685 462 y Fo(0)2723 450 y Fu(\))p Fs(.)48 b(We)32 b(r)l(e)l(c)l(al)t(l)i(that)e(to)456 550 y(take)26 b Fq(\024)676 562 y Fo(0)740 550 y Fs(bigger)i(implies)g (to)e(take)h Fq(\016)1567 562 y Fo(0)1630 550 y Fs(smal)t(ler.)39 b(Given)27 b Fq(\016)2236 520 y Fj(0)2233 570 y Fo(0)2293 550 y Fq(>)c Fu(0)j Fs(such)h(that)f Fq(\016)2841 520 y Fj(0)2838 570 y Fo(0)2898 550 y Fq(<)d(\016)3023 562 y Fo(0)3060 550 y Fs(,)k(we)g(de\014ne)1210 730 y Fq(r)1247 742 y Fo(0)1308 730 y Fu(=)1405 674 y(1)p 1405 711 42 4 v 1405 787 a(2)1457 638 y Fg(\020)1525 730 y Fp(\000)18 b Fq(C)1667 742 y Fo(1)1705 730 y Fq(\016)1745 690 y Fi(\014)1742 752 y Fo(0)1808 730 y Fu(+)1891 622 y Fg(q)p 1974 622 642 4 v 108 x Fq(C)2039 702 y Fo(2)2033 752 y(1)2076 730 y Fq(\016)2116 690 y Fo(2)p Fi(\014)2113 752 y Fo(0)2213 730 y Fu(+)g Fq(C)2361 702 y Fo(2)2355 752 y(0)2398 730 y Fu(\()p Fq(\016)2470 702 y Fj(0)2467 752 y Fo(0)2505 730 y Fu(\))2537 706 y Fo(2)p Fi(\014)2615 638 y Fg(\021)2664 730 y Fs(.)605 895 y(In)29 b(the)h(fol)t(lowing)j (we)d(take)g Fq(\016)1543 865 y Fj(0)1540 915 y Fo(0)1607 895 y Fs(\(and)g(c)l(onse)l(quently)f Fq(r)2316 907 y Fo(0)2354 895 y Fs(\))g(smal)t(l)i(enough.)605 1046 y Fu(F)-7 b(or)25 b Fq(\024)800 1011 y Fj(\000)800 1068 y Fo(1)879 1046 y Fq(>)e(\024)1015 1058 y Fo(0)1052 1046 y Fu(,)j Fq(\024)1149 1011 y Fo(+)1149 1068 y(1)1227 1046 y Fq(>)d(\024)1363 1011 y Fj(\000)1363 1068 y Fo(1)1418 1046 y Fu(,)k Fq(\024)1516 1058 y Fo(2)1576 1046 y Fq(>)22 b Fu(0,)k Fq(\024)1802 1058 y Fo(3)1862 1046 y Fq(>)d Fu(0)i(suc)n(h)g(that)h Fq(\024)2428 1011 y Fj(\000)2428 1068 y Fo(1)2499 1046 y Fp(\000)14 b Fq(\024)2626 1058 y Fo(3)2686 1046 y Fq(>)23 b(\024)2822 1058 y Fo(0)2884 1046 y Fu(and)j(0)d Fq(<)f(r)k(<)d(r)3384 1058 y Fo(0)3421 1046 y Fu(,)456 1146 y(w)n(e)k(de\014ne)h(the)g(op)r(en)f(sets)629 1282 y Fq(D)698 1294 y Fo(0)735 1282 y Fu(\()p Fq(\024)815 1247 y Fj(\006)815 1304 y Fo(1)871 1282 y Fq(;)14 b(\024)956 1294 y Fo(2)993 1282 y Fq(;)g(\024)1078 1294 y Fo(3)1115 1282 y Fu(\))83 b(=)g Fp(f)p Fq(s)22 b Fp(2)i Fl(C)44 b Fu(:)23 b Fq(\024)1737 1247 y Fj(\000)1737 1304 y Fo(1)1811 1282 y Fp(\000)18 b Fq(\024)1942 1294 y Fo(3)2002 1282 y Fq(<)23 b Fu(Re)14 b Fq(s)23 b(<)g(\024)2400 1247 y Fo(+)2400 1304 y(1)2473 1282 y Fu(+)18 b Fq(\024)2604 1294 y Fo(3)2641 1282 y Fu(,)28 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b Fq(<)f(\024)3062 1294 y Fo(2)3118 1282 y Fu(+)c Fq(\024)3249 1294 y Fo(3)3286 1282 y Fp(g)626 1409 y Fq(D)697 1375 y Fj(\003)735 1409 y Fu(\()p Fq(\024)815 1373 y Fj(\006)815 1431 y Fo(1)871 1409 y Fq(;)c(\024)956 1421 y Fo(2)993 1409 y Fq(;)g(\024)1078 1421 y Fo(3)1115 1409 y Fu(\))83 b(=)g Fq(\013)1431 1421 y Fo(0)1468 1409 y Fu(\()p Fq(D)1569 1421 y Fo(0)1607 1409 y Fu(\()p Fq(\024)1687 1373 y Fj(\006)1687 1431 y Fo(1)1743 1409 y Fq(;)14 b(\024)1828 1421 y Fo(2)1865 1409 y Fq(;)g(\024)1950 1421 y Fo(3)1987 1409 y Fu(\)\))573 1536 y Fq(V)621 1548 y Fo(0)658 1536 y Fu(\()p Fq(\024)738 1501 y Fj(\006)738 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b(\024)2874 1821 y Fo(2)2911 1809 y Fq(;)g(\024)2996 1821 y Fo(3)3033 1809 y Fq(;)g(r)r Fu(\))p Fp(g)p Fu(.)456 1950 y(In)30 b(the)g(follo)n(wing)f(w)n(e)g(will)h(consider)f(some)g(c)n(hanges)g (of)g(v)-5 b(ariables)29 b(dep)r(ending)h(on)g(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\),)456 2050 y(and)33 b(w)n(e)h(will)g(refer)f(to)g (them)i(as)e(canonical)f(c)n(hanges,)i(meaning)f(that)h(are)f (canonical)g(as)g(a)456 2149 y(function)28 b(of)f(\()p Fq(x;)14 b(y)s Fu(\))29 b(with)f Fq(\022)i Fu(\014xed.)605 2249 y(The)e(main)f(result)h(of)f(this)h(Chapter)f(is:)605 2400 y Ft(Theorem)35 b Fu(4.1)p Ft(.)42 b Fs(\(Flow)34 b(b)l(ox)e(c)l(o)l(or)l(dinates\).)48 b(L)l(et)33 b Fq(\024)2279 2364 y Fj(\000)2279 2422 y Fo(1)2363 2400 y Fq(>)28 b Fu(2)p Fq(\024)2546 2412 y Fo(0)2582 2400 y Fs(.)47 b(If)33 b(the)g(hyp)l(otheses)h Ff(H1)p Fs(,)456 2501 y Ff(H2)e Fs(hold,)j(for)f(any)f Fq(\024)1164 2466 y Fo(+)1164 2524 y(1)1246 2501 y Fq(>)28 b(\024)1387 2466 y Fj(\000)1387 2524 y Fo(1)1475 2501 y Fs(and)33 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3250 y Fq(I)90 b Fu(=)83 b(0)456 3387 y Fs(and)30 b(satis\014es)841 3523 y Fp(T)907 3489 y Fo(1)944 3523 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fp(T)1371 3535 y Fo(0)1408 3523 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")1888 3489 y Fi(q)1926 3523 y Fu(\))p Fs(,)119 b Fp(I)2153 3489 y Fo(1)2191 3523 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)e Fp(I)2617 3535 y Fo(0)2655 3523 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")3135 3489 y Fi(q)3172 3523 y Fu(\))p Fs(,)456 3665 y(wher)l(e)36 b(the)f(change)h Fu(\()p Fq(x;)14 b(y)s Fu(\))34 b Fp(7!)f Fu(\()p Fp(T)1539 3677 y Fo(0)1577 3665 y Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fq(;)g Fp(I)1851 3677 y Fo(0)1889 3665 y Fu(\()p Fq(x;)g(y)s Fu(\)\))36 b Fs(is)g(the)f(c)l(orr)l(esp)l (onding)i(change)f(for)g(the)456 3764 y(unp)l(erturb)l(e)l(d)22 b(system.)36 b(Mor)l(e)l(over)24 b(the)f(change)i(is)e(c)l(ontinuous)f (in)h Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))24 b Fs(and)f(analytic)456 3864 y(in)29 b Fu(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\))p Fs(.)605 4014 y Fu(T)-7 b(o)30 b(study)g(the)g(splitting)g(of)g(separatrices)d(w)n(e)j(will)g (use)g(the)g(next)g(theorem)f(whic)n(h)h(is)g(a)456 4114 y(sp)r(ecialized)d(v)n(ersion)f(of)h(Theorem)g(4.1)g(when)h(w)n(e)f (apply)g(it)h(to)g(a)f(system)g(of)h(the)g(form)1029 4250 y Fq(x)1076 4216 y Fj(0)1183 4250 y Fu(=)83 b Fq(y)21 b Fu(+)d Fq(\026")1565 4216 y Fi(p)p Fo(+3)1687 4250 y Fq(@)1731 4262 y Fi(y)1771 4250 y Fq(F)1824 4262 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2006 4250 y Fu(+)g Fq(\026)2139 4216 y Fo(2)2177 4250 y Fq(")2216 4216 y Fi(p)p Fo(+2)2338 4250 y Fq(@)2382 4262 y Fi(y)2422 4250 y Fq(R)2485 4262 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)1033 4379 y Fq(y)1077 4345 y Fj(0)1183 4379 y Fu(=)83 b Fp(\000)p Fq(V)1462 4345 y Fj(0)1486 4379 y Fu(\()p Fq(x)p Fu(\))19 b Fp(\000)f Fq(\026")1788 4345 y Fi(p)p Fo(+3)1910 4379 y Fq(@)1954 4391 y Fi(x)1996 4379 y Fq(F)2049 4391 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)2231 4379 y Fp(\000)g Fq(\026)2364 4345 y Fo(2)2402 4379 y Fq(")2441 4345 y Fi(p)p Fo(+2)2563 4379 y Fq(@)2607 4391 y Fi(x)2649 4379 y Fq(R)2712 4391 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)456 4515 y Fu(whic)n(h)33 b(comes)g(from)h(the)g(system)g(\(1.1\))f(of)h(Chapter)f(1)h(b)n(y)f (the)h(a)n(v)n(eraging)d(pro)r(cedure)i(de-)456 4615 y(scrib)r(ed)d(in)g(Chapter)g(3.)44 b(If)31 b(the)g(system)f(\(1.1\))g (satis\014es)f(the)i(h)n(yp)r(otheses)f Fk(HP1-HP4)p Fu(,)g(the)456 4715 y(results)d(of)g(Chapter)g(3)g(apply)h(and)f(w)n(e) g(ha)n(v)n(e)g(that)g(the)h(stable)g(manifold)f(exists)g(and)h(can)f(b) r(e)456 4814 y(parameterized)f(b)n(y)h Fq(\015)1159 4784 y Fi(s)1154 4835 y(\026;")1250 4814 y Fu(\()p Fq(t;)14 b(s)p Fu(\).)605 4914 y(Next)23 b(theorem)f(giv)n(es)f(a)h(new)h(\015o) n(w)e(b)r(o)n(x)h(co)r(ordinates)f(and)i(additional)e(information)h(o)n (v)n(er)456 5014 y(the)27 b(v)-5 b(alues)27 b(of)h(these)f(\015o)n(w)g (b)r(o)n(x)g(co)r(ordinates)f(on)h(the)g(stable)g(manifold)h Fq(\015)2814 4983 y Fi(s)2809 5034 y(\026;")2904 5014 y Fu(\()p Fq(t;)14 b(s)p Fu(\).)38 b(Let)27 b Fp(C)32 b Fu(b)r(e)456 5116 y(the)26 b(c)n(hange)f(whic)n(h)h(transforms)f(the) h(system)g(\(1.1\))g(of)g(Chapter)f(1)h(to)g(the)g(a)n(v)n(eraged)e (system)456 5216 y(whic)n(h)j(is)h(giv)n(en)e(in)i(Prop)r(osition)e (3.1.)p eop %%Page: 38 46 38 45 bop 456 251 a Fo(38)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Ft(Theorem)31 b Fu(4.2)p Ft(.)40 b Fs(Given)29 b Fq(\024)1472 415 y Fj(\000)1472 472 y Fo(1)1551 450 y Fq(>)23 b Fu(2)p Fq(\024)1729 462 y Fo(0)1766 450 y Fs(,)29 b Fq(\024)1868 415 y Fo(+)1868 472 y(1)1946 450 y Fq(>)23 b(\024)2082 415 y Fj(\000)2082 472 y Fo(1)2167 450 y Fs(and)29 b Fq(\024)2375 462 y Fo(2)2435 450 y Fq(>)23 b(a)p Fs(,)30 b(ther)l(e)f(exist)f Fq(r)e(>)d Fu(0)28 b Fs(and)i(a)456 550 y(c)l(anonic)l(al)h(change)f(of)h(variables)563 786 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)26 b Fu(=)c Fq(t=")p Fu(\))h Fp(2)g(C)5 b Fu(\()p Fq(V)19 b Fu(\()p Fq(\024)1384 751 y Fj(\006)1384 808 y Fo(1)1440 786 y Fq(;)14 b(\024)1525 798 y Fo(2)1562 786 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\)\))24 b Fp(7!)f Fu(\()p Fq(S;)14 b(E)5 b(;)14 b(\022)r Fu(\))24 b(=)e(\()p Fp(S)6 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))p Fq(;)g Fp(E)7 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))p Fq(;)g(\022)r Fu(\))26 b Fp(2)e(V)3300 798 y Fo(1)456 937 y Fs(of)32 b(class)h Fq(C)822 907 y Fo(1)859 937 y Fs(,)g Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)h(in)e Fq(\022)i Fs(and)e(analytic)i(in)e(the)g Fq(x;)14 b(y)35 b Fs(variables,)f(such)e(that)g(it)g(tr)l(ans-)456 1036 y(forms)e(the)g(system)g(\(1.1\))g(of)h(Chapter)g(1)f(to)1815 1162 y Fu(_)1791 1183 y Fq(S)88 b Fu(=)83 b(1)1809 1286 y(_)1781 1307 y Fq(E)88 b Fu(=)83 b(0)456 1448 y Fs(and)30 b(satis\014es)808 1588 y Fp(S)6 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fp(S)1296 1600 y Fo(0)1334 1588 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")1814 1554 y Fi(p)p Fo(+1)1937 1588 y Fu(\))p Fs(,)119 b Fp(E)7 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fp(E)2590 1600 y Fo(0)2627 1588 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")3107 1554 y Fi(p)p Fo(+1)3230 1588 y Fu(\))456 1733 y Fs(wher)l(e)30 b Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b Fp(7!)f Fu(\()p Fp(S)1094 1745 y Fo(0)1132 1733 y Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fq(;)g Fp(E)1405 1745 y Fo(0)1442 1733 y Fu(\()p Fq(x;)g(y)s Fu(\)\))31 b Fs(is)f(the)g(c)l(orr)l(esp)l(onding)h(change)g(when)f Fq(\026)23 b Fu(=)g(0)p Fs(.)38 b(In)29 b(fact)1606 1874 y Fp(E)1650 1886 y Fo(0)1687 1874 y Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)f Fq(h)2039 1886 y Fo(0)2076 1874 y Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fs(.)605 2028 y(Mor)l(e)l(over,)31 b(given)g Fq(T)j Fp(\025)22 b Fu(0)29 b Fs(big)i(enough)f(ther)l(e)f (exists)h Fq(\024)2349 1993 y Fj(\000)2349 2050 y Fo(1)2427 2028 y Fq(>)23 b Fu(2)p Fq(\024)2605 2040 y Fo(0)2642 2028 y Fs(,)30 b Fq(\024)2745 1993 y Fo(+)2745 2050 y(1)2823 2028 y Fq(>)22 b(\024)2958 1993 y Fj(\000)2958 2050 y Fo(1)3044 2028 y Fs(and)30 b Fq(\024)3253 2040 y Fo(2)3313 2028 y Fq(>)23 b(a)456 2128 y Fs(such)k(that)h(for)g(al)t(l)h Fu(\()p Fq(t;)14 b(s)p Fu(\))28 b Fs(such)g(that)f Fq(T)35 b Fp(\024)22 b(j)p Fq(t)14 b Fu(+)f(Re)h Fq(s)p Fp(j)23 b(\024)g Fu(2)p Fq(T)38 b Fs(and)28 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b Fq(<)g(a)p Fs(,)28 b(the)g(p)l(ar)l(ameter-)456 2228 y(ization)i Fq(\015)777 2197 y Fi(s)772 2248 y(\026;")868 2228 y Fu(\()p Fq(t;)14 b(s)p Fu(\))30 b Fs(of)h(the)e(lo)l(c)l(al)i (stable)f(manifold)i(given)f(in)e(Chapter)j(3)e(satis\014es)1432 2380 y Fq(\015)1480 2346 y Fi(s)1475 2400 y(\026;")1571 2380 y Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)g(C)5 b Fu(\()p Fq(V)19 b Fu(\()p Fq(\024)2070 2344 y Fj(\006)2070 2402 y Fo(1)2126 2380 y Fq(;)14 b(\024)2211 2392 y Fo(2)2248 2380 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\)\))456 2521 y Fs(and,)30 b(for)h(any)f Fq(t)964 2533 y Fo(0)1024 2521 y Fp(2)24 b Fl(R)702 2661 y Fp(S)6 b Fu(\()p Fq(\015)838 2627 y Fi(s)833 2682 y(\026;")929 2661 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))23 b(=)g Fq(t)18 b Fp(\000)g Fq(t)1551 2673 y Fo(0)1607 2661 y Fu(+)g Fq(s)g Fu(+)g Fq(\026")1919 2627 y Fi(p)p Fo(+1)2041 2661 y Fp(X)12 b Fu(\()p Fq(s)p Fu(\))150 b Fs(and)f Fp(E)7 b Fu(\()p Fq(\015)2776 2627 y Fi(s)2771 2682 y(\026;")2867 2661 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))23 b(=)f(0)456 2807 y Fs(and)32 b Fp(X)12 b Fu(\()p Fq(s)761 2819 y Fo(0)799 2807 y Fu(\))27 b(=)g(0)k Fs(for)i(some)g Fq(s)1412 2819 y Fo(0)1449 2807 y Fs(,)g(which)g(we)g(c)l(an)f(cho)l (ose)h(fr)l(e)l(ely,)h(dep)l(ending)f(on)f(initial)i(c)l(on-)456 2906 y(ditions)c(on)g(the)g(stable)g(curve.)39 b(Mor)l(e)l(over)31 b Fp(X)12 b Fu(\()p Fq(s)p Fu(\))31 b Fs(is)f(analytic)h(and)f Fu(2)p Fq(\031)s(")p Fs(-p)l(erio)l(dic.)605 3006 y(In)k(addition)j (the)e(change)h Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))32 b Fp(7!)h Fu(\()p Fq(S;)14 b(E)5 b(;)14 b(\022)r Fu(\))35 b Fs(is)g(c)l(ontinuous)f(in)h Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r(;)g (\026;)g(")p Fu(\))35 b Fs(and)456 3105 y(analytic)c(in)e Fu(\()p Fq(x;)14 b(y)s(;)g(\026)p Fu(\))p Fs(.)605 3258 y Ft(Remark)41 b Fu(4.3)p Ft(.)46 b Fs(T)-6 b(o)38 b(\014x)g(ide)l(as)h (we)f(c)l(onsider)i(only)e(the)h(p)l(ar)l(ab)l(olic)h(c)l(ase,)h(which) f(is)e(the)456 3358 y(obje)l(ct)d(of)h(this)f(memoir,)i(but)d(the)h(pr) l(o)l(of)h(also)f(would)h(b)l(e)f(adapte)l(d)h(to)f(the)f(hyp)l(erb)l (olic)j(c)l(ase,)456 3457 y(with)30 b(some)g(smal)t(l)h(changes.)605 3610 y Fu(In)h(this)f(c)n(hapter,)h(at)f(some)f(places,)i(w)n(e)f(omit) g(the)h(dep)r(endence)g(on)f Fq(\026)g Fu(and)g Fq(")g Fu(whic)n(h)h(is)456 3710 y(assumed)27 b(that)g(it)i(is)e(analytical)g (and)g(con)n(tin)n(uous)g(resp)r(ectiv)n(ely)-7 b(.)1147 3896 y Fk(4.3.)46 b(A)32 b(preliminary)f(c)m(hange)h(of)g(v)-5 b(ariables)605 4046 y Fu(Since)32 b(the)g(stable)f(manifold)h(of)f(the) h(unp)r(erturb)r(ed)g(system)g(can)f(b)r(e)h(expressed)e(as)h(the)456 4145 y(graph)26 b(of)i(an)f(analytic)g(function,)h(w)n(e)f(can)h (easily)e(mo)n(v)n(e)h(it)h(to)f(the)h Fq(x)p Fu(-axis.)605 4245 y(F)-7 b(or)28 b(an)n(y)g Fq(\024)961 4209 y Fj(\000)961 4267 y Fo(1)1041 4245 y Fq(>)c Fu(2)p Fq(\024)1220 4257 y Fo(0)1257 4245 y Fu(,)29 b Fq(\024)1357 4209 y Fo(+)1357 4267 y(1)1436 4245 y Fq(>)24 b(\024)1573 4209 y Fj(\000)1573 4267 y Fo(1)1629 4245 y Fu(,)29 b Fq(\024)1729 4257 y Fo(2)1791 4245 y Fq(>)24 b Fu(0)k(and)g Fq(\024)2160 4257 y Fo(3)2222 4245 y Fq(>)c Fu(0)k(suc)n(h)g(that)h Fq(\024)2798 4209 y Fj(\000)2798 4267 y Fo(1)2873 4245 y Fp(\000)19 b Fu(3)p Fq(\024)3047 4257 y Fo(3)3108 4245 y Fq(>)24 b(\024)3245 4257 y Fo(0)3311 4245 y Fu(and)456 4344 y Fq(r)h Fp(\024)e Fq(r)643 4356 y Fo(0)681 4344 y Fu(,)k(w)n(e)h(de\014ne)g(the)g(sets)799 4485 y Fq(U)856 4497 y Fo(0)893 4485 y Fu(\()p Fq(\024)973 4449 y Fj(\006)973 4507 y Fo(1)1029 4485 y Fq(;)14 b(\024)1114 4497 y Fo(2)1151 4485 y Fq(;)g(\024)1236 4497 y Fo(3)1273 4485 y Fq(;)g(r)r Fu(\))24 b(=)1493 4418 y Fg(\010)1541 4485 y Fu(\()p Fq(x;)14 b(v)s Fu(\))24 b Fp(2)g Fl(C)1889 4451 y Fo(2)1955 4485 y Fu(:)f Fq(x)h Fp(2)f Fq(D)2221 4451 y Fj(\003)2259 4485 y Fu(\()p Fq(\024)2339 4449 y Fj(\006)2339 4507 y Fo(1)2395 4485 y Fq(;)14 b(\024)2480 4497 y Fo(2)2517 4485 y Fq(;)g(\024)2602 4497 y Fo(3)2639 4485 y Fq(;)g(r)r Fu(\),)29 b Fp(j)p Fq(v)s Fp(j)23 b Fq(<)g(r)3038 4418 y Fg(\011)456 4626 y Fu(and)848 4766 y Fq(U)9 b Fu(\()p Fq(\024)994 4730 y Fj(\006)994 4788 y Fo(1)1049 4766 y Fq(;)14 b(\024)1134 4778 y Fo(2)1171 4766 y Fq(;)g(\024)1256 4778 y Fo(3)1293 4766 y Fq(;)g(r)r Fu(\))24 b(=)1513 4699 y Fg(\010)1562 4766 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))24 b Fp(2)f Fl(C)1987 4732 y Fo(2)2049 4766 y Fp(\002)18 b Fl(R)29 b Fu(:)23 b(\()p Fq(x;)14 b(v)s Fu(\))24 b Fp(2)g Fq(U)2612 4778 y Fo(0)2649 4766 y Fu(\()p Fq(\024)2729 4730 y Fj(\006)2729 4788 y Fo(1)2785 4766 y Fq(;)14 b(\024)2870 4778 y Fo(2)2907 4766 y Fq(;)g(\024)2992 4778 y Fo(3)3029 4766 y Fq(;)g(r)r Fu(\))3137 4699 y Fg(\011)3200 4766 y Fu(.)605 4921 y(W)-7 b(e)26 b(p)r(erform)e(the)i(c)n(hange)e(of)h(v) -5 b(ariables)23 b Fq(C)30 b Fu(:)23 b Fq(V)c Fu(\()p Fq(\024)2185 4885 y Fj(\006)2185 4943 y Fo(1)2241 4921 y Fq(;)14 b(\024)2326 4933 y Fo(2)2363 4921 y Fq(;)g Fu(3)p Fq(\024)2490 4933 y Fo(3)2526 4921 y Fq(;)g(r)2600 4933 y Fo(0)2638 4921 y Fu(\))23 b Fp(!)g Fl(C)2853 4890 y Fo(2)2910 4921 y Fp(\002)13 b Fl(R)31 b Fu(de\014ned)26 b(b)n(y)1395 5061 y(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))23 b Fp(7!)h Fu(\()p Fq(x;)14 b(v)26 b Fu(=)d Fq(y)e Fp(\000)d Fq(f)9 b Fu(\()p Fq(x)p Fu(\))p Fq(;)14 b(\022)r Fu(\))p Fq(:)-2048 b Fu(\(4.2\))456 5216 y(This)27 b(c)n(hange)g(is)g (canonical.)36 b(It)28 b(maps)f Fq(V)19 b Fu(\()p Fq(\024)1854 5180 y Fj(\006)1854 5238 y Fo(1)1910 5216 y Fq(;)14 b(\024)1995 5228 y Fo(2)2032 5216 y Fq(;)g Fu(3)p Fq(\024)2159 5228 y Fo(3)2195 5216 y Fq(;)g(r)2269 5228 y Fo(0)2307 5216 y Fu(\))28 b(on)n(to)f Fq(U)9 b Fu(\()p Fq(\024)2700 5180 y Fj(\006)2700 5238 y Fo(1)2756 5216 y Fq(;)14 b(\024)2841 5228 y Fo(2)2878 5216 y Fq(;)g Fu(3)p Fq(\024)3005 5228 y Fo(3)3041 5216 y Fq(;)g(r)3115 5228 y Fo(0)3153 5216 y Fu(\).)p eop %%Page: 39 47 39 46 bop 1437 251 a Fo(4.4.)29 b(THE)g(UNPER)-5 b(TURBED)29 b(CASE)915 b(39)605 450 y Fu(W)-7 b(e)26 b(note)f(that,)h(in)g (general,)e(w)n(e)h(can)g(not)g(extend)h Fq(C)31 b Fu(in)26 b(suc)n(h)f(a)g(w)n(a)n(y)f(that)h(it)h(is)f(analytic)456 550 y(at)i Fq(x)d Fu(=)e(0)27 b(b)r(ecause)h Fq(f)36 b Fu(is)27 b(not)h(analytic)f(at)g(0.)37 b(The)27 b(equations)g(in)h (these)g(new)f(v)-5 b(ariables)27 b(are)1385 685 y(_)-37 b Fq(x)83 b Fu(=)g Fq(v)22 b Fu(+)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\))19 b(+)f Fq(\026")2146 650 y Fi(q)2185 685 y Fu(~)-45 b Fq(g)2222 697 y Fo(1)2259 685 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))1387 809 y(_)-35 b Fq(v)86 b Fu(=)d Fp(\000)p Fq(v)s(f)1807 775 y Fj(0)1830 809 y Fu(\()p Fq(x)p Fu(\))19 b(+)f Fq(\026")2132 775 y Fi(q)2171 809 y Fu(~)-45 b Fq(g)2208 821 y Fo(2)2245 809 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))-2058 b(\(4.3\))1393 918 y(_)1377 940 y Fq(\022)85 b Fu(=)e(1)p Fq(=")456 1075 y Fu(where)752 1210 y(~)-44 b Fq(g)790 1222 y Fo(1)826 1210 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))85 b(=)d Fq(@)1371 1222 y Fi(y)1411 1210 y Fq(H)1480 1222 y Fo(1)1518 1210 y Fu(\()p Fq(x;)14 b(v)22 b Fu(+)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\))p Fq(;)14 b(\022)r Fu(\))752 1335 y(~)-44 b Fq(g)790 1347 y Fo(2)826 1335 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))85 b(=)d Fp(\000)p Fq(@)1436 1347 y Fi(x)1478 1335 y Fq(H)1547 1347 y Fo(1)1584 1335 y Fu(\()p Fq(x;)14 b(v)22 b Fu(+)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\))p Fq(;)14 b(\022)r Fu(\))20 b Fp(\000)e Fq(f)2269 1300 y Fj(0)2292 1335 y Fu(\()p Fq(x)p Fu(\))p Fq(@)2447 1347 y Fi(y)2488 1335 y Fq(H)2557 1347 y Fo(1)2594 1335 y Fu(\()p Fq(x;)c(v)22 b Fu(+)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\))p Fq(;)14 b(\022)r Fu(\).)605 1473 y(W)-7 b(e)21 b(denote)g(b)n(y)g Fq(X)29 b Fu(:)24 b Fq(U)9 b Fu(\()p Fq(\024)1402 1438 y Fj(\006)1402 1495 y Fo(1)1457 1473 y Fq(;)14 b(\024)1542 1485 y Fo(2)1579 1473 y Fq(;)g Fu(3)p Fq(\024)1706 1485 y Fo(3)1743 1473 y Fq(;)g(r)1817 1485 y Fo(0)1854 1473 y Fu(\))24 b Fp(!)f Fl(C)2070 1443 y Fo(2)2118 1473 y Fp(\002)5 b Fl(R)p Fu(,)29 b(the)21 b(v)n(ector)f(\014eld)h Fq(X)29 b Fu(=)23 b Fq(X)3101 1485 y Fo(0)3143 1473 y Fu(+)5 b Fq(\026")3302 1443 y Fi(q)3338 1473 y Fq(X)3407 1485 y Fo(1)456 1573 y Fu(with)1003 1787 y Fq(X)1072 1799 y Fo(0)1132 1787 y Fu(=)1220 1620 y Fg(0)1220 1769 y(@)1334 1686 y Fq(v)22 b Fu(+)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\))1334 1786 y Fp(\000)p Fq(v)s(f)1492 1756 y Fj(0)1515 1786 y Fu(\()p Fq(x)p Fu(\))1334 1885 y(1)p Fq(=")1682 1620 y Fg(1)1682 1769 y(A)1851 1787 y Fu(and)83 b Fq(X)2137 1799 y Fo(1)2197 1787 y Fu(=)2285 1620 y Fg(0)2285 1769 y(@)2402 1686 y Fu(~)-45 b Fq(g)2439 1698 y Fo(1)2476 1686 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))2402 1786 y(~)-45 b Fq(g)2439 1798 y Fo(2)2476 1786 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))2399 1885 y(0)2787 1620 y Fg(1)2787 1769 y(A)2874 1787 y Fu(.)456 2016 y(Along)27 b(this)h(section,)f Fq(K)33 b Fu(will)28 b(denote)f(a)h(generic)e(constan)n(t)h(indep)r(enden)n(t)h (of)g Fq(\026;)14 b(")p Fu(.)605 2116 y(Some)27 b(preliminary)g(b)r (ounds)h(of)f(the)h(v)n(ector)f(\014eld)g Fq(X)34 b Fu(are)27 b(necessary)-7 b(.)605 2266 y Ft(Lemma)31 b Fu(4.1)p Ft(.)40 b Fs(We)30 b(have)h(that)679 2383 y Fu(1\))41 b Fs(F)-6 b(or)32 b(al)t(l)h Fu(\()p Fq(x;)14 b(v)s Fu(\))28 b Fp(2)g Fq(U)1431 2395 y Fo(0)1468 2383 y Fu(\()p Fq(\024)1548 2347 y Fj(\006)1548 2405 y Fo(1)1604 2383 y Fq(;)14 b(\024)1689 2395 y Fo(2)1726 2383 y Fq(;)g Fu(3)p Fq(\024)1853 2395 y Fo(3)1889 2383 y Fq(;)g(r)1963 2395 y Fo(0)2001 2383 y Fu(\))32 b Fs(and)h Fq(h)26 b Fp(2)i Fl(R)2440 2353 y Fo(2)2515 2383 y Fs(so)k(smal)t(l)h(that)f(the)g(se)l(gment)p 794 2420 571 4 v 794 2492 a Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fu(\()p Fq(x;)g(v)s Fu(\))21 b(+)d Fq(h)23 b Fp(2)g Fq(U)1523 2504 y Fo(0)1560 2492 y Fu(\()p Fq(\024)1640 2457 y Fj(\006)1640 2514 y Fo(1)1696 2492 y Fq(;)14 b(\024)1781 2504 y Fo(2)1818 2492 y Fq(;)g Fu(3)p Fq(\024)1945 2504 y Fo(3)1982 2492 y Fq(;)g(r)2056 2504 y Fo(0)2093 2492 y Fu(\))p Fs(,)1131 2634 y Fp(k)p Fq(X)1242 2646 y Fo(1)1279 2634 y Fu(\()p Fq(x)19 b Fu(+)f Fq(h)1508 2646 y Fo(1)1545 2634 y Fq(;)c(v)21 b Fu(+)e Fq(h)1775 2646 y Fo(2)1812 2634 y Fu(\))f Fp(\000)g Fq(X)2014 2646 y Fo(1)2051 2634 y Fu(\()p Fq(x;)c(v)s Fu(\))p Fp(k)24 b(\024)f Fq(K)6 b(\016)2513 2598 y Fi(k)q Fj(\000)p Fo(2)2510 2656 y(0)2638 2634 y Fp(k)p Fq(h)p Fp(k)794 2769 y Fs(and)936 2904 y Fp(k)p Fq(X)1047 2916 y Fo(0)1084 2904 y Fu(\()p Fq(x)19 b Fu(+)f Fq(h)1313 2916 y Fo(1)1350 2904 y Fq(;)c(v)22 b Fu(+)c Fq(h)1580 2916 y Fo(2)1617 2904 y Fu(\))g Fp(\000)g Fq(X)1819 2916 y Fo(0)1856 2904 y Fu(\()p Fq(x;)c(v)s Fu(\))20 b Fp(\000)e Fq(D)r(X)2290 2916 y Fo(0)2327 2904 y Fu(\()p Fq(x;)c(v)s Fu(\))p Fq(h)p Fp(k)24 b(\024)e Fq(K)6 b Fp(k)p Fq(h)p Fp(k)2928 2870 y Fo(2)794 3053 y Fs(wher)l(e)31 b Fq(K)k Fs(dep)l(ends)c(on)e(the)h(domain)h Fq(U)2055 3065 y Fo(0)2092 3053 y Fu(\()p Fq(\024)2172 3017 y Fj(\006)2172 3075 y Fo(1)2228 3053 y Fq(;)14 b(\024)2313 3065 y Fo(2)2350 3053 y Fq(;)g Fu(3)p Fq(\024)2477 3065 y Fo(3)2514 3053 y Fq(;)g(r)2588 3065 y Fo(0)2625 3053 y Fu(\))p Fs(.)679 3171 y Fu(2\))41 b Fs(Ther)l(e)31 b(exists)e(a)h(c)l(onstant)f Fq(K)35 b Fs(such)30 b(that)g(for)h(al)t (l)f Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))25 b Fp(2)p 2751 3090 632 4 v 23 w Fq(U)9 b Fu(\()p Fq(\024)2897 3136 y Fj(\006)2897 3193 y Fo(1)2953 3171 y Fq(;)14 b(\024)3038 3183 y Fo(2)3075 3171 y Fq(;)g Fu(3)p Fq(\024)3202 3183 y Fo(3)3239 3171 y Fq(;)g(r)3313 3183 y Fo(0)3350 3171 y Fu(\))1414 3312 y Fp(k)p Fq(X)1525 3324 y Fo(1)1561 3312 y Fu(\()p Fq(x;)g(v)s(;)g(\022)r Fu(\))p Fp(k)24 b(\024)f Fq(K)6 b Fp(k)p Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fp(k)2336 3277 y Fi(k)q Fj(\000)p Fo(1)2461 3312 y Fs(.)605 3462 y Ft(Pr)n(oof.)41 b Fu(The)k(pro)r(of)f(is)h(straigh)n(tforw)n (ard.)87 b(It)45 b(is)g(only)g(necessary)e(to)i(use)g(that)g(if)456 3574 y Fq(x)33 b Fp(2)p 624 3493 V 33 w Fq(U)9 b Fu(\()p Fq(\024)770 3539 y Fj(\006)770 3596 y Fo(1)826 3574 y Fq(;)14 b(\024)911 3586 y Fo(2)948 3574 y Fq(;)g Fu(3)p Fq(\024)1075 3586 y Fo(3)1111 3574 y Fq(;)g(r)1185 3586 y Fo(0)1223 3574 y Fu(\),)35 b(then)f Fp(j)p Fq(x)p Fp(j)f(\024)f Fu(2)p Fq(\016)1810 3586 y Fo(0)1847 3574 y Fu(,)j(that)e Fq(f)9 b Fu(\()p Fq(x)p Fu(\))34 b(=)e Fp(\000)2447 3503 y Fg(p)p 2529 3503 285 4 v 2529 3574 a Fp(\000)p Fu(2)p Fq(V)18 b Fu(\()p Fq(x)p Fu(\))34 b(=)e Fq(O)r Fu(\()p Fq(x)3088 3544 y Fi(n=)p Fo(2)3202 3574 y Fu(\),)j(that)456 3674 y Fq(H)525 3686 y Fo(1)562 3674 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))22 b(is)f(a)f(function)i(of)f(order)e Fq(k)24 b Fu(and)d(\014nally)-7 b(,)22 b(w)n(e)f(use)g(that)g(the)g (origin)f(do)r(es)h(not)g(b)r(elong)456 3791 y(to)28 b(the)h(domain)p 999 3709 632 4 v 29 w Fq(U)9 b Fu(\()p Fq(\024)1145 3755 y Fj(\006)1145 3813 y Fo(1)1201 3791 y Fq(;)14 b(\024)1286 3803 y Fo(2)1323 3791 y Fq(;)g Fu(3)p Fq(\024)1450 3803 y Fo(3)1486 3791 y Fq(;)g(r)1560 3803 y Fo(0)1598 3791 y Fu(\).)41 b(The)28 b(condition)h(on)f(the)i (segmen)n(t,)e(p)r(ermits)h(to)g(apply)456 3891 y(the)f(mean)f(v)-5 b(alue)27 b(theorem.)2027 b Fe(\003)1390 4061 y Fk(4.4.)46 b(The)32 b(unp)s(erturb)s(ed)g(case)605 4210 y Fu(When)c Fq(\026)23 b Fu(=)g(0,)k(the)h(system)f(\(4.3\))h(is)f(Hamiltonian)h (with)g(Hamiltonian)1535 4398 y Fp(F)1595 4410 y Fo(0)1632 4398 y Fu(\()p Fq(x;)14 b(v)s Fu(\))24 b(=)1945 4342 y Fq(v)1988 4312 y Fo(2)p 1945 4379 81 4 v 1964 4455 a Fu(2)2054 4398 y(+)18 b Fq(v)s(f)9 b Fu(\()p Fq(x)p Fu(\).)-1908 b(\(4.4\))456 4566 y(Then)32 b(for)f(an)n(y)g(initial)h (condition)g Fq(z)1625 4536 y Fo(0)1692 4566 y Fu(=)e(\()p Fq(x)1866 4536 y Fo(0)1904 4566 y Fq(;)14 b(v)1984 4536 y Fo(0)2021 4566 y Fq(;)g(\022)2099 4536 y Fo(0)2136 4566 y Fu(\))33 b(the)f(corresp)r(onding)e(solution)h(is)h(con-)456 4666 y(tained)27 b(in)h(the)g(curv)n(e)1315 4801 y Fq(v)e Fu(=)d Fp(\000)p Fq(f)9 b Fu(\()p Fq(x)p Fu(\))19 b Fp(\006)1797 4726 y Fg(p)p 1880 4726 706 4 v 75 x Fq(f)1930 4777 y Fo(2)1966 4801 y Fu(\()p Fq(x)p Fu(\))h(+)e(2)p Fp(F)2282 4813 y Fo(0)2318 4801 y Fu(\()p Fq(x)2397 4777 y Fo(0)2435 4801 y Fq(;)c(v)2515 4777 y Fo(0)2553 4801 y Fu(\))456 4936 y(when)28 b(one)g(has)f(to)i(c)n(ho)r(ose)e(the)h(sign)g(in)g(suc) n(h)g(a)g(w)n(a)n(y)f(that)h(the)h(relation)e(is)h(satis\014ed)g(b)n(y) g(the)456 5036 y(initial)f(condition.)37 b(F)-7 b(rom)27 b(system)h(\(4.3\))f(it)h(is)f(clear)g(that)1337 5182 y(_)-38 b Fq(x)24 b Fu(=)f Fp(\006)1546 5106 y Fg(p)p 1628 5106 927 4 v 1628 5182 a Fu(\()p Fq(v)1703 5158 y Fo(0)1741 5182 y Fu(\))1773 5158 y Fo(2)1829 5182 y Fu(+)18 b(2)p Fq(v)1997 5158 y Fo(0)2034 5182 y Fq(f)9 b Fu(\()p Fq(x)2163 5158 y Fo(0)2200 5182 y Fu(\))19 b Fp(\000)f Fu(2)p Fq(V)h Fu(\()p Fq(x)p Fu(\))q(.)-2122 b(\(4.5\))p eop %%Page: 40 48 40 47 bop 456 251 a Fo(40)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(Let)h Fq( )654 462 y Fo(0)691 450 y Fu(\()p Fq(t;)14 b(x;)g(v)s Fu(\))25 b(b)r(e)f(the)g(\015o)n(w)e(of)i(the)f(unp)r (erturb)r(ed)h(Hamiltonian)f(system)h(and)f(let)h Fq(\016)i Fu(b)r(e)e(suc)n(h)456 550 y(that)1549 693 y Fq(\013)1602 705 y Fo(0)1640 693 y Fu(\()p Fq(\024)1720 657 y Fo(+)1720 715 y(1)1775 693 y Fu(\))f Fq(<)g(\016)j(<)d(\013)2122 705 y Fo(0)2159 693 y Fu(\()p Fq(\024)2239 657 y Fj(\000)2239 715 y Fo(1)2295 693 y Fu(\).)456 841 y(F)-7 b(rom)22 b(no)n(w)h(on)g Fq(\016)j Fu(will)e(b)r(e)f(\014xed.)36 b(In)n(tegrating)22 b(equation)g(\(4.5\))h(w)n(e)g(\014nd)h(the)f(time) h(\(in)g(general)456 941 y(complex)k(time\))i(to)g(arriv)n(e)d(from)i (\()p Fq(x)1623 911 y Fj(\003)1662 941 y Fq(;)14 b(v)1742 911 y Fj(\003)1780 941 y Fu(\))30 b(to)f(\()p Fq(x;)14 b(v)s Fu(\))30 b(where)f Fq(x)2455 911 y Fj(\003)2520 941 y Fu(=)c Fq(\016)32 b Fu(and)d Fq(v)2885 911 y Fj(\003)2953 941 y Fu(is)g(determined)456 1040 y(b)n(y)24 b(the)i(energy)e(conserv) -5 b(ation.)34 b(In)25 b(this)h(w)n(a)n(y)e(w)n(e)g(get)h(that)g(the)h (functions)f Fp(T)2881 1052 y Fo(0)2919 1040 y Fu(\()p Fq(x;)14 b(v)s Fu(\),)27 b Fp(Y)3215 1052 y Fo(0)3252 1040 y Fu(\()p Fq(x;)14 b(v)s Fu(\))456 1141 y(de\014ned)28 b(in)f Fq(U)895 1153 y Fo(0)932 1141 y Fu(\()p Fq(\024)1012 1105 y Fj(\006)1012 1163 y Fo(1)1069 1141 y Fq(;)14 b(\024)1154 1153 y Fo(2)1190 1141 y Fq(;)g Fu(3)p Fq(\024)1317 1153 y Fo(3)1354 1141 y Fq(;)g(r)1428 1153 y Fo(0)1466 1141 y Fu(\))28 b(b)n(y)1189 1348 y Fp(T)1234 1360 y Fo(0)1272 1348 y Fu(\()p Fq(x;)14 b(v)s Fu(\))84 b(=)e Fp(\000)1773 1235 y Fg(Z)1856 1255 y Fi(\016)1819 1423 y(x)2238 1292 y Fq(ds)p 1916 1329 727 4 v 1916 1345 a Fg(p)p 1999 1345 644 4 v 71 x Fu(2)p Fp(F)2101 1428 y Fo(0)2137 1416 y Fu(\()p Fq(x;)14 b(v)s Fu(\))20 b Fp(\000)e Fu(2)p Fq(V)g Fu(\()p Fq(s)p Fu(\))456 1348 y(\(4.6\))1179 1556 y Fp(Y)1234 1568 y Fo(0)1272 1556 y Fu(\()p Fq(x;)c(v)s Fu(\))84 b(=)e Fp(\000)p Fq(f)9 b Fu(\()p Fq(\016)s Fu(\))19 b(+)2015 1481 y Fg(p)p 2098 1481 624 4 v 75 x Fq(f)2148 1532 y Fo(2)2184 1556 y Fu(\()p Fq(\016)s Fu(\))h(+)e(2)p Fp(F)2493 1568 y Fo(0)2529 1556 y Fu(\()p Fq(x;)c(v)s Fu(\))-2264 b(\(4.7\))456 1700 y(are)29 b(suc)n(h)h(that)h Fq( )1024 1712 y Fo(0)1061 1700 y Fu(\()p Fp(T)1138 1712 y Fo(0)1176 1700 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g(\016)o(;)g Fp(Y)1532 1712 y Fo(0)1570 1700 y Fu(\()p Fq(x;)g(v)s Fu(\)\))29 b(=)e(\()p Fq(x;)14 b(v)s Fu(\).)46 b(W)-7 b(e)31 b(c)n(ho)r(ose)e(the)i(sign)f(min)n(us)g(in)h(\(4.5\),)456 1799 y(b)r(ecause)c(it)g(is)h(ob)n(vious)e(that,)i(in)f(the)h(real)f (case)f(and)i(o)n(v)n(er)d(the)j(stable)f(manifold,)h Fq(x)p Fu(\()p Fq(t)p Fu(\))g(m)n(ust)456 1899 y(decrease)e(as)h Fq(t)g Fu(go)r(es)g(to)g(+)p Fp(1)p Fu(.)37 b(In)28 b(the)g(co)r (ordinates)1452 2042 y(\()p Fq(T)7 b(;)14 b(Y)19 b Fu(\))k(=)g(\()p Fp(T)1864 2054 y Fo(0)1901 2042 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(Y)2184 2054 y Fo(0)2223 2042 y Fu(\()p Fq(x;)g(v)s Fu(\)\))456 2186 y(the)28 b(equations)e(of)i(the)g(unp)r (erturb)r(ed)g(system)f(b)r(ecome:)1801 2309 y(_)1776 2330 y Fq(T)94 b Fu(=)82 b(1)1791 2434 y(_)1769 2455 y Fq(Y)102 b Fu(=)82 b(0.)456 2598 y(Also)27 b(w)n(e)g(can)g(consider)g (the)h(c)n(hange)985 2741 y(\()p Fq(x;)14 b(v)s Fu(\))24 b Fp(2)g Fq(U)1336 2753 y Fo(0)1373 2741 y Fu(\()p Fq(\024)1453 2705 y Fj(\006)1453 2763 y Fo(1)1509 2741 y Fq(;)14 b(\024)1594 2753 y Fo(2)1631 2741 y Fq(;)g Fu(3)p Fq(\024)1758 2753 y Fo(3)1794 2741 y Fq(;)g(r)1868 2753 y Fo(0)1906 2741 y Fu(\))23 b Fp(7!)g Fu(\()p Fp(T)2144 2753 y Fo(0)2182 2741 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(F)2470 2753 y Fo(0)2508 2741 y Fu(\()p Fq(x;)g(v)s Fu(\)\))25 b Fp(2)e(V)7 b Fu(,)456 2884 y(where)27 b Fp(F)756 2896 y Fo(0)820 2884 y Fu(is)h(the)g(Hamiltonian.)36 b(The)28 b(equations)f(in)h(the)g(co)r(ordinates)1451 3028 y(\()p Fq(T)7 b(;)14 b(F)e Fu(\))23 b(=)g(\()p Fp(T)1861 3040 y Fo(0)1898 3028 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(F)2186 3040 y Fo(0)2224 3028 y Fu(\()p Fq(x;)g(v)s Fu(\)\))456 3171 y(also)26 b(are)1800 3293 y(_)1775 3314 y Fq(T)94 b Fu(=)82 b(1)1798 3418 y(_)1770 3439 y Fq(F)95 b Fu(=)82 b(0.)456 3582 y(This)27 b(second)g(c)n(hange)g(is)g (canonical,)g(i.e.)36 b Fq(@)1845 3594 y Fi(x)1887 3582 y Fp(T)1932 3594 y Fo(0)1970 3582 y Fq(@)2014 3594 y Fi(v)2053 3582 y Fp(F)2113 3594 y Fo(0)2169 3582 y Fp(\000)18 b Fq(@)2296 3594 y Fi(v)2335 3582 y Fp(T)2380 3594 y Fo(0)2418 3582 y Fq(@)2462 3594 y Fi(x)2504 3582 y Fp(F)2564 3594 y Fo(0)2624 3582 y Fu(=)k(1.)954 3776 y Fk(4.5.)46 b(Flo)m(w)32 b(b)s(o)m(x)g(co)s(ordinates)f(in)g(a)i(complex)d(domain) 605 3926 y(4.5.1.)46 b(In)m(tro)s(duction)d(and)g(de\014nitions.)c Fu(Let)f Fq(z)2337 3938 y Fo(0)2374 3926 y Fu(\()p Fq(u;)14 b(Y)k Fu(\))37 b(b)r(e)h(the)f(solution)g(of)g(the)456 4025 y(unp)r(erturb)r(ed)28 b(Hamiltonian)f(system)g Fp(F)1748 4037 y Fo(0)1785 4025 y Fu(,)h(giv)n(en)f(in)h(\(4.4\))f(suc) n(h)g(that)h Fq(z)2754 4037 y Fo(0)2791 4025 y Fu(\(0)p Fq(;)14 b(Y)k Fu(\))24 b(=)e(\()p Fq(\016)o(;)14 b(Y)19 b Fu(\).)605 4125 y(When)26 b Fq(Y)42 b Fu(=)23 b(0,)j Fq(z)1153 4137 y Fo(0)1189 4125 y Fu(\()p Fq(u;)14 b Fu(0\))26 b(parametrizes)e(a)h(piece)h(of)g(the)g(stable)g(manifold)g (near)f(\()p Fq(x;)14 b(v)s Fu(\))24 b(=)456 4225 y(\()p Fq(\016)o(;)14 b Fu(0\).)36 b(Therefore,)26 b(writing)i Fq(z)1419 4237 y Fo(0)1479 4225 y Fu(=)22 b(\()p Fq(x)1645 4237 y Fo(0)1683 4225 y Fq(;)14 b(v)1760 4237 y Fo(0)1798 4225 y Fu(\),)28 b(there)f(exists)g Fq(\034)33 b Fp(2)23 b Fl(R)34 b Fu(suc)n(h)27 b(that)717 4389 y Fq(x)764 4401 y Fo(0)801 4389 y Fu(\()p Fq(u;)14 b Fu(0\))23 b(=)f Fq(\013)1155 4401 y Fo(0)1193 4389 y Fu(\()p Fq(u)c Fu(+)g Fq(\034)9 b Fu(\))24 b(=)1704 4333 y Fq(c)p 1573 4370 298 4 v 1573 4446 a Fu(\()p Fq(u)18 b Fu(+)g Fq(\034)9 b Fu(\))1831 4422 y Fi(p)1899 4389 y Fu(+)18 b Fq(O)r Fu(\()p Fp(j)p Fq(u)h Fu(+)f Fq(\034)9 b Fp(j)2320 4355 y Fj(\000)p Fi(\027)2414 4389 y Fu(\),)120 b Fq(p)23 b Fu(=)f(2)p Fq(=)p Fu(\()p Fq(n)c Fp(\000)g Fu(2\),)28 b Fq(\027)g(>)23 b Fu(0.)-2899 b(\(4.8\))456 4590 y(Note)27 b(that)h Fq(\016)e Fu(=)d Fq(\013)1040 4602 y Fo(0)1077 4590 y Fu(\()p Fq(\034)9 b Fu(\).)38 b(The)28 b(condition)f Fq(\016)g(<)22 b(\013)1986 4602 y Fo(0)2024 4590 y Fu(\()p Fq(\034)2101 4555 y Fj(\000)2092 4612 y Fo(1)2158 4590 y Fu(\))28 b(means)f(that)h Fq(\034)k(>)23 b(\024)2856 4555 y Fj(\000)2856 4612 y Fo(1)2912 4590 y Fu(.)605 4692 y(F)-7 b(or)27 b(an)n(y)g Fq(\034)956 4657 y Fj(\006)947 4714 y Fo(1)1013 4692 y Fu(,)g Fq(\034)1099 4704 y Fo(2)1137 4692 y Fu(,)h Fq(\034)1224 4704 y Fo(3)1289 4692 y Fu(and)f Fq(r)1487 4704 y Fo(1)1548 4692 y Fq(<)c(r)1673 4704 y Fo(0)1710 4692 y Fu(,)28 b(w)n(e)f(de\014ne)h(the)g(sets)609 4838 y Fq(D)r Fu(\()p Fq(\034)757 4802 y Fj(\006)748 4860 y Fo(1)814 4838 y Fq(;)14 b(\034)887 4850 y Fo(2)925 4838 y Fq(;)g(\034)998 4850 y Fo(3)1035 4838 y Fu(\))83 b(=)g Fp(f)p Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)g Fl(R)i Fp(\002)18 b Fl(C)44 b Fu(:)23 b Fq(s;)14 b(t)k Fu(+)g Fq(s)23 b Fp(2)h Fq(D)2319 4850 y Fo(0)2356 4838 y Fu(\()p Fq(\034)2433 4802 y Fj(\006)2424 4860 y Fo(1)2490 4838 y Fq(;)14 b(\034)2563 4850 y Fo(2)2600 4838 y Fq(;)g(\034)2673 4850 y Fo(3)2711 4838 y Fu(\))27 b(and)h Fp(j)p Fq(t)p Fp(j)23 b Fq(<)g Fu(4)p Fq(\031)s(")p Fp(g)838 4962 y Fq(W)12 b Fu(\()p Fq(r)997 4974 y Fo(1)1035 4962 y Fu(\))83 b(=)g Fp(f)p Fq(Y)41 b Fp(2)24 b Fl(C)44 b Fu(:)23 b Fp(j)p Fq(Y)c Fp(j)k Fq(<)f(r)1897 4974 y Fo(1)1935 4962 y Fp(g)p Fu(.)456 5114 y(W)-7 b(e)35 b(observ)n(e)f(that,)j(if)f(\()p Fq(t;)14 b(s)p Fu(\))36 b Fp(2)g Fq(D)r Fu(\()p Fq(\034)1651 5078 y Fj(\006)1642 5136 y Fo(1)1708 5114 y Fq(;)14 b(\034)1781 5126 y Fo(2)1819 5114 y Fq(;)g(\034)1892 5126 y Fo(3)1929 5114 y Fu(\))36 b(the)f(segmen)n(t)p 2476 5041 390 4 v 35 w(\(0)p Fq(;)14 b(s)p Fu(\))p Fq(;)g Fu(\()p Fq(t;)g(s)p Fu(\))36 b(is)f(con)n(tained)f(in)456 5216 y Fq(D)r Fu(\()p Fq(\034)604 5180 y Fj(\006)595 5238 y Fo(1)661 5216 y Fq(;)14 b(\034)734 5228 y Fo(2)771 5216 y Fq(;)g(\034)844 5228 y Fo(3)882 5216 y Fu(\).)p eop %%Page: 41 49 41 48 bop 1006 251 a Fo(4.5.)30 b(FLO)n(W)f(BO)n(X)g(COORDINA)-5 b(TES)29 b(IN)g(A)g(COMPLEX)g(DOMAIN)484 b(41)605 450 y Ft(Lemma)39 b Fu(4.2)p Ft(.)44 b Fs(L)l(et)36 b Fq(\024)1298 415 y Fj(\000)1298 472 y Fo(1)1389 450 y Fq(>)f Fu(2)p Fq(\024)1579 462 y Fo(0)1652 450 y Fs(and)i Fq(\024)1868 462 y Fo(2)1940 450 y Fq(>)d Fu(0)p Fs(.)58 b(Then,)39 b(given)e Fq(\024)2685 415 y Fo(+)2685 472 y(1)2775 450 y Fq(>)e(\024)2923 415 y Fj(\000)2923 472 y Fo(1)3015 450 y Fs(and)i Fq(\024)3231 462 y Fo(3)3303 450 y Fq(>)e Fu(0)456 552 y Fs(such)42 b(that)g Fq(\024)887 517 y Fj(\000)887 574 y Fo(1)971 552 y Fp(\000)27 b Fu(3)p Fq(\024)1153 564 y Fo(3)1235 552 y Fq(>)46 b(\024)1394 564 y Fo(0)1431 552 y Fs(,)g(ther)l(e)c(exists)g Fq(r)49 b Fp(\024)c Fq(r)2194 564 y Fo(1)2277 552 y Fq(<)h(r)2425 564 y Fo(0)2463 552 y Fq(=)p Fu(2)41 b Fs(such)h(that)h(the)f(image)i (of)456 654 y Fu(\001)23 b(=)f Fq(D)704 666 y Fo(0)741 654 y Fu(\()p Fq(\024)821 618 y Fj(\006)821 676 y Fo(1)896 654 y Fp(\000)c Fq(\034)5 b(;)14 b(\024)1105 666 y Fo(2)1142 654 y Fq(;)g(\024)1227 666 y Fo(3)1264 654 y Fu(\))19 b Fp(\002)f Fq(W)12 b Fu(\()p Fq(r)1557 666 y Fo(1)1595 654 y Fu(\))30 b Fs(by)g Fq(z)1804 666 y Fo(0)1871 654 y Fs(c)l(ontains)1680 830 y Fq(U)1737 842 y Fo(0)1774 830 y Fu(\()p Fq(\024)1854 795 y Fj(\006)1854 852 y Fo(1)1910 830 y Fq(;)14 b(\024)1995 842 y Fo(2)2032 830 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))456 1004 y Fs(and)30 b(is)g(c)l(ontaine)l(d)g (in)1566 1178 y Fq(U)1623 1190 y Fo(0)1660 1178 y Fu(\()p Fq(\024)1740 1143 y Fj(\006)1740 1200 y Fo(1)1796 1178 y Fq(;)14 b(\024)1881 1190 y Fo(2)1918 1178 y Fq(;)g Fu(2)p Fq(\024)2045 1190 y Fo(3)2082 1178 y Fq(;)g(r)2156 1190 y Fo(0)2193 1178 y Fq(=)p Fu(2\))p Fs(.)605 1364 y Ft(Pr)n(oof.)41 b Fu(By)24 b(\(4.8\),)h(the)f(image)g(of)g Fq(D)1815 1376 y Fo(0)1852 1364 y Fu(\()p Fq(\024)1932 1329 y Fj(\006)1932 1386 y Fo(1)2000 1364 y Fp(\000)12 b Fq(\034)5 b(;)14 b(\024)2203 1376 y Fo(2)2240 1364 y Fq(;)g Fu(0\))e Fp(\002)g(f)p Fu(0)p Fp(g)22 b Fu(b)n(y)i Fq(z)2739 1376 y Fo(0)2800 1364 y Fu(is)g Fq(U)2937 1376 y Fo(0)2974 1364 y Fu(\()p Fq(\024)3054 1329 y Fj(\006)3054 1386 y Fo(1)3110 1364 y Fq(;)14 b(\024)3195 1376 y Fo(2)3232 1364 y Fq(;)g Fu(0)p Fq(;)g Fu(0\).)456 1464 y(By)31 b(con)n(tin)n(uit)n(y)h(of)g Fq(z)1125 1476 y Fo(0)1194 1464 y Fu(it)g(follo)n(ws)f(that)i(if)j(~)-46 b Fq(r)1860 1476 y Fo(1)1929 1464 y Fq(>)30 b Fu(0)h(is)h(small)g(enough,)h(the)f Fq(x)h Fu(comp)r(onen)n(t)f(of)456 1563 y(the)26 b(image)f(of)h(\001)g (b)n(y)f Fq(z)1175 1575 y Fo(0)1238 1563 y Fu(con)n(tains)g Fq(D)1633 1533 y Fj(\003)1671 1563 y Fu(\()p Fq(\024)1751 1528 y Fj(\006)1751 1586 y Fo(1)1808 1563 y Fq(;)14 b(\024)1893 1575 y Fo(2)1930 1563 y Fq(;)g Fu(0\))25 b(and)h(it)g(is)g(con)n (tained)f(in)h Fq(D)2929 1533 y Fj(\003)2967 1563 y Fu(\()p Fq(\024)3047 1528 y Fj(\006)3047 1586 y Fo(1)3104 1563 y Fq(;)14 b(\024)3189 1575 y Fo(2)3226 1563 y Fq(;)g Fu(2)p Fq(\024)3353 1575 y Fo(3)3389 1563 y Fu(\).)605 1663 y(On)21 b(the)h(other)e(hand)i(the)f(solutions)g Fq(z)1802 1675 y Fo(0)1839 1663 y Fu(\()p Fq(u;)14 b(Y)k Fu(\))24 b(=)e(\()p Fq(x)p Fu(\()p Fq(u;)14 b(Y)19 b Fu(\))p Fq(;)14 b(v)s Fu(\()p Fq(u;)g(Y)19 b Fu(\)\))j(sta)n(y)e(in)i (the)g(energy)456 1763 y(lev)n(el)27 b Fp(F)705 1775 y Fo(0)742 1763 y Fu(\()p Fq(z)t Fu(\(0)p Fq(;)14 b(Y)k Fu(\)\))23 b(=)g Fq(Y)1235 1733 y Fo(2)1273 1763 y Fq(=)p Fu(2)17 b(+)h Fq(Y)h(f)9 b Fu(\()p Fq(\016)s Fu(\).)37 b(Therefore,)960 1948 y Fq(v)s Fu(\()p Fq(u;)14 b(Y)19 b Fu(\))k(=)g Fp(\000)p Fq(f)9 b Fu(\()p Fq(x)p Fu(\()p Fq(u;)14 b(Y)k Fu(\)\))h(+)1873 1873 y Fg(p)p 1956 1873 984 4 v 75 x Fq(Y)2023 1924 y Fo(2)2078 1948 y Fu(+)f(2)p Fq(Y)h(f)9 b Fu(\()p Fq(\016)s Fu(\))18 b(+)g Fq(f)2575 1924 y Fo(2)2612 1948 y Fu(\()p Fq(x)p Fu(\()p Fq(u;)c(Y)19 b Fu(\)\))456 2136 y(and)27 b(hence,)h(while)f Fq(x)p Fu(\()p Fq(u;)14 b(Y)19 b Fu(\))28 b(b)r(elongs)f(to)h Fq(D)1849 2106 y Fj(\003)1887 2136 y Fu(\()p Fq(\024)1967 2100 y Fj(\006)1967 2158 y Fo(1)2023 2136 y Fq(;)14 b(\024)2108 2148 y Fo(2)2145 2136 y Fq(;)g Fu(2)p Fq(\024)2272 2148 y Fo(3)2309 2136 y Fu(\),)1466 2310 y Fq(K)1537 2322 y Fo(1)1574 2310 y Fp(j)p Fq(Y)19 b Fp(j)k(\024)g(j)p Fq(v)s Fu(\()p Fq(u;)14 b(Y)k Fu(\))p Fp(j)24 b(\024)e Fq(K)2284 2322 y Fo(2)2321 2310 y Fp(j)p Fq(Y)d Fp(j)456 2484 y Fu(and)27 b(th)n(us)h(there)f(exists)g Fq(r)n(;)14 b(r)1350 2496 y Fo(1)1411 2484 y Fp(\024)26 b Fu(~)-45 b Fq(r)1536 2496 y Fo(1)1601 2484 y Fu(satisfying)27 b(the)h(stated)g(prop)r(erties.)629 b Fe(\003)605 2689 y Fu(Our)35 b(goal)f(is)i(to)f(\014nd)h(\015o)n(w)f(b)r(o)n(x)g(co)r (ordinates)f(in)i Fq(U)9 b Fu(\()p Fq(\024)2390 2654 y Fj(\006)2390 2711 y Fo(1)2446 2689 y Fq(;)14 b(\024)2531 2701 y Fo(2)2568 2689 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\).)61 b(W)-7 b(e)36 b(will)g(\014nd)g(the)456 2789 y(solutions)26 b(of)i(the)g(equations)f(\(4.3\))g(parameterized)f(in)i(the)g(form)1686 2963 y(\()p Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))p Fq(;)c(t=")p Fu(\))456 3137 y(with)1230 3311 y Fq(z)t Fu(\()p Fq(t;)g(s;)g(Y)j Fu(\))24 b(=)e Fq(z)1695 3323 y Fo(0)1732 3311 y Fu(\()p Fq(t)d Fu(+)f Fq(s;)c(Y)k Fu(\))h(+)f Fq(\026")2261 3277 y Fi(q)2298 3311 y Fq(z)2337 3323 y Fo(1)2373 3311 y Fu(\()p Fq(t;)c(s;)g(Y)19 b Fu(\),)456 3490 y Fq(z)t Fu(\(0)p Fq(;)14 b Fu(0)p Fq(;)g(Y)j Fu(\))23 b(=)g(\()p Fq(\016)o(;)14 b(Y)k Fu(\))28 b(and)f(the)h(additional)f (prop)r(ert)n(y)1335 3663 y Fq(z)t Fu(\()p Fq(t)18 b Fu(+)g(2)p Fq(\031)s(";)c(s;)g(Y)k Fu(\))23 b(=)g Fq(z)t Fu(\()p Fq(t;)14 b(s)k Fu(+)g(2)p Fq(\031)s(";)c(Y)k Fu(\).)456 3838 y(This)33 b(relation)f(p)r(ermits)i(to)f(giv)n(e)f(a)h (dynamical)g(in)n(terpretation)f(of)h(the)h(parameter)e Fq(s)p Fu(:)48 b(the)456 3937 y(iterations)27 b(of)h(the)h(P)n(oincar)n (\023)-39 b(e)25 b(map)k(simply)f(consists)g(in)g(increasing)f(the)i(v) -5 b(ariable)27 b Fq(s)i Fu(b)n(y)f(2)p Fq(\031)s(")p Fu(.)456 4037 y(T)-7 b(o)27 b(get)g(the)h(solutions)f(in)h(this)g(form) f(w)n(e)g(will)h(rewrite)f(\(4.3\))g(in)h(the)g(form)1451 4211 y(_)-37 b Fq(z)86 b Fu(=)d Fq(A)p Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))p Fq(z)j Fu(+)d Fq(b)p Fu(\()p Fq(z)t Fu(\)\()p Fq(t;)c(s)p Fu(\))1454 4320 y(_)1438 4342 y Fq(\022)85 b Fu(=)e(1)p Fq(=")456 4516 y Fu(and)23 b(w)n(e)h(will)f(apply)h(the)g(\014xed)g(p)r(oin)n(t)g(theorem)f(to)g (a)h(suitable)f(op)r(erator)f(in)i(a)g(Banac)n(h)e(space.)456 4616 y(T)-7 b(o)29 b(construct)g(this)g(op)r(erator)f(w)n(e)h(will)h (need)g(another)e(op)r(erator)g(whic)n(h)h(w)n(e)h(call)f(incremen)n(t) 456 4715 y(op)r(erator.)35 b(This)27 b(op)r(erator)f(w)n(as)h(in)n(tro) r(duced)g(b)n(y)g(Lazutkin)h(in)g([)p Fk(La2)p Fu(].)605 4815 y(Next,)g(w)n(e)f(will)g(pro)n(v)n(e)f(that,)h(as)g(in)g(the)h (unp)r(erturb)r(ed)g(case,)e(the)i(solutions)e(with)i(initial)456 4914 y(condition)i(in)h Fq(U)9 b Fu(\()p Fq(\024)1069 4879 y Fj(\006)1069 4937 y Fo(1)1125 4914 y Fq(;)14 b(\024)1210 4926 y Fo(2)1247 4914 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))30 b(arriv)n(e)f(at)h Fq(x)f Fu(=)e Fq(\016)s Fu(.)46 b(Then)31 b(w)n(e)f(will)g(pro)n(v)n(e)f(that)i(the)g(\015o)n(w)f(can)456 5016 y(b)r(e)e(straigh)n(tened)e(in)i Fq(U)9 b Fu(\()p Fq(\024)1285 4981 y Fj(\006)1285 5038 y Fo(1)1341 5016 y Fq(;)14 b(\024)1426 5028 y Fo(2)1463 5016 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\).)605 5116 y(Finally)-7 b(,)28 b(w)n(e)g(will)g(construct)f(another)g(c)n(hange)g(in)h(order)f(to)h (get)f(that)i(the)f(comp)r(osition)456 5216 y(of)f(c)n(hanges)f(is)i (canonical.)p eop %%Page: 42 50 42 49 bop 456 251 a Fo(42)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fk(4.5.2.)46 b(Incremen)m(t)25 b(op)s(erator)h(and)g(analytic)g(con)m (tin)m(uation.)42 b Fu(Let)22 b Fq(h)p Fu(,)i Fq(\034)3112 415 y Fj(\006)3103 472 y Fo(1)3168 450 y Fu(,)g Fq(\034)3251 462 y Fo(2)3311 450 y Fu(and)456 552 y Fq(D)525 564 y Fo(0)586 552 y Fu(=)h Fq(D)745 564 y Fo(0)782 552 y Fu(\()p Fq(\034)859 517 y Fj(\006)850 574 y Fo(1)916 552 y Fq(;)14 b(\034)989 564 y Fo(2)1026 552 y Fq(;)g Fu(0\))29 b(and)f Fq(W)37 b Fu(=)24 b Fq(W)12 b Fu(\()p Fq(r)1691 564 y Fo(1)1729 552 y Fu(\))29 b(as)f(in)h(the)g(previous)e(subsection.)40 b(W)-7 b(e)29 b(de\014ne)g Fq(D)i Fu(is)456 652 y(a)c(sligh)n(tly)g (more)g(general)f(w)n(a)n(y:)624 789 y Fq(D)85 b Fu(=)e Fq(D)r Fu(\()p Fq(h;)14 b(\034)1159 753 y Fj(\006)1150 811 y Fo(1)1216 789 y Fq(;)g(\034)1289 801 y Fo(2)1326 789 y Fu(\))778 916 y(=)83 b Fp(f)p Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)g Fl(R)i Fp(\002)18 b Fl(C)44 b Fu(:)23 b Fp(j)p Fq(t)p Fp(j)g Fq(<)g Fu(2)p Fq(h;)41 b(\034)1916 880 y Fj(\000)1907 938 y Fo(1)1995 916 y Fq(<)23 b Fu(Re)14 b Fq(s;)g(t)k Fu(+)g(Re)c Fq(s)23 b(<)g(\034)2709 880 y Fo(+)2700 938 y(1)2764 916 y Fu(,)28 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b Fq(<)g(\034)3174 928 y Fo(2)3211 916 y Fp(g)p Fu(.)456 1053 y(W)-7 b(e)28 b(consider)e(the)i(equation)1512 1190 y(_)-37 b Fq(z)27 b Fu(=)22 b Fq(A)p Fu(\()p Fq(t)d Fu(+)f Fq(s)p Fu(\))p Fq(z)k Fu(+)c Fq(b)p Fu(\()p Fq(t;)c(s;)g(Y)k Fu(\))-1945 b(\(4.9\))456 1333 y(\(here)31 b Fp(\001)h Fu(denotes)f(deriv)-5 b(ativ)n(e)31 b(with)h(resp)r(ect)f(to)h Fq(t)p Fu(\),)h(where)e Fq(A)p Fu(\()p Fq(u)p Fu(\))h(is)f(a)h(2)20 b Fp(\002)h Fu(2)31 b(matrix)g(whose)456 1435 y(elemen)n(ts)i(are)f(analytic)g(in)i Fq(D)1438 1447 y Fo(0)1507 1435 y Fu(=)e Fq(D)1673 1447 y Fo(0)1710 1435 y Fu(\()p Fq(\034)1787 1399 y Fj(\006)1778 1457 y Fo(1)1844 1435 y Fq(;)14 b(\034)1917 1447 y Fo(2)1954 1435 y Fq(;)g Fu(0\))33 b(and)g(con)n(tin)n(uous)f(in)p 2788 1368 106 4 v 33 w Fq(D)2857 1447 y Fo(0)2894 1435 y Fu(.)54 b(The)33 b(function)456 1539 y Fq(b)22 b Fu(:)p 560 1472 72 4 v 23 w Fq(D)c Fp(\002)d Fq(W)35 b Fp(!)23 b Fl(C)1000 1509 y Fo(2)1069 1539 y Fu(is)j(con)n(tin)n(uous)f(and,)h (for)g(an)n(y)f Fq(t)h Fu(suc)n(h)g(that)g Fp(j)p Fq(t)p Fp(j)d(\024)g Fu(2)p Fq(h)p Fu(,)j Fq(b)p Fu(\()p Fq(t;)14 b(:;)g(:)p Fu(\))26 b(is)g(analytic.)456 1638 y(W)-7 b(e)28 b(assume)e(that)i Fq(b)g Fu(v)n(eri\014es)1436 1776 y Fq(b)p Fu(\()p Fq(t)18 b Fu(+)g Fq(h;)c(s;)g(Y)k Fu(\))24 b(=)e Fq(b)p Fu(\()p Fq(t;)14 b(s)19 b Fu(+)f Fq(h;)c(Y)k Fu(\))-2008 b(\(4.10\))456 1918 y(and)30 b(w)n(e)g(lo)r(ok)g(for)g(solutions)g Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))30 b(of)h(\(4.9\))f(analytic)g(with)h(resp)r (ect)f(to)h Fq(s)f Fu(and)h Fq(Y)18 b Fu(,)32 b(and)456 2018 y(satisfying)1418 2155 y Fq(z)t Fu(\()p Fq(t)18 b Fu(+)g Fq(h;)c(s;)g(Y)k Fu(\))23 b(=)g Fq(z)t Fu(\()p Fq(t;)14 b(s)k Fu(+)g Fq(h;)c(Y)k Fu(\))p Fq(:)456 2297 y Fu(Let)27 b Fq(M)9 b Fu(\()p Fq(u)p Fu(\))28 b(b)r(e)g(a)f(fundamen)n (tal)h(matrix)f(of)g(the)h(homogeneous)e(equation)1732 2421 y Fq(d)p 1708 2459 91 4 v 1708 2535 a(du)1809 2478 y(\020)j Fu(=)23 b Fq(A)p Fu(\()p Fq(u)p Fu(\))p Fq(\020)6 b Fu(.)456 2638 y(By)31 b(the)g(general)f(theory)h(of)g(linear)g (equations,)g Fq(M)40 b Fu(is)31 b(analytic)g(in)g Fq(D)2720 2650 y Fo(0)2789 2638 y Fu(and)g(there)g(exists)g(a)456 2738 y(constan)n(t)26 b Fq(C)849 2750 y Fi(M)951 2738 y Fu(suc)n(h)h(that)1017 2875 y Fp(j)p Fq(M)9 b Fu(\()p Fq(u)p Fu(\))p Fp(j)22 b(\024)h Fq(C)1434 2887 y Fi(M)1508 2875 y Fu(,)194 b Fp(j)p Fq(M)1838 2841 y Fj(\000)p Fo(1)1927 2875 y Fu(\()p Fq(u)p Fu(\))p Fp(j)23 b(\024)g Fq(C)2232 2887 y Fi(M)2306 2875 y Fu(,)276 b Fq(u)23 b Fp(2)g Fq(D)2823 2887 y Fo(0)2860 2875 y Fu(.)-2427 b(\(4.11\))456 3017 y(By)27 b(the)h(v)-5 b(ariation)26 b(of)i(parameters)e(metho)r(d,)i (the)g(solutions)f(of)g(\(4.9\))h(can)f(b)r(e)h(expressed)e(as)844 3212 y Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))24 b(=)e Fq(M)9 b Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))1596 3120 y Fg(h)1635 3212 y Fq(M)1725 3178 y Fj(\000)p Fo(1)1814 3212 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(s;)c(Y)19 b Fu(\))g(+)2262 3099 y Fg(Z)2345 3120 y Fi(t)2308 3288 y Fo(0)2388 3212 y Fq(M)2478 3178 y Fj(\000)p Fo(1)2567 3212 y Fu(\()p Fq(\030)j Fu(+)c Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(\030)t(;)c(s;)g(Y)19 b Fu(\))14 b Fq(d\030)3228 3120 y Fg(i)456 3212 y Fu(\(4.12\))456 3396 y(where)25 b Fq(c)p Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))26 b(is)g(an)g(arbitrary)e (function.)37 b(Therefore,)25 b(if)h(the)h(function)f Fq(c)p Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))26 b(is)g(analytic)456 3499 y(in)36 b Fq(D)630 3511 y Fo(0)690 3499 y Fp(\002)24 b Fq(W)12 b Fu(,)38 b Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))36 b(giv)n(en)f(in)h(\(4.12\))f(is)g(con)n(tin)n(uous)g(in)p 2480 3432 72 4 v 36 w Fq(D)26 b Fp(\002)e Fq(W)47 b Fu(and)36 b(analytic)f(with)456 3598 y(resp)r(ect)27 b(to)g(\()p Fq(s;)14 b(Y)19 b Fu(\).)605 3698 y(W)-7 b(e)28 b(write)533 3976 y Fq(z)t Fu(\()p Fq(t)18 b Fu(+)g Fq(h;)c(s;)g(Y)k Fu(\))23 b(=)g Fq(M)9 b Fu(\()p Fq(t)18 b Fu(+)g Fq(h)g Fu(+)g Fq(s)p Fu(\))1582 3884 y Fg(h)1622 3976 y Fq(M)1712 3942 y Fj(\000)p Fo(1)1801 3976 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(s;)c(Y)19 b Fu(\))f(+)2248 3863 y Fg(Z)2331 3884 y Fi(t)p Fo(+)p Fi(h)2295 4052 y Fo(0)2464 3976 y Fq(M)2554 3942 y Fj(\000)p Fo(1)2643 3976 y Fu(\()p Fq(\030)23 b Fu(+)18 b Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(\030)t(;)c(s;)g(Y)19 b Fu(\))14 b Fq(d\030)3305 3884 y Fg(i)3344 3976 y Fu(.)456 4156 y(Also)910 4297 y Fq(z)t Fu(\()p Fq(t;)g(s)k Fu(+)g Fq(h;)c(Y)19 b Fu(\))83 b(=)g Fq(M)9 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)g Fu(+)g Fq(h)p Fu(\))2080 4205 y Fg(h)2120 4297 y Fq(M)2210 4263 y Fj(\000)p Fo(1)2298 4297 y Fu(\()p Fq(s)h Fu(+)f Fq(h)p Fu(\))p Fq(c)p Fu(\()p Fq(s)h Fu(+)f Fq(h;)c(Y)k Fu(\))1607 4512 y(+)1686 4399 y Fg(Z)1768 4420 y Fi(t)1731 4588 y Fo(0)1811 4512 y Fq(M)1901 4478 y Fj(\000)p Fo(1)1990 4512 y Fu(\()p Fq(\030)23 b Fu(+)18 b Fq(s)g Fu(+)g Fq(h)p Fu(\))p Fq(b)p Fu(\()p Fq(\030)t(;)c(s)k Fu(+)h Fq(h;)14 b(Y)k Fu(\))c Fq(d\030)2950 4420 y Fg(i)1459 4717 y Fu(=)83 b Fq(M)9 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)g Fu(+)g Fq(h)p Fu(\))2080 4625 y Fg(h)2120 4717 y Fq(M)2210 4682 y Fj(\000)p Fo(1)2298 4717 y Fu(\()p Fq(s)h Fu(+)f Fq(h)p Fu(\))p Fq(c)p Fu(\()p Fq(s)h Fu(+)f Fq(h;)c(Y)k Fu(\))1607 4937 y(+)1686 4824 y Fg(Z)1768 4844 y Fi(t)p Fo(+)p Fi(h)1731 5012 y(h)1901 4937 y Fq(M)1991 4902 y Fj(\000)p Fo(1)2080 4937 y Fu(\()p Fq(\030)k Fu(+)c Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(\030)t(;)c(s;)g(Y)19 b Fu(\))14 b Fq(d\030)2741 4845 y Fg(i)456 5116 y Fu(where)30 b(w)n(e)g(ha)n(v)n(e)g(made)g(the)h(ob)n(vious)f(c)n(hange)f(of)i(v)-5 b(ariables)29 b(in)i(the)g(in)n(tegral,)g(and)f(w)n(e)h(ha)n(v)n(e)456 5216 y(used)c(\(4.10\).)p eop %%Page: 43 51 43 50 bop 1006 251 a Fo(4.5.)30 b(FLO)n(W)f(BO)n(X)g(COORDINA)-5 b(TES)29 b(IN)g(A)g(COMPLEX)g(DOMAIN)484 b(43)605 450 y Fu(W)-7 b(e)28 b(in)n(tro)r(duce)f(the)h(auxiliary)e(function)j Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)k Fu(\))23 b(=)g Fq(M)2389 420 y Fj(\000)p Fo(1)2478 450 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(s;)14 b(Y)19 b Fu(\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(that)1429 587 y Fq(z)t Fu(\()p Fq(t)18 b Fu(+)g Fq(h;)c(s;)g(Y)k Fu(\))24 b(=)e Fq(z)t Fu(\()p Fq(t;)14 b(s)k Fu(+)g Fq(h;)c(Y)19 b Fu(\))456 725 y(if)28 b(and)f(only)g(if)972 919 y Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)k Fu(\))h Fp(\000)f Fq(f)9 b Fu(\()p Fq(s)18 b Fu(+)g Fq(h;)c(Y)k Fu(\))24 b(=)e Fp(\000)1925 806 y Fg(Z)2008 827 y Fi(h)1971 995 y Fo(0)2065 919 y Fq(M)2155 885 y Fj(\000)p Fo(1)2243 919 y Fu(\()p Fq(\030)h Fu(+)18 b Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(\030)t(;)c(s;)g(Y)19 b Fu(\))14 b Fq(d\030)t Fu(.)456 1098 y(Therefore,)26 b(it)i(is)g(natural)e(to)i(study)g(the)g(op)r(erator)1314 1235 y Fp(4)1389 1247 y Fi(h)1431 1235 y Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))k(=)g Fq(f)9 b Fu(\()p Fq(s)18 b Fu(+)g Fq(h;)c(Y)k Fu(\))h Fp(\000)f Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)k Fu(\).)456 1373 y(W)-7 b(e)28 b(w)n(an)n(t)f(to)g (\014nd)h(analytic)f(solutions)g(of)g(the)h(equation)1071 1510 y Fp(4)1146 1522 y Fi(h)1188 1510 y Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)k Fu(\))23 b(=)g Fq(g)s Fu(\()p Fq(s;)14 b(Y)k Fu(\),)222 b Fq(s;)14 b(s)k Fu(+)g Fq(h)23 b Fp(2)g Fq(D)2483 1522 y Fo(0)2520 1510 y Fu(,)28 b Fq(Y)42 b Fp(2)23 b Fq(W)-2361 b Fu(\(4.13\))456 1663 y(where)27 b Fq(g)j Fu(is)d(analytic)g(in)h Fq(D)1331 1675 y Fo(0)1387 1663 y Fp(\002)18 b Fq(W)39 b Fu(and)28 b(con)n(tin)n(uous)e(in)p 2261 1596 106 4 v 28 w Fq(D)2330 1675 y Fo(0)2386 1663 y Fp(\002)18 b Fq(W)12 b Fu(.)605 1762 y(W)-7 b(e)28 b(de\014ne)g(the)g(auxiliary)e(op)r(en)i(sets)1250 1900 y Fq(D)1321 1864 y Fj(\000)1319 1922 y Fo(0)1400 1900 y Fu(=)23 b Fp(f)p Fq(s)g Fp(2)g Fl(C)44 b Fu(:)23 b(Re)14 b Fq(s)23 b(<)g(\034)2106 1864 y Fo(+)2097 1922 y(1)2161 1900 y Fu(,)28 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b Fq(<)g(\034)2571 1912 y Fo(2)2608 1900 y Fp(g)456 2037 y Fu(and)1225 2174 y Fq(D)1296 2138 y Fo(+)1294 2196 y(0)1374 2174 y Fu(=)g Fp(f)p Fq(s)f Fp(2)i Fl(C)44 b Fu(:)23 b Fq(\034)1818 2138 y Fj(\000)1809 2196 y Fo(1)1898 2174 y Fq(<)f Fu(Re)14 b Fq(s)p Fu(,)55 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b Fq(<)g(\034)2573 2186 y Fo(2)2611 2174 y Fp(g)p Fu(.)456 2325 y(It)32 b(is)f(clear)g(that)h Fq(D)1093 2337 y Fo(0)1160 2325 y Fu(=)d Fq(D)1325 2290 y Fo(+)1323 2347 y(0)1401 2325 y Fp(\\)22 b Fq(D)1549 2290 y Fj(\000)1547 2347 y Fo(0)1605 2325 y Fu(.)49 b(F)-7 b(or)31 b(an)n(y)g(op)r(en)h(set)g (\012)d Fp(\032)h Fl(C)15 b Fu(,)39 b(w)n(e)31 b(de\014ne)h(the)g (function)456 2425 y(space)456 2660 y Fp(A)p Fu(\(\012)p Fq(;)14 b(W)e Fu(\))23 b(=)g Fp(f)p Fq(H)29 b Fu(:)p 1070 2593 60 4 v 23 w(\012)19 b Fp(\002)f Fq(W)35 b Fp(!)23 b Fl(C)44 b Fu(:)23 b Fq(H)35 b Fu(is)27 b(analytic)g(in)h(\012)18 b Fp(\002)g Fq(W)40 b Fu(and)27 b(con)n(tin)n(uous)g(in)p 3133 2593 V 28 w(\012)18 b Fp(\002)g Fq(W)12 b Fp(g)p Fu(.)456 2802 y(The)28 b(main)h(idea)f(of)h(what)g(w)n(as)e(dev)n(elop) r(ed)h(b)n(y)h(Lazutkin)f(in)h([)p Fk(La2)q Fu(])g(is)f(the)h(follo)n (wing.)39 b(Con-)456 2902 y(struct)27 b(t)n(w)n(o)g(analytic)g (functions)h Fq(g)1570 2872 y Fo(+)1652 2902 y Fp(2)c(A)p Fu(\()p Fq(D)1900 2867 y Fo(+)1898 2924 y(0)1955 2902 y Fq(;)14 b(W)e Fu(\))28 b(and)g Fq(g)2347 2872 y Fj(\000)2430 2902 y Fp(2)23 b(A)p Fu(\()p Fq(D)2677 2867 y Fj(\000)2675 2924 y Fo(0)2734 2902 y Fq(;)14 b(W)e Fu(\))28 b(suc)n(h)f(that)1474 3040 y Fq(g)f Fu(=)d Fq(g)1671 3006 y Fo(+)1744 3040 y Fu(+)18 b Fq(g)1870 3006 y Fj(\000)2008 3040 y Fu(in)28 b Fq(D)2174 3052 y Fo(0)2230 3040 y Fp(\002)18 b Fq(W)12 b Fu(.)-1970 b(\(4.14\))456 3182 y(Then,)21 b(b)r(ecause)f(of)f(the)i (linearit)n(y)e(of)g(equation)h(\(4.13\),)g(the)h(problem)e(of)h (\014nding)g(the)g(function)456 3282 y Fq(f)44 b Fu(can)34 b(b)r(e)i(reduced)f(to)g(t)n(w)n(o)g(simpler)g(problems:)51 b(to)35 b(\014nd)h(t)n(w)n(o)f(functions)g Fq(f)2977 3252 y Fo(+)3067 3282 y Fu(and)h Fq(f)3287 3252 y Fj(\000)3378 3282 y Fu(of)456 3382 y Fp(A)p Fu(\()p Fq(D)625 3346 y Fo(+)623 3404 y(0)680 3382 y Fq(;)14 b(W)e Fu(\))28 b(and)g Fp(A)p Fu(\()p Fq(D)1198 3346 y Fj(\000)1196 3404 y Fo(0)1254 3382 y Fq(;)14 b(W)e Fu(\))28 b(resp)r(ectiv)n(ely)f (suc)n(h)g(that)1723 3520 y Fp(4)1798 3532 y Fi(h)1839 3520 y Fq(f)1889 3485 y Fj(\006)1968 3520 y Fu(=)c Fq(g)2099 3485 y Fj(\006)2154 3520 y Fu(.)456 3657 y(Therefore,)j(since)h(the)h (op)r(erator)e Fp(4)1611 3669 y Fi(h)1681 3657 y Fu(is)h(linear,)g(the) h(function)1702 3794 y Fq(f)k Fu(=)23 b Fq(f)1913 3760 y Fo(+)1986 3794 y Fu(+)18 b Fq(f)2119 3760 y Fj(\000)2175 3794 y Fu(,)456 3945 y(whic)n(h)27 b(is)h(de\014ned)g(in)f(\()p Fq(D)1262 3910 y Fo(+)1260 3968 y(0)1336 3945 y Fp(\002)18 b Fq(W)12 b Fu(\))19 b Fp(\\)g Fu(\()p Fq(D)1737 3910 y Fj(\000)1735 3968 y Fo(0)1812 3945 y Fp(\002)f Fq(W)12 b Fu(\))23 b(=)g Fq(D)2197 3957 y Fo(0)2252 3945 y Fp(\002)18 b Fq(W)12 b Fu(,)28 b(satis\014es)f(the)h(equation:)743 4084 y Fp(4)818 4096 y Fi(h)860 4084 y Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)k Fu(\))23 b(=)g Fp(4)1302 4096 y Fi(h)1344 4084 y Fq(f)1394 4049 y Fo(+)1448 4084 y Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))g(+)f Fp(4)1832 4096 y Fi(h)1874 4084 y Fq(f)1924 4049 y Fj(\000)1979 4084 y Fu(\()p Fq(s;)c(Y)19 b Fu(\))24 b(=)e Fq(g)2340 4049 y Fo(+)2395 4084 y Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))f(+)g Fq(g)2746 4049 y Fj(\000)2802 4084 y Fu(\()p Fq(s;)c(Y)19 b Fu(\))k(=)g Fq(g)s Fu(\()p Fq(s;)14 b(Y)k Fu(\))456 4221 y(if)28 b Fq(s;)14 b(s)k Fu(+)g Fq(h)23 b Fp(2)g Fq(D)966 4233 y Fo(0)1003 4221 y Fu(,)28 b Fq(Y)42 b Fp(2)23 b Fq(W)12 b Fu(.)605 4321 y(T)-7 b(o)41 b(follo)n(w)f(the)h(previous)f(program)f(the)j(\014rst)e (thing)i(w)n(e)e(m)n(ust)h(do)g(is)g(to)g(construct)456 4420 y(functions)31 b Fq(g)860 4390 y Fj(\006)916 4420 y Fu(,)h(de\014ned)f(in)h(the)f(corresp)r(onding)e(extended)j(domain)e (and)h(v)n(erifying)f(\(4.14\).)456 4520 y(This)36 b(is)g(done)f(b)n(y) h(using)g(the)g(next)h(lemma)f(whic)n(h)g(also)f(pro)n(vides)f(useful)j (b)r(ounds)f(of)g(the)456 4620 y(norm)27 b(of)g Fq(g)810 4589 y Fj(\006)893 4620 y Fu(in)h(terms)g(of)f(the)h(norm)f(of)h Fq(g)s Fu(.)605 4770 y Ft(Lemma)j Fu(4.3)p Ft(.)40 b Fs(L)l(et)30 b Fq(\037)23 b Fu(:)g Fl(C)44 b Fp(\000)-14 b(!)23 b Fl(C)50 b Fs(b)l(e)30 b(a)g(Lipschitz)i(b)l(ounde)l(d)e (function)f(such)h(that)1430 4907 y Fu(supp)14 b Fq(\037)23 b Fu(=)g Fp(f)p Fq(\030)j Fp(2)e Fl(C)44 b Fu(:)23 b(Re)14 b Fq(\030)27 b Fp(\024)c Fq(\033)s Fp(g)p Fs(.)456 5044 y(L)l(et)1176 5182 y Fu(\012)g(=)f Fp(f)p Fq(\030)27 b Fp(2)c Fl(C)44 b Fu(:)37 b Fq(s)1711 5194 y Fo(1)1772 5182 y Fq(<)22 b Fu(Re)14 b Fq(\030)27 b(<)c(s)2161 5194 y Fo(2)2198 5182 y Fq(;)28 b Fp(j)14 b Fu(Im)g Fq(\030)t Fp(j)23 b Fq(<)g(\034)2609 5194 y Fo(2)2646 5182 y Fp(g)p Fq(;)p eop %%Page: 44 52 44 51 bop 456 251 a Fo(44)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 471 y Fs(with)35 b Fq(s)680 483 y Fo(1)750 471 y Fp(2)f Fl(R)p Fs(,)42 b Fq(s)999 483 y Fo(1)1069 471 y Fq(<)33 b(\033)38 b Fs(and)e Fq(s)1458 483 y Fo(2)1528 471 y Fp(2)d Fu([)p Fq(\033)n(;)14 b Fp(1)p Fu(\))24 b Fp([)e(f1g)p Fs(.)54 b(L)l(et)35 b Fu(\012)2392 441 y Fj(\003)2463 471 y Fu(=)d(\012)23 b Fp(\\)2822 402 y Fj(\016)p 2720 426 237 4 v 2720 471 a Fu(supp)14 b Fq(\037)35 b Fs(\(smal)t(l)h(cir)l(cle)456 571 y(denotes)30 b(top)l(olo)l(gic)l(al)i(interior\))e(and)g(let)g Fq(g)c Fp(2)d(A)p Fu(\(\012)2075 541 y Fj(\003)2114 571 y Fq(;)14 b(W)e Fu(\))p Fs(.)39 b(We)30 b(de\014ne)844 763 y Fq(h)p Fu(\()p Fq(\030)t(;)14 b(\021)s Fu(\))24 b(=)1238 706 y(1)p 1198 743 121 4 v 1198 819 a(2)p Fq(\031)s(i)1342 650 y Fg(Z)1389 838 y Fi(@)t Fo(\012)p Fj(\\)p Fo(supp)10 b Fi(\037)1749 706 y Fq(\037)p Fu(\()p Fq(\020)c Fu(\))p 1737 743 185 4 v 1737 819 a Fq(\020)25 b Fp(\000)18 b Fq(\030)1931 763 y(g)s Fu(\()p Fq(\020)6 b(;)14 b(\021)s Fu(\))g Fq(d\020)30 b Fu(=)2421 706 y(1)p 2382 743 121 4 v 2382 819 a(2)p Fq(\031)s(i)2526 650 y Fg(Z)2572 838 y Fi(@)t Fo(\012)2658 822 y Fb(\003)2734 706 y Fq(\037)p Fu(\()p Fq(\020)6 b Fu(\))p 2721 743 185 4 v 2721 819 a Fq(\020)25 b Fp(\000)18 b Fq(\030)2915 763 y(g)s Fu(\()p Fq(\020)6 b(;)14 b(\021)s Fu(\))g Fq(d\020)6 b(:)456 959 y Fs(Then)679 1081 y Fu(1\))41 b Fq(h)30 b Fs(is)g(analytic)h(on)f Fu(\012)18 b Fp(\002)g Fq(W)12 b Fs(,)679 1184 y Fu(2\))41 b Fq(h)30 b Fs(extends)f(c)l(ontinuously)g(to)p 1746 1117 60 4 v 30 w Fu(\012)19 b Fp(\002)f Fq(W)12 b Fs(,)679 1283 y Fu(3\))41 b Fs(if)31 b Fu(\()p Fq(\030)943 1295 y Fo(0)981 1283 y Fq(;)14 b(\021)1059 1295 y Fo(0)1096 1283 y Fu(\))23 b Fp(2)h Fu(\()p Fq(@)5 b Fu(\012)18 b Fp(\\)h Fu(supp)14 b Fq(\037)p Fu(\))19 b Fp(\002)f Fq(W)920 1475 y Fu(lim)877 1532 y Fo(\()p Fi(\030)933 1540 y Fh(0)966 1532 y Fi(;\021)1020 1540 y Fh(0)1052 1532 y Fo(\))1092 1475 y Fq(h)p Fu(\()p Fq(\030)t(;)c(\021)s Fu(\))24 b(=)e Fq(\037)p Fu(\()p Fq(\030)1556 1487 y Fo(0)1594 1475 y Fu(\))p Fq(g)s Fu(\()p Fq(\030)1737 1487 y Fo(0)1775 1475 y Fq(;)14 b(\021)1853 1487 y Fo(0)1890 1475 y Fu(\))19 b(+)2073 1419 y(1)p 2034 1456 121 4 v 2034 1532 a(2)p Fq(\031)s(i)2178 1362 y Fg(Z)2224 1550 y Fi(@)t Fo(\012)2310 1534 y Fb(\003)2373 1419 y Fq(\037)p Fu(\()p Fq(\020)6 b Fu(\))20 b Fp(\000)e Fq(\037)p Fu(\()p Fq(\030)2754 1431 y Fo(0)2791 1419 y Fu(\))p 2373 1456 451 4 v 2490 1532 a Fq(\020)25 b Fp(\000)18 b Fq(\030)2670 1544 y Fo(0)2834 1475 y Fq(g)s Fu(\()p Fq(\020)6 b(;)14 b(\021)3029 1487 y Fo(0)3066 1475 y Fu(\))g Fq(d\020)6 b(;)794 1670 y Fs(and)30 b(if)h Fu(\()p Fq(\030)1104 1682 y Fo(0)1142 1670 y Fq(;)14 b(\021)1220 1682 y Fo(0)1257 1670 y Fu(\))24 b Fp(2)f Fu(\()p Fq(@)5 b Fu(\012)18 b Fp(\\)h Fu(\(supp)14 b Fq(\037)p Fu(\))1925 1639 y Fi(c)1959 1670 y Fu(\))19 b Fp(\002)f Fq(W)1125 1861 y Fu(lim)1082 1918 y Fo(\()p Fi(\030)1138 1926 y Fh(0)1171 1918 y Fi(;\021)1225 1926 y Fh(0)1257 1918 y Fo(\))1297 1861 y Fq(h)p Fu(\()p Fq(\030)t(;)c(\021)s Fu(\))23 b(=)1690 1805 y(1)p 1651 1842 121 4 v 1651 1918 a(2)p Fq(\031)s(i)1795 1748 y Fg(Z)1841 1937 y Fi(@)t Fo(\012)1927 1920 y Fb(\003)1990 1805 y Fq(\037)p Fu(\()p Fq(\020)6 b Fu(\))20 b Fp(\000)e Fq(\037)p Fu(\()p Fq(\030)2371 1817 y Fo(0)2409 1805 y Fu(\))p 1990 1842 451 4 v 2107 1918 a Fq(\020)25 b Fp(\000)18 b Fq(\030)2287 1930 y Fo(0)2451 1861 y Fq(g)s Fu(\()p Fq(\020)6 b(;)14 b(\021)s Fu(\))g Fq(d\020)6 b(;)679 2056 y Fu(4\))41 b Fs(if)31 b Fu(\()p Fq(\030)t(;)14 b(\021)s Fu(\))24 b Fp(2)f Fu(\012)18 b Fp(\002)g Fq(K)6 b Fs(,)30 b(wher)l(e)h Fq(K)k Fs(is)30 b(a)g(c)l(omp)l(act)g(subset)f (of)i Fq(W)42 b Fs(we)30 b(have)1070 2249 y Fp(j)p Fq(h)p Fu(\()p Fq(\030)t(;)14 b(\021)s Fu(\))p Fp(j)24 b(\024)1474 2132 y Fg(\022)1536 2249 y Fp(k)p Fq(\037)p Fp(k)17 b Fu(+)1807 2193 y(1)p 1782 2230 92 4 v 1782 2306 a(2)p Fq(\031)1898 2249 y Fu(Lip)c Fq(\037)44 b Fu(length\()p Fq(@)5 b Fu(\012)2495 2215 y Fj(\003)2533 2249 y Fu(\))2565 2132 y Fg(\023)2640 2249 y Fp(k)p Fq(g)s Fp(k)2767 2261 y Fi(K)794 2443 y Fs(wher)l(e)39 b Fp(k)p Fq(\037)p Fp(k)f Fu(=)g(sup)p Fp(fj)p Fq(\037)p Fu(\()p Fq(\030)t Fu(\))p Fp(j)g Fu(:)53 b Fq(\030)42 b Fp(2)d Fl(C)15 b Fp(g)44 b Fs(and)39 b Fp(k)p Fq(g)s Fp(k)2406 2455 y Fi(K)2507 2443 y Fu(=)f(sup)p Fp(fj)p Fq(g)s Fu(\()p Fq(\030)t(;)14 b(\021)s Fu(\))p Fp(j)38 b Fu(:)53 b(\()p Fq(\030)t(;)14 b(\021)s Fu(\))39 b Fp(2)794 2543 y Fq(@)5 b Fu(\012)903 2512 y Fj(\003)959 2543 y Fp(\002)18 b Fq(K)6 b Fp(g)p Fs(.)605 2664 y(The)34 b(same)f(r)l(esults)f(hold)i(in)e(the)h(c)l(ase) g Fu(supp)14 b Fq(\037)29 b Fu(=)e Fp(f)p Fq(\030)32 b Fp(2)d Fl(C)15 b Fu(;)34 b(Re)14 b Fq(\030)32 b Fp(\025)c Fq(\033)s Fp(g)p Fs(,)34 b Fq(s)2989 2676 y Fo(1)3054 2664 y Fp(2)29 b(f\0001g)19 b([)456 2764 y Fu(\()p Fp(\0001)p Fq(;)14 b(\033)s Fu(])p Fs(,)30 b Fq(s)840 2776 y Fo(2)900 2764 y Fp(2)24 b Fl(R)p Fs(,)36 b Fq(s)1133 2776 y Fo(2)1193 2764 y Fq(>)23 b(\033)s Fs(.)605 2918 y Ft(Remark)31 b Fu(4.4)p Ft(.)39 b Fs(We)29 b(observe)g(that,)h(in)f(or)l(der)g(to)g (apply)i(this)e(r)l(esult,)f(we)h(only)h(ne)l(e)l(d)e(that)456 3018 y(the)i(function)f Fq(g)k Fs(to)c(b)l(e)h(analytic)h(in)f(a)g(b)l (ounde)l(d)g(c)l(omplex)h(r)l(e)l(ctangle.)605 3173 y Fu(This)21 b(Lemma)h(is)f(a)g(parameter)f(\(with)i(resp)r(ect)f(to)g Fq(\021)s Fu(\))h(v)n(ersion)e(of)h(a)g(Lemma)h(b)n(y)f(Lazutkin)456 3272 y(in)33 b([)p Fk(La2)q Fu(].)53 b(The)33 b(pro)r(of)f(of)h(the)h (presen)n(t)e(v)n(ersion)g(of)h(the)g(Lemma)g(can)g(b)r(e)g(found)g(in) h([)p Fk(F)-8 b(o3)o Fu(].)456 3372 y(Using)20 b(the)h(previous)e(tec)n (hnical)h(lemma,)i(w)n(e)e(will)h(construct)f(a)g(righ)n(t)g(in)n(v)n (erse)f(of)h(the)h(op)r(erator)456 3472 y Fp(4)531 3484 y Fi(h)572 3472 y Fu(.)605 3626 y Ft(Lemma)31 b Fu(4.4)p Ft(.)40 b Fs(L)l(et)30 b Fq(D)1301 3638 y Fo(0)1361 3626 y Fu(=)22 b Fq(D)1517 3638 y Fo(0)1554 3626 y Fu(\()p Fq(\034)1631 3591 y Fj(\006)1622 3648 y Fo(1)1688 3626 y Fq(;)14 b(\034)1761 3638 y Fo(2)1799 3626 y Fq(;)g Fu(0\))p Fs(.)38 b(Then)31 b(ther)l(e)e(is)i(a)f(c)l(ontinuous)f(op)l (er)l(ator)1406 3774 y Fp(4)1481 3738 y Fj(\000)p Fo(1)1481 3799 y Fi(h)1592 3774 y Fu(:)23 b Fp(A)p Fu(\()p Fq(D)1805 3786 y Fo(0)1842 3774 y Fq(;)14 b(W)e Fu(\))24 b Fp(!)f(A)p Fu(\()p Fq(D)2298 3786 y Fo(0)2335 3774 y Fq(;)14 b(W)e Fu(\))456 3933 y Fs(such)29 b(that)h(given)h Fq(g)25 b Fp(2)f(A)p Fu(\()p Fq(D)1342 3945 y Fo(0)1379 3933 y Fq(;)14 b(W)e Fu(\))p Fs(,)30 b Fq(f)i Fu(=)23 b Fp(4)1829 3898 y Fj(\000)p Fo(1)1829 3958 y Fi(h)1917 3933 y Fq(g)32 b Fs(is)e(a)g(solution)g(of)h(the)f(e)l(quation)902 4077 y Fq(f)9 b Fu(\()p Fq(s)18 b Fu(+)g Fq(h;)c(Y)k Fu(\))h Fp(\000)f Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)k Fu(\))24 b(=)e Fq(g)s Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))178 b Fs(for)31 b Fq(s;)14 b(s)k Fu(+)g Fq(h)23 b Fp(2)g Fq(D)2771 4089 y Fo(0)2808 4077 y Fs(,)30 b Fq(Y)42 b Fp(2)24 b Fq(W)-2654 b Fu(\(4.15\))456 4237 y Fs(and)26 b(its)h(op)l(er)l(ator)g(norm)f (veri\014es)h Fp(k4)1667 4201 y Fj(\000)p Fo(1)1667 4262 y Fi(h)1754 4237 y Fp(k)c(\024)f Fq(C)1965 4249 y Fi(D)2019 4257 y Fh(0)2057 4237 y Fq(e)2096 4207 y Fi(h=\034)2200 4215 y Fh(2)2235 4237 y Fq(h)2283 4207 y Fj(\000)p Fo(1)2372 4237 y Fs(,)28 b(wher)l(e)f(the)f(c)l(onstant)f Fq(C)3176 4249 y Fi(D)3230 4257 y Fh(0)3294 4237 y Fs(only)456 4337 y(dep)l(ends)30 b(on)g(the)g(size)g(of)h(the)f(domain)h Fq(D)1785 4349 y Fo(0)1822 4337 y Fs(.)605 4491 y Ft(Pr)n(oof.)41 b Fu(Let)28 b Fq(\037)23 b Fu(:)g Fl(R)29 b Fp(!)p Fu([0)p Fq(;)14 b Fu(1],)26 b(b)r(e)i(the)g(Lipsc)n(hitz)g(function)g (de\014ned)g(b)n(y)1214 4764 y Fq(\037)p Fu(\()p Fq(u)p Fu(\))23 b(=)1489 4569 y Fg(8)1489 4644 y(>)1489 4669 y(<)1489 4818 y(>)1489 4843 y(:)1604 4636 y Fu(1)427 b(if)28 b Fq(u)22 b Fp(\024)h Fq(\034)2352 4600 y Fj(\000)2343 4658 y Fo(1)1604 4764 y Fu(1)18 b Fp(\000)1780 4720 y Fi(u)p Fj(\000)p Fi(\034)1909 4693 y Fb(\000)1902 4738 y Fh(1)p 1757 4745 223 4 v 1757 4803 a Fi(\034)1795 4776 y Fh(+)1788 4821 y(1)1841 4803 y Fj(\000)p Fi(\034)1931 4776 y Fb(\000)1924 4821 y Fh(1)2073 4764 y Fu(if)28 b Fq(\034)2194 4729 y Fj(\000)2185 4786 y Fo(1)2274 4764 y Fq(<)22 b(u)h(<)g(\034)2565 4729 y Fo(+)2556 4786 y(1)1604 4892 y Fu(0)427 b(if)28 b Fq(u)22 b Fp(\025)h Fq(\034)2352 4857 y Fo(+)2343 4915 y(1)2408 4892 y Fu(.)456 5037 y(Let)k Fq(\037)656 5049 y Fo(+)711 5037 y Fu(\()p Fq(s)p Fu(\))d(=)f Fq(\037)p Fu(\(Re)14 b Fq(s)p Fu(\))28 b(and)f Fq(\037)1434 5049 y Fj(\000)1490 5037 y Fu(\()p Fq(s)p Fu(\))d(=)e(1)c Fp(\000)g Fq(\037)1899 5049 y Fo(+)1954 5037 y Fu(\()p Fq(s)p Fu(\),)28 b(de\014ned)g(in)g Fl(C)15 b Fu(.)43 b(W)-7 b(e)28 b(observ)n(e)e(that)1391 5182 y(supp)14 b Fq(\037)1628 5194 y Fo(+)1706 5182 y Fu(=)23 b Fp(f)p Fq(s)f Fp(2)i Fl(C)44 b Fu(:)23 b(Re)14 b Fq(s)23 b Fp(\024)f Fq(\034)2411 5146 y Fo(+)2402 5204 y(1)2467 5182 y Fp(g)p eop %%Page: 45 53 45 52 bop 1006 251 a Fo(4.5.)30 b(FLO)n(W)f(BO)n(X)g(COORDINA)-5 b(TES)29 b(IN)g(A)g(COMPLEX)g(DOMAIN)484 b(45)456 450 y Fu(and)1379 602 y(supp)14 b Fq(\037)1616 614 y Fj(\000)1695 602 y Fu(=)22 b Fp(f)p Fq(s)h Fp(2)g Fl(C)44 b Fu(:)23 b(Re)14 b Fq(s)23 b Fp(\025)g Fq(\034)2400 567 y Fj(\000)2391 625 y Fo(1)2457 602 y Fp(g)p Fu(.)605 802 y(Moreo)n(v)n(er)30 b(it)k(is)e(clear)g(that)i Fq(D)1611 814 y Fo(0)1680 802 y Fu(=)d Fq(D)1847 766 y Fo(+)1845 824 y(0)1924 802 y Fp(\\)2131 733 y Fj(\016)p 2002 756 293 4 v 2002 802 a Fu(supp)14 b Fq(\037)2239 814 y Fo(+)2327 802 y Fu(and)32 b Fq(D)2562 814 y Fo(0)2631 802 y Fu(=)g Fq(D)2799 766 y Fj(\000)2797 824 y Fo(0)2877 802 y Fp(\\)3084 733 y Fj(\016)p 2954 756 V 2954 802 a Fu(supp)14 b Fq(\037)3191 814 y Fj(\000)3247 802 y Fu(.)53 b(Let)456 905 y Fq(\032)23 b Fu(=)f Fq(\034)654 870 y Fj(\000)p Fo(1)645 927 y(2)772 905 y Fu(and)27 b Fq(g)f Fp(2)d(A)p Fu(\()p Fq(D)1244 917 y Fo(0)1282 905 y Fq(;)14 b(W)e Fu(\).)37 b(By)27 b(Lemma)g(4.3,)g(the)h(functions)960 1107 y Fq(g)1000 1119 y Fj(\006)1055 1107 y Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))24 b(=)1423 1051 y(1)p 1383 1088 121 4 v 1383 1164 a(2)p Fq(\031)s(i)1655 1051 y Fu(1)p 1523 1088 304 4 v 1523 1164 a(cosh\()p Fq(\032s)p Fu(\))1851 994 y Fg(Z)1897 1182 y Fi(@)t(D)1990 1190 y Fh(0)2051 1051 y Fq(\037)2103 1063 y Fj(\006)2159 1051 y Fu(\()p Fq(\030)t Fu(\))14 b(cosh\()p Fq(\032\030)t Fu(\))p 2051 1088 532 4 v 2226 1164 a Fq(\030)23 b Fp(\000)18 b Fq(s)2592 1107 y(g)s Fu(\()p Fq(\030)t(;)c(Y)19 b Fu(\))14 b Fq(d\030)456 1323 y Fu(b)r(elong)27 b(to)g Fp(A)p Fu(\()p Fq(D)991 1288 y Fj(\006)989 1345 y Fo(0)1048 1323 y Fq(;)14 b(W)e Fu(\))28 b(resp)r(ectiv)n(ely)-7 b(.)36 b(Moreo)n(v)n(er,)25 b(b)n(y)i(4\))g(of)h(Lemma)f(4.3,)g(w)n(e)g(ha)n(v)n(e)588 1525 y Fp(j)p Fq(g)651 1537 y Fj(\006)707 1525 y Fu(\()p Fq(s;)14 b(Y)k Fu(\))p Fp(j)84 b(\024)e Fu([)p Fp(k)p Fq(\037)1284 1537 y Fj(\006)1340 1525 y Fp(k)18 b Fu(+)1518 1469 y(1)p 1493 1506 92 4 v 1493 1582 a(2)p Fq(\031)1608 1525 y Fu(Lip)c Fq(\037)1795 1537 y Fj(\006)1865 1525 y Fu(length\()p Fq(@)5 b(D)2241 1537 y Fo(0)2278 1525 y Fu(\)])2462 1469 y Fp(k)p Fq(g)s Fp(k)p 2343 1506 364 4 v 2343 1582 a(j)14 b Fu(cosh\()p Fq(\032s)p Fu(\))p Fp(j)2755 1525 y Fu(max)2731 1579 y Fi(\030)r Fj(2)p Fi(@)t(D)2901 1587 y Fh(0)2947 1525 y Fp(j)g Fu(cosh)o(\()p Fq(\032\030)t Fu(\))p Fp(j)1020 1745 y(\024)82 b Fq(C)1226 1757 y Fi(D)1280 1765 y Fh(0)1317 1745 y Fp(k)p Fq(g)s Fp(k)1614 1689 y Fu(1)p 1454 1726 V 1454 1802 a Fp(j)14 b Fu(cosh)n(\()p Fq(\032s)p Fu(\))p Fp(j)2159 1745 y Fu(for)27 b(\()p Fq(s;)14 b(Y)k Fu(\))24 b Fp(2)f Fq(D)2665 1710 y Fj(\006)2663 1768 y Fo(0)2740 1745 y Fp(\002)18 b Fq(W)-2445 b Fu(\(4.16\))456 1946 y(where)27 b Fp(k)p Fq(g)s Fp(k)f Fu(means)h(sup)1228 1967 y Fi(D)1282 1975 y Fh(0)1315 1967 y Fj(\002)p Fi(W)1456 1946 y Fp(j)p Fq(g)s Fu(\()p Fq(s;)14 b(Y)k Fu(\))p Fp(j)p Fu(.)605 2055 y(No)n(w)27 b(w)n(e)g(construct)g(the)h(in)n(v)n(erse)f(of)g Fp(4)1869 2067 y Fi(h)1911 2055 y Fu(.)37 b(Giv)n(en)27 b(\()p Fq(s;)14 b(Y)19 b Fu(\))k Fp(2)h Fq(D)2591 2019 y Fo(+)2589 2077 y(0)2664 2055 y Fp(\002)18 b Fq(W)12 b Fu(,)28 b(w)n(e)f(de\014ne)1376 2225 y Fq(f)1417 2237 y Fo(+)1472 2225 y Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))k(=)g Fp(\000)1869 2146 y Fg(X)1869 2325 y Fi(k)q Fj(\025)p Fo(0)2004 2225 y Fq(g)2044 2237 y Fo(+)2098 2225 y Fu(\()p Fq(s)c Fu(+)f Fq(k)s(h;)c(Y)k Fu(\).)456 2477 y(A)29 b(direct)g(substitution)h(sho)n(ws)e(that)i Fq(f)1711 2489 y Fo(+)1795 2477 y Fu(satis\014es)f(\(4.15\))f(in)i Fq(D)2514 2441 y Fo(+)2512 2499 y(0)2589 2477 y Fp(\002)19 b Fq(W)12 b Fu(.)42 b(In)29 b(the)h(same)f(w)n(a)n(y)-7 b(,)456 2579 y(if)28 b(\()p Fq(s;)14 b(Y)k Fu(\))24 b Fp(2)f Fq(D)911 2543 y Fj(\000)909 2601 y Fo(0)986 2579 y Fp(\002)18 b Fq(W)12 b Fu(,)1426 2749 y Fq(f)1467 2761 y Fj(\000)1523 2749 y Fu(\()p Fq(s;)i(Y)19 b Fu(\))k(=)1841 2670 y Fg(X)1841 2849 y Fi(k)q Fj(\025)p Fo(1)1976 2749 y Fq(g)2016 2761 y Fj(\000)2072 2749 y Fu(\()p Fq(s)18 b Fp(\000)g Fq(k)s(h;)c(Y)19 b Fu(\))456 3000 y(satis\014es)26 b(\(4.15\))h(in)h Fq(D)1169 2965 y Fj(\000)1167 3023 y Fo(0)1244 3000 y Fp(\002)18 b Fq(W)12 b Fu(.)605 3100 y(Un)n(til)22 b(the)f(end)h(of)f(the)g(pro)r(of,)h Fq(C)1616 3112 y Fi(D)1670 3120 y Fh(0)1729 3100 y Fu(will)f(b)r(e)g(a)g(generic) f(constan)n(t)h(whic)n(h)g(ma)n(y)f(tak)n(e)h(di\013er-)456 3201 y(en)n(t)27 b(v)-5 b(alues)27 b(in)g(di\013eren)n(t)h(form)n(ulas) e(but)i(only)f(dep)r(ends)g(on)g Fq(D)2442 3213 y Fo(0)2507 3201 y Fu(and)g Fq(\034)2704 3213 y Fo(1)2764 3201 y Fu(=)c(max)p Fp(fj)p Fq(\034)3117 3165 y Fo(+)3108 3223 y(1)3172 3201 y Fp(j)p Fq(;)14 b Fp(j)p Fq(\034)3300 3165 y Fj(\000)3291 3223 y Fo(1)3357 3201 y Fp(jg)p Fu(.)605 3302 y(These)25 b(series)f(are)h(con)n(v)n(ergen)n(t.)34 b(Indeed,)26 b(from)f(\(4.16\))g(w)n(e)g(ha)n(v)n(e,)f(for)h(\()p Fq(s;)14 b(Y)19 b Fu(\))k Fp(2)h Fq(D)3207 3267 y Fo(+)3205 3325 y(0)3276 3302 y Fp(\002)13 b Fq(W)791 3495 y Fp(j)p Fq(f)855 3507 y Fo(+)910 3495 y Fu(\()p Fq(s;)h(Y)19 b Fu(\))p Fp(j)k(\024)1252 3416 y Fg(X)1251 3595 y Fi(k)q Fj(\025)p Fo(0)1387 3495 y Fp(j)p Fq(g)1450 3507 y Fo(+)1504 3495 y Fu(\()p Fq(s)c Fu(+)f Fq(k)s(h;)c(Y)k Fu(\))p Fp(j)24 b(\024)e Fq(C)2099 3507 y Fi(D)2153 3515 y Fh(0)2190 3495 y Fp(k)p Fq(g)s Fp(k)2331 3416 y Fg(X)2331 3595 y Fi(k)q Fj(\025)p Fo(0)2766 3439 y Fu(1)p 2475 3476 624 4 v 2475 3552 a Fp(j)14 b Fu(cosh)o(\()p Fq(\032)p Fu(\()p Fq(s)19 b Fu(+)f Fq(k)s(h)p Fu(\)\))p Fp(j)1163 3757 y(\024)23 b Fq(C)1310 3769 y Fi(D)1364 3777 y Fh(0)1401 3757 y Fp(k)p Fq(g)s Fp(k)1541 3678 y Fg(X)1542 3857 y Fi(k)q Fj(\025)p Fo(0)2225 3701 y Fu(1)p 1686 3738 1120 4 v 1686 3814 a(cosh)o(\()p Fq(\032)p Fu(\(Re)15 b Fq(s)j Fu(+)g Fq(k)s(h)p Fu(\)\))p Fp(j)c Fu(cos)o(\()p Fq(\032)g Fu(Im)g Fq(s)p Fu(\))p Fp(j)1163 4019 y(\024)23 b Fq(C)1310 4031 y Fi(D)1364 4039 y Fh(0)1401 4019 y Fp(k)p Fq(g)s Fp(k)1625 3963 y Fu(2)p 1538 4000 218 4 v 1538 4076 a(cos)n(\(1\))1764 4019 y Fq(e)1803 3985 y Fj(\000)p Fi(\032)11 b Fo(Re)g Fi(s)2040 3940 y Fg(X)2039 4119 y Fi(k)q Fj(\025)p Fo(0)2174 4019 y Fq(e)2213 3985 y Fj(\000)p Fi(\032hk)1163 4281 y Fp(\024)23 b Fq(C)1310 4293 y Fi(D)1364 4301 y Fh(0)1401 4281 y Fp(k)p Fq(g)s Fp(k)1671 4225 y Fu(1)p 1538 4262 311 4 v 1538 4338 a(1)17 b Fp(\000)h Fq(e)1719 4314 y Fj(\000)p Fi(\032h)1858 4281 y Fq(e)1897 4246 y Fj(\000)p Fi(\032)11 b Fo(Re)g Fi(s)1163 4452 y Fp(\024)23 b Fq(C)1310 4464 y Fi(D)1364 4472 y Fh(0)1401 4452 y Fp(k)p Fq(g)s Fp(k)p Fq(h)1576 4418 y Fj(\000)p Fo(1)1664 4452 y Fq(e)1703 4418 y Fi(\032h)1780 4452 y Fq(e)1819 4418 y Fj(\000)p Fi(\032)11 b Fo(Re)g Fi(s)2041 4452 y Fu(.)456 4619 y(In)27 b(the)h(same)f(w)n(a)n(y)g(w)n (e)g(obtain,)g(for)g(\()p Fq(s;)14 b(Y)19 b Fu(\))k Fp(2)h Fq(D)1989 4584 y Fj(\000)1987 4641 y Fo(0)2063 4619 y Fp(\002)18 b Fq(W)1339 4777 y Fp(j)p Fq(f)1403 4789 y Fj(\000)1459 4777 y Fu(\()p Fq(s;)c(Y)19 b Fu(\))p Fp(j)k(\024)g Fq(C)1859 4789 y Fi(D)1913 4797 y Fh(0)1950 4777 y Fp(k)p Fq(g)s Fp(k)p Fq(h)2125 4743 y Fj(\000)p Fo(1)2213 4777 y Fq(e)2252 4743 y Fi(\032h)2329 4777 y Fq(e)2368 4743 y Fi(\032)11 b Fo(Re)g Fi(s)2538 4777 y Fu(.)456 4930 y(No)n(w)24 b(w)n(e)g(consider)f(the)i(function)h Fq(f)31 b Fu(:)23 b Fq(D)1731 4942 y Fo(0)1781 4930 y Fp(\002)12 b Fq(W)35 b Fp(!)23 b Fl(C)15 b Fu(,)31 b(de\014ned)25 b(b)n(y)49 b Fq(f)32 b Fu(=)22 b Fq(f)2806 4942 y Fo(+)2873 4930 y Fu(+)12 b Fq(f)2991 4942 y Fj(\000)3047 4930 y Fu(.)36 b(It)25 b(is)g(clear)456 5029 y(that)755 5182 y Fp(4)830 5194 y Fi(h)872 5182 y Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)k Fu(\))23 b(=)g Fp(4)1314 5194 y Fi(h)1356 5182 y Fq(f)1397 5194 y Fo(+)1451 5182 y Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))g(+)f Fp(4)1835 5194 y Fi(h)1877 5182 y Fq(f)1918 5194 y Fj(\000)1973 5182 y Fu(\()p Fq(s;)c(Y)19 b Fu(\))24 b(=)e Fq(g)2331 5194 y Fo(+)2386 5182 y Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))f(+)g Fq(g)2734 5194 y Fj(\000)2790 5182 y Fu(\()p Fq(s;)c(Y)19 b Fu(\))k(=)g Fq(g)s Fu(\()p Fq(s;)14 b(Y)k Fu(\))p eop %%Page: 46 54 46 53 bop 456 251 a Fo(46)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(for)27 b Fq(s;)14 b(s)k Fu(+)g Fq(h)23 b Fp(2)g Fq(D)1017 462 y Fo(0)1054 450 y Fu(,)28 b Fq(Y)42 b Fp(2)23 b Fq(W)12 b Fu(.)37 b(Moreo)n(v)n(er,)25 b(since)i Fq(\032)c Fu(=)g Fq(\034)2213 415 y Fj(\000)p Fo(1)2204 472 y(2)2303 450 y Fu(,)k(on)h Fq(D)2538 462 y Fo(0)2593 450 y Fp(\002)18 b Fq(W)12 b Fu(,)1002 612 y Fp(j)p Fq(f)d Fu(\()p Fq(s;)14 b(Y)k Fu(\))p Fp(j)24 b(\024)e Fq(C)1474 624 y Fi(D)1528 632 y Fh(0)1566 612 y Fp(k)p Fq(g)s Fp(k)p Fq(h)1741 578 y Fj(\000)p Fo(1)1828 612 y Fq(e)1867 578 y Fo(\()p Fi(\034)1924 586 y Fh(1)1956 578 y Fo(+)p Fi(h)p Fo(\))p Fi(=\034)2137 586 y Fh(2)2196 612 y Fu(=)g Fq(C)2342 624 y Fi(D)2396 632 y Fh(0)2434 612 y Fp(k)p Fq(g)s Fp(k)p Fq(h)2609 578 y Fj(\000)p Fo(1)2696 612 y Fq(e)2735 578 y Fi(h=\034)2839 586 y Fh(2)2875 612 y Fu(.)456 771 y(Then)27 b(the)h Fq(f)37 b Fu(so)27 b(constructed)g(solv)n(es)f(\(4.15\).)1465 b Fe(\003)605 947 y Fk(4.5.3.)46 b(A)27 b(useful)g(parameterization)g (of)g(the)h(solutions)d(of)j(the)f(system)f(\(4.3\).)456 1047 y Fu(In)c(this)h(section)f(w)n(e)g(giv)n(e)f(a)h(go)r(o)r(d)g (parameterization)e(of)j(a)f(set)g(of)g(solutions)g(passing)f(through) 456 1146 y Fq(x)26 b Fu(=)f Fq(\016)s Fu(,)30 b(of)f(the)h(system)f (asso)r(ciated)f(to)h(the)h(v)n(ector)e(\014eld)h Fq(X)7 b Fu(.)42 b(W)-7 b(e)29 b(in)n(tro)r(duce)g(an)g(additional)456 1246 y(parameter,)i Fq(s)f Fp(2)g Fl(C)15 b Fu(,)38 b(to)32 b(b)r(e)g(able)f(to)h(reac)n(h)e Fp(f)p Fq(x)g Fu(=)f Fq(\016)s Fp(g)i Fu(and)h(to)f(obtain)h(useful)g(prop)r(erties)f(of)456 1346 y(the)d(parameterization.)605 1445 y(W)-7 b(e)28 b(recall)f(that)h(w)n(e)f(denote)g(b)n(y)h Fq(z)1694 1457 y Fo(0)1730 1445 y Fu(\()p Fq(u;)14 b(Y)19 b Fu(\))28 b(the)g(solution)f(of)g(the)h(unp)r(erturb)r(ed)h(system)1672 1599 y(_)-37 b Fq(x)83 b Fu(=)g Fq(v)22 b Fu(+)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\))1674 1723 y(_)-35 b Fq(v)86 b Fu(=)d Fp(\000)p Fq(v)s(f)2094 1689 y Fj(0)2117 1723 y Fu(\()p Fq(x)p Fu(\))456 1877 y(suc)n(h)22 b(that)g Fq(z)851 1889 y Fo(0)888 1877 y Fu(\(0)p Fq(;)14 b(Y)19 b Fu(\))k(=)g(\()p Fq(\016)o(;)14 b(Y)k Fu(\).)35 b(In)23 b(a)f(similar)f(w)n(a)n(y)h(as)f(in)i(Lemma)f(4.2)g(w)n(e)g(can)g(pro)n (v)n(e)e(that)j(the)456 1977 y(image)d(b)n(y)h Fq(z)837 1989 y Fo(0)896 1977 y Fu(of)g(\001)1053 1989 y Fo(1)1113 1977 y Fu(=)i Fq(D)1270 1989 y Fo(0)1307 1977 y Fu(\()p Fq(\024)1387 1942 y Fj(\006)1387 1999 y Fo(1)1449 1977 y Fp(\000)6 b Fq(\034)f(;)14 b(\024)1646 1989 y Fo(2)1683 1977 y Fq(;)g Fu(3)p Fq(\024)1810 1989 y Fo(3)1846 1977 y Fu(\))6 b Fp(\002)g Fq(W)12 b Fu(\(2)p Fq(r)2156 1989 y Fo(1)2194 1977 y Fu(\))21 b(is)h(con)n(tained)e(in)i Fq(U)2842 1989 y Fo(0)2879 1977 y Fu(\()p Fq(\024)2959 1942 y Fj(\006)2959 1999 y Fo(1)3015 1977 y Fq(;)14 b(\024)3100 1989 y Fo(2)3137 1977 y Fq(;)g Fu(3)p Fq(\024)3264 1989 y Fo(3)3301 1977 y Fq(;)g(r)3375 1989 y Fo(0)3412 1977 y Fu(\))456 2077 y(\(w)n(e)31 b(recall)g(that)h Fq(r)1061 2089 y Fo(1)1129 2077 y Fq(<)d(r)1260 2089 y Fo(0)1298 2077 y Fq(=)p Fu(2\).)49 b(Moreo)n(v)n(er,)30 b Fq(z)1918 2089 y Fo(0)1986 2077 y Fu(is)i(analytic)f(on)g(\001)2582 2089 y Fo(1)2652 2077 y Fu(and)g(con)n(tin)n(uous)g(on)g(its)456 2178 y(b)r(oundary)-7 b(.)39 b(Therefore,)28 b(since)g(the)h(v)n(ector) f(\014eld)h Fq(X)2108 2190 y Fo(0)2173 2178 y Fu(is)g(analytic)f(on)g Fq(U)2748 2190 y Fo(0)2785 2178 y Fu(\()p Fq(\024)2865 2142 y Fj(\006)2865 2200 y Fo(1)2921 2178 y Fq(;)14 b(\024)3006 2190 y Fo(2)3043 2178 y Fq(;)g Fu(3)p Fq(\024)3170 2190 y Fo(3)3207 2178 y Fq(;)g(r)3281 2190 y Fo(0)3319 2178 y Fu(\),)29 b(a)456 2277 y(fundamen)n(tal)e(matrix)g Fq(M)9 b Fu(\()p Fq(u)p Fu(\))28 b(of)f(the)h(system)1564 2418 y Fq(d)p 1540 2455 91 4 v 1540 2531 a(du)1641 2474 y(z)e Fu(=)d Fq(D)r(X)1934 2486 y Fo(0)1971 2474 y Fu(\()p Fq(z)2042 2486 y Fo(0)2079 2474 y Fu(\()p Fq(u;)14 b(Y)19 b Fu(\)\))p Fq(z)456 2664 y Fu(is)25 b(analytic)h(on)f Fq(D)1034 2676 y Fo(0)1071 2664 y Fu(\()p Fq(\024)1151 2629 y Fj(\006)1151 2687 y Fo(1)1222 2664 y Fp(\000)15 b Fq(\034)5 b(;)14 b(\024)1428 2676 y Fo(2)1465 2664 y Fq(;)g Fu(2)p Fq(\024)1592 2676 y Fo(3)1628 2664 y Fu(\))27 b(and)e(con)n(tin)n(uous)g(on)h(its)g(b)r(oundary)-7 b(.)36 b(Moreo)n(v)n(er)23 b(as)i(w)n(e)456 2766 y(p)r(oin)n(ted)i(out) h(in)g(\(4.11\),)f Fq(M)9 b Fu(\()p Fq(u)p Fu(\))27 b(and)h Fq(M)1744 2736 y Fj(\000)p Fo(1)1832 2766 y Fu(\()p Fq(u)p Fu(\))g(are)f(b)r(ounded)h(in)g Fq(D)2616 2778 y Fo(0)2653 2766 y Fu(\()p Fq(\024)2733 2731 y Fj(\006)2733 2789 y Fo(1)2807 2766 y Fp(\000)18 b Fq(\034)5 b(;)14 b(\024)3016 2778 y Fo(2)3053 2766 y Fq(;)g Fu(2)p Fq(\024)3180 2778 y Fo(3)3217 2766 y Fu(\).)605 2866 y(No)n(w)27 b(w)n(e)g(presen)n(t)g (the)h(parameterization)e(of)h(the)h(solutions)f(of)h(the)g(system)f (\(4.3\).)605 3026 y Ft(Pr)n(oposition)35 b Fu(4.1)p Ft(.)43 b Fs(If)33 b Fq(")g Fs(and)h Fq(\026)g Fs(ar)l(e)f(smal)t(l)h (enough)g(then)f(a)h(set)f(of)h(solutions)g(of)g(the)456 3126 y(e)l(quation)c(\(4.3\))h(c)l(an)e(b)l(e)h(expr)l(esse)l(d)g(as)g (p)l(ar)l(ameterize)l(d)h(curves)1172 3279 y Fu(\()p Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))p Fq(;)14 b(t=")p Fu(\))22 b(=)h(\()p Fq(x)p Fu(\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))p Fq(;)14 b(v)s Fu(\()p Fq(t;)g(s;)g(Y)19 b Fu(\))p Fq(;)14 b(t=")p Fu(\))456 3452 y Fs(with)30 b Fu(\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))k Fp(2)1026 3431 y Fu(~)1011 3452 y Fq(U)39 b Fs(de\014ne)l(d)30 b(by)1344 3590 y Fu(~)1329 3611 y Fq(U)i Fu(=)23 b Fq(D)r Fu(\()p Fq(\024)1657 3576 y Fj(\006)1657 3633 y Fo(1)1731 3611 y Fp(\000)18 b Fq(\034)5 b(;)14 b(\024)1940 3623 y Fo(2)1977 3611 y Fq(;)g Fu(2)p Fq(\024)2104 3623 y Fo(3)2141 3611 y Fu(\))19 b Fp(\002)f Fq(W)12 b Fu(\(2)p Fq(r)2476 3623 y Fo(1)2513 3611 y Fu(\))p Fs(,)456 3765 y(satisfying)31 b(the)f(fol)t(lowing)i(pr)l(op)l(erties:)679 3892 y Fu(1\))41 b Fq(t)23 b Fp(7!)g Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))30 b Fs(is)g(a)g(solution)g(of)h(system)f(\(4.3\).)679 3991 y Fu(2\))41 b Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))30 b Fs(is)g Fq(C)1294 3961 y Fo(1)1361 3991 y Fs(and)h(analytic)g(in)e Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))p Fs(.)679 4091 y Fu(3\))41 b Fq(z)t Fu(\()p Fq(t)18 b Fu(+)g(2)p Fq(\031)s(";)c(s;)g(Y)k Fu(\))23 b(=)g Fq(z)t Fu(\()p Fq(t;)14 b(s)k Fu(+)g(2)p Fq(\031)s(";)c(Y)k Fu(\))p Fq(:)679 4191 y Fu(4\))41 b Fs(The)31 b(solution)f(of)g(the)g (system)g(\(4.3\))h(is)f(of)h(the)f(form)1241 4344 y Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))23 b(=)g Fq(z)1707 4356 y Fo(0)1744 4344 y Fu(\()p Fq(t)c Fu(+)f Fq(s;)c(Y)k Fu(\))h(+)f Fq(\026")2273 4310 y Fi(q)2309 4344 y Fq(z)2348 4356 y Fo(1)2385 4344 y Fu(\()p Fq(t;)c(s;)g(Y)19 b Fu(\))824 4498 y Fs(with)1580 4651 y Fu(sup)1628 4720 y Fo(~)1617 4735 y Fi(U)1719 4651 y Fp(j)p Fq(z)1781 4663 y Fo(1)1818 4651 y Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))p Fp(j)24 b(\024)e Fq(K)q(:)679 4868 y Fu(5\))41 b Fs(F)-6 b(or)30 b(al)t(l)h Fq(Y)42 b Fp(2)23 b Fq(W)12 b Fu(\(2)p Fq(r)1437 4880 y Fo(1)1474 4868 y Fu(\))p Fs(,)31 b Fq(z)t Fu(\(0)p Fq(;)14 b Fu(0)p Fq(;)g(Y)j Fu(\))23 b(=)g(\()p Fq(\016)o(;)14 b(Y)k Fu(\))p Fs(.)605 5028 y Ft(Pr)n(oof.)41 b Fu(If)1029 5182 y(\()p Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))p Fq(;)14 b(t=")p Fu(\))22 b(=)h(\()p Fq(z)1739 5194 y Fo(0)1776 5182 y Fu(\()p Fq(t)c Fu(+)f Fq(s;)c(Y)k Fu(\))h(+)f Fq(\026")2305 5147 y Fi(q)2341 5182 y Fq(z)2380 5194 y Fo(1)2417 5182 y Fu(\()p Fq(t;)c(s;)g(Y)19 b Fu(\))p Fq(;)14 b(t=")p Fu(\))p eop %%Page: 47 55 47 54 bop 1006 251 a Fo(4.5.)30 b(FLO)n(W)f(BO)n(X)g(COORDINA)-5 b(TES)29 b(IN)g(A)g(COMPLEX)g(DOMAIN)484 b(47)456 450 y Fu(is)27 b(a)g(solution)g(of)h(the)g(equation)f(\(4.3\),)g(where)g Fq(z)1999 462 y Fo(0)2064 450 y Fu(is)g(a)h(solution)f(of)g(the)h(unp)r (erturb)r(ed)g(equa-)456 550 y(tion,)f(it)h(is)g(clear)e(that)1229 692 y(_)-37 b Fq(z)1254 704 y Fo(1)1314 692 y Fu(=)23 b Fq(D)r(X)1542 704 y Fo(0)1579 692 y Fu(\()p Fq(z)1650 704 y Fo(0)1687 692 y Fu(\()p Fq(t)c Fu(+)f Fq(s;)c(Y)k Fu(\)\))p Fq(z)2096 704 y Fo(1)2152 692 y Fu(+)g Fq(b)p Fu(\()p Fq(z)2342 704 y Fo(1)2379 692 y Fu(\)\()p Fq(t;)c(s;)g(Y)19 b Fu(\))-2229 b(\(4.17\))456 835 y(with)933 996 y Fq(b)p Fu(\()p Fq(z)1040 1008 y Fo(1)1077 996 y Fu(\)\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))83 b(=)1666 940 y(1)p 1624 977 126 4 v 1624 1053 a Fq(\026")1713 1029 y Fi(q)1759 929 y Fg(\002)1794 996 y Fq(X)1863 1008 y Fo(0)1900 996 y Fu(\()p Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\)\))h Fp(\000)f Fq(X)2451 1008 y Fo(0)2488 996 y Fu(\()p Fq(z)2559 1008 y Fo(0)2596 996 y Fu(\()p Fq(t)h Fu(+)f Fq(s;)c(Y)k Fu(\)\))1614 1173 y Fp(\000)p Fq(\026")1768 1139 y Fi(q)1804 1173 y Fq(D)r(X)1944 1185 y Fo(0)1981 1173 y Fu(\()p Fq(z)2052 1185 y Fo(0)2089 1173 y Fu(\()p Fq(t)h Fu(+)f Fq(s;)c(Y)k Fu(\)\))p Fq(z)2498 1185 y Fo(1)2536 1173 y Fu(\()p Fq(t;)c(s;)g(Y)k Fu(\))2809 1106 y Fg(\003)1614 1298 y Fu(+)p Fq(X)1748 1310 y Fo(1)1784 1298 y Fu(\()p Fq(z)t Fu(\()p Fq(t;)c(s;)g(Y)19 b Fu(\))p Fq(;)14 b(t=")p Fu(\).)-1880 b(\(4.18\))456 1440 y(Th)n(us,)27 b Fq(z)728 1452 y Fo(1)793 1440 y Fu(is)g(a)g(solution)g(of)h(\(4.17\))f(if)h(and)f(only)g(if)726 1640 y Fq(z)765 1652 y Fo(1)802 1640 y Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))24 b(=)e Fq(M)9 b Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))1525 1523 y Fg(\024)1569 1640 y Fq(M)1659 1606 y Fj(\000)p Fo(1)1747 1640 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(s;)c(Y)19 b Fu(\))g(+)2195 1527 y Fg(Z)2278 1548 y Fi(t)2241 1716 y Fo(0)2321 1640 y Fq(M)2411 1606 y Fj(\000)p Fo(1)2500 1640 y Fu(\()p Fq(\033)j Fu(+)c Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(z)2862 1652 y Fo(1)2899 1640 y Fu(\)\()p Fq(\033)n(;)c(s;)g(Y)20 b Fu(\))14 b Fq(d\033)3328 1523 y Fg(\025)456 1640 y Fu(\(4.19\))456 1833 y(where)33 b Fq(M)9 b Fu(\()p Fq(u)p Fu(\))34 b(is)g(a)g(fundamen) n(tal)h(matrix)e(of)i(the)f(homogeneous)f(system.)56 b(A)n(t)35 b(this)f(p)r(oin)n(t)456 1932 y Fq(c)p Fu(\()p Fq(s;)14 b(Y)k Fu(\))37 b(is)f(an)g(arbitrary)f(function.)64 b(W)-7 b(e)37 b(c)n(ho)r(ose)e(the)i(function)g Fq(c)p Fu(\()p Fq(s;)14 b(Y)k Fu(\))37 b(as)f(follo)n(ws.)62 b(W)-7 b(e)456 2032 y(consider)1054 2213 y Fq(g)s Fu(\()p Fq(z)1168 2225 y Fo(1)1204 2213 y Fu(\)\()p Fq(s;)14 b(Y)19 b Fu(\))24 b(=)e Fp(\000)1633 2100 y Fg(Z)1716 2121 y Fo(2)p Fi(\031)r(")1679 2289 y Fo(0)1839 2213 y Fq(M)1929 2179 y Fj(\000)p Fo(1)2018 2213 y Fu(\()p Fq(\033)g Fu(+)c Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(z)2380 2225 y Fo(1)2417 2213 y Fu(\)\()p Fq(\033)n(;)c(s;)g(Y)20 b Fu(\))14 b Fq(d\033)-2387 b Fu(\(4.20\))456 2398 y(and)27 b(w)n(e)g(tak)n(e)1143 2540 y Fq(c)p Fu(\()p Fq(z)1250 2552 y Fo(1)1287 2540 y Fu(\)\()p Fq(s;)14 b(Y)19 b Fu(\))k Fp(\021)g Fq(c)p Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))k(=)g Fq(M)9 b Fu(\()p Fq(s)p Fu(\))p Fp(4)2259 2505 y Fj(\000)p Fo(1)2259 2562 y(2)p Fi(\031)r(")2367 2540 y Fq(g)s Fu(\()p Fq(z)2481 2552 y Fo(1)2518 2540 y Fu(\)\()p Fq(s;)14 b(Y)19 b Fu(\))-2301 b(\(4.21\))456 2699 y(where)25 b Fp(4)769 2663 y Fj(\000)p Fo(1)769 2721 y(2)p Fi(\031)r(")904 2699 y Fu(is)h(the)g(op)r(erator)f(de\014ned)h(in)g(Lemma)g(4.4.)36 b(This)26 b(c)n(hoice)f(of)h Fq(c)p Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))26 b(is)g(the)h(one)456 2798 y(whic)n(h)h(will)h(p)r (ermit)g(us)g(to)g(c)n(hec)n(k)e(that)i(an)g(op)r(erator)e(to)h(b)r(e)i (de\014ned)f(b)r(elo)n(w)f(is)h(w)n(ell)f(de\014ned)456 2898 y(in)f(its)h(domain.)605 2997 y(W)-7 b(e)29 b(de\014ne)g(\006)g (to)g(b)r(e)g(the)h(space)e(of)g(functions)i Fq(z)2158 3009 y Fo(1)2220 2997 y Fu(:)2282 2976 y(~)2268 2997 y Fq(U)k Fp(!)25 b Fl(C)2521 2967 y Fo(2)2593 2997 y Fu(suc)n(h)k(that)g Fq(z)3002 3009 y Fo(1)3064 2997 y Fp(2)c Fu(\006)k(if)h(and)456 3097 y(only)d(if)h Fq(z)753 3109 y Fo(1)817 3097 y Fu(satis\014es)647 3218 y(\(a\))41 b Fq(z)833 3230 y Fo(1)870 3218 y Fu(\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))28 b(is)f Fq(C)1320 3188 y Fo(0)1385 3218 y Fu(and)h(analytic)f (on)g(\()p Fq(s;)14 b(Y)19 b Fu(\).)642 3324 y(\(b\))42 b(F)-7 b(or)30 b(all)f(\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))28 b Fp(2)1462 3303 y Fu(~)1447 3324 y Fq(U)39 b Fu(suc)n(h)30 b(that)g(\()p Fq(t)21 b Fu(+)f(2)p Fq(\031)s(";)14 b(s)p Fu(\))26 b Fp(2)i Fq(D)r Fu(\()p Fq(\024)2582 3288 y Fj(\006)2582 3346 y Fo(1)2658 3324 y Fp(\000)20 b Fq(\034)5 b(;)14 b(\024)2869 3336 y Fo(2)2906 3324 y Fq(;)g Fu(2)p Fq(\024)3033 3336 y Fo(3)3070 3324 y Fu(\),)31 b(w)n(e)f(ha)n(v)n(e)794 3423 y(that)1301 3566 y Fq(z)1340 3578 y Fo(1)1377 3566 y Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s;)g(Y)k Fu(\))23 b(=)g Fq(z)2033 3578 y Fo(1)2070 3566 y Fu(\()p Fq(t;)14 b(s)k Fu(+)g(2)p Fq(\031)s(";)c(Y)k Fu(\))p Fq(:)651 3713 y Fu(\(c\))42 b Fp(k)p Fq(z)875 3725 y Fo(1)912 3713 y Fp(k)22 b Fu(=)h(sup)1200 3723 y Fo(~)1189 3738 y Fi(U)1258 3713 y Fp(j)p Fq(z)1320 3725 y Fo(1)1357 3713 y Fu(\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))p Fp(j)k Fq(<)g Fu(+)p Fp(1)p Fu(.)605 3835 y(W)-7 b(e)34 b(endo)n(w)e(\006)h (with)h(the)g(suprem)n(um)e(norm)h(and)g(it)h(b)r(ecomes)f(a)f(Banac)n (h)g(space.)53 b(F)-7 b(or)456 3934 y(an)n(y)21 b Fq(\032)i(>)g Fu(0,)g(w)n(e)f(de\014ne)g(\006\()p Fq(\032)p Fu(\))h(as)f(the)h (closed)e(ball)i(of)f(radius)f Fq(\032)i Fu(of)f(\006.)35 b(W)-7 b(e)23 b(de\014ne)f(the)h(op)r(erator)456 4034 y Fp(G)28 b Fu(:)23 b(\006\()p Fq(\032)p Fu(\))h Fp(!)f Fu(\006\()p Fq(\032)p Fu(\))28 b(to)f(b)r(e)h(the)g(righ)n(t)f(hand)h (side)f(of)h(\(4.19\):)490 4334 y Fp(G)5 b Fu(\()p Fq(z)615 4346 y Fo(1)653 4334 y Fu(\)\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))k(=)f Fq(M)9 b Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))1408 4217 y Fg(\024)1452 4334 y Fq(M)1542 4299 y Fj(\000)p Fo(1)1630 4334 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(z)1840 4346 y Fo(1)1878 4334 y Fu(\)\()p Fq(s;)c(Y)19 b Fu(\))g(+)2219 4221 y Fg(Z)2302 4241 y Fi(t)2265 4409 y Fo(0)2345 4334 y Fq(M)2435 4299 y Fj(\000)p Fo(1)2524 4334 y Fu(\()p Fq(\033)j Fu(+)c Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(z)2886 4346 y Fo(1)2923 4334 y Fu(\)\()p Fq(\033)n(;)c(s;)g (Y)19 b Fu(\))14 b Fq(d\033)3351 4217 y Fg(\025)456 4526 y Fu(with)26 b Fq(c)p Fu(\()p Fq(z)750 4538 y Fo(1)787 4526 y Fu(\))f(c)n(hosen)g(as)g(\(4.21\).)35 b(Our)25 b(goal)f(is)i(to)f(pro)n(v)n(e)f(that)h(if)h Fq(\032)g Fu(is)f(suitably)h(c)n(hosen)e(then)i Fp(G)456 4626 y Fu(has)h(a)h(\014xed)g(p)r(oin)n(t)g(in)g(\006\()p Fq(\032)p Fu(\).)39 b(F)-7 b(or)27 b(that)h(w)n(e)g(will)g(see)g(that)g Fp(G)33 b Fu(is)28 b(w)n(ell)g(de\014ned)g(and)g(that)g(it)h(is)456 4725 y(a)e(con)n(traction)f(in)i(\006\()p Fq(\032)p Fu(\).)605 4825 y(First,)g(w)n(e)f(pro)n(v)n(e)f(that)i Fp(G)33 b Fu(is)27 b(w)n(ell)g(de\014ned.)38 b(Let)27 b Fq(z)2196 4837 y Fo(1)2256 4825 y Fp(2)d Fu(\006\()p Fq(\032)p Fu(\).)37 b(If)28 b Fq(\032)g Fu(is)f(small,)1426 4968 y Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)j Fu(\))24 b Fp(2)f Fq(U)9 b Fu(\()p Fq(\024)1989 4933 y Fj(\006)1989 4991 y Fo(1)2045 4968 y Fq(;)14 b(\024)2130 4980 y Fo(2)2167 4968 y Fq(;)g Fu(3)p Fq(\024)2294 4980 y Fo(3)2331 4968 y Fq(;)g(r)2405 4980 y Fo(0)2442 4968 y Fu(\))456 5116 y(and)33 b(th)n(us,)j(the)e(function)g Fq(b)p Fu(\()p Fq(z)1424 5128 y Fo(1)1461 5116 y Fu(\))g(giv)n(en)f(in)h(\(4.18\))f (is)g(w)n(ell)h(de\014ned.)55 b(Moreo)n(v)n(er,)33 b(it)h(is)g(clear) 456 5216 y(that,)28 b(since)h Fq(M)9 b Fu(\()p Fq(t)18 b Fu(+)h Fq(s)p Fu(\),)29 b Fq(M)1331 5185 y Fj(\000)p Fo(1)1419 5216 y Fu(\()p Fq(t)20 b Fu(+)e Fq(s)p Fu(\))29 b(and)f Fq(z)1885 5228 y Fo(1)1922 5216 y Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))29 b(are)e Fq(C)2428 5185 y Fo(0)2494 5216 y Fu(and)h(analytic)g(in)g(\()p Fq(s;)14 b(Y)19 b Fu(\),)29 b(the)p eop %%Page: 48 56 48 55 bop 456 251 a Fo(48)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(function)g Fq(g)i Fu(de\014ned)e(in)g(\(4.20\))e(is)i(analytic)f(on) 1955 429 y(~)1940 450 y Fq(U)9 b Fu(.)35 b(Therefore,)21 b(b)n(y)h(Lemma)f(4.4,)h(the)g(function)456 556 y Fq(c)p Fu(\()p Fq(z)563 568 y Fo(1)600 556 y Fu(\)\()p Fq(s;)14 b(Y)19 b Fu(\))28 b(is)f(analytic)g(in)1377 535 y(~)1363 556 y Fq(U)9 b Fu(.)37 b(Th)n(us)27 b Fp(G)5 b Fu(\()p Fq(z)1824 568 y Fo(1)1861 556 y Fu(\)\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))28 b(is)g(also)e Fq(C)2510 525 y Fo(0)2575 556 y Fu(and)i(analytic)f(in)h(\()p Fq(s;)14 b(Y)k Fu(\).)605 655 y(No)n(w)34 b(w)n(e)h(pro)n(v)n(e)e(that)i(the)h(prop)r(ert)n(y)d (\(b\))j(holds)f(for)f Fp(G)5 b Fu(\()p Fq(z)2476 667 y Fo(1)2514 655 y Fu(\).)94 b(It)35 b(is)g(clear)f(that,)j(since)456 755 y Fq(z)495 767 y Fo(1)555 755 y Fp(2)23 b Fu(\006)28 b(and)f Fq(X)951 767 y Fo(1)988 755 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))29 b(is)e(2)p Fq(\031)k Fu(-p)r(erio)r(dic)c(in) h Fq(\022)r Fu(,)1201 897 y Fq(b)p Fu(\()p Fq(z)1308 909 y Fo(1)1345 897 y Fu(\)\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s;)g(Y)k Fu(\))23 b(=)g Fq(b)p Fu(\()p Fq(z)2101 909 y Fo(1)2138 897 y Fu(\)\()p Fq(t;)14 b(s)k Fu(+)g(2)p Fq(\031)s(";)c(Y)19 b Fu(\).)456 1040 y(Then)579 1187 y Fp(G)5 b Fu(\()p Fq(z)704 1199 y Fo(1)742 1187 y Fu(\)\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s;)g(Y)k Fu(\))83 b(=)f Fq(M)9 b Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(")g Fu(+)g Fq(s)p Fu(\))2067 1095 y Fg(h)2106 1187 y Fq(M)2196 1153 y Fj(\000)p Fo(1)2285 1187 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(s;)c(Y)19 b Fu(\))1510 1404 y(+)1589 1291 y Fg(Z)1672 1312 y Fi(t)p Fo(+2)p Fi(\031)r(")1635 1480 y Fo(0)1871 1404 y Fq(M)1961 1370 y Fj(\000)p Fo(1)2050 1404 y Fu(\()p Fq(\033)j Fu(+)c Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(z)2412 1416 y Fo(1)2449 1404 y Fu(\)\()p Fq(\033)n(;)c(s;)g(Y)20 b Fu(\))14 b Fq(d\033)2878 1312 y Fg(i)1363 1608 y Fu(=)82 b Fq(M)9 b Fu(\()p Fq(t)19 b Fu(+)f Fq(s)g Fu(+)g(2)p Fq(\031)s(")p Fu(\))2067 1516 y Fg(h)2106 1608 y Fq(M)2196 1574 y Fj(\000)p Fo(1)2285 1608 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(s;)c(Y)19 b Fu(\))1510 1824 y(+)1589 1711 y Fg(Z)1672 1731 y Fi(t)1635 1899 y Fj(\000)p Fo(2)p Fi(\031)r(")1810 1824 y Fq(M)1900 1789 y Fj(\000)p Fo(1)1989 1824 y Fu(\()p Fq(\033)j Fu(+)c(2)p Fq(\031)s(")g Fu(+)g Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(z)2583 1836 y Fo(1)2620 1824 y Fu(\)\()p Fq(\033)n(;)c(s)20 b Fu(+)e(2)p Fq(\031)s(";)c(Y)k Fu(\))c Fq(d\033)3281 1732 y Fg(i)456 2015 y Fu(and)615 2162 y Fp(G)5 b Fu(\()p Fq(z)740 2174 y Fo(1)778 2162 y Fu(\)\()p Fq(t;)14 b(s)19 b Fu(+)f(2)p Fq(\031)s(";)c(Y)k Fu(\))83 b(=)g Fq(M)9 b Fu(\()p Fq(t)18 b Fu(+)g Fq(s)g Fu(+)g(2)p Fq(\031)s(")p Fu(\))2103 2070 y Fg(h)2142 2162 y Fq(M)2232 2128 y Fj(\000)p Fo(1)2321 2162 y Fu(\()p Fq(s)h Fu(+)f(2)p Fq(\031)s(")p Fu(\))p Fq(c)p Fu(\()p Fq(s)g Fu(+)g(2)p Fq(\031)s(";)c(Y)k Fu(\))1547 2377 y(+)1626 2264 y Fg(Z)1708 2285 y Fi(t)1671 2453 y Fo(0)1751 2377 y Fq(M)1841 2343 y Fj(\000)p Fo(1)1930 2377 y Fu(\()p Fq(\033)k Fu(+)c(2)p Fq(\031)s(")g Fu(+)g Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(z)2524 2389 y Fo(1)2561 2377 y Fu(\)\()p Fq(\033)n(;)c(s)20 b Fu(+)e(2)p Fq(\031)s(";)c(Y)k Fu(\))c Fq(d\033)3222 2285 y Fg(i)3262 2377 y Fu(.)456 2566 y(Th)n(us,)27 b Fp(G)5 b Fu(\()p Fq(z)814 2578 y Fo(1)851 2566 y Fu(\)\()p Fq(t;)14 b(s)19 b Fu(+)f(2)p Fq(\031)s(";)c(Y)k Fu(\))24 b(=)e Fp(G)5 b Fu(\()p Fq(z)1625 2578 y Fo(1)1663 2566 y Fu(\)\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s;)g(Y)k Fu(\))28 b(if)g(and)f(only)g(if)667 2710 y Fq(M)757 2676 y Fj(\000)p Fo(1)845 2710 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(s;)14 b(Y)20 b Fu(\))e Fp(\000)g Fq(M)1383 2676 y Fj(\000)p Fo(1)1472 2710 y Fu(\()p Fq(s)h Fu(+)f(2)p Fq(\031)s(")p Fu(\))o Fq(c)p Fu(\()p Fq(s)h Fu(+)f(2)p Fq(\031)s(";)c(Y)k Fu(\))1831 2907 y(=)k Fp(\000)1997 2794 y Fg(Z)2080 2815 y Fo(2)p Fi(\031)r(")2043 2983 y Fo(0)2203 2907 y Fq(M)2293 2873 y Fj(\000)p Fo(1)2382 2907 y Fu(\()p Fq(\033)g Fu(+)c Fq(s)p Fu(\))p Fq(b)p Fu(\()p Fq(z)2744 2919 y Fo(1)2781 2907 y Fu(\)\()p Fq(\033)n(;)c(s;)g (Y)20 b Fu(\))14 b Fq(d\033)s Fu(.)456 3096 y(This)27 b(last)g(equalit)n(y)g(holds)h(b)n(y)f(de\014nition)h(of)f Fq(c)h Fu(in)g(\(4.21\).)605 3196 y(Next)d(w)n(e)f(will)h(see)f(that)h (if)h(w)n(e)e(c)n(ho)r(ose)f Fq(\032)i Fu(in)g(a)f(suitable)h(w)n(a)n (y)-7 b(,)24 b Fp(G)5 b Fu(\(\006\()p Fq(\032)p Fu(\)\))24 b Fp(\032)f Fu(\006\()p Fq(\032)p Fu(\).)36 b(Indeed,)456 3296 y(let)28 b Fq(C)635 3308 y Fi(M)736 3296 y Fu(b)r(e)g(a)f(constan) n(t)g(suc)n(h)h(that)f Fp(k)p Fq(M)9 b Fu(\()p Fq(u)p Fu(\))p Fp(k)p Fu(,)27 b Fp(k)p Fq(M)2088 3265 y Fj(\000)p Fo(1)2176 3296 y Fu(\()p Fq(u)p Fu(\))p Fp(k)c(\024)f Fq(C)2499 3308 y Fi(M)2573 3296 y Fu(.)37 b(W)-7 b(e)28 b(recall)f(that,)1492 3439 y Fq(f)9 b Fu(\()p Fq(s;)14 b(Y)19 b Fu(\))k(=)g Fq(M)1950 3405 y Fj(\000)p Fo(1)2038 3439 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(s;)14 b(Y)19 b Fu(\).)456 3582 y(By)27 b(Lemmas)g(4.1)g(and)g(4.4)g(w)n(e)g (obtain)536 3879 y Fp(k)p Fq(f)9 b Fp(k)21 b(\024)i Fq(C)838 3891 y Fi(D)892 3899 y Fh(0)939 3823 y Fq(e)978 3793 y Fo(2)p Fi(\031)r("=\034)1148 3801 y Fh(2)p 939 3860 246 4 v 996 3936 a Fu(2)p Fq(\031)s(")1194 3879 y Fp(k)p Fq(g)s Fu(\()p Fq(z)1350 3891 y Fo(1)1387 3879 y Fu(\))p Fp(k)f(\024)h Fq(C)1630 3891 y Fi(M)1704 3879 y Fq(C)1763 3891 y Fi(D)1817 3899 y Fh(0)1854 3879 y Fp(k)p Fq(b)p Fu(\()p Fq(z)2003 3891 y Fo(1)2040 3879 y Fu(\))p Fp(k)f(\024)h Fq(C)2283 3891 y Fi(M)2357 3879 y Fq(C)2416 3891 y Fi(D)2470 3899 y Fh(0)2507 3879 y Fu([)p Fq(K)6 b Fp(j)p Fq(\026)p Fp(j)p Fq(")2742 3845 y Fi(q)2778 3879 y Fp(k)p Fq(z)2859 3891 y Fo(1)2896 3879 y Fp(k)2938 3845 y Fo(2)2993 3879 y Fu(+)18 b Fq(K)6 b(\016)3193 3843 y Fi(k)q Fj(\000)p Fo(1)3190 3901 y(0)3318 3879 y Fu(].)456 4045 y(Th)n(us)635 4187 y Fp(kG)f Fu(\()p Fq(z)802 4199 y Fo(1)839 4187 y Fu(\))p Fp(k)83 b(\024)g Fq(C)1209 4153 y Fo(2)1203 4208 y Fi(M)1277 4187 y Fq(C)1336 4199 y Fi(D)1390 4207 y Fh(0)1427 4187 y Fu(\()p Fq(K)6 b Fp(j)p Fq(\026)p Fp(j)p Fq(")1671 4153 y Fi(q)1707 4187 y Fq(\032)1750 4153 y Fo(2)1806 4187 y Fu(+)18 b Fq(K)6 b(\016)2006 4152 y Fi(k)q Fj(\000)p Fo(1)2003 4209 y(0)2131 4187 y Fu(\))19 b(+)f Fq(C)2330 4153 y Fo(2)2324 4208 y Fi(M)2398 4187 y Fu(4)p Fq(\031)s(")p Fu(\()p Fq(K)6 b Fp(j)p Fq(\026)p Fp(j)p Fq(")2773 4153 y Fi(q)2809 4187 y Fq(\032)2852 4153 y Fo(2)2907 4187 y Fu(+)18 b Fq(K)6 b(\016)3107 4152 y Fi(k)q Fj(\000)p Fo(1)3104 4209 y(0)3233 4187 y Fu(\))996 4312 y Fp(\024)83 b Fq(\032)456 4455 y Fu(if)28 b Fq(\032)23 b Fu(=)f(2)p Fq(C)792 4425 y Fo(2)786 4478 y Fi(M)860 4455 y Fq(K)6 b Fu(\()p Fq(C)1028 4467 y Fi(D)1082 4475 y Fh(0)1137 4455 y Fu(+)18 b(4)p Fq(\031)s(")p Fu(\))p Fq(\016)1423 4419 y Fi(k)q Fj(\000)p Fo(1)1420 4477 y(0)1577 4455 y Fu(and)27 b Fp(j)p Fq(\026)p Fp(j)p Fq(")1873 4425 y Fi(q)1937 4455 y Fu(is)h(small)f(enough.)605 4555 y(Therefore,)f Fp(G)5 b Fu(\()p Fq(\033)s Fu(\))25 b Fp(2)e Fu(\006\()p Fq(\032)p Fu(\),)28 b(and)g(the)g(op)r(erator)e Fp(G)33 b Fu(is)27 b(w)n(ell)h(de\014ned.)605 4654 y(Finally)h(w)n(e)g (pro)n(v)n(e)f(that)i Fp(G)35 b Fu(is)29 b(a)g(con)n(traction.)41 b(Let)29 b Fq(z)2324 4666 y Fo(1)2390 4654 y Fu(and)h Fq(z)2593 4666 y Fo(2)2659 4654 y Fu(b)r(e)g(t)n(w)n(o)e(functions)i (that)456 4754 y(b)r(elong)d(to)g(\006\()p Fq(\032)p Fu(\):)731 4921 y Fp(j)p Fu(\()p Fp(G)5 b Fu(\()p Fq(z)911 4933 y Fo(1)948 4921 y Fu(\))19 b Fp(\000)f(G)5 b Fu(\()p Fq(z)1207 4933 y Fo(2)1245 4921 y Fu(\)\)\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))p Fp(j)k(\024)1717 4826 y Fg(\014)1717 4876 y(\014)1717 4925 y(\014)1744 4921 y Fq(M)9 b Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))2069 4829 y Fg(h)2108 4921 y Fq(M)2198 4887 y Fj(\000)p Fo(1)2287 4921 y Fu(\()p Fq(s)p Fu(\))2390 4829 y Fg(\020)2440 4921 y Fq(c)p Fu(\()p Fq(z)2547 4933 y Fo(1)2584 4921 y Fu(\)\()p Fq(s;)c(Y)19 b Fu(\))g Fp(\000)f Fq(c)p Fu(\()p Fq(z)3032 4933 y Fo(2)3069 4921 y Fu(\)\()p Fq(s;)c(Y)19 b Fu(\))3308 4829 y Fg(\021)456 4921 y Fu(\(4.22\))1333 5137 y(+)1412 5024 y Fg(Z)1494 5044 y Fi(t)1457 5212 y Fo(0)1537 5137 y Fq(M)1627 5102 y Fj(\000)p Fo(1)1716 5137 y Fu(\()p Fq(\033)j Fu(+)c Fq(s)p Fu(\))1971 5045 y Fg(\020)2021 5137 y Fq(b)p Fu(\()p Fq(z)2128 5149 y Fo(1)2165 5137 y Fu(\)\()p Fq(\033)n(;)c(s;)g(Y)20 b Fu(\))e Fp(\000)g Fq(b)p Fu(\()p Fq(z)2695 5149 y Fo(2)2732 5137 y Fu(\)\()p Fq(\033)n(;)c(s;)g(Y)20 b Fu(\))3054 5045 y Fg(\021)3118 5137 y Fq(d\033)3211 5045 y Fg(i)3250 5016 y(\014)3250 5066 y(\014)3250 5116 y(\014)3250 5166 y(\014)3292 5137 y Fu(.)p eop %%Page: 49 57 49 56 bop 1006 251 a Fo(4.5.)30 b(FLO)n(W)f(BO)n(X)g(COORDINA)-5 b(TES)29 b(IN)g(A)g(COMPLEX)g(DOMAIN)484 b(49)456 450 y Fu(W)-7 b(e)28 b(observ)n(e)d(that,)j(since)g(the)g(op)r(erator)e Fp(4)1854 415 y Fj(\000)p Fo(1)1854 472 y(2)p Fi(\031)r(")1990 450 y Fu(is)h(linear)1016 632 y Fq(M)1106 597 y Fj(\000)p Fo(1)1195 632 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(z)1405 644 y Fo(1)1442 632 y Fu(\))19 b Fp(\000)f Fq(M)1666 597 y Fj(\000)p Fo(1)1755 632 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(z)1965 644 y Fo(2)2002 632 y Fu(\))24 b(=)e Fp(4)2220 596 y Fj(\000)p Fo(1)2220 654 y(2)p Fi(\031)r(")2329 632 y Fu(\()p Fq(g)s Fu(\()p Fq(z)2475 644 y Fo(1)2512 632 y Fu(\))c Fp(\000)h Fq(g)s Fu(\()p Fq(z)2760 644 y Fo(2)2796 632 y Fu(\)\).)456 810 y(By)27 b(Lemma)g(4.4)g(w)n(e)g(ha)n(v)n(e)955 1042 y Fp(k)p Fq(M)1087 1008 y Fj(\000)p Fo(1)1176 1042 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(z)1386 1054 y Fo(1)1423 1042 y Fu(\))19 b Fp(\000)f Fq(M)1647 1008 y Fj(\000)p Fo(1)1736 1042 y Fu(\()p Fq(s)p Fu(\))p Fq(c)p Fu(\()p Fq(z)1946 1054 y Fo(2)1983 1042 y Fu(\))p Fp(k)23 b(\024)g Fq(C)2227 1054 y Fi(D)2281 1062 y Fh(0)2328 986 y Fq(e)2367 956 y Fo(2)p Fi(\031)r("=\034)2537 964 y Fh(2)p 2328 1023 246 4 v 2385 1099 a Fu(2)p Fq(\031)s(")2583 1042 y Fp(k)p Fq(g)s Fu(\()p Fq(z)2739 1054 y Fo(1)2775 1042 y Fu(\))c Fp(\000)f Fq(g)s Fu(\()p Fq(z)3023 1054 y Fo(2)3060 1042 y Fu(\))p Fp(k)p Fu(.)-2701 b(\(4.23\))456 1248 y(No)n(w)26 b(w)n(e)g(b)r(ound)h Fp(k)p Fq(b)p Fu(\()p Fq(z)1169 1260 y Fo(1)1206 1248 y Fu(\))17 b Fp(\000)f Fq(b)p Fu(\()p Fq(z)1443 1260 y Fo(2)1480 1248 y Fu(\))p Fp(k)p Fu(.)36 b(Un)n(til)28 b(the)f(end)g(of)f(the)h(pro)r(of,)g Fq(z)2637 1260 y Fo(0)2674 1248 y Fu(,)g Fq(z)2763 1260 y Fo(1)2826 1248 y Fu(and)g Fq(z)3026 1260 y Fo(2)3090 1248 y Fu(will)g(stand)456 1347 y(for)g Fq(z)622 1359 y Fo(0)659 1347 y Fu(\()p Fq(t)18 b Fu(+)g Fq(s;)c(Y)19 b Fu(\),)28 b Fq(z)1087 1359 y Fo(1)1124 1347 y Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))28 b(and)g Fq(z)1626 1359 y Fo(2)1662 1347 y Fu(\()p Fq(t;)14 b(s;)g(Y)19 b Fu(\))28 b(resp)r(ectiv)n(ely)-7 b(.)605 1447 y(It)28 b(is)f(clear)g(that)h(w)n (e)f(can)g(write)h Fq(b)p Fu(\()p Fq(z)1751 1459 y Fo(1)1788 1447 y Fu(\))18 b Fp(\000)g Fq(b)p Fu(\()p Fq(z)2028 1459 y Fo(2)2065 1447 y Fu(\))28 b(as)481 1664 y Fq(b)p Fu(\()p Fq(z)588 1676 y Fo(1)625 1664 y Fu(\))19 b Fp(\000)f Fq(b)p Fu(\()p Fq(z)866 1676 y Fo(2)902 1664 y Fu(\))84 b(=)1217 1607 y(1)p 1175 1645 126 4 v 1175 1721 a Fq(\026")1264 1697 y Fi(q)1311 1664 y Fu([)p Fq(X)1403 1676 y Fo(0)1440 1664 y Fu(\()p Fq(z)1511 1676 y Fo(0)1566 1664 y Fu(+)18 b Fq(\026")1738 1629 y Fi(q)1775 1664 y Fq(z)1814 1676 y Fo(1)1851 1664 y Fu(\))g Fp(\000)g Fq(X)2053 1676 y Fo(0)2090 1664 y Fu(\()p Fq(z)2161 1676 y Fo(0)2217 1664 y Fu(+)g Fq(\026")2389 1629 y Fi(q)2425 1664 y Fq(z)2464 1676 y Fo(2)2501 1664 y Fu(\))h Fp(\000)f Fq(\026")2724 1629 y Fi(q)2760 1664 y Fq(D)r(X)2900 1676 y Fo(0)2938 1664 y Fu(\()p Fq(z)3009 1676 y Fo(0)3046 1664 y Fu(\)\()p Fq(z)3149 1676 y Fo(1)3205 1664 y Fp(\000)g Fq(z)3327 1676 y Fo(2)3363 1664 y Fu(\)])1165 1832 y(+)p Fq(X)1299 1844 y Fo(1)1336 1832 y Fu(\()p Fq(z)1407 1844 y Fo(0)1463 1832 y Fu(+)g Fq(\026")1635 1798 y Fi(q)1671 1832 y Fq(z)1710 1844 y Fo(1)1747 1832 y Fq(;)c(t=")p Fu(\))k Fp(\000)g Fq(X)2097 1844 y Fo(1)2134 1832 y Fu(\()p Fq(z)2205 1844 y Fo(0)2260 1832 y Fu(+)g Fq(\026")2432 1798 y Fi(q)2469 1832 y Fq(z)2508 1844 y Fo(2)2545 1832 y Fq(;)c(t=")p Fu(\))1018 1996 y(=)1217 1940 y(1)p 1175 1977 V 1175 2053 a Fq(\026")1264 2029 y Fi(q)1311 1996 y Fu(\()p Fq(X)1412 2008 y Fo(0)1449 1996 y Fu(\()p Fq(z)1520 2008 y Fo(0)1576 1996 y Fu(+)k Fq(\026")1748 1962 y Fi(q)1784 1996 y Fq(z)1823 2008 y Fo(1)1860 1996 y Fu(\))h Fp(\000)f Fq(X)2063 2008 y Fo(0)2100 1996 y Fu(\()p Fq(z)2171 2008 y Fo(0)2226 1996 y Fu(+)g Fq(\026")2398 1962 y Fi(q)2435 1996 y Fq(z)2474 2008 y Fo(2)2511 1996 y Fu(\)\))1165 2164 y Fp(\000)p Fq(D)r(X)1370 2176 y Fo(0)1407 2164 y Fu(\()p Fq(z)1478 2176 y Fo(0)1534 2164 y Fu(+)g Fq(\026")1706 2130 y Fi(q)1742 2164 y Fq(z)1781 2176 y Fo(1)1818 2164 y Fu(\)\()p Fq(z)1921 2176 y Fo(1)1977 2164 y Fp(\000)g Fq(z)2099 2176 y Fo(2)2136 2164 y Fu(\))1165 2289 y(+)p Fq(D)r(X)1370 2301 y Fo(0)1407 2289 y Fu(\()p Fq(z)1478 2301 y Fo(0)1534 2289 y Fu(+)g Fq(\026")1706 2255 y Fi(q)1742 2289 y Fq(z)1781 2301 y Fo(1)1818 2289 y Fu(\)\()p Fq(z)1921 2301 y Fo(1)1977 2289 y Fp(\000)g Fq(z)2099 2301 y Fo(2)2136 2289 y Fu(\))g Fp(\000)g Fq(D)r(X)2409 2301 y Fo(0)2446 2289 y Fu(\()p Fq(z)2517 2301 y Fo(0)2555 2289 y Fu(\)\()p Fq(z)2658 2301 y Fo(1)2713 2289 y Fp(\000)h Fq(z)2836 2301 y Fo(2)2872 2289 y Fu(\))1165 2413 y(+)p Fq(X)1299 2425 y Fo(1)1336 2413 y Fu(\()p Fq(z)1407 2425 y Fo(0)1463 2413 y Fu(+)f Fq(\026")1635 2379 y Fi(q)1671 2413 y Fq(z)1710 2425 y Fo(1)1747 2413 y Fq(;)c(t=")p Fu(\))k Fp(\000)g Fq(X)2097 2425 y Fo(1)2134 2413 y Fu(\()p Fq(z)2205 2425 y Fo(0)2260 2413 y Fu(+)g Fq(\026")2432 2379 y Fi(q)2469 2413 y Fq(z)2508 2425 y Fo(2)2545 2413 y Fq(;)c(t=")p Fu(\).)456 2591 y(Using)27 b(the)h(b)r(ounds)g(of)f(Lemma)h(4.1)e(w)n (e)i(get)886 2769 y Fp(k)p Fq(b)p Fu(\()p Fq(z)1035 2781 y Fo(1)1071 2769 y Fu(\))19 b Fp(\000)f Fq(b)p Fu(\()p Fq(z)1312 2781 y Fo(2)1349 2769 y Fu(\))p Fp(k)83 b(\024)f(j)p Fq(\026)p Fp(j)p Fq(")1788 2735 y Fi(q)1825 2769 y Fq(K)6 b Fu(\()p Fp(k)p Fq(z)2015 2781 y Fo(1)2069 2769 y Fp(\000)18 b Fq(z)2191 2781 y Fo(2)2228 2769 y Fp(k)2270 2735 y Fo(2)2325 2769 y Fu(+)g Fp(k)p Fq(z)2489 2781 y Fo(1)2526 2769 y Fp(k)c(k)p Fq(z)2663 2781 y Fo(1)2717 2769 y Fp(\000)k Fq(z)2839 2781 y Fo(2)2876 2769 y Fp(k)p Fu(\))1653 2895 y(+)p Fp(j)p Fq(\026)p Fp(j)p Fq(")1853 2860 y Fi(q)1889 2895 y Fq(K)6 b Fu(\()p Fp(k)p Fq(z)2079 2907 y Fo(1)2134 2895 y Fp(\000)18 b Fq(z)2256 2907 y Fo(2)2293 2895 y Fp(k)2335 2860 y Fo(2)2390 2895 y Fu(+)g Fp(k)p Fq(z)2554 2907 y Fo(1)2590 2895 y Fp(k)c(k)p Fq(z)2727 2907 y Fo(1)2781 2895 y Fp(\000)k Fq(z)2903 2907 y Fo(2)2940 2895 y Fp(k)p Fu(\))-2558 b(\(4.24\))1506 3019 y(=)82 b Fq(K)6 b Fp(j)p Fq(\026)p Fp(j)p Fq(")1865 2985 y Fi(q)1901 3019 y Fp(k)p Fq(z)1982 3031 y Fo(1)2037 3019 y Fp(\000)18 b Fq(z)2159 3031 y Fo(2)2196 3019 y Fp(k)p Fu(.)456 3197 y(Moreo)n(v)n(er,)25 b(it)j(is)f(clear)g(that)1208 3374 y Fp(k)p Fq(g)s Fu(\()p Fq(z)1364 3386 y Fo(1)1400 3374 y Fu(\))19 b Fp(\000)f Fq(g)s Fu(\()p Fq(z)1648 3386 y Fo(2)1684 3374 y Fu(\))p Fp(k)23 b(\024)g Fq(C)1928 3386 y Fi(M)2002 3374 y Fu(2)p Fq(\031)s(")p Fp(k)p Fq(b)p Fu(\()p Fq(z)2282 3386 y Fo(1)2318 3374 y Fu(\))c Fp(\000)f Fq(b)p Fu(\()p Fq(z)2559 3386 y Fo(2)2595 3374 y Fu(\))p Fp(k)p Fu(.)456 3557 y(Then,)27 b(using)h(\(4.23\))f(and)g(\(4.24\))g(in)h(\(4.22\),)f(w)n (e)g(obtain)547 3794 y Fp(kG)5 b Fu(\()p Fq(z)714 3806 y Fo(1)751 3794 y Fu(\))19 b Fp(\000)f(G)5 b Fu(\()p Fq(z)1010 3806 y Fo(2)1048 3794 y Fu(\))p Fp(k)82 b(\024)h Fq(C)1411 3806 y Fi(M)1485 3794 y Fq(C)1544 3806 y Fi(D)1598 3814 y Fh(0)1645 3738 y Fq(e)1684 3708 y Fo(2)p Fi(\031)r("=\034)1854 3716 y Fh(2)p 1645 3775 246 4 v 1703 3851 a Fu(2)p Fq(\031)s(")1900 3794 y Fp(k)p Fq(g)s Fu(\()p Fq(z)2056 3806 y Fo(1)2093 3794 y Fu(\))18 b Fp(\000)g Fq(g)s Fu(\()p Fq(z)2340 3806 y Fo(2)2377 3794 y Fu(\))p Fp(k)g Fu(+)g(4)p Fq(\031)s("C)2748 3760 y Fo(2)2742 3815 y Fi(M)2816 3794 y Fp(k)p Fq(b)p Fu(\()p Fq(z)2965 3806 y Fo(1)3001 3794 y Fu(\))h Fp(\000)f Fq(b)p Fu(\()p Fq(z)3242 3806 y Fo(2)3279 3794 y Fu(\))p Fp(k)1204 3946 y(\024)83 b Fq(K)6 b Fu(\()p Fq(D)1530 3958 y Fo(0)1567 3946 y Fq(;)14 b(M)9 b Fu(\))p Fp(k)p Fq(b)p Fu(\()p Fq(z)1875 3958 y Fo(1)1911 3946 y Fu(\))19 b Fp(\000)f Fq(b)p Fu(\()p Fq(z)2152 3958 y Fo(2)2188 3946 y Fu(\))p Fp(k)1204 4071 y(\024)83 b Fq(K)6 b Fu(\()p Fq(D)1530 4083 y Fo(0)1567 4071 y Fq(;)14 b(M)9 b Fu(\))p Fp(j)p Fq(\026)p Fp(j)p Fq(")1861 4037 y Fi(q)1897 4071 y Fp(k)p Fq(z)1978 4083 y Fo(1)2033 4071 y Fp(\000)18 b Fq(z)2155 4083 y Fo(2)2192 4071 y Fp(k)1204 4235 y(\024)1362 4178 y Fu(1)p 1362 4216 42 4 v 1362 4292 a(2)1413 4235 y Fp(k)p Fq(z)1494 4247 y Fo(1)1549 4235 y Fp(\000)g Fq(z)1671 4247 y Fo(2)1708 4235 y Fp(k)456 4440 y Fu(if)28 b Fp(j)p Fq(\026)p Fp(j)p Fq(")667 4410 y Fi(q)731 4440 y Fu(is)f(small)h(enough.)605 4540 y(Therefore,)i(since)g Fp(G)36 b Fu(is)30 b(a)g(con)n(traction,)g(b)n(y)g(the)g(\014xed)h(p)r (oin)n(t)f(theorem,)h(there)f(exists)g(a)456 4639 y(unique)e Fq(z)765 4651 y Fo(1)826 4639 y Fp(2)d Fu(\006\()p Fq(\032)p Fu(\))k(suc)n(h)f(that)h Fq(z)1510 4651 y Fo(0)1546 4639 y Fu(\()p Fq(t)20 b Fu(+)e Fq(s;)c(Y)19 b Fu(\))g(+)f Fq(\026")2077 4609 y Fi(q)2114 4639 y Fq(z)2153 4651 y Fo(1)2190 4639 y Fu(\()p Fq(t;)c(s;)g(Y)k Fu(\))29 b(satis\014es)f(the)g(conclusions)g(of)456 4739 y(the)g(prop)r (osition,)e(except)i(that)g Fq(z)j Fu(is)c Fq(C)1717 4709 y Fo(0)1755 4739 y Fu(.)37 b(Since)27 b Fq(z)k Fu(satis\014es)c (the)h(equation)1055 4970 y Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))23 b(=)g Fq(z)t Fu(\(0)p Fq(;)14 b(s;)g(Y)j Fu(\))i(+)1911 4857 y Fg(Z)1994 4877 y Fi(t)1957 5045 y Fo(0)2037 4970 y Fq(X)2106 4982 y Fi(\026)2150 4970 y Fu(\()p Fq(z)t Fu(\()p Fq(\033)n(;)14 b(s;)g(Y)19 b Fu(\))p Fq(;)14 b(\033)s(=")p Fu(\))g Fq(d\033)s Fu(,)456 5199 y(it)28 b(is)f Fq(C)687 5169 y Fo(1)725 5199 y Fu(,)g(and)h(the)g(prop)r (osition)e(holds.)1651 b Fe(\003)p eop %%Page: 50 58 50 57 bop 456 251 a Fo(50)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fk(4.5.4.)46 b(Pro)s(of)25 b(of)g(Theorem)g(4.1.)40 b Fu(The)22 b(pro)r(of)f(of)h(Theorem)g(4.1)f(has)h(t)n(w)n(o)f(parts.)34 b(The)456 550 y(\014rst)23 b(one)g(consists)f(on)h(constructing)g (\015o)n(w)g(b)r(o)n(x)g(co)r(ordinates)f(in)h Fq(D)2577 562 y Fo(0)2624 550 y Fp(\002)10 b Fq(W)35 b Fu(using)23 b(Prop)r(osition)456 649 y(4.1.)46 b(The)32 b(c)n(hange)e(of)h(co)r (ordinates)e(so)i(obtained)g(ma)n(y)f(b)r(e)i(non)f(canonical.)46 b(In)32 b(the)f(second)456 749 y(step)23 b(w)n(e)h(mo)r(dify)g(these)g (\015o)n(w)f(b)r(o)n(x)g(co)r(ordinates)f(in)i(suc)n(h)g(w)n(a)n(y)e (that)i(they)g(b)r(ecome)g(canonical.)605 849 y(W)-7 b(e)28 b(b)r(egin)g(b)n(y)f(de\014ning)1427 985 y Fq(w)r Fu(\()p Fq(u;)14 b(\022)r(;)g(Y)19 b Fu(\))24 b(=)e Fq(z)t Fu(\()p Fq("\022)r(;)14 b(u)k Fp(\000)g Fq("\022)r(;)c(Y)k Fu(\).)456 1131 y(Note)37 b(that)i Fq(w)h Fu(is)e Fq(C)1115 1101 y Fo(1)1190 1131 y Fu(and)g(analytic)f(with)h(resp)r(ect)g(to)f (its)h(\014rst)g(and)g(third)f(v)-5 b(ariables)37 b(for)456 1231 y(\()p Fq(u;)14 b(Y)k Fu(\))29 b Fp(2)g Fq(D)853 1243 y Fo(0)890 1231 y Fu(\()p Fq(\024)970 1195 y Fj(\006)970 1253 y Fo(1)1047 1231 y Fp(\000)21 b Fq(\034)5 b(;)14 b(\024)1259 1243 y Fo(2)1296 1231 y Fq(;)g(\024)1381 1243 y Fo(3)1418 1231 y Fu(\))21 b Fp(\002)f Fq(W)12 b Fu(\()p Fq(r)1715 1243 y Fo(1)1753 1231 y Fu(\).)48 b(Moreo)n(v)n(er,)29 b(since)i(the)h(solutions)e(of)h(the)h(system)456 1330 y(\(4.3\))c(satisfy)g(that)h Fq(z)t Fu(\()p Fq(t)19 b Fu(+)g(2)p Fq(\031)s(";)14 b(s;)g(Y)k Fu(\))25 b(=)f Fq(z)t Fu(\()p Fq(t;)14 b(s)19 b Fu(+)g(2)p Fq(\031)s(";)14 b(Y)k Fu(\),)29 b(w)n(e)g(ha)n(v)n(e)e(that)i Fq(w)j Fu(is)c(2)p Fq(\031)s Fu(-p)r(erio)r(dic)456 1430 y(resp)r(ect)g(to)g (its)g(second)g(v)-5 b(ariable.)37 b(This)29 b(is)f(a)f(v)n(ery)g(imp)r (ortan)n(t)h(prop)r(ert)n(y)f(b)r(ecause)h(it)h(allo)n(ws)456 1529 y(us)22 b(to)g(extend)g(the)h(domain)f(of)g Fq(w)j Fu(with)e(resp)r(ect)f(to)g(the)g Fq(\022)j Fu(v)-5 b(ariable,)22 b(that)h(is,)g(w)n(e)f(can)g(consider)456 1629 y Fq(w)30 b Fu(in)e(the)g(domain)1320 1765 y Fq(D)1389 1777 y Fo(0)1426 1765 y Fu(\()p Fq(\024)1506 1729 y Fj(\006)1506 1787 y Fo(1)1581 1765 y Fp(\000)18 b Fq(\034)5 b(;)14 b(\024)1790 1777 y Fo(2)1827 1765 y Fq(;)g Fu(2)p Fq(\024)1954 1777 y Fo(3)1991 1765 y Fu(\))k Fp(\002)g Fl(R)25 b Fp(\002)18 b Fq(W)12 b Fu(\(2)p Fq(r)2487 1777 y Fo(1)2524 1765 y Fu(\).)456 1901 y(W)-7 b(e)28 b(ha)n(v)n(e)e(that)1571 2037 y(\()p Fq(w)r Fu(\()p Fq(t)19 b Fu(+)f Fq(s;)c(t=";)g(Y)k Fu(\))p Fq(;)c(t=")p Fu(\))-1874 b(\(4.25\))456 2178 y(is)27 b(a)g(new)h(parameterization)e(of)h(the)h(solutions)f(of)g (\(4.3\).)37 b(Indeed,)590 2353 y Fq(@)634 2365 y Fi(t)663 2353 y Fu([)p Fq(w)r Fu(\()p Fq(t)20 b Fu(+)e Fq(s;)c(t=";)g(Y)j Fu(\)])84 b(=)e Fq(@)1531 2365 y Fi(u)1575 2353 y Fq(w)r Fu(\()p Fq(t)19 b Fu(+)f Fq(s;)c(t=";)g(Y)k Fu(\))g(+)2233 2297 y(1)p 2233 2334 42 4 v 2234 2410 a Fq(")2285 2353 y(@)2329 2365 y Fi(\022)2366 2353 y Fq(w)r Fu(\()p Fq(t)i Fu(+)e Fq(s;)c(t=";)g(Y)j Fu(\))1340 2506 y(=)82 b Fq(@)1531 2518 y Fi(s)1567 2506 y Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)k Fu(\))g(+)g([)p Fq(@)2051 2518 y Fi(t)2081 2506 y Fq(z)t Fu(\()p Fq(t;)c(s;)g(Y)k Fu(\))p Fq(")g Fu(+)g Fq(@)2581 2518 y Fi(s)2617 2506 y Fq(z)t Fu(\()p Fq(t;)c(s;)g(Y)k Fu(\)\()p Fp(\000)p Fq(")p Fu(\)]\(1)p Fq(=")p Fu(\))1340 2630 y(=)82 b Fq(@)1531 2642 y Fi(t)1561 2630 y Fq(z)t Fu(\()p Fq(t;)14 b(s;)g(Y)j Fu(\).)456 2766 y(W)-7 b(e)28 b(observ)n(e)f(that,)h(if)h Fq(\026)24 b Fu(=)f(0,)28 b Fq(w)r Fu(\()p Fq(u;)14 b(\022)r(;)g(Y)19 b Fu(\))25 b Fp(\021)e Fq(w)1958 2778 y Fo(0)1996 2766 y Fu(\()p Fq(u;)14 b(Y)k Fu(\))24 b(=)g Fq(z)2363 2778 y Fo(0)2400 2766 y Fu(\()p Fq(u;)14 b(Y)k Fu(\).)39 b(W)-7 b(e)28 b(denote)g Fq(z)3127 2778 y Fo(1)3164 2766 y Fu(\()p Fq("\022)r(;)14 b(u)19 b Fp(\000)456 2866 y Fq("\022)r(;)14 b(Y)k Fu(\))28 b(b)n(y)f Fq(w)873 2878 y Fo(1)911 2866 y Fu(\()p Fq(u;)14 b(\022)r(;)g(Y)19 b Fu(\))28 b(and)f(hence)1241 3002 y Fq(w)r Fu(\()p Fq(u;)14 b(\022)r(;)g(Y)19 b Fu(\))k(=)g Fq(w)1766 3014 y Fo(0)1803 3002 y Fu(\()p Fq(u;)14 b(Y)19 b Fu(\))f(+)h Fq(\026")2210 2968 y Fi(q)2246 3002 y Fq(w)2305 3014 y Fo(1)2343 3002 y Fu(\()p Fq(u;)14 b(\022)r(;)g(Y)k Fu(\).)456 3138 y(In)37 b(the)h(previous)e(argumen)n(ts)g(w)n(e)h(ha)n (v)n(e)f(not)h(men)n(tioned)h(explicitly)f(the)h(dep)r(endence)g(on)456 3238 y(the)28 b(parameters,)f(but)i(it)g(is)f(clear)g(that)g(the)h(con) n(tin)n(uit)n(y)f(on)g Fq(")g Fu(and)h(the)f(analyticit)n(y)g(on)g Fq(\026)h Fu(is)456 3338 y(main)n(tained,)e(and)g(in)h(particular)e Fq(w)31 b Fu(is)c Fq(C)1789 3307 y Fo(0)1855 3338 y Fu(in)g Fq(")h Fu(and)f(analytic)g(in)h Fq(\026)p Fu(.)605 3488 y Ft(Lemma)34 b Fu(4.5)p Ft(.)41 b Fs(L)l(et)32 b Fq(\024)1286 3452 y Fj(\000)1286 3510 y Fo(1)1369 3488 y Fq(>)26 b Fu(2)p Fq(\024)1550 3500 y Fo(0)1587 3488 y Fs(.)45 b(Under)32 b(the)g(hyp)l(otheses)i Ff(H1,H2)p Fs(,)f(for)g(any)f Fq(\024)3141 3452 y Fo(+)3141 3510 y(1)3223 3488 y Fq(>)27 b(\024)3363 3452 y Fj(\000)3363 3510 y Fo(1)3419 3488 y Fs(,)456 3590 y Fq(\024)504 3602 y Fo(2)568 3590 y Fq(>)g Fu(0)p Fs(,)33 b Fq(\024)808 3602 y Fo(3)873 3590 y Fq(>)27 b Fu(0)32 b Fs(such)g(that)g Fq(\024)1450 3554 y Fj(\000)1450 3612 y Fo(1)1527 3590 y Fp(\000)19 b Fu(3)p Fq(\024)1701 3602 y Fo(3)1766 3590 y Fq(>)27 b(\024)1906 3602 y Fo(0)1975 3590 y Fs(ther)l(e)32 b(exists)g Fq(r)f(>)c Fu(0)32 b Fs(smal)t(l)h(enough)f(and)h(two)456 3692 y(unique)c (functions)h Fp(T)51 b Fs(and)30 b Fp(Y)37 b Fs(de\014ne)l(d)30 b(in)g Fq(U)9 b Fu(\()p Fq(\024)1961 3656 y Fj(\006)1961 3714 y Fo(1)2017 3692 y Fq(;)14 b(\024)2102 3704 y Fo(2)2139 3692 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))30 b Fs(such)g(that)1331 3828 y Fq(w)r Fu(\()p Fp(T)22 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g(\022)r(;)g Fp(Y)7 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\)\))26 b(=)d(\()p Fq(x;)14 b(v)s Fu(\))p Fs(.)-2112 b Fu(\(4.26\))456 3974 y Fs(The)32 b(functions)f Fp(T)53 b Fs(and)32 b Fp(Y)39 b Fs(ar)l(e)32 b Fq(C)1552 3944 y Fo(1)1589 3974 y Fs(,)g(analytic)h(in)f(the)f Fu(\()p Fq(x;)14 b(v)s Fu(\))33 b Fs(variables)g(and)f Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)h(in)456 4074 y Fq(\022)r Fs(.)39 b(Mor)l(e)l(over)871 4210 y Fp(T)21 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fp(T)1364 4222 y Fo(0)1401 4210 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)g Fq(O)r Fu(\()p Fq(\026")1882 4175 y Fi(q)1919 4210 y Fu(\))p Fs(,)179 b Fp(Y)7 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fp(Y)2654 4222 y Fo(0)2691 4210 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")3171 4175 y Fi(q)3209 4210 y Fu(\))456 4351 y Fs(wher)l(e)30 b Fp(T)735 4363 y Fo(0)802 4351 y Fs(and)g Fp(Y)1018 4363 y Fo(0)1086 4351 y Fs(ar)l(e)g(de\014ne)l(d)g(in)g(\(4.6\))h(and)f (\(4.7\).)605 4501 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)28 b(de\014ne)g(the)g(function)1145 4637 y Fq(G)p Fu(\()p Fq(S;)14 b(Y)5 b(;)14 b(x;)g(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))23 b(=)g Fq(w)r Fu(\()p Fq(S;)14 b(\022)r(;)g(Y)5 b(;)14 b(\026;)g(")p Fu(\))19 b Fp(\000)f Fu(\()p Fq(x;)c(v)s Fu(\).)456 4773 y(on)27 b(the)h(set)928 4909 y Fq(D)997 4921 y Fo(0)1034 4909 y Fu(\()p Fq(\024)1114 4874 y Fj(\006)1114 4931 y Fo(1)1189 4909 y Fp(\000)18 b Fq(\034)5 b(;)14 b(\024)1398 4921 y Fo(2)1435 4909 y Fq(;)g Fu(2)p Fq(\024)1562 4921 y Fo(3)1599 4909 y Fu(\))k Fp(\002)g Fq(W)12 b Fu(\(2)p Fq(r)1933 4921 y Fo(1)1971 4909 y Fu(\))18 b Fp(\002)g Fq(U)2161 4921 y Fo(0)2198 4909 y Fu(\()p Fq(\024)2278 4874 y Fj(\006)2278 4931 y Fo(1)2335 4909 y Fq(;)c(\024)2420 4921 y Fo(2)2457 4909 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))k Fp(\002)g Fl(R)25 b Fp(\002)18 b Fq(P)456 5046 y Fu(where)1099 5182 y Fq(P)35 b Fu(=)23 b Fp(f)p Fu(\()p Fq(\026;)14 b(")p Fu(\))23 b Fp(2)g Fl(C)39 b Fp(\002)18 b Fl(R)29 b Fu(:)23 b Fp(j)p Fq(\026)p Fp(j)h Fq(<)e(\026)2155 5194 y Fo(0)2220 5182 y Fu(and)28 b(0)22 b Fq(<)h(")g(<)f(")2722 5194 y Fo(0)2759 5182 y Fp(g)p eop %%Page: 51 59 51 58 bop 1006 251 a Fo(4.5.)30 b(FLO)n(W)f(BO)n(X)g(COORDINA)-5 b(TES)29 b(IN)g(A)g(COMPLEX)g(DOMAIN)484 b(51)456 450 y Fu(with)26 b Fq(\026)693 462 y Fo(0)757 450 y Fu(and)g Fq(")956 462 y Fo(0)1019 450 y Fu(small)g(enough.)36 b(Here)25 b(w)n(e)h(put)h(explicitly)f(the)h(dep)r(endence)f(on)g Fq(\026)h Fu(and)f Fq(")g Fu(of)456 550 y(the)i(solutions.)605 649 y(By)f(the)h(de\014nitions)g(of)g Fp(T)1420 661 y Fo(0)1485 649 y Fu(and)f Fp(Y)1701 661 y Fo(0)1767 649 y Fu(in)g(\(4.6\))h(and)f(\(4.7\))h(w)n(e)f(ha)n(v)n(e)f(that,)i(when)g Fq(\026)23 b Fu(=)g(0,)963 789 y Fq(z)1002 801 y Fo(0)1038 789 y Fu(\()p Fp(T)1115 801 y Fo(0)1153 789 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(Y)1436 801 y Fo(0)1475 789 y Fu(\()p Fq(x;)g(v)s Fu(\)\))24 b(=)f Fq( )1864 801 y Fo(0)1901 789 y Fu(\()p Fp(T)1978 801 y Fo(0)2016 789 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g(\016)o(;)g Fp(Y)2372 801 y Fo(0)2410 789 y Fu(\()p Fq(x;)g(v)s Fu(\)\))25 b(=)d(\()p Fq(x;)14 b(v)s Fu(\))-2480 b(\(4.27\))456 928 y(where)27 b Fq( )750 940 y Fo(0)815 928 y Fu(is)g(in)n(tro)r (duced)g(in)h(Section)g(4.4.)36 b(Then)649 1067 y Fq(G)p Fu(\()p Fp(T)791 1079 y Fo(0)829 1067 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(Y)1112 1079 y Fo(0)1151 1067 y Fu(\()p Fq(x;)g(v)s Fu(\))p Fq(;)g(x;)g(v)s(;)g(\022)r(;)g Fu(0)p Fq(;)g(")p Fu(\))84 b(=)f Fq(w)r Fu(\()p Fp(T)2141 1079 y Fo(0)2179 1067 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g(\022)r(;)g Fp(Y)2540 1079 y Fo(0)2579 1067 y Fu(\()p Fq(x;)g(v)s Fu(\))p Fq(;)g Fu(0)p Fq(;)g(")p Fu(\))19 b Fp(\000)f Fu(\()p Fq(x;)c(v)s Fu(\))1855 1192 y(=)83 b Fq(z)2042 1204 y Fo(0)2078 1192 y Fu(\()p Fp(T)2155 1204 y Fo(0)2193 1192 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(Y)2476 1204 y Fo(0)2515 1192 y Fu(\()p Fq(x;)g(v)s Fu(\)\))20 b Fp(\000)e Fu(\()p Fq(x;)c(v)s Fu(\))1855 1316 y(=)83 b(0.)456 1456 y(W)-7 b(e)22 b(observ)n(e)e(that,)k(b)n(y)e (Prop)r(osition)e(4.1,)j Fq(G)f Fu(is)g(analytic)f(on)h(\()p Fq(S;)14 b(Y)19 b Fu(\))24 b Fp(2)f Fq(D)2761 1468 y Fo(0)2798 1456 y Fu(\()p Fq(\024)2878 1420 y Fj(\006)2878 1478 y Fo(1)2942 1456 y Fp(\000)7 b Fq(\034)e(;)14 b(\024)3140 1468 y Fo(2)3177 1456 y Fq(;)g Fu(2)p Fq(\024)3304 1468 y Fo(3)3340 1456 y Fu(\))7 b Fp(\002)456 1555 y Fq(W)12 b Fu(\(2)p Fq(r)657 1567 y Fo(1)694 1555 y Fu(\).)605 1655 y(Next)20 b(w)n(e)e(study)i(the)f(matrix)g Fq(D)1599 1667 y Fi(S;Y)1716 1655 y Fq(G)p Fu(.)34 b(Since)20 b(the)f(unp)r (erturb)r(ed)h(system)f(is)g(Hamiltonian)456 1754 y(with)28 b(Hamiltonian)1547 1926 y Fp(F)1607 1938 y Fo(0)1644 1926 y Fu(\()p Fq(x;)14 b(v)s Fu(\))24 b(=)1956 1870 y Fq(v)1999 1840 y Fo(2)p 1956 1907 81 4 v 1976 1983 a Fu(2)2065 1926 y(+)18 b Fq(v)s(f)9 b Fu(\()p Fq(x)p Fu(\))456 2098 y(the)28 b(solution)f Fq(z)952 2110 y Fo(0)989 2098 y Fu(\()p Fq(u;)14 b(Y)k Fu(\))23 b(=)g(\()p Fq(z)1390 2068 y Fo(1)1386 2119 y(0)1427 2098 y Fu(\()p Fq(u;)14 b(Y)k Fu(\))p Fq(;)c(z)1722 2068 y Fo(2)1718 2119 y(0)1759 2098 y Fu(\()p Fq(u;)g(Y)k Fu(\)\))28 b(satis\014es)1059 2235 y Fq(Y)1125 2204 y Fo(2)p 1059 2272 104 4 v 1090 2348 a Fu(2)1191 2291 y(+)18 b Fq(Y)h(f)9 b Fu(\()p Fq(\016)s Fu(\))23 b(=)1615 2235 y(\()p Fq(z)1690 2204 y Fo(2)1686 2255 y(0)1727 2235 y Fu(\()p Fq(u;)14 b(Y)19 b Fu(\)\))1975 2204 y Fo(2)p 1615 2272 397 4 v 1793 2348 a Fu(2)2041 2291 y(+)f Fq(z)2167 2257 y Fo(2)2163 2311 y(0)2203 2291 y Fu(\()p Fq(u;)c(Y)19 b Fu(\))p Fq(f)9 b Fu(\()p Fq(z)2544 2257 y Fo(1)2540 2311 y(0)2580 2291 y Fu(\()p Fq(u;)14 b(Y)19 b Fu(\)\).)-2395 b(\(4.28\))456 2458 y(Di\013eren)n(tiating)27 b(with)h(resp)r(ect)g(to)f Fq(Y)46 b Fu(in)28 b(\(4.28\))f(w)n(e)g (obtain)676 2597 y Fq(Y)37 b Fu(+)18 b Fq(f)9 b Fu(\()p Fq(\016)s Fu(\))766 2736 y(=)22 b(\()p Fq(z)928 2702 y Fo(2)924 2757 y(0)965 2736 y Fu(\()p Fq(u;)14 b(Y)k Fu(\))h(+)f Fq(f)9 b Fu(\()p Fq(z)1407 2702 y Fo(1)1403 2757 y(0)1444 2736 y Fu(\()p Fq(u;)14 b(Y)k Fu(\)\)\))p Fq(@)1767 2748 y Fi(Y)1825 2736 y Fq(z)1868 2702 y Fo(2)1864 2757 y(0)1905 2736 y Fu(\()p Fq(u;)c(Y)k Fu(\))h(+)f Fq(z)2265 2702 y Fo(2)2261 2757 y(0)2302 2736 y Fu(\()p Fq(u;)c(Y)k Fu(\))p Fq(f)2567 2702 y Fj(0)2590 2736 y Fu(\()p Fq(z)2665 2702 y Fo(1)2661 2757 y(0)2702 2736 y Fu(\()p Fq(u;)c(Y)k Fu(\)\))p Fq(@)2993 2748 y Fi(Y)3051 2736 y Fq(z)3094 2702 y Fo(1)3090 2757 y(0)3131 2736 y Fu(\()p Fq(u;)c(Y)k Fu(\).)-2913 b(\(4.29\))605 2880 y(Ev)-5 b(aluating)27 b(\(4.29\))f(at)i(\()p Fq(u;)14 b(Y)k Fu(\))24 b(=)e(\()p Fp(T)1768 2892 y Fo(0)1806 2880 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(Y)2089 2892 y Fo(0)2128 2880 y Fu(\()p Fq(x;)g(v)s Fu(\)\),)29 b(w)n(e)e(get)901 3021 y Fp(Y)956 3033 y Fo(0)1012 3021 y Fu(+)18 b Fq(f)9 b Fu(\()p Fq(\016)s Fu(\))23 b(=)g(\()p Fq(v)f Fu(+)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\)\))p Fq(@)1774 3033 y Fi(Y)1832 3021 y Fq(z)1875 2986 y Fo(2)1871 3041 y(0)1912 3021 y Fu(\()p Fp(T)1989 3033 y Fo(0)2027 3021 y Fq(;)14 b Fp(Y)2119 3033 y Fo(0)2156 3021 y Fu(\))19 b(+)f Fq(v)s(f)2383 2986 y Fj(0)2406 3021 y Fu(\()p Fq(x)p Fu(\))p Fq(@)2561 3033 y Fi(Y)2620 3021 y Fq(z)2663 2986 y Fo(1)2659 3041 y(0)2699 3021 y Fu(\()p Fp(T)2776 3033 y Fo(0)2814 3021 y Fq(;)c Fp(Y)2906 3033 y Fo(0)2944 3021 y Fu(\).)605 3165 y(No)n(w)33 b(w)n(e)g(pro)n(v)n(e)f(that)i(the)g (deriv)-5 b(ativ)n(e)32 b Fq(@)1923 3177 y Fi(S)s(Y)2025 3165 y Fq(G)i Fu(at)f Fq(\026)g Fu(=)g(0)g(is)g(in)n(v)n(ertible.)54 b(Using)33 b(\(4.27\))456 3264 y(and)27 b(\(4.29\),)g(the)h(determinan) n(t)f(of)h Fq(D)1655 3276 y Fi(S;Y)1772 3264 y Fq(G)g Fu(ev)-5 b(aluated)27 b(at)h(\()p Fq(S;)14 b(Y)19 b Fu(\))k(=)g(\()p Fp(T)2745 3276 y Fo(0)2783 3264 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(Y)3066 3276 y Fo(0)3104 3264 y Fu(\()p Fq(x;)g(v)s Fu(\)\))29 b(is)670 3408 y(det)q(\()p Fq(D)887 3420 y Fi(S)s(Y)988 3408 y Fq(G)p Fu(\))84 b(=)e Fq(@)1360 3420 y Fi(S)1408 3408 y Fq(z)1451 3373 y Fo(1)1447 3428 y(0)1488 3408 y Fu(\()p Fp(T)1565 3420 y Fo(0)1603 3408 y Fq(;)14 b Fp(Y)1695 3420 y Fo(0)1732 3408 y Fu(\))p Fq(@)1808 3420 y Fi(Y)1866 3408 y Fq(z)1909 3373 y Fo(2)1905 3428 y(0)1946 3408 y Fu(\()p Fp(T)2023 3420 y Fo(0)2060 3408 y Fq(;)g Fp(Y)2152 3420 y Fo(0)2190 3408 y Fu(\))19 b Fp(\000)f Fq(@)2368 3420 y Fi(Y)2425 3408 y Fq(z)2468 3373 y Fo(1)2464 3428 y(0)2505 3408 y Fu(\()p Fp(T)2582 3420 y Fo(0)2620 3408 y Fq(;)c Fp(Y)2712 3420 y Fo(0)2749 3408 y Fu(\))p Fq(@)2825 3420 y Fi(S)2874 3408 y Fq(z)2917 3373 y Fo(2)2913 3428 y(0)2953 3408 y Fu(\()p Fp(T)3030 3420 y Fo(0)3068 3408 y Fq(;)g Fp(Y)3160 3420 y Fo(0)3197 3408 y Fu(\))1169 3533 y(=)82 b Fq(X)1392 3499 y Fo(1)1385 3554 y(0)1429 3533 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(@)1664 3545 y Fi(Y)1722 3533 y Fq(z)1765 3499 y Fo(2)1761 3554 y(0)1802 3533 y Fu(\()p Fp(T)1879 3545 y Fo(0)1917 3533 y Fq(;)g Fp(Y)2009 3545 y Fo(0)2046 3533 y Fu(\))19 b Fp(\000)f Fq(X)2256 3499 y Fo(2)2249 3554 y(0)2293 3533 y Fu(\()p Fq(x;)c(v)s Fu(\))p Fq(@)2528 3545 y Fi(Y)2586 3533 y Fq(z)2629 3499 y Fo(1)2625 3554 y(0)2666 3533 y Fu(\()p Fp(T)2743 3545 y Fo(0)2781 3533 y Fq(;)g Fp(Y)2873 3545 y Fo(0)2910 3533 y Fu(\))1169 3658 y(=)82 b Fp(Y)1371 3670 y Fo(0)1427 3658 y Fu(+)18 b Fq(f)9 b Fu(\()p Fq(\016)s Fu(\))456 3797 y(and)27 b(b)n(y)g(de\014nition)h(\(4.7\))g(of)f Fp(Y)1449 3809 y Fo(0)1487 3797 y Fu(,)h(w)n(e)f(obtain)1241 3947 y(det\()p Fq(D)1457 3959 y Fi(S)s(Y)1559 3947 y Fq(G)p Fu(\))c(=)g Fp(\000)1832 3872 y Fg(p)p 1915 3872 722 4 v 75 x Fq(f)1965 3923 y Fo(2)2002 3947 y Fu(\()p Fq(\016)s Fu(\))c(+)f Fq(v)2251 3923 y Fo(2)2307 3947 y Fu(+)g(2)p Fq(v)s(f)9 b Fu(\()p Fq(x)p Fu(\).)456 4100 y(W)-7 b(e)30 b(recall)g(that)g Fp(j)p Fq(v)s Fp(j)e Fq(<)g(r)i(<)d(r)1414 4112 y Fo(0)1452 4100 y Fu(,)k(and)g(that)f Fq(C)1912 4112 y Fo(0)1950 4100 y Fq(\016)1990 4070 y Fi(\014)2062 4100 y Fp(\024)e(j)p Fq(f)9 b Fu(\()p Fq(\016)s Fu(\))p Fp(j)28 b(\024)f Fq(C)2534 4112 y Fo(1)2572 4100 y Fq(\016)2612 4070 y Fi(\014)2656 4100 y Fu(,)32 b(hence,)f(b)n(y)f (De\014nition)456 4199 y(4.1,)c(taking)h Fq(\016)908 4169 y Fj(0)905 4220 y Fo(0)966 4199 y Fu(=)22 b Fq(\016)1172 4340 y Fp(j)14 b Fu(det\()p Fq(D)1425 4352 y Fi(S)s(Y)1526 4340 y Fq(G)p Fu(\))p Fp(j)1646 4305 y Fo(2)1767 4340 y Fp(\025)83 b Fq(f)9 b Fu(\()p Fq(\016)s Fu(\))2069 4305 y Fo(2)2125 4340 y Fp(\000)18 b(j)p Fq(v)s Fp(j)2297 4305 y Fo(2)2353 4340 y Fp(\000)g Fu(2)p Fp(j)p Fq(v)s(f)9 b Fu(\()p Fq(x)p Fu(\))p Fp(j)1767 4474 y(\025)83 b Fq(C)1980 4440 y Fo(2)1974 4494 y(0)2017 4474 y Fq(\016)2057 4440 y Fo(2)p Fi(\014)2154 4474 y Fp(\000)18 b Fq(r)2276 4440 y Fo(2)2274 4494 y(0)2332 4474 y Fp(\000)g Fu(2)p Fq(r)2494 4486 y Fo(0)2531 4474 y Fq(C)2590 4486 y Fo(1)2628 4474 y Fq(\016)2668 4434 y Fi(\014)2665 4496 y Fo(0)1767 4598 y Fq(>)83 b Fu(0.)456 4738 y(A)n(t)36 b(this)g(p)r(oin)n(t)h(it)f(w)n (ould)g(b)r(e)g(natural)g(to)g(apply)f(the)i(implicit)g(function)g (theorem)e(to)h(the)456 4837 y(equation)1524 4976 y Fq(G)p Fu(\()p Fq(S;)14 b(Y)5 b(;)14 b(x;)g(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))24 b(=)f(0.)456 5116 y(Ho)n(w)n(ev)n(er)j(to)j(ha)n(v)n(e)e(a)h (go)r(o)r(d)g(con)n(trol)f(on)i(the)g(domains)f(in)g(whic)n(h)h(w)n(e)f (will)h(\014nd)g(the)g(solution)456 5216 y Fp(T)21 b Fu(,)29 b Fp(Y)36 b Fu(in)29 b(terms)f(of)g(\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))30 b(w)n(e)e(follo)n(w)g(the)h (pro)r(of)f(of)g(the)h(implicit)h(function)f(theorem)p eop %%Page: 52 60 52 59 bop 456 251 a Fo(52)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(using)29 b(the)i(sp)r(ecial)f(structure)f(of)i(the)f(equation)g(w)n (e)f(deal)h(with.)45 b(W)-7 b(e)31 b(will)f(w)n(ork)f(in)h(a)g(space) 456 550 y(of)d(functions)h(of)g(the)g(form)1533 692 y(\()p Fq(S;)14 b(Y)19 b Fu(\))k(=)g Fq(h)p Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\).)456 836 y(In)27 b(the)h(rest)f(of)h(this)g(pro)r(of)f(w)n(e)g(tak)n(e)g(the)h(norm)1535 978 y Fp(k)p Fu(\()p Fq(\030)t(;)14 b(\021)s Fu(\))p Fp(k)23 b Fu(=)f(max)p Fp(fj)p Fq(\030)t Fp(j)p Fq(;)14 b Fp(j)p Fq(\021)s Fp(jg)456 1131 y Fu(for)32 b(\()p Fq(\030)t(;)14 b(\021)s Fu(\))33 b Fp(2)g Fl(C)948 1101 y Fo(2)991 1131 y Fu(.)53 b(W)-7 b(e)34 b(de\014ne)f(the)g(space)g (\000)g(of)g(functions)g Fq(h)f Fu(:)h Fq(U)2578 1143 y Fo(0)2637 1131 y Fp(\002)21 b Fl(R)28 b Fp(\002)22 b Fq(P)44 b Fp(!)32 b Fl(C)3158 1101 y Fo(2)3235 1131 y Fu(whic)n(h)456 1231 y(satisfy)27 b(\(w)n(e)g(call)h(\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))28 b(the)g(v)-5 b(ariables)26 b(of)i Fq(h)p Fu(\))647 1352 y(\(a\))41 b Fq(h)28 b Fu(is)f Fq(C)1018 1322 y Fo(0)1056 1352 y Fu(.)642 1453 y(\(b\))42 b Fq(h)28 b Fu(is)f(analytic)g(in)h(\()p Fq(x;)14 b(v)s(;)g(\026)p Fu(\))24 b Fp(2)f Fq(U)1803 1465 y Fo(0)1840 1453 y Fu(\()p Fq(\024)1920 1417 y Fj(\006)1920 1475 y Fo(1)1977 1453 y Fq(;)14 b(\024)2062 1465 y Fo(2)2099 1453 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))k Fp(\002)g(f)p Fq(\026)23 b Fp(2)g Fl(C)44 b Fu(:)24 b Fp(j)p Fq(\026)p Fp(j)f(\024)f Fq(\026)2966 1465 y Fo(0)3004 1453 y Fp(g)p Fu(.)651 1552 y(\(c\))42 b Fq(h)28 b Fu(is)f Fq(C)1018 1522 y Fo(1)1083 1552 y Fu(and)h(2)p Fq(\031)s Fu(-p)r(erio)r(dic)f (with)h(resp)r(ect)f(to)h Fq(\022)r Fu(.)642 1652 y(\(d\))42 b(The)28 b(norm)803 1795 y Fp(k)p Fq(h)p Fp(k)935 1807 y Fo(\000)1062 1795 y Fu(=)157 b(sup)1210 1865 y Fi(U)1255 1873 y Fh(0)1287 1865 y Fj(\002)p Fd(R)p Fj(\002)-5 b Fi(P)1498 1795 y Fp(k)p Fq(h)p Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g (\026;)g(")p Fu(\))p Fp(k)k Fu(+)93 b(sup)2163 1865 y Fi(U)2208 1873 y Fh(0)2241 1865 y Fj(\002)p Fd(R)p Fj(\002)-5 b Fi(P)2452 1795 y Fp(k)p Fq(@)2538 1807 y Fi(\022)2575 1795 y Fq(h)p Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))p Fp(k)1062 1977 y(\021)83 b(k)p Fq(h)p Fp(k)1342 1989 y Fj(1)1429 1977 y Fu(+)18 b Fp(k)p Fq(@)1598 1989 y Fi(\022)1635 1977 y Fq(h)p Fp(k)1725 1989 y Fj(1)794 2119 y Fu(is)28 b(b)r(ounded.)605 2241 y(W)-7 b(e)27 b(endo)n(w)f(\000)g(with)h(the)g(norm)f Fp(k)p Fq(:)p Fp(k)1733 2253 y Fo(\000)1804 2241 y Fu(and)g(it)h(b)r(ecomes)f(a)g (Banac)n(h)f(space.)36 b(W)-7 b(e)27 b(call)f(\000\()p Fq(\032)p Fu(\))456 2340 y(the)34 b(closed)f(ball)h(of)g(radius)f Fq(\032)h Fu(of)f(\000,)j(cen)n(tered)d(at)h(\()p Fp(T)2190 2352 y Fo(0)2228 2340 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(Y)2511 2352 y Fo(0)2549 2340 y Fu(\()p Fq(x;)g(v)s Fu(\)\))35 b Fp(2)f Fu(\000.)56 b(W)-7 b(e)34 b(observ)n(e)456 2440 y(that,)28 b(since)f(\()p Fp(T)939 2452 y Fo(0)977 2440 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(Y)1260 2452 y Fo(0)1298 2440 y Fu(\()p Fq(x;)g(v)s Fu(\)\))30 b(do)r(es)d(not)g(dep)r(end)i(on)e Fq(\022)r Fu(,)h(for)f(an)n(y)g Fq(h)c Fp(2)g Fu(\000\()p Fq(\032)p Fu(\),)1124 2583 y Fp(k)p Fq(h)18 b Fp(\000)g Fu(\()p Fp(T)1392 2595 y Fo(0)1430 2583 y Fq(;)c Fp(Y)1522 2595 y Fo(0)1559 2583 y Fu(\))p Fp(k)1633 2595 y Fo(\000)1701 2583 y Fu(=)23 b Fp(k)p Fq(h)18 b Fp(\000)g Fu(\()p Fp(T)2057 2595 y Fo(0)2095 2583 y Fq(;)c Fp(Y)2187 2595 y Fo(0)2224 2583 y Fu(\))p Fp(k)2298 2595 y Fj(1)2387 2583 y Fu(+)k Fp(k)p Fq(@)2556 2595 y Fi(\022)2593 2583 y Fq(h)p Fp(k)2683 2595 y Fj(1)2753 2583 y Fu(.)456 2730 y(W)-7 b(e)28 b(de\014ne)f(the)h (op)r(erator)e Fp(G)j Fu(:)23 b(\000\()p Fq(\032)p Fu(\))g Fp(!)g Fu(\000\()p Fq(\032)p Fu(\))28 b(b)n(y)784 2874 y Fp(G)5 b Fu(\()p Fq(h)p Fu(\)\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g (\026;)g(")p Fu(\))25 b(=)d Fq(h)d Fp(\000)f Fu(\()p Fq(D)1745 2886 y Fi(S;Y)1862 2874 y Fq(G)p Fu(\()p Fp(T)2004 2886 y Fo(0)2042 2874 y Fq(;)c Fp(Y)2134 2886 y Fo(0)2172 2874 y Fq(;)g(x;)g(v)s(;)g(\022)r(;)g Fu(0)p Fq(;)g(")p Fu(\)\))2633 2840 y Fj(\000)p Fo(1)2722 2874 y Fq(G)p Fu(\()p Fq(h;)g(x;)g(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\),)456 3022 y(where)29 b(in)i(the)g(righ)n(t)e(hand)h(side,)h Fq(h)d Fu(=)f Fq(h)p Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\),)32 b Fp(T)2298 3034 y Fo(0)2363 3022 y Fu(=)27 b Fp(T)2500 3034 y Fo(0)2537 3022 y Fu(\()p Fq(x;)14 b(v)s Fu(\))32 b(and)e Fp(Y)2979 3034 y Fo(0)3044 3022 y Fu(=)d Fp(Y)3191 3034 y Fo(0)3229 3022 y Fu(\()p Fq(x;)14 b(v)s Fu(\).)456 3122 y Fp(G)33 b Fu(is)27 b(w)n(ell)g(de\014ned.)38 b(Indeed,)28 b(let)f Fq(h)c Fp(2)h Fu(\000\()p Fq(\032)p Fu(\))k(with)g Fq(\032)g Fu(small)f(enough.)36 b(By)27 b(Section)h(4.5.1)1254 3265 y(\()p Fp(T)1331 3277 y Fo(0)1368 3265 y Fq(;)14 b Fp(Y)1460 3277 y Fo(0)1498 3265 y Fu(\))23 b Fp(2)h Fq(D)1701 3277 y Fo(0)1738 3265 y Fu(\()p Fq(\024)1818 3230 y Fj(\006)1818 3287 y Fo(1)1892 3265 y Fp(\000)18 b Fq(\034)5 b(;)14 b(\024)2101 3277 y Fo(2)2138 3265 y Fq(;)g(\024)2223 3277 y Fo(3)2260 3265 y Fu(\))19 b Fp(\002)f Fq(W)12 b Fu(\()p Fq(r)2553 3277 y Fo(1)2591 3265 y Fu(\),)456 3422 y(th)n(us)30 b(,)h(if)g Fq(\032)f Fu(is)g(small)g(enough,)h Fq(h)c Fp(2)h Fq(D)1693 3434 y Fo(0)1730 3422 y Fu(\()p Fq(\024)1810 3387 y Fj(\006)1810 3444 y Fo(1)1886 3422 y Fp(\000)20 b Fq(\034)5 b(;)14 b(\024)2097 3434 y Fo(2)2134 3422 y Fq(;)g Fu(2)p Fq(\024)2261 3434 y Fo(3)2298 3422 y Fu(\))20 b Fp(\002)g Fq(W)12 b Fu(\(2)p Fq(r)2636 3434 y Fo(1)2674 3422 y Fu(\))30 b(and)g(then)h Fp(G)5 b Fu(\()p Fq(h)p Fu(\))29 b Fp(2)e Fu(\000.)456 3522 y(Next)h(w)n(e)f(will)h(c)n(hec)n(k)e(that)i Fp(G)5 b Fu(\()p Fq(h)p Fu(\))24 b Fp(2)f Fu(\000\()p Fq(\032)p Fu(\).)605 3621 y(T)-7 b(o)39 b(shorten)g(the)h(notation)f(w) n(e)h(will)f(not)h(write)f(the)h(dep)r(endence)g(on)g(the)g(v)-5 b(ariables)456 3721 y(\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g(")p Fu(\),)28 b(and)g(w)n(e)f(will)h(denote)f(\()p Fp(T)1637 3733 y Fo(0)1675 3721 y Fq(;)14 b Fp(Y)1767 3733 y Fo(0)1805 3721 y Fu(\))28 b(b)n(y)f Fq(h)2028 3733 y Fo(0)2065 3721 y Fu(.)37 b(By)27 b(T)-7 b(a)n(ylor's)26 b(theorem,)679 3865 y Fp(G)5 b Fu(\()p Fq(h)p Fu(\)\()p Fq(\026)p Fu(\))85 b(=)d Fq(h)18 b Fp(\000)g Fu(\()p Fq(D)1441 3877 y Fi(S;Y)1559 3865 y Fq(G)p Fu(\()p Fq(h)1704 3877 y Fo(0)1742 3865 y Fq(;)c Fu(0\)\))1885 3831 y Fj(\000)p Fo(1)1974 3865 y Fq(G)p Fu(\()p Fq(h;)g(\026)p Fu(\))1044 4017 y(=)82 b Fq(h)18 b Fp(\000)g Fu(\()p Fq(D)1441 4029 y Fi(S;Y)1559 4017 y Fq(G)p Fu(\()p Fq(h)1704 4029 y Fo(0)1742 4017 y Fq(;)c Fu(0\)\))1885 3983 y Fj(\000)p Fo(1)1974 3925 y Fg(\020)2023 4017 y Fq(G)p Fu(\()p Fq(h)2168 4029 y Fo(0)2206 4017 y Fq(;)g Fu(0\))k(+)g Fq(D)r(G)p Fu(\()p Fq(h)2634 4029 y Fo(0)2672 4017 y Fq(;)c Fu(0\)\()p Fq(h)k Fp(\000)g Fq(h)3012 4029 y Fo(0)3049 4017 y Fq(;)c(\026)p Fu(\))3168 3983 y Fi(T)1191 4234 y Fu(+)1270 4121 y Fg(Z)1353 4142 y Fo(1)1316 4310 y(0)1390 4234 y Fu(\()p Fq(D)r(G)p Fu(\()p Fp(Z)7 b Fu(\()p Fq(\020)f Fu(\)\))20 b Fp(\000)e Fq(D)r(G)p Fu(\()p Fq(h)2114 4246 y Fo(0)2152 4234 y Fq(;)c Fu(0\)\)\()p Fq(h)19 b Fp(\000)f Fq(h)2525 4246 y Fo(0)2562 4234 y Fq(;)c(\026)p Fu(\))2681 4200 y Fi(T)2747 4234 y Fq(d\020)2832 4142 y Fg(\021)456 4424 y Fu(where)29 b Fq(D)r(G)h Fp(\021)c Fu(\()p Fq(@)1031 4436 y Fi(S)1079 4424 y Fq(G;)14 b(@)1225 4436 y Fi(Y)1283 4424 y Fq(G;)g(@)1429 4436 y Fi(\026)1474 4424 y Fq(G)p Fu(\))30 b(and)g Fp(Z)7 b Fu(\()p Fq(\020)f Fu(\))27 b(=)f(\()p Fq(h)2136 4436 y Fo(0)2193 4424 y Fu(+)19 b Fq(\020)6 b Fu(\()p Fq(h)21 b Fp(\000)e Fq(h)2552 4436 y Fo(0)2589 4424 y Fu(\))p Fq(;)14 b(\020)6 b(\026)p Fu(\).)44 b(W)-7 b(e)30 b(observ)n(e)e(that) 456 4523 y Fq(G)g Fu(is)f(w)n(ell)g(de\014ned)h(in)g Fp(Z)7 b Fu(\()p Fq(\020)f Fu(\))29 b(for)e(all)g Fq(\020)j Fp(2)23 b Fu([0)p Fq(;)14 b Fu(1].)36 b(Then,)28 b(using)f(that)1450 4666 y Fq(G)p Fu(\()p Fq(h)1595 4678 y Fo(0)1633 4666 y Fq(;)14 b Fu(0\))23 b(=)f Fq(G)p Fu(\()p Fp(T)1996 4678 y Fo(0)2034 4666 y Fq(;)14 b Fp(Y)2126 4678 y Fo(0)2164 4666 y Fq(;)g Fu(0\))23 b(=)f(0,)456 4809 y(w)n(e)27 b(obtain)837 4956 y Fp(G)5 b Fu(\()p Fq(h)p Fu(\)\()p Fq(\026)p Fu(\))84 b(=)f Fq(h)1397 4968 y Fo(0)1452 4956 y Fp(\000)18 b Fu(\()p Fq(D)1636 4968 y Fi(S;Y)1754 4956 y Fq(G)p Fu(\()p Fq(h)1899 4968 y Fo(0)1937 4956 y Fq(;)c Fu(0\)\))2080 4922 y Fj(\000)p Fo(1)2169 4864 y Fg(\020)2218 4956 y Fq(\026@)2312 4968 y Fi(\026)2357 4956 y Fq(G)p Fu(\()p Fq(h)2502 4968 y Fo(0)2540 4956 y Fq(;)g Fu(0\))1349 5173 y Fp(\000)1428 5060 y Fg(Z)1510 5081 y Fo(1)1473 5249 y(0)1548 5173 y Fu(\()p Fq(D)r(G)p Fu(\()p Fp(Z)7 b Fu(\()p Fq(\020)f Fu(\)\))20 b Fp(\000)e Fq(D)r(G)p Fu(\()p Fq(h)2272 5185 y Fo(0)2310 5173 y Fq(;)c Fu(0\)\)\()p Fq(h)k Fp(\000)g Fq(h)2682 5185 y Fo(0)2719 5173 y Fq(;)c(\026)p Fu(\))2838 5139 y Fi(T)2905 5173 y Fq(d\020)2990 5081 y Fg(\021)3040 5173 y Fu(.)p eop %%Page: 53 61 53 60 bop 1006 251 a Fo(4.5.)30 b(FLO)n(W)f(BO)n(X)g(COORDINA)-5 b(TES)29 b(IN)g(A)g(COMPLEX)g(DOMAIN)484 b(53)605 450 y Fu(W)-7 b(e)28 b(observ)n(e)e(that)984 586 y Fq(@)1028 598 y Fi(\026)1073 586 y Fq(G)p Fu(\()p Fq(h)1218 598 y Fo(0)1256 586 y Fq(;)14 b(\026)p Fu(\))23 b(=)g Fq(")1525 552 y Fi(q)1561 586 y Fq(w)1620 598 y Fo(1)1658 586 y Fu(\()p Fp(T)1735 598 y Fo(0)1772 586 y Fq(;)14 b(\022)r(;)g Fp(Y)1942 598 y Fo(0)1980 586 y Fq(;)g(\026;)g(")p Fu(\))19 b(+)f Fq(\026")2366 552 y Fi(q)2402 586 y Fq(@)2446 598 y Fi(\026)2491 586 y Fq(w)2550 598 y Fo(1)2587 586 y Fu(\()p Fp(T)2664 598 y Fo(0)2702 586 y Fq(;)c(\022)r(;)g Fp(Y)2872 598 y Fo(0)2910 586 y Fq(;)g(\026;)g(")p Fu(\).)-2672 b(\(4.30\))605 723 y(Since)26 b Fq(G)f Fu(is)h(analytic)e(in)i Fq(h)p Fu(,)g Fq(G)g Fu(has)f(its)g(second)g(deriv)-5 b(ativ)n(e)24 b(with)i(resp)r(ect)f(to)h Fq(h)f Fu(b)r(ounded)456 823 y(in)i Fq(D)621 835 y Fo(0)658 823 y Fu(\()p Fq(\024)738 787 y Fj(\006)738 845 y Fo(1)813 823 y Fp(\000)18 b Fq(\034)5 b(;)14 b(\024)1022 835 y Fo(2)1059 823 y Fq(;)g Fu(2)p Fq(\024)1186 835 y Fo(3)1223 823 y Fu(\))k Fp(\002)g Fq(W)12 b Fu(\(2)p Fq(r)1557 835 y Fo(1)1595 823 y Fu(\),)28 b(restricting)e Fq(\024)2123 835 y Fo(3)2188 823 y Fu(if)i(necessary)-7 b(.)35 b(Therefore)1156 962 y Fp(kG)5 b Fu(\()p Fq(h)p Fu(\))18 b Fp(\000)g Fq(h)1513 974 y Fo(0)1550 962 y Fp(k)1592 974 y Fj(1)1685 962 y Fp(\024)23 b Fq(K)6 b Fu(\()p Fp(j)p Fq(\026)p Fp(j)p Fq(")2017 927 y Fi(q)2072 962 y Fu(+)18 b Fq(\032)2198 927 y Fo(2)2253 962 y Fu(+)g Fq(\032)p Fp(j)p Fq(\026)p Fp(j)p Fu(\))24 b Fp(\024)e Fq(\032=)p Fu(2)456 1103 y(if)27 b Fq(\032)g Fu(and)g Fp(j)p Fq(\026)p Fp(j)p Fq(")897 1073 y Fi(q)961 1103 y Fu(are)f(small)g(enough.)36 b(Here)27 b(w)n(e)g(ha)n(v)n(e)f(used)h (that)g Fp(k)p Fq(@)2596 1115 y Fi(\026)2640 1103 y Fq(G)p Fu(\()p Fq(h)2785 1115 y Fo(0)2823 1103 y Fq(;)14 b Fu(0\))p Fp(k)2976 1115 y Fj(1)3068 1103 y Fu(=)23 b Fq(O)r Fu(\()p Fq(")3292 1073 y Fi(q)3329 1103 y Fu(\))28 b(is)456 1203 y(b)r(ounded)g(and)f(that,)h(b)n(y)f(the)h(mean)g(v)-5 b(alue)27 b(theorem,)981 1339 y Fp(k)1023 1272 y Fg(\002)1057 1339 y Fq(D)1126 1351 y Fi(S;Y)1243 1339 y Fq(G)p Fu(\()p Fp(Z)7 b Fu(\()p Fq(\020)f Fu(\)\))20 b Fp(\000)e Fq(D)1717 1351 y Fi(S;Y)1834 1339 y Fq(G)p Fu(\()p Fq(h)1979 1351 y Fo(0)2017 1339 y Fq(;)c Fu(0\))2128 1272 y Fg(\003)2162 1339 y Fu(\()p Fq(h)19 b Fp(\000)f Fq(h)2392 1351 y Fo(0)2429 1339 y Fu(\))p Fp(k)2503 1351 y Fj(1)2596 1339 y Fp(\024)23 b Fq(K)6 b(\032)p Fu(\()p Fq(\032)18 b Fu(+)g Fp(j)p Fq(\026)p Fp(j)p Fu(\).)605 1480 y(Using)27 b(\(4.30\),)g(w)n(e)h(ha)n (v)n(e)e(that)1687 1617 y Fq(@)1731 1629 y Fi(\022)r(\026)1809 1617 y Fq(G)d Fu(=)g Fq(O)r Fu(\()p Fq(")2121 1582 y Fi(q)2158 1617 y Fu(\).)456 1753 y(Moreo)n(v)n(er,)i(since)i Fq(h)1095 1765 y Fo(0)1160 1753 y Fu(do)r(es)g(not)h(dep)r(end)g(on)f Fq(\022)r Fu(,)h(and)g(that)1547 1890 y Fq(@)1591 1902 y Fi(\022)1628 1890 y Fq(w)r Fu(\()p Fq(S;)14 b(\022)r(;)g(Y)20 b Fu(\))j(=)g Fq(O)r Fu(\()p Fq(\026")2284 1855 y Fi(q)2321 1890 y Fu(\))456 2031 y(and)k(consequen)n(tly)g Fq(@)1153 2043 y Fi(\022)1190 2031 y Fq(D)1259 2043 y Fi(S)s(Y)1361 2031 y Fq(G)c Fu(=)f Fq(O)r Fu(\()p Fq(\026")1722 2001 y Fi(q)1760 2031 y Fu(\),)28 b(w)n(e)f(ha)n(v)n(e)f(that)655 2227 y Fp(k)p Fq(@)741 2239 y Fi(\022)778 2227 y Fp(G)5 b Fu(\()p Fq(h)p Fu(\)\()p Fq(\026)p Fu(\))p Fp(k)84 b(\024)e(j)p Fq(\026)p Fp(jj)p Fq(K)1541 2114 y Fg(Z)1624 2134 y Fo(1)1587 2302 y(0)1675 2227 y Fp(j)p Fq(@)1742 2239 y Fi(\022)1780 2227 y Fu(\()p Fq(D)r(G)p Fu(\()p Fp(Z)7 b Fu(\()p Fq(\020)f Fu(\)\))20 b Fp(\000)e Fq(D)r(G)p Fu(\()p Fq(h)2504 2239 y Fo(0)2542 2227 y Fq(;)c Fu(0\)\)\()p Fq(h)k Fp(\000)g Fq(h)2914 2239 y Fo(0)2951 2227 y Fq(;)c(\026)p Fu(\))3070 2192 y Fi(T)3123 2227 y Fp(j)g Fq(d\020)1184 2465 y Fp(\024)82 b(j)p Fq(\026)p Fp(j)p Fq(K)6 b Fp(j)1527 2373 y Fg(\020)1590 2352 y(Z)1673 2373 y Fo(1)1636 2541 y(0)1724 2465 y Fq(\020)g Fp(j)p Fq(@)1833 2477 y Fi(\022)1872 2465 y Fq(D)r(G)p Fu(\()p Fp(Z)h Fu(\()p Fq(\020)f Fu(\)\)\()p Fq(@)2321 2477 y Fi(\022)2360 2465 y Fq(h;)14 b Fu(0\))2519 2431 y Fi(T)2571 2465 y Fu(\()p Fq(h)19 b Fp(\000)f Fq(h)2801 2477 y Fo(0)2838 2465 y Fq(;)c(\026)p Fu(\))2957 2431 y Fi(T)3009 2465 y Fp(j)g Fq(d\020)1331 2704 y Fu(+)1410 2591 y Fg(Z)1493 2612 y Fo(1)1456 2780 y(0)1544 2704 y Fp(j)p Fu(\()p Fq(D)r(G)p Fu(\()p Fp(Z)7 b Fu(\()p Fq(\020)f Fu(\)\))20 b Fp(\000)e Fq(D)r(G)p Fu(\()p Fq(h)2291 2716 y Fo(0)2329 2704 y Fq(;)c Fu(0\)\)\()p Fq(@)2548 2716 y Fi(\022)2586 2704 y Fq(h;)g Fu(0\))2745 2670 y Fi(T)2810 2704 y Fq(d\020)2895 2612 y Fg(\021)1184 2885 y Fp(\024)82 b(j)p Fq(\026)p Fp(j)p Fq(K)6 b Fp(j)p Fu(\()p Fp(j)p Fq(\026)p Fp(j)p Fq(")1694 2851 y Fi(q)1731 2885 y Fq(\032)1774 2851 y Fo(2)1829 2885 y Fu(+)18 b Fp(j)p Fq(\026)p Fp(j)p Fq(")2047 2851 y Fi(q)2102 2885 y Fu(+)g Fq(\032)2228 2851 y Fo(3)2266 2885 y Fu(\))1184 3009 y Fp(\024)82 b Fq(\032=)p Fu(2)456 3146 y(if)28 b Fp(j)p Fq(\026)p Fp(j)p Fq(")667 3116 y Fi(q)731 3146 y Fu(is)f(small)h (enough.)36 b(In)28 b(fact)f(w)n(e)h(can)f(tak)n(e)g Fq(\032)c Fu(=)f Fq(O)r Fu(\()p Fq(\026")2411 3116 y Fi(q)2449 3146 y Fu(\).)605 3245 y(The)29 b(op)r(erator)f Fp(G)35 b Fu(is)29 b(a)g(con)n(traction,)f(th)n(us)h(the)h(\014xed)f(p) r(oin)n(t)h(theorem)f(can)g(b)r(e)g(applied,)456 3345 y(and)24 b(w)n(e)g(\014nd)g(functions)h Fp(T)45 b Fu(and)25 b Fp(Y)31 b Fu(suc)n(h)24 b(that)h(for)e(an)n(y)h(\()p Fq(x;)14 b(v)s(;)g(\022)r(;)g(\026;)g(")p Fu(\))24 b Fp(2)f Fq(U)9 b Fu(\()p Fq(\024)2903 3309 y Fj(\006)2903 3367 y Fo(1)2959 3345 y Fq(;)14 b(\024)3044 3357 y Fo(2)3081 3345 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))e Fp(\002)g Fq(P)g Fu(,)1521 3481 y Fq(w)r Fu(\()p Fp(T)22 b Fq(;)14 b(\022)r(;)g Fp(Y)7 b Fq(;)14 b(\026;)g(")p Fu(\))23 b(=)g(\()p Fq(x;)14 b(v)s Fu(\).)3380 3618 y Fe(\003)605 3770 y Fu(No)n(w)27 b(w)n(e)g(pro)n(v)n(e)f(that)i(the)g(\015o)n(w)f(can)g(b)r(e)h(straigh) n(tened)f(in)h Fq(U)9 b Fu(\()p Fq(\024)2618 3734 y Fj(\006)2618 3792 y Fo(1)2674 3770 y Fq(;)14 b(\024)2759 3782 y Fo(2)2796 3770 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\).)605 3922 y Ft(Pr)n(oposition)41 b Fu(4.2)p Ft(.)46 b Fs(L)l(et)39 b Fq(\024)1530 3887 y Fj(\000)1530 3945 y Fo(1)1625 3922 y Fq(>)g Fu(2)p Fq(\024)1819 3934 y Fo(0)1856 3922 y Fs(.)65 b(If)40 b(the)f(hyp)l(otheses)h Ff(H1)p Fs(,)i Ff(H2)d Fs(hold,)k(for)d(any)456 4024 y Fq(\024)504 3989 y Fo(+)504 4047 y(1)582 4024 y Fq(>)22 b(\024)717 3989 y Fj(\000)717 4047 y Fo(1)773 4024 y Fs(,)30 b Fq(\024)876 4036 y Fo(2)936 4024 y Fq(>)23 b Fu(0)p Fs(,)30 b(ther)l(e)g(exists)f Fq(r)d(>)c Fu(0)30 b Fs(and)g(a)g(change)h(of)f(variables)753 4199 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)26 b Fu(=)1116 4143 y Fq(t)p 1112 4180 39 4 v 1112 4256 a(")1161 4199 y Fu(\))d Fp(2)g Fq(U)9 b Fu(\()p Fq(\024)1440 4163 y Fj(\006)1440 4221 y Fo(1)1496 4199 y Fq(;)14 b(\024)1581 4211 y Fo(2)1618 4199 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))24 b Fp(7!)f Fu(\()p Fq(T)7 b(;)14 b(F)r(;)g(\022)r Fu(\))23 b(=)g(\()p Fp(T)f Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g Fp(F)8 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g(\022)r Fu(\))26 b Fp(2)3305 4178 y Fu(~)3292 4199 y Fq(V)456 4368 y Fs(analytic)31 b(in)e(the)h Fq(x)p Fs(,)h Fq(v)i Fs(variables,)f Fq(C)1620 4338 y Fo(1)1687 4368 y Fs(and)f Fu(2)p Fq(\031)s Fs(-p)l(erio)l(dic)g(in)f Fq(\022)r Fs(,)g(such)g(that)g(it)f(tr)l(ansforms)h(the)456 4468 y(system)f(\(4.3\))i(to)1772 4589 y Fu(_)1746 4610 y Fq(T)94 b Fu(=)83 b(1)1769 4714 y(_)1741 4735 y Fq(F)95 b Fu(=)83 b(0)1781 4837 y(_)1765 4859 y Fq(\022)i Fu(=)e(1)p Fq(=")456 4996 y Fs(and)31 b(satis\014es)h Fp(T)21 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))27 b(=)e Fp(T)1429 5008 y Fo(0)1467 4996 y Fu(\()p Fq(x;)14 b(v)s Fu(\))20 b(+)f Fq(O)r Fu(\()p Fq(\026")1948 4965 y Fi(q)1986 4996 y Fu(\))31 b Fs(,)h Fp(F)8 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))27 b(=)e Fp(F)2620 5008 y Fo(0)2657 4996 y Fu(\()p Fq(x;)14 b(v)s Fu(\))21 b(+)e Fq(O)r Fu(\()p Fq(\026")3139 4965 y Fi(q)3176 4996 y Fu(\))32 b Fs(wher)l(e)456 5095 y Fu(\()p Fq(x;)14 b(v)s Fu(\))24 b Fp(7!)f Fu(\()p Fp(T)854 5107 y Fo(0)892 5095 y Fu(\()p Fq(x;)14 b(v)s Fu(\))p Fq(;)g Fp(F)1180 5107 y Fo(0)1218 5095 y Fu(\()p Fq(x;)g(v)s Fu(\)\))29 b Fs(is)f(the)g(c)l(orr)l(esp)l(onding)h(change)f(for)h(the) f(unp)l(erturb)l(e)l(d)f(system)456 5195 y(and)j(is)g(given)g(in)g (\(4.6\))h(and)f(\(4.4\).)p eop %%Page: 54 62 54 61 bop 456 251 a Fo(54)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)34 b(\014x)g(\()p Fq(x;)14 b(v)s Fu(\))35 b Fp(2)g Fq(U)1579 462 y Fo(0)1615 450 y Fu(\()p Fq(\024)1695 415 y Fj(\006)1695 472 y Fo(1)1752 450 y Fq(;)14 b(\024)1837 462 y Fo(2)1874 450 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))34 b(and)g(w)n(e)g(consider)f(the)h(solution)g Fq( )s Fu(\()p Fq(t)p Fu(\))h(of)456 550 y(the)d(system)g(\(4.3\))f (suc)n(h)h(that)g Fq( )s Fu(\(0\))e(=)g(\()p Fq(x;)14 b(v)s(;)g Fu(0\).)51 b(By)31 b(Lemma)h(4.5,)g(there)g(exist)g Fp(T)21 b Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\))456 649 y(and)27 b Fp(Y)7 b Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\))29 b(suc)n(h)e(that)1331 797 y Fq(w)r Fu(\()p Fp(T)c Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\))p Fq(;)g Fu(0)p Fq(;)g Fp(Y)7 b Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\)\))24 b(=)e(\()p Fq(x;)14 b(v)s Fu(\).)456 950 y(Moreo)n(v)n(er)24 b(since)j(the)h(solutions)e(of)h(\(4.3\))g(can) g(b)r(e)h(parameterized)e(as)g(\(4.25\),)h(taking)f Fq(s)d Fu(=)g Fp(T)456 1050 y Fu(and)k Fq(Y)42 b Fu(=)22 b Fp(Y)35 b Fu(in)28 b(\(4.25\))f(w)n(e)g(obtain)h(that)1163 1183 y(~)1146 1205 y Fq( )s Fu(\()p Fq(t)p Fu(\))23 b Fp(\021)g Fu(\()p Fq(w)r Fu(\()p Fp(T)g Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\))19 b(+)f Fq(t;)c(t=";)g Fp(Y)7 b Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\)\))p Fq(;)g(t=")p Fu(\),)-2299 b(\(4.31\))456 1373 y(also)26 b(is)h(a)g(solution)g(of)g(\(4.3\))g(suc)n(h)g(that)1764 1351 y(~)1747 1373 y Fq( )s Fu(\(0\))c(=)g(\()p Fq(x;)14 b(v)s(;)g Fu(0\))24 b(=)e Fq( )s Fu(\(0\).)37 b(By)27 b(uniqueness,)3214 1351 y(~)3197 1373 y Fq( )f Fu(=)c Fq( )s Fu(.)456 1473 y(On)h(the)i(other)f(hand,)g(if)h Fq(t)f Fu(is)g(suc)n(h)g(that,)h Fq( )s Fu(\()p Fq(t)p Fu(\))f Fp(2)f Fq(U)9 b Fu(\()p Fq(\024)2161 1438 y Fj(\006)2161 1495 y Fo(1)2217 1473 y Fq(;)14 b(\024)2302 1485 y Fo(2)2339 1473 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\),)25 b(b)n(y)f(Lemma)g(4.5,)g (applying)456 1573 y(\(4.26\))i(with)j(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))24 b(=)f Fq( )s Fu(\()p Fq(t)p Fu(\),)28 b(w)n(e)f(obtain)1266 1721 y Fq( )s Fu(\()p Fq(t)p Fu(\))d(=)e(\()p Fq(w)r Fu(\()p Fp(T)h Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))p Fq(;)14 b(t=";)g Fp(Y)7 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\)\))p Fq(;)14 b(t=")p Fu(\).)-2177 b(\(4.32\))456 1874 y(Therefore)25 b(\(4.31\))g(and)i (\(4.32\))e(giv)n(e)h(us)g(t)n(w)n(o)g(expressions)e(for)i(the)h(same)f (solution)g Fq( )s Fu(\()p Fq(t)p Fu(\).)37 b(W)-7 b(e)456 1973 y(observ)n(e)25 b(that,)1525 2121 y Fp(T)1570 2133 y Fo(0)1608 2121 y Fu(\()p Fq(x;)14 b(v)s Fu(\))19 b(+)f Fq(t)23 b Fu(=)g Fp(T)2087 2133 y Fo(0)2124 2121 y Fu(\()p Fq( )2210 2133 y Fo(0)2248 2121 y Fu(\()p Fq(t)p Fu(\)\))456 2270 y(therefore,)g(b)n(y)h(the)g(uniqueness)g(of)g(the)g(functions)h Fp(T)45 b Fu(and)24 b Fp(Y)31 b Fu(giv)n(en)23 b(in)h(Lemma)g(4.5,)g(w) n(e)g(ha)n(v)n(e)1459 2418 y Fp(T)e Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))84 b(=)e Fp(T)22 b Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\))19 b(+)f Fq(t)-1985 b Fu(\(4.33\))1463 2542 y Fp(Y)7 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))85 b(=)d Fp(Y)7 b Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\))456 2690 y(and)27 b(then)1640 2804 y Fq(d)p 1625 2842 74 4 v 1625 2918 a(dt)1708 2861 y Fp(T)21 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))84 b(=)f(1)1644 3008 y Fq(d)p 1629 3046 V 1629 3122 a(dt)1712 3065 y Fp(Y)7 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))84 b(=)f(0.)456 3236 y(W)-7 b(e)28 b(de\014ne)f(a)h(new) f(function)1442 3384 y Fp(F)8 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))25 b(=)d Fp(F)1951 3396 y Fo(0)1988 3384 y Fu(\()p Fq(\016)o(;)14 b Fp(Y)7 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\)\))456 3537 y(where)30 b Fp(F)759 3549 y Fo(0)826 3537 y Fu(is)h(the)g(Hamiltonian)f(of)h(the)g(unp)r(erturb)r(ed)g (system)g(giv)n(en)f(in)h(\(4.4\).)45 b(W)-7 b(e)31 b(recall)456 3636 y(that)1550 3784 y Fp(kY)26 b(\000)18 b(Y)1811 3796 y Fo(0)1848 3784 y Fp(k)1890 3796 y Fj(1)1983 3784 y Fu(=)23 b Fq(O)r Fu(\()p Fq(\026")2257 3750 y Fi(q)2294 3784 y Fu(\),)456 3933 y(then,)28 b(since)f Fp(F)931 3945 y Fo(0)996 3933 y Fu(is)g(constan)n(t)g(along)f(the)i(tra)5 b(jectories)26 b(of)i(the)g(unp)r(erturb)r(ed)g(system,)1228 4081 y Fp(F)8 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))84 b(=)f Fp(F)1857 4093 y Fo(0)1893 4081 y Fu(\()p Fq(\016)o(;)14 b Fp(Y)2053 4093 y Fo(0)2091 4081 y Fu(\()p Fq(x;)g(v)s Fu(\)\))20 b(+)e Fq(O)r Fu(\()p Fq(\026")2603 4046 y Fi(q)2640 4081 y Fu(\))1649 4205 y(=)83 b Fp(F)1857 4217 y Fo(0)1893 4205 y Fu(\()p Fq(x;)14 b(v)s Fu(\))20 b(+)e Fq(O)r Fu(\()p Fq(\026")2373 4171 y Fi(q)2410 4205 y Fu(\).)456 4353 y(Therefore,)26 b(from)h(\(4.33\),)g(it)h(is)g(easily)e (seen)i(that)1322 4501 y(\()p Fq(T)7 b(;)14 b(F)r(;)g(\022)r Fu(\))24 b(=)e(\()p Fp(T)g Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g Fp(F)8 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g(\022)r Fu(\))456 4663 y(transforms)26 b(\(4.3\))h(in)h Fq(U)9 b Fu(\()p Fq(\024)1311 4628 y Fj(\006)1311 4685 y Fo(1)1367 4663 y Fq(;)14 b(\024)1452 4675 y Fo(2)1489 4663 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))28 b(to)1772 4798 y(_)1746 4819 y Fq(T)94 b Fu(=)83 b(1)1769 4922 y(_)1741 4943 y Fq(F)95 b Fu(=)83 b(0)1781 5046 y(_)1765 5068 y Fq(\022)i Fu(=)e(1)p Fq(=")456 5216 y Fu(and)27 b(the)h(statemen)n(t)g(holds.)2021 b Fe(\003)p eop %%Page: 55 63 55 62 bop 1006 251 a Fo(4.5.)30 b(FLO)n(W)f(BO)n(X)g(COORDINA)-5 b(TES)29 b(IN)g(A)g(COMPLEX)g(DOMAIN)484 b(55)605 450 y Fu(No)n(w)28 b(w)n(e)g(turn)h(to)g(mo)r(dify)g(the)g(c)n(hange)e(of)i (v)-5 b(ariables)27 b(to)i(get)f(a)g(canonical)g(one.)39 b(Before)456 550 y(starting)26 b(the)i(result)g(w)n(e)f(need)h(some)f (preliminary)f(calculations.)605 649 y(W)-7 b(e)34 b(denote)f(b)n(y)g Fq( )s Fu(\()p Fq(t;)14 b(x;)g(v)s Fu(\))34 b(the)f(solution)g(of)g (the)h(system)f(\(4.3\))g(suc)n(h)f(that)i(\()p Fq(x;)14 b(v)s(;)g Fu(0\))33 b(=)456 749 y Fq( )s Fu(\(0\).)k(In)27 b(the)h(pro)r(of)f(of)h(the)g(Prop)r(osition)e(4.2,)h(concretely)f(in)i (\(4.33\),)f(w)n(e)g(ha)n(v)n(e)g(seen)g(that)1436 895 y Fp(T)22 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))i(=)e Fp(T)g Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\))19 b(+)f Fq(t)-1842 b Fu(\(4.34\))1541 1019 y Fp(F)8 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))25 b(=)d Fp(F)8 b Fu(\()p Fq(x;)14 b(v)s(;)g Fu(0\).)456 1165 y(W)-7 b(e)28 b(in)n(tro)r(duce)f(the)h(matrix)1286 1340 y(\010\()p Fq(t)p Fu(\))c(=)1551 1223 y Fg(\022)1655 1290 y Fq(@)1699 1302 y Fi(x)1741 1290 y Fp(T)d Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))86 b Fq(@)2152 1302 y Fi(v)2191 1290 y Fp(T)21 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))1654 1389 y Fq(@)1698 1401 y Fi(x)1740 1389 y Fp(F)8 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))84 b Fq(@)2151 1401 y Fi(v)2190 1389 y Fp(F)8 b Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))2516 1223 y Fg(\023)2591 1340 y Fu(.)456 1536 y(Di\013eren)n(tiating)27 b(with)h(resp)r(ect)g(to)f(\()p Fq(x;)14 b(v)s Fu(\))29 b(in)f(b)r(oth)g(sides)f(of)h(\(4.35\),)f(w)n(e)g(obtain)1301 1732 y(\010\()p Fq(t)p Fu(\))1469 1615 y Fg(\022)1572 1681 y Fq(@)1616 1693 y Fi(x)1658 1681 y Fq( )1712 1693 y Fo(1)1749 1681 y Fu(\()p Fq(t)p Fu(\))84 b Fq(@)1971 1693 y Fi(v)2010 1681 y Fq( )2064 1693 y Fo(1)2102 1681 y Fu(\()p Fq(t)p Fu(\))1572 1781 y Fq(@)1616 1793 y Fi(x)1658 1781 y Fq( )1712 1793 y Fo(2)1749 1781 y Fu(\()p Fq(t)p Fu(\))g Fq(@)1971 1793 y Fi(v)2010 1781 y Fq( )2064 1793 y Fo(2)2102 1781 y Fu(\()p Fq(t)p Fu(\))2238 1615 y Fg(\023)2322 1732 y Fu(=)23 b(\010\(0\).)456 1927 y(Since)k(system)h(\(4.3\))f(is)h (Hamiltonian,)1395 2123 y(det)1524 2006 y Fg(\022)1626 2072 y Fq(@)1670 2084 y Fi(x)1712 2072 y Fq( )1766 2084 y Fo(1)1804 2072 y Fu(\()p Fq(t)p Fu(\))83 b Fq(@)2025 2084 y Fi(v)2065 2072 y Fq( )2119 2084 y Fo(1)2156 2072 y Fu(\()p Fq(t)p Fu(\))1626 2172 y Fq(@)1670 2184 y Fi(x)1712 2172 y Fq( )1766 2184 y Fo(2)1804 2172 y Fu(\()p Fq(t)p Fu(\))g Fq(@)2025 2184 y Fi(v)2065 2172 y Fq( )2119 2184 y Fo(2)2156 2172 y Fu(\()p Fq(t)p Fu(\))2292 2006 y Fg(\023)2376 2123 y Fu(=)23 b(1)456 2314 y(and)k(therefore)1605 2460 y(det)14 b(\010\()p Fq(t)p Fu(\))24 b(=)f(det)14 b(\010\(0\))-1839 b(\(4.35\))456 2606 y(for)37 b(all)h Fq(t)g Fu(for)g(whic)n(h)g(the)g (solution)g(is)g(de\014ned.)69 b(Moreo)n(v)n(er,)38 b(w)n(e)g(kno)n(w)f (that)h(for)g Fq(\026)j Fu(=)f(0,)456 2706 y(det)14 b(\010\()p Fq(t)p Fu(\))23 b(=)g(det)14 b(\010\(0\))23 b(=)g(1,)k(th)n(us)1489 2852 y(det)14 b(\010\()p Fq(t)p Fu(\))24 b(=)f(1)18 b(+)g Fq(\026")2116 2818 y Fi(q)2155 2852 y Fu(~)-45 b Fq(g)s Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))456 3008 y(where)35 b(~)-44 b Fq(g)35 b Fu(=)g(~)-45 b Fq(g)s Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))35 b(is)e(some)g Fq(C)1589 2978 y Fo(1)1660 3008 y Fu(function,)j(analytic)d(in)g(\()p Fq(x;)14 b(v)s Fu(\))35 b(and)e(2)p Fq(\031)s Fu(-p)r(erio)r(dic)g(in)h Fq(\022)r Fu(.)456 3108 y(Moreo)n(v)n(er)24 b(from)k(\(4.35\))f(it)h (is)f(clear)g(that)1711 3241 y Fq(d)p 1696 3278 74 4 v 1696 3354 a(dt)1782 3297 y Fu(~)-44 b Fq(g)r Fu(\()p Fq( )s Fu(\()p Fq(t)p Fu(\)\))24 b(=)f(0.)-1758 b(\(4.36\))456 3466 y(W)-7 b(e)28 b(de\014ne)f(the)h(function)h Fq(g)c Fu(:)e Fq(U)1475 3478 y Fi(g)1537 3466 y Fp(!)g Fl(C)15 b Fu(,)33 b(b)n(y)1106 3626 y Fq(g)s Fu(\()p Fq(T)7 b(;)14 b(F)r(;)g(\022)r Fu(\))24 b(=)h(~)-45 b Fq(g)s Fu(\()p Fq(w)r Fu(\()p Fq(T)7 b(;)14 b(\022)r(;)g(f)9 b Fu(\()p Fq(\016)s Fu(\))19 b Fp(\000)2145 3551 y Fg(p)p 2228 3551 400 4 v 75 x Fq(f)2278 3602 y Fo(2)2315 3626 y Fu(\()p Fq(\016)s Fu(\))g(+)f(2)p Fq(F)12 b Fu(\))p Fq(;)i(\022)r Fu(\).)456 3782 y(The)33 b(function)g Fq(g)j Fu(is)d Fq(C)1192 3752 y Fo(1)1229 3782 y Fu(,)i(analytic)d(in)h(\()p Fq(T)7 b(;)14 b(F)e Fu(\))33 b(and)g(2)p Fq(\031)s Fu(-p)r(erio)r(dic)f (in)h Fq(\022)r Fu(.)54 b(If)33 b(w)n(e)g(di\013eren)n(tiate)456 3889 y(with)22 b(resp)r(ect)f(to)g(the)h(time,)h Fq(t)p Fu(,)g(in)e Fq(g)k Fu(ev)-5 b(aluated)21 b(on)g(the)h(solutions)e(of) 2652 3868 y(_)2627 3889 y Fq(T)34 b Fu(=)22 b(1,)2913 3868 y(_)2885 3889 y Fq(F)35 b Fu(=)22 b(0,)3164 3867 y(_)3148 3889 y Fq(\022)j Fu(=)e(1)p Fq(=")p Fu(,)456 3988 y(b)n(y)k(\(4.36\),)g(w)n(e)g(get)g(the)h(follo)n(wing)f(equalit)n (y:)1649 4173 y(0)c(=)g Fq(@)1846 4185 y Fi(T)1898 4173 y Fq(g)e Fu(+)2052 4117 y(1)p 2052 4154 42 4 v 2053 4230 a Fq(")2103 4173 y(@)2147 4185 y Fi(\022)2185 4173 y Fq(g)s Fu(.)-1795 b(\(4.37\))456 4347 y(T)-7 b(o)32 b(deal)g(with)h (equation)f(\(4.37\),)g(w)n(e)g(de\014ne)h(the)g(c)n(hange)e(\()p Fq(\030)t(;)14 b(\021)s Fu(\))32 b(=)f(\()p Fq(T)i Fu(+)21 b Fq("\022)r(;)14 b(T)33 b Fp(\000)21 b Fq("\022)r Fu(\))33 b(and)456 4447 y(the)28 b(function)1256 4593 y Fq(h)p Fu(\()p Fq(\030)t(;)14 b(\021)s(;)g(F)e Fu(\))23 b(=)g Fq(g)s Fu(\(\()p Fq(\030)g Fu(+)18 b Fq(\021)s Fu(\))p Fq(=)p Fu(2)p Fq(;)c(F)r(;)g Fu(\()p Fq(\030)23 b Fp(\000)18 b Fq(\021)s Fu(\))p Fq(=)p Fu(2)p Fq(")p Fu(\))456 4739 y(de\014ned)28 b(in)1071 4885 y Fp(f)p Fu(\()p Fq(\030)t(;)14 b(\021)s(;)g(F)e Fu(\))23 b Fp(2)h Fl(C)1556 4850 y Fo(3)1622 4885 y Fu(:)f(\(\()p Fq(\030)g Fu(+)18 b Fq(\021)s Fu(\))p Fq(=)p Fu(2)p Fq(;)c(F)r(;)g Fu(\()p Fq(\030)23 b Fp(\000)18 b Fq(\021)s Fu(\))p Fq(=)p Fu(2)p Fq(")p Fu(\))23 b Fp(2)g Fq(U)2726 4897 y Fi(g)2764 4885 y Fp(g)p Fu(.)456 5036 y(Then,)k(b)n(y)h(\(4.37\))1810 5182 y Fq(@)1854 5194 y Fi(\030)1890 5182 y Fq(h)23 b Fu(=)g(0)p eop %%Page: 56 64 56 63 bop 456 251 a Fo(56)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(therefore,)1038 601 y Fq(g)s Fu(\()p Fq(T)7 b(;)14 b(F)r(;)g(\022)r Fu(\))24 b(=)f Fq(h)p Fu(\()p Fq(T)29 b Fu(+)18 b Fq("\022)r(;)c(T)30 b Fp(\000)18 b Fq("\022)r(;)c(F)e Fu(\))23 b(=)f Fq(h)p Fu(\(0)p Fq(;)14 b(T)29 b Fp(\000)18 b Fq("\022)r(;)c(F)e Fu(\))456 758 y(is)25 b(a)f(function)i(that)f (only)g(dep)r(ends)h(on)e(\014rst)h(in)n(tegrals)f(of)h(the)h(system)e (\(4.3\).)36 b(W)-7 b(e)26 b(de\014ne)f(the)456 857 y(function)j Fq(\032)p Fu(\()p Fq(F)r(;)14 b(S)5 b Fu(\))28 b(b)n(y)g(the)g (condition)1318 1048 y(1)18 b(+)g Fq(@)1505 1060 y Fi(F)1560 1048 y Fq(\032)p Fu(\()p Fq(F)r(;)c(S)5 b Fu(\))24 b(=)2222 992 y(1)p 1937 1029 613 4 v 1937 1105 a(1)18 b(+)g Fq(\026")2169 1081 y Fi(q)2205 1105 y Fq(h)p Fu(\(0)p Fq(;)c(S;)g(F)e Fu(\))2559 1048 y(.)456 1248 y(W)-7 b(e)28 b(remark)e(that,)i(since)f Fq(h)h Fu(is)f(analytic)g(in)h Fq(F)39 b Fu(and)28 b Fq(S)5 b Fu(,)27 b(the)h(function)g Fq(\032)g Fu(is)g Fq(O)r Fu(\()p Fq(\026")3030 1218 y Fi(q)3067 1248 y Fu(\).)605 1407 y Ft(Pr)n(oposition)j Fu(4.3)p Ft(.)40 b Fs(The)31 b(change)g(of)f(variables)i(de\014ne)l(d)e(by)742 1558 y Fu(\()p Fq(x;)14 b(v)s(;)g(\022)26 b Fu(=)d Fq(t=")p Fu(\))f Fp(2)i Fq(U)9 b Fu(\()p Fq(\024)1481 1522 y Fj(\006)1481 1580 y Fo(1)1537 1558 y Fq(;)14 b(\024)1622 1570 y Fo(2)1659 1558 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))23 b Fp(!)g Fu(\()p Fq(T)7 b(;)14 b(I)7 b(;)14 b(\022)r Fu(\))23 b(=)g(\()p Fp(T)e Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g Fp(I)6 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g(\022)r Fu(\))27 b Fp(2)c Fq(V)456 1714 y Fs(with)33 b Fp(I)6 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))31 b(=)e Fp(F)8 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))22 b(+)f Fq(\032)p Fu(\()p Fp(F)8 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))p Fq(;)g Fp(T)22 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))22 b Fp(\000)f Fq(\022)r(")p Fu(\))33 b Fs(is)h(c)l(anonic)l(al)g(and)g (is)f(such)456 1814 y(that)c(tr)l(ansforms)h(the)g(system)g(\(4.3\))h (to)1826 1950 y Fu(_)1800 1971 y Fq(T)k Fu(=)22 b(1)1820 2056 y(_)1800 2077 y Fq(I)30 b Fu(=)23 b(0)1816 2163 y(_)1800 2185 y Fq(\022)j Fu(=)c(1)p Fq(=")p Fs(.)456 2337 y(Mor)l(e)l(over,)33 b Fp(T)21 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))26 b(=)f Fp(T)1344 2349 y Fo(0)1381 2337 y Fu(\()p Fq(x;)14 b(v)s Fu(\))21 b(+)d Fq(O)r Fu(\()p Fq(\026")1862 2307 y Fi(q)1900 2337 y Fu(\))31 b Fs(and)g Fp(I)6 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))27 b(=)d Fp(F)2621 2349 y Fo(0)2658 2337 y Fu(\()p Fq(x;)14 b(v)s Fu(\))20 b(+)f Fq(O)r Fu(\()p Fq(\026")3139 2307 y Fi(q)3177 2337 y Fu(\))31 b Fs(wher)l(e)456 2436 y Fp(T)501 2448 y Fo(0)568 2436 y Fs(and)f Fp(F)789 2448 y Fo(0)856 2436 y Fs(is)g(the)g(c)l(orr)l(esp)l(onding)h(change)g(in)e(the)h(unp)l (erturb)l(e)l(d)f(c)l(ase.)605 2595 y Ft(Pr)n(oof.)41 b Fu(Let)24 b Fp(I)29 b Fu(=)23 b Fp(I)6 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))25 b(b)r(e)f(as)f(in)h(the)g(statemen)n(t.)35 b(Along)23 b(the)h(solutions)f(of)h(\(4.3\),)1109 2732 y(_)1090 2753 y Fq(I)30 b Fu(=)1271 2732 y(_)1244 2753 y Fq(F)g Fu(+)18 b Fq(D)1479 2765 y Fi(F)1534 2753 y Fq(\032)p Fu(\()p Fq(F)r(;)c(S)5 b Fu(\))1818 2732 y(_)1789 2753 y Fq(F)31 b Fu(+)18 b Fq(D)2025 2765 y Fi(S)2073 2753 y Fq(\032)p Fu(\()p Fq(F)r(;)c(S)5 b Fu(\)\()2386 2732 y(_)2360 2753 y Fq(T)31 b Fp(\000)18 b Fq(")2577 2732 y Fu(_)2562 2753 y Fq(\022)r Fu(\))23 b(=)g(0.)456 2925 y(Th)n(us,)k(this)h(c)n(hange)e(transforms)h(the)g(system)h (\(4.3\))f(to)2283 2904 y(_)2258 2925 y Fq(T)34 b Fu(=)22 b(1,)2540 2904 y(_)2521 2925 y Fq(I)30 b Fu(=)22 b(1,)2783 2903 y(_)2767 2925 y Fq(\022)j Fu(=)e(1)p Fq(=")p Fu(.)605 3024 y(T)-7 b(o)28 b(see)g(that)h(the)g(c)n(hange)e(is)h(canonical)f(w) n(e)i(only)f(ha)n(v)n(e)f(to)h(calculate)g(the)h(determinan)n(t)456 3124 y(of)1413 3304 y Fq(C)6 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))25 b(=)1860 3187 y Fg(\022)1962 3254 y Fq(@)2006 3266 y Fi(x)2048 3254 y Fp(T)104 b Fq(@)2241 3266 y Fi(v)2281 3254 y Fp(T)1970 3353 y Fq(@)2014 3365 y Fi(x)2056 3353 y Fp(I)g Fq(@)2249 3365 y Fi(v)2288 3353 y Fp(I)2389 3187 y Fg(\023)2464 3304 y Fu(.)456 3501 y(W)-7 b(e)28 b(ha)n(v)n(e)571 3652 y(det)14 b Fq(C)6 b Fu(\()p Fq(x;)14 b(v)s(;)g(\022)r Fu(\))85 b(=)d Fq(@)1310 3664 y Fi(x)1352 3652 y Fp(T)22 b Fq(@)1463 3664 y Fi(v)1502 3652 y Fp(I)6 b(\000)p Fq(@)1662 3664 y Fi(v)1702 3652 y Fp(T)21 b Fq(@)1812 3664 y Fi(x)1854 3652 y Fp(I)1119 3777 y Fu(=)82 b([)p Fq(@)1333 3789 y Fi(v)1373 3777 y Fp(F)26 b Fu(+)18 b Fq(@)1586 3789 y Fi(F)1642 3777 y Fq(\032)p Fu(\()p Fp(F)8 b Fq(;)14 b Fp(T)39 b(\000)18 b Fq(\022)r(")p Fu(\))p Fq(@)2145 3789 y Fi(v)2185 3777 y Fp(F)27 b Fu(+)18 b Fq(@)2399 3789 y Fi(S)2447 3777 y Fq(\032)p Fu(\()p Fp(F)8 b Fq(;)14 b Fp(T)40 b(\000)18 b Fq(\022)r(")p Fu(\))p Fq(@)2951 3789 y Fi(v)2990 3777 y Fp(T)k Fu(])14 b Fq(@)3138 3789 y Fi(x)3180 3777 y Fp(T)1266 3901 y(\000)g Fu([)p Fq(@)1412 3913 y Fi(x)1454 3901 y Fp(F)26 b Fu(+)18 b Fq(@)1667 3913 y Fi(F)1722 3901 y Fq(\032)p Fu(\()p Fp(F)8 b Fq(;)14 b Fp(T)40 b(\000)18 b Fq(\022)r(")p Fu(\))p Fq(@)2226 3913 y Fi(x)2268 3901 y Fp(F)27 b Fu(+)18 b Fq(@)2482 3913 y Fi(S)2530 3901 y Fq(\032)p Fu(\()p Fp(F)8 b Fq(;)14 b Fp(T)40 b(\000)18 b Fq(\022)r(")p Fu(\))p Fq(@)3034 3913 y Fi(x)3076 3901 y Fp(T)j Fu(])14 b Fq(@)3223 3913 y Fi(v)3263 3901 y Fp(T)1119 4026 y Fu(=)82 b(\()p Fq(@)1342 4038 y Fi(v)1382 4026 y Fp(F)8 b Fq(@)1494 4038 y Fi(x)1536 4026 y Fp(T)40 b(\000)18 b Fq(@)1748 4038 y Fi(x)1790 4026 y Fp(F)8 b Fq(@)1902 4038 y Fi(v)1941 4026 y Fp(T)21 b Fu(\)\(1)e(+)f Fq(@)2259 4038 y Fi(F)2314 4026 y Fq(\032)p Fu(\()p Fp(F)8 b Fq(;)14 b Fp(T)40 b(\000)18 b Fq(\022)r(")p Fu(\)\))1119 4150 y(=)82 b(\(1)19 b(+)f Fq(\026")1531 4116 y Fi(q)1567 4150 y Fq(h)p Fu(\(0)p Fq(;)c Fp(T)39 b(\000)18 b Fq(\022)r(";)c Fp(F)8 b Fu(\)\)\(1)19 b(+)f Fq(@)2362 4162 y Fi(F)2417 4150 y Fq(\032)p Fu(\()p Fp(F)8 b Fq(;)14 b Fp(T)40 b(\000)18 b Fq(\022)r(")p Fu(\)\))1119 4275 y(=)82 b(1.)3380 4426 y Fe(\003)605 4600 y Fu(No)n(w)27 b(w)n(e)g(turn)h(to)g(the)g(system)f (in)h(the)g(original)e(v)-5 b(ariables)26 b(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\).)38 b(W)-7 b(e)28 b(de\014ne)1414 4752 y Fp(T)1480 4718 y Fo(1)1518 4752 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))83 b(=)g Fp(T)21 b Fu(\()p Fq(C)6 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\)\))1429 4878 y Fp(I)1480 4843 y Fo(1)1518 4878 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r Fu(\))83 b(=)g Fp(I)6 b Fu(\()p Fq(C)g Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\)\))456 5029 y(where)27 b Fq(C)34 b Fu(de\014ned)28 b(in)f(\(4.2\).)37 b(It)28 b(is)f(clear)g(that)h(the) g(c)n(hange)819 5182 y(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))23 b Fp(2)h Fq(V)19 b Fu(\()p Fq(\024)1338 5146 y Fj(\006)1338 5204 y Fo(1)1394 5182 y Fq(;)14 b(\024)1479 5194 y Fo(2)1516 5182 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))23 b Fp(7!)g Fu(\()p Fq(T)7 b(;)14 b(I)7 b(;)14 b(\022)r Fu(\))23 b(=)g(\()p Fp(T)2320 5147 y Fo(1)2357 5182 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))p Fq(;)g Fp(I)2715 5147 y Fo(1)2753 5182 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r Fu(\))p Fq(;)g(\022)r Fu(\))25 b Fp(2)e(V)p eop %%Page: 57 65 57 64 bop 1375 251 a Fo(4.6.)29 b(PR)n(OOF)g(OF)g(THE)g(THEOREM)g(4.2) 853 b(57)456 450 y Fu(is)27 b(canonical,)g(since)g(it)h(is)f(the)h (comp)r(osition)f(of)h(t)n(w)n(o)f(canonical)f(c)n(hanges.)36 b(Moreo)n(v)n(er)1809 582 y(_)1784 603 y Fq(T)94 b Fu(=)83 b(1)1820 706 y(_)1801 727 y Fq(I)90 b Fu(=)83 b(0)456 879 y(and)1193 1030 y Fp(T)1259 996 y Fo(1)1296 1030 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))84 b(=)f Fp(T)1843 1042 y Fo(0)1880 1030 y Fu(\()p Fq(x;)14 b(y)22 b Fp(\000)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\)\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")2623 996 y Fi(q)2661 1030 y Fu(\))1208 1156 y Fp(I)1259 1122 y Fo(1)1297 1156 y Fu(\()p Fq(x;)c(v)s(;)g(\022)r Fu(\))84 b(=)f Fp(I)1843 1168 y Fo(0)1880 1156 y Fu(\()p Fq(x;)14 b(y)22 b Fp(\000)c Fq(f)9 b Fu(\()p Fq(x)p Fu(\)\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")2623 1122 y Fi(q)2661 1156 y Fu(\))1650 1280 y(=)83 b Fp(F)1858 1292 y Fo(0)1895 1280 y Fu(\()p Fq(x;)14 b(y)21 b Fp(\000)d Fq(f)9 b Fu(\()p Fq(x)p Fu(\)\))20 b(+)e Fq(O)r Fu(\()p Fq(\026")2638 1246 y Fi(q)2675 1280 y Fu(\))456 1432 y(where)29 b Fp(T)743 1444 y Fo(0)780 1432 y Fu(\()p Fq(x;)14 b(y)23 b Fp(\000)d Fq(f)9 b Fu(\()p Fq(x)p Fu(\)\))30 b(and)f Fp(F)1491 1444 y Fo(0)1528 1432 y Fu(\()p Fq(x;)14 b(y)23 b Fp(\000)d Fq(f)9 b Fu(\()p Fq(x)p Fu(\)\))27 b(=)f Fq(h)2152 1444 y Fo(0)2189 1432 y Fu(\()p Fq(x;)14 b(y)s Fu(\))30 b(is)g(the)g(c)n (hange)e(when)i Fq(\026)c Fu(=)g(0)j(in)456 1531 y(the)f(original)e(v) -5 b(ariables.)35 b(This)28 b(ends)f(the)h(pro)r(of)f(of)h(Theorem)f (4.1.)1319 1749 y Fk(4.6.)46 b(Pro)s(of)32 b(of)g(the)g(Theorem)e(4.2) 605 1899 y Fu(W)-7 b(e)40 b(consider)e(the)i(a)n(v)n(eraged)d(system)i (obtained)g(in)h(Prop)r(osition)e(3.1)g(in)i(Chapter)f(3)456 1998 y(whose)27 b(equation)g(is:)824 2150 y(_)-38 b Fq(x)84 b Fu(=)f Fq(y)21 b Fu(+)d Fq(\026")1322 2115 y Fi(p)p Fo(+3)1444 2150 y Fq(@)1488 2162 y Fi(y)1528 2150 y Fq(F)1581 2162 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1744 2150 y Fu(\()p Fq(x;)c(y)s(;)g(\022)r Fu(\))20 b(+)e Fq(\026)2167 2115 y Fo(2)2204 2150 y Fq(")2243 2115 y Fi(p)p Fo(+2)2365 2150 y Fq(@)2409 2162 y Fi(y)2449 2150 y Fq(R)2512 2162 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2671 2150 y Fu(\()p Fq(x;)c(y)s Fu(\))828 2278 y(_)-38 b Fq(y)86 b Fu(=)d Fp(\000)p Fq(V)1219 2244 y Fj(0)1242 2278 y Fu(\()p Fq(x)p Fu(\))20 b Fp(\000)e Fq(\026")1545 2244 y Fi(p)p Fo(+3)1667 2278 y Fq(@)1711 2290 y Fi(x)1753 2278 y Fq(F)1806 2290 y Fo(2)p Fi(n)p Fj(\000)p Fo(2)1969 2278 y Fu(\()p Fq(x;)c(y)s(;)g(\022)r Fu(\))20 b Fp(\000)e Fq(\026)2392 2244 y Fo(2)2429 2278 y Fq(")2468 2244 y Fi(p)p Fo(+2)2590 2278 y Fq(@)2634 2290 y Fi(x)2676 2278 y Fq(R)2739 2290 y Fo(2)p Fi(k)q Fj(\000)p Fo(2)2898 2278 y Fu(\()p Fq(x;)c(y)s Fu(\))-2634 b(\(4.38\))832 2388 y(_)816 2410 y Fq(\022)85 b Fu(=)e(1)p Fq(=")p Fu(.)456 2561 y(W)-7 b(e)28 b(denote)f(b)n(y)1376 2715 y(~)-45 b Fq(\015)1421 2680 y Fi(s)1416 2735 y(\026;")1512 2715 y Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b(=)g(\()8 b(~)-50 b Fq(\013)1878 2680 y Fi(s)1878 2735 y(\026;")1974 2715 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)2193 2693 y Fu(~)2181 2715 y Fq(\014)2232 2680 y Fi(s)2228 2735 y(\026;")2324 2715 y Fu(\()p Fq(t;)g(s)p Fu(\)\))456 2866 y(the)28 b(stable)f(curv)n(e)g(of)g(this)h(system)f(and)h(b)n(y)1364 3018 y Fq(\015)1407 3030 y Fo(0)1444 3018 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))23 b(=)g(\()p Fq(\013)1875 3030 y Fo(0)1913 3018 y Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\))p Fq(;)c(\014)2231 3030 y Fo(0)2269 3018 y Fu(\()p Fq(t)k Fu(+)g Fq(s)p Fu(\)\))456 3170 y(the)28 b(homo)r(clinic)f(orbit)g(of)h (the)g(unp)r(erturb)r(ed)g(system.)605 3269 y(W)-7 b(e)27 b(\014x)f Fq(T)34 b(>)23 b Fu(0)i(big)h(enough,)g(then)h(for)e Fq(\024)1907 3234 y Fj(\000)1907 3291 y Fo(1)1986 3269 y Fu(=)e Fq(T)k Fp(\000)15 b Fu(1,)26 b Fq(\024)2369 3234 y Fo(+)2369 3291 y(1)2447 3269 y Fu(=)c(2)p Fq(T)27 b Fu(+)15 b(1)25 b(and)h Fq(\024)3007 3281 y Fo(2)3068 3269 y Fu(=)c(3)p Fq(a=)p Fu(2)j(w)n(e)456 3369 y(ha)n(v)n(e)h(that) 1341 3520 y Fq(\015)1384 3532 y Fo(0)1422 3520 y Fu(\()p Fq(t)18 b Fu(+)g Fq(s)p Fu(\))24 b Fp(2)f Fq(V)1806 3532 y Fo(0)1844 3520 y Fu(\()p Fq(\024)1924 3485 y Fj(\006)1924 3542 y Fo(1)1998 3520 y Fp(\007)18 b Fu(1)p Fq(;)c Fu(2)p Fq(\024)2250 3532 y Fo(2)2286 3520 y Fq(=)p Fu(3)p Fq(;)g Fu(0)p Fq(;)g Fu(0\))456 3677 y(for)27 b Fq(T)34 b Fp(\024)22 b Fq(t)d Fu(+)f(Re)c Fq(s)23 b Fp(\024)f Fu(2)p Fq(T)39 b Fu(and)27 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)g Fq(a)p Fu(.)605 3776 y(W)-7 b(e)31 b(observ)n(e)d(that)i(the)h(a)n(v)n (eraged)c(system)j(\(4.38\))f(satis\014es)h(the)g(h)n(yp)r(otheses)g Fk(H1)p Fu(,)g Fk(H2)p Fu(,)456 3876 y(of)h(Theorem)g(4.1)g(with)i Fq(q)g Fu(=)c Fq(p)21 b Fu(+)g(2,)33 b(therefore)e(there)g(exists)h Fq(r)g(<)e(r)2656 3888 y Fo(0)2726 3876 y Fu(small)h(enough)g(and)h(a) 456 3975 y(canonical)22 b(c)n(hange)h(of)g(v)-5 b(ariables,)23 b(de\014ned)i(in)e(the)i(set)e Fq(V)c Fu(\()p Fq(\024)2328 3940 y Fj(\006)2328 3998 y Fo(1)2384 3975 y Fq(;)14 b(\024)2469 3987 y Fo(2)2506 3975 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\),)25 b(whic)n(h)f(w)n(e)f(denote)h(b)n(y)456 4075 y(\()p Fq(T)7 b(;)14 b(I)7 b Fu(\))23 b(=)f(\()p Fp(T)865 4045 y Fo(1)902 4075 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r Fu(\))p Fq(;)g Fp(I)1260 4045 y Fo(1)1298 4075 y Fu(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\)\),)28 b(suc)n(h)f(that)h (transforms)e(the)i(system)g(\(4.38\))f(to)1798 4211 y(_)1772 4232 y Fq(T)94 b Fu(=)83 b(1)1809 4336 y(_)1789 4357 y Fq(I)90 b Fu(=)83 b(0.)456 4508 y(Moreo)n(v)n(er,)24 b Fp(T)910 4478 y Fo(1)947 4508 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r Fu(\))24 b(=)e Fp(T)1373 4520 y Fo(0)1411 4508 y Fu(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r Fu(\))18 b(+)g Fq(O)r Fu(\()p Fq(\026")1890 4478 y Fi(p)p Fo(+2)2013 4508 y Fu(\))28 b(and)f Fp(I)2285 4478 y Fo(1)2323 4508 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s(;)14 b(\022)r Fu(\))24 b(=)e Fp(I)2749 4520 y Fo(0)2787 4508 y Fu(\()5 b(\026)-47 b Fq(x;)20 b Fu(\026)-48 b Fq(y)s Fu(\))18 b(+)g Fq(O)r Fu(\()p Fq(\026")3266 4478 y Fi(p)p Fo(+2)3389 4508 y Fu(\).)605 4608 y(W)-7 b(e)20 b(m)n(ust)g(see)g(that)g(the)g (parametric)e(represen)n(tation)h(of)g(the)i(stable)e(manifold)h(of)g (system)456 4708 y(\(4.38\))26 b(en)n(ters)h(the)h(domain)f(of)h (analyticit)n(y)f(of)g(\()p Fp(T)2077 4677 y Fo(1)2114 4708 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))p Fq(;)g Fp(I)2472 4677 y Fo(1)2510 4708 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r Fu(\)\).)605 4807 y(Using)33 b(the)h(ab)r(o)n(v)n(e)e(de\014nition,)j (it)f(is)f(clear)g(that,)i(b)n(y)e(con)n(tin)n(uit)n(y)-7 b(,)34 b(for)f(\()p Fq(t;)14 b(s)p Fu(\))34 b(suc)n(h)f(that)456 4907 y Fq(T)h Fp(\024)22 b Fq(t)d Fu(+)f(Re)c Fq(s)23 b Fp(\024)f Fu(2)p Fq(T)38 b Fu(and)28 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)f Fq(a)28 b Fu(w)n(e)f(ha)n(v)n(e)g(that)1482 5059 y(~)-45 b Fq(\015)1527 5025 y Fi(s)1522 5080 y(\026;")1618 5059 y Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)h Fq(V)1938 5071 y Fo(0)1975 5059 y Fu(\()p Fq(\024)2055 5024 y Fj(\006)2055 5081 y Fo(1)2111 5059 y Fq(;)14 b(\024)2196 5071 y Fo(2)2233 5059 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\))456 5216 y(if)28 b Fp(j)p Fq(\026)p Fp(j)g Fu(is)f(small)g(enough.)p eop %%Page: 58 66 58 65 bop 456 251 a Fo(58)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(W)-7 b(e)22 b(call)g Fp(C)k Fu(the)d(c)n(hange)d(from)i(the)g (initial)g(to)g(the)g(a)n(v)n(eraged)d(systems,)k(de\014ned)f(in)g (Prop)r(o-)456 550 y(sition)34 b(3.1)f(in)i(Chapter)e(3.)57 b(W)-7 b(e)35 b(write)f(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))35 b(=)e Fp(C)5 b Fu(\()g(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r(;)14 b(\022)r Fu(\).)58 b(Where)34 b(\()5 b(\026)-47 b Fq(x)q(;)19 b Fu(\026)-47 b Fq(y)r Fu(\))35 b(denote)f(the)456 649 y(v)-5 b(ariables)26 b(of)h(the)h(a)n(v)n(eraged)d(systems.)37 b(Moreo)n(v)n(er,)25 b(w)n(e)i(kno)n(w)g(that)1469 789 y(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e(\()5 b(\026)-47 b Fq(x)q(;)20 b Fu(\026)-48 b Fq(y)s Fu(\))18 b(+)g Fq(O)r Fu(\()p Fq(\026")2252 755 y Fi(p)p Fo(+1)2376 789 y Fu(\).)456 929 y(W)-7 b(e)28 b(de\014ne)f(new)h(\015o)n(w)f(b)r(o)n(x)g(co)r(ordinates)952 1068 y(\()p Fp(T)1050 1034 y Fo(2)1088 1068 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))p Fq(;)g Fp(I)1446 1034 y Fo(2)1484 1068 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r Fu(\)\))24 b(=)f(\()p Fp(T)1996 1034 y Fo(1)2033 1068 y Fu(\()p Fp(C)2114 1034 y Fj(\000)p Fo(1)2203 1068 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\)\))p Fq(;)g Fp(I)2593 1034 y Fo(1)2632 1068 y Fu(\()p Fp(C)2713 1034 y Fj(\000)p Fo(1)2802 1068 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r Fu(\)\)\).)456 1218 y(Since)25 b(the)h(c)n(hange)e Fp(C)29 b Fu(is)c Fq(O)r Fu(\()p Fq(\026")1423 1188 y Fi(p)p Fo(+1)1547 1218 y Fu(\))g(close)f(to)h(the)h(iden)n(tit)n(y)f(and)g(canonical,)g(the)g (new)h(c)n(hange)456 1318 y(is)h(also)g(canonical)f(and)h(satis\014es) 905 1457 y(\()p Fp(T)1003 1423 y Fo(2)1041 1457 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))p Fq(;)g Fp(I)1399 1423 y Fo(2)1437 1457 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r Fu(\)\))24 b(=)f(\()p Fp(T)1949 1423 y Fo(2)1928 1478 y(0)1986 1457 y Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fq(;)g Fp(I)2266 1423 y Fo(2)2260 1478 y(0)2305 1457 y Fu(\()p Fq(x;)g(y)s Fu(\)\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")2817 1423 y Fi(p)p Fo(+1)2940 1457 y Fu(\).)605 1607 y(The)25 b(c)n(hange)e Fp(T)1111 1577 y Fo(2)1089 1628 y(0)1148 1607 y Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fq(;)g Fp(I)1428 1577 y Fo(2)1422 1628 y(0)1466 1607 y Fu(\()p Fq(x;)g(y)s Fu(\))24 b(=)e Fq(h)1817 1619 y Fo(0)1854 1607 y Fu(\()p Fq(x;)14 b(y)s Fu(\))26 b(is)e(the)i(corresp)r(onding)c(c)n(hange)i(for)g Fq(\026)f Fu(=)g(0.)456 1708 y(The)30 b(domain)h(of)f(the)h(new)g(c)n (hange)f(is)g Fp(C)5 b Fu(\()p Fq(V)19 b Fu(\()p Fq(\024)1937 1672 y Fj(\006)1937 1730 y Fo(1)1993 1708 y Fq(;)14 b(\024)2078 1720 y Fo(2)2115 1708 y Fq(;)g Fu(0)p Fq(;)g(r)r Fu(\)\).)47 b(Moreo)n(v)n(er)28 b(it)j(is)f(clear)g(that)h(the)456 1807 y(c)n(hange)g(\()p Fq(T)7 b(;)14 b(I)7 b Fu(\))30 b(=)g(\()p Fp(T)1159 1777 y Fo(2)1196 1807 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))p Fq(;)g Fp(I)1554 1777 y Fo(2)1592 1807 y Fu(\()p Fq(x;)g(y)s(;)g(\022)r Fu(\)\))33 b(transforms)e(the)i(system)e(\(1.1\))h(of)g(Chapter)g(1)456 1907 y(to)1798 2026 y(_)1772 2047 y Fq(T)94 b Fu(=)83 b(1)1809 2150 y(_)1789 2171 y Fq(I)90 b Fu(=)83 b(0.)456 2311 y(Let)40 b Fq(\015)665 2280 y Fi(s)660 2331 y(\026;")755 2311 y Fu(\()p Fq(t;)14 b(s)p Fu(\))41 b(b)r(e)f(the)h (parameterization)d(of)i(the)g(stable)g(manifold)g(of)g(system)g (\(1.1\))g(of)456 2413 y(Chapter)26 b(1)g(giv)n(en)g(in)h(Theorem)f (3.1.)36 b(Since)27 b Fq(\015)1938 2383 y Fi(s)1933 2434 y(\026;")2051 2413 y Fu(=)c Fp(C)5 b Fu(\()s(~)-45 b Fq(\015)2268 2383 y Fi(s)2263 2434 y(\026;")2358 2413 y Fu(\))27 b(w)n(e)g(ha)n(v)n(e)e(that)i Fq(\015)2956 2383 y Fi(s)2951 2434 y(\026;")3074 2413 y Fu(b)r(elongs)f(to)456 2513 y(the)i(domain)f(of)g(the)h(new)g(\015o)n(w)f(b)r(o)n(x)g(co)r (ordinates.)605 2612 y(W)-7 b(e)37 b(de\014ne)h(a)e(new)i(c)n(hange)d (of)i(v)-5 b(ariables.)64 b(Let)38 b Fq(s)2256 2624 y Fo(0)2330 2612 y Fu(b)r(e)f(suc)n(h)g(that)g Fp(j)14 b Fu(Im)g Fq(s)3027 2624 y Fo(0)3064 2612 y Fp(j)39 b(\024)g Fq(a)e Fu(and)456 2712 y Fq(t)486 2724 y Fo(0)546 2712 y Fp(2)23 b Fl(R)p Fu(.)43 b(W)-7 b(e)28 b(de\014ne)1427 2851 y(\()p Fq(x)1506 2817 y Fj(\003)1545 2851 y Fq(;)14 b(y)1626 2817 y Fj(\003)1664 2851 y Fu(\))23 b(=)g Fq(\015)1855 2817 y Fi(s)1850 2872 y(\026;")1946 2851 y Fu(\()p Fq(T)29 b Fp(\000)18 b Fu(Re)c Fq(s)2290 2863 y Fo(0)2327 2851 y Fq(;)g(s)2403 2863 y Fo(0)2440 2851 y Fu(\))456 2991 y(and)27 b(the)h(parameter)1235 3131 y Fq(\034)33 b Fu(=)23 b Fq(s)1431 3143 y Fo(0)1486 3131 y Fp(\000)18 b(T)1636 3097 y Fo(2)1673 3131 y Fu(\()p Fq(x)1752 3097 y Fj(\003)1791 3131 y Fq(;)c(y)1872 3097 y Fj(\003)1909 3131 y Fq(;)g Fu(\()p Fq(T)30 b Fp(\000)18 b Fu(Re)c Fq(s)2291 3143 y Fo(0)2328 3131 y Fu(\))p Fq(=")p Fu(\))k Fp(\000)g Fq(t)2604 3143 y Fo(0)2642 3131 y Fu(.)456 3271 y(W)-7 b(e)28 b(observ)n(e)d(that,)j(if)h Fq(\026)23 b Fu(=)f(0,)27 b(the)h(constan)n(t)f Fq(\034)38 b Fu(is)1157 3479 y Fq(\034)33 b Fu(=)1314 3366 y Fg(Z)1397 3387 y Fj(C)1436 3362 y Fh(1)1468 3387 y Fo(\()p Fi(\016)o(;f)7 b Fo(\()p Fi(\016)r Fo(\))p Fi(;)p Fo(0\))1360 3555 y Fi(x)1398 3563 y Fh(0)2127 3423 y Fq(ds)p 1772 3460 792 4 v 1772 3477 a Fg(p)p 1855 3477 709 4 v 71 x Fu(2)p Fq(h)1945 3560 y Fo(0)1982 3548 y Fu(\()p Fq(x)2061 3524 y Fj(\003)2100 3548 y Fq(;)14 b(y)2181 3524 y Fj(\003)2218 3548 y Fu(\))19 b Fp(\000)f Fu(2)p Fq(V)h Fu(\()p Fq(s)p Fu(\))2592 3479 y Fp(\000)g Fq(t)2706 3491 y Fo(0)456 3689 y Fu(where)27 b Fp(C)745 3659 y Fo(1)810 3689 y Fu(denotes)h(the)h(\014rst)e(comp)r (onen)n(t)h(of)g(the)h(c)n(hange)e(and)h Fq(x)2574 3701 y Fo(0)2640 3689 y Fu(is)g(the)g(initial)h(condition)456 3789 y(of)c Fq(\015)591 3801 y Fo(0)653 3789 y Fu(as)g(is)g(de\014ned)h (in)g(h)n(yp)r(othesis)e Fk(HP1)p Fu(.)36 b Fq(\034)f Fu(do)r(es)25 b(not)h(dep)r(end)g(on)f(\()p Fq(x)2746 3759 y Fj(\003)2785 3789 y Fq(;)14 b(y)2866 3759 y Fj(\003)2903 3789 y Fu(\))26 b(while)f(\()p Fq(x)3254 3759 y Fj(\003)3293 3789 y Fq(;)14 b(y)3374 3759 y Fj(\003)3412 3789 y Fu(\))456 3888 y(b)r(elongs)27 b(to)g(the)h(stable)f(manifold)h(of)f(the)h(unp)r (erturb)r(ed)h(system.)605 3988 y(W)-7 b(e)28 b(de\014ne)g(the)g (functions)1211 4127 y Fp(S)6 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))85 b(=)d Fp(T)1835 4093 y Fo(2)1873 4127 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))19 b(+)f Fq(\034)1217 4253 y Fp(E)7 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))84 b(=)e Fp(I)1820 4219 y Fo(2)1858 4253 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))19 b Fp(\000)f(I)2281 4219 y Fo(2)2319 4253 y Fu(\()p Fq(x)2398 4219 y Fj(\003)2436 4253 y Fq(;)c(y)2517 4219 y Fj(\003)2555 4253 y Fq(;)g(\022)r Fu(\).)456 4399 y(Next)28 b(w)n(e)f(will)h(see)f(that,)h(since)1504 4378 y(_)1478 4399 y Fq(T)1539 4368 y Fo(2)1599 4399 y Fu(=)22 b(1)27 b(and)1936 4378 y(_)1917 4399 y Fq(I)1960 4368 y Fo(2)2020 4399 y Fu(=)c(0,)k(w)n(e)g(ha)n(v)n(e)g(that)g(for)h Fq(s)23 b Fu(=)f Fq(s)3009 4411 y Fo(0)3046 4399 y Fu(,)1388 4538 y Fp(S)6 b Fu(\()p Fq(\015)1524 4504 y Fi(s)1519 4559 y(\026;")1615 4538 y Fu(\()p Fq(t;)14 b(s)1753 4550 y Fo(0)1790 4538 y Fu(\))p Fq(;)g(t=")p Fu(\))23 b(=)g Fq(t)18 b Fp(\000)g Fq(t)2274 4550 y Fo(0)2330 4538 y Fu(+)g Fq(s)2452 4550 y Fo(0)2489 4538 y Fu(.)456 4678 y(Indeed,)27 b(using)h(the)g(de\014nition)g(of)f Fq(\034)706 4818 y Fp(S)6 b Fu(\()p Fq(\015)842 4783 y Fi(s)837 4838 y(\026;")933 4818 y Fu(\()p Fq(t;)14 b(s)1071 4830 y Fo(0)1108 4818 y Fu(\))p Fq(;)g(t=")p Fu(\))83 b(=)g Fp(T)1617 4783 y Fo(2)1654 4818 y Fu(\()p Fq(\015)1734 4783 y Fi(s)1729 4838 y(\026;")1825 4818 y Fu(\()p Fq(t;)14 b(s)1963 4830 y Fo(0)2000 4818 y Fu(\))p Fq(;)g(t=")p Fu(\))19 b(+)f Fq(\034)1403 4954 y Fu(=)83 b Fq(t)18 b Fu(+)g Fp(T)1748 4920 y Fo(2)1786 4954 y Fu(\()p Fq(\015)1866 4920 y Fi(s)1861 4975 y(\026;")1957 4954 y Fu(\()p Fq(T)29 b Fp(\000)18 b Fu(Re)c Fq(s)2301 4966 y Fo(0)2338 4954 y Fq(;)g(s)2414 4966 y Fo(0)2451 4954 y Fu(\))p Fq(;)g Fu(\()p Fq(T)30 b Fp(\000)18 b Fu(Re)c Fq(s)2865 4966 y Fo(0)2902 4954 y Fu(\))p Fq(=")p Fu(\))19 b(+)f Fq(\034)1403 5091 y Fu(=)83 b Fq(t)18 b Fu(+)g Fp(T)1748 5057 y Fo(2)1786 5091 y Fu(\()p Fq(x)1865 5057 y Fj(\003)1904 5091 y Fq(;)c(y)1985 5057 y Fj(\003)2022 5091 y Fq(;)g Fu(\()p Fq(T)30 b Fp(\000)18 b Fu(Re)c Fq(s)2404 5103 y Fo(0)2441 5091 y Fu(\))p Fq(=")p Fu(\))k(+)g Fq(\034)1403 5216 y Fu(=)83 b Fq(t)18 b Fu(+)g Fq(s)1721 5228 y Fo(0)1777 5216 y Fp(\000)g Fq(t)1890 5228 y Fo(0)1927 5216 y Fu(.)p eop %%Page: 59 67 59 66 bop 1375 251 a Fo(4.6.)29 b(PR)n(OOF)g(OF)g(THE)g(THEOREM)g(4.2) 853 b(59)456 450 y Fu(Let)31 b Fq(s)f Fp(2)f Fl(C)15 b Fu(,)39 b Fq(s)29 b Fp(6)p Fu(=)h Fq(s)1079 462 y Fo(0)1116 450 y Fu(.)48 b(Using)32 b(that,)g(for)f(the)h(unp)r(erturb)r(ed)g (Hamiltonian)f(system)h(w)n(e)f(ha)n(v)n(e)456 550 y(that)d Fp(S)686 562 y Fo(0)724 550 y Fu(\()p Fq(\015)799 562 y Fo(0)836 550 y Fu(\()p Fq(t)19 b Fu(+)g Fq(s)p Fu(\)\))24 b(=)g Fq(t)19 b Fp(\000)f Fq(t)1379 562 y Fo(0)1436 550 y Fu(+)g Fq(s)28 b Fu(and)g(the)h(estimates)f Fp(S)6 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))25 b(=)f Fp(S)2748 562 y Fo(0)2785 550 y Fu(\()p Fq(x;)14 b(v)s Fu(\))21 b(+)d Fq(O)r Fu(\()p Fq(\026")3266 520 y Fi(p)p Fo(+1)3389 550 y Fu(\),)456 649 y Fq(\015)504 619 y Fi(s)499 670 y(\026;")594 649 y Fu(\()p Fq(t;)c(s)p Fu(\))24 b(=)e Fq(\015)918 661 y Fo(0)955 649 y Fu(\()p Fq(t)d Fu(+)f Fq(s)p Fu(\))h(+)f Fq(O)r Fu(\()p Fq(\026")1478 619 y Fi(p)p Fo(+1)1601 649 y Fu(\),)28 b(w)n(e)f(obtain)1181 796 y Fp(S)6 b Fu(\()p Fq(\015)1317 762 y Fi(s)1312 817 y(\026;")1409 796 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))22 b(=)h Fq(t)c Fp(\000)f Fq(t)2031 808 y Fo(0)2086 796 y Fu(+)g Fq(s)h Fu(+)f Fq(\026")2399 762 y Fi(p)p Fo(+1)2521 796 y Fp(X)12 b Fu(\()p Fq(s)p Fu(\).)456 936 y(W)-7 b(e)30 b(note)g(that)h(w)n(e)f(ha)n(v)n(e)f Fp(X)12 b Fu(\()p Fq(s)1432 948 y Fo(0)1470 936 y Fu(\))28 b(=)f(0,)j(and)g(that)h(w)n(e)f(ha)n(v)n(e)f(a)h(lot)g(of)g(freedom)g (to)g(c)n(ho)r(ose)f Fq(s)3407 948 y Fo(0)456 1035 y Fu(and)e Fq(t)647 1047 y Fo(0)684 1035 y Fq(:)h Fu(Eac)n(h)e(c)n(hoice) h(giv)n(es)g(a)g(sligh)n(tly)g(di\013eren)n(t)g(de\014nition)h(of)g Fp(S)6 b Fu(.)605 1135 y(W)-7 b(e)28 b(also)f(note)g(that)h Fp(X)12 b Fu(\()p Fq(s)p Fu(\))28 b(is)g(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)f(in)g Fq(s)p Fu(.)37 b(Indeed,)28 b(w)n(e)f(ha)n(v)n(e)g(that)785 1270 y Fp(S)6 b Fu(\()p Fq(\015)921 1236 y Fi(s)916 1291 y(\026;")1012 1270 y Fu(\()p Fq(t;)14 b(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\))p Fq(;)c(t=")p Fu(\))82 b(=)h Fq(t)18 b Fp(\000)g Fq(t)1986 1282 y Fo(0)2042 1270 y Fu(+)g Fq(s)g Fu(+)g(2)p Fq(\031)s(")g Fu(+)g Fq(\026")2586 1236 y Fi(p)p Fo(+1)2708 1270 y Fp(X)12 b Fu(\()p Fq(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\))785 1407 y Fp(S)6 b Fu(\()p Fq(\015)921 1373 y Fi(s)916 1428 y(\026;")1012 1407 y Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\))p Fq(;)g(t=")p Fu(\))82 b(=)h Fq(t)18 b Fp(\000)g Fq(t)1986 1419 y Fo(0)2042 1407 y Fu(+)g(2)p Fq(\031)s(")g Fu(+)g Fq(s)g Fu(+)g Fq(\026")2586 1373 y Fi(p)p Fo(+1)2708 1407 y Fp(X)12 b Fu(\()p Fq(s)p Fu(\))456 1544 y(and,)25 b(since)g Fq(\015)887 1514 y Fi(s)882 1565 y(\026;")978 1544 y Fu(\()p Fq(t;)14 b(s)g Fu(+)g(2)p Fq(\031)s(")p Fu(\))22 b(=)g Fq(\015)1529 1514 y Fi(s)1524 1565 y(\026;")1620 1544 y Fu(\()p Fq(t)14 b Fu(+)g(2)p Fq(\031)s(";)g(s)p Fu(\),)25 b(from)g(the)g(previous)g (equations)f(w)n(e)h(obtain)1592 1682 y Fp(X)12 b Fu(\()p Fq(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\))23 b(=)g Fp(X)12 b Fu(\()p Fq(s)p Fu(\).)456 1817 y(Finally)-7 b(,)975 1951 y Fp(E)7 b Fu(\()p Fq(\015)1106 1917 y Fi(s)1101 1972 y(\026;")1198 1951 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))82 b(=)h Fp(I)1829 1917 y Fo(2)1867 1951 y Fu(\()p Fq(\015)1947 1917 y Fi(s)1942 1972 y(\026;")2037 1951 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))19 b Fp(\000)f(I)2540 1917 y Fo(2)2577 1951 y Fu(\()p Fq(x)2656 1917 y Fj(\003)2695 1951 y Fq(;)c(y)2776 1917 y Fj(\003)2814 1951 y Fq(;)g Fu(0\))1630 2076 y(=)83 b(0.)456 2210 y(This)27 b(ends)h(the)g(pro)r(of)f(of)g(Theorem)g(4.2.)p eop %%Page: 60 68 60 67 bop 1156 566 a Fw(5.)60 b(The)44 b(Extension)i(Theorem)605 849 y Fu(This)25 b(short)f(Chapter)h(is)g(dev)n(oted)f(to)h(recall)f (the)i(statemen)n(t)f(of)g(the)g(extension)g(theorem)456 948 y(whic)n(h)i(is)h(giv)n(en)e(in)i([)p Fk(DS2)p Fu(].)37 b(This)28 b(theorem)f(is)g(stated)h(for)f(systems)g(of)h(the)g(form) 1322 1088 y(_)-37 b Fq(x)83 b Fu(=)g Fq(y)21 b Fu(+)d Fq(\026")1820 1054 y Fi(p)1858 1088 y Fq(@)1902 1100 y Fi(y)1942 1088 y Fq(h)1990 1100 y Fo(1)2027 1088 y Fu(\()p Fq(x;)c(y)s(;)g(t=")p Fu(\))1327 1213 y(_)-38 b Fq(y)85 b Fu(=)e Fp(\000)p Fq(V)1717 1178 y Fj(0)1741 1213 y Fu(\()p Fq(x)p Fu(\))19 b Fp(\000)f Fq(\026")2043 1178 y Fi(p)2081 1213 y Fq(@)2125 1225 y Fi(x)2167 1213 y Fq(h)2215 1225 y Fo(1)2252 1213 y Fu(\()p Fq(x;)c(y)s(;)g(t=")p Fu(\))456 1353 y(suc)n(h)32 b(that)g(the)h(unp)r(erturb)r(ed)g(system)f (has)f(a)h(homo)r(clinic)g(orbit,)i Fq(\015)2669 1365 y Fo(0)2706 1353 y Fu(\()p Fq(u)p Fu(\))d(=)f(\()p Fq(\013)3029 1365 y Fo(0)3067 1353 y Fu(\()p Fq(u)p Fu(\))p Fq(;)14 b(\014)3263 1365 y Fo(0)3300 1353 y Fu(\()p Fq(u)p Fu(\)\))456 1452 y(and)25 b Fq(\014)662 1464 y Fo(0)724 1452 y Fu(is)g(an)g (analytic)g(function)h(in)f Fp(j)14 b Fu(Im)g Fq(u)p Fp(j)23 b Fq(<)g(a)i Fu(and)g(has)g(singularities)f(at)h Fq(u)d Fu(=)h Fp(\006)p Fq(ia)i Fu(whic)n(h)456 1552 y(are)h(p)r(oles.)605 1652 y(F)-7 b(ollo)n(wing)32 b(the)i(pro)r(of)f (in)h([)p Fk(DS2)p Fu(])g(one)f(can)h(see)f(that)h(one)f(can)g(replace) g(the)h(condition)456 1751 y(of)e Fq(u)g Fu(=)f Fp(\006)p Fq(ia)h Fu(b)r(eing)h(p)r(oles)f(b)n(y)g Fq(u)g Fu(=)f Fp(\006)p Fq(ia)h Fu(b)r(eing)h(branc)n(hing)e(p)r(oin)n(ts)i(in)g(the) g(sense)f(w)n(e)h(ha)n(v)n(e)456 1851 y(in)n(tro)r(duced)g(in)i Fk(HP1)f Fu(in)g(Chapter)g(1.)56 b(F)-7 b(or)33 b(this)h(reason)f(here) g(w)n(e)h(do)g(not)g(repro)r(duce)f(the)456 1950 y(pro)r(of)27 b(of)g(the)h(extension)f(theorem.)605 2050 y(The)42 b(goal)e(of)i(this) g(theorem)f(is)h(to)f(extend)h(the)g(domain)g(of)f(the)h (parameterization)456 2150 y Fq(\015)504 2120 y Fi(u)499 2170 y(\026;")594 2150 y Fu(\()p Fq(t;)14 b(s)p Fu(\))33 b(of)g(the)g(unstable)g(manifold)g(un)n(til)g(\(in)g(our)f(case)g(pro)r (duced)h(in)g(Chapter)f(3\))h Fq(\015)3354 2120 y Fi(u)3349 2170 y(\026;")456 2249 y Fu(en)n(ters)d(the)i(domain)f(of)g(the)h (\015o)n(w)e(b)r(o)n(x)h(co)r(ordinates.)47 b(T)-7 b(o)31 b(do)g(this,)h(the)g(parameterizations)456 2349 y Fq(\015)504 2319 y Fi(u)499 2370 y(\026;")594 2349 y Fu(\()p Fq(t;)14 b(s)p Fu(\))28 b(and)g Fq(\015)997 2361 y Fo(0)1034 2349 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))28 b(are)e(compared)h(in)g(the) h(complex)g(domain)f Fq(D)2745 2319 y Fo(ext)2743 2370 y Fi(")2839 2349 y Fu(:)915 2499 y Fq(D)986 2465 y Fo(ext)984 2520 y Fi(")1103 2499 y Fp(\021)c(f)p Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)g Fl(R)i Fp(\002)18 b Fl(C)44 b Fu(:)23 b Fp(j)p Fq(t)18 b Fu(+)g(Re)c Fq(s)p Fp(j)23 b(\024)g Fu(2)p Fq(T)38 b Fu(,)28 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)g Fq(a)18 b Fp(\000)g Fq(")p Fp(g)p Fu(.)456 2640 y(The)27 b(extension)g(theorem)g(giv)n(es)g(a)g(useful)h (b)r(ound)g(for)f(the)h(distance)f(b)r(et)n(w)n(een)h(the)g(unstable) 456 2739 y(manifold)33 b Fq(\015)853 2709 y Fi(u)848 2760 y(\026;")977 2739 y Fu(and)g(the)g(homo)r(clinic)h(orbit)e Fq(\015)1967 2751 y Fo(0)2038 2739 y Fu(of)h(the)h(unp)r(erturb)r(ed)g (system)f(for)f(\()p Fq(t;)14 b(s)p Fu(\))33 b Fp(2)456 2845 y Fq(D)527 2815 y Fo(ext)525 2866 y Fi(")621 2845 y Fu(.)k(That)28 b(is,)1390 2985 y Fq(\015)1438 2951 y Fi(u)1433 3006 y(\026;")1529 2985 y Fu(\()p Fq(t;)14 b(s)p Fu(\))19 b Fp(\000)f Fq(\015)1844 2997 y Fo(0)1881 2985 y Fu(\()p Fq(t)h Fu(+)f Fq(s)p Fu(\))23 b(=)f Fq(O)r Fu(\()p Fq(\026")2412 2951 y Fi(\027)2454 2985 y Fu(\),)456 3126 y(where)27 b Fq(\027)33 b Fu(is)27 b(a)g(parameter)f(whic)n(h)i (dep)r(ends)g(on)f(the)h(system.)605 3225 y(Therefore,)g(if)h Fq(\027)h Fp(\025)25 b Fu(0)j(and)h Fq(\026)g Fu(and)f Fq(")h Fu(are)e(small)i(enough)f Fq(\015)2479 3195 y Fi(u)2474 3246 y(\026;")2569 3225 y Fu(\()p Fq(t;)14 b(s)p Fu(\),)30 b(for)e(some)g(v)-5 b(alues)29 b(of)456 3328 y(\()p Fq(t;)14 b(s)p Fu(\),)28 b(b)r(elongs)f(to)g(the)h(domain)f (of)h(the)g(\015o)n(w)f(b)r(o)n(x)g(co)r(ordinates.)605 3427 y(The)h(extension)f(theorem)g(is)605 3580 y Ft(Theorem)32 b Fu(5.1)p Ft(.)40 b Fs(L)l(et)29 b Fq(z)t Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b(=)g(\()p Fq(x)p Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(y)s Fu(\()p Fq(t;)g(s)p Fu(\)\))31 b Fs(b)l(e)f(a)g (family)h(of)g(solutions)f(of)1241 3720 y Fu(_)-37 b Fq(x)83 b Fu(=)g Fq(y)21 b Fu(+)d Fq(\026")1739 3686 y Fi(p)1777 3720 y Fq(@)1821 3732 y Fi(y)1861 3720 y Fq(h)1909 3732 y Fo(1)1946 3720 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g(\026;) g(")p Fu(\))1245 3844 y(_)-38 b Fq(y)86 b Fu(=)d Fp(\000)p Fq(V)1636 3810 y Fj(0)1660 3844 y Fu(\()p Fq(x)p Fu(\))19 b Fp(\000)f Fq(\026")1962 3810 y Fi(p)2000 3844 y Fq(@)2044 3856 y Fi(x)2086 3844 y Fq(h)2134 3856 y Fo(1)2171 3844 y Fu(\()p Fq(x;)c(y)s(;)g(t=";)g(\026;)g(")p Fu(\))456 3984 y Fs(de\014ne)l(d)30 b(for)g Fq(t)898 3996 y Fo(0)954 3984 y Fu(+)18 b(Re)c Fq(s)23 b Fu(=)f Fp(\000)p 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y Fu(\()p Fq(x;)g(y)s(;)g(\022)r(;)g (\026;)g(")p Fu(\)\))456 4553 y Fs(and)30 b(has)g(zer)l(o)g(me)l(an)g (with)g(r)l(esp)l(e)l(ct)g(to)g Fq(\022)r Fs(,)g(and)g Fu(\()p Fq(t)2031 4565 y Fo(0)2069 4553 y Fq(;)14 b(s)p Fu(\))23 b Fp(2)g Fq(D)2349 4523 y Fo(ext)2347 4574 y Fi(")2474 4553 y Fs(veri\014es)30 b Fq(t)2789 4565 y Fo(0)2845 4553 y Fu(+)18 b(Re)c Fq(s)23 b Fu(=)f Fp(\000)p Fu(2)p Fq(T)12 b Fs(.)605 4653 y(L)l(et)29 b Fq(`)h Fs(b)l(e)f (de\014ne)l(d)i(by)f(\(1.2\).)40 b(We)29 b(assume)h(that)1694 4793 y Fq(\027)e Fp(\021)23 b Fq(p)18 b Fp(\000)g Fq(`)23 b Fp(\025)f Fu(0)p Fs(.)456 4938 y(Then,)40 b(ther)l(e)d(exist)g Fq(")1161 4950 y Fo(0)1198 4938 y Fs(,)j Fq(\026)1313 4950 y Fo(0)1387 4938 y Fs(and)e Fq(K)k Fs(such)c(that)f(the)g (solution)h Fq(z)t Fu(\()p Fq(t;)14 b(s)p Fu(\))37 b Fs(c)l(an)g(b)l(e)g(extende)l(d)g(to)456 5037 y(values)30 b(of)g Fq(t)g Fp(2)24 b Fu([)p Fq(t)993 5049 y Fo(0)1030 5037 y Fq(;)14 b Fu(2)p Fq(T)29 b Fp(\000)18 b Fu(Re)c Fq(s)p Fu(])p Fs(,)30 b(with)g(the)g(b)l(ound)1415 5182 y Fp(j)p Fq(z)t Fu(\()p Fq(t;)14 b(s)p Fu(\))k Fp(\000)g Fq(\015)1795 5194 y Fo(0)1833 5182 y Fu(\()p Fq(t)g Fu(+)g Fq(s)p Fu(\))p Fp(j)24 b(\024)e Fq(K)6 b(\026")2367 5147 y Fi(p)p Fj(\000)p Fi(`)456 5182 y Fu(\(5.1\))1917 5365 y Fo(60)p eop %%Page: 61 69 61 68 bop 1440 251 a Fo(5.)29 b(THE)g(EXTENSION)g(THEOREM)917 b(61)456 450 y Fs(for)30 b Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b Fp(2)f Fq(D)931 420 y Fo(ext)929 471 y Fi(")1026 450 y Fs(,)30 b Fu(0)22 b Fq(<)h(")g Fp(\024)f Fq(")1421 462 y Fo(0)1488 450 y Fs(and)30 b Fp(j)p Fq(\026)p Fp(j)23 b(\024)g Fq(\026)1906 462 y Fo(0)1943 450 y Fs(.)605 600 y Ft(Remark)28 b Fu(5.1)p Ft(.)37 b Fs(The)l(or)l(em)27 b(5.1)g(is)g(also)g(valid)h(if)g Fq(V)45 b Fs(is)26 b(a)h (trigonometric)g(p)l(olynomial)i(and)456 699 y(we)i(assume)g(that)g Fq(h)1091 711 y Fo(1)1159 699 y Fs(is)h(also)g(a)f(trigonometric)i(p)l (olynomial)g(in)e Fq(x)h Fs(and)f(a)h(p)l(olynomial)h(in)e Fq(y)s Fs(.)456 799 y(We)k(observe)h(that)f(in)g(this)h(c)l(ase)f Fq(\013)1594 811 y Fo(0)1632 799 y Fu(\()p Fq(u)p Fu(\))e Fp(\030)f Fq(iC)20 b Fu(log\()p Fq(u)i Fp(\007)g Fq(ia)p Fu(\))35 b Fs(ne)l(ar)g(of)h(singularity)g Fq(u)c Fu(=)g Fp(\006)p Fq(ia)456 898 y Fs(with)g Fq(C)37 b Fs(a)32 b(c)l(onstant)f(which)i(dep)l(ends)g(on)e(the)h(de)l(gr)l(e)l(e)g(of)g Fq(V)19 b Fs(.)44 b(\(Se)l(e)31 b Fu([)p Fk(DS2)p Fu(])h Fs(for)g(mor)l(e)g(details)456 998 y(ab)l(out)d(this)h(c)l(ase\).)605 1147 y Ft(Remark)43 b Fu(5.2)p Ft(.)k Fs(We)40 b(note)f(that,)k(if)d Fq(s)h Fp(2)h Fl(R)p Fs(,)48 b(the)40 b(estimate)g(\(5.1\))h(c)l(an)f (b)l(e)f(impr)l(ove)l(d.)456 1247 y(Concr)l(etely,)31 b(for)f Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b Fp(2)f Fl(R)1352 1217 y Fo(2)1425 1247 y Fs(such)30 b(that)g Fp(\000)p Fu(2)p Fq(T)j Fp(\024)22 b Fq(t)d Fu(+)f Fq(s)23 b Fp(\024)g Fu(2)p Fq(T)40 b Fs(we)30 b(have)h(that)1359 1383 y Fq(z)t Fu(\()p Fq(t)1464 1395 y Fo(0)1501 1383 y Fq(;)14 b(s)p Fu(\))19 b Fp(\000)f Fq(\015)1754 1395 y Fo(0)1791 1383 y Fu(\()p Fq(t)1853 1395 y Fo(0)1909 1383 y Fu(+)g Fq(s)p Fu(\))23 b(=)g Fq(O)r Fu(\()p Fq(\026")2360 1348 y Fi(p)p Fo(+1)2483 1383 y Fu(\))p Fs(.)p eop %%Page: 62 70 62 69 bop 1168 566 a Fw(6.)60 b(Splitting)46 b(of)f(separatrices)1589 849 y Fk(6.1.)h(In)m(tro)s(duction)605 998 y Fu(This)21 b(c)n(hapter)g(is)g(dev)n(oted)g(to)g(pro)n(v)n(e)f(Theorem)h(1.1)f (and)i(Corollary)d(1.1.)34 b(T)-7 b(o)21 b(pro)n(v)n(e)e(these)456 1098 y(results)i(w)n(e)h(will)h(use)f(the)h(results)e(established)h(in) h(the)f(previous)g(c)n(hapters.)34 b(In)22 b(particular)f(the)456 1197 y(parameterization)h(of)j(the)f(stable)h(manifold,)g(in)g(Chapter) f(3)g(and)g(the)h(\015o)n(w)f(b)r(o)n(x)g(co)r(ordinates)456 1297 y(dev)n(elop)r(ed)j(in)h(Chapter)f(4)g(will)h(pla)n(y)f(an)g(imp)r (ortan)n(t)g(role.)36 b(Let)1311 1445 y(_)-38 b Fq(x)84 b Fu(=)e Fq(y)22 b Fu(+)c Fq(\026")1809 1411 y Fi(p)1847 1445 y Fq(@)1891 1457 y Fi(y)1931 1445 y Fq(h)1979 1457 y Fo(1)2016 1445 y Fu(\()p 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b(in)h(Chapter)g(4)f(\(see)h(in)h(particular)d(Theorem)h(4.2\),)h(whic) n(h)g(is)g(a)g(neigh)n(b)r(orho)r(o)r(d)456 4349 y(of)1161 4488 y Fp(f)p Fq(\015)1246 4500 y Fo(0)1283 4488 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))23 b(:)g Fq(T)34 b Fp(\024)23 b Fq(t)18 b Fu(+)g(Re)c Fq(s)23 b Fp(\024)g Fu(2)p Fq(T)12 b Fu(,)54 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)f Fq(a)p Fp(g)456 4642 y Fu(and)c(it)h(is)f(indep)r(enden)n(t)h(of)g Fq(\026;)14 b(")p Fu(.)33 b(In)19 b(fact,)h(w)n(e)e(ha)n(v)n(e)f(c)n (hosen)h Fq(\024)2359 4606 y Fj(\006)2359 4664 y Fo(1)2415 4642 y Fu(,)i(and)e Fq(\024)2658 4654 y Fo(2)2714 4642 y Fu(suc)n(h)g(that,)j Fq(\015)3131 4654 y Fo(0)3168 4642 y Fu(\()p Fq(t)p Fu(+)p Fq(s)p Fu(\))i Fp(2)456 4744 y Fq(V)504 4756 y Fo(0)541 4744 y Fu(\()p Fq(\024)621 4708 y Fj(\006)621 4766 y Fo(1)696 4744 y Fp(\007)18 b Fu(1)p Fq(;)c Fu(2)p Fq(\024)948 4756 y Fo(2)984 4744 y Fq(=)p Fu(3)p Fq(;)g Fu(0)p Fq(;)g Fu(0\).)605 4843 y(By)26 b(the)g(extension)g(theorem)f(of)h(Chapter)g(5,)g(the)g(domain) g(of)g(the)g(parameterization)f(of)456 4943 y(the)31 b(unstable)g(manifold)g Fq(\015)1333 4913 y Fi(u)1328 4963 y(\026;")1424 4943 y Fu(\()p Fq(t;)14 b(s)p Fu(\))31 b(can)g(b)r(e)g(extended)g(to)g(v)-5 b(alues)31 b(of)g(\()p Fq(t;)14 b(s)p Fu(\))29 b Fp(2)g Fq(D)3061 4913 y Fo(ext)3059 4963 y Fi(")3155 4943 y Fu(,)j(and)f(in)456 5042 y(this)c(extended)h (set)g(v)n(eri\014es)1402 5182 y Fq(\015)1450 5147 y Fi(u)1445 5202 y(\026;")1540 5182 y Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b(=)e Fq(\015)1864 5194 y Fo(0)1902 5182 y Fu(\()p Fq(t)c Fu(+)g Fq(s)p Fu(\))h(+)f Fq(O)r Fu(\()p Fq(\026")2424 5147 y Fi(\027)2466 5182 y Fu(\))p eop %%Page: 64 72 64 71 bop 456 251 a Fo(64)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(where)27 b Fq(\027)h Fu(=)23 b Fq(p)18 b Fp(\000)g Fq(`)p Fu(.)605 550 y(Since)29 b Fq(\027)h Fp(\025)24 b Fu(0)k(and)h Fq(\015)1259 562 y Fo(0)1296 550 y Fu(\()p Fq(t)19 b Fu(+)g Fq(s)p Fu(\))25 b Fp(2)g Fq(V)1685 562 y Fo(0)1723 550 y Fu(\()p Fq(\024)1803 514 y Fj(\006)1803 572 y Fo(1)1878 550 y Fp(\007)19 b Fu(1)p Fq(;)14 b Fu(2)p Fq(\024)2131 562 y Fo(2)2167 550 y Fq(=)p Fu(3)p Fq(;)g Fu(0)p Fq(;)g Fu(0\),)27 b(there)h(exist)h Fq(")2941 562 y Fo(0)3006 550 y Fu(and)g Fq(\026)3219 562 y Fo(0)3285 550 y Fu(suc)n(h)456 649 y(that,)38 b(for)e(all)f(0)i Fq(<)g(")f(<)h(")1325 661 y Fo(0)1362 649 y Fu(,)h Fp(j)p Fq(\026)p Fp(j)g Fq(<)e(\026)1708 661 y Fo(0)1782 649 y Fu(and)f(\()p Fq(t;)14 b(s)p Fu(\))37 b(suc)n(h)f(that)g Fq(T)48 b Fp(\024)37 b Fq(t)24 b Fu(+)f(Re)14 b Fq(s)37 b Fp(\024)g Fu(2)p Fq(T)47 b Fu(and)456 749 y Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)27 b(\024)f Fq(a)20 b Fp(\000)g Fq(")p Fu(,)30 b(the)h(manifolds)e Fq(\015)1599 719 y Fi(u)1594 770 y(\026;")1690 749 y Fu(\()p Fq(t;)14 b(s)p Fu(\))30 b(and)g Fq(\015)2102 719 y Fi(s)2097 770 y(\026;")2193 749 y Fu(\()p Fq(t;)14 b(s)p Fu(\))30 b(are)f(so)h(close)f(to)h Fq(\015)2987 761 y Fo(0)3024 749 y Fu(\()p Fq(t)20 b Fu(+)g Fq(s)p Fu(\))30 b(that)456 851 y(b)r(oth)e(b)r(elong)f(to)g Fq(U)37 b Fu(for)27 b(those)g(v)-5 b(alues)27 b(of)h(\()p Fq(t;)14 b(s)p Fu(\).)605 951 y(The)24 b(expressions)e Fp(S)6 b Fu(\()p Fq(\015)1337 921 y Fi(u)1332 972 y(\026;")1428 951 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))c Fp(\000)g Fu(\()p Fq(t)g Fp(\000)g Fq(t)2040 963 y Fo(0)2078 951 y Fu(\))24 b(and)g Fp(E)7 b Fu(\()p Fq(\015)2423 921 y Fi(u)2418 972 y(\026;")2514 951 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))23 b(are)g(w)n(ell)g(de\014ned)456 1053 y(for)36 b Fq(s)k Fp(2)f Fl(C)58 b Fu(suc)n(h)37 b(that)h Fq(T)50 b Fp(\024)39 b Fq(t)25 b Fu(+)f(Re)14 b Fq(s)39 b Fp(\024)g Fu(2)p Fq(T)48 b Fu(and)37 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)39 b(\024)g Fq(a)25 b Fp(\000)f Fq(")p Fu(.)66 b(Then)38 b(c)n(ho)r(osing)d Fq(t)456 1153 y Fu(arbitrarily)25 b(w)n(e)i(can)h(de\014ne)806 1305 y Fp(S)862 1270 y Fi(u)905 1305 y Fu(\()p Fq(s)p Fu(\))c(=)f Fp(S)6 b Fu(\()p Fq(\015)1256 1270 y Fi(u)1251 1325 y(\026;")1347 1305 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))k Fp(\000)g Fu(\()p Fq(t)h Fp(\000)f Fq(t)1992 1317 y Fo(0)2029 1305 y Fu(\),)277 b Fp(E)2412 1270 y Fi(u)2456 1305 y Fu(\()p Fq(s)p Fu(\))24 b(=)e Fp(E)7 b Fu(\()p Fq(\015)2801 1270 y Fi(u)2796 1325 y(\026;")2892 1305 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\).)-2809 b(\(6.4\))456 1456 y(Moreo)n(v)n(er,)19 b(as)g(a)h(consequence)g(of)g (Theorem)f(4.2,)i(they)g(do)f(not)g(dep)r(end)h(on)f(time.)35 b(W)-7 b(e)21 b(c)n(ho)r(ose)456 1556 y(it)28 b(in)f(suc)n(h)h(a)f(w)n (a)n(y)f(that)i Fq(T)34 b Fp(\024)23 b Fq(t)18 b Fu(+)g(Re)c Fq(s)23 b Fp(\024)g Fu(2)p Fq(T)12 b Fu(.)605 1715 y Ft(Lemma)31 b Fu(6.1)p Ft(.)40 b Fs(The)31 b(functions)f Fp(S)1675 1685 y Fi(u)1748 1715 y Fs(and)h Fp(E)1961 1685 y Fi(u)2034 1715 y Fs(satisfy)g(the)f(fol)t(lowing)i(pr)l(op)l (erties)673 1841 y(a\))45 b(The)d(functions)f Fp(S)1403 1811 y Fi(u)1447 1841 y Fu(\()p Fq(s)p Fu(\))27 b Fp(\000)g Fq(s)41 b Fs(and)g Fp(E)1972 1811 y Fi(u)2016 1841 y Fu(\()p Fq(s)p Fu(\))g Fs(ar)l(e)h Fu(2)p Fq(\031)s(")p Fs(-p)l(erio)l(dic)g(with)g(r)l(esp)l(e)l(ct)f(to)g Fq(s)p Fs(.)794 1940 y(Henc)l(e)e Fp(S)1108 1910 y Fi(u)1191 1940 y Fs(and)g Fp(E)1412 1910 y Fi(u)1495 1940 y Fs(c)l(an)g(b)l(e)g (analytic)l(al)t(ly)i(extende)l(d)e(for)h(al)t(l)g Fq(s)g Fp(2)g Fl(C)60 b Fs(such)39 b(that)794 2040 y Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)g Fq(a)18 b Fp(\000)g Fq(")p Fs(.)677 2140 y(b\))45 b(Mor)l(e)l(over,)40 b(for)c Fq(s)f Fp(2)f Fl(R)p Fs(,)44 b Fq(S)39 b Fu(=)34 b Fp(S)1862 2110 y Fi(u)1906 2140 y Fu(\()p Fq(s)p Fu(\))i Fs(is)h(r)l(e)l(al)f (analytic)h(and)g(invertible,)i(and)d(its)794 2239 y(inverse)e Fq(s)c Fu(=)f Fq(s)1285 2209 y Fi(u)1329 2239 y Fu(\()p Fq(S)5 b Fu(\))34 b Fs(satis\014es)f(that)h Fq(s)2011 2209 y Fi(u)2054 2239 y Fu(\()p Fq(S)5 b Fu(\))21 b Fp(\000)g Fq(S)38 b Fs(is)c Fq(O)r Fu(\()p Fq(\026")2649 2209 y Fi(p)p Fo(+1)2772 2239 y Fu(\))g Fs(and)g Fu(2)p Fq(\031)s(")p Fs(-p)l(erio)l(dic)794 2339 y(in)c Fq(S)5 b Fs(.)605 2498 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)26 b(pro)n(v)n(e)e(it)i(for)g Fp(S)1559 2468 y Fi(u)1602 2498 y Fu(\()p Fq(s)p Fu(\))16 b Fp(\000)e Fq(s)p Fu(.)36 b(By)26 b(prop)r(ert)n(y)f(\(6.3\))g(and)h (since)f Fp(S)2978 2468 y Fi(u)3022 2498 y Fu(\()p Fq(s)p Fu(\))16 b Fp(\000)e Fq(s)26 b Fu(do)r(es)456 2598 y(not)h(dep)r(end)h (on)g Fq(t)p Fu(,)f(w)n(e)h(ha)n(v)n(e)e(that)456 2749 y Fp(S)512 2715 y Fi(u)556 2749 y Fu(\()p Fq(s)18 b Fu(+)g(2)p Fq(\031)s(")p Fu(\))g Fp(\000)g Fu(\()p Fq(s)h Fu(+)f(2)p Fq(\031)s(")p Fu(\))83 b(=)g Fp(S)6 b Fu(\()p Fq(\015)1695 2715 y Fi(u)1690 2770 y(\026;")1786 2749 y Fu(\()p Fq(t;)14 b(s)k Fu(+)h(2)p Fq(\031)s(")p Fu(\))p Fq(;)14 b(t=")p Fu(\))j Fp(\000)h Fu(\()p Fq(t)h Fp(\000)f Fq(t)2663 2761 y Fo(0)2700 2749 y Fu(\))h Fp(\000)f Fu(\()p Fq(s)h Fu(+)f(2)p Fq(\031)s(")p Fu(\))1411 2877 y(=)83 b Fp(S)6 b Fu(\()p Fq(\015)1695 2842 y Fi(u)1690 2897 y(\026;")1786 2877 y Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\))p Fq(;)g Fu(\()p Fq(t)k Fu(+)g(2)p Fq(\031)s(")p Fu(\))p Fq(=")p Fu(\))g Fp(\000)g Fu([\()p Fq(t)h Fu(+)f(2)p Fq(\031)s(")p Fu(\))g Fp(\000)g Fq(t)3247 2889 y Fo(0)3284 2877 y Fu(])h Fp(\000)f Fq(s)1411 3004 y Fu(=)83 b Fp(S)1615 2970 y Fi(u)1658 3004 y Fu(\()p Fq(s)p Fu(\))19 b Fp(\000)f Fq(s)p Fu(.)456 3155 y(Analogously)-7 b(,)26 b Fp(E)994 3125 y Fi(u)1037 3155 y Fu(\()p Fq(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\))23 b(=)f Fp(E)1534 3125 y Fi(u)1578 3155 y Fu(\()p Fq(s)p Fu(\).)605 3255 y(Next)k(w)n(e)g(pro)n(v)n(e)f (the)h(second)g(part)f(of)h(this)h(lemma.)36 b(W)-7 b(e)26 b(recall)f(that,)i(b)n(y)f(Theorem)f(4.2:)1193 3407 y Fp(S)6 b Fu(\()p Fq(\015)1329 3372 y Fi(s)1324 3427 y(\026;")1420 3407 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))23 b(=)g Fq(t)18 b Fp(\000)g Fq(t)2042 3419 y Fo(0)2098 3407 y Fu(+)g Fq(s)g Fu(+)g Fq(\026")2410 3372 y Fi(p)p Fo(+1)2532 3407 y Fp(X)12 b Fu(\()p Fq(s)p Fu(\))456 3563 y(and)27 b(that,)h(for)f(\()p Fq(x;)14 b(y)s Fu(\))24 b Fp(2)f Fq(U)1370 3716 y Fp(S)6 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fp(S)1858 3728 y Fo(0)1895 3716 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")2375 3682 y Fi(p)p Fo(+1)2498 3716 y Fu(\))456 3868 y(where)27 b Fp(S)746 3880 y Fo(0)811 3868 y Fu(is)g(a)h(\015o)n(w)f(b)r(o)n(x)g (co)r(ordinate)f(when)i Fq(\026)23 b Fu(=)g(0.)36 b(Also,)27 b(b)n(y)h(Theorem)e(5.1,)1402 4019 y Fq(\015)1450 3985 y Fi(u)1445 4040 y(\026;")1540 4019 y Fu(\()p Fq(t;)14 b(s)p Fu(\))19 b Fp(\000)f Fq(\015)1855 4031 y Fo(0)1892 4019 y Fu(\()p Fq(t)h Fu(+)f Fq(s)p Fu(\))23 b(=)g Fq(O)r Fu(\()p Fq(\026")2424 3985 y Fi(\027)2466 4019 y Fu(\))456 4176 y(for)k(an)n(y)f Fq(t)d Fp(2)h Fl(R)33 b Fu(and)27 b Fq(s)c Fp(2)h Fl(C)48 b Fu(suc)n(h)27 b(that)h Fq(T)34 b Fp(\024)23 b Fq(t)18 b Fu(+)g(Re)13 b Fq(s)23 b Fp(\024)g Fu(2)p Fq(T)38 b Fu(and)27 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)g Fq(a)18 b Fp(\000)g Fq(")p Fu(.)36 b(Then)28 b(w)n(e)456 4275 y(obtain)f(that)872 4427 y Fp(S)928 4393 y Fi(u)972 4427 y Fu(\()p Fq(s)p Fu(\))19 b Fp(\000)f Fq(s)83 b Fu(=)f Fp(S)6 b Fu(\()p Fq(\015)1582 4393 y Fi(u)1577 4448 y(\026;")1673 4427 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))19 b Fp(\000)f Fu(\()p Fq(t)g Fp(\000)g Fq(t)2318 4439 y Fo(0)2356 4427 y Fu(\))g Fp(\000)g Fq(s)1299 4564 y Fu(=)82 b Fp(S)1496 4576 y Fo(0)1534 4564 y Fu(\()p Fq(\015)1609 4576 y Fo(0)1646 4564 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\)\))h Fp(\000)f Fu(\()p Fq(t)h Fp(\000)f Fq(t)2209 4576 y Fo(0)2246 4564 y Fu(\))h Fp(\000)f Fq(s)g Fu(+)g Fq(O)r Fu(\()p Fq(\026")2706 4529 y Fi(\027)2748 4564 y Fq(;)c(\026")2874 4529 y Fi(p)p Fo(+1)2996 4564 y Fu(\))1299 4688 y(=)82 b Fq(O)r Fu(\()p Fq(\026")1632 4654 y Fi(\027)1674 4688 y Fu(\).)-1273 b(\(6.5\))605 4840 y(Since)27 b Fp(S)877 4810 y Fi(u)921 4840 y Fu(\()p Fq(s)p Fu(\))18 b Fp(\000)f Fq(s)27 b Fu(is)g(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)f(in)h Fq(s)g Fu(and)g(analytic)g(in)g(the)g(complex)g(strip)g Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)456 4939 y Fq(a)18 b Fp(\000)g Fq(")p Fu(,)27 b(w)n(e)h(expand)f(it)h(in)g(F)-7 b(ourier)26 b(series,)1454 5103 y Fp(S)1510 5069 y Fi(u)1554 5103 y Fu(\()p Fq(s)p Fu(\))19 b Fp(\000)f Fq(s)23 b Fu(=)1909 5024 y Fg(X)1909 5203 y Fi(k)q Fj(2)p Fd(Z)2043 5103 y Fp(S)2099 5069 y Fi(u)2093 5124 y(k)2143 5103 y Fu(\()p Fq(")p Fu(\))p Fq(e)2285 5069 y Fi(ik)q(s=")p eop %%Page: 65 73 65 72 bop 1421 251 a Fo(6.2.)30 b(THE)f(SPLITTING)g(FUNCTION)900 b(65)456 450 y Fu(with)1225 627 y Fp(S)1281 593 y Fi(u)1275 648 y(k)1325 627 y Fu(\()p Fq(")p Fu(\))23 b(=)1594 571 y(1)p 1549 608 131 4 v 1549 684 a(2)p Fq(\031)s(")1703 514 y Fg(Z)1786 535 y Fo(2)p Fi(\031)r(")1749 703 y Fo(0)1896 627 y Fu(\()p Fp(S)1984 593 y Fi(u)2028 627 y Fu(\()p Fq(s)p Fu(\))c Fp(\000)f Fq(s)p Fu(\))p Fq(e)2343 593 y Fj(\000)p Fi(ik)q(s=")2570 627 y Fq(ds)p Fu(.)456 808 y(Moreo)n(v)n(er,)25 b(for)i Fq(s)c Fp(2)g Fl(R)34 b Fu(w)n(e)27 b(can)g(write,)757 1001 y Fp(S)813 967 y Fi(u)807 1022 y(k)857 1001 y Fu(\()p Fq(")p Fu(\))c(=)1126 945 y(1)p 1081 982 V 1081 1058 a(2)p Fq(\031)s(")1235 888 y Fg(Z)1318 909 y Fo(2)p Fi(\031)r(")1281 1077 y Fo(0)1442 1001 y Fu([)p Fp(S)1521 967 y Fi(u)1565 1001 y Fu(\()p Fq(s)18 b Fp(\006)g Fq(i)p Fu(\()p Fq(a)h Fp(\000)f Fq(")p Fu(\)\))g Fp(\000)g Fu(\()p Fq(s)h Fp(\006)f Fq(i)p Fu(\()p Fq(a)g Fp(\000)g Fq(")p Fu(\)\)])c Fq(e)2706 967 y Fj(\000)p Fi(ik)q Fo(\()p Fi(s)p Fj(\006)p Fi(i)p Fo(\()p Fi(a)p Fj(\000)p Fi(")p Fo(\)\))p Fi(=")3231 1001 y Fq(ds)456 1186 y Fu(Th)n(us,)27 b(using)g(the)h(estimate)g (\(6.5\),)f(for)g Fq(k)f Fp(6)p Fu(=)d(0,)k(w)n(e)g(obtain)611 1384 y Fp(S)667 1350 y Fi(u)661 1405 y(k)711 1384 y Fu(\()p Fq(")p Fu(\))83 b(=)1055 1328 y Fq(e)1094 1298 y Fj(\000j)p Fi(k)q Fj(j)p Fo(\()p Fi(a)p Fj(\000)p Fi(")p Fo(\))p Fi(=")p 1055 1365 408 4 v 1193 1441 a Fu(2)p Fq(\031)s(")1486 1271 y Fg(Z)1569 1292 y Fo(2)p Fi(\031)r(")1532 1460 y Fo(0)1692 1384 y Fu([)q Fp(S)1772 1350 y Fi(u)1815 1384 y Fu(\()p Fq(s)19 b Fp(\006)f Fq(i)p Fu(\()p Fq(a)g Fp(\000)g Fq(")p Fu(\)\))h Fp(\000)f Fu(\()p Fq(s)h Fp(\006)f Fq(i)p Fu(\()p Fq(a)g Fp(\000)g Fq(")p Fu(\)\)])c Fq(e)2957 1350 y Fj(\000)p Fi(ik)q(s=")3184 1384 y Fq(ds)p Fu(.)897 1571 y(=)83 b Fq(O)r Fu(\()p Fq(\026")1231 1537 y Fi(\027)1273 1571 y Fu(\))p Fq(e)1344 1537 y Fj(\000j)p Fi(k)q Fj(j)p Fi(a=")1577 1571 y Fu(.)456 1710 y(where)31 b(w)n(e)h(consider)f(the)h (sign)g(+)g(for)f Fq(k)j(<)c Fu(0)i(and)g(the)g(sign)g Fp(\000)f Fu(for)h Fq(k)h(<)e Fu(0.)50 b(Summing)32 b(the)456 1809 y(F)-7 b(ourier)26 b(series)h(and)g(applying)g(the)h(ab)r(o)n(v)n (e)e(equalit)n(y)-7 b(,)27 b(w)n(e)h(deduce)1315 1951 y Fp(S)1371 1916 y Fi(u)1415 1951 y Fu(\()p Fq(s)p Fu(\))19 b Fp(\000)f Fq(s)83 b Fu(=)g Fp(S)1946 1916 y Fi(u)1940 1971 y Fo(0)1989 1951 y Fu(\()p Fq(")p Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")2380 1916 y Fi(\027)2422 1951 y Fu(\))p Fq(e)2493 1916 y Fj(\000)p Fi(a=")1259 2062 y Fq(d)p Fp(S)1358 2032 y Fi(u)p 1259 2099 144 4 v 1290 2175 a Fq(ds)1412 2118 y Fu(\()p Fq(s)p Fu(\))h Fp(\000)f Fu(1)83 b(=)g Fq(O)r Fu(\()p Fq(\026")2076 2084 y Fi(\027)2118 2118 y Fu(\))p Fq(e)2189 2084 y Fj(\000)p Fi(a=")2346 2118 y Fu(,)456 2298 y(\(taking)27 b(in)n(to)g(accoun)n(t)g(that)1409 2266 y Fi(d)p Fj(S)1489 2241 y Fc(u)p 1409 2280 119 4 v 1435 2327 a Fi(ds)1538 2298 y Fu(\()p Fq(s)p Fu(\))19 b Fp(\000)f Fu(1)27 b(has)g(zero)f(mean\).)605 2398 y(This)35 b(implies)h(that)g Fq(S)k Fu(=)c Fp(S)1528 2368 y Fi(u)1572 2398 y Fu(\()p Fq(s)p Fu(\))g(is)f(in)n(v)n(ertible.)59 b(On)35 b(the)h(other)f(hand,)i(if)f Fq(s)g Fp(2)g Fl(R)p Fu(,)44 b(b)n(y)456 2498 y(Remark)26 b(5.2)h(w)n(e)g(ha)n(v)n(e)g(that) 1350 2636 y Fq(\015)1398 2602 y Fi(u)1393 2657 y(\026;")1488 2636 y Fu(\()p Fq(t;)14 b(s)p Fu(\))19 b Fp(\000)f Fq(\015)1803 2648 y Fo(0)1840 2636 y Fu(\()p Fq(t)h Fu(+)f Fq(s)p Fu(\))23 b(=)g Fq(O)r Fu(\()p Fq(\026")2372 2602 y Fi(p)p Fo(+1)2495 2636 y Fu(\),)456 2775 y(hence)1541 2914 y Fp(S)1597 2879 y Fi(u)1640 2914 y Fu(\()p Fq(s)p Fu(\))c Fp(\000)f Fq(s)23 b Fu(=)g Fq(O)r Fu(\()p Fq(\026")2181 2879 y Fi(p)p Fo(+1)2304 2914 y Fu(\).)456 3063 y(Therefore)31 b Fp(S)893 3033 y Fi(u)887 3083 y Fo(0)937 3063 y Fu(\()p Fq(")p Fu(\))g(=)f Fq(O)r Fu(\()p Fq(\026")1352 3033 y Fi(p)p Fo(+1)1475 3063 y Fu(\).)52 b(W)-7 b(e)33 b(denote)f Fq(s)f Fu(=)f Fq(s)2206 3033 y Fi(u)2250 3063 y Fu(\()p Fq(S)5 b Fu(\))32 b(its)h(in)n(v)n(erse)e(whic)n(h)h(is)g(analytic.)456 3162 y(Moreo)n(v)n(er)22 b Fq(s)857 3132 y Fi(u)900 3162 y Fu(\()p Fq(S)5 b Fu(\))13 b Fp(\000)g Fq(S)27 b Fu(=)c Fq(O)r Fu(\()p Fq(\026")1463 3132 y Fi(p)p Fo(+1)1586 3162 y Fu(\).)36 b(T)-7 b(o)25 b(see)f(that)h Fq(s)2144 3132 y Fi(u)2188 3162 y Fu(\()p Fq(S)5 b Fu(\))13 b Fp(\000)g Fq(S)29 b Fu(is)24 b(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic,)h(w)n(e)f (observ)n(e)456 3262 y(that,)k(b)n(y)f(a\),)g Fp(S)954 3232 y Fi(u)998 3262 y Fu(\()p Fq(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\))23 b(=)f Fp(S)1500 3232 y Fi(u)1544 3262 y Fu(\()p Fq(s)p Fu(\))d(+)f(2)p Fq(\031)s(")p Fu(,)28 b(th)n(us)873 3400 y Fq(s)912 3366 y Fi(u)955 3400 y Fu(\()p Fq(S)23 b Fu(+)18 b(2)p Fq(\031)s(")p Fu(\))h Fp(\000)f Fu(\()p Fq(S)23 b Fu(+)18 b(2)p Fq(\031)s(")p Fu(\))83 b(=)f Fq(s)2030 3366 y Fi(u)2074 3400 y Fu(\()p Fp(S)2162 3366 y Fi(u)2206 3400 y Fu(\()p Fq(s)p Fu(\))19 b(+)f(2)p Fq(\031)s(")p Fu(\))g Fp(\000)g Fu(\()p Fq(S)24 b Fu(+)18 b(2)p Fq(\031)s(")p Fu(\))1844 3525 y(=)82 b Fq(s)2030 3491 y Fi(u)2074 3525 y Fu(\()p Fp(S)2162 3491 y Fi(u)2206 3525 y Fu(\()p Fq(s)19 b Fu(+)f(2)p Fq(\031)s(")p Fu(\)\))g Fp(\000)g Fu(\()p Fq(S)24 b Fu(+)18 b(2)p Fq(\031)s(")p Fu(\))1844 3650 y(=)82 b Fq(s)19 b Fu(+)f(2)p Fq(\031)s(")g Fp(\000)g Fu(\()p Fq(S)23 b Fu(+)18 b(2)p Fq(\031)s(")p Fu(\))1844 3774 y(=)82 b Fq(s)2030 3740 y Fi(u)2074 3774 y Fu(\()p Fq(S)5 b Fu(\))18 b Fp(\000)h Fq(S)456 3913 y Fu(as)27 b(w)n(e)g(w)n(an)n(ted.)2418 b Fe(\003)605 4068 y Fu(No)n(w)20 b(w)n(e)g(de\014ne)g(the)h(splitting)g(function.)35 b(F)-7 b(rom)20 b(Theorem)f(4.2)h(it)g(follo)n(ws)g(that)g(the)h(lo)r (cal)456 4168 y(stable)27 b(manifold)h Fq(\015)1088 4138 y Fi(s)1083 4188 y(\026)1127 4168 y Fu(\()p Fq(t;)14 b(s)p Fu(\))28 b(\(for)g(\()p Fq(t;)14 b(s)p Fu(\))28 b(suc)n(h)f(that)h Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)g Fq(a)18 b Fp(\000)g Fq(")28 b Fu(and)f Fq(T)35 b Fp(\024)22 b Fq(t)d Fu(+)f(Re)c Fq(s)23 b Fp(\024)g Fu(2)p Fq(T)12 b Fu(\))456 4270 y(can)27 b(b)r(e)h(written)g(in)f(the)h(\()p Fq(S;)14 b(E)5 b Fu(\))29 b(co)r(ordinates:)705 4410 y(\()p Fq(S;)14 b(E)5 b Fu(\))24 b(=)f(\()p Fp(S)6 b Fu(\()p Fq(\015)1203 4376 y Fi(s)1198 4430 y(\026;")1294 4410 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))p Fq(;)g Fp(E)7 b Fu(\()p Fq(\015)1812 4376 y Fi(s)1807 4430 y(\026;")1903 4410 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\)\))23 b(=)g(\()p Fq(t)c Fp(\000)f Fq(t)2590 4422 y Fo(0)2645 4410 y Fu(+)g Fq(s)h Fu(+)f Fq(\026")2958 4376 y Fi(p)p Fo(+1)3080 4410 y Fp(X)12 b Fu(\()p Fq(s)p Fu(\))p Fq(;)i Fu(0\))-2909 b(\(6.6\))456 4553 y(and)32 b(the)i(lo)r(cal)e(unstable)h(manifold)g Fq(\015)1707 4523 y Fi(u)1702 4574 y(\026;")1797 4553 y Fu(\()p Fq(t;)14 b(s)p Fu(\))34 b(\(for)e(\()p Fq(t;)14 b(s)p Fu(\))34 b(suc)n(h)e(that)h Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)32 b(\024)g Fq(a)21 b Fp(\000)h Fq(")33 b Fu(and)456 4656 y Fq(T)h Fp(\024)22 b Fq(t)d Fu(+)f(Re)c Fq(s)23 b Fp(\024)f Fu(2)p Fq(T)12 b Fu(\))27 b(can)g(b)r(e)h(expressed)e(as)777 4794 y(\()p Fq(S;)14 b(E)5 b Fu(\))24 b(=)e(\()p Fp(S)6 b Fu(\()p Fq(\015)1274 4760 y Fi(u)1269 4815 y(\026;")1366 4794 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))p Fq(;)g Fp(E)7 b Fu(\()p Fq(\015)1884 4760 y Fi(u)1879 4815 y(\026;")1975 4794 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\)\))23 b(=)f(\()p Fq(t)d Fp(\000)f Fq(t)2661 4806 y Fo(0)2717 4794 y Fu(+)g Fp(S)2856 4760 y Fi(u)2900 4794 y Fu(\()p Fq(s)p Fu(\))p Fq(;)c Fp(E)3091 4760 y Fi(u)3135 4794 y Fu(\()p Fq(s)p Fu(\)\).)605 4933 y(W)-7 b(e)28 b(consider)f(the)h(P)n(oincar)n(\023)-39 b(e)24 b(map)1321 5072 y Fq(P)1386 5038 y Fi(t)1411 5046 y Fh(0)1374 5093 y Fi(\026;")1470 5072 y Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e Fq(')1827 5084 y Fi(\026;")1923 5072 y Fu(\(2)p Fq(\031)s(")c Fu(+)g Fq(t)2217 5084 y Fo(0)2255 5072 y Fq(;)c(t)2322 5084 y Fo(0)2359 5072 y Fq(;)g(x;)g(y)s Fu(\),)456 5216 y(where)27 b Fq(')750 5228 y Fi(\026)795 5216 y Fu(\()p Fq(t;)14 b(t)924 5228 y Fo(0)961 5216 y Fq(;)g(x;)g(y)s Fu(\))28 b(is)f(the)h(solution)f(of)h(system)f (\(6.1\).)p eop %%Page: 66 74 66 73 bop 456 251 a Fo(66)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(The)34 b(restriction)f(to)h Fq(U)43 b Fu(of)34 b(the)h(unstable)f (curv)n(e)f Fq(C)2277 420 y Fi(u)2354 450 y Fu(of)h Fq(P)2520 420 y Fi(t)2545 428 y Fh(0)2508 471 y Fi(\026;")2604 450 y Fu(,)i(is)e(giv)n(en)f(b)n(y)h Fq(\015)3146 420 y Fi(u)3141 471 y(\026;")3237 450 y Fu(\()p Fq(t)3299 462 y Fo(0)3336 450 y Fq(;)14 b(s)p Fu(\))456 552 y(parametrized)35 b(b)n(y)h Fq(s)i Fp(2)g Fl(C)58 b Fu(suc)n(h)36 b(that)h Fq(T)49 b Fp(\024)37 b Fu(Re)14 b Fq(s)38 b Fp(\024)g Fu(2)p Fq(T)47 b Fu(and)36 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)38 b(\024)f Fq(a)25 b Fp(\000)f Fq(")p Fu(.)63 b(Indeed,)456 652 y(let)31 b Fq(W)669 622 y Fj(\003)p Fi(;)p Fo(+)778 652 y Fu(\()p Fq(P)875 622 y Fi(t)900 630 y Fh(0)863 673 y Fi(\026;")959 652 y Fq(;)14 b Fu(0\))32 b(\()p Fp(\003)d Fu(=)g Fq(s;)14 b(u)p Fu(\))32 b(b)r(e)g(the)g(righ)n(t)e (hand)i(side)f(of)h(the)g(stable)f(and)h(the)g(unstable)456 758 y(in)n(v)-5 b(arian)n(t)32 b(curv)n(es)h(of)g(the)i(origin)d(of)i (the)g(map)g Fq(P)2068 728 y Fi(t)2093 736 y Fh(0)2056 779 y Fi(\026;")2152 758 y Fu(.)55 b(Since)34 b(the)g (parameterizations)e Fq(\015)3331 728 y Fj(\003)3326 779 y Fi(\026;")3421 758 y Fu(,)456 861 y Fp(\003)22 b Fu(=)h Fq(s;)14 b(u)p Fu(,)27 b(as)g(functions)h(of)f Fq(t)h Fu(are)f(solutions)f(of)i(the)g(system)f(\(6.1\),)h(w)n(e)f(ha)n (v)n(e)f(that)627 1016 y Fq(C)692 981 y Fi(s)810 1016 y Fp(\021)83 b(f)p Fq(\015)1048 981 y Fi(s)1043 1036 y(\026;")1138 1016 y Fu(\()p Fq(t)1200 1028 y Fo(0)1238 1016 y Fq(;)14 b(s)p Fu(\))23 b(:)g(Re)14 b Fq(s)k Fu(+)g Fq(t)1697 1028 y Fo(0)1757 1016 y Fp(\025)23 b Fq(T)12 b Fu(,)110 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)f Fq(a)d Fp(\000)f Fq(")p Fp(g)k(\032)h Fq(W)2788 981 y Fi(s;)p Fo(+)2894 1016 y Fu(\()p Fq(P)2991 981 y Fi(t)3016 989 y Fh(0)2979 1036 y Fi(\026;")3075 1016 y Fq(;)14 b Fu(0\))619 1151 y Fq(C)684 1117 y Fi(u)810 1151 y Fp(\021)83 b(f)p Fq(\015)1048 1117 y Fi(u)1043 1171 y(\026;")1138 1151 y Fu(\()p Fq(t)1200 1163 y Fo(0)1238 1151 y Fq(;)14 b(s)p Fu(\))23 b(:)g(Re)14 b Fq(s)k Fu(+)g Fq(t)1697 1163 y Fo(0)1757 1151 y Fp(\024)23 b(\000)p Fq(T)12 b Fu(,)109 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)g Fq(a)18 b Fp(\000)g Fq(")p Fp(g)23 b(\032)f Fq(W)2852 1117 y Fi(u;)p Fo(+)2966 1151 y Fu(\()p Fq(P)3063 1117 y Fi(t)3088 1125 y Fh(0)3051 1171 y Fi(\026;")3148 1151 y Fq(;)14 b Fu(0\).)456 1306 y(Moreo)n(v)n(er,)25 b(since)1124 1461 y Fq(\015)1172 1427 y Fj(\003)1167 1482 y Fi(\026;")1262 1461 y Fu(\()p Fq(t)19 b Fu(+)f(2)p Fq(\031)s(";)c(s)p Fu(\))23 b(=)f Fq(\015)1823 1427 y Fj(\003)1818 1482 y Fi(\026;")1914 1461 y Fu(\()p Fq(t;)14 b(s)k Fu(+)g(2)p Fq(\031)s(")p Fu(\),)139 b Fp(\003)22 b Fu(=)h Fq(s;)14 b(u)p Fu(,)456 1617 y(in)27 b(their)h(resp)r(ectiv)n(e)f(domains,)g(w)n (e)g(ha)n(v)n(e)f(that)970 1772 y Fq(P)1035 1738 y Fi(t)1060 1746 y Fh(0)1023 1792 y Fi(\026;")1119 1772 y Fu(\()p Fq(\015)1199 1738 y Fj(\003)1194 1792 y Fi(\026;")1290 1772 y Fu(\()p Fq(t)1352 1784 y Fo(0)1389 1772 y Fq(;)14 b(s)p Fu(\)\))23 b(=)g Fq(\015)1688 1738 y Fj(\003)1683 1792 y Fi(\026;")1779 1772 y Fu(\(2)p Fq(\031)s(")18 b Fu(+)g Fq(t)2073 1784 y Fo(0)2110 1772 y Fq(;)c(s)p Fu(\))23 b(=)g Fq(\015)2377 1738 y Fj(\003)2372 1792 y Fi(\026;")2468 1772 y Fu(\()p Fq(t)2530 1784 y Fo(0)2567 1772 y Fq(;)14 b(s)k Fu(+)g(2)p Fq(\031)s(")p Fu(\),)456 1936 y(whic)n(h)33 b(means)f(that)h(if)h(w)n(e)e(consider)g Fq(s)h Fu(as)g(the)g(v)-5 b(ariable)32 b(in)h Fq(C)2493 1906 y Fi(s)2561 1936 y Fp(\032)e Fq(W)2747 1906 y Fi(s;)p Fo(+)2853 1936 y Fu(\()p Fq(P)2950 1906 y Fi(t)2975 1914 y Fh(0)2938 1956 y Fi(\026;")3035 1936 y Fq(;)14 b Fu(0\))32 b(the)i(dy-)456 2042 y(namics)27 b(of)g Fq(P)892 2012 y Fi(t)917 2020 y Fh(0)880 2062 y Fi(\026;")1004 2042 y Fu(on)g Fq(C)1184 2012 y Fi(s)1248 2042 y Fu(is)g(just)1719 2197 y Fq(s)c Fp(7!)g Fq(s)18 b Fu(+)g(2)p Fq(\031)s(")p Fu(.)605 2357 y(Therefore)26 b(in)i(the)g(\()p Fq(S;)14 b(E)5 b Fu(\))29 b(v)-5 b(ariables,)26 b Fq(C)1900 2327 y Fi(u)1971 2357 y Fu(is)i(represen)n(ted)e(b)n(y)853 2512 y(\()p Fq(S;)14 b(E)5 b Fu(\))23 b(=)g(\()p Fp(S)6 b Fu(\()p Fq(\015)1350 2478 y Fi(u)1345 2533 y(\026;")1442 2512 y Fu(\()p Fq(t)1504 2524 y Fo(0)1541 2512 y Fq(;)14 b(s)p Fu(\))p Fq(;)g(t)1716 2524 y Fo(0)1753 2512 y Fq(=")p Fu(\))p Fq(;)g Fp(E)7 b Fu(\()p Fq(\015)2034 2478 y Fi(u)2029 2533 y(\026;")2125 2512 y Fu(\()p Fq(t)2187 2524 y Fo(0)2224 2512 y Fq(;)14 b(s)p Fu(\))p Fq(;)g(t)2399 2524 y Fo(0)2437 2512 y Fq(=")p Fu(\)\))22 b(=)h(\()p Fp(S)2780 2478 y Fi(u)2824 2512 y Fu(\()p Fq(s)p Fu(\))p Fq(;)14 b Fp(E)3015 2478 y Fi(u)3059 2512 y Fu(\()p Fq(s)p Fu(\)\).)456 2672 y(By)26 b(prop)r(ert)n(y)f(b\))h(of)h(Lemma)f(6.1,)f(the)i(equalit)n(y) f Fq(S)h Fu(=)c Fp(S)2250 2642 y Fi(u)2294 2672 y Fu(\()p Fq(s)p Fu(\))k(can)f(b)r(e)g(in)n(v)n(erted)g(for)g(v)-5 b(alues)26 b(of)456 2772 y Fq(s)h Fu(suc)n(h)g(that)h Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b Fq(<)g(a)18 b Fp(\000)g Fq(")p Fu(,)27 b Fq(s)c Fu(=)g Fq(s)1635 2742 y Fi(u)1678 2772 y Fu(\()p Fq(S)5 b Fu(\),)28 b(th)n(us)g(the)g(function)g Fq(\036)g Fu(whic)n(h)f(giv)n(es)g(the)h(v)-5 b(ariable)456 2872 y Fq(E)32 b Fu(as)27 b(a)g(function)i(of)e Fq(S)5 b Fu(,)28 b(is,)f(in)h(fact,)g(de\014ned)g(explicitly)g(b)n(y:)1617 3027 y Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))24 b(=)f Fp(E)1949 2992 y Fi(u)1992 3027 y Fu(\()p Fq(s)2063 2992 y Fi(u)2107 3027 y Fu(\()p Fq(S)5 b Fu(\)\).)-1826 b(\(6.7\))456 3182 y(W)-7 b(e)28 b(observ)n(e)d(that)j(the)g(splitting)g(function)g (is)g(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)f(and)g(hence)h(is)f (de\014ned)h(on)f Fl(R)p Fu(.)605 3282 y(The)i(parameterization)e(for)i (the)h(unstable)f(manifold,)h Fq(\015)2445 3252 y Fi(u)2440 3302 y(\026;")2535 3282 y Fu(\()p Fq(t;)14 b(s)p Fu(\),)30 b(in)n(tro)r(duced)f(in)h(The-)456 3384 y(orem)35 b(3.1)g(is)h(not)g (uniquely)h(determined.)62 b(Indeed,)39 b(if)e(w)n(e)e(tak)n(e)h Fq(s)h Fu(=)g Fq(S)29 b Fu(+)23 b Fq(\045)p Fu(\()p Fq(S)5 b Fu(\))37 b(where)e Fq(\045)456 3484 y Fu(is)g(a)h(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)e(function)j(whic)n(h)e(is)h Fq(O)r Fu(\()p Fq(\026")1966 3454 y Fi(p)p Fo(+1)2089 3484 y Fu(\),)i(then)i(\026)-45 b Fq(\015)2428 3454 y Fi(u)2423 3504 y(\026;")2518 3484 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\))37 b(=)f Fq(\015)2891 3454 y Fi(u)2886 3504 y(\026;")2982 3484 y Fu(\()p Fq(t;)14 b(S)28 b Fu(+)c Fq(\045)p Fu(\()p Fq(S)5 b Fu(\)\))456 3583 y(is)31 b(another)f (parameterization)f(whic)n(h)i(also)f(satis\014es)h(has)f(all)h(prop)r (erties)f(w)n(e)h(ha)n(v)n(e)f(pro)n(v)n(ed)456 3683 y(un)n(til)e(no)n(w.)605 3783 y(Since)i Fq(s)863 3753 y Fi(u)907 3783 y Fu(\()p Fq(S)5 b Fu(\))20 b Fp(\000)g Fq(S)35 b Fu(is)30 b Fq(O)r Fu(\()p Fq(\026")1490 3753 y Fi(p)p Fo(+1)1613 3783 y Fu(\))h(and)f(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)f(in)h Fq(S)35 b Fu(w)n(e)30 b(can)g(in)n(tro)r (duce)g(the)g(new)456 3882 y(parameterization)25 b(for)i(the)h (unstable)g(manifold)1487 4038 y(~)-45 b Fq(\015)1532 4003 y Fi(u)1527 4058 y(\026;")1623 4038 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\))23 b(=)f Fq(\015)1968 4003 y Fi(u)1963 4058 y(\026;")2059 4038 y Fu(\()p Fq(t;)14 b(s)2197 4003 y Fi(u)2240 4038 y Fu(\()p Fq(S)5 b Fu(\)\).)456 4193 y(W)-7 b(e)28 b(do)f(not)g(c)n(hange)g(the)h(parameterization)d(for)j (the)g(stable)f(manifold)1560 4348 y(~)-45 b Fq(\015)1605 4314 y Fi(s)1600 4369 y(\026;")1696 4348 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\))23 b(=)g Fq(\015)2042 4314 y Fi(s)2037 4369 y(\026;")2132 4348 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\).)456 4508 y(Finally)-7 b(,)31 b(after)g(this)g(c)n(hange)f(of)h(parameter,)f (the)h(splitting)g(function)h(de\014ned)f(in)g(\(6.7\))g(can)456 4608 y(b)r(e)d(also)e(represen)n(ted)h(in)g(the)h(form)1504 4763 y Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))24 b(=)e Fp(E)7 b Fu(\()s(~)-45 b Fq(\015)1915 4729 y Fi(u)1910 4784 y(\026;")2006 4763 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\))p Fq(;)14 b(t=")p Fu(\).)-1940 b(\(6.8\))605 4927 y(No)n(w)40 b(w)n(e)h(state)g(t)n(w)n(o)f(tec)n(hnical)g(lemmas)h(from)f([)p Fk(DS2)p Fu(].)77 b(Giv)n(en)41 b(\()p Fq(t;)14 b(s)p Fu(\))46 b Fp(2)f Fq(D)3175 4897 y Fo(ext)3173 4947 y Fi(")3311 4927 y Fu(and)456 5026 y Fq(\030)27 b Fu(:)c Fq(D)636 4996 y Fo(ext)634 5047 y Fi(")754 5026 y Fp(!)g Fl(C)914 4996 y Fo(2)957 5026 y Fu(,)28 b(w)n(e)f(in)n(tro)r(duce)g Fq(\034)33 b Fu(=)22 b Fp(j)p Fq(t)d Fu(+)f Fq(s)g Fp(\000)g Fq(ia)p Fp(j)28 b Fu(and)1370 5182 y Fp(j)p Fq(\030)t Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fp(j)1626 5194 y Fi(\034)1692 5182 y Fu(=)22 b Fp(j)p Fq(\030)1838 5194 y Fo(1)1876 5182 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fp(j)19 b Fu(+)f Fq(\034)9 b Fp(j)p Fq(\030)2275 5194 y Fo(2)2313 5182 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fp(j)p Fu(.)p eop %%Page: 67 75 67 74 bop 1421 251 a Fo(6.2.)30 b(THE)f(SPLITTING)g(FUNCTION)900 b(67)605 450 y Ft(Lemma)31 b Fu(6.2)p Ft(.)40 b Fs(F)-6 b(or)30 b Fq(t)p Fs(,)h Fq(t)1359 462 y Fo(0)1396 450 y Fs(,)f Fq(l)h Fs(r)l(e)l(al)f(and)g Fq(s)g Fs(c)l(omplex,)h(such)f (that)f Fu(0)23 b Fp(\024)g Fu(Im)14 b Fq(s)23 b(<)f(a)30 b Fs(and)1012 589 y Fp(\000)p Fu(2)p Fq(T)j Fp(\024)22 b Fq(t)1318 601 y Fo(0)1374 589 y Fu(+)c(Re)c Fq(s)23 b Fp(\024)g Fq(t)18 b Fu(+)g(Re)c Fq(s)23 b Fp(\024)f Fu(2)p Fq(T)12 b Fs(,)178 b Fq(t)2447 601 y Fo(0)2502 589 y Fu(+)18 b(Re)c Fq(s)23 b(<)g Fu(0)456 728 y Fs(we)30 b(denote)1089 945 y Fq(\032)1132 910 y Fj(\000)p Fi(l)1132 973 y Fo([)p Fi(t)1176 981 y Fh(0)1208 973 y Fi(;t)p Fo(])1276 945 y Fu(\()p Fq(s)p Fu(\))23 b Fp(\021)1490 775 y Fg(8)1490 850 y(<)1490 999 y(:)1605 891 y Fu(sup)1951 834 y(1)p 1754 871 437 4 v 1754 947 a Fp(j)p Fq(\033)f Fu(+)c Fq(s)g Fp(\000)g Fq(ia)p Fp(j)2165 924 y Fi(l)2200 891 y Fs(,)262 b(if)30 b Fq(l)25 b Fp(6)p Fu(=)d(0)1605 1038 y(sup)14 b Fp(j)g Fu(ln\()p Fp(j)p Fq(\033)22 b Fu(+)c Fq(s)h Fp(\000)f Fq(ia)p Fp(j)p Fu(\))p Fp(j)p Fs(,)113 b(if)30 b Fq(l)25 b Fu(=)d(0)456 1183 y Fs(wher)l(e)30 b(the)g(supr)l(emum)f(is)h(taken)g(for)g Fq(\033)d Fp(2)c Fu([)p Fq(t)1877 1195 y Fo(0)1915 1183 y Fq(;)14 b(t)p Fu(])p Fs(.)605 1283 y(Then)30 b(ther)l(e)g(exists)g(a)g(c)l(onstant)f Fq(K)35 b Fs(which)c(only)g(dep)l(ends)f(on)g Fq(l)h Fs(such)f(that)1372 1362 y Fg(Z)1455 1382 y Fi(t)1418 1550 y(t)1443 1558 y Fh(0)1680 1418 y Fq(d\033)p 1508 1456 V 1508 1532 a Fp(j)p Fq(\033)22 b Fu(+)c Fq(s)h Fp(\000)f Fq(ia)p Fp(j)1920 1508 y Fi(l)1978 1475 y Fp(\024)k Fq(K)6 b(\032)2185 1431 y Fj(\000)p Fo(\()p Fi(l)p Fj(\000)p Fo(1\))2185 1503 y([)p Fi(t)2229 1511 y Fh(0)2261 1503 y Fi(;t)p Fo(])2399 1475 y Fu(\()p Fq(s)p Fu(\))p Fs(.)-2071 b Fu(\(6.9\))605 1681 y Ft(Lemma)37 b Fu(6.3)p Ft(.)43 b Fs(L)l(et)34 b Fq(\016)1282 1693 y Fo(0)1351 1681 y Fp(2)f Fu(\(0)p Fq(;)14 b Fu(1\))34 b Fs(and)h(let)g Fq(\016)g Fu(:)d([0)p Fq(;)14 b Fu(+)p Fp(1)p Fu(\))31 b Fp(!)h Fl(R)40 b Fs(b)l(e)35 b(a)g(function)g(such)f(that)456 1781 y Fq(\016)s Fu(\()p Fq(\034)9 b Fu(\))27 b Fp(\024)e Fq(\016)759 1793 y Fo(0)796 1781 y Fq(=\034)883 1751 y Fi(r)r Fj(\000)p Fo(1)1005 1781 y Fs(.)43 b(Supp)l(ose)31 b(that)h Fq(\030)t Fu(\()p Fq(t;)14 b(s)p Fu(\))31 b Fs(and)1975 1759 y Fu(\026)1966 1781 y Fq(\030)t Fu(\()p Fq(t;)14 b(s)p Fu(\))32 b Fs(ar)l(e)f(two)g(functions)h(de\014ne)l(d)f (in)g Fq(D)3324 1751 y Fo(ext)3322 1801 y Fi(")3419 1781 y Fs(.)456 1881 y(We)e(wil)t(l)i(write)g Fq(\030)t Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b(=)g Fq(\030)k Fu(=)c(\()p Fq(\030)1515 1893 y Fo(1)1552 1881 y Fq(;)14 b(\030)1625 1893 y Fo(2)1663 1881 y Fu(\))30 b Fs(and)1895 1859 y Fu(\026)1886 1881 y Fq(\030)t Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b(=)2216 1859 y(\026)2207 1881 y Fq(\030)k Fu(=)22 b(\()2399 1859 y(\026)2390 1881 y Fq(\030)2426 1893 y Fo(1)2464 1881 y Fq(;)2509 1859 y Fu(\026)2501 1881 y Fq(\030)2537 1893 y Fo(2)2574 1881 y Fu(\))p Fs(.)39 b(Assume)29 b(that)1661 2019 y Fp(j)p Fq(\030)t Fp(j)1747 2031 y Fi(\034)1788 2019 y Fq(;)14 b Fp(j)1857 1997 y Fu(\026)1848 2019 y Fq(\030)t Fp(j)1911 2031 y Fi(\034)1976 2019 y Fp(\024)23 b Fq(\016)s Fu(\()p Fq(\034)9 b Fu(\))p Fs(.)456 2158 y(Then,)30 b(we)h(have)f(that)647 2347 y Fp(j)p Fq(V)737 2312 y Fj(0)760 2347 y Fu(\()p Fq(\013)845 2359 y Fo(0)882 2347 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))h(+)f Fq(\030)1255 2359 y Fo(1)1292 2347 y Fu(\))h Fp(\000)f Fq(V)1493 2312 y Fj(0)1516 2347 y Fu(\()p Fq(\013)1601 2359 y Fo(0)1639 2347 y Fu(\()p Fq(t)h Fu(+)f Fq(s)p Fu(\))g(+)1984 2325 y(\026)1975 2347 y Fq(\030)2011 2359 y Fo(1)2049 2347 y Fu(\))p Fp(j)83 b(\024)g Fq(K)2421 2291 y Fp(j)p Fq(\030)2480 2303 y Fo(1)2536 2291 y Fp(\000)2628 2269 y Fu(\026)2619 2291 y Fq(\030)2655 2303 y Fo(1)2693 2291 y Fp(j)p 2421 2328 295 4 v 2527 2404 a Fq(\034)2572 2380 y Fo(2)2726 2347 y Fs(,)179 b Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)h Fq(D)3273 2312 y Fo(ext)3271 2367 y Fi(")3367 2347 y Fs(,)493 2562 y Fp(j)p Fq(g)s Fu(\()p Fq(\015)634 2574 y Fo(0)671 2562 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))h(+)f Fq(\030)t(;)c(t=")p Fu(\))k Fp(\000)g Fq(g)s Fu(\()p Fq(\015)1447 2574 y Fo(0)1484 2562 y Fu(\()p Fq(t)h Fu(+)f Fq(s)p Fu(\))g(+)1829 2540 y(\026)1820 2562 y Fq(\030)t(;)c(t=")p Fu(\))p Fp(j)2063 2574 y Fi(\034)2187 2562 y Fp(\024)83 b Fq(K)2426 2506 y Fp(j)p Fq(\030)23 b Fp(\000)2600 2484 y Fu(\026)2591 2506 y Fq(\030)t Fp(j)2654 2518 y Fi(\034)p 2421 2543 280 4 v 2421 2619 a Fq(\034)2466 2595 y Fi(`)p Fj(\000)p Fo(2)p Fi(r)r Fo(+1)2711 2562 y Fs(,)208 b Fu(\()p Fq(t;)14 b(s)p Fu(\))23 b Fp(2)h Fq(D)3287 2528 y Fo(ext)3285 2583 y Fi(")3381 2562 y Fs(,)456 2729 y(wher)l(e)30 b Fq(g)s Fu(\()p Fq(x;)14 b(y)s(;)g(t=")p Fu(\))22 b(=)h(\()p Fq(@)1259 2741 y Fi(y)1299 2729 y Fq(h)1347 2741 y Fo(1)1384 2729 y Fu(\()p Fq(x;)14 b(y)s(;)g(t=")p Fu(\))p Fq(;)g Fp(\000)p Fq(@)1870 2741 y Fi(x)1911 2729 y Fq(h)1959 2741 y Fo(1)1997 2729 y Fu(\()p Fq(x;)g(y)s(;)g(t=")p Fu(\)\))p Fs(.)605 2880 y Ft(Remark)32 b Fu(6.1)p Ft(.)40 b Fs(Sinc)l(e)1429 3019 y Fp(\000)p Fq(V)1560 2985 y Fj(0)1584 3019 y Fu(\()p Fq(\013)1669 3031 y Fo(0)1706 3019 y Fu(\()p Fq(u)p Fu(\)\))24 b(=)1983 2997 y(_)1962 3019 y Fq(\014)2009 3031 y Fo(0)2046 3019 y Fu(\()p Fq(u)p Fu(\))f(=)31 b(\177)-50 b Fq(\013)2322 3031 y Fo(0)2359 3019 y Fu(\()p Fq(u)p Fu(\))456 3166 y Fs(has)30 b(a)h(singularity)g (of)f(or)l(der)h Fq(r)22 b Fu(+)c(1)29 b Fs(at)h Fq(u)23 b Fu(=)g Fq(ia)p Fs(,)30 b(we)h(have)g(that,)f(for)h Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b Fp(2)g Fq(D)2991 3136 y Fo(ext)2989 3187 y Fi(")3115 3166 y Fs(such)30 b(that)456 3266 y Fu(0)22 b Fp(\024)h Fu(Im)14 b Fq(s)23 b(<)f(a)p Fs(:)1027 3434 y Fp(j)p Fq(V)1117 3400 y Fo(\()p Fi(j)s Fo(+1\))1288 3434 y Fu(\()p Fq(\013)1373 3446 y Fo(0)1410 3434 y Fu(\()p Fq(t)d Fu(+)f Fq(s)p Fu(\)\))p Fp(j)24 b(\024)e Fq(K)2113 3378 y Fu(1)p 1898 3415 473 4 v 1898 3492 a Fq(\034)1943 3468 y Fo(2)p Fj(\000)p Fo(\()p Fi(j)s Fj(\000)p Fo(1\)\()p Fi(r)r Fj(\000)p Fo(1\))2380 3434 y Fs(,)119 b(for)31 b Fq(j)d Fp(\025)22 b Fu(0)p Fs(.)-2417 b Fu(\(6.10\))456 3602 y Fs(By)32 b(hyp)l(othesis)h Ff(HP3)p Fs(,)h Fq(h)1277 3614 y Fo(1)1314 3602 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))32 b Fs(is)g(a)g(p)l(olynomial)i(in)e Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fs(.)45 b(When)31 b(we)h(evaluate)h Fq(h)3306 3614 y Fo(1)3374 3602 y Fs(at)456 3702 y Fu(\()p Fq(x;)14 b(y)s Fu(\))23 b(=)g Fq(\015)802 3714 y Fo(0)839 3702 y Fu(\()p Fq(u)p Fu(\))p Fs(,)29 b(by)g(the)f(de\014nition)h(of)g Fq(`)e Fs(in)i(Chapter)g(1,)h(the)e(function)g(has)h(a)g(singularity)g (of)456 3801 y(or)l(der)h(at)g(most)f Fq(`)h Fs(at)f Fq(u)23 b Fu(=)g Fq(ia)p Fs(,)29 b(henc)l(e)h(for)h Fu(\()p Fq(t;)14 b(s)p Fu(\))30 b Fs(as)g(b)l(efor)l(e)908 3979 y Fp(j)p Fq(@)980 3945 y Fi(k)1015 3953 y Fh(1)975 3999 y Fi(x)1052 3979 y Fq(@)1101 3945 y Fi(k)1136 3953 y Fh(2)1096 3999 y Fi(y)1172 3979 y Fq(h)1220 3991 y Fo(1)1257 3979 y Fu(\()p Fq(\015)1332 3991 y Fo(0)1370 3979 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))p Fp(j)22 b(\024)h Fq(K)2243 3923 y Fu(1)p 2004 3960 519 4 v 2004 4037 a Fq(\034)2049 4013 y Fi(`)p Fj(\000)p Fi(k)2164 4021 y Fh(1)2198 4013 y Fo(\()p Fi(r)r Fj(\000)p Fo(1\))p Fj(\000)p Fi(k)2455 4021 y Fh(2)2487 4013 y Fi(r)2533 3979 y Fs(,)120 b(for)30 b Fq(k)2853 3991 y Fo(1)2891 3979 y Fs(,)g Fq(k)2989 3991 y Fo(2)3049 3979 y Fp(\025)23 b Fu(0)p Fs(.)-2748 b Fu(\(6.11\))605 4160 y(Since)28 b Fq(h)870 4172 y Fo(1)907 4160 y Fu(\()p Fq(x;)14 b(y)s(;)g(t=")p Fu(\))27 b(is)h(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)e(in)i Fq(t)p Fu(,)g(the)g(Melnik)n(o)n(v)e (function,)1195 4357 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))22 b(=)1574 4244 y Fg(Z)1657 4264 y Fo(+)p Fj(1)1620 4432 y(\0001)1778 4357 y Fp(f)p Fq(h)1868 4369 y Fo(0)1905 4357 y Fq(;)14 b(h)1990 4369 y Fo(1)2027 4357 y Fp(g)p Fu(\()p Fq(\015)2144 4369 y Fo(0)2181 4357 y Fu(\()p Fq(t)19 b Fu(+)f Fq(s)p Fu(\))p Fq(;)c(t=")p Fu(\))g Fq(dt)p Fu(,)456 4549 y(has)39 b(the)i(same)f(p)r(erio)r(dicit)n(y)g (with)h(resp)r(ect)f(to)g Fq(s)p Fu(.)75 b(W)-7 b(e)41 b(denote)f(b)n(y)g Fq(M)2816 4561 y Fi(k)2856 4549 y Fu(\()p Fq(")p Fu(\))h(its)g(F)-7 b(ourier's)456 4648 y(co)r(e\016cien)n(ts,)27 b(i.e.,)1470 4799 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))22 b(=)1849 4720 y Fg(X)1849 4899 y Fi(k)q Fj(2)p Fd(Z)1983 4799 y Fq(M)2064 4811 y Fi(k)2104 4799 y Fu(\()p Fq(")p Fu(\))p Fq(e)2246 4765 y Fi(ik)q(s=")2407 4799 y Fu(.)605 5016 y(The)28 b(next)g(prop)r (osition)f(asserts)g(that)h(the)g(Melnik)n(o)n(v)f(function)i(is)f(a)f (go)r(o)r(d)h(appro)n(xima-)456 5116 y(tion)37 b(of)h Fp(E)792 5086 y Fi(u)835 5116 y Fu(\()p Fq(s)p Fu(\))h(for)e Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)39 b(\024)h Fq(a)25 b Fp(\000)f Fq(")38 b Fu(and)f(giv)n(es)g(that,)j(in)e(particular)e(when)i Fq(s)i Fp(2)g Fl(R)p Fu(,)46 b(the)456 5216 y(appro)n(ximation)25 b(is)j(exp)r(onen)n(tially)f(small.)p eop %%Page: 68 76 68 75 bop 456 251 a Fo(68)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Ft(Pr)n(oposition)k Fu(6.1)p Ft(.)35 b Fs(Under)25 b(hyp)l(otheses)i Ff(HP1)p Fs(-)p Ff(HP5)p Fs(,)h Fp(S)2470 420 y Fi(u)2539 450 y Fs(and)d Fp(E)2746 420 y Fi(u)2815 450 y Fs(satisfy)h(the)g(fol)t (low-)456 550 y(ing)k(estimates:)673 707 y(a\))45 b(F)-6 b(or)30 b Fq(s)23 b Fp(2)h Fl(C)50 b Fs(such)30 b(that)g Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)f Fq(a)d Fp(\000)f Fq(")p Fs(,)1161 922 y Fp(E)1212 888 y Fi(u)1255 922 y Fu(\()p Fq(s)p Fu(\))24 b(=)e Fq(\026")1558 888 y Fi(p)1597 922 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(\026)2114 888 y Fo(2)2152 922 y Fq(")2191 888 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(1)2434 922 y Fq(;)c(\026")2560 888 y Fi(p)p Fo(+1)2682 922 y Fu(\))p Fs(.)677 1163 y(b\))45 b(L)l(et)29 b Fp(E)988 1133 y Fi(u)981 1183 y Fo(0)1032 1163 y Fu(\()p Fq(")p Fu(\))23 b(=)1292 1130 y Fo(1)p 1256 1144 106 4 v 1256 1191 a(2)p Fi(\031)r(")1385 1096 y Fg(R)1441 1116 y Fo(2)p Fi(\031)r(")1424 1192 y Fo(0)1564 1163 y Fp(E)1615 1133 y Fi(u)1659 1163 y Fu(\()p Fq(s)p Fu(\))14 b Fq(ds)p Fs(.)39 b(F)-6 b(or)30 b Fq(s)23 b Fp(2)g Fl(R)p Fs(,)913 1393 y Fp(E)964 1359 y Fi(u)1007 1393 y Fu(\()p Fq(s)p Fu(\))c Fp(\000)f(E)1263 1359 y Fi(u)1256 1414 y Fo(0)1307 1393 y Fu(\()p Fq(")p Fu(\))23 b(=)g Fq(\026")1610 1359 y Fi(p)1648 1393 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(\026)2165 1359 y Fo(2)2204 1393 y Fq(")2243 1359 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(1)2485 1393 y Fq(;)c(\026")2611 1359 y Fi(p)p Fo(+1)2733 1393 y Fu(\))p Fq(e)2804 1359 y Fj(\000)p Fi(a=")2962 1393 y Fs(.)605 1588 y Ft(Pr)n(oof.)41 b Fu(In)28 b(Theorem)e(4.2)h(w)n(e)g (ha)n(v)n(e)g(pro)n(v)n(ed)f(that)1588 1803 y Fp(E)7 b Fu(\()p Fq(\015)1719 1768 y Fi(s)1714 1823 y(\026;")1810 1803 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))23 b(=)g(0)-1857 b(\(6.12\))456 2017 y(and)1362 2232 y Fp(E)7 b Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)f Fq(h)1843 2244 y Fo(0)1880 2232 y Fu(\()p Fq(x;)14 b(y)s Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")2360 2197 y Fi(p)p Fo(+1)2483 2232 y Fu(\).)-2082 b(\(6.13\))456 2451 y(Since)29 b Fp(E)725 2421 y Fi(u)769 2451 y Fu(\()p Fq(s)p Fu(\))h(do)r(es)f(not)h (dep)r(end)g(on)f Fq(t)p Fu(,)h(for)f(an)n(y)g Fq(s)g Fu(w)n(e)h(c)n(ho)r(ose)e Fq(t)e Fu(=)g Fq(T)2673 2463 y Fi(s)2737 2451 y Fu(with)k Fq(T)2977 2463 y Fi(s)3039 2451 y Fu(=)c Fq(T)k Fp(\000)20 b Fu(Re)13 b Fq(s)456 2550 y Fu(and)31 b(therefore,)i(for)e(\()p Fq(t;)14 b(s)p Fu(\))31 b(=)f(\()p Fq(T)1507 2562 y Fi(s)1542 2550 y Fq(;)14 b(s)p Fu(\),)34 b Fq(\015)1755 2520 y Fi(u)1750 2571 y(\026;")1845 2550 y Fu(\()p Fq(t;)14 b(s)p Fu(\))33 b(and)f Fq(\015)2262 2520 y Fi(s)2257 2571 y(\026;")2352 2550 y Fu(\()p Fq(t;)14 b(s)p Fu(\))33 b(b)r(elong)f(to)f(the)i(domain) f(of)456 2653 y(the)f(\015o)n(w)g(b)r(o)n(x)g(co)r(ordinates)f Fq(U)9 b Fu(.)49 b(Then,)32 b(from)g(the)f(de\014nition)h(\(6.4\))f(of) h Fp(E)2841 2623 y Fi(u)2916 2653 y Fu(and)f(prop)r(erties)456 2752 y(\(6.12\))26 b(and)i(\(6.13\):)993 2967 y Fp(E)1044 2932 y Fi(u)1088 2967 y Fu(\()p Fq(s)p Fu(\))83 b(=)g Fp(E)7 b Fu(\()p Fq(\015)1553 2932 y Fi(u)1548 2987 y(\026;")1644 2967 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))k Fp(\000)g(E)7 b Fu(\()p Fq(\015)2226 2932 y Fi(s)2221 2987 y(\026;")2318 2967 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(t=")p Fu(\))1274 3103 y(=)83 b Fq(h)1470 3115 y Fo(0)1507 3103 y Fu(\()p Fq(\015)1587 3069 y Fi(u)1582 3124 y(\026;")1678 3103 y Fu(\()p Fq(t;)14 b(s)p Fu(\)\))19 b Fp(\000)f Fq(h)2030 3115 y Fo(0)2067 3103 y Fu(\()p Fq(\015)2147 3069 y Fi(s)2142 3124 y(\026;")2238 3103 y Fu(\()p Fq(t;)c(s)p Fu(\)\))19 b(+)f Fq(O)r Fu(\()p Fq(\026")2728 3069 y Fi(p)p Fo(+1)2851 3103 y Fu(\),)-2450 b(\(6.14\))456 3320 y(if)28 b Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)f Fq(a)d Fp(\000)f Fq(")p Fu(.)605 3420 y(Since,)28 b(for)f(an)n(y)g Fq(s)g Fu(suc)n(h)h(that)g Fp(j)14 b Fu(Im)f Fq(s)p Fp(j)23 b(\024)g Fq(a)18 b Fp(\000)g Fq(")p Fu(,)28 b(w)n(e)f(ha)n(v)n(e)g(that)1264 3634 y Fq(\015)1312 3600 y Fi(s)1307 3655 y(\026;")1403 3634 y Fu(\()p Fq(t;)14 b(s)p Fu(\))83 b Fp(!)g Fu(0)249 b(when)27 b Fq(t)c Fp(!)g Fu(+)p Fp(1)1264 3762 y Fq(\015)1312 3727 y Fi(u)1307 3782 y(\026;")1403 3762 y Fu(\()p Fq(t;)14 b(s)p Fu(\))83 b Fp(!)g Fu(0)249 b(when)27 b Fq(t)c Fp(!)g(\0001)456 3976 y Fu(w)n(e)k(deduce)1172 4190 y(lim)1126 4240 y Fi(t)p Fj(!)p Fo(+)p Fj(1)1348 4190 y Fq(h)1396 4202 y Fo(0)1433 4190 y Fu(\()p Fq(\015)1513 4156 y Fi(s)1508 4211 y(\026;")1604 4190 y Fu(\()p Fq(t;)14 b(s)p Fu(\)\))24 b(=)69 b(lim)1917 4240 y Fi(t)p Fj(!\0001)2140 4190 y Fq(h)2188 4202 y Fo(0)2226 4190 y Fu(\()p Fq(\015)2306 4156 y Fi(u)2301 4211 y(\026;")2396 4190 y Fu(\()p Fq(t;)14 b(s)p Fu(\)\))24 b(=)f(0.)456 4436 y(Then)580 4650 y Fq(h)628 4662 y Fo(0)665 4650 y Fu(\()p Fq(\015)745 4616 y Fi(u)740 4670 y(\026;")836 4650 y Fu(\()p Fq(T)917 4662 y Fi(s)952 4650 y Fq(;)14 b(s)p Fu(\)\))19 b Fp(\000)f Fq(h)1242 4662 y Fo(0)1279 4650 y Fu(\()p Fq(\015)1359 4616 y Fi(s)1354 4670 y(\026;")1450 4650 y Fu(\()p Fq(T)1531 4662 y Fi(s)1566 4650 y Fq(;)c(s)p Fu(\)\))688 4849 y(=)776 4736 y Fg(Z)859 4757 y Fi(T)898 4765 y Fc(s)822 4925 y Fj(\0001)958 4849 y Fq(@)1002 4861 y Fi(t)1045 4782 y Fg(\002)1080 4849 y Fq(h)1128 4861 y Fo(0)1165 4849 y Fu(\()p Fq(\015)1245 4815 y Fi(u)1240 4870 y(\026;")1336 4849 y Fu(\()p Fq(t;)g(s)p Fu(\)\))1538 4782 y Fg(\003)1601 4849 y Fq(dt)k Fp(\000)1775 4736 y Fg(Z)1858 4757 y Fo(+)p Fj(1)1821 4925 y Fi(T)1860 4933 y Fc(s)1993 4849 y Fq(@)2037 4861 y Fi(t)2080 4782 y Fg(\002)2115 4849 y Fq(h)2163 4861 y Fo(0)2200 4849 y Fu(\()p Fq(\015)2280 4815 y Fi(s)2275 4870 y(\026;")2371 4849 y Fu(\()p Fq(t;)c(s)p Fu(\)\))2573 4782 y Fg(\003)2636 4849 y Fq(dt)688 5112 y Fu(=)23 b Fq(\026")865 5078 y Fi(p)917 4970 y Fg(")965 4999 y(Z)1048 5019 y Fi(T)1087 5027 y Fc(s)1011 5187 y Fj(\0001)1133 5112 y Fp(f)p Fq(h)1223 5124 y Fo(0)1260 5112 y Fq(;)14 b(h)1345 5124 y Fo(1)1382 5112 y Fp(g)p Fu(\()p Fq(\015)1504 5078 y Fi(u)1499 5132 y(\026;")1594 5112 y Fu(\()p Fq(t;)g(s)p Fu(\))p Fq(;)g(t=")p Fu(\))g Fq(dt)19 b Fu(+)2133 4999 y Fg(Z)2216 5019 y Fo(+)p Fj(1)2179 5187 y Fi(T)2218 5195 y Fc(s)2337 5112 y Fp(f)p Fq(h)2427 5124 y Fo(0)2464 5112 y Fq(;)14 b(h)2549 5124 y Fo(1)2586 5112 y Fp(g)p Fu(\()p Fq(\015)2708 5078 y Fi(s)2703 5132 y(\026;")2798 5112 y Fu(\()p Fq(t;)g(s)p Fu(\))p Fq(;)g(t=")p Fu(\))g Fq(dt)3235 4970 y Fg(#)3297 5112 y Fu(.)p eop %%Page: 69 77 69 76 bop 1421 251 a Fo(6.2.)30 b(THE)f(SPLITTING)g(FUNCTION)900 b(69)456 450 y Fu(Adding)27 b(and)h(subtracting)f(the)h(Melnik)n(o)n(v) e(function)i(w)n(e)g(can)f(write)737 602 y Fq(h)785 614 y Fo(0)822 602 y Fu(\()p Fq(\015)902 568 y Fi(u)897 623 y(\026;")993 602 y Fu(\()p Fq(T)1074 614 y Fi(s)1109 602 y Fq(;)14 b(s)p Fu(\)\))19 b Fp(\000)f Fq(h)1399 614 y Fo(0)1436 602 y Fu(\()p Fq(\015)1516 568 y Fi(s)1511 623 y(\026;")1607 602 y Fu(\()p Fq(T)1688 614 y Fi(s)1724 602 y Fq(;)c(s)p Fu(\)\))1273 802 y(=)23 b Fq(\026")1450 767 y Fi(p)1502 689 y Fg(Z)1585 709 y Fj(\000)p Fi(T)1676 717 y Fc(s)1548 877 y Fj(\0001)1712 802 y Fp(f)p Fq(h)1802 814 y Fo(0)1838 802 y Fq(;)14 b(h)1923 814 y Fo(1)1960 802 y Fp(g)p Fu(\()p Fq(\015)2082 767 y Fi(u)2077 822 y(\026;")2173 802 y Fq(;)g(t=")p Fu(\))k Fp(\000)g(f)p Fq(h)2544 814 y Fo(0)2580 802 y Fq(;)c(h)2665 814 y Fo(1)2702 802 y Fp(g)p Fu(\()p Fq(\015)2819 814 y Fo(0)2856 802 y Fq(;)g(t=")p Fu(\))g Fq(dt)1324 1050 y Fu(+)k Fq(\026")1496 1015 y Fi(p)1548 937 y Fg(Z)1631 957 y Fi(T)1670 965 y Fc(s)1594 1125 y Fj(\000)p Fi(T)1685 1133 y Fc(s)1721 1050 y Fp(f)p Fq(h)1811 1062 y Fo(0)1848 1050 y Fq(;)c(h)1933 1062 y Fo(1)1970 1050 y Fp(g)p Fu(\()p Fq(\015)2092 1015 y Fi(u)2087 1070 y(\026;")2182 1050 y Fq(;)g(t=")p Fu(\))k Fp(\000)g(f)p Fq(h)2553 1062 y Fo(0)2589 1050 y Fq(;)c(h)2674 1062 y Fo(1)2711 1050 y Fp(g)p Fu(\()p Fq(\015)2828 1062 y Fo(0)2865 1050 y Fq(;)g(t=")p Fu(\))g Fq(dt)-2676 b Fu(\(6.15\))1324 1295 y(+)18 b Fq(\026")1496 1261 y Fi(p)1548 1182 y Fg(Z)1631 1203 y Fo(+)p Fj(1)1594 1371 y Fi(T)1633 1379 y Fc(s)1752 1295 y Fp(f)p Fq(h)1842 1307 y Fo(0)1879 1295 y Fq(;)c(h)1964 1307 y Fo(1)2001 1295 y Fp(g)p Fu(\()p Fq(\015)2123 1261 y Fi(s)2118 1316 y(\026;")2213 1295 y Fq(;)g(t=")p Fu(\))k Fp(\000)g(f)p Fq(h)2584 1307 y Fo(0)2620 1295 y Fq(;)c(h)2705 1307 y Fo(1)2742 1295 y Fp(g)p Fu(\()p Fq(\015)2859 1307 y Fo(0)2896 1295 y Fq(;)g(t=")p Fu(\))g Fq(dt)1324 1541 y Fu(+)k Fq(\026")1496 1506 y Fi(p)1548 1428 y Fg(Z)1631 1448 y Fo(+)p Fj(1)1594 1616 y(\0001)1752 1541 y Fp(f)p Fq(h)1842 1553 y Fo(0)1879 1541 y Fq(;)c(h)1964 1553 y Fo(1)2001 1541 y Fp(g)p Fu(\()p Fq(\015)2118 1553 y Fo(0)2155 1541 y Fq(;)g(t=")p Fu(\))g Fq(dt)456 1746 y Fu(where)22 b Fq(\015)739 1716 y Fi(u)734 1767 y(\026;")830 1746 y Fu(,)i Fq(\015)925 1716 y Fi(s)920 1767 y(\026;")1039 1746 y Fu(and)e Fq(\015)1238 1758 y Fo(0)1299 1746 y Fu(denote)h Fq(\015)1610 1716 y Fi(u)1605 1767 y(\026;")1700 1746 y Fu(\()p Fq(t;)14 b(s)p Fu(\),)25 b Fq(\015)1966 1716 y Fi(s)1961 1767 y(\026;")2056 1746 y Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b(and)f Fq(\015)2450 1758 y Fo(0)2487 1746 y Fu(\()p Fq(t)9 b Fu(+)g Fq(s)p Fu(\))24 b(resp)r(ectiv)n(ely)-7 b(.)34 b(By)23 b(the)456 1854 y(conclusion)g(4\))i(of)f(Theorem)g(3.1,)g(the)h(\014rst)f(and)h (the)f(third)h(lines)g(in)f(\(6.15\))g(are)g Fq(O)r Fu(\()p Fq(\026)3157 1824 y Fo(2)3195 1854 y Fq(")3234 1824 y Fo(2)p Fi(p)p Fo(+1)3389 1854 y Fu(\).)456 1954 y(It)k(remains)e(to)i (b)r(ound)g(the)g(second)f(line.)605 2053 y(It)34 b(is)f(not)g (di\016cult)i(to)e(see)g(that,)i(if)f(w)n(e)f(write)g Fq(\015)2187 2023 y Fj(\003)2182 2074 y Fi(\026;")2278 2053 y Fu(\()p Fq(t;)14 b(s)p Fu(\))33 b(=)f(\()p Fq(\013)2663 2023 y Fj(\003)2663 2074 y Fi(\026;")2759 2053 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)g(\014)3017 2023 y Fj(\003)3013 2074 y Fi(\026;")3109 2053 y Fu(\()p Fq(t;)g(s)p Fu(\)\))34 b(for)456 2153 y Fp(\003)22 b Fu(=)h Fq(u;)14 b(s)p Fu(,)529 2305 y Fp(f)o Fq(h)618 2317 y Fo(0)656 2305 y Fq(;)g(h)741 2317 y Fo(1)777 2305 y Fp(g)p Fu(\()p Fq(\015)899 2271 y Fj(\003)894 2325 y Fi(\026;")990 2305 y Fq(;)g(t=")p Fu(\))k Fp(\000)g(f)p Fq(h)1361 2317 y Fo(0)1397 2305 y Fq(;)c(h)1482 2317 y Fo(1)1519 2305 y Fp(g)p Fu(\()p Fq(\015)1636 2317 y Fo(0)1673 2305 y Fq(;)g(t=")p Fu(\))594 2439 y(=)22 b Fq(V)748 2404 y Fj(0)771 2439 y Fu(\()p Fq(\013)856 2404 y Fj(\003)856 2459 y Fi(\026;")952 2439 y Fu(\)[)p Fq(@)1051 2451 y Fi(y)1092 2439 y Fq(h)1140 2451 y Fo(1)1177 2439 y Fu(\()p Fq(\015)1257 2404 y Fj(\003)1252 2459 y Fi(\026;")1348 2439 y Fq(;)14 b(t=")p Fu(\))j Fp(\000)h Fq(@)1672 2451 y Fi(y)1712 2439 y Fq(h)1760 2451 y Fo(1)1798 2439 y Fu(\()p Fq(\015)1873 2451 y Fo(0)1910 2439 y Fq(;)c(t=")p Fu(\)])k(+)g([)p Fq(V)2304 2404 y Fj(0)2327 2439 y Fu(\()p Fq(\013)2412 2404 y Fj(\003)2412 2459 y Fi(\026;")2508 2439 y Fu(\))h Fp(\000)f Fq(V)2709 2404 y Fj(0)2732 2439 y Fu(\()p Fq(\013)2817 2451 y Fo(0)2855 2439 y Fu(\)])p Fq(@)2954 2451 y Fi(y)2994 2439 y Fq(h)3042 2451 y Fo(1)3079 2439 y Fu(\()p Fq(\015)3154 2451 y Fo(0)3192 2439 y Fq(;)c(t=")p Fu(\))644 2568 y Fp(\000)k Fq(\014)778 2534 y Fj(\003)774 2589 y Fi(\026;")870 2568 y Fu([)p Fq(@)937 2580 y Fi(x)979 2568 y Fq(h)1027 2580 y Fo(1)1064 2568 y Fu(\()p Fq(\015)1144 2534 y Fj(\003)1139 2589 y Fi(\026;")1235 2568 y Fq(;)c(t=")p Fu(\))k Fp(\000)g Fq(@)1560 2580 y Fi(x)1601 2568 y Fq(h)1649 2580 y Fo(1)1687 2568 y Fu(\()p Fq(\015)1762 2580 y Fo(0)1799 2568 y Fq(;)c(t=")p Fu(\)])k Fp(\000)g Fu(\()p Fq(\014)2186 2534 y Fj(\003)2182 2589 y Fi(\026;")2296 2568 y Fp(\000)g Fq(\014)2426 2580 y Fo(0)2463 2568 y Fu(\))p Fq(@)2539 2580 y Fi(x)2582 2568 y Fq(h)2630 2580 y Fo(1)2667 2568 y Fu(\()p Fq(\015)2742 2580 y Fo(0)2779 2568 y Fq(;)c(t=")p Fu(\).)456 2725 y(Using)30 b(b)r(ounds)i(\(6.10\),)f(\(6.11\))f(and)h(Lemma)g(6.3)f (and)h(taking)f(in)n(to)h(accoun)n(t)f(that,)i(b)n(y)f(the)456 2825 y(extension)c(theorem,)g Fq(\015)1217 2795 y Fi(u)1212 2845 y(\026;")1326 2825 y Fp(\000)18 b Fq(\015)1452 2837 y Fo(0)1512 2825 y Fu(=)23 b Fq(O)r Fu(\()p Fq(\026")1786 2795 y Fi(\027)1828 2825 y Fu(\),)28 b(w)n(e)f(get)1038 3029 y Fp(jf)p Fq(h)1151 3041 y Fo(0)1188 3029 y Fq(;)14 b(h)1273 3041 y Fo(1)1310 3029 y Fp(g)p Fu(\()p Fq(\015)1432 2994 y Fi(u)1427 3049 y(\026;")1522 3029 y Fq(;)g(t=")p Fu(\))k Fp(\000)g(f)p Fq(h)1893 3041 y Fo(0)1930 3029 y Fq(;)c(h)2015 3041 y Fo(1)2052 3029 y Fp(g)p Fu(\()p Fq(\015)2169 3041 y Fo(0)2206 3029 y Fq(;)g(t=")p Fu(\))p Fp(j)22 b(\024)h Fq(K)2664 2973 y(\026")2753 2942 y Fi(\027)p 2605 3010 247 4 v 2605 3086 a Fq(\034)2650 3062 y Fi(`)p Fj(\000)p Fi(r)r Fo(+2)456 3209 y Fu(\(w)n(e)h(recall)f(that)i Fq(\034)33 b Fu(=)22 b Fp(j)p Fq(t)12 b Fu(+)g Fq(s)g Fp(\000)g Fq(ia)p Fp(j)p Fu(\).)35 b(Then)25 b(applying)f(the)h (estimate)f(\(6.9\))g(with)h Fq(l)g Fu(=)d Fq(`)12 b Fp(\000)g Fq(r)i Fu(+)e(2)456 3308 y(w)n(e)27 b(obtain)g(that)h(the)g (second)f(line)h(in)g(\(6.15\))e(is)i Fq(O)r Fu(\()p Fq(\026")2190 3278 y Fi(p)p Fo(+)p Fi(\027)t Fj(\000)p Fi(`)p Fo(+)p Fi(r)r Fj(\000)p Fo(1)2565 3308 y Fu(\).)38 b(Th)n(us)768 3462 y Fq(h)816 3474 y Fo(0)853 3462 y Fu(\()p Fq(\015)933 3427 y Fi(u)928 3482 y(\026;")1024 3462 y Fu(\()p Fq(T)1105 3474 y Fi(s)1141 3462 y Fq(;)14 b(s)p Fu(\)\))k Fp(\000)h Fq(h)1431 3474 y Fo(0)1468 3462 y Fu(\()p Fq(\015)1548 3427 y Fi(s)1543 3482 y(\026;")1638 3462 y Fu(\()p Fq(T)1719 3474 y Fi(s)1755 3462 y Fq(;)14 b(s)p Fu(\)\))23 b(=)g Fq(\026")2095 3427 y Fi(p)2133 3462 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(\026)2650 3427 y Fo(2)2688 3462 y Fq(")2727 3427 y Fo(2)p Fi(p)p Fo(+1)2883 3462 y Fq(;)c(\026)2970 3427 y Fo(2)3007 3462 y Fq(")3046 3427 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(1)3288 3462 y Fu(\).)456 3619 y(No)n(w)37 b(a\))h(follo)n(ws)f(from)g(\(6.14\))g(and)h(from)g(the)g (previous)f(expression.)66 b(Note)38 b(that,)j(since)456 3718 y Fq(`)22 b Fp(\025)h Fq(r)e Fp(\000)d Fu(1,)27 b(one)h(has)f(2)p Fq(p)17 b Fu(+)h(1)23 b Fp(\025)g Fu(2)p Fq(\027)g Fu(+)18 b Fq(r)j Fp(\000)d Fu(1.)605 3818 y(T)-7 b(o)27 b(pro)n(v)n(e)f(b\),)j(since)e Fp(E)1335 3788 y Fi(u)1379 3818 y Fu(\()p Fq(s)p Fu(\))h(is)g(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic)f(in)h Fq(s)g Fu(and)f(analytic)g(in)h(the)g (complex)g(strip)456 3917 y Fp(j)14 b Fu(Im)g Fq(s)p Fp(j)23 b(\024)f Fq(a)d Fp(\000)f Fq(")p Fu(,)27 b(w)n(e)g(expand)g(it) i(in)e(F)-7 b(ourier)27 b(series)1518 4086 y Fp(E)1569 4052 y Fi(u)1613 4086 y Fu(\()p Fq(s)p Fu(\))c(=)1827 4007 y Fg(X)1827 4186 y Fi(k)q Fj(2)p Fd(Z)1961 4086 y Fp(E)2012 4052 y Fi(u)2005 4107 y(k)2056 4086 y Fu(\()p Fq(")p Fu(\))p Fq(e)2198 4052 y Fi(ik)q(s=")2359 4086 y Fu(.)456 4317 y(It)28 b(is)f(clear)g(that,)h(for)f Fq(s)c Fp(2)g Fl(R)34 b Fu(w)n(e)27 b(can)g(write)987 4524 y Fp(E)1038 4490 y Fi(u)1031 4544 y(k)1082 4524 y Fu(\()p Fq(")p Fu(\))c(=)1350 4468 y(1)p 1306 4505 131 4 v 1306 4581 a(2)p Fq(\031)s(")1460 4411 y Fg(Z)1543 4431 y Fo(2)p Fi(\031)r(")1506 4599 y Fo(0)1667 4524 y Fp(E)1718 4490 y Fi(u)1761 4524 y Fu(\()p Fq(s)c Fp(\006)f Fq(i)p Fu(\()p Fq(a)g Fp(\000)g Fq(")p Fu(\)\))p Fq(e)2282 4490 y Fj(\000)p Fi(ik)q Fo(\()p Fi(s)p Fj(\006)p Fi(i)p Fo(\()p Fi(a)p Fj(\000)p Fi(")p Fo(\)\))p Fi(=")2807 4524 y Fq(ds)p Fu(.)456 4722 y(Th)n(us,)32 b(b)n(y)f(the)h(conclusion)f (a\))g(of)g(this)h(prop)r(osition)f(ab)r(out)g(the)h(estimate)f(of)h Fp(E)3050 4692 y Fi(u)3093 4722 y Fu(\()p Fq(s)p Fu(\))g(in)g(the)456 4822 y(complex)27 b(domain,)g(for)g Fq(k)f Fp(6)p Fu(=)d(0)k(w)n(e)g (obtain)920 5029 y Fp(E)971 4995 y Fi(u)964 5049 y(k)1015 5029 y Fu(\()p Fq(")p Fu(\))83 b(=)1359 4973 y Fq(e)1398 4943 y Fj(\000j)p Fi(k)q Fj(j)p Fo(\()p Fi(a)p Fj(\000)p Fi(")p Fo(\))p Fi(=")p 1359 5010 408 4 v 1497 5086 a Fu(2)p Fq(\031)s(")1790 4916 y Fg(Z)1873 4936 y Fo(2)p Fi(\031)r(")1836 5104 y Fo(0)1996 5029 y Fp(E)2047 4995 y Fi(u)2091 5029 y Fu(\()p Fq(s)19 b Fp(\006)f Fq(i)p Fu(\()p Fq(a)g Fp(\000)g Fq(")p Fu(\)\))p Fq(e)2612 4995 y Fj(\000)p Fi(ik)q(s=")2839 5029 y Fq(ds)1201 5216 y Fu(=)83 b Fq(\026")1438 5181 y Fi(p)1476 5216 y Fq(M)1557 5228 y Fi(k)1597 5216 y Fu(\()p Fq(")p Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026)1949 5181 y Fo(2)1987 5216 y Fq(")2026 5181 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(1)2269 5216 y Fq(;)c(\026")2395 5181 y Fi(p)p Fo(+1)2517 5216 y Fu(\))p Fq(e)2588 5181 y Fj(\000j)p Fi(k)q Fj(j)p Fo(\()p Fi(a)p Fj(\000)p Fi(")p Fo(\))p Fi(=")2957 5216 y Fu(,)p eop %%Page: 70 78 70 77 bop 456 251 a Fo(70)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(where)34 b(w)n(e)g(consider)g(the)h(sign)f(+)h(for)f Fq(k)k(<)d Fu(0)f(and)h(the)g(sign)f Fp(\000)h Fu(for)f Fq(k)k(>)c Fu(0.)58 b(Here)35 b Fq(M)3301 462 y Fi(k)3341 450 y Fu(\()p Fq(")p Fu(\))456 550 y(are)27 b(the)h(F)-7 b(ourier)28 b(co)r(e\016cien)n(ts)f(of)h(the)h(Melnik)n(o)n(v)e (function.)39 b(No)n(w)28 b(b\))h(follo)n(ws)e(summing)h(the)456 649 y(F)-7 b(ourier)26 b(series)h(and)g(applying)g(the)h(ab)r(o)n(v)n (e)e(equalit)n(y)-7 b(.)1231 b Fe(\003)1044 837 y Fk(6.3.)46 b(Pro)s(of)32 b(of)g(Theorem)e(1.1)h(and)h(its)g(corollary)605 986 y Fu(First)g(w)n(e)f(will)h(sho)n(w)f(that)h(the)g(function)h Fq(\036)f Fu(giv)n(en)f(in)h(\(6.7\))g(can)f(b)r(e)h(used)g(to)g (measure)456 1086 y(some)40 b(magnitudes)h(related)f(to)h(the)g (splitting.)77 b(Then)41 b(w)n(e)g(will)g(pro)n(v)n(e)e(the)i(form)n (ulas)f(in)456 1185 y(Theorem)23 b(1.1.)35 b(In)24 b(the)g(next)g(prop) r(osition)f(w)n(e)h(pro)n(v)n(e)e(the)i(existence)g(of)g(primary)f (homo)r(clinic)456 1285 y(p)r(oin)n(ts)j(and)g(w)n(e)g(relate)g(the)g (angle)g(b)r(et)n(w)n(een)g(the)h(in)n(v)-5 b(arian)n(t)25 b(manifolds)h(and)g(the)h(area)e(of)h(the)456 1384 y(lob)r(es)20 b(with)i(the)f(splitting)h(function.)35 b(Moreo)n(v)n(er,)20 b(w)n(e)g(relate)h(it)g(with)h(the)f(Melnik)n(o)n(v)f(function.)456 1484 y(First)27 b(w)n(e)g(state)h(a)f(tec)n(hnical)g(lemma)h(whic)n(h)f (w)n(e)g(will)h(pro)n(v)n(e)e(in)i(Section)g(6.4.)605 1637 y Ft(Lemma)j Fu(6.4)p Ft(.)40 b Fs(Under)30 b(the)g(standing)g(c)l (onditions)h(we)f(have)1179 1790 y Fq(\026")1268 1756 y Fi(p)1306 1790 y Fq(M)9 b Fu(\()p Fq(S;)14 b(")p Fu(\))23 b(=)f Fq(\026")1786 1756 y Fi(\027)1918 1711 y Fg(X)1841 1893 y Fi(k)q Fj(2)p Fd(Z)-10 b Fj(\000f)p Fo(0)p Fj(g)2128 1790 y Fq(e)2167 1756 y Fj(\000j)p Fi(k)q Fj(j)p Fi(a=")2400 1790 y Fq(M)2481 1802 y Fi(k)2522 1790 y Fq(e)2561 1756 y Fi(ik)q(s=")456 2027 y Fs(with)30 b Fq(M)717 2039 y Fi(k)780 2027 y Fu(=)23 b Fq(O)r Fu(\(1\))30 b Fs(uniformily)i(in)d Fq(k)s Fs(.)39 b(Ther)l(efor)l(e,)1376 2211 y Fq(\026")1465 2177 y Fi(p)1513 2155 y Fq(dM)p 1513 2192 133 4 v 1530 2268 a(dS)1656 2211 y Fu(\()p Fq(S;)14 b(")p Fu(\))23 b(=)g Fq(O)r Fu(\()p Fq(\026")2144 2177 y Fi(\027)t Fj(\000)p Fo(1)2270 2211 y Fu(\))p Fq(e)2341 2177 y Fj(\000)p Fi(a=")2499 2211 y Fs(.)605 2388 y Ft(Pr)n(oposition)35 b Fu(6.2)p Ft(.)43 b Fs(The)34 b(function)g Fq(\036)c Fu(:)g Fl(R)f Fp(!)24 b Fl(R)39 b Fs(is)34 b Fu(2)p Fq(\031)s(")p Fs(-p)l(erio)l (dic,)i(r)l(e)l(al)e(analytic)h(and)456 2487 y(satis\014es)29 b(the)h(fol)t(lowing)j(pr)l(op)l(erties:)673 2607 y(a\))45 b(Ther)l(e)33 b(exists)e Fq(h)1312 2577 y Fi(u)1382 2607 y Fp(2)c Fl(R)37 b Fs(such)32 b(that)g Fq(\015)1966 2577 y Fi(u)1961 2628 y(\026;")2056 2607 y Fu(\()p Fq(t;)14 b(h)2203 2577 y Fi(u)2247 2607 y Fu(\))26 b(=)g Fq(\015)2444 2577 y Fi(s)2439 2628 y(\026;")2535 2607 y Fu(\()p Fq(t;)14 b(h)2682 2577 y Fi(s)2718 2607 y Fu(\))p Fs(,)32 b(for)h(al)t(l)g Fq(t)e Fs(\(giving)i(a)794 2710 y(homo)l(clinic)f(orbit\),)f(with)f Fq(h)1696 2680 y Fi(s)1754 2710 y Fu(=)23 b Fp(S)1898 2680 y Fi(u)1942 2710 y Fu(\()p Fq(h)2022 2680 y Fi(u)2066 2710 y Fu(\))p Fs(.)38 b(F)-6 b(or)30 b Fq(n)23 b Fp(2)h Fl(N)t Fs(,)36 b(we)30 b(de\014ne)1666 2851 y Fq(h)1714 2816 y Fi(s)1714 2871 y(n)1782 2851 y Fu(=)22 b Fq(h)1917 2816 y Fi(s)1971 2851 y Fu(+)c(2)p Fq(\031)s("n)794 2996 y Fs(which)38 b(give)e(homo)l(clinic)i(p)l(oints.)58 b(Cle)l(arly,)39 b(for)e(al)t(l)g Fq(n)p Fs(,)g Fq(\036)p Fu(\()p Fq(h)2749 2966 y Fi(s)2749 3017 y(n)2795 2996 y Fu(\))d(=)g(0)p Fs(.)57 b(Mor)l(e)l(over,)794 3096 y Fq(\036)843 3066 y Fj(0)867 3096 y Fu(\()p Fq(h)947 3066 y Fi(s)947 3116 y(n)992 3096 y Fu(\))30 b Fs(is)g(indep)l(endent)h (of)f Fq(n)p Fs(,)g(and)678 3238 y Fq(\036)727 3203 y Fj(0)750 3238 y Fu(\()p Fq(h)830 3203 y Fi(s)830 3258 y(n)876 3238 y Fu(\))83 b(=)g Fq(@)1183 3250 y Fi(S)1234 3238 y Fu(~)-45 b Fq(\015)1279 3203 y Fi(s)1274 3258 y(\026;")1369 3238 y Fu(\()p Fq(t;)14 b(h)1516 3203 y Fi(s)1516 3258 y(n)1562 3238 y Fu(\))k Fp(^)h Fq(@)1730 3250 y Fi(S)1781 3238 y Fu(~)-45 b Fq(\015)1826 3203 y Fi(u)1821 3258 y(\026;")1917 3238 y Fu(\()p Fq(t;)14 b(h)2064 3203 y Fi(s)2064 3258 y(n)2109 3238 y Fu(\)\(1)19 b(+)f Fq(O)r Fu(\()p Fq(\026")2503 3203 y Fi(p)p Fo(+1)2626 3238 y Fu(\)\))991 3375 y(=)83 b Fp(k)p Fq(@)1225 3387 y Fi(S)1275 3375 y Fu(~)-45 b Fq(\015)1320 3340 y Fi(s)1315 3395 y(\026;")1411 3375 y Fu(\()p Fq(t;)14 b(h)1558 3340 y Fi(s)1558 3395 y(n)1603 3375 y Fu(\))p Fp(k)30 b(k)p Fq(@)1793 3387 y Fi(S)1843 3375 y Fu(~)-45 b Fq(\015)1888 3340 y Fi(u)1883 3395 y(\026;")1979 3375 y Fu(\()p Fq(t;)14 b(h)2126 3340 y Fi(s)2126 3395 y(n)2171 3375 y Fu(\))p Fp(k)g Fu(sin)f Fq(#)p Fu(\()p Fq(t;)h(h)2570 3340 y Fi(s)2570 3395 y(n)2616 3375 y Fu(\)\(1)k(+)g Fq(O)r Fu(\()p Fq(\026")3009 3340 y Fi(p)p Fo(+1)3132 3375 y Fu(\)\))p Fs(,)794 3523 y(for)32 b(al)t(l)g Fq(t)p Fs(,)g(wher)l(e)g Fp(^)g Fs(denotes)f(the)g(exterior)h(pr)l(o)l(duct)f(on)g Fl(R)2678 3493 y Fo(2)2721 3523 y Fs(,)h(and)g Fq(#)p Fu(\()p Fq(t;)14 b(h)3137 3493 y Fi(s)3137 3544 y(n)3182 3523 y Fu(\))32 b Fs(is)f(the)794 3623 y(angle)g(b)l(etwe)l(en)e Fq(@)1358 3635 y Fi(s)1397 3623 y Fu(~)-45 b Fq(\015)1442 3593 y Fi(u)1437 3643 y(\026;")1532 3623 y Fu(\()p Fq(t;)14 b(h)1679 3593 y Fi(s)1679 3643 y(n)1724 3623 y Fu(\))30 b Fs(and)h Fq(@)1992 3635 y Fi(s)2030 3623 y Fu(~)-45 b Fq(\015)2075 3593 y Fi(s)2070 3643 y(\026;")2166 3623 y Fu(\()p Fq(t;)14 b(h)2313 3593 y Fi(s)2313 3643 y(n)2358 3623 y Fu(\))p Fs(.)677 3725 y(b\))45 b(The)31 b(ar)l(e)l(a)f(of)h(the) f(lob)l(e)g(b)l(etwe)l(en)g(the)f(invariant)i(curves)f(is)g(given)g(by) 1605 3941 y Fq(A)23 b Fu(=)1778 3795 y Fg(\014)1778 3845 y(\014)1778 3895 y(\014)1778 3945 y(\014)1778 3995 y(\014)1806 3828 y(Z)1890 3833 y Fo(\026)1889 3848 y Fi(h)1852 4016 y(h)1946 3941 y Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))14 b Fq(dS)2228 3795 y Fg(\014)2228 3845 y(\014)2228 3895 y(\014)2228 3945 y(\014)2228 3995 y(\014)2270 3941 y Fs(,)794 4163 y(wher)l(e)31 b Fq(h)e Fs(and)1268 4141 y Fu(\026)1267 4163 y Fq(h)h Fs(ar)l(e)g(two)g(c)l(onse)l(cutive)g(zer) l(os)g(of)g Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))p Fs(.)677 4279 y(c\))45 b Fq(\036)843 4291 y Fo(0)904 4279 y Fu(=)992 4212 y Fg(R)1047 4232 y Fi(h)1086 4240 y Fc(n)1126 4232 y Fo(+2)p Fi(\031)r(")1031 4308 y(h)1070 4316 y Fc(n)1301 4279 y Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))14 b Fq(dS)28 b Fu(=)23 b(0)p Fs(.)673 4387 y(d\))45 b(F)-6 b(or)30 b Fq(S)e Fp(2)23 b Fl(R)p Fs(,)36 b Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))31 b Fs(satis\014es)f(the)g(estimate)796 4535 y Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))24 b Fp(\021)e(E)1127 4501 y Fi(u)1171 4535 y Fu(\()p Fq(s)1242 4501 y Fi(u)1285 4535 y Fu(\()p Fq(S)5 b Fu(\)\))24 b(=)f Fq(\026")1638 4501 y Fi(p)1676 4535 y Fq(M)9 b Fu(\()p Fq(S;)14 b(")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(\026)2205 4501 y Fo(2)2243 4535 y Fq(")2282 4501 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(1)2525 4535 y Fq(;)c(\026)2612 4501 y Fo(2)2649 4535 y Fq(")2688 4501 y Fi(\027)t Fo(+)p Fi(p)2814 4535 y Fq(;)g(\026")2940 4501 y Fi(p)p Fo(+1)3063 4535 y Fu(\))p Fq(e)3134 4501 y Fj(\000)p Fi(a=")3291 4535 y Fs(.)605 4689 y Ft(Remark)35 b Fu(6.2)p Ft(.)42 b Fs(In)33 b(p)l(articular)h (the)e(splitting)i(function)e Fq(\036)i Fs(is)f(an)g(instrument)e(to)i (study)456 4788 y(the)d(tr)l(ansversality)g(of)h(the)e(interse)l (ctions.)605 4941 y Ft(Pr)n(oof.)41 b Fu(W)-7 b(e)31 b(b)r(egin)g(b)n(y)g(pro)n(ving)e(the)i(existence)g(of)g(homo)r(clinic) f(orbits.)46 b(Let)31 b Fq(P)3244 4911 y Fi(t)3269 4919 y Fh(0)3232 4962 y Fi(\026;")3359 4941 y Fu(b)r(e)456 5041 y(the)d(P)n(oincar)n(\023)-39 b(e)24 b(map)1321 5182 y Fq(P)1386 5147 y Fi(t)1411 5155 y Fh(0)1374 5202 y Fi(\026;")1470 5182 y Fu(\()p Fq(x;)14 b(y)s Fu(\))24 b(=)e Fq(')1827 5194 y Fi(\026;")1923 5182 y Fu(\(2)p Fq(\031)s(")c Fu(+)g Fq(t)2217 5194 y Fo(0)2255 5182 y Fq(;)c(t)2322 5194 y Fo(0)2359 5182 y Fq(;)g(x;)g(y)s Fu(\).)p eop %%Page: 71 79 71 78 bop 1079 251 a Fo(6.3.)29 b(PR)n(OOF)g(OF)g(THEOREM)g(1.1)g(AND)g (ITS)g(COR)n(OLLAR)-5 b(Y)558 b(71)605 450 y Fu(Since)32 b Fq(P)891 420 y Fi(t)916 428 y Fh(0)879 471 y Fi(\026;")1007 450 y Fu(is)g(area)e(preserving)g(and)i Fq(P)1914 413 y Fi(t)1939 421 y Fh(0)1902 472 y Fo(0)p Fi(;")2023 450 y Fu(has)f(a)g(homo)r(clinic)h(connexion)f(\(whic)n(h)h(co-)456 554 y(incides)e(with)g(the)g(homo)r(clinic)g(orbit)g(of)f(the)i(unp)r (erturb)r(ed)f(di\013eren)n(tial)g(equation\),)g(a)g(w)n(ell)456 654 y(kno)n(wn)e(geometric)g(argumen)n(t,)h(applied)g(to)g Fq(P)1957 624 y Fi(t)1982 632 y Fh(0)1945 674 y Fi(\026;")2070 654 y Fu(restricted)g(to)g(the)g(reals,)g(giv)n(es)f(that)h Fq(P)3360 624 y Fi(t)3385 632 y Fh(0)3348 674 y Fi(\026;")456 756 y Fu(has)j(\(real\))g(primary)g(homo)r(clinic)g(p)r(oin)n(ts.)52 b(Since)33 b(the)g(iterates)f(of)h(the)g(homo)r(clinic)g(p)r(oin)n(ts) 456 856 y(also)26 b(are)h(homo)r(clinic)g(p)r(oin)n(ts)h(there)f(will)h (b)r(e)g(suc)n(h)f(p)r(oin)n(ts)h(in)g Fq(U)9 b Fu(.)605 955 y(Then)28 b(there)f(exist)h Fq(h)1279 925 y Fi(u)1322 955 y Fu(,)g Fq(h)1421 925 y Fi(s)1479 955 y Fp(2)23 b Fl(R)p Fu(,)34 b Fq(T)g Fp(\024)23 b Fq(h)1887 925 y Fi(u)1949 955 y Fu(+)18 b Fq(t)2062 967 y Fo(0)2099 955 y Fq(;)c(h)2184 925 y Fi(s)2238 955 y Fu(+)k Fq(t)2351 967 y Fo(0)2411 955 y Fp(\024)k Fu(2)p Fq(T)12 b Fu(,)27 b(suc)n(h)g(that)1391 1121 y Fq(z)1434 1086 y Fi(h)1499 1121 y Fu(=)c Fq(\015)1635 1086 y Fi(u)1630 1141 y(\026;")1725 1121 y Fu(\()p Fq(t)1787 1133 y Fo(0)1825 1121 y Fq(;)14 b(h)1910 1086 y Fi(u)1953 1121 y Fu(\))23 b(=)g Fq(\015)2144 1086 y Fi(s)2139 1141 y(\026;")2234 1121 y Fu(\()p Fq(t)2296 1133 y Fo(0)2334 1121 y Fq(;)14 b(h)2419 1086 y Fi(s)2454 1121 y Fu(\).)456 1286 y(Hence)1537 1451 y Fq(\015)1585 1417 y Fi(u)1580 1472 y(\026;")1676 1451 y Fu(\()p Fq(t;)g(h)1823 1417 y Fi(u)1866 1451 y Fu(\))24 b(=)e Fq(\015)2057 1417 y Fi(s)2052 1472 y(\026;")2148 1451 y Fu(\()p Fq(t;)14 b(h)2295 1417 y Fi(s)2330 1451 y Fu(\))456 1622 y(are)26 b(de\014ned)i(for)f(all)h Fq(t)23 b Fp(2)g Fl(R)34 b Fu(and)27 b(are)g(a)g(homo)r(clinic)g(solution)g(of)h(\(6.1\).)605 1721 y(By)33 b(Theorem)f(4.2,)i(w)n(e)e(can)h(c)n(ho)r(ose)f Fq(s)1854 1733 y Fo(0)1923 1721 y Fu(=)g Fq(h)2068 1691 y Fi(s)2103 1721 y Fu(.)53 b(Therefore,)33 b(taking)g Fq(t)g Fu(suc)n(h)f(that)i Fq(T)43 b Fp(\024)456 1821 y Fq(t)18 b Fu(+)g Fq(h)635 1791 y Fi(u)678 1821 y Fq(;)c(t)19 b Fu(+)f Fq(h)895 1791 y Fi(s)953 1821 y Fp(\024)23 b Fu(2)p Fq(T)12 b Fu(,)26 b(w)n(e)h(can)g(write)1381 1986 y Fp(S)6 b Fu(\()p Fq(\015)1517 1952 y Fi(s)1512 2007 y(\026;")1608 1986 y Fu(\()p Fq(t;)14 b(h)1755 1952 y Fi(s)1790 1986 y Fu(\))p Fq(;)g(t=")p Fu(\))23 b(=)g Fq(t)18 b Fp(\000)g Fq(t)2274 1998 y Fo(0)2330 1986 y Fu(+)g Fq(h)2461 1952 y Fi(s)2496 1986 y Fu(,)456 2152 y(and)719 2317 y Fq(h)767 2283 y Fi(s)825 2317 y Fu(=)23 b Fp(S)6 b Fu(\()p Fq(\015)1049 2283 y Fi(s)1044 2338 y(\026;")1140 2317 y Fu(\()p Fq(t;)14 b(h)1287 2283 y Fi(s)1323 2317 y Fu(\))p Fq(;)g(t=")p Fu(\))k Fp(\000)g Fu(\()p Fq(t)h Fp(\000)f Fq(t)1830 2329 y Fo(0)1867 2317 y Fu(\))23 b(=)g Fp(S)6 b Fu(\()p Fq(\015)2146 2283 y Fi(u)2141 2338 y(\026;")2237 2317 y Fu(\()p Fq(t;)14 b(h)2384 2283 y Fi(u)2428 2317 y Fu(\))p Fq(;)g(t=")p Fu(\))k Fp(\000)g Fu(\()p Fq(t)g Fp(\000)g Fq(t)2934 2329 y Fo(0)2972 2317 y Fu(\))23 b(=)g Fp(S)3171 2283 y Fi(u)3215 2317 y Fu(\()p Fq(h)3295 2283 y Fi(u)3338 2317 y Fu(\).)456 2487 y(Moreo)n(v)n(er,)i(b)n(y)i(the)h(de\014nition)g (of)f Fq(\036)h Fu(in)g(\(6.7\))g(and)f(the)h(de\014nition)g(of)g Fp(E)2757 2457 y Fi(u)2828 2487 y Fu(in)g(\(6.4\))f(w)n(e)g(ha)n(v)n(e) 1215 2653 y Fq(\036)p Fu(\()p Fq(h)1344 2618 y Fi(s)1380 2653 y Fu(\))83 b(=)g Fq(\036)p Fu(\()p Fp(S)1780 2618 y Fi(u)1825 2653 y Fu(\()p Fq(h)1905 2618 y Fi(u)1948 2653 y Fu(\)\))24 b(=)e Fp(E)2174 2618 y Fi(u)2218 2653 y Fu(\()p Fq(s)2289 2618 y Fi(u)2333 2653 y Fu(\()p Fp(S)2421 2618 y Fi(u)2465 2653 y Fu(\()p Fq(h)2545 2618 y Fi(u)2588 2653 y Fu(\)\)\))1495 2777 y(=)83 b Fp(E)1694 2743 y Fi(u)1737 2777 y Fu(\()p Fq(h)1817 2743 y Fi(u)1861 2777 y Fu(\))23 b(=)g Fp(E)7 b Fu(\()p Fq(\015)2135 2743 y Fi(u)2130 2798 y(\026;")2226 2777 y Fu(\()p Fq(t;)14 b(h)2373 2743 y Fi(u)2417 2777 y Fu(\))p Fq(;)g(t=")p Fu(\))1495 2905 y(=)83 b Fp(E)7 b Fu(\()p Fq(\015)1774 2870 y Fi(s)1769 2925 y(\026;")1865 2905 y Fu(\()p Fq(t;)14 b(h)2012 2870 y Fi(s)2047 2905 y Fu(\))p Fq(;)g(t=")p Fu(\))23 b(=)g(0.)456 3073 y(By)k(the)h(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dicit)n(y)e(of)i Fq(\036)p Fu(,)g Fq(\036)p Fu(\()p Fq(h)1626 3042 y Fi(s)1626 3093 y(n)1672 3073 y Fu(\))23 b(=)g Fq(\036)p Fu(\()p Fq(h)1944 3042 y Fi(s)1998 3073 y Fu(+)18 b(2)p Fq(\031)s("n)p Fu(\))23 b(=)g Fq(\036)p Fu(\()p Fq(h)2534 3042 y Fi(s)2570 3073 y Fu(\))g(=)g(0.)605 3172 y(Ob)n(viously)28 b Fq(\036)1046 3142 y Fj(0)1099 3172 y Fu(is)h(also)g(2)p Fq(\031)s(")p Fu(-p)r(erio)r(dic,)g(th)n(us)g Fq(\036)2088 3142 y Fj(0)2112 3172 y Fu(\()p Fq(h)2192 3142 y Fi(s)2192 3193 y(n)2237 3172 y Fu(\))h(do)r(es)f(not)g(dep)r (end)h(on)f Fq(n)p Fu(.)42 b(No)n(w)29 b(w)n(e)456 3272 y(compute)e Fq(\036)841 3242 y Fj(0)865 3272 y Fu(\()p Fq(h)945 3242 y Fi(s)981 3272 y Fu(\).)37 b(W)-7 b(e)28 b(recall)f(that)1220 3437 y Fq(\015)1268 3403 y Fi(u)1263 3458 y(\026;")1359 3437 y Fu(\()p Fq(t;)14 b(h)1506 3403 y Fi(u)1549 3437 y Fu(\))24 b(=)e Fq(\015)1740 3403 y Fi(u)1735 3458 y(\026;")1831 3437 y Fu(\()p Fq(t;)14 b(s)1969 3403 y Fi(u)2012 3437 y Fu(\()p Fq(h)2092 3403 y Fi(s)2128 3437 y Fu(\)\))24 b(=)h(~)-45 b Fq(\015)2351 3403 y Fi(u)2346 3458 y(\026;")2442 3437 y Fu(\()p Fq(t;)14 b(h)2589 3403 y Fi(s)2624 3437 y Fu(\).)456 3607 y(Di\013eren)n (tiating)27 b(in)h(\(6.8\))f(w)n(e)h(obtain)736 3780 y Fq(\036)785 3745 y Fj(0)808 3780 y Fu(\()p Fq(h)888 3745 y Fi(s)924 3780 y Fu(\))23 b(=)g Fq(@)1111 3792 y Fi(x)1153 3780 y Fp(E)7 b Fu(\()s(~)-45 b Fq(\015)1284 3745 y Fi(u)1279 3800 y(\026;")1375 3780 y Fu(\()p Fq(t;)14 b(h)1522 3745 y Fi(s)1558 3780 y Fu(\))p Fq(;)g(t=")p Fu(\))p Fq(@)1814 3792 y Fi(S)1870 3780 y Fu(~)-51 b Fq(\013)1914 3745 y Fi(u)1958 3780 y Fu(\()p Fq(t;)14 b(h)2105 3745 y Fi(s)2141 3780 y Fu(\))k(+)g Fq(@)2318 3792 y Fi(y)2358 3780 y Fp(E)7 b Fu(\()s(~)-45 b Fq(\015)2489 3745 y Fi(u)2484 3800 y(\026;")2581 3780 y Fu(\()p Fq(t;)14 b(h)2728 3745 y Fi(s)2763 3780 y Fu(\))p Fq(;)g(t=")p Fu(\))p Fq(@)3019 3792 y Fi(S)3079 3758 y Fu(~)3067 3780 y Fq(\014)3118 3745 y Fi(u)3162 3780 y Fu(\()p Fq(t;)g(h)3309 3745 y Fi(s)3344 3780 y Fu(\))-2920 b(\(6.16\))456 3965 y(where)30 b(~)-45 b Fq(\015)744 3935 y Fi(u)739 3985 y(\026;")834 3965 y Fu(\()p Fq(t;)14 b(s)p Fu(\))24 b(=)f(\()8 b(~)-50 b Fq(\013)1201 3935 y Fi(u)1245 3965 y Fu(\()p Fq(t;)14 b(s)p Fu(\))p Fq(;)1464 3943 y Fu(~)1452 3965 y Fq(\014)1503 3935 y Fi(u)1547 3965 y Fu(\()p Fq(t;)g(s)p Fu(\)\).)38 b(Moreo)n(v)n(er,)25 b(di\013eren)n(tiating)i(with)h(resp)r (ect)g(to)g Fq(s)f Fu(in)456 4067 y(\(6.6\))g(w)n(e)g(obtain)538 4233 y(1)18 b(+)g Fq(O)r Fu(\()p Fq(\026")867 4198 y Fi(p)906 4233 y Fu(\))83 b(=)g Fq(@)1213 4245 y Fi(x)1255 4233 y Fp(S)6 b Fu(\()p Fq(\015)1391 4198 y Fi(s)1386 4253 y(\026;")1482 4233 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\))p Fq(;)14 b(t=")p Fu(\))p Fq(@)1893 4245 y Fi(S)1941 4233 y Fq(\013)1994 4198 y Fi(s)2030 4233 y Fu(\()p Fq(t;)g(S)5 b Fu(\))18 b(+)g Fq(@)2362 4245 y Fi(y)2402 4233 y Fp(S)6 b Fu(\()p Fq(\015)2538 4198 y Fi(s)2533 4253 y(\026;")2630 4233 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\))p Fq(;)14 b(t=")p Fu(\))p Fq(@)3041 4245 y Fi(S)3088 4233 y Fq(\014)3139 4198 y Fi(s)3175 4233 y Fu(\()p Fq(t;)g(S)5 b Fu(\))897 4360 y(0)82 b(=)h Fq(@)1213 4372 y Fi(x)1255 4360 y Fp(E)7 b Fu(\()p Fq(\015)1386 4326 y Fi(s)1381 4380 y(\026;")1477 4360 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\))p Fq(;)14 b(t=")p Fu(\))p Fq(@)1888 4372 y Fi(S)1935 4360 y Fq(\013)1988 4326 y Fi(s)2024 4360 y Fu(\()p Fq(t;)g(S)5 b Fu(\))19 b(+)f Fq(@)2357 4372 y Fi(y)2397 4360 y Fp(E)7 b Fu(\()p Fq(\015)2528 4326 y Fi(s)2523 4380 y(\026;")2619 4360 y Fu(\()p Fq(t;)14 b(S)5 b Fu(\))p Fq(;)14 b(t=")p Fu(\))p Fq(@)3030 4372 y Fi(S)3078 4360 y Fq(\014)3129 4326 y Fi(s)3165 4360 y Fu(\()p Fq(t;)g(S)5 b Fu(\))456 4528 y(and)26 b(from)g(this,)g(taking)g(in)n(to)g(accoun)n(t)f(that)i(the)g (c)n(hange)e(\()p Fq(x;)14 b(y)s Fu(\))23 b Fp(7!)g Fu(\()p Fq(S;)14 b(E)5 b Fu(\))27 b(is)f(canonical,)g(w)n(e)456 4628 y(get,)h(when)h Fq(S)g Fu(=)22 b Fq(h)1048 4597 y Fi(s)571 4793 y Fq(@)615 4805 y Fi(S)663 4793 y Fq(\013)716 4759 y Fi(s)752 4793 y Fu(\()p Fq(t;)14 b(h)899 4759 y Fi(s)934 4793 y Fu(\)\(1)19 b(+)f Fq(O)r Fu(\()p Fq(\026")1328 4759 y Fi(p)1367 4793 y Fu(\)\))83 b(=)g Fq(@)1706 4805 y Fi(y)1746 4793 y Fp(E)7 b Fu(\()p Fq(\015)1877 4759 y Fi(s)1872 4813 y(\026;")1968 4793 y Fu(\()p Fq(t;)14 b(h)2115 4759 y Fi(s)2151 4793 y Fu(\))p Fq(;)g(t=")p Fu(\))22 b(=)h Fq(@)2517 4805 y Fi(y)2557 4793 y Fp(E)7 b Fu(\()p Fq(\015)2688 4759 y Fi(u)2683 4813 y(\026;")2779 4793 y Fu(\()p Fq(t;)14 b(h)2926 4759 y Fi(u)2970 4793 y Fu(\))p Fq(;)g(t=")p Fu(\))1514 4920 y(=)83 b Fq(@)1706 4932 y Fi(y)1746 4920 y Fp(E)7 b Fu(\()s(~)-45 b Fq(\015)1877 4886 y Fi(u)1872 4941 y(\026;")1968 4920 y Fu(\()p Fq(t;)14 b(h)2115 4886 y Fi(s)2151 4920 y Fu(\))p Fq(;)g(t=")p Fu(\))573 5047 y Fq(@)617 5059 y Fi(S)665 5047 y Fq(\014)716 5013 y Fi(s)752 5047 y Fu(\()p Fq(t;)g(h)899 5013 y Fi(s)934 5047 y Fu(\)\(1)19 b(+)f Fq(O)r Fu(\()p Fq(\026")1328 5013 y Fi(p)1367 5047 y Fu(\)\))83 b(=)g Fp(\000)p Fq(@)1771 5059 y Fi(x)1813 5047 y Fp(E)7 b Fu(\()p Fq(\015)1944 5013 y Fi(s)1939 5068 y(\026;")2035 5047 y Fu(\()p Fq(t;)14 b(h)2182 5013 y Fi(s)2217 5047 y Fu(\))p Fq(;)g(t=")p Fu(\))23 b(=)f Fp(\000)p Fq(@)2648 5059 y Fi(x)2690 5047 y Fp(E)7 b Fu(\()s(~)-45 b Fq(\015)2821 5013 y Fi(u)2816 5068 y(\026;")2912 5047 y Fu(\()p Fq(t;)14 b(h)3059 5013 y Fi(s)3095 5047 y Fu(\))p Fq(;)g(t=")p Fu(\).)456 5216 y(Substituting)28 b(the)g(deriv)-5 b(ativ)n(es)27 b(of)g Fp(E)35 b Fu(in)28 b(\(6.16\))f(w)n(e)g(obtain)g(the)h(form)n(ula)f (stated)g(in)h(a\).)p eop %%Page: 72 80 72 79 bop 456 251 a Fo(72)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(In)29 b(order)e(to)h(pro)n(v)n(e)f(b\))i(w)n(e)g(recall)e(that)i (the)g(c)n(hange)2324 429 y(~)2309 450 y Fp(C)t Fu(,)g(giv)n(en)f(in)h (Theorem)e(4.2,)i(whic)n(h)456 550 y(transforms)19 b(the)i(initial)g (co)r(ordinates)f(\()p Fq(x;)14 b(y)s Fu(\))21 b(to)g(the)g(\015o)n(w)f (b)r(o)n(x)h(co)r(ordinates)e(\()p Fq(S;)14 b(E)5 b Fu(\),)23 b(is)e(canon-)456 649 y(ical.)36 b(Therefore.)1168 849 y Fq(A)24 b Fu(=)1341 728 y Fg(\014)1341 778 y(\014)1341 828 y(\014)1341 878 y(\014)1369 736 y(Z)51 b(Z)1512 924 y Fi(Lobe)1683 849 y Fq(dx)14 b(dy)1874 728 y Fg(\014)1874 778 y(\014)1874 828 y(\014)1874 878 y(\014)1925 849 y Fu(=)2013 703 y Fg(\014)2013 753 y(\014)2013 803 y(\014)2013 853 y(\014)2013 903 y(\014)2040 736 y(Z)51 b(Z)2195 910 y Fo(~)2183 925 y Fj(C)2222 924 y Fo(\()p Fi(Lobe)p Fo(\))2445 849 y Fq(dS)19 b(dE)2667 703 y Fg(\014)2667 753 y(\014)2667 803 y(\014)2667 853 y(\014)2667 903 y(\014)2709 849 y Fu(.)456 1072 y(Moreo)n(v)n(er,)g(since)h(the)g(P)n(oincar)n(\023)-39 b(e)17 b(map)j Fq(P)1744 1042 y Fi(t)1769 1050 y Fh(0)1732 1093 y Fi(\026;")1849 1072 y Fu(is)g(orien)n(tation)f(preserving,)h (there)g(exists)g(at)g(least)456 1183 y(one)30 b(primary)f(homo)r (clinic)h(p)r(oin)n(t)h(of)g Fq(P)1733 1153 y Fi(t)1758 1161 y Fh(0)1721 1204 y Fi(\026;")1847 1183 y Fu(b)r(et)n(w)n(een)f Fq(z)2213 1153 y Fi(h)2284 1183 y Fu(=)d Fq(\015)2424 1153 y Fi(s)2419 1204 y(\026;")2515 1183 y Fu(\()p Fq(t)2577 1195 y Fo(0)2614 1183 y Fq(;)14 b(h)2699 1153 y Fi(s)2734 1183 y Fu(\))31 b(and)g Fq(P)3027 1153 y Fi(t)3052 1161 y Fh(0)3015 1204 y Fi(\026;")3111 1183 y Fu(\()p Fq(z)3186 1153 y Fi(h)3228 1183 y Fu(\).)46 b(W)-7 b(e)456 1292 y(denote)25 b(this)g(homo)r(clinic)g(p)r(oin)n(t)g(b)n(y)g Fq(\015)1670 1262 y Fi(s)1665 1313 y(\026;")1761 1292 y Fu(\()p Fq(t)1823 1304 y Fo(0)1860 1292 y Fq(;)1898 1270 y Fu(\026)1897 1292 y Fq(h)1945 1262 y Fi(s)1980 1292 y Fu(\).)37 b(By)24 b(de\014nition)i(of)f(the)g(splitting)h (function,)456 1395 y(the)21 b(area)f(of)g(a)h(lob)r(e,)h(in)g(\()p Fq(S;)14 b(E)5 b Fu(\))21 b(co)r(ordinates,)g(is)g(the)h(area)d(of)i (the)h(splitting)f(function)h(b)r(et)n(w)n(een)456 1494 y(t)n(w)n(o)k(consecutiv)n(e)h(zeros)f(of)i Fq(\036)p Fu(,)g(hence)1602 1710 y Fq(A)23 b Fu(=)1775 1564 y Fg(\014)1775 1614 y(\014)1775 1664 y(\014)1775 1714 y(\014)1775 1764 y(\014)1802 1597 y(Z)1887 1602 y Fo(\026)1885 1617 y Fi(h)1924 1592 y Fc(s)1849 1785 y Fi(h)1888 1769 y Fc(s)1974 1710 y Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))14 b Fq(dS)2256 1564 y Fg(\014)2256 1614 y(\014)2256 1664 y(\014)2256 1714 y(\014)2256 1764 y(\014)456 1937 y Fu(with)28 b Fq(h)693 1907 y Fi(s)756 1937 y Fu(and)918 1915 y(\026)917 1937 y Fq(h)965 1907 y Fi(s)1028 1937 y Fu(are)f(t)n(w)n(o)g (consecutiv)n(e)f(zeros)g(of)i Fq(\036)p Fu(.)605 2036 y(The)23 b(conclusion)g(c\))g(asserts)f(that)i(the)f(splitting)h (function)g(has)e(zero)g(mean.)36 b(T)-7 b(o)22 b(pro)n(v)n(e)g(it)456 2136 y(w)n(e)e(note)h(that)g(since)g Fq(P)1184 2106 y Fi(t)1209 2114 y Fh(0)1172 2156 y Fi(\026;")1289 2136 y Fu(is)f(area)g(preserving,)g(a)h(standard)f(geometric)g(argumen)n(t)f (giv)n(es)h(that)456 2236 y(the)j(area)e(of)h(t)n(w)n(o)g(consecutiv)n (e)g(lob)r(es)g(one)h(inner)f(and)h(the)g(other)f(outer,)h(coincide.)35 b(Therefore)456 2336 y(c\))27 b(follo)n(ws)g(from)g(b\))h(and)g(the)g (fact)g(that)f(the)h(c)n(hange)2208 2315 y(~)2193 2336 y Fp(C)k Fu(is)c(canonical.)605 2436 y(No)n(w)e(w)n(e)g(pro)n(v)n(e)f (estimate)i(d\).)37 b(By)26 b(estimate)h(b\))g(of)g(Prop)r(osition)e (6.1)g(as)h(w)n(ell)h(as)f(b)n(y)g(the)456 2536 y(de\014nition)i(of)f Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))24 b(=)f Fp(E)1251 2506 y Fi(u)1294 2536 y Fu(\()p Fq(s)1365 2506 y Fi(u)1409 2536 y Fu(\()p Fq(S)5 b Fu(\)\),)28 b(w)n(e)f(ha)n(v)n(e)g(that)h(for)f (real)f(v)-5 b(alues)28 b(of)f Fq(S)716 2688 y(\036)p Fu(\()p Fq(S)5 b Fu(\))24 b Fp(\021)f(E)1048 2654 y Fi(u)1091 2688 y Fu(\()p Fq(s)1162 2654 y Fi(u)1206 2688 y Fu(\()p Fq(S)5 b Fu(\)\))24 b(=)e Fp(E)1520 2654 y Fi(u)1513 2709 y Fo(0)1564 2688 y Fu(\()p Fq(")p Fu(\))d(+)f Fq(\026")1858 2654 y Fi(p)1896 2688 y Fq(M)9 b Fu(\()p Fq(s)2057 2654 y Fi(u)2100 2688 y Fu(\()p Fq(S)c Fu(\))p Fq(;)14 b(")p Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(\026)2577 2654 y Fo(2)2615 2688 y Fq(")2654 2654 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(1)2896 2688 y Fq(;)c(\026")3022 2654 y Fi(p)p Fo(+1)3144 2688 y Fu(\))p Fq(e)3215 2654 y Fj(\000)p Fi(a=")3373 2688 y Fu(,)456 2839 y(where)25 b Fp(E)745 2809 y Fi(u)738 2859 y Fo(0)788 2839 y Fu(\()p Fq(")p Fu(\))h(is)g(the)g(0-F)-7 b(ourier)23 b(co)r(e\016cien)n(t)j(of)f Fp(E)2023 2809 y Fi(u)2067 2839 y Fu(.)36 b(Th)n(us,)26 b(in)g(order)e(to)i(pro)n(v)n(e)e(Theorem)g(1.1)456 2938 y(w)n(e)c(need)h(to)g(estimate)g Fp(E)1230 2908 y Fi(u)1223 2959 y Fo(0)1273 2938 y Fu(\()p Fq(")p Fu(\).)35 b(By)21 b(T)-7 b(a)n(ylor's)19 b(Theorem)h(and)h(Lemma)g(6.4)f(w)n(e)g(ha)n(v)n (e)g(for)g Fq(S)28 b Fp(2)23 b Fl(R)p Fu(,)723 3085 y Fq(M)9 b Fu(\()p Fq(s)884 3051 y Fi(u)927 3085 y Fu(\()p Fq(S)c Fu(\))p Fq(;)14 b(")p Fu(\))83 b(=)g Fq(M)9 b Fu(\()p Fq(S)23 b Fu(+)18 b Fq(O)r Fu(\()p Fq(\026")1851 3051 y Fi(p)p Fo(+1)1974 3085 y Fu(\))p Fq(;)c(")p Fu(\))1238 3269 y(=)83 b Fq(M)9 b Fu(\()p Fq(S;)14 b(")p Fu(\))k(+)1768 3156 y Fg(Z)1851 3176 y Fo(1)1814 3344 y(0)1912 3213 y Fq(dM)p 1912 3250 133 4 v 1929 3326 a(dS)2055 3269 y Fu(\()p Fq(S)23 b Fu(+)18 b Fq(\020)6 b(O)r Fu(\()p Fq(\026")2472 3235 y Fi(p)p Fo(+1)2596 3269 y Fu(\))p Fq(;)14 b(")p Fu(\))p Fq(O)r Fu(\()p Fq(\026")2922 3235 y Fi(p)p Fo(+1)3045 3269 y Fu(\))g Fq(d\020)1238 3456 y Fu(=)83 b Fq(M)9 b Fu(\()p Fq(S;)14 b(")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(\026")1954 3421 y Fi(\027)1996 3456 y Fu(\))p Fq(e)2067 3421 y Fj(\000)p Fi(a=")2225 3456 y Fu(.)456 3601 y(Therefore,)26 b(w)n(e)h(ha)n(v)n(e)g(that)884 3749 y Fq(\036)p Fu(\()p Fq(S)5 b Fu(\))23 b(=)g Fp(E)1215 3715 y Fi(u)1208 3770 y Fo(0)1259 3749 y Fu(\()p Fq(")p Fu(\))18 b(+)g Fq(\026")1552 3715 y Fi(p)1591 3749 y Fq(M)9 b Fu(\()p Fq(S;)14 b(")p Fu(\))k(+)g Fq(O)r Fu(\()p Fq(\026)2120 3715 y Fo(2)2158 3749 y Fq(")2197 3715 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(1)2440 3749 y Fq(;)c(\026)2527 3715 y Fo(2)2564 3749 y Fq(")2603 3715 y Fi(\027)t Fo(+)p Fi(p)2729 3749 y Fq(;)g(\026")2855 3715 y Fi(p)p Fo(+1)2977 3749 y Fu(\))p Fq(e)3048 3715 y Fj(\000)p Fi(a=")3206 3749 y Fu(.)456 3899 y(Finally)27 b(since)h(b)n(y)f(c\),)h Fq(\036)1226 3911 y Fo(0)1287 3899 y Fu(=)22 b(0,)27 b(and)h(the)g(a)n(v)n(erage)d(of)i Fq(M)36 b Fu(is)28 b(zero)e(w)n(e)i(get)1179 4052 y Fp(E)1230 4018 y Fi(u)1223 4073 y Fo(0)1274 4052 y Fu(\()p Fq(")p Fu(\))23 b(=)g Fq(O)r Fu(\()p Fq(\026)1635 4018 y Fo(2)1673 4052 y Fq(")1712 4018 y Fo(2)p Fi(\027)t Fo(+)p Fi(r)r Fj(\000)p Fo(1)1955 4052 y Fq(;)14 b(\026)2042 4018 y Fo(2)2079 4052 y Fq(")2118 4018 y Fi(\027)t Fo(+)p Fi(p)2244 4052 y Fq(;)g(\026")2370 4018 y Fi(p)p Fo(+1)2492 4052 y Fu(\))p Fq(e)2563 4018 y Fj(\000)p Fi(a=")456 4202 y Fu(and)27 b(the)h(estimate)g(d\))g(is)f(pro)n(v)n(ed.)1831 b Fe(\003)605 4367 y Fu(The)28 b(pro)r(of)f(of)g(Theorem)g(1.1)g(is)g (an)h(immediate)f(consequence)g(of)h(Prop)r(osition)e(6.2.)1435 4568 y Fk(6.4.)47 b(Pro)s(of)31 b(of)h(Lemma)e(6.4)605 4717 y Fu(The)41 b(pro)r(of)f(of)g(this)h(lemma)g(has)f(signi\014cativ) n(e)g(di\013erences)g(from)g(the)h(pro)r(of)f(of)h(the)456 4817 y(corresp)r(onding)28 b(lemma)i(in)g([)p Fk(DS2)p Fu(].)45 b(As)30 b(w)n(e)g(p)r(oin)n(ted)g(out)g(b)r(efore,)h(w)n(e)e (are)g(considering)g(the)456 4917 y(case)23 b(suc)n(h)i(that)g(the)g (parameterization)d(of)j(the)g(homo)r(clinic)g(orbit)f(has)g(a)g (singularit)n(y)g(whic)n(h)456 5016 y(is)31 b(a)h(branc)n(hing)e(p)r (oin)n(t.)50 b(The)32 b(pro)r(of)f(of)h(Lemma)f(6.4)g(is)h(the)g(place) g(where)f(this)h(h)n(yp)r(othesis)456 5116 y(has)c(to)h(b)r(e)g(tak)n (en)g(in)n(to)f(accoun)n(t.)40 b(Since)30 b Fq(u)24 b Fu(=)h Fp(\006)p Fq(ia)j Fu(are)g(branc)n(hing)g(p)r(oin)n(ts,)h(the)h (homo)r(clinic)456 5216 y(orbit)35 b(is)g(de\014ned)h(in)g(a)f(neigh)n (b)r(orho)r(o)r(d)g(of)g(the)h(singularities)f(except)g(a)g(segmen)n(t) g(starting)p eop %%Page: 73 81 73 80 bop 1505 251 a Fo(6.4.)30 b(PR)n(OOF)f(OF)f(LEMMA)h(6.4)983 b(73)456 450 y Fu(at)29 b(them,)h(and)g(therefore)e(w)n(e)h(can)g(not)g (use)h(the)f(residue)g(theory)g(in)g(order)f(to)i(estimate)f(the)456 550 y(Melnik)n(o)n(v)d(in)n(tegral.)605 649 y(The)e(case)f(that)i(the)f (singularit)n(y)f(is)h(a)g(p)r(ole)g(also)f(follo)n(ws)g(from)g(this)i (pro)r(of)e(taking)h(b)r(elo)n(w)456 749 y Fq(q)i Fu(=)c(1.)605 849 y(W)-7 b(e)28 b(recall)f(the)h(de\014nition)g(of)f Fq(J)36 b Fu(giv)n(en)27 b(in)h(Chapter)f(1:)1017 1010 y Fq(J)8 b Fu(\()p Fq(x;)14 b(y)s(;)g(t=")p Fu(\))23 b Fp(\021)g(f)p Fq(h)1612 1022 y Fo(0)1648 1010 y Fq(;)14 b(h)1733 1022 y Fo(1)1770 1010 y Fp(g)p Fu(\()p Fq(x;)g(y)s(;)g(t=")p Fu(\))23 b Fp(\030)2265 931 y Fg(X)2262 1110 y Fi(n)p Fj(6)p Fo(=0)2401 1010 y Fq(J)2447 1022 y Fi(n)2492 1010 y Fu(\()p Fq(x;)14 b(y)s Fu(\))p Fq(e)2723 976 y Fi(int=")456 1250 y Fu(and)26 b(that)h Fq(J)8 b Fu(\()p Fq(\015)924 1262 y Fo(0)961 1250 y Fu(\()p Fq(t)17 b Fu(+)f Fq(s)p Fu(\))p Fq(;)e(t=")p Fu(\))26 b(has)g(a)g(singularit)n(y)f(of)i(order)e (at)h(most)h Fq(`)16 b Fu(+)g(1.)36 b(W)-7 b(e)27 b(also)e(observ)n(e) 456 1350 y(that)i(the)h(p)r(erturbation)g Fq(h)1314 1362 y Fo(1)1351 1350 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))28 b(can)f(b)r(e)h(written)g(as)1307 1516 y Fq(h)1355 1528 y Fo(1)1392 1516 y Fu(\()p Fq(x;)14 b(y)s(;)g(\022)r Fu(\))24 b(=)1914 1437 y Fg(X)1774 1619 y Fi(k)q Fj(\024j)p Fi(l)p Fj(j\024)p Fi(\024;l)p Fj(2)p Fd(N)2143 1603 y Fh(2)2188 1516 y Fq(a)2232 1528 y Fi(l)2257 1516 y Fu(\()p Fq(\022)r Fu(\))p Fq(x)2409 1482 y Fi(l)2430 1490 y Fh(1)2468 1516 y Fq(y)2512 1482 y Fi(l)2533 1490 y Fh(2)2570 1516 y Fu(.)456 1758 y(W)-7 b(e)28 b(recall)e(that,)i(b)n(y)f(its)h (de\014nition,)g Fq(`)g Fu(can)f(b)r(e)h(expressed)e(in)i(the)g(form) 1273 1925 y Fq(`)23 b Fu(=)f Fq(j)1452 1937 y Fo(1)1490 1925 y Fu(\()p Fq(r)f Fp(\000)d Fu(1\))h(+)f Fq(j)1873 1937 y Fo(2)1910 1925 y Fq(r)26 b Fu(=)c Fq(j)2094 1937 y Fo(2)2150 1925 y Fu(+)c(\()p Fq(j)2299 1937 y Fo(1)2355 1925 y Fu(+)h Fq(j)2473 1937 y Fo(2)2510 1925 y Fu(\))2552 1868 y Fq(p)p 2552 1906 42 4 v 2553 1982 a(q)2604 1925 y Fu(,)-2171 b(\(6.17\))456 2114 y(where)27 b Fq(j)730 2126 y Fo(1)795 2114 y Fu(and)g Fq(j)990 2126 y Fo(2)1055 2114 y Fu(are)g(suc)n(h)g(that)566 2350 y Fq(j)600 2362 y Fo(1)637 2350 y Fu(\()p Fq(r)22 b Fp(\000)c Fu(1\))g(+)g Fq(j)1020 2362 y Fo(2)1057 2350 y Fq(r)26 b Fu(=)d(max)o Fp(f)p Fq(l)1429 2362 y Fo(1)1466 2350 y Fu(\()p Fq(r)e Fp(\000)d Fu(1\))h(+)f Fq(l)1840 2362 y Fo(2)1877 2350 y Fq(r)53 b Fu(:)e Fq(l)2066 2362 y Fo(1)2121 2350 y Fu(+)18 b Fq(l)2229 2362 y Fo(2)2289 2350 y Fp(\025)23 b Fq(k)31 b Fu(,)c Fq(l)e Fu(=)d(\()p Fq(l)2695 2362 y Fo(1)2733 2350 y Fq(;)14 b(l)2795 2362 y Fo(2)2832 2350 y Fu(\))28 b(,)f Fq(a)2986 2362 y Fi(l)3012 2350 y Fu(\()p Fq(\022)r Fu(\))d Fp(6)p Fu(=)e(0)p Fp(g)p Fu(.)605 2505 y(No)n(w)41 b(w)n(e)h(write)f(the)h(F)-7 b(ourier's)41 b(co)r(e\016cien)n(ts)g(of)h Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\),)45 b Fq(M)2647 2517 y Fi(n)2692 2505 y Fu(\()p Fq(")p Fu(\),)h(in)c(terms)f(of)h(the)456 2604 y(F)-7 b(ourier's)29 b(co)r(e\016cien)n(ts)h(of)g Fq(J)38 b Fu(ev)-5 b(aluated)30 b(at)h Fq(\015)1928 2616 y Fo(0)1965 2604 y Fu(\()p Fq(u)p Fu(\):)42 b Fq(J)2188 2616 y Fi(n)2234 2604 y Fu(\()p Fq(\015)2309 2616 y Fo(0)2346 2604 y Fu(\()p Fq(u)p Fu(\)\).)k(W)-7 b(e)30 b(note)h(that,)g(since)f Fq(J)39 b Fu(is)456 2704 y(only)21 b(con)n(tin)n(uous)h(with)g(resp)r (ect)h(to)f Fq(t=")p Fu(,)g(the)h(F)-7 b(ourier's)21 b(series)g(ma)n(y)h(not)g(con)n(v)n(erge,)f(although)456 2804 y(their)29 b(F)-7 b(ourier's)29 b(co)r(e\016cien)n(ts)g(are)f(w)n 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4867 y Fq(du)p Fu(.)-2761 b(\(6.20\))456 5116 y(T)-7 b(o)28 b(ev)-5 b(aluate)28 b(the)g(in)n(tegrals)f(in)i(the)f(righ)n(t)g (hand)g(side)h(of)f(\(6.20\))g(w)n(e)g(distinguish)g(t)n(w)n(o)g (cases:)456 5216 y Fq(m=q)34 b(=)-51 b Fp(2)24 b Fl(N)37 b Fu(and)28 b Fq(m=q)d Fp(2)f Fl(N)s Fu(.)43 b(First)28 b(w)n(e)f(deal)g(with)i(the)e(case)g Fq(m=q)35 b(=)-51 b Fp(2)23 b Fl(N)t Fu(:)p eop %%Page: 75 83 75 82 bop 1505 251 a Fo(6.4.)30 b(PR)n(OOF)f(OF)f(LEMMA)h(6.4)983 b(75)605 450 y Fu(1.)42 b(In)n(tegral)28 b(o)n(v)n(er)g(\000)1255 462 y Fo(5)1292 450 y Fu(\()p Fq(\032)p Fu(\).)43 b(W)-7 b(e)30 b(parameterize)e(this)i(path)f(b)n(y)h Fq(g)2623 462 y Fo(5)2660 450 y Fu(\()p Fq(\022)r Fu(\))d(=)e Fq(ia)20 b Fu(+)f Fq(\032e)3141 420 y Fj(\000)p Fi(i\022)3283 450 y Fu(with)456 550 y Fq(\022)25 b Fp(2)e Fu([0)p Fq(;)14 b(\031)s Fu(].)37 b(Using)28 b(the)g(series)e(expansion)h(of)g(the)h (exp)r(onen)n(tial)f(w)n(e)h(ha)n(v)n(e)e(that)468 643 y Fg(Z)515 832 y Fo(\000)556 840 y Fh(5)588 832 y Fo(\()p Fi(\032)p Fo(\))848 700 y Fq(e)887 670 y Fj(\000)p Fi(inu=")p 702 737 557 4 v 702 824 a Fu(\()p Fq(u)19 b Fp(\000)f Fq(ia)p Fu(\))999 765 y Fc(m)p 998 774 51 3 v 1009 807 a(q)1059 787 y Fo(+)p Fi(j)1137 795 y Fh(2)1170 787 y Fo(+1)1282 756 y Fq(du)83 b Fu(=)f Fp(\000)p Fq(ie)1736 722 y Fi(na=")1882 756 y Fq(\032)1939 643 y Fg(Z)2022 664 y Fi(\031)1985 832 y Fo(0)2091 700 y Fq(e)2130 670 y Fj(\000)p Fi(in\032e)2311 645 y Fb(\000)p Fc(i\022)2412 670 y Fi(=")2482 700 y Fq(e)2521 670 y Fj(\000)p Fi(i\022)p 2091 737 543 4 v 2098 824 a Fu(\()p Fq(\032e)2212 800 y Fj(\000)p Fi(i\022)2324 824 y Fu(\))2366 765 y Fc(m)p 2367 774 51 3 v 2378 807 a(q)2427 787 y Fo(+)p Fi(j)2505 795 y Fh(2)2538 787 y Fo(+1)2657 756 y Fq(d\022)1456 1020 y Fu(=)g Fp(\000)p Fq(ie)1736 986 y Fi(na=")1882 1020 y Fq(\032)1925 983 y Fj(\000)1987 961 y Fc(m)p 1987 970 V 1998 1003 a(q)2048 983 y Fj(\000)p Fi(j)2127 991 y Fh(2)2177 941 y Fg(X)2184 1120 y Fi(l)p Fj(\025)p Fo(0)2311 907 y Fg(Z)2394 927 y Fi(\031)2357 1096 y Fo(0)2467 964 y Fu(1)p 2463 1001 50 4 v 2463 1077 a Fq(l)r Fu(!)2536 903 y Fg(\022)2607 964 y Fp(\000)p Fq(in\032)p 2607 1001 186 4 v 2681 1077 a(")2803 903 y Fg(\023)2864 920 y Fi(l)2904 1020 y Fq(e)2943 983 y Fi(i\022)r Fo(\()3035 961 y Fc(m)p 3035 970 51 3 v 3046 1003 a(q)3096 983 y Fo(+)p Fi(j)3174 991 y Fh(2)3207 983 y Fj(\000)p Fi(l)p Fo(\))3324 1020 y Fq(d\022)r Fu(.)456 1247 y(Therefore,)28 b(since)g Fq(m=q)37 b(=)-51 b Fp(2)26 b Fl(N)t Fu(,)35 b(the)30 b(in)n(tegral)d(o)n(v)n(er)h(\000)2114 1259 y Fo(5)2151 1247 y Fu(\()p Fq(\032)p Fu(\))h(has)g(no)f(constan)n(t)g(term)h(in)h Fq(\032)p Fu(,)f(then)456 1346 y(this)e(in)n(tegral)g(has)g(not)g(con)n (tribution)g(to)h(the)g Fq(\032)1976 1316 y Fo(0)2041 1346 y Fu(term)f(of)h(its)g(expansion.)605 1446 y(2.)67 b(In)n(tegral)36 b(o)n(v)n(er)g(\000)1296 1458 y Fo(4)1333 1446 y Fu(\()p Fq(\032)p Fu(\).)68 b(W)-7 b(e)39 b(parameterize)d(this) i(path)g(b)n(y)f Fq(g)2730 1458 y Fo(4)2767 1446 y Fu(\()p Fq(\022)r Fu(\))k(=)e Fq(\032)25 b Fp(\000)g Fq(i\022)40 b Fu(with)456 1546 y Fq(\022)25 b Fp(2)e Fu([)p Fp(\000)p Fq(b;)14 b Fp(\000)p Fq(a)p Fu(].)36 b(Then)884 1632 y Fg(Z)930 1821 y Fo(\000)971 1829 y Fh(4)1004 1821 y Fo(\()p Fi(\032)p Fo(\))1264 1689 y Fq(e)1303 1659 y Fj(\000)p Fi(inu=")p 1118 1726 557 4 v 1118 1813 a Fu(\()p Fq(u)18 b Fp(\000)g Fq(ia)p Fu(\))1414 1754 y Fc(m)p 1414 1763 51 3 v 1425 1796 a(q)1475 1776 y Fo(+)p Fi(j)1553 1784 y Fh(2)1586 1776 y Fo(+1)1698 1745 y Fq(du)k Fu(=)h Fp(\000)p Fq(i)2007 1632 y Fg(Z)2089 1652 y Fi(b)2052 1821 y(a)2295 1689 y Fq(e)2334 1659 y Fj(\000)p Fi(\032ni=")2554 1689 y Fq(e)2593 1659 y Fi(n\022)r(=")p 2146 1726 739 4 v 2146 1830 a Fu(\()p Fq(\032)c Fu(+)f(\()p Fq(\022)j Fp(\000)d Fq(a)p Fu(\))p Fq(i)p Fu(\))2645 1754 y Fc(m)p 2645 1761 51 3 v 2656 1792 a(q)2706 1770 y Fh(+)p Fc(j)2773 1782 y Fh(2)2805 1770 y(+1)2908 1745 y Fq(d\022)r Fu(.)456 1963 y(Giv)n(en)27 b Fq(l)d Fp(2)g Fl(Z)885 1933 y Fo(+)962 1963 y Fu(and)j Fq(\021)g Fp(2)c Fu(\(0)p Fq(;)14 b Fu(1\),)27 b(w)n(e)h(in)n(tro)r(duce)f(the)h(notation)1390 2162 y Fq(I)1426 2174 y Fi(l)1452 2162 y Fu(\()p Fq(\032)p Fu(\))23 b(=)1670 2049 y Fg(Z)1753 2069 y Fi(b)1716 2238 y(a)2015 2106 y Fq(e)2054 2076 y Fi(n\022)r(=")p 1810 2143 593 4 v 1810 2222 a Fu(\()p Fq(\032)c Fu(+)f(\()p Fq(\022)j Fp(\000)d Fq(a)p Fu(\))p Fq(i)p Fu(\))2299 2182 y Fc(\021)q Fh(+)p Fc(l)2426 2162 y Fq(d\022)456 2346 y Fu(and)1335 2522 y Fq(f)1376 2534 y Fi(l)1401 2522 y Fu(\()p Fq(\032)p Fu(\))24 b(=)1629 2466 y Fq(e)1668 2436 y Fi(na=")p 1629 2503 186 4 v 1644 2579 a Fq(\032)1687 2555 y Fi(\021)r Fo(+)p Fi(l)1843 2522 y Fp(\000)2145 2466 y Fq(e)2184 2436 y Fi(nb=")p 1936 2503 596 4 v 1936 2579 a Fu(\()p Fq(\032)19 b Fu(+)f Fq(i)p Fu(\()p Fq(b)g Fp(\000)g Fq(a)p Fu(\)\))2419 2555 y Fi(\021)r Fo(+)p Fi(l)2542 2522 y Fu(.)456 2707 y(In)n(tegrating)26 b(b)n(y)h(parts)g (in)h Fq(I)1346 2719 y Fi(l)1372 2707 y Fu(\()p Fq(\032)p Fu(\))g(w)n(e)f(obtain)g(a)h(recurrence)e(form)n(ula)g(for)h Fq(I)2826 2719 y Fi(l)2852 2707 y Fu(\()p Fq(\032)p Fu(\):)1258 2884 y Fq(I)1294 2896 y Fi(l)1320 2884 y Fu(\()p Fq(\032)p Fu(\))c(=)1731 2827 y(1)p 1548 2864 409 4 v 1548 2940 a Fq(i)p Fu(\()p Fq(\021)f Fu(+)c Fq(l)h Fp(\000)f Fu(1\))1966 2791 y Fg(\020)2016 2884 y Fq(f)2057 2896 y Fi(l)p Fj(\000)p Fo(1)2167 2884 y Fu(\()p Fq(\032)p Fu(\))h(+)2386 2827 y Fq(n)p 2386 2864 50 4 v 2392 2940 a(")2446 2884 y(I)2482 2896 y Fi(l)p Fj(\000)p Fo(1)2593 2791 y Fg(\021)456 3065 y Fu(and)27 b(therefore)911 3263 y Fq(I)947 3275 y Fi(l)973 3263 y Fu(\()p Fq(\032)p Fu(\))84 b(=)1361 3159 y Fi(l)1311 3184 y Fg(X)1314 3361 y Fi(j)s Fo(=1)1445 3170 y Fg(\020)1505 3206 y Fq(n)p 1505 3244 V 1511 3320 a(")1564 3170 y Fg(\021)1614 3188 y Fi(j)s Fj(\000)p Fo(1)1769 3206 y Fu(1)p 1758 3244 64 4 v 1758 3320 a Fq(i)1787 3296 y Fi(j)2261 3206 y Fu(1)p 1841 3244 882 4 v 1841 3320 a(\()p Fq(\021)22 b Fu(+)c Fq(l)i Fp(\000)e Fu(1\))c Fp(\001)g(\001)g(\001)f Fu(\()p Fq(\021)22 b Fu(+)c Fq(l)i Fp(\000)e Fq(j)5 b Fu(\))2732 3263 y Fq(f)2773 3275 y Fi(l)p Fj(\000)p Fi(j)2881 3263 y Fu(\()p Fq(\032)p Fu(\))-2532 b(\(6.21\))1311 3529 y(+)1390 3437 y Fg(\020)1449 3473 y Fq(n)p 1449 3510 50 4 v 1455 3586 a(")1509 3437 y Fg(\021)1559 3454 y Fi(l)1614 3473 y Fu(1)p 1608 3510 55 4 v 1608 3586 a Fq(i)1637 3562 y Fi(l)1935 3473 y Fu(1)p 1682 3510 549 4 v 1682 3586 a(\()p Fq(\021)22 b Fu(+)c Fq(l)i Fp(\000)e Fu(1\))c Fp(\001)g(\001)g(\001)f Fq(\021)2240 3529 y(I)2276 3541 y Fo(0)2314 3529 y Fu(\()p Fq(\032)p Fu(\),)456 3720 y(The)27 b(con)n(tribution)g(of)h(the)g Fq(j)5 b Fu(-term)27 b(in)h(the)g(sum)g(\(6.21\))e(to)i(the)g Fq(\032)2545 3690 y Fo(0)2610 3720 y Fu(term)f(is)977 3916 y Fp(\000)1056 3824 y Fg(\020)1115 3860 y Fq(n)p 1115 3897 50 4 v 1121 3973 a(")1175 3824 y Fg(\021)1225 3841 y Fi(j)s Fj(\000)p Fo(1)1380 3860 y Fu(1)p 1368 3897 64 4 v 1368 3973 a Fq(i)1397 3949 y Fi(j)1872 3860 y Fu(1)p 1452 3897 882 4 v 1452 3973 a(\()p Fq(\021)22 b Fu(+)c Fq(l)i Fp(\000)e Fu(1\))c Fp(\001)g(\001)g(\001)f Fu(\()p Fq(\021)22 b Fu(+)c Fq(l)i Fp(\000)e Fq(j)5 b Fu(\))2532 3860 y Fq(e)2571 3830 y Fi(nb=")p 2353 3897 537 4 v 2353 3973 a Fu(\()p Fq(i)p Fu(\()p Fq(b)18 b Fp(\000)g Fq(a)p Fu(\)\))2691 3949 y Fi(\021)r Fo(+)p Fi(l)p Fj(\000)p Fo(1)2900 3916 y Fu(.)-2467 b(\(6.22\))456 4102 y(No)n(w)22 b(w)n(e)h(analyze)g Fq(I)1088 4114 y Fo(0)1125 4102 y Fu(\()p Fq(\032)p Fu(\).)36 b(Note)24 b(that,)g(since)f Fq(\032)g(>)g Fu(0,)h(w)n(e)f(ha)n(v)n(e)f(that)i (arg)o(\()p Fq(\032)10 b Fu(+)g Fq(i)p Fu(\()p Fq(z)k Fp(\000)c Fq(a)p Fu(\)\))22 b Fp(!)h Fq(\031)s(=)p Fu(2)456 4201 y(when)k Fq(\032)c Fp(!)g Fu(0)28 b(for)f Fq(z)f(>)d(a)p Fu(.)37 b(Therefore,)26 b(using)h(the)h(dominated)g(con)n(v)n(ergence)d (theorem)739 4396 y Fq(I)775 4408 y Fo(0)812 4396 y Fu(\()p Fq(\032)p Fu(\))f(=)1031 4283 y Fg(Z)1114 4303 y Fi(b)1077 4472 y(a)1344 4340 y Fq(e)1383 4310 y Fi(n\022)r(=")p 1171 4377 530 4 v 1171 4453 a Fu(\()p Fq(\032)19 b Fu(+)f Fq(i)p Fu(\()p Fq(\022)i Fp(\000)e Fq(a)p Fu(\)\))1659 4429 y Fi(\021)1724 4396 y Fq(d\022)25 b Fp(!)f Fq(e)1977 4362 y Fj(\000)p Fi(\021)r(i\031)r(=)p Fo(2)2214 4283 y Fg(Z)2297 4303 y Fi(b)2260 4472 y(a)2409 4340 y Fq(e)2448 4310 y Fi(n\022)r(=")p 2354 4377 292 4 v 2354 4453 a Fu(\()p Fq(\022)d Fp(\000)d Fq(a)p Fu(\))2605 4429 y Fi(\021)2669 4396 y Fq(d\022)169 b Fu(when)28 b Fq(\032)23 b Fp(!)g Fu(0.)456 4577 y(With)28 b(elemen)n(tary)f(c)n(hanges)f(of)h (v)-5 b(ariables)27 b(w)n(e)g(get)940 4663 y Fg(Z)1023 4684 y Fi(b)986 4852 y(a)1135 4720 y Fq(e)1174 4690 y Fi(n\022)r(=")p 1080 4757 V 1080 4833 a Fu(\()p Fq(\022)21 b Fp(\000)d Fq(a)p Fu(\))1331 4809 y Fi(\021)1396 4776 y Fq(d\022)25 b Fu(=)1591 4659 y Fg(\022)1691 4720 y Fq(")p 1662 4757 96 4 v 1662 4833 a Fp(j)p Fq(n)p Fp(j)1768 4659 y Fg(\023)1829 4676 y Fo(1)p Fj(\000)p Fi(\021)1968 4776 y Fq(e)2007 4742 y Fi(na=")2168 4663 y Fg(Z)2251 4684 y Fj(j)p Fi(n)p Fj(j)p Fo(\()p Fi(b)p Fj(\000)p Fi(a)p Fo(\))p Fi(=")2214 4852 y Fo(0)2583 4776 y Fq(s)2622 4742 y Fj(\000)p Fi(\021)2715 4776 y Fq(e)2754 4742 y Fj(\000)p Fi(s)2855 4776 y Fq(ds)p Fu(.)-2504 b(\(6.23\))456 4958 y(Moreo)n(v)n(er)24 b(w)n(e)k(ha)n(v)n(e)e(that)1227 5027 y Fg(Z)1310 5047 y Fj(j)p Fi(n)p Fj(j)p Fo(\()p Fi(b)p Fj(\000)p Fi(a)p Fo(\))p Fi(=")1273 5216 y Fo(0)1643 5140 y Fq(s)1682 5106 y Fj(\000)p Fi(\021)1774 5140 y Fq(e)1813 5106 y Fj(\000)p Fi(s)1914 5140 y Fq(ds)e Fu(=)e(\000\(1)c Fp(\000)g Fq(\021)s Fu(\))h Fp(\000)f Fq( )s Fu(\()p Fq(")p Fu(\))-2216 b(\(6.24\))p eop %%Page: 76 84 76 83 bop 456 251 a Fo(76)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fu(where)27 b(\000)g(is)h(the)g(Gamma)f(function)h(and)919 647 y Fq( )s Fu(\()p Fq(")p Fu(\))c(=)1190 534 y Fg(Z)1273 555 y Fo(+)p Fj(1)1236 723 y(j)p Fi(n)p Fj(j)p Fo(\()p Fi(b)p Fj(\000)p Fi(a)p Fo(\))p Fi(=")1569 647 y Fq(s)1608 613 y Fj(\000)p Fi(\021)1701 647 y Fq(e)1740 613 y Fj(\000)p Fi(s)1840 647 y Fq(ds)g Fp(\024)2033 530 y Fg(\022)2256 591 y Fq(")p 2104 628 342 4 v 2104 704 a Fp(j)p Fq(n)p Fp(j)p Fu(\()p Fq(b)19 b Fp(\000)f Fq(a)p Fu(\))2456 530 y Fg(\023)2517 547 y Fi(\021)2571 647 y Fq(e)2610 613 y Fj(\000j)p Fi(n)p Fj(j)p Fo(\()p Fi(b)p Fj(\000)p Fi(a)p Fo(\))p Fi(=")456 868 y Fu(whic)n(h)32 b(is)g(exp)r(onen)n (tially)g(small.)51 b(Using)32 b(\(6.22\),)h(\(6.23\))f(and)g (\(6.24\),)h(w)n(e)f(obtain)g(that)h(the)456 968 y(constan)n(t)26 b(term)i(in)g Fq(\032)f Fu(of)h Fq(I)1287 980 y Fi(l)1313 968 y Fu(\()p Fq(\032)p Fu(\))g(is)605 1209 y Fq(I)648 1175 y Fo(0)641 1230 y Fi(l)768 1209 y Fp(\021)965 1106 y Fi(l)916 1131 y Fg(X)918 1307 y Fi(j)s Fo(=1)1049 1117 y Fg(\020)1109 1153 y Fq(n)p 1109 1190 50 4 v 1115 1266 a(")1169 1117 y Fg(\021)1218 1134 y Fi(j)s Fj(\000)p Fo(1)1373 1153 y Fu(1)p 1362 1190 64 4 v 1362 1266 a Fq(i)1391 1242 y Fi(j)2038 1153 y Fp(\000)p Fq(e)2142 1123 y Fi(nb=")p 1446 1190 1428 4 v 1446 1266 a Fu(\()p Fq(\021)22 b Fu(+)c Fq(l)h Fp(\000)f Fu(1\))c Fp(\001)g(\001)g(\001)g Fu(\()p Fq(\021)22 b Fu(+)c Fq(l)i Fp(\000)e Fq(j)5 b Fu(\)\()p Fq(i)p Fu(\()p Fq(b)18 b Fp(\000)g Fq(a)p Fu(\)\))2665 1242 y Fi(\021)r Fo(+)p Fi(j)s Fj(\000)p Fo(1)916 1498 y Fu(+)995 1405 y Fg(\020)1054 1441 y Fq(n)p 1054 1478 50 4 v 1060 1554 a(")1113 1405 y Fg(\021)1163 1423 y Fi(l)1219 1441 y Fu(1)p 1212 1478 55 4 v 1212 1554 a Fq(i)1241 1530 y Fi(l)1540 1441 y Fu(1)p 1286 1478 549 4 v 1286 1554 a(\()p Fq(\021)k Fu(+)c Fq(l)i Fp(\000)e Fu(1\))c Fp(\001)g(\001)g(\001)f Fq(\021)1858 1380 y Fg(\022)1958 1441 y Fq(")p 1930 1478 96 4 v 1930 1554 a Fp(j)p Fq(n)p Fp(j)2035 1380 y Fg(\023)2097 1398 y Fo(1)p Fj(\000)p Fi(\021)2236 1498 y Fq(e)2275 1463 y Fi(na=")2421 1498 y Fq(e)2460 1463 y Fj(\000)p Fi(\021)r(i\031)r(=)p Fo(2)2684 1498 y Fu([\000\(1)18 b Fp(\000)g Fq(\021)s Fu(\))h Fp(\000)f Fq( )s Fu(\()p Fq(")p Fu(\)])456 1703 y(and)27 b(therefore)868 1911 y Fq(I)911 1877 y Fo(0)904 1932 y Fi(l)972 1911 y Fu(=)1069 1855 y(\()p Fp(\000)p Fu(1\))1240 1825 y Fi(l)p 1069 1892 197 4 v 1140 1968 a Fq(i)1169 1944 y Fi(l)1289 1794 y Fg(\022)1360 1855 y Fp(j)p Fq(n)p Fp(j)p 1360 1892 96 4 v 1389 1968 a Fq(")1466 1794 y Fg(\023)1527 1811 y Fi(l)p Fj(\000)p Fo(1+)p Fi(\021)1739 1911 y Fq(e)1778 1877 y Fj(\000)p Fi(\021)r(i\031)r(=)p Fo(2)2117 1855 y Fq(\031)p 2011 1892 262 4 v 2011 1968 a Fu(sin\()p Fq(\031)s(\021)s Fu(\))2416 1855 y(1)p 2292 1892 289 4 v 2292 1968 a(\000\()p Fq(l)20 b Fu(+)e Fq(\021)s Fu(\))2591 1911 y Fq(e)2630 1877 y Fj(\000j)p Fi(n)p Fj(j)p Fi(a=")2867 1911 y Fu(\(1)h(+)f Fq(O)r Fu(\()p Fq(")p Fu(\)\))456 2121 y(where)k(w)n(e)g(ha)n(v)n(e)g(used)h(the)g (form)n(ula)f(\000\(1)9 b Fp(\000)g Fq(\021)s Fu(\)\000\()p Fq(\021)s Fu(\))24 b(=)e Fq(\031)s(=)14 b Fu(sin\()p Fq(\021)s(\031)s Fu(\).)36 b(Th)n(us)23 b(the)g(constan)n(t)g(term)456 2221 y(in)i Fq(\032)g Fu(of)g(the)h(in)n(tegral)e(o)n(v)n(er)f(\000) 1378 2233 y Fo(4)1415 2221 y Fu(\()p Fq(\032)p Fu(\))j(can)f(b)r(e)g (calculated)g(with)h(the)f(previous)f(estimates)h(taking)456 2320 y Fq(l)f Fu(=)f Fq(j)627 2332 y Fo(2)676 2320 y Fu(+)12 b(1)g(+)g([)m Fq(m=q)s Fu(])24 b(and)g Fq(\021)i Fu(=)d Fq(m=q)14 b Fp(\000)e Fu([)o Fq(m=q)s Fu(],)25 b(where)e([)p Fp(\001)p Fu(])i(denotes)e(the)i(in)n(teger)e(part)h (function,)456 2420 y(and)j(it)h(is)906 2629 y Fp(\000)p Fq(i)1010 2573 y Fu(\()p Fp(\000)p Fu(1\))1181 2542 y Fi(l)p 1009 2610 197 4 v 1080 2686 a Fq(i)1109 2662 y Fi(l)1229 2512 y Fg(\022)1300 2573 y Fp(j)p Fq(n)p Fp(j)p 1300 2610 96 4 v 1328 2686 a Fq(")1406 2512 y Fg(\023)1467 2529 y Fi(l)p Fj(\000)p Fo(1+)p Fi(\021)1678 2629 y Fq(e)1717 2594 y Fj(\000)p Fi(\021)r(i\031)r(=)p Fo(2)2056 2573 y Fq(\031)p 1951 2610 262 4 v 1951 2686 a Fu(sin\()p Fq(\031)s(\021)s Fu(\))2355 2573 y(1)p 2232 2610 289 4 v 2232 2686 a(\000\()p Fq(l)20 b Fu(+)e Fq(\021)s Fu(\))2530 2629 y Fq(e)2569 2594 y Fj(\000j)p Fi(n)p Fj(j)p Fi(a=")2807 2629 y Fu(\(1)g(+)g Fq(O)r Fu(\()p Fq(")p Fu(\)\),)-2749 b(\(6.25\))456 2838 y(where)27 b(the)h Fq(O)r Fu(\()p Fq(")p Fu(\))g(term)g(is)f(uniform)h(with)g(resp)r(ect)f(to)h Fq(n)p Fu(.)605 2938 y(3.)74 b(In)n(tegral)38 b(o)n(v)n(er)h(\000)1308 2950 y Fo(6)1345 2938 y Fu(.)74 b(W)-7 b(e)40 b(parameterize)f(this)h (path)g(b)n(y)g Fq(g)2652 2950 y Fo(6)2689 2938 y Fu(\()p Fq(z)t Fu(\))j(=)h Fp(\000)p Fq(\032)26 b Fu(+)g Fq(i\022)42 b Fu(with)456 3038 y Fq(\022)25 b Fp(2)e Fu([)p Fq(a;)14 b(b)p Fu(].)37 b(Therefore,)884 3147 y Fg(Z)930 3335 y Fo(\000)971 3343 y Fh(6)1004 3335 y Fo(\()p Fi(\032)p Fo(\))1264 3204 y Fq(e)1303 3173 y Fj(\000)p Fi(inu=")p 1118 3241 557 4 v 1118 3328 a Fu(\()p Fq(u)18 b Fp(\000)g Fq(ia)p Fu(\))1414 3268 y Fc(m)p 1414 3277 51 3 v 1425 3311 a(q)1475 3291 y Fo(+)p Fi(j)1553 3299 y Fh(2)1586 3291 y Fo(+1)1698 3260 y Fq(du)k Fu(=)h Fq(i)1942 3147 y Fg(Z)2025 3167 y Fi(b)1988 3335 y(a)2288 3204 y Fq(e)2327 3173 y Fi(\032ni=")2495 3204 y Fq(e)2534 3173 y Fi(n\022)r(=")p 2082 3241 803 4 v 2082 3345 a Fu(\()p Fp(\000)p Fq(\032)18 b Fu(+)g(\()p Fq(\022)j Fp(\000)d Fq(a)p Fu(\))p Fq(i)p Fu(\))2645 3268 y Fc(m)p 2645 3275 51 3 v 2656 3307 a(q)2706 3285 y Fh(+)p Fc(j)2773 3297 y Fh(2)2805 3285 y(+1)2908 3260 y Fq(d\022)r Fu(.)456 3492 y(Th)n(us,)27 b(if)h(w)n(e)f(de\014ne) 1340 3709 y Fq(K)1411 3721 y Fi(l)1436 3709 y Fu(\()p Fq(\032)p Fu(\))d(=)1655 3596 y Fg(Z)1738 3617 y Fi(b)1701 3785 y(a)2019 3653 y Fq(e)2058 3623 y Fi(\032ni=")p 1795 3690 657 4 v 1795 3770 a Fu(\()p Fp(\000)p Fq(\032)18 b Fu(+)g(\()p Fq(\022)j Fp(\000)d Fq(a)p Fu(\))p Fq(i)p Fu(\))2348 3729 y Fc(l)p Fh(+)p Fc(\021)2475 3709 y Fq(d\022)456 3921 y Fu(w)n(e)30 b(ha)n(v)n(e)h(that)g Fq(K)1031 3933 y Fi(l)1056 3921 y Fu(\()p Fq(\032)p Fu(\))f(=)e Fq(I)1322 3933 y Fi(l)1348 3921 y Fu(\()p Fp(\000)p Fq(\032)p Fu(\))k(and)f(for)g Fq(\021)h(>)d Fu(0,)i(b)n(y)g(using)g(the)h(previous)e(computations,) 456 4021 y(w)n(e)d(obtain)862 4242 y Fq(K)933 4254 y Fi(l)958 4242 y Fu(\()p Fq(\032)p Fu(\))83 b(=)1345 4138 y Fi(l)1296 4163 y Fg(X)1299 4340 y Fi(j)s Fo(=1)1430 4150 y Fg(\020)1489 4186 y Fq(n)p 1489 4223 50 4 v 1495 4299 a(")1549 4150 y Fg(\021)1599 4167 y Fi(j)s Fj(\000)p Fo(1)1754 4186 y Fu(1)p 1742 4223 64 4 v 1742 4299 a Fq(i)1771 4275 y Fi(j)2246 4186 y Fu(1)p 1826 4223 882 4 v 1826 4299 a(\()p Fq(\021)22 b Fu(+)c Fq(l)i Fp(\000)e Fu(1\))c Fp(\001)g(\001)g(\001)f Fu(\()p Fq(\021)22 b Fu(+)c Fq(l)i Fp(\000)e Fq(j)5 b Fu(\))2717 4242 y Fq(f)2758 4254 y Fi(l)p Fj(\000)p Fi(j)2866 4242 y Fu(\()p Fp(\000)p Fq(\032)p Fu(\))1296 4508 y(+)1375 4416 y Fg(\020)1434 4452 y Fq(n)p 1434 4489 50 4 v 1440 4565 a(")1494 4416 y Fg(\021)1543 4433 y Fi(l)1599 4452 y Fu(1)p 1593 4489 55 4 v 1593 4565 a Fq(i)1622 4541 y Fi(l)1920 4452 y Fu(1)p 1667 4489 549 4 v 1667 4565 a(\()p Fq(\021)22 b Fu(+)c Fq(l)h Fp(\000)f Fu(1\))c Fp(\001)g(\001)g(\001)g Fq(\021)2225 4508 y(K)2296 4520 y Fo(0)2333 4508 y Fu(.)456 4716 y(As)28 b(b)r(efore)h(w)n(e)f(calculate)g(the)h(constan)n(t)e (term)i Fq(K)2050 4728 y Fo(0)2115 4716 y Fu(of)g Fq(K)2282 4728 y Fo(0)2319 4716 y Fu(\()p Fq(\032)p Fu(\).)40 b(In)29 b(this)g(case,)f(the)h(argumen)n(t)456 4816 y(of)e Fp(\000)p Fq(\032)18 b Fu(+)g(\()p Fq(z)k Fp(\000)c Fq(a)p Fu(\))p Fq(i)28 b Fu(b)r(elongs)f(to)g(\()p Fp(\000)p Fu(3)p Fq(\031)s(=)p Fu(2)p Fq(;)14 b Fp(\000)p Fq(\031)s Fu(\))26 b(and)i(therefore,)524 5138 y Fq(K)595 5150 y Fo(0)631 5138 y Fu(\()p Fq(\032)p Fu(\))c(=)850 5025 y Fg(Z)933 5045 y Fi(b)896 5214 y(a)1195 5082 y Fq(e)1234 5052 y Fi(n\022)r(=")p 990 5119 594 4 v 990 5195 a Fu(\()p Fp(\000)p Fq(\032)18 b Fu(+)g Fq(i)p Fu(\()p Fq(\022)j Fp(\000)d Fq(a)p Fu(\)\))1543 5171 y Fi(\021)1607 5138 y Fq(d\022)26 b Fp(!)d Fq(e)1860 5104 y Fi(\021)r(i)p Fo(3)p Fi(\031)r(=)p Fo(2)2078 5025 y Fg(Z)2161 5045 y Fi(b)2125 5214 y(a)2273 5082 y Fq(e)2312 5052 y Fi(n\022)r(=")p 2219 5119 292 4 v 2219 5195 a Fu(\()p Fq(\022)e Fp(\000)d Fq(a)p Fu(\))2470 5171 y Fi(\021)2534 5138 y Fq(d\022)307 b Fu(when)28 b Fq(\032)23 b Fp(!)g Fu(0.)p eop %%Page: 77 85 77 84 bop 1338 251 a Fo(6.5.)29 b(PR)n(OOF)g(OF)g(THE)g(COR)n(OLLAR)-5 b(Y)30 b(1.1)816 b(77)456 450 y Fu(Consequen)n(tly)-7 b(,)26 b(the)i(constan)n(t)f(term)h(in)g Fq(\032)f Fu(of)h(the)g(in)n (tegral)e(o)n(v)n(er)g(\000)2600 462 y Fo(6)2637 450 y Fu(\()p Fq(\032)p Fu(\))i(is)947 659 y Fq(i)986 603 y Fu(\()p Fp(\000)p Fu(1\))1157 573 y Fi(l)p 986 640 197 4 v 1057 716 a Fq(i)1086 692 y Fi(l)1206 542 y Fg(\022)1277 603 y Fp(j)p Fq(n)p Fp(j)p 1277 640 96 4 v 1306 716 a Fq(")1383 542 y Fg(\023)1444 559 y Fi(l)p Fj(\000)p Fo(1+)p Fi(\021)1656 659 y Fq(e)1695 625 y Fi(\021)r(i)p Fo(3)p Fi(\031)r(=)p Fo(2)2015 603 y Fq(\031)p 1909 640 262 4 v 1909 716 a Fu(sin\()p Fq(\031)s(\021)s Fu(\))2314 603 y(1)p 2190 640 289 4 v 2190 716 a(\000\()p Fq(l)20 b Fu(+)e Fq(\021)s Fu(\))2489 659 y Fq(e)2528 625 y Fj(\000j)p Fi(n)p Fj(j)p Fi(a=")2765 659 y Fu(\(1)g(+)h Fq(O)r Fu(\()p Fq(")p Fu(\)\),)-2708 b(\(6.26\))456 848 y(where)27 b(the)h Fq(O)r Fu(\()p Fq(")p Fu(\))g(term)g(is)f(uniform)h(with)g(resp)r(ect)f (to)h Fq(n)p Fu(.)605 948 y(No)n(w)40 b(w)n(e)h(consider)f(the)h(case)f (suc)n(h)g(that)h Fq(m=q)48 b Fp(2)d Fl(N)t Fu(.)83 b(T)-7 b(aking)40 b(in)n(to)g(accoun)n(t)g(that,)456 1047 y Fq(m=q)20 b Fu(+)f Fq(j)746 1059 y Fo(2)801 1047 y Fu(+)f(1)23 b Fp(2)g Fl(N)t Fu(,)34 b(the)28 b(functions)1818 1190 y Fq(e)1857 1160 y Fj(\000)p Fi(inu=")p 1672 1227 557 4 v 1672 1314 a Fu(\()p Fq(u)18 b Fp(\000)g Fq(ia)p Fu(\))1968 1255 y Fc(m)p 1968 1264 51 3 v 1979 1297 a(q)2029 1277 y Fo(+)p Fi(j)2107 1285 y Fh(2)2140 1277 y Fo(+1)456 1445 y Fu(ha)n(v)n(e)28 b(a)h(p)r(ole)h(of)g(order)e Fq(m=q)23 b Fu(+)c Fq(j)1509 1457 y Fo(2)1566 1445 y Fu(+)h(1.)43 b(Therefore)28 b(w)n(e)i(can)f(apply)g(the)h(residue)g (theory)-7 b(.)42 b(In)456 1544 y(this)27 b(case)g(the)h(in)n(tegral)f (o)n(v)n(er)e(\000)1468 1556 y Fo(1)1533 1544 y Fu(turns)j(out)f(to)h (b)r(e)1058 1752 y(2)p Fq(\031)s(i)1375 1696 y Fu(1)p 1189 1733 415 4 v 1189 1809 a(\()p Fq(m=q)20 b Fu(+)e Fq(j)1510 1821 y Fo(2)1548 1809 y Fu(\)!)1627 1635 y Fg(\022)1698 1696 y Fp(\000)p Fq(in)p 1698 1733 144 4 v 1750 1809 a(")1851 1635 y Fg(\023)1912 1652 y Fi(m=q)r Fo(+)p Fi(j)2115 1660 y Fh(2)2166 1752 y Fq(e)2205 1718 y Fj(\000j)p Fi(n)p Fj(j)p Fi(a=")2443 1752 y Fu(\(1)g(+)g Fq(O)r Fu(\()p Fq(")p Fu(\)\).)-2385 b(\(6.27\))456 1941 y(No)n(w)27 b(w)n(e)g(compute)g(the)h(dominan)n(t)g(term)f(of)h Fq(M)1992 1953 y Fi(n)2036 1941 y Fu(\()p Fq(")p Fu(\))g(for)f Fq(n)c(<)g Fu(0.)36 b(Let)28 b Fq(\021)2746 1953 y Fi(`)2801 1941 y Fu(=)23 b Fq(`)18 b Fp(\000)g Fu([)p Fq(`)p Fu(].)36 b(If)28 b Fq(`)k(=)-51 b Fp(2)23 b Fl(N)456 2041 y Fu(and)k Fq(n)c(<)g Fu(0,)k(b)n(y)g(\(6.25\))g(and)g(\(6.26\))g(w)n(e)g(obtain)h (that)768 2239 y Fq(M)849 2251 y Fi(n)894 2239 y Fu(\()p Fq(")p Fu(\))84 b(=)1352 2160 y Fg(X)1228 2342 y Fi(m)p Fj(\024)p Fo(\()p Fi(j)1392 2350 y Fh(1)1425 2342 y Fo(+)p Fi(j)1503 2350 y Fh(2)1536 2342 y Fo(\))p Fi(p)1610 2239 y Fq(J)1664 2204 y Fj(\000)1656 2268 y(\000)p Fi(n;)p Fo(\()p Fi(j)1822 2276 y Fh(1)1854 2268 y Fo(+)p Fi(j)1932 2276 y Fh(2)1965 2268 y Fo(\))p Fi(p)p Fj(\000)p Fi(m)2154 2126 y Fg(Z)2237 2147 y Fj(1)2200 2315 y(\0001)2519 2183 y Fq(e)2558 2153 y Fj(\000)p Fi(inu=")p 2346 2220 611 4 v 2346 2298 a Fu(\()p Fq(u)18 b Fp(\000)g Fq(ia)p Fu(\))2632 2274 y Fi(m=q)r Fo(+)p Fi(j)2835 2282 y Fh(2)2868 2274 y Fo(+1)2980 2239 y Fq(du)1081 2539 y Fu(=)82 b Fq(i)1257 2504 y Fo([)p Fi(`)p Fo(]+1)1424 2422 y Fg(\022)1495 2482 y Fp(j)p Fq(n)p Fp(j)p 1495 2519 96 4 v 1524 2596 a Fq(")1601 2422 y Fg(\023)1662 2439 y Fi(`)1819 2482 y Fu(2)p Fq(\031)p 1718 2519 295 4 v 1718 2596 a Fu(\000\()p Fq(`)18 b Fu(+)g(1\))2022 2539 y Fq(J)2076 2503 y Fj(\000)2068 2561 y(\000)p Fi(n;)p Fo(0)2218 2539 y Fq(e)2257 2504 y Fi(\021)2291 2513 y Fc(`)2320 2504 y Fi(i\031)r(=)p Fo(2)2455 2539 y Fq(e)2494 2504 y Fj(\000j)p Fi(n)p Fj(j)p Fi(a=")2732 2539 y Fu(\(1)g(+)g Fq(O)r Fu(\()p Fq(")p Fu(\)\).)-2674 b(\(6.28\))456 2739 y(Note)27 b(that)h Fq(i)865 2709 y Fo([)p Fi(`)p Fo(])934 2739 y Fq(e)973 2709 y Fi(\021)1007 2718 y Fc(`)1036 2709 y Fi(i\031)r(=)p Fo(2)1195 2739 y Fu(=)22 b Fq(i)1311 2709 y Fi(`)1343 2739 y Fu(.)37 b(If)28 b Fq(`)23 b Fp(2)g Fl(N)t Fu(,)34 b(using)27 b(the)h(expression)e(\(6.27\),)1067 2948 y Fq(M)1148 2960 y Fi(n)1193 2948 y Fu(\()p Fq(")p Fu(\))d(=)g Fq(i)1436 2914 y Fi(`)p Fo(+1)1565 2831 y Fg(\022)1636 2892 y Fp(j)p Fq(n)p Fp(j)p 1636 2929 96 4 v 1665 3005 a Fq(")1742 2831 y Fg(\023)1803 2848 y Fi(`)1859 2892 y Fu(2)p Fq(\031)p 1859 2929 92 4 v 1876 3005 a(`)p Fu(!)1961 2948 y Fq(J)2015 2912 y Fj(\000)2007 2970 y(\000)p Fi(n;)p Fo(0)2157 2948 y Fq(e)2196 2914 y Fj(\000j)p Fi(n)p Fj(j)p Fi(a=")2433 2948 y Fu(\(1)c(+)f Fq(O)r Fu(\()p Fq(")p Fu(\)\).)-2376 b(\(6.29\))605 3150 y Ft(Remark)43 b Fu(6.3)p Ft(.)j Fs(The)41 b(expr)l(essions)f(of)g Fq(M)1968 3162 y Fi(n)2013 3150 y Fu(\()p Fq(")p Fu(\))g Fs(in)f(\(6.28\))i(and)f (\(6.29\))h(ar)l(e)f(c)l(ompute)l(d)456 3250 y(indep)l(endently.)g(We) 30 b(note)f(that)h(\(6.28\))i(go)l(es)e(to)g(\(6.29\))h(when)f Fq(\021)2549 3262 y Fi(`)2611 3250 y Fs(go)l(es)h(to)e(zer)l(o.)40 b(Ther)l(efor)l(e)456 3349 y(we)30 b(c)l(an)f(use)h(\(6.28\))h(for)g (al)t(l)g Fq(`)p Fs(.)605 3501 y Fu(F)-7 b(or)27 b(the)h(case)f Fq(n)c(>)f Fu(0,)28 b(it)g(is)f(su\016cien)n(t)h(to)f(observ)n(e)f (that)1216 3654 y Fq(M)1297 3666 y Fi(n)1342 3654 y Fu(\()p Fq(")p Fu(\))d(=)p 1556 3581 281 4 v 23 w Fq(M)1637 3666 y Fj(\000)p Fi(n)1733 3654 y Fu(\()p Fq(")p Fu(\))121 b(and)e Fq(J)2264 3618 y Fj(\000)2256 3676 y(\000)p Fi(n;)p Fo(0)2429 3654 y Fu(=)p 2517 3572 145 4 v 23 w Fq(J)2571 3618 y Fo(+)2563 3676 y Fi(n;)p Fo(0)2661 3654 y Fq(:)-2228 b Fu(\(6.30\))456 3793 y(The)29 b(result)h(follo)n(ws)e(summing)i(the)g (F)-7 b(ourier)29 b(series,)g(taking)g(in)n(to)g(accoun)n(t)g(that)h (the)g(terms)456 3893 y Fq(O)r Fu(\()p Fq(")p Fu(\))e(are)f(uniform)g (in)h Fq(n)p Fu(.)1309 4076 y Fk(6.5.)47 b(Pro)s(of)31 b(of)h(the)g(Corollary)g(1.1)605 4225 y Fu(Assume)25 b(the)g(same)f(h)n(yp)r(otheses)g(as)g(Theorem)g(1.1)f(and)i(the)g (further)f(h)n(yp)r(othesis)h Fk(HP6)p Fq(:)456 4338 y Fu(Then,)i Fq(J)749 4302 y Fo(+)741 4360 y(1)p Fi(;)p Fo(0)855 4338 y Fu(=)p 942 4256 188 4 v 22 w Fq(J)996 4302 y Fj(\000)988 4360 y(\000)p Fo(1)p Fi(;)p Fo(0)1158 4338 y Fu(are)f(di\013eren)n(t)i(from)f(zero.)36 b(W)-7 b(e)28 b(write)g Fq(J)2438 4302 y Fj(\000)2430 4360 y Fo(1)p Fi(;)p Fo(0)2543 4338 y Fu(=)22 b Fp(j)p Fq(J)2707 4302 y Fj(\000)2699 4360 y Fo(1)p Fi(;)p Fo(0)2790 4338 y Fp(j)p Fq(e)2852 4308 y Fi(i\022)2912 4338 y Fu(.)605 4442 y(F)-7 b(rom)27 b(\(6.28\))g(and)g(\(6.30\))g(w)n(e)h(ha)n(v)n(e) 489 4581 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))602 4763 y(=)692 4685 y Fg(X)689 4860 y Fi(n>)p Fo(0)940 4707 y Fu(2)p Fq(\031)p 839 4744 295 4 v 839 4820 a Fu(\000\()p Fq(`)k Fu(+)g(1\))1157 4671 y Fg(\020)1216 4707 y Fq(n)p 1216 4744 50 4 v 1222 4820 a(")1276 4671 y Fg(\021)1326 4689 y Fi(`)1372 4763 y Fq(e)1411 4729 y Fj(\000)p Fi(na=")1623 4671 y Fg(h)1662 4763 y Fu(\()p Fp(\000)p Fq(i)p Fu(\))1820 4729 y Fi(`)p Fo(+1)1936 4763 y Fq(J)1990 4728 y Fo(+)1982 4786 y Fj(\000)p Fi(n;)p Fo(0)2131 4763 y Fq(e)2170 4729 y Fi(ins=")2354 4763 y Fu(+)g Fq(i)2466 4729 y Fi(`)p Fo(+1)2582 4763 y Fq(J)2636 4728 y Fj(\000)2628 4786 y Fi(n;)p Fo(0)2726 4763 y Fq(e)2765 4729 y Fj(\000)p Fi(ins=")2982 4671 y Fg(i)3035 4763 y Fu(\(1)g(+)g Fq(O)r Fu(\()p Fq(")p Fu(\)\))602 5016 y(=)k Fq(")728 4982 y Fj(\000)p Fi(`)812 5016 y Fq(e)851 4982 y Fj(\000)p Fi(a=")1119 4960 y Fu(2)p Fq(\031)p 1018 4997 295 4 v 1018 5073 a Fu(\000\()p Fq(`)c Fu(+)g(1\))1322 5016 y(2)c(Re)o(\()p Fq(i)1536 4982 y Fi(`)p Fo(+1)1652 5016 y Fq(J)1706 4980 y Fj(\000)1698 5038 y Fo(1)p Fi(;)p Fo(0)1788 5016 y Fq(e)1827 4982 y Fj(\000)p Fi(is=")2003 5016 y Fu(\))19 b(+)f Fq(O)r Fu(\()p Fq(")2273 4982 y Fj(\000)p Fi(`)p Fo(+1)2441 5016 y Fq(e)2480 4982 y Fj(\000)p Fi(a=")2637 5016 y Fu(\).)456 5215 y(Since)27 b Fq(i)701 5185 y Fi(`)p Fo(+1)817 5215 y Fq(J)871 5180 y Fj(\000)863 5237 y Fo(1)p Fi(;)p Fo(0)953 5215 y Fq(e)992 5185 y Fj(\000)p Fi(is=")1191 5215 y Fu(=)c Fp(j)p Fq(J)1356 5180 y Fj(\000)1348 5237 y Fo(1)p Fi(;)p Fo(0)1438 5215 y Fp(j)p Fq(e)1500 5185 y Fi(i)p Fo(\()p Fi(\022)r Fo(+\()p Fi(`)p Fo(+1\))p Fi(\031)r(=)p Fo(2\))1935 5215 y Fq(e)1974 5185 y Fj(\000)p Fi(is=")2178 5215 y Fu(the)28 b(form)n(ula)e(for)h Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))28 b(follo)n(ws.)p eop %%Page: 78 86 78 85 bop 456 251 a Fo(78)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)605 450 y Fu(F)-7 b(rom)27 b(Theorem)g(1.1,)g(to)g(compute)h(the)g(area)e(w)n(e)h (ha)n(v)n(e)g(to)g(estimate)624 522 y Fg(Z)710 543 y Fo(\026)-36 b Fi(s)738 551 y Fh(0)671 711 y Fi(s)702 719 y Fh(0)775 635 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))g Fq(ds)798 878 y Fu(=)23 b Fq(")925 844 y Fj(\000)p Fi(`)1009 878 y Fq(e)1048 844 y Fj(\000)p Fi(a=")1218 761 y Fg(\024)1373 822 y Fu(4)p Fq(\031)p 1272 859 295 4 v 1272 935 a Fu(\000\()p Fq(`)18 b Fu(+)g(1\))1576 878 y Fp(j)p Fq(J)1653 843 y Fj(\000)1645 900 y Fo(1)p Fi(;)p Fo(0)1735 878 y Fp(j)1772 765 y Fg(Z)1858 786 y Fo(\026)-36 b Fi(s)1886 794 y Fh(0)1818 954 y Fi(s)1849 962 y Fh(0)1937 878 y Fu(cos)o(\()p Fq(\022)21 b Fu(+)d(\()p Fq(`)h Fu(+)f(1\))p Fq(\031)s(=)p Fu(2)f Fp(\000)h Fq(s=")p Fu(\))c Fq(ds)k Fu(+)g Fq(O)r Fu(\()p Fq(")p Fu(\))3217 761 y Fg(\025)798 1116 y Fu(=)23 b Fq(")925 1082 y Fj(\000)p Fi(`)p Fo(+1)1093 1116 y Fq(e)1132 1082 y Fj(\000)p Fi(a=")1303 999 y Fg(\024)1457 1060 y Fu(8)p Fq(\031)p 1356 1097 V 1356 1173 a Fu(\000\()p Fq(`)c Fu(+)f(1\))1660 1116 y Fp(j)p Fq(J)1737 1080 y Fj(\000)1729 1138 y Fo(1)p Fi(;)p Fo(0)1819 1116 y Fp(j)h Fu(+)f Fq(O)r Fu(\()p Fq(")p Fu(\))2112 999 y Fg(\025)2171 1116 y Fu(.)456 1300 y(Summing)42 b(the)g(F)-7 b(ourier)40 b(series)h(corresp)r(onding) e(to)j Fq(M)2287 1270 y Fj(0)2310 1300 y Fu(\()p Fq(s;)14 b(")p Fu(\),)45 b(in)d(the)g(same)f(w)n(a)n(y)f(as)h(for)456 1400 y Fq(M)9 b Fu(\()p Fq(s;)14 b(")p Fu(\))27 b(w)n(e)g(ha)n(v)n(e) 529 1673 y Fp(j)p Fq(M)642 1639 y Fj(0)665 1673 y Fu(\()p Fq(s;)14 b(")p Fu(\))p Fp(j)24 b Fu(=)e Fq(")1017 1639 y Fj(\000)p Fi(`)p Fj(\000)p Fo(1)1297 1617 y Fu(4)p Fq(\031)p 1196 1654 V 1196 1730 a Fu(\000\()p Fq(`)c Fu(+)g(1\))1500 1673 y Fp(j)p Fq(J)1577 1638 y Fj(\000)1569 1695 y Fo(1)p Fi(;)p Fo(0)1659 1673 y Fp(j)p Fq(e)1721 1639 y Fj(\000)p Fi(a=")1892 1673 y Fu(sin\()p Fq(\022)j Fu(+)d(\()p Fq(`)g Fu(+)g(1\))p Fq(\031)s(=)p Fu(2)g Fp(\000)g Fq(s=")p Fu(\))g(+)g Fq(O)r Fu(\()p Fq(")3035 1639 y Fj(\000)p Fi(`)3119 1673 y Fq(e)3158 1639 y Fj(\000)p Fi(a=")3315 1673 y Fu(\).)456 1856 y(If)24 b Fq(M)9 b Fu(\()p Fq(s)696 1868 y Fo(0)733 1856 y Fq(;)14 b(")p Fu(\))23 b(=)f(0,)i(then)g Fq(s)1264 1868 y Fo(0)1325 1856 y Fu(is)g(suc)n(h)f(that)h Fq(s)1803 1868 y Fo(0)1840 1856 y Fq(=")e Fu(=)h Fq(\022)13 b Fu(+)d(\()p Fq(`)g Fu(+)g(1\))p Fq(\031)s(=)p Fu(2)g(+)g Fq(\031)s(=)p Fu(2)g(+)g Fq(k)s(\031)j Fu(+)d Fq(O)r Fu(\()p Fq(")p Fu(\),)25 b Fq(k)h Fp(2)d Fl(Z)o Fu(,)456 1956 y(and)31 b(therefore,)i(sin\()p Fq(\022)24 b Fu(+)d(\()p Fq(`)g Fu(+)g(1\))p Fq(\031)s(=)p Fu(2)g Fp(\000)g Fq(s=")p Fu(\))29 b(=)h Fp(\006)p Fu(1)21 b(+)g Fq(O)r Fu(\()p Fq(")p Fu(\).)51 b(Using)32 b(the)g(form)n(ula)g (for)f(the)456 2056 y(angle)26 b(in)i(Theorem)f(1.1)g(w)n(e)g(get)g (the)h(result.)p eop %%Page: 79 87 79 86 bop 1635 1114 a Fw(References)456 1380 y Fn([An])190 b(S.)34 b(Angenen)n(t,)39 b Fm(A)c(variational)i(interpr)l(etation)f (of)f(Melnikov's)h(function)f(and)i(exp)l(onential)t(ly)778 1463 y(smal)t(l)26 b(sep)l(ar)l(atrix)h(splitting)p Fn(,)c(Symplectic)g (geometry)-6 b(,)24 b(London)g(Math.)g(So)r(c.)f(Lecture)i(Note)f (Ser.,)778 1546 y(v)n(ol)g(192,)g(Cam)n(bridge)e(Univ.)h(Press,)g(Cam)n (bridge)g(1993,)h(pp.)g(5{35.)456 1629 y([Ar1])166 b(V.I.)29 b(Arnold,)h Fm(Instability)h(of)h(dynamic)l(al)h(systems)e(with)g (sever)l(al)h(de)l(gr)l(e)l(es)g(of)g(fr)l(e)l(e)l(dom)p Fn(,)g(So)n(viet)778 1712 y(Math.)24 b(Dokl.)f Fa(5)g Fn(\(1964\),)i(581{585.)456 1795 y([AKN])121 b(V.I.)21 b(Arnold,)h(V.V.)f(Koslo)n(v)h(and)h(A.I.)e(Neish)n(tadt,)j Fm(Dynamic)l(al)h(systems)f(III)p Fn(,)e(Encyclopaedia)j(of)778 1878 y(Mathematical)f(Sciences,)g(v)n(ol.)f(3,)h(Springer,)f(Berl)-8 b(\023)-27 b(\020n,)22 b(1988.)456 1961 y([CFN])132 b(J.)23 b(Casasa)n(y)n(as,)h(E.)g(F)-6 b(on)n(tic)n(h)24 b(and)h(A.)e(Nunes,)g Fm(Homo)l(clinic)k(orbits)f(to)f(p)l(ar)l(ab)l(olic)j(p)l(oints)p Fn(,)d(NoDEA)778 2044 y(Nonlinear)e(Di\013eren)n(tial)h(Equations)h (Appl.)e Fa(4)g Fn(\(1997\),)i(201{216.)456 2127 y([Ch])192 b(V.)34 b(Cherno)n(v,)j Fm(On)e(sep)l(ar)l(atrix)i(splitting)e(of)h (some)g(quadr)l(atic)g(ar)l(e)l(a-pr)l(eserving)h(maps)g(of)e(the)778 2210 y(plane)p Fn(,)24 b(Regular)g(&)f(Chaotic)i(Dynamics)d Fa(3)i Fn(\()p Fa(1)p Fn(\))g(\(1998\),)h(49{65.)456 2293 y([DG])173 b(A.)27 b(Delshams)f(and)i(P)-6 b(.)27 b(Guti)n(\023)-33 b(errez,)29 b Fm(Splitting)g(p)l(otential)h(and)g (the)f(Poinc)l(ar)n(\023)-35 b(e-Melnikov)31 b(metho)l(d)778 2376 y(for)26 b(whisker)l(e)l(d)h(tori)e(in)h(Hamiltonian)g(systems)p Fn(,)d(Nonlinear)h(Science)h Fa(10)e Fn(\(2000\),)i(433{476.)456 2459 y([DGJS1])63 b(A.)21 b(Delshams,)f(V.G.)i(Gelfreic)n(h,)f(A.)g (Jorba)i(and)f(T.M.Seara,)f Fm(Exp)l(onential)t(ly)k(smal)t(l)g (splitting)e(of)778 2542 y(sep)l(ar)l(atric)l(es)k(under)f(fast)g (quasip)l(erio)l(dic)h(for)l(cing)p Fn(,)d(Comm.)d(Math.)i(Ph)n(ys.)h Fa(189)e Fn(\(1997\),)k(35{71.)456 2625 y([DGJS2])63 b(A.)18 b(Delshams,)g(V.G.)g(Gelfreic)n(h,)h(A.)f(Jorba)h(and)g(T.M.)e (Seara,)j Fm(L)l(ower)i(and)g(upp)l(er)g(b)l(ounds)g(for)f(the)778 2708 y(splitting)29 b(of)g(sep)l(ar)l(atric)l(es)h(of)g(the)e(p)l (endulum)j(under)f(a)f(fast)g(quasip)l(erio)l(dic)i(for)l(cing)p Fn(,)d(Electron.)778 2791 y(Res.)23 b(Announc.)i(Amer.)c(Math.)j(So)r (c.)g Fa(3)f Fn(\(1997\),)i(1{10.)456 2874 y([DR1])141 b(A.)22 b(Delshams)g(and)i(R.)e(Ram)-8 b(\023)-27 b(\020rez-Ros,)21 b Fm(Poinc)l(ar)n(\023)-35 b(e-Melnikov-A)n(rnold)27 b(metho)l(d)f(for)g(analytic)f(pla-)778 2957 y(nar)h(maps)p Fn(,)e(Nonlinearit)n(y)h Fa(9)e Fn(\(1996\),)j(1{26.)456 3040 y([DR2])141 b(A.)15 b(Delshams)g(and)h(R.)f(Ram)-8 b(\023)-27 b(\020rez-Ros,)16 b Fm(Melnikov)i(p)l(otential)h(for)g (exact)e(symple)l(ctic)i(maps)p Fn(,)f(Comm.)778 3123 y(Math.)24 b(Ph)n(ys.)f Fa(190)g Fn(\(1997\),)i(231{245.)456 3206 y([DS1])154 b(A.)19 b(Delshams)g(and)h(T.M.)e(Seara,)j Fm(A)n(n)h(asymptotic)h(expr)l(ession)g(for)f(the)g(splitting)f(of)i (sep)l(ar)l(atric)l(es)778 3289 y(of)j(the)f(r)l(apid)t(ly)i(for)l(c)l (e)l(d)g(p)l(endulum)p Fn(,)e(Comm.)c(Math.)j(Ph)n(ys.)f Fa(150)g Fn(\(1992\),)i(433{163.)456 3372 y([DS2])154 b(A.)18 b(Delshams)g(and)h(T.M.)f(Seara,)i Fm(Splitting)g(of)i(sep)l (ar)l(atric)l(es)g(in)f(Hamiltonian)i(systems)e(with)g(one)778 3456 y(and)26 b(a)g(half)h(de)l(gr)l(e)l(es)f(of)g(fr)l(e)l(e)l(dom)p Fn(,)f(Math.)f(Ph)n(ys.)f(Electron.)h(J.)f Fa(3)g Fn(\(1997\),)j(4{40.) 456 3539 y([FS])197 b(B.)24 b(Fiedler)f(and)i(J.)f(Sc)n(heurle,)h Fm(Discr)l(etization)h(of)g(homo)l(clinic)i(orbits,)e(r)l(apid)h(for)l (cing)g(and)f(\\ho-)778 3622 y(mo)l(clinic")g(chaos)p Fn(,)e(Mem.)e(Amer.)g(Math.)h(So)r(c.)h Fa(119)f Fn(\()p Fa(570)p Fn(\))g(\(1996\),)i(viii+79)e(pp.)456 3705 y([F)-6 b(o1])172 b(E.)25 b(F)-6 b(on)n(tic)n(h,)27 b Fm(Exp)l(onential)t(ly)h (smal)t(l)g(upp)l(er)h(b)l(ounds)f(for)g(the)f(splitting)g(of)g(sep)l (ar)l(atric)l(es)i(for)f(high)778 3788 y(fr)l(e)l(quency)d(p)l(erio)l (dic)i(p)l(erturb)l(ations)p Fn(,)e(Nonlinear)e(Anal.)g Fa(20)g Fn(\(1993\),)i(733{744.)456 3871 y([F)-6 b(o2])172 b(E.)21 b(F)-6 b(on)n(tic)n(h,)22 b Fm(R)l(apid)t(ly)j(for)l(c)l(e)l(d) g(planar)g(ve)l(ctor)e(\014elds)h(and)h(splitting)e(of)h(sep)l(ar)l (atric)l(es)p Fn(,)f(J.)e(Di\013eren-)778 3954 y(tial)i(Equations)i Fa(119)e Fn(\(1995\),)i(310{335.)456 4037 y([F)-6 b(o3])172 b(E.)23 b(F)-6 b(on)n(tic)n(h,)24 b Fm(A)i(\015ow)g(b)l(ox)g(the)l(or)l (em)h(for)f(di\013e)l(omorphisms)p Fn(,)g(preprin)n(t,)d(\(2000\).)456 4120 y([FS1])162 b(E.)32 b(F)-6 b(on)n(tic)n(h)34 b(and)g(C.)e(Sim\023) -35 b(o,)34 b Fm(The)g(splitting)g(of)g(sep)l(ar)l(atric)l(es)i(for)f (analytic)f(di\013e)l(omorphisms)p Fn(,)778 4203 y(Ergo)r(dic)24 b(Theory)g(Dynam.)e(Systems)i Fa(10)e Fn(\(1990\),)k(295{318.)456 4286 y([FS2])162 b(E.)23 b(F)-6 b(on)n(tic)n(h)24 b(and)g(C.)f(Sim\023) -35 b(o,)22 b Fm(Invariant)k(manifolds)i(for)d(ne)l(ar)h(identity)f (di\013er)l(entiable)g(maps)h(and)778 4369 y(splitting)f(of)h(sep)l(ar) l(atric)l(es)p Fn(,)f(Ergo)r(dic)f(Theory)g(Dynam.)e(Systems)h Fa(10)g Fn(\(1990\),)i(319{346.)456 4452 y([GGM])107 b(G.)31 b(Galla)n(v)n(otti,)i(G.)d(Gen)n(tile)i(and)f(V.)g (Mastropietro,)h Fm(Sep)l(ar)l(atrix)i(splitting)e(for)g(systems)g (with)778 4535 y(thr)l(e)l(e)26 b(times)f(sc)l(ales)p Fn(,)f(Comm.)d(Math.)j(Ph)n(ys.)f Fa(202)g Fn(\()p Fa(1)p Fn(\))h(\(1999\))i(197{236.)456 4618 y([Ge1])161 b(V.G.)23 b(Gelfreic)n(h,)h Fm(Sep)l(ar)l(atric)l(es)j(splitting)f(for)g(the)g(r) l(apid)t(ly)h(for)l(c)l(e)l(d)h(p)l(endulum)p Fn(,)d(Seminar)e(on)i (Dy-)778 4701 y(namical)32 b(Systems)g(\(St.)i(P)n(etersburg\),)i (Progress)c(in)h(Nonlinear)g(Di\013eren)n(tial)g(Equations)h(and)778 4784 y(their)24 b(Applications,)f(v)n(ol.)g(12,)h(Birkh\177)-35 b(auser,)23 b(1991,)h(pp.)g(47{67.)456 4867 y([Ge2])161 b(V.G.)16 b(Gelfreic)n(h,)i Fm(Melnikov)h(metho)l(d)h(and)h(exp)l (onential)t(ly)e(smal)t(l)i(splitting)e(of)g(sep)l(ar)l(atric)l(es)p Fn(,)g(Ph)n(ys.)778 4950 y(D)k Fa(101)g Fn(\(1997\),)i(227{248.)456 5033 y([Ge3])161 b(V.G.)22 b(Gelfreic)n(h,)g Fm(R)l(efer)l(enc)l(e)j (systems)f(for)h(splitting)f(of)h(sep)l(ar)l(atric)l(es)p Fn(,)f(Nonlinearit)n(y)g Fa(10)e Fn(\(1997\),)778 5116 y(175{193.)1917 5365 y Fo(79)p eop %%Page: 80 88 80 87 bop 456 251 a Fo(80)929 b(I.)22 b(BALDOM)1821 236 y(\023)1813 251 y(A)g(AND)h(E.)f(F)n(ONTICH)456 450 y Fn([Ge4])161 b(V.G.)21 b(Gelfreic)n(h,)h Fm(A)h(pr)l(o)l(of)j(of)e(the) g(exp)l(onential)t(ly)h(smal)t(l)f(tr)l(ansversality)h(of)f(the)f(sep)l (ar)l(atric)l(es)j(for)778 533 y(the)f(standar)l(d)j(map)p Fn(,)c(Comm.)d(Math.)j(Ph)n(ys.)f Fa(201)g Fn(\()p Fa(1)p Fn(\))h(\(1999\),)h(155{216.)456 616 y([Ge5])161 b(V.G.)24 b(Gelfreic)n(h,)h Fm(Exp)l(onential)t(ly)i(smal)t(l)h(splitting)e(of)h (sep)l(ar)l(atric)l(es)i(for)e(ar)l(e)l(a-pr)l(eserving)g(maps)p Fn(,)778 699 y(Chaos,)d(solitons)f(and)h(fractals)g Fa(11)f Fn(\(2000\))i(241{243.)456 782 y([GS])188 b(V.G.)21 b(Gelfreic)n(h)h (and)h(D.)e(Sauzin,)i Fm(Bor)l(el)i(summation)h(and)f(the)f(splitting)g (of)g(sep)l(ar)l(atric)l(es)i(of)e(the)778 865 y(Hen\023)-36 b(on)26 b(map)p Fn(,)e(Ann.)f(Inst.)h(F)-6 b(ourier)24 b(\(Grenoble\))h Fa(51)e Fn(\()p Fa(2)p Fn(\))h(\(2001\),)h(513{567.) 456 948 y([HMS])125 b(P)-6 b(.)30 b(Holmes,)g(J.)g(Marsden)g(and)h(J.)f (Sc)n(heurle,)j Fm(Exp)l(onential)t(ly)f(smal)t(l)h(splitting)f(of)g (sep)l(ar)l(atric)l(es)778 1031 y(with)24 b(applic)l(ations)h(to)f(KAM) f(the)l(ory)g(and)i(de)l(gener)l(ate)e(bifur)l(c)l(ations)p Fn(,)g(Con)n(temp.)e(Math.)g(v)n(ol.)g(81,)778 1114 y(1988,)j(pp.)g (213{244.)456 1197 y([La1])168 b(V.F.)39 b(Lazutkin,)45 b Fm(Splitting)40 b(of)i(sep)l(ar)l(atric)l(es)g(for)f(the)g (Chirikov's)g(standar)l(d)i(map)p Fn(,)h(VINITI)778 1280 y Fa(6372/84)21 b Fn(\(1984\).)k(\(English)f(v)n(ersion)g(in)f Fm(www.maia.ub.es/mp)p 2553 1280 22 4 v 29 w(ar)l(c)j(#98-421)p Fn(.\))456 1363 y([La2])168 b(V.F.)34 b(Lazutkin,)39 b Fm(A)n(n)e(analytic)f(inte)l(gr)l(al)h(along)h(the)e(sep)l(ar)l (atrix)i(of)f(the)f(semistandar)l(d)i(map:)778 1446 y(existenc)l(e)24 b(and)i(an)f(exp)l(onential)h(estimate)f(for)g(the)g(distanc)l(e)g(b)l (etwe)l(en)g(the)g(stable)g(and)h(unstable)778 1529 y(sep)l(ar)l(atric) l(es)p Fn(,)f(St.)f(P)n(etersburg)g(Math.)f(J.)h Fa(4)f Fn(\()p Fa(4)p Fn(\))h(\(1993\),)h(721{748.)456 1612 y([LST])148 b(V.F.)17 b(Lazutkin,)k(I.G.)d(Sc)n(hac)n(hmanski)h(and)g (M.B.)e(T)-6 b(abano)n(v,)21 b Fm(Splitting)f(of)h(sep)l(ar)l(atric)l (es)i(for)e(stan-)778 1695 y(dar)l(d)27 b(and)g(semistandar)l(d)g (mappings)p Fn(,)e(Ph)n(ysica)f(D)f Fa(40)g Fn(\(1989\),)i(235{348.)456 1778 y([Me])186 b(V.F.)29 b(Melnik)n(o)n(v,)i Fm(On)g(the)g(stability)g (of)g(the)g(c)l(enter)g(for)h(time)f(p)l(erio)l(dic)i(p)l(erturb)l (ations)p Fn(,)f(T)-6 b(rans.)778 1861 y(Mosco)n(w)24 b(Math.)f(So)r(c.)h Fa(12)f Fn(\(1963\),)i(3{56.)456 1944 y([Ne])198 b(Neish)n(tadt,)25 b Fm(The)h(sep)l(ar)l(ation)j(of)d (motions)h(in)f(systems)g(with)g(r)l(apid)t(ly)h(r)l(otating)g(phase)p Fn(,)e(J.)f(Appl.)778 2028 y(Math.)g(Mec)n(h.)f Fa(48)g Fn(\()p Fa(2)p Fn(\))h(\(1984\),)h(133{139.)456 2111 y([P)n(o])201 b(H.)28 b(P)n(oincar)n(\023)-33 b(e,)31 b Fm(Sur)g(le)g(pr)l(obl)n(\022)-35 b(eme)33 b(des)e(tr)l(ois)g(c)l (orps)h(et)e(les)f(\023)-35 b(equations)32 b(de)f(la)h(dynamique)p Fn(,)e(Acta)778 2194 y(Math.)24 b Fa(13)e Fn(\(1890\),)k(1{271.)456 2277 y([Sa])208 b(D.)25 b(Sauzin,)i Fm(A)g(new)h(metho)l(d)h(for)f(me)l (asuring)g(the)g(splitting)f(of)h(invariant)f(manifolds)p Fn(,)h(Annales)778 2364 y(Scien)n(ti\014ques)d(de)f(l')1345 2348 y(\023)1339 2364 y(Ecole)g(Normale)e(Sup)n(\023)-33 b(eriore.)24 b(Quatri)n(\022)-33 b(eme)23 b(S)n(\023)-33 b(erie)23 b Fa(34)g Fn(\()p Fa(2)p Fn(\))h(\(2001\),)h(159{221.)456 2447 y([SMH])125 b(J.)29 b(Sc)n(heurle,)i(J.)e(Marsden)g(and)h(P)-6 b(.)29 b(Holmes,)g Fm(Exp)l(onential)t(ly)i(smal)t(l)h(estimates)f(for) g(sep)l(ar)l(atrix)778 2530 y(splitting)p Fn(,)e(Asymptotics)f(b)r(ey)n (ond)h(all)f(orders)g(\(La)h(Jolla,)g(CA,)e(1991\),)k(NA)-6 b(TO)28 b(Adv.)g(Sci.)g(Inst.)778 2613 y(Ser.)23 b(B)h(Ph)n(ys.,)f(v)n (ol.)g(284,)h(Plen)n(um,)e(New)i(Y)-6 b(ork,)23 b(1991,)h(pp.)g (187{195.)456 2696 y([Su])204 b(Y.B.)29 b(Suris,)h Fm(On)i(the)f(c)l (omplex)i(sep)l(ar)l(atric)l(es)g(of)f(some)g(standar)l(d-like)h(maps)p Fn(,)f(Nonlinearit)n(y)g Fa(7)778 2780 y Fn(\(1994\),)25 b(1225{1236.)456 2863 y([T)-6 b(r])209 b(D.V.)24 b(T)-6 b(resc)n(hev,)25 b Fm(Sep)l(ar)l(atrix)j(splitting)f(for)f(a)i(p)l (endulum)g(with)f(r)l(apid)t(ly)h(oscil)t(lating)f(susp)l(ension)778 2946 y(p)l(oint)p Fn(,)d(Russ.)f(J.)g(Math.)h(Ph)n(ys.)f Fa(5)g Fn(\()p Fa(2)p Fn(\))i(\(1997\),)g(63{98.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0204031251900--