Content-Type: multipart/mixed; boundary="-------------9910080857181" This is a multi-part message in MIME format. ---------------9910080857181 Content-Type: text/plain; name="99-377.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-377.keywords" Generalized standard maps, Lindstedt series, Invariant curves, Radius of convergence, Complexified rotation number, Scaling ---------------9910080857181 Content-Type: application/postscript; name="bg3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bg3.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: bg3.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips bg3 %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.10.08:1531 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N 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b(anish\).)54 b(F)-7 b(or)33 b(eac)n(h)330 1955 y(sequence)684 1934 y(~)674 1955 y Fr(I)710 1967 y Fq(c)744 1955 y Ft(\()p Fr(f)9 b Ft(\),)28 b Fr(c)23 b Ft(=)g(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(q)s Ft(,)28 b(let)1328 2143 y Fr(I)1364 2155 y Fq(c)1399 2143 y Ft(\()p Fr(f)9 b Ft(\))23 b(=)g Fn(f)p Fr(f)1707 2155 y Fq(\027)1770 2143 y Fn(2)1858 2122 y Ft(~)1848 2143 y Fr(I)1884 2155 y Fq(c)1919 2143 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)p Fr(f)2097 2155 y Fq(\027)2161 2143 y Fn(6)p Ft(=)23 b(0)p Fn(g)330 2331 y Ft(b)r(e)28 b(the)g(set)g(of)f (nonzero)f(v)-5 b(alues)28 b(of)f(the)h(sequence)1958 2310 y(~)1948 2331 y Fr(I)1984 2343 y Fq(c)2019 2331 y Ft(\()p Fr(f)9 b Ft(\).)430 2471 y(W)-7 b(e)28 b(de\014ne)f(the)h (follo)n(wing)f(sets)g(of)h(in)n(tegers:)1158 2659 y Fr(A)1220 2674 y Fq(p=q)1325 2659 y Ft(\()p Fr(f)9 b Ft(\))23 b(=)g Fn(f)p Fr(c)f Fn(2)i(f)p Ft(1)p Fr(;)14 b(:)g(:)g(:)e(;)i(q)s Fn(gj)p Fr(I)2137 2671 y Fq(c)2171 2659 y Ft(\()p Fr(f)9 b Ft(\))23 b Fn(6)p Ft(=)g Fn(;g)p Fr(:)615 b Ft(\(1.12\))330 2847 y(and)1320 3036 y Fr(B)1383 3048 y Fq(c)1416 3036 y Ft(\()p Fr(f)9 b Ft(\))24 b(=)e Fn(f)p Fr(l)i Fn(2)g Fh(Z)1872 3048 y Fl(+)1921 3036 y Fn(j)p Fr(f)1985 3048 y Fq(lq)r Fl(+)p Fq(c)2147 3036 y Fn(6)p Ft(=)e(0)p Fn(g)p Fr(:)777 b Ft(\(1.13\))330 3224 y(Of)36 b(course)e Fr(A)780 3239 y Fq(p=q)885 3224 y Ft(\()p Fr(f)9 b Ft(\))37 b(=)f Fn(f)p Fr(c)1215 3236 y Fl(1)1252 3224 y Fr(;)14 b(:)g(:)g(:)g(;)g(c)1473 3236 y Fq(M)1546 3224 y Fn(g)p Ft(,)38 b(where)d(1)h Fn(\024)g Fr(c)2112 3236 y Fl(1)2186 3224 y Fr(<)g Fn(\001)14 b(\001)g(\001)36 b Fr(<)h(c)2558 3236 y Fq(M)2668 3224 y Fn(\024)f Fr(q)s Ft(,)i(and)e Fr(M)45 b Fn(\024)36 b Fr(q)s Ft(.)330 3363 y(Note)26 b(that)h Fr(A)770 3378 y Fq(p=q)875 3363 y Ft(\()p Fr(f)9 b Ft(\))27 b(is)f(the)h(set)f(of)h(equiv)-5 b(alence)26 b(classes)f(mo)r(dulo)h Fr(q)k Ft(of)c(frequencies)g (actually)330 3503 y(app)r(earing)g(in)i(the)g(F)-7 b(ourier)27 b(expansion)f(of)i(the)g(p)r(erturbation,)330 3710 y Fx(Lemma)i(1.)40 b Fj(If)30 b Fr(q)36 b(=)-51 b Fn(2)23 b Fr(A)1075 3725 y Fq(p=q)1180 3710 y Ft(\()p Fr(f)9 b Ft(\))30 b Fj(and)g Fn(j)p Fr(I)1544 3722 y Fq(c)1579 3710 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)23 b Ft(=)g(1)29 b Fn(8)p Fr(c)22 b Fn(2)h Fr(A)2143 3725 y Fq(p=q)2248 3710 y Ft(\()p Fr(f)9 b Ft(\))p Fj(,)31 b(de\014ne)1392 3898 y Fn(A)p Ft(\()p Fr(f)9 b Ft(\))24 b(=)e Fn(f)p Fr(\027)28 b Fn(2)23 b Fh(N)t Fn(j)p Fr(f)1990 3910 y Fq(\027)2060 3898 y Fn(6)p Ft(=)g(0)p Fn(g)p Ft(;)863 b(\(1.14\))330 4087 y Fj(then)30 b Fn(A)p Ft(\()p Fr(f)9 b Ft(\))23 b(=)g Fn(f)p Fr(\027)889 4099 y Fl(1)925 4087 y Fr(;)14 b(:)g(:)g(:)28 b(;)14 b(\027)1165 4099 y Fq(M)1239 4087 y Fn(g)p Fj(,)29 b(with)i Ft(1)22 b Fn(\024)h Fr(M)31 b Fn(\024)23 b Fr(q)f Fn(\000)c Ft(1)29 b Fj(and)h Fr(\027)2283 4099 y Fq(i)2329 4087 y Fn(\000)18 b Fr(\027)2453 4099 y Fq(j)2520 4087 y Fr(=)-51 b Fn(2)24 b Fr(q)s Fh(Z)f Fn(8)p Fr(i)e Fn(6)p Ft(=)i Fr(j)5 b Fj(.)330 4294 y(Pr)l(o)l(of.)43 b Ft(If)34 b Fr(q)46 b(=)-51 b Fn(2)34 b Fr(A)904 4309 y Fq(p=q)1009 4294 y Ft(\()p Fr(f)9 b Ft(\))34 b(then)h Fr(f)1394 4306 y Fq(\027)1468 4294 y Ft(=)e(0)h(for)f(all)h Fr(\027)39 b Fn(2)33 b Fr(q)s Fh(Z)p Ft(,)c(so)34 b(that)g Fr(I)2550 4306 y Fq(q)2587 4294 y Ft(\()p Fr(f)9 b Ft(\))34 b(=)f Fn(;)p Ft(:)49 b(then)34 b Fr(M)42 b Fn(\021)330 4433 y(j)p Fr(A)415 4448 y Fq(p=q)520 4433 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)29 b(\024)g Fr(q)23 b Fn(\000)e Ft(1.)47 b(As)31 b Fn(j)p Fr(I)1223 4445 y Fq(c)1257 4433 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)29 b Ft(=)g(1)h Fn(8)p Fr(c)e Fn(2)h Fr(A)1846 4448 y Fq(p=q)1951 4433 y Ft(\()p Fr(f)9 b Ft(\),)33 b(for)d(an)n(y)g Fr(c)f Fn(2)g Fr(A)2622 4448 y Fq(p=q)2727 4433 y Ft(\()p Fr(f)9 b Ft(\))32 b(there)f(is)f(only)330 4573 y(one)g Fr(\027)k Fn(2)29 b Fh(N)40 b Ft(of)31 b(the)g(form)g Fr(\027)i Ft(=)28 b Fr(c)20 b Ft(+)h Fr(l)r(q)s Ft(,)31 b(with)g Fr(l)f Fn(2)e Fh(Z)2000 4585 y Fl(+)2049 4573 y Ft(,)j(suc)n(h)g(that)g Fr(f)2518 4585 y Fq(\027)2587 4573 y Fn(6)p Ft(=)d(0.)46 b(So)30 b(the)h(n)n(um)n(b)r(er)330 4712 y(of)f(F)-7 b(ourier)29 b(lab)r(els)h Fr(\027)5 b Ft('s)31 b(for)e(whic)n(h)h Fr(f)1496 4724 y Fq(\027)1565 4712 y Fn(6)p Ft(=)d(0)j(is)g(giv)n(en)f (b)n(y)h Fr(M)36 b Fn(\024)27 b Fr(q)c Fn(\000)d Ft(1;)31 b(of)f(course)f(the)i(b)r(ound)330 4852 y Fr(M)h Fn(\025)22 b Ft(1)27 b(is)h(ob)n(vious.)p 3269 4852 4 57 v 3273 4799 50 4 v 3273 4852 V 3323 4852 4 57 v 430 5041 a(If)j Fr(q)40 b(=)-51 b Fn(2)28 b Fr(A)729 5056 y Fq(p=q)834 5041 y Ft(\()p Fr(f)9 b Ft(\))31 b(and)g Fn(j)p Fr(I)1203 5053 y Fq(c)1237 5041 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)28 b Ft(=)g(1)i Fn(8)p Fr(c)d Fn(2)i Fr(A)1823 5056 y Fq(p=q)1928 5041 y Ft(\()p Fr(f)9 b Ft(\),)31 b(w)n(e)g(de\014ne)f(2)p Fr(M)39 b Ft(in)n(tegers)30 b Fr(r)2974 5053 y Fl(1)3011 5041 y Fr(;)14 b(:)g(:)g(:)g(;)g(r)3233 5053 y Fq(M)3307 5041 y Ft(,)330 5180 y Fr(r)369 5150 y Fo(0)367 5201 y Fl(1)405 5180 y Fr(;)g(:)g(:)g(:)f(;)h(r)628 5150 y Fo(0)626 5203 y Fq(M)700 5180 y Ft(,)27 b(with)g Fr(r)975 5192 y Fq(i)1003 5180 y Fr(;)14 b(r)1079 5150 y Fo(0)1077 5202 y Fq(i)1128 5180 y Fn(\025)23 b Ft(0,)j(and)g(an)g(in)n(teger)g Fr(R)e(>)e Ft(0)k(as)g(those)g(in)n(tegers)f(whic)n(h)i(satisfy)f(the) 330 5320 y(follo)n(wing)h(conditions:)1092 5508 y(\()p Fr(r)1161 5520 y Fl(1)1218 5508 y Fn(\000)18 b Fr(r)1340 5474 y Fo(0)1338 5529 y Fl(1)1375 5508 y Ft(\))c Fr(\027)1462 5520 y Fl(1)1518 5508 y Ft(+)k Fn(\001)c(\001)g(\001)19 b Ft(+)f(\()p Fr(r)1869 5520 y Fq(M)1961 5508 y Fn(\000)g Fr(r)2083 5474 y Fo(0)2081 5529 y Fq(M)2156 5508 y Ft(\))c Fr(\027)2243 5520 y Fq(M)2340 5508 y Ft(=)22 b Fr(R)q(q)s(;)522 b Ft(\(1.15a\))p eop %%Page: 4 4 4 3 bop 450 42 a Fl(4)735 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f(GUIDO)g (GENTILE)1212 243 y Fr(r)1249 255 y Fl(1)1305 243 y Ft(+)18 b Fr(r)1427 209 y Fo(0)1425 263 y Fl(1)1482 243 y Ft(+)g Fn(\001)c(\001)g(\001)k Ft(+)g Fr(r)1800 255 y Fq(M)1893 243 y Ft(+)g Fr(r)2015 209 y Fo(0)2013 263 y Fq(M)2110 243 y Ft(=)k Fr(r)2234 255 y Fl(0)2295 243 y Fn(\025)h Ft(2)p Fr(;)744 b Ft(\(1.15b\))1212 417 y Fr(r)1249 429 y Fl(0)1315 417 y Ft(is)27 b(minimal)p Fr(;)1485 b Ft(\(1.15c\))450 606 y(where)27 b Fn(f)p Fr(\027)773 618 y Fl(1)810 606 y Fr(;)14 b(:)g(:)g(:)27 b(;)14 b(\027)1049 618 y Fq(M)1123 606 y Fn(g)22 b Ft(=)h Fn(A)p Ft(\()p Fr(f)9 b Ft(\).)550 745 y(W)-7 b(e)22 b(observ)n(e)f(that)i(this)g(problem)e(is)i (reminiscen)n(t)f(of)g(the)h(so)f(called)g(\\knapsac)n(k)e(problem",) 450 885 y(whic)n(h)j(is)g(b)r(eliev)n(ed)g(to)f(b)r(e)i (computationally)e(\\hard")f(\(see)i([8)o(])h(for)e(details)h(on)f (knapsac)n(ks)f(and)450 1024 y(on)27 b(what)h(\\hard")e(really)g (means\).)550 1163 y(The)37 b(meaning)g(of)g(the)h(ab)r(o)n(v)n(e)e (de\014nition)i(will)f(b)r(e)h(clear)e(after)h(prop)r(osition)g(1)f(in) i(sect.)450 1303 y(2,)32 b(while)g(its)g(relev)-5 b(ance)31 b(for)g(the)h(problem)f(w)n(e)g(are)g(studying)g(will)h(app)r(ear)f(in) h(the)g(theorem)450 1442 y(b)r(elo)n(w.)550 1582 y(As)j(w)n(e)h(no)n(w) f(sho)n(w,)i(in)n(tegers)e(satisfying)g(the)h(conditions)f(\(1.15\))g (alw)n(a)n(ys)f(exist.)61 b(Some)450 1721 y(prop)r(erties)27 b(of)g(the)h(solutions)f(will)h(b)r(e)g(explicitly)g(discussed)f(in)h (sect.)36 b(6.)450 1931 y Fx(Lemma)30 b(2.)40 b Fj(The)c(Diophantine)f (pr)l(oblem)42 b Ft(\(1.15\))33 b Fj(has)j(always)f(a)g(\014nite,)g (nonzer)l(o)g(numb)l(er)450 2070 y(of)c(solutions)f(and)g Fr(r)1094 2082 y Fl(0)1155 2070 y Fn(\024)22 b Fr(q)s Fj(.)450 2279 y(Pr)l(o)l(of.)43 b Ft(Solutions)23 b(to)g(\(1.15a\))e (and)i(\(1.15b\))f(do)h(exist:)35 b(one)22 b(is)h(giv)n(en)f(for)h (example)f(b)n(y)h Fr(R)h Ft(=)e Fr(\027)3422 2291 y Fq(i)450 2419 y Ft(for)31 b(some)g Fr(i)f Ft(and)i Fr(r)1055 2431 y Fq(i)1112 2419 y Ft(=)d Fr(q)s Ft(,)j(while)g Fr(r)1559 2431 y Fq(j)1623 2419 y Ft(=)d(0)i Fn(8)p Fr(j)i Fn(6)p Ft(=)c Fr(i)i Ft(and)g Fr(r)2262 2389 y Fo(0)2260 2440 y Fq(j)2325 2419 y Ft(=)e(0)i Fn(8)p Fr(j)5 b Ft(.)47 b(Note)31 b(that)h(in)f(this)h(case)450 2558 y(\(1.15b\))f(yields)g Fr(r)1014 2570 y Fl(0)1081 2558 y Ft(=)d Fr(q)s Ft(.)48 b(As)32 b(the)g(set)f(of)g(solutions)g(to)g(\(1.15a\))f(and)h (\(1.15b\))g(is)g(not)g(empt)n(y)450 2698 y(and)23 b(as)g(eac)n(h)g (solution)g(has)f(asso)r(ciated)h(a)g Fj(p)l(ositive)31 b Ft(v)-5 b(alue)23 b(of)g Fr(r)2434 2710 y Fl(0)2472 2698 y Ft(,)i(there)e(m)n(ust)g(b)r(e)h(at)f(least)g(one)450 2837 y(suc)n(h)i(that)h Fr(r)850 2849 y Fl(0)914 2837 y Ft(is)g(minimal.)36 b(On)26 b(the)g(other)f(hand,)h(a)g(minimal)g (solution)f(has)g Fr(r)2956 2849 y Fl(0)3017 2837 y Fn(\024)e Fr(q)28 b Ft(b)r(ecause)450 2977 y(of)g(the)g(ab)r(o)n(v)n(e)e(note,)h (so)g(there)h(can)f(b)r(e)h(only)f(a)g(\014nite)h(n)n(um)n(b)r(er)g(of) f(them.)p 3389 2977 4 57 v 3393 2924 50 4 v 3393 2977 V 3443 2977 4 57 v 450 3186 a Fj(R)l(emark)39 b Ft(1)p Fj(.)i Ft(Of)28 b(course)e(one)i(can)f(exhibit)h(functions)g(in)g(whic) n(h)g Fr(r)2561 3198 y Fl(0)2622 3186 y Ft(=)23 b Fr(q)s Ft(:)37 b(an)28 b(example)f(is)h(the)450 3325 y(standard)f(map)g (itself)h(\(for)g(whic)n(h)f Fr(M)32 b Ft(=)22 b(1)28 b(and)f Fr(\027)2051 3337 y Fl(1)2111 3325 y Ft(=)c(1\).)550 3535 y(De\014ne)28 b Fr(r)846 3504 y Fo(\003)885 3535 y Ft(\()p Fr(f)9 b Ft(\))23 b(=)g Fr(r)1149 3504 y Fo(\003)1215 3535 y Ft(as)604 3884 y Fr(r)643 3850 y Fo(\003)705 3884 y Ft(=)792 3614 y Fp(8)792 3689 y(>)792 3714 y(>)792 3739 y(>)792 3764 y(>)792 3789 y(<)792 3938 y(>)792 3963 y(>)792 3988 y(>)792 4013 y(>)792 4038 y(:)866 3730 y Ft(1)199 b(if)28 b Fr(q)e Fn(2)d Fr(A)1386 3745 y Fq(p=q)1492 3730 y Ft(\()p Fr(f)9 b Ft(\))p Fr(;)866 3897 y Ft(2)199 b(if)28 b Fr(q)35 b(=)-51 b Fn(2)23 b Fr(A)1386 3912 y Fq(p=q)1492 3897 y Ft(\()p Fr(f)9 b Ft(\))27 b(and)h Fn(9)r Ft(\026)-44 b Fr(c)23 b Fn(2)g Fr(A)2040 3912 y Fq(p=q)2146 3897 y Ft(\()p Fr(f)9 b Ft(\))27 b(suc)n(h)h(that)g Fn(j)p Fr(I)2716 3909 y Fl(\026)-35 b Fq(c)2748 3897 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)23 b(\025)g Ft(2)p Fr(;)866 4064 y(r)903 4076 y Fl(0)1107 4064 y Ft(otherwise)o Fr(;)3238 3884 y Ft(\(1.16\))450 4234 y(where)k Fr(r)727 4246 y Fl(0)792 4234 y Ft(is)h(de\014ned)g(as)f(in)h(\(1.15\))o(.)37 b(Note)27 b(that)h(1)23 b Fn(\024)g Fr(r)2205 4204 y Fo(\003)2267 4234 y Fn(\024)f Fr(q)s Ft(.)550 4373 y(W)-7 b(e)28 b(can)f(no)n(w)g(state)g(our)g(main)h(result.)450 4583 y Fx(Theorem.)40 b Fj(Consider)25 b(the)g(c)l(one)f Fn(C)1601 4598 y Fq(p=q)r(;\014)1789 4583 y Ft(=)f Fn(f)p Fr(z)j Fn(2)d Fh(C)44 b Ft(:)i Fn(j)14 b Ft(Im)g Fr(z)t Fn(j)23 b Fr(>)f Ft(0)p Fr(;)27 b Fn(j)14 b Ft(Re)g Fr(z)c Fn(\000)c Fr(p=q)s Fn(j)22 b(\024)h Fr(\014)t Fn(j)14 b Ft(Im)g Fr(z)t Fn(j)p Fj(,)450 4722 y Fr(\014)27 b Fn(\025)c Ft(0)p Fn(g)p Fj(;)29 b(let)h Fr(!)c Fn(2)d(C)1067 4737 y Fq(p=q)r(;\014)1232 4722 y Fj(.)39 b(Then)30 b(the)36 b Ft(rescaled)27 b(conjugating)f(function)1258 4929 y(\026)-47 b Fr(u)1301 4944 y Fq(p=q)1405 4929 y Ft(\()p Fr(\013;)14 b(";)g(!)s Ft(\))24 b(=)e Fr(u)1849 4812 y Fp(\022)1910 4929 y Fr(\013;)14 b(")2039 4837 y Fp(\020)2088 4929 y Fr(!)22 b Fn(\000)2255 4873 y Fr(p)p 2255 4910 42 4 v 2256 4986 a(q)2306 4837 y Fp(\021)2356 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b(series;)f(in)h(particular)1360 2701 y Fr(\032)p Ft(\()p Fr(!)s Ft(\))c(=)e Fr(O)1698 2584 y Fp(\022)1760 2609 y(\020)1810 2701 y Fr(!)f Fn(\000)1976 2644 y Fr(p)p 1976 2682 42 4 v 1977 2758 a(q)2027 2609 y Fp(\021)2077 2626 y Fl(2)p Fq(=r)2177 2601 y Fk(\003)2215 2584 y Fp(\023)2277 2701 y Fr(:)818 b Ft(\(1.19\))430 2915 y(The)22 b(pro)r(of)g(of)h(the)g(theorem)f(is)g(ac)n(hiev)n(ed)g (through)g(a)g(series)f(of)i(lemmata)f(in)h(the)g(follo)n(wing)330 3055 y(sections.)749 3264 y(2.)46 b Fs(Trees,)31 b(Clusters)h(and)f (other)g(combina)-6 b(torial)32 b(tools)430 3453 y Ft(By)e(inserting)h (\(1.10\))f(in)n(to)h(the)g(functional)f(equation)g(\(1.8\))h(w)n(e)f (can)g(\014nd)i(the)f(recursion)330 3593 y(relation)c(satis\014ed)g(b)n (y)g(the)h(co)r(e\016cien)n(ts)f Fr(u)1680 3550 y Fl(\()p Fq(k)q Fl(\))1680 3602 y Fq(\027)1796 3593 y Fn(\021)22 b Fr(u)1931 3550 y Fl(\()p Fq(k)q Fl(\))1931 3602 y Fq(\027)2024 3593 y Ft(\()p Fr(!)s Ft(\))505 3808 y Fr(u)553 3773 y Fl(\()p Fq(k)q Fl(\))553 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Ft(is)g(a)f(maximal)g(connected)g(set)h (of)g(no)r(des)f(connected)450 5312 y(b)n(y)23 b(lines)g(on)g(scale)f Fr(n)h Ft(=)g(1;)h(suc)n(h)f(lines)g(are)f(called)h Fj(internal)32 b Ft(lines,)24 b(and)f(w)n(e)g(shall)g(write)g Fr(`)f Fn(2)i Fr(T)12 b Ft(.)450 5451 y(The)27 b(line)g(exiting)g(and)g(the)g (lines)g(en)n(tering)f(a)g(cluster)h(are)f(all)g(on)h(scale)f Fr(n)d Ft(=)g(0)j(and)h(they)g(are)p eop %%Page: 7 7 7 6 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)913 b(7)330 241 y Ft(called)24 b(the)h Fj(external)h(lines)32 b Ft(of)24 b Fr(T)12 b Ft(.)35 b(Let)24 b Fr(k)1611 253 y Fq(T)1688 241 y Ft(b)r(e)h(the)g(n)n(um)n(b)r(er)f(of)g(no)r(des)g (in)h Fr(T)12 b Ft(:)34 b(the)25 b(case)e Fr(k)3125 253 y Fq(T)3201 241 y Ft(=)f(1)330 380 y(is)31 b(p)r(ossible)g(and)h (corresp)r(onds)d(to)j(the)f(case)g(of)g(a)g(single)g(no)r(de)h(suc)n (h)f(that)h(the)g(exiting)f(line)330 520 y(and)e(the)h(en)n(tering)f (ones)g(are)g(all)g(on)h(scale)e Fr(n)f Ft(=)f(0.)42 b(W)-7 b(e)30 b(de\014ne)g(also)f Fr(m)2628 532 y Fq(T)2710 520 y Ft(as)g(the)h(n)n(um)n(b)r(er)f(of)330 659 y(lines)34 b(en)n(tering)e(the)i(cluster)g Fr(T)44 b Ft(\(of)34 b(course)e(there)i(is)f(alw)n(a)n(ys)f(a)h(single)g(line)h(exiting)f Fr(T)12 b Ft(\).)55 b(A)330 799 y(no)r(de)28 b Fr(u)f Ft(is)g(said)g(to)h(b)r(e)g Fj(internal)36 b Ft(to)28 b Fr(T)39 b Ft(if)28 b Fr(u)22 b Fn(2)i Fr(T)12 b Ft(.)330 1008 y Fj(De\014nition.)42 b Ft(A)28 b Fj(r)l(esonanc)l(e)34 b Ft(\(or)27 b Fj(r)l(esonant)i(cluster)9 b Ft(\))28 b(is)f(a)h(cluster)f Fr(V)46 b Ft(suc)n(h)28 b(that:)1626 1117 y Fp(X)1618 1295 y Fq(u)p Fo(2)p Fq(V)1769 1196 y Fr(\027)1810 1208 y Fq(u)1876 1196 y Ft(=)23 b(0;)1130 b(\(2.7\))330 1423 y(in)28 b(this)g(case,)f(w)n(e)g(shall)g(sa)n(y)f (that)i(the)g(line)g(exiting)g(the)f(resonance)f(is)i(a)f Fj(r)l(esonant)i(line)p Ft(.)430 1632 y(Giv)n(en)24 b(a)g(resonance)f Fr(V)44 b Ft(in)25 b(the)g(tree)f Fr(#)p Ft(,)h(with)g(resonan)n(t)e (line)i Fr(`)2402 1644 y Fq(V)2460 1632 y Ft(,)g(w)n(e)f(de\014ne)h (its)g Fj(r)l(esonanc)l(e)330 1772 y(factor)38 b Ft(as)1163 1960 y Fn(V)1214 1972 y Fq(V)1271 1960 y Ft(\()p Fr(#)p Ft(\))24 b(=)1496 1868 y Fp(h)1551 1881 y(Y)1535 2059 y Fq(u)p Fo(2)p Fq(V)1696 1904 y Fr(f)1737 1916 y Fq(\027)1770 1924 y Fg(u)1814 1904 y Fr(\027)1860 1874 y Fq(m)1919 1882 y Fg(u)1855 1924 y Fq(u)p 1696 1941 266 4 v 1760 2017 a Fr(m)1833 2029 y Fq(u)1876 2017 y Ft(!)1972 1868 y Fp(ih)2061 1881 y(Y)2051 2060 y Fq(`)p Fo(2)p Fq(V)2292 1904 y Ft(1)p 2200 1941 225 4 v 2200 2017 a Fr(\015)5 b Ft(\()p Fr(\027)2321 2029 y Fq(`)2349 2037 y Fg(u)2393 2017 y Ft(\))2435 1868 y Fp(i)2474 1960 y Fr(:)662 b Ft(\(2.8\))430 2187 y(W)-7 b(e)28 b(no)n(w)f(pro)n(v)n(e)f(a)h(simple)g (but)i(imp)r(ortan)n(t)e(fact)h(ab)r(out)f(the)h(minimal)g(size)f(of)h (clusters.)330 2396 y Fx(Prop)s(osition)i(1.)41 b Fj(A)f(nonr)l (esonant)g(cluster)g(has)h(at)f(le)l(ast)h Fr(r)2336 2366 y Fo(\003)2415 2396 y Fj(no)l(des.)71 b(If)41 b Fr(q)k(>)d(N)3064 2408 y Fq(f)3107 2396 y Fj(,)h(then)330 2536 y Fr(r)369 2506 y Fo(\003)431 2536 y Ft(=)23 b Fr(r)556 2548 y Fl(0)593 2536 y Fj(.)330 2745 y(Pr)l(o)l(of.)43 b Ft(A)20 b(cluster)f Fr(T)31 b Ft(has)18 b Fr(m)1224 2757 y Fq(T)1296 2745 y Ft(en)n(tering)h(lines,)i(carrying)c(momen)n (ta)i Fr(\027)2527 2757 y Fq(i)2578 2745 y Ft(=)k Fr(s)2705 2757 y Fq(i)2732 2745 y Fr(q)s Ft(,)e Fr(i)i Ft(=)g(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(m)3255 2757 y Fq(T)3307 2745 y Ft(,)330 2885 y(as)31 b(they)h(are)f(on)g(scale)g(0,)h(and)g(one)f (exiting)h(line)g(carrying)e(a)h(momen)n(tum)h Fr(\027)2835 2897 y Fl(0)2902 2885 y Ft(=)e Fr(s)3036 2897 y Fl(0)3073 2885 y Fr(q)s Ft(,)j(as)e(it)330 3024 y(is)f(also)f(on)h(scale)f(0;)i (of)f(course)f Fr(s)1393 3036 y Fq(i)1421 3024 y Ft(,)i Fr(s)1514 3036 y Fl(0)1581 3024 y Ft(are)e(all)h(di\013eren)n(t)g(from) g(0.)44 b(Eac)n(h)29 b(no)r(de)h Fr(u)d Fn(2)h Fr(T)41 b Ft(has)29 b(a)330 3164 y(mo)r(de)j(lab)r(el)g Fr(\027)804 3176 y Fq(u)878 3164 y Ft(=)d Fr(s)1011 3176 y Fq(u)1055 3164 y Fr(q)24 b Ft(+)d(sign)o(\()p Fr(\027)1418 3176 y Fq(u)1462 3164 y Ft(\))14 b Fr(c)1544 3176 y Fq(u)1587 3164 y Ft(,)34 b(where)d Fr(s)1927 3176 y Fq(u)2000 3164 y Fn(2)g Fh(Z)25 b Ft(and)32 b Fr(c)2375 3176 y Fq(u)2449 3164 y Fn(2)e Fr(A)2596 3179 y Fq(p=q)2701 3164 y Ft(\()p Fr(f)9 b Ft(\);)35 b(note)c(that)i Fr(\027)3287 3176 y Fq(u)330 3303 y Ft(uniquely)28 b(iden)n(ti\014es)f(the)h(in)n(tegers) f Fr(s)1505 3315 y Fq(u)1548 3303 y Ft(,)h Fr(c)1635 3315 y Fq(u)1678 3303 y Ft(.)37 b(Then)621 3522 y Fr(\027)662 3534 y Fl(0)722 3522 y Ft(=)816 3443 y Fp(X)810 3621 y Fq(u)p Fo(2)p Fq(T)956 3522 y Fr(\027)997 3534 y Fq(u)1059 3522 y Ft(+)1149 3417 y Fq(m)1208 3425 y Fg(T)1142 3443 y Fp(X)1148 3619 y Fq(i)p Fl(=1)1275 3522 y Fr(\027)1316 3534 y Fq(i)1367 3522 y Ft(=)1455 3380 y Fp( )1527 3443 y(X)1520 3621 y Fq(u)p Fo(2)p Fq(T)1666 3522 y Fr(s)1705 3534 y Fq(u)1767 3522 y Ft(+)1858 3417 y Fq(m)1917 3425 y Fg(T)1850 3443 y Fp(X)1856 3619 y Fq(i)p Fl(=1)1984 3522 y Fr(s)2023 3534 y Fq(i)2050 3380 y 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Ft(,)27 b Fr(s)1763 5167 y Fl(1)1800 5155 y Fr(;)14 b(s)1876 5167 y Fl(2)1936 5155 y Fn(\025)23 b Ft(0,)j Fr(s)2154 5167 y Fl(1)2214 5155 y Fn(6)p Ft(=)d Fr(s)2341 5167 y Fl(2)2404 5155 y Ft(and)i(a)h(cluster)f(can)h(b)r(e)g(built)330 5294 y(with)e(only)f(t)n(w)n(o)g(no)r(des)h Fr(u)1124 5306 y Fl(1)1160 5294 y Ft(,)h Fr(u)1256 5306 y Fl(2)1316 5294 y Ft(with)f(mo)r(de)g(lab)r(els)g Fr(\027)1992 5306 y Fq(u)2031 5314 y Fi(1)2090 5294 y Ft(=)f Fr(\027)2219 5306 y Fl(1)2280 5294 y Ft(and)g Fr(\027)2478 5306 y Fq(u)2517 5314 y Fi(2)2577 5294 y Ft(=)g Fn(\000)p Fr(\027)2771 5306 y Fl(2)2808 5294 y Ft(:)35 b(it)24 b(is)f(a)g(cluster)330 5434 y(since)33 b(the)g(sum)g Fr(\027)909 5446 y Fl(1)969 5434 y Fn(\000)21 b Fr(\027)1096 5446 y Fl(2)1167 5434 y Ft(is)33 b(a)f(m)n(ultiple)i(of)f Fr(q)s Ft(,)h(while)f(neither)g Fr(\027)2408 5446 y Fl(1)2479 5434 y Ft(nor)f Fr(\027)2673 5446 y Fl(2)2743 5434 y Ft(are)g(\(so)h(the)g(only)p eop %%Page: 8 8 8 7 bop 450 42 a Fl(8)735 b(ALBER)-5 b(TO)29 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Fq(i)1988 758 y Fn(\000)18 b Fr(r)2110 724 y Fo(0)2108 779 y Fq(i)2136 758 y Ft(\))c Fr(\027)2223 770 y Fq(i)2274 758 y Ft(=)23 b Fr(R)q(q)s(;)450 990 y Ft(where)30 b Fr(r)730 1002 y Fq(i)789 990 y Ft(is)h(the)g(n)n (um)n(b)r(er)f(of)h Fr(\027)1466 1002 y Fq(u)1509 990 y Ft('s)g(equal)f(to)h(+)p Fr(\027)2029 1002 y Fq(i)2087 990 y Ft(and)f Fr(r)2290 959 y Fo(0)2288 1011 y Fq(i)2347 990 y Ft(is)h(the)g(n)n(um)n(b)r(er)f(of)h Fr(\027)3024 1002 y Fq(u)3067 990 y Ft('s)g(equal)f(to)450 1129 y Fn(\000)p Fr(\027)556 1141 y Fq(i)583 1129 y Ft(.)38 b(Of)28 b(course)f Fr(R)e Fn(6)p Ft(=)e(0)k(as)h Fr(T)39 b Ft(is)28 b(not)g(a)f(resonance.)37 b(Then)28 b(\(1.15a\))f(m)n(ust)h (hold.)38 b(Moreo)n(v)n(er,)450 1269 y(a)d(cluster)h(m)n(ust)g(ha)n(v)n (e)e(at)i(least)f(t)n(w)n(o)g(no)r(des)h(in)g(it)g(if)h Fr(q)49 b(=)-52 b Fn(2)38 b Fr(A)2448 1284 y Fq(p=q)2553 1269 y Ft(\()p Fr(f)9 b Ft(\),)38 b(so)d(that)h Fr(r)3063 1281 y Fl(0)3138 1269 y Fn(\025)g Ft(2)f(and)450 1408 y(also)29 b(\(1.15b\))g(holds;)i(and)f(of)g(course,)g(a)f(cluster)h (has)g(the)g(least)g(p)r(ossible)f(n)n(um)n(b)r(er)h(of)g(no)r(des)450 1547 y(if)35 b(\(1.15c\))26 b(is)i(sati\014ed.)550 1687 y(Finally)-7 b(,)40 b(if)e Fr(q)43 b(>)d(N)1199 1699 y Fq(f)1241 1687 y Ft(,)h(then)d Fr(q)52 b(=)-51 b Fn(2)40 b Fr(A)1741 1702 y Fq(p=q)1846 1687 y Ft(\()p Fr(f)9 b Ft(\))38 b(and)g Fn(j)p Fr(I)2229 1699 y Fq(c)2263 1687 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)40 b Ft(=)g(1)d Fn(8)p Fr(c)i Fn(2)h Fr(A)2903 1702 y Fq(p=q)3008 1687 y Ft(\()p Fr(f)9 b Ft(\),)41 b(so)c(that)450 1826 y Fr(r)489 1796 y Fo(\003)553 1826 y Ft(=)25 b Fr(r)680 1838 y Fl(0)718 1826 y Ft(.)42 b(In)29 b(fact)g(the)g(in)n(tegers)f Fr(c)1542 1838 y Fq(i)1599 1826 y Ft(are)g(the)h(actual)g(frequencies)g(app)r (earing)e(in)j(the)f(F)-7 b(ourier)450 1966 y(expansion)27 b(of)g(the)h(p)r(erturbation)f Fr(f)9 b Ft(\()p Fr(x)p Ft(\),)29 b(that)e(is)h Fr(s)2075 1978 y Fq(u)2141 1966 y Ft(=)23 b(0)k(and)h(sign)o(\()p Fr(\027)2676 1978 y Fq(u)2720 1966 y Ft(\))14 b Fr(c)2802 1978 y Fq(u)2868 1966 y Ft(=)23 b Fr(\027)2997 1978 y Fq(u)3040 1966 y Ft(.)p 3389 1966 4 57 v 3393 1913 50 4 v 3393 1966 V 3443 1966 4 57 v 1170 2175 a(3.)46 b Fs(Existence)31 b(of)h(the)f(f)n(ormal)h(solution)550 2364 y Ft(The)j(\014rst)f(step)h (is,)i(of)e(course,)g(to)g(sho)n(w)f(that)h(\(1.8\))g(actually)f(has)g (a)h Fj(formal)45 b Ft(solution,)450 2504 y Fj(i.e.)39 b Ft(that)d(a)f(T)-7 b(a)n(ylor)34 b(series)h(in)h Fr(")p Ft(,)i(whose)d(co)r(e\016cien)n(ts)g(admit)h(a)g(F)-7 b(ourier)35 b(expansion)f(in)j Fr(\013)p Ft(,)450 2643 y(satis\014es)27 b(order)f(b)n(y)h(order)g(eq.)36 b(\(1.8\),)28 b(disregarding)d(an)n(y)i(issue)g(of)h(actual)f(con)n(v)n(ergence.)550 2783 y(This)35 b(fact)h(w)n(as)f(essen)n(tially)g(pro)n(v)n(ed)f(b)n(y) h(P)n(oincar)n(\023)-39 b(e,)35 b([14)o(],)j(in)e(the)g(case)f(of)g (Hamiltonian)450 2922 y(\015o)n(ws)28 b(and)i(it)f(w)n(as)g(explicitly) g(w)n(ork)n(ed)f(out)h(b)n(y)g([15)o(])h(in)f(the)h(case)e(of)i (area-preserving)25 b(maps.)450 3062 y(W)-7 b(e)29 b(b)r(eliev)n(e)g (that)g(a)f(direct,)i(purely)e(com)n(binatorial)f(pro)r(of)h(of)h(the)g (existence)g(of)g(the)g(formal)450 3201 y(solution)h(is)g(of)g(some)f (in)n(terest,)i(esp)r(ecially)f(as)f(it)i(helps)f(to)g(understand)g (the)g(mec)n(hanism)g(of)450 3341 y(the)e(renormalization)d(of)j (resonances)e(in)h(the)h(follo)n(wing)f(sections.)450 3550 y Fx(Prop)s(osition)j(2.)41 b Fj(If)e Fr(f)1219 3562 y Fl(0)1294 3550 y Ft(=)f Fn(h)p Fr(f)9 b Fn(i)38 b Ft(=)g(0)p Fj(,)i(then)k Ft(\(1.8\))38 b Fj(has)h(a)g(formal)g (solution)g Fr(u)p Fj(,)h(such)e(that)450 3689 y Fn(h)p Fr(u)p Fn(i)23 b Ft(=)g(1)p Fr(=)p Ft(\(2)p Fr(\031)s Ft(\))927 3623 y Fp(R)981 3643 y Fl(2)p Fq(\031)965 3719 y Fl(0)1073 3689 y Ft(d)q Fr(\013)14 b(u)p Ft(\()p Fr(\013;)g(";)g(!)s Ft(\))23 b(=)f(0)p Fj(.)450 3899 y(Pr)l(o)l(of.)43 b Ft(First,)21 b(note)e(that)h(it)f(is)h(alw)n(a)n(ys)d(p)r(ossible)i(to) g(c)n(ho)r(ose)f Fr(u)2384 3856 y Fl(\(0\))2384 3921 y(0)2496 3899 y Ft(=)k(0,)f(since)e(\(1.8\))g(is)g(in)n(v)-5 b(arian)n(t)450 4038 y(with)28 b(resp)r(ect)f(to)h(the)g (transformation)1668 4111 y Fp(8)1668 4186 y(>)1668 4211 y(<)1668 4360 y(>)1668 4385 y(:)1742 4236 y Fr(\013)c Fn(7!)f Fr(\013)c Ft(+)f Fr(c;)1742 4403 y(u)23 b Fn(7!)g Fr(u)18 b Fn(\000)g Fr(c:)450 4575 y Ft(Next,)28 b(for)f Fr(k)f Ft(=)d(1,)k(w)n(e)g(let)1538 4763 y Fr(u)1586 4720 y Fl(\(1\))1586 4785 y(0)1698 4763 y Ft(=)22 b(0)p Fr(;)1443 b Ft(\(3.1\))1538 4938 y Fr(u)1586 4903 y Fl(\(1\))1586 4958 y Fq(\027)1698 4938 y Ft(=)22 b Fr(g)s Ft(\()p Fr(\027)5 b Ft(\))p Fr(f)1979 4950 y Fq(\027)2021 4938 y Fr(;)97 b(\027)28 b Fn(6)p Ft(=)23 b(0)p Fr(;)930 b Ft(\(3.2\))450 5126 y(since)27 b Fr(D)r(u)772 5096 y Fl(\(1\))861 5126 y Ft(\()p Fr(\013;)14 b(!)s Ft(\))24 b(=)e Fr(f)9 b Ft(\()p Fr(\013)p Ft(\))29 b(and)e Fr(f)1579 5138 y Fl(0)1639 5126 y Ft(=)c(0.)550 5265 y(T)-7 b(o)27 b(order)f Fr(k)31 b Ft(in)c Fr(")p Ft(,)h(equation)f(\(1.8\))g(b)r(ecomes)1405 5454 y Fr(\015)5 b Ft(\()p Fr(\027)g Ft(\))p Fr(u)1611 5419 y Fl(\()p Fq(k)q Fl(\))1611 5474 y Fq(\027)1726 5454 y Ft(=)23 b([)p Fr(f)9 b Ft(\()p Fr(\013)19 b Ft(+)f Fr(u)p Ft(\()p Fr(\013)p Ft(\)\)])2294 5419 y Fl(\()p Fq(k)q Fo(\000)p Fl(1\))2294 5474 y Fq(\027)2472 5454 y Fr(:)p eop %%Page: 9 9 9 8 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)913 b(9)330 241 y Ft(F)-7 b(or)27 b(this)h(to)f(b)r(e)h (soluble,)g(one)f(m)n(ust)h(ha)n(v)n(e)1361 429 y([)p Fr(f)9 b Ft(\()p Fr(\013)19 b Ft(+)f Fr(u)p Ft(\()p Fr(\013)p Ft(\)\)])1841 386 y Fl(\()p Fq(k)q Fl(\))1841 451 y(0)1958 429 y Ft(=)k(0)83 b Fn(8)p Fr(k)s(;)873 b Ft(\(3.3\))330 617 y(since)27 b Fr(\015)5 b Ft(\(0\))22 b(=)h(0.)36 b(W)-7 b(e)28 b(pro)n(v)n(e)d(this)i(b)n(y)g(induction)g(on)g Fr(k)s Ft(,)g(considering)f(that)h(the)g(case)f Fr(k)g Ft(=)d(1)j(is)330 757 y(trivial)h(since)g([)p Fr(f)9 b Ft(\()p Fr(\013)19 b Ft(+)f Fr(u)p Ft(\()p Fr(\013)p Ft(\)\)])1260 714 y Fl(\(0\))1260 779 y(0)1373 757 y Ft(=)23 b Fr(f)1502 769 y Fl(0)1562 757 y Ft(=)f(0)28 b(\(see)f(ab)r(o)n(v)n(e\).)430 896 y(Of)e(course,)f(once)h(\(3.3\))g (has)g(b)r(een)g(pro)n(v)n(ed,)g(the)g(formal)g(solubilit)n(y)f(of)32 b(\(1.8\))25 b(immediately)330 1036 y(follo)n(ws,)36 b(as)e Fn(j)p Fr(\015)5 b Ft(\()p Fr(\027)g Ft(\))p Fn(j)35 b(\025)g Fr(C)1148 1006 y Fo(\000)p Fl(1)1237 1036 y Fn(j)p Fr(\020)6 b Fn(j)1325 1006 y Fl(2)1398 1036 y Fn(8)p Fr(\027)40 b Fn(2)35 b Fh(Z)o Ft(,)c(for)j Fr(\020)42 b Ft(=)34 b Fr(!)26 b Fn(\000)d Fr(p=q)37 b Ft(\014xed)e(with)g Fn(j)14 b Ft(Im)g Fr(\020)6 b Fn(j)36 b(6)p Ft(=)e(0;)k(see)330 1175 y(lemma)28 b(3)f(in)h(sect.)36 b(2.)430 1315 y(W)-7 b(e)33 b(therefore)f(assume)h(that)g(\(3.3\))g(holds)g Fn(8)p Fr(k)1930 1285 y Fo(0)1985 1315 y Fr(<)e(k)s Ft(.)54 b(Note)33 b(that)h([)p Fr(f)9 b Ft(\()p Fr(\013)22 b Ft(+)g Fr(u)p Ft(\()p Fr(\013)p Ft(\)\)])3083 1272 y Fl(\()p Fq(k)q Fl(\))3083 1324 y Fq(\027)3210 1315 y Ft(has)330 1454 y(the)33 b(same)e(represen)n(tation,)h(in)g(term)h(of)f (trees,)h(of)f Fr(u)2041 1411 y Fl(\()p Fq(k)q Fl(+1\))2041 1464 y Fq(\027)2217 1454 y Ft(,)i(with)e(the)h(only)f(di\013erence)g (that)330 1594 y(the)27 b(propagator)c(asso)r(ciated)i(to)h(the)h(ro)r (ot)f(line)g(is)h(neglected)f({)g Fj(i.e.)k Ft(tak)n(en)25 b(to)i(b)r(e)f(iden)n(tically)330 1733 y(1.)36 b(W)-7 b(e)28 b(denote)g(with)1031 1711 y Fp(g)1031 1733 y Ft(V)-7 b(al\()p Fr(#)p Ft(\))28 b(the)g(v)-5 b(alue)28 b(of)f(a)h(tree)f Fr(#)h Ft(according)d(to)j(this)g(new)f(rule;)h(then)1250 1921 y([)p Fr(f)9 b Ft(\()p Fr(\013)18 b Ft(+)g Fr(u)p Ft(\()p Fr(\013)p Ft(\)\)])1729 1878 y Fl(\()p Fq(k)q Fl(\))1729 1944 y(0)1846 1921 y Ft(=)1977 1843 y Fp(X)1934 2021 y Fq(#)p Fo(2T)2057 2030 y Fg(k)q(;)p Fi(0)2154 1900 y Fp(g)2154 1921 y Ft(V)-7 b(al\()p Fr(#)p Ft(\))p Fr(;)330 2161 y Ft(and,)27 b(b)n(y)h(the)g(induction)g(h)n(yp)r (othesis,)f(all)g(lines)h(of)f Fr(#)h Ft(ha)n(v)n(e)f(nonzero)f(momen)n (tum.)430 2301 y(W)-7 b(e)28 b(no)n(w)g(de\014ne)g(a)g(group)f Fn(G)33 b Ft(of)c(transformations)d(acting)i(on)f Fn(T)2480 2313 y Fq(k)q(;)p Fl(0)2574 2301 y Ft(.)39 b(A)28 b(generator)f(of)h Fn(G)33 b Ft(is)330 2440 y(a)27 b(transformation)f(of)i(the)g(follo)n (wing)e(t)n(yp)r(e:)37 b(detac)n(h)27 b(the)h(ro)r(ot)f(line)h(of)g Fr(#)f Ft(and)h(reattac)n(h)e(it)i(to)330 2580 y(a)f(no)r(de)h(of)f Fr(#)p Ft(.)37 b(The)28 b(action)f(of)h Fn(G)33 b Ft(on)27 b(a)g(tree)g Fr(#)h Ft(de\014nes)g(the)g Fn(G)5 b Fj(-family)37 b Fn(F)2645 2592 y Fo(G)2692 2580 y Ft(\()p Fr(#)p Ft(\))28 b(of)g(the)g(tree)f Fr(#)p Ft(.)430 2719 y(Then)g(w)n(e)h(ha)n(v)n(e)e (the)i(follo)n(wing.)430 2876 y(1.)41 b Fn(G)33 b Ft(lea)n(v)n(es)25 b Fn(T)900 2888 y Fq(k)q(;)p Fl(0)1022 2876 y Ft(in)n(v)-5 b(arian)n(t:)35 b(that)28 b(is,)g(all)f(trees)g(obtained)g(from)g (trees)g(in)h Fn(T)2881 2888 y Fq(k)q(;)p Fl(0)3002 2876 y Ft(b)n(y)f(trans-)536 3016 y(formations)f(of)i Fn(G)33 b Ft(still)28 b(b)r(elong)f(to)g Fn(T)1699 3028 y Fq(k)q(;)p Fl(0)1793 3016 y Ft(.)430 3155 y(2.)41 b(If)31 b Fr(G)e Fn(2)g(G)5 b Ft(,)32 b Fr(#)958 3125 y Fo(0)1010 3155 y Ft(=)c Fr(G#)p Ft(,)k(then)g(on)f(some)f(lines)h(the)g(arro)n(ws)d (indicating)j(the)h(\015o)n(w)e(of)h(the)536 3295 y(momen)n(tum)i(c)n (hange)g(direction)f(so)h(that)h(the)f(partial)g(ordering)f(relation)g (de\014ned)i(in)536 3434 y Fr(#)585 3404 y Fo(0)638 3434 y Ft(is)c(di\013eren)n(t)h(than)f(the)h(one)e(in)i Fr(#)p Ft(;)h(if)e Fr(`)d Fn(2)h Fr(#)p Ft(,)j(denote)f(b)n(y)g Fr(`)2504 3404 y Fo(0)2557 3434 y Ft(the)h(same)e(line)i(seen)f(as)536 3573 y(an)25 b(elemen)n(t)i(of)f Fr(#)1097 3543 y Fo(0)1120 3573 y Ft(.)36 b(Consequen)n(tly)-7 b(,)26 b(on)g(lines)g Fr(`)f Ft(where)h(the)g(direction)g(did)g(not)g(c)n(hange)536 3713 y Fr(\027)577 3725 y Fq(`)605 3709 y Fk(0)654 3713 y Ft(=)d Fr(\027)783 3725 y Fq(`)815 3713 y Ft(,)h(while)g(on)f(lines)g Fr(`)h Ft(where)f(the)g(direction)g(did)h(c)n(hange)f Fr(\027)2597 3725 y Fq(`)2625 3709 y Fk(0)2674 3713 y Ft(=)g Fn(\000)p Fr(\027)2868 3725 y Fq(`)2899 3713 y Ft(;)i(in)f(fact,)g(if)g Fr(`)536 3852 y Ft(is)g(the)g(exiting)g(line)g (of)g Fr(u)p Ft(,)h Fr(\027)1407 3864 y Fq(`)1462 3852 y Ft(=)1550 3790 y Fp(P)1637 3877 y Fq(w)r Ff(4)p Fq(u)1796 3852 y Fr(\027)1837 3864 y Fq(w)1915 3852 y Ft(and)f Fr(\027)2114 3864 y Fq(`)2142 3848 y Fk(0)2191 3852 y Ft(=)2279 3790 y Fp(P)2367 3877 y Fq(w)r Fo(\037)p Fq(u)2525 3852 y Fr(\027)2566 3864 y Fq(w)2644 3852 y Ft(\(using)g(the)h (ordering)536 3992 y(of)i Fr(#)h Ft(in)g(b)r(oth)g(cases\),)f(so)g (that)1075 4180 y Fr(\027)1116 4192 y Fq(`)1167 4180 y Ft(+)18 b Fr(\027)1291 4192 y Fq(`)1319 4176 y Fk(0)1368 4180 y Ft(=)1466 4101 y Fp(X)1456 4280 y Fq(w)r Ff(4)p Fq(u)1610 4180 y Fr(\027)1651 4192 y Fq(w)1723 4180 y Ft(+)1817 4101 y Fp(X)1807 4276 y Fq(w)r Fo(\037)p Fq(u)1961 4180 y Fr(\027)2002 4192 y Fq(w)2079 4180 y Ft(=)2174 4101 y Fp(X)2167 4280 y Fq(w)r Fo(2)p Fq(#)2315 4180 y Fr(\027)2356 4192 y Fq(w)2433 4180 y Ft(=)k(0)p Fr(:)430 4414 y Ft(3.)41 b(Supp)r(ose)34 b(that)g(the)g(ro)r(ot)g(line)g(is)g (detac)n(hed)f(from)h(the)h(no)r(de)f Fr(u)2603 4426 y Fl(0)2674 4414 y Ft(and)f(reattac)n(hed)g(to)536 4553 y(the)i(no)r(de)h Fr(u)943 4565 y Fl(1)979 4553 y Ft(:)52 b(set)36 b Fr(P)12 b Ft(\()p Fr(u)1337 4565 y Fl(0)1373 4553 y Fr(;)i(u)1458 4565 y Fl(1)1495 4553 y Ft(\))36 b(=)f Fn(f)p Fr(w)j Fn(2)e Fr(#)p Fn(j)p Fr(u)2013 4565 y Fl(1)2086 4553 y Fe(4)f Fr(w)k Fe(4)c Fr(u)2432 4565 y Fl(0)2469 4553 y Fn(g)p Ft(.)59 b(Then,)37 b(forgetting)e(for)536 4693 y(a)d(momen)n(t)g(the)h(com)n(binatorial)e(factor,)i(the)g(v)-5 b(alue)32 b(of)g(the)h(tree)f(c)n(hanges)f(b)r(ecause)h(a)536 4832 y(factor)g Fr(\027)820 4844 y Fq(u)859 4852 y Fi(0)928 4832 y Ft(is)h(replaced)f(with)h(a)g(factor)f Fr(\027)1903 4844 y Fq(u)1942 4852 y Fi(1)1979 4832 y Ft(.)52 b(This)33 b(follo)n(ws)f(from)g(the)i(fact)e(that)i(for)536 4972 y(eac)n(h)28 b(no)r(de)h Fr(u)c Fn(2)h Fr(#)k Ft(there)e(is)h(a)g (factor)f(\()p Fr(i\027)1869 4984 y Fq(u)1913 4972 y Ft(\))1945 4942 y Fq(m)2004 4950 y Fg(u)2077 4972 y Ft(and,)h(b)n(y)g (shifting)g(the)h(ro)r(ot)e(line)h(from)536 5111 y Fr(u)584 5123 y Fl(0)646 5111 y Ft(to)24 b Fr(u)792 5123 y Fl(1)829 5111 y Ft(,)i(for)e(eac)n(h)g(no)r(de)h Fr(w)h Fn(2)d Fr(P)12 b Ft(\()p Fr(u)1692 5123 y Fl(0)1729 5111 y Fr(;)i(u)1814 5123 y Fl(1)1851 5111 y Ft(\))25 b(an)g(en)n(tering)f(line)h(b)r (ecomes)g(an)g(exiting)f(one)536 5251 y(and)34 b Fj(vic)l(e)i(versa)p Ft(,)h(except)d(for)f Fr(u)1570 5263 y Fl(0)1641 5251 y Ft(and)h Fr(u)1857 5263 y Fl(1)1894 5251 y Ft(:)50 b(the)34 b(no)r(de)g Fr(u)2371 5263 y Fl(0)2442 5251 y Ft(simply)g(loses)f(an)h(en)n(tering)536 5390 y(line,)27 b(while)g(the)g(no)r(de)g Fr(u)1321 5402 y Fl(1)1385 5390 y Ft(turns)g(out)g(to)g(ha)n(v)n(e)e(an)i(extra)f(en)n(tering)g (line.)37 b(So)27 b Fr(m)3066 5402 y Fq(u)3136 5390 y Ft(k)n(eeps)p eop %%Page: 10 10 10 9 bop 450 42 a Fl(10)702 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f (GUIDO)g(GENTILE)656 241 y Ft(the)35 b(same)f(v)-5 b(alue)34 b(for)g(all)h Fr(u)f Fn(2)h Fr(#)p Ft(,)i(except)d(for)h Fr(u)2230 253 y Fl(0)2301 241 y Ft(and)f Fr(u)2517 253 y Fl(1)2554 241 y Ft(:)51 b Fr(m)2701 253 y Fq(u)2740 261 y Fi(0)2812 241 y Ft(b)r(ecomes)34 b Fr(m)3222 253 y Fq(u)3261 261 y Fi(0)3321 241 y Fn(\000)22 b Ft(1)656 380 y(and)29 b Fr(m)892 392 y Fq(u)931 400 y Fi(1)998 380 y Ft(b)r(ecomes)g Fr(m)1403 392 y Fq(u)1442 400 y Fi(1)1499 380 y Ft(+)20 b(1.)43 b(Therefore)28 b(when)i(w)n(e)g(sum)g (o)n(v)n(er)e(all)h(the)h(trees)g(in)g(the)656 520 y(same)24 b Fn(G)5 b Ft(-family)-7 b(,)26 b Fj(assuming)i(that)f(al)t(l)i(c)l (ombinatorial)g(factors)g(ar)l(e)f(the)f(same)p Ft(,)g(w)n(e)e(get)f(a) 656 659 y(factor)893 597 y Fp(P)981 684 y Fq(u)p Fo(2)p Fq(#)1123 659 y Fr(\027)1164 671 y Fq(u)1231 659 y Ft(=)e(0)27 b(and)h(therefore)f(w)n(e)g(obtain)g(0:)1077 776 y Fp(X)1033 955 y Fq(#)p Fo(2T)1156 964 y Fg(k)q(;)p Fi(0)1253 833 y Fp(g)1253 855 y Ft(V)-7 b(al\()p Fr(#)p Ft(\))24 b(=)1641 776 y Fp(X)1598 955 y Fq(#)p Fo(2T)1721 964 y Fg(k)q(;)p Fi(0)1940 799 y Ft(1)p 1828 836 267 4 v 1828 912 a Fn(jF)1911 924 y Fo(G)1958 912 y Ft(\()p Fr(#)p Ft(\))p Fn(j)2202 776 y Fp(X)2118 958 y Fq(#)2158 942 y Fk(0)2181 958 y Fo(2F)2274 966 y Fk(G)2315 958 y Fl(\()p Fq(#)p Fl(\))2421 833 y Fp(g)2421 855 y Ft(V)-7 b(al)o(\()p Fr(#)2621 821 y Fo(0)2645 855 y Ft(\))2118 1016 y Fp(|)p 2155 1016 205 10 v 205 w({z)p 2434 1016 V 205 w(})2356 1092 y Fl(=0)2700 855 y Ft(=)23 b(0)p Fr(;)426 b Ft(\(3.4\))656 1215 y(and)21 b(this)g(concludes)g(the)h(pro)r(of)e(of)h(the)h(theorem,)g Fj(pr)l(ovide)l(d)k(everything)f(go)l(es)f(wel)t(l)h(with)656 1355 y(the)30 b(c)l(ombinatorial)h(factors)p Ft(.)550 1512 y(W)-7 b(e)22 b(are)e(therefore)h(left)h(with)h(the)f(task)f(of)g (pro)n(ving)f(that)i(all)g(com)n(binatorial)d(factors)i(inside)450 1651 y(a)32 b Fn(G)5 b Ft(-family)33 b(do)g(matc)n(h.)52 b(First,)33 b(w)n(e)g(note)f(that)h(it)h(is)e(con)n(v)n(enien)n(t)g(to) g(use)h Fj(top)l(olo)l(gic)l(al)44 b Ft(trees)450 1791 y(instead)36 b(of)g(the)h(usual)e Fj(semitop)l(olo)l(gic)l(al)48 b Ft(ones)35 b(used)h(throughout)g(the)g(pap)r(er)g(\(in)h([1)o(])f (only)450 1930 y(semitop)r(ological)f(trees)i(w)n(ere)f(used;)41 b(see)c([11)o(])g(and)g([9)o(])h(for)e(the)h(de\014nition)h(of)f(top)r (ological)450 2070 y(trees)28 b(and)h(for)f(a)g(detailed)g(discussion)g (of)h(the)g(di\013erences)f(b)r(et)n(w)n(een)h(what)f(\014nally)h (amoun)n(ts)450 2209 y(to)h(a)g(di\013eren)n(t)g(w)n(a)n(y)f(to)h(coun) n(t)g(trees\).)45 b(W)-7 b(e)30 b(de\014ne)h Fj(e)l(quivalent)38 b Ft(t)n(w)n(o)30 b(trees)f(whic)n(h)h(are)g(equal)450 2349 y(as)d(top)r(ological)f(trees.)550 2488 y(Then)g(w)n(e)g(can)f (still)i(write)e Fr(u)1455 2445 y Fl(\()p Fq(k)q Fl(\))1455 2498 y Fq(\027)1574 2488 y Ft(in)h(the)h(form)e(\(2.3\),)i(restricting) e(the)h(sum)g(o)n(v)n(er)f(the)h(set)g(of)450 2628 y(all)h(nonequiv)-5 b(alen)n(t)27 b(top)r(ological)f(trees)h(with)h Fr(k)i Ft(no)r(des)d(and)h(momen)n(tum)f Fr(\027)33 b Ft(\015o)n(wing)27 b(through)450 2767 y(the)34 b(ro)r(ot)f(line)h({)g(w)n(e)f(can)h (denote)g(it)g(b)n(y)g Fn(T)1860 2727 y Fl(top)1839 2792 y Fq(\027;k)1994 2767 y Ft({,)h Fj(pr)l(ovide)l(d)j(that)d(to)h(e)l (ach)g(no)l(de)g Fr(u)d Fn(2)h Fr(#)i Fj(we)450 2906 y(asso)l(ciate)31 b(a)f(c)l(ombinatorial)i(factor)f(which)g(is)f(not)g (the)g Ft(1)p Fr(=m)2398 2918 y Fq(u)2440 2906 y Ft(!)g Fj(app)l(e)l(aring)h(in)36 b Ft(\(2.4\))p Fj(.)550 3046 y Ft(In)31 b(fact)g(for)f(top)r(ological)g(trees)g(the)h(com)n (binatorial)e(factor)i(asso)r(ciated)e(to)i(eac)n(h)f(no)r(de)h(is)450 3185 y(di\013eren)n(t,)39 b(b)r(ecause)e(w)n(e)f(ha)n(v)n(e)g(to)g(lo)r (ok)g(no)n(w)h(to)f(ho)n(w)g(the)i(subtrees)e(emerging)f(from)i(eac)n (h)450 3325 y(no)r(de)i(di\013er.)70 b(F)-7 b(or)38 b(semitop)r (ological)e(trees)i(w)n(e)h(ha)n(v)n(e)e(a)h(factor)g(1)p Fr(=m)2728 3337 y Fq(u)2770 3325 y Ft(!)h(for)f(eac)n(h)g(no)r(de)g Fr(u)p Ft(,)450 3464 y(where)33 b Fr(m)769 3476 y Fq(u)845 3464 y Ft(is)g(the)h(n)n(um)n(b)r(er)f(of)g(lines)g(en)n(tering)f Fr(u)h Fj(disr)l(e)l(gar)l(ding)k(the)e(kinds)g(of)h(the)f(subtr)l(e)l (es)450 3604 y(entering)26 b Fr(u)p Ft(,)e(so)f(coun)n(ting)g(as)f (di\013eren)n(t)i(man)n(y)f(trees)g(otherwise)f(iden)n(tical,)j(while)e (in)h(the)g(case)450 3743 y(of)i(top)r(ological)e(trees)h(w)n(e)g (consider)g(one)g(and)g(the)i(same)e(tree)g(those)g(trees)g(that)h(are) f(di\013eren)n(t)450 3883 y(as)34 b(semitop)r(ological)g(trees,)i(but)g (ha)n(v)n(e)d(the)j(same)e(v)-5 b(alue)35 b(b)r(ecause)g(they)g(just)g (di\013er)g(in)h(the)450 4022 y(order)d(in)i(whic)n(h)g Fj(identic)l(al)44 b Ft(subtrees)34 b(en)n(ter)h(eac)n(h)e(no)r(de)i Fr(u)p Ft(:)51 b(therefore,)35 b(if)g Fr(s)2943 4034 y Fq(u;)p Fl(1)3039 4022 y Ft(,)g Fr(:)14 b(:)g(:)28 b Ft(,)36 b Fr(s)3320 4034 y Fq(u;j)3406 4042 y Fg(u)450 4162 y Ft(are)d(the)h(n)n(um)n(b)r(er)f(of)h(en)n(tering)e(lines)i(to)f (whic)n(h)h(are)f(attac)n(hed)g(subtrees)g(of)g(a)g(giv)n(en)g(shap)r (e)450 4301 y(and)k(with)h(the)g(same)e(lab)r(els)h(\(so)g(that)h Fr(s)1806 4313 y Fq(u;)p Fl(1)1927 4301 y Ft(+)24 b Fn(\001)14 b(\001)g(\001)25 b Ft(+)f Fr(s)2266 4313 y Fq(u;j)2352 4321 y Fg(u)2435 4301 y Ft(=)39 b Fr(m)2612 4313 y Fq(u)2655 4301 y Ft(,)h(1)f Fn(\024)g Fr(j)2937 4313 y Fq(u)3020 4301 y Fn(\024)f Fr(m)3196 4313 y Fq(u)3240 4301 y Ft(\),)i(the)450 4441 y(com)n(binatorial)26 b(factor,)h(for)g(eac)n(h)f(no)r(de,)i(b)r (ecomes)1402 4573 y(1)p 1354 4610 140 4 v 1354 4686 a Fr(m)1427 4698 y Fq(u)1470 4686 y Ft(!)1521 4629 y Fn(\001)1708 4573 y Fr(m)1781 4585 y Fq(u)1824 4573 y Ft(!)p 1573 4610 410 4 v 1573 4686 a Fr(s)1612 4698 y Fq(u;)p Fl(1)1708 4686 y Ft(!)18 b Fn(\001)h Fr(s)1830 4698 y Fq(u;j)1916 4706 y Fg(u)1960 4686 y Ft(!)2016 4629 y(=)2298 4573 y(1)p 2113 4610 V 2113 4686 a Fr(s)2152 4698 y Fq(u;)p Fl(1)2249 4686 y Ft(!)f Fn(\001)h Fr(s)2371 4698 y Fq(u;j)2457 4706 y Fg(u)2500 4686 y Ft(!)2533 4629 y(;)450 4824 y(note)32 b(in)h(the)f(second)g(factor)f(in)i(the)f(ab)r(o)n(v)n(e)f(form)n(ula)g (the)i(m)n(ultinomial)f(co)r(e\016cien)n(t)g(corres-)450 4964 y(p)r(onding)26 b(to)f(the)i(n)n(um)n(b)r(er)e(of)h(di\013eren)n (t)g Fj(semitop)l(olo)l(gic)l(al)37 b Ft(trees)25 b(corresp)r(onding)f (to)i(the)g(same)450 5103 y Fj(top)l(olo)l(gic)l(al)39 b Ft(tree,)27 b(for)g(eac)n(h)g(no)r(de.)p eop %%Page: 11 11 11 10 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)880 b(11)430 244 y Ft(So)27 b(in)h(terms)f(of)h(top)r(ological)e (trees)h Fr(u)1637 201 y Fl(\()p Fq(k)q Fl(\))1637 254 y Fq(\027)1757 244 y Ft(can)g(b)r(e)h(expressed)e(as)1450 433 y Fr(u)1498 398 y Fl(\()p Fq(k)q Fl(\))1498 453 y Fq(\027)1614 433 y Ft(=)1753 354 y Fp(X)1701 546 y Fq(#)p Fo(2T)1839 518 y Fi(top)1824 566 y Fg(\027;k)p 1939 365 120 4 v 1939 433 a Ft(V)-7 b(al\()p Fr(#)p Ft(\))p Fr(;)964 b Ft(\(3.5\))330 697 y(where)p 1021 818 V 1021 885 a(V)-7 b(al\()p Fr(#)p Ft(\))24 b(=)f Fr(i)1395 851 y Fq(k)q Fo(\000)p Fl(1)1520 793 y Fp(h)1568 807 y(Y)1559 985 y Fq(u)p Fo(2)p Fq(#)1811 829 y Fr(f)1852 841 y Fq(\027)1885 849 y Fg(u)1929 829 y Fr(\027)1975 799 y Fq(m)2034 807 y Fg(u)1970 850 y Fq(u)p 1707 866 475 4 v 1707 942 a Fr(s)1746 954 y Fq(u;)p Fl(1)1842 942 y Ft(!)14 b Fr(:)g(:)g(:)g(s)2029 954 y Fq(u;j)2115 962 y Fg(u)2159 942 y Ft(!)2192 793 y Fp(ih)2273 807 y(Y)2270 985 y Fq(`)p Fo(2)p Fq(#)2396 885 y Fr(g)s Ft(\()p Fr(\027)2512 897 y Fq(`)2544 885 y Ft(\))2576 793 y Fp(i)2616 885 y Fr(:)520 b Ft(\(3.6\))430 1113 y(Still,)39 b(when)d(computing)g(the)g(com)n(binatorial)e(factors) h(inside)h(eac)n(h)g Fn(G)5 b Ft(-family)-7 b(,)38 b(they)e(do)330 1252 y(di\013er.)64 b(But)36 b(this)h(is)g(actually)e(an)i(apparen)n (t,)g(not)g(a)f(real)f(discrepancy)-7 b(.)63 b(In)37 b(fact,)h(due)f(to)330 1392 y(symmetries)31 b(in)g(the)h(tree)e(\(that) i(is,)g(to)f(the)h(fact)f(that)h(the)f(subtrees)g(emerging)f(from)h (some)330 1531 y(no)r(de)22 b(are)f(sometimes)h(equal,)h Fj(i.e.)j Ft(that)c(some)g Fr(s)1848 1543 y Fq(u;i)1957 1531 y Ft(are)f(greater)f(than)i(1\),)h(the)g(actual)e(n)n(um)n(b)r(er) 330 1671 y(of)f(top)r(ological)e(trees)i(in)g(a)f(giv)n(en)g 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b(but)f(then)g(this)330 2368 y(means)23 b(that)i(the)f(same)f(top)r(ological)f(tree)i(could)f(b)r(e) i(formed)e(b)n(y)h(the)g(action)f(of)h Fr(s)2909 2380 y Fq(w)r(;i)3030 2368 y Ft(di\013eren)n(t)330 2508 y(transformations)35 b(of)j(the)f(group)f Fn(G)5 b Ft(:)57 b(eac)n(h)36 b(of)h(the)h Fr(s)2066 2520 y Fq(w)r(;i)2200 2508 y Ft(equiv)-5 b(alen)n(t)37 b(subtrees)g(en)n(tering)f Fr(w)330 2647 y Ft(con)n(tains)22 b(a)h(no)r(de)g(suc)n(h)g(that,)h(b)n(y)f(attac)n(hing)g(to)g(it)g(the) h(ro)r(ot)e(line,)j(the)e(same)g(top)r(ological)e(tree)330 2786 y(is)g(obtained.)34 b(Therefore,)21 b Fj(by)i(c)l(ounting)g(al)t (l)i(tr)l(e)l(es)d(obtaine)l(d)j(by)f(the)f(action)h(of)g(the)g(gr)l (oup)f Fn(G)j Ft(\(as)330 2926 y(done)g(to)g(implemen)n(t)h(the)g (cancellation)e(\(3.4\)\),)i(the)f(corresp)r(onding)f(top)r(ological)f (tree)i(v)-5 b(alue)330 3065 y(is)34 b(in)h(fact)f(coun)n(ted)h Fr(s)1052 3077 y Fq(w)r(;i)1183 3065 y Ft(times,)h(so)e(to)g(a)n(v)n (oid)f(o)n(v)n(ercoun)n(ting)f(one)i(needs)g(a)g(factor)g(1)p Fr(=s)3211 3077 y Fq(w)r(;i)3307 3065 y Ft(:)330 3205 y(this)h(giv)n(es)e(bac)n(k)h(the)h(same)f(com)n(binatorial)e(factor)i (1)p Fr(=s)2173 3217 y Fq(w)r(;i)2269 3205 y Ft(!.)58 b(Analogously)33 b(one)h(discusses)330 3344 y(the)i(case)e(of)h(a)g (factor)g(1)p Fr(=s)1212 3356 y Fq(w)r(;i)1307 3344 y Ft(!)h(turning)f(in)n(to)g(1)p Fr(=)p Ft(\()p Fr(s)2000 3356 y Fq(w)r(;i)2120 3344 y Ft(+)23 b(1\)!,)37 b(simply)e(b)n(y)g (noting)g(that)h(the)330 3484 y(same)29 b(argumen)n(t)f(as)h(ab)r(o)n (v)n(e)f(can)h(b)r(e)h(follo)n(w)n(ed)f(also)f(in)i(this)g(case)f(b)n (y)g(c)n(hanging)f(the)i(r^)-42 b(oles)28 b(of)330 3623 y(the)g(t)n(w)n(o)f(no)r(des)g Fr(u)911 3635 y Fl(0)976 3623 y Ft(and)g Fr(u)1185 3635 y Fl(1)1222 3623 y Ft(.)430 3763 y(Considering)f(the)i(argumen)n(t)e(in)i(pts.)37 b(1,)28 b(2,)f(3,)g(the)h(pro)r(of)f(is)h(complete.)p 3269 3763 4 57 v 3273 3710 50 4 v 3273 3763 V 3323 3763 4 57 v 330 3972 a Fj(R)l(emark)39 b Ft(4)p Fj(.)i Ft(If)f(one)f(w)n(an) n(ts)f(to)h(pro)n(v)n(e)f(the)i(formal)e(solubilit)n(y)h(of)g(the)h (equation)f(\(1.8\))g(for)330 4111 y Fr(!)26 b Fn(2)d Fh(R)p Ft(,)34 b(one)27 b(has)g(to)h(imp)r(ose)f(some)g(condition)g(on) h Fr(!)s Ft(.)36 b(The)28 b(w)n(eak)n(est)e(one)h(is)1517 4300 y(lim)1488 4350 y Fq(n)p Fo(!1)1685 4244 y Ft(log)14 b Fr(q)1843 4256 y Fq(n)p Fl(+1)p 1685 4281 288 4 v 1788 4357 a Fr(q)1825 4369 y Fq(n)2006 4300 y Ft(=)22 b(0)p Fr(;)1001 b Ft(\(3.7\))330 4492 y(where)35 b Fn(f)p Fr(q)657 4504 y Fq(n)702 4492 y Fn(g)g Ft(are)f(the)i(denominators)e(of)i(the)g (con)n(v)n(ergen)n(ts)d(arising)h(from)h(the)h(con)n(tin)n(uous)330 4632 y(fraction)29 b(expansion)f(for)h Fr(!)s Ft(.)42 b(Note)29 b(that)h(suc)n(h)f(a)g(condition)g(is)h(guaran)n(teed)d(b)n (y)i(the)h(Bryuno)330 4771 y(condition,)1521 4878 y Fo(1)1494 4903 y Fp(X)1492 5079 y Fq(n)p Fl(=0)1640 4925 y Ft(log)15 b Fr(q)1799 4937 y Fq(n)p Fl(+1)p 1640 4962 V 1743 5039 a Fr(q)1780 5051 y Fq(n)1961 4982 y Fr(<)23 b Fn(1)p Fr(;)1004 b Ft(\(3.8\))330 5202 y(under)32 b(whic)n(h)h(the)g(formal)f (series)f(de\014ning)i(the)g(conjugating)f(function)h(can)f(b)r(e)h (pro)n(v)n(ed)e(to)330 5341 y(con)n(v)n(erge)25 b(\(for)j Fr(")f Ft(small)g(enough\).)p eop %%Page: 12 12 12 11 bop 450 42 a Fl(12)702 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f (GUIDO)g(GENTILE)1030 241 y Ft(4.)46 b Fs(Estima)-6 b(tes)31 b(on)h(the)f(radius)g(of)h(conver)n(gence)550 430 y Ft(Let)25 b Fr(N)763 442 y Fq(n)808 430 y Ft(\()p Fr(#)p Ft(\),)h Fr(n)d Ft(=)g(0)p Fr(;)k Ft(1,)e(b)r(e)h(the)g(n)n(um)n(b)r(er)f(of)g (lines)g(of)g Fr(#)g Ft(on)g(scale)g Fr(n)g Ft(and)g(let)g Fr(N)3047 400 y Fo(\003)3038 451 y Fl(0)3085 430 y Ft(\()p Fr(#)p Ft(\))h(b)r(e)g(the)450 570 y(n)n(um)n(b)r(er)f(of)g(lines)f(of) h Fr(#)g Ft(on)g(scale)f(0)h(whic)n(h)g(are)f(not)h(resonan)n(t)e (lines,)i Fj(i.e.)k Ft(exiting)c(lines)g(of)g(an)n(y)450 709 y(resonance.)35 b(W)-7 b(e)28 b(then)g(ha)n(v)n(e)f(the)h(follo)n (wing)e(fundamen)n(tal)i(estimate.)450 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b(pro)r(of)f(go)r(es)g(m)n(uc)n(h)g(along)g(the)h(w)n(a)n(y)e (of)i(the)g(analogous)d(pro)r(of)i(in)h([1],)g(except)450 3837 y(that)d(no)n(w)f(w)n(e)h(ha)n(v)n(e)e(to)i(tak)n(e)f(in)n(to)h (accoun)n(t)f(that)h(the)g(p)r(erturbation)f(is)h(no)f(longer)g (necessarily)450 3977 y(o)r(dd)k(in)h(the)f(angle)f(v)-5 b(ariable)26 b Fr(\013)p Ft(,)h(so)e(one)h(cannot)f(use)h(symmetry)g (argumen)n(ts)f(but)i(rather)e(has)450 4116 y(to)h(enlarge)f(the)h (group)f(of)h(renormalization)e(transformations)g(to)i(include)h (transformations)450 4256 y(of)g(the)g(t)n(yp)r(e)h(used)f(to)f(sho)n (w)h(the)g(existence)g(of)g(the)g(formal)f(series)g(\(where)h(the)h (problem)e(w)n(as)450 4395 y(just)i(the)g(absence)f(of)h(parit)n(y)e (in)i(the)g(p)r(erturbation\).)550 4535 y(Lik)n(e)37 b(in)h(sect.)68 b(3,)40 b(to)e(b)r(etter)g(implemen)n(t)h(the)f (cancellation)f(mec)n(hanisms,)j(let)e(us)g(use)450 4674 y(top)r(ological)26 b(trees,)h(so)g(that)h(a)f(factor)1929 4806 y(1)p 1713 4843 475 4 v 1713 4919 a 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5481 y Ft(+)16 b Fr(:)e(:)g(:)i Ft(+)f Fr(s)1381 5493 y Fq(u;j)1470 5477 y Fk(0)1467 5510 y Fg(u)1507 5493 y Fl(+)p Fq(j)1588 5477 y Fk(0)q(0)1585 5510 y Fg(u)1660 5481 y Ft(is)27 b(the)g(n)n(um)n(b)r(er)f(of)g (non-equiv)-5 b(alen)n(t)26 b(subtrees)g(en)n(tering)p eop %%Page: 13 13 13 12 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)880 b(13)330 241 y Fr(u)32 b Ft(and)g(en)n(tering)g Fr(V)19 b Ft(;)35 b(in)d(fact)h(a)f(subtree)g(en)n(tering)g Fr(V)51 b Ft(cannot)32 b(b)r(e)h(equiv)-5 b(alen)n(t)32 b(to)g(a)g(subtree)330 380 y(in)n(ternal)27 b(to)g Fr(V)19 b Ft(.)430 520 y(This)27 b(allo)n(ws)f(us)i(to)f(rewrite)g(the)h (resonance)e(factor)h Fn(V)2167 532 y Fq(V)2224 520 y Ft(\()p Fr(#)p Ft(\))i(in)f(\(2.8\))f(as)1070 847 y Fn(V)1121 859 y Fq(V)1179 847 y Ft(\()p Fr(#)p Ft(\))c(=)1403 755 y Fp(h)1458 769 y(Y)1442 947 y Fq(u)p Fo(2)p Fq(V)1696 791 y Fr(f)1737 803 y Fq(\027)1770 811 y Fg(u)1814 791 y Fr(\027)1860 761 y Fq(m)1919 769 y Fg(u)1855 812 y Fq(u)p 1604 828 452 4 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Ft(!)14 b Fr(:)g(:)g(:)f(s)2335 1332 y Fq(u;j)2421 1340 y Fg(u)2475 1171 y Fp(ih)2600 1184 y(Y)2553 1366 y Fq(`)p Fo(2)p Fq(#)p Fo(n)p Fq(V)2869 1207 y Ft(1)p 2777 1244 225 4 v 2777 1320 a Fr(\015)5 b Ft(\()p Fr(\027)2898 1332 y Fq(`)2926 1340 y Fg(u)2969 1320 y Ft(\))3012 1171 y Fp(i)3051 1263 y Fr(:)85 b Ft(\(4.4\))430 1504 y(Giv)n(en)24 b(a)h(tree)f Fr(#)h Ft(and)g(a)f(resonance)f Fr(V)44 b Ft(with)25 b Fr(m)1923 1516 y Fq(V)2006 1504 y Ft(incoming)f(lines)h Fr(`)2580 1516 y Fl(1)2616 1504 y Ft(,)g Fr(:)14 b(:)g(:)28 b Ft(,)d Fr(`)2872 1516 y Fq(m)2931 1524 y Fg(V)3010 1504 y Ft(and)g(with)330 1643 y Fr(k)373 1655 y Fq(V)462 1643 y Ft(no)r(des,)31 b(w)n(e)g(de\014ne)g(the)g(family)g Fn(F)1554 1655 y Fq(V)1612 1643 y Ft(\()p Fr(#)p Ft(\))g(of)g Fr(V)50 b Ft(in)31 b Fr(#)g Ft(as)f(the)i(set)f(of)f(trees)h(obtained)f (from)330 1783 y Fr(#)e Ft(b)n(y)g(the)g(action)g(of)g(a)f(group)g(of)h (transformations)e Fn(P)2068 1795 y Fq(V)2153 1783 y Ft(on)i Fr(#)p Ft(,)g(generated)f(b)n(y)h(the)g(follo)n(wing)330 1922 y(op)r(erations.)430 2079 y(1.)41 b(F)-7 b(or)25 b(eac)n(h)g(en)n(tering)g(line,)i(detac)n(h)f(it)g(and)g(reattac)n(h)f (it)h(to)g(all)g(no)r(des)f(of)h(the)h(resonance.)430 2219 y(2.)41 b(W)-7 b(e)25 b(include)g(in)h(the)f(group)f Fn(P)1485 2231 y Fq(V)1567 2219 y Ft(transformations)f(lik)n(e)i(those) f(of)h(the)h(group)d Fn(G)31 b Ft(de\014ned)536 2358 y(in)d(sect.)36 b(3:)h(detac)n(h)27 b(the)h(exiting)f(line)h(of)g Fr(V)46 b Ft(and)28 b(reattac)n(h)e(it)i(to)g(all)f(no)r(des)g(of)h Fr(V)19 b Ft(.)430 2515 y(W)-7 b(e)37 b(shall)h(call)f(transformation)e (of)j(t)n(yp)r(e)g(1)f(and)g(2)g(the)h(op)r(erations)e(describ)r(ed)h (ab)r(o)n(v)n(e.)330 2654 y(W)-7 b(e)40 b(note)g(also)f(that)i(the)f (idea)g(of)g(including)g(the)h(transformations)d(of)i(t)n(yp)r(e)g(2)g (in)g(order)330 2794 y(to)f(implemen)n(t)h(the)f(cancellation)g(mec)n (hanism)f(comes)h(bac)n(k)f(to)h([16)o(],)j(in)e(the)f(con)n(text)g(of) 330 2933 y(con)n(tin)n(uous)27 b(Hamiltonian)g(systems.)430 3073 y(W)-7 b(e)32 b(then)g(ha)n(v)n(e)f(the)h(follo)n(wing)f(lemma,)h (where)g(the)g(cancellations)e(are)h(exhibited)h(for)f(a)330 3212 y(resonance,)25 b(and)h(its)g(corollary)-7 b(,)24 b(for)i(trees)f(whic)n(h)i(con)n(tain)e(more)g(than)i(one)e(resonance)g (\(as)h(it)330 3352 y(generally)g(happ)r(ens\).)330 3561 y Fx(Lemma)k(5.)40 b Fj(Given)34 b(a)g(tr)l(e)l(e)e Fr(#)h Fj(with)h(a)f(r)l(esonanc)l(e)h Fr(V)18 b Fj(,)35 b(let)e Fr(\027)2284 3573 y Fl(1)2321 3561 y Fj(,)h Fr(:)14 b(:)g(:)27 b Fj(,)34 b Fr(\027)2604 3573 y Fq(m)2663 3581 y Fg(V)2751 3561 y Fj(b)l(e)f(the)g(momenta)330 3701 y(\015owing)28 b(thr)l(ough)f(the)g(entering)g(lines)g Fr(`)1594 3713 y Fl(1)1631 3701 y Fj(,)g Fr(:)14 b(:)g(:)27 b Fj(,)h Fr(`)1895 3713 y Fq(m)1954 3721 y Fg(V)2035 3701 y Fj(of)g Fr(V)18 b Fj(;)29 b(then)d(we)h(have,)j(for)d(any)g Fr(\030)h(>)22 b(\016)k(>)330 3840 y Ft(0)p Fj(,)541 4007 y Ft(1)p 423 4044 278 4 v 423 4120 a Fn(jF)506 4132 y Fq(V)563 4120 y Ft(\()p Fr(#)p Ft(\))p Fn(j)710 3968 y Fp(\014)710 4018 y(\014)710 4067 y(\014)826 3984 y(X)738 4166 y Fq(#)778 4150 y Fk(0)800 4166 y Fo(2F)893 4174 y Fg(V)943 4166 y Fl(\()p Fq(#)p Fl(\))1048 4063 y Fn(V)1099 4075 y Fq(V)1157 4063 y Ft(\()p Fr(#)1238 4029 y Fo(0)1262 4063 y Ft(\))1294 3968 y Fp(\014)1294 4018 y(\014)1294 4067 y(\014)1345 4063 y Fn(\024)768 4370 y(\024)c Fr(q)895 4336 y Fl(2\()p Fq(k)989 4344 y Fg(V)1040 4336 y Fl(+1\))1154 4370 y Fr(D)1223 4382 y Fl(0)1260 4370 y Fr(C)1325 4333 y Fq(k)1360 4341 y Fg(V)1319 4392 y Fl(1)1415 4370 y Fr(D)1486 4333 y Fq(k)1521 4341 y Fg(V)1571 4333 y Fo(\000)p Fl(1+)p Fq(m)1766 4341 y Fg(V)1484 4392 y Fl(1)1820 4370 y Ft(\()p Fr(\016)s Ft(\))p Fr(k)1970 4336 y Fl(2)1967 4391 y Fq(V)2026 4370 y Fr(e)2065 4336 y Fo(\000)p Fl(\()p Fq(\030)r Fo(\000)p Fq(\016)r Fl(\))p Fo(j)p Fq(\027)2338 4344 y Fg(V)2387 4336 y Fo(j)2493 4266 y Fq(m)2552 4274 y Fg(V)2487 4291 y Fp(X)2425 4470 y Fq(m;m)2563 4453 y Fk(0)2585 4470 y Fl(=1)2683 4370 y Fn(j)p Fr(\027)2747 4382 y Fq(m)2810 4370 y Fr(\027)2851 4382 y Fq(m)2910 4366 y Fk(0)2937 4370 y Fr(\020)2979 4336 y Fl(2)3016 4370 y Fn(j)p Fr(;)97 b Ft(\(4.5\))330 4614 y Fj(for)31 b(some)f(c)l(onstant)f Fr(D)1075 4626 y Fl(0)1141 4614 y Fj(and)i(with)f Fr(D)1552 4626 y Fl(1)1589 4614 y Ft(\()p Fr(\016)s Ft(\))g Fj(diver)l(ging)i(as) e Fr(\016)2223 4584 y Fo(\000)p Fl(1)2342 4614 y Fj(for)g Fr(\016)c Fn(!)d Ft(0)2685 4584 y Fl(+)2740 4614 y Fj(.)430 4823 y Ft(The)j(pro)r(of)g(of)g(this)h(lemma)f(is)h(fairly)e(standard)h (but)h(rather)e(tec)n(hnical,)h(so)g(w)n(e)g(defer)g(it)h(to)330 4963 y(the)h(next)g(section.)330 5172 y Fx(Corollary)-8 b(.)41 b Fj(F)-6 b(or)36 b(a)h(tr)l(e)l(e)e Fr(#)g Fn(2)f(T)1407 5184 y Fq(\027:k)1538 5172 y Fj(with)i(r)l(esonanc)l(es)g Fr(V)2195 5184 y Fl(1)2233 5172 y Fj(,)h Fr(:)14 b(:)g(:)27 b Fj(,)38 b Fr(V)2530 5184 y Fq(s)2566 5172 y Fj(,)g(c)l(onsider)f(the) f(family)330 5311 y Fn(F)8 b Ft(\()p Fr(#)p Ft(\))36 b Fj(obtaine)l(d)g(by)f(the)h(action)f(on)g Fr(#)h Fj(of)f(al)t(l)i (the)e(gr)l(oups)g Fn(P)2299 5323 y Fq(V)2338 5331 y Fi(1)2375 5311 y Fj(,)g Fr(:)14 b(:)g(:)28 b Fj(,)36 b Fn(P)2679 5323 y Fq(V)2718 5331 y Fg(s)2754 5311 y Fj(;)j(the)c(numb)l(er)f(of)p eop %%Page: 14 14 14 13 bop 450 42 a Fl(14)702 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f (GUIDO)g(GENTILE)450 241 y Fj(tr)l(e)l(es)g(in)h(this)g(family)i(is)e (then)f(given)i(by)1581 451 y Fr(k)1624 466 y Fo(F)6 b Fl(\()p Fq(#)p Fl(\))1797 451 y Ft(=)1923 347 y Fq(s)1886 372 y Fp(Y)1885 549 y Fq(i)p Fl(=1)2006 451 y Fn(jF)2089 463 y Fq(V)2128 471 y Fg(i)2159 451 y Ft(\()p Fr(#)p Ft(\))p Fn(j)p Fr(;)450 672 y Fj(and)30 b(we)g(have)h(the)f(estimate:) 636 804 y Ft(1)p 560 841 193 4 v 560 917 a Fr(k)603 932 y Fo(F)6 b Fl(\()p Fq(#)p Fl(\))763 765 y Fp(\014)763 815 y(\014)763 865 y(\014)857 782 y(X)791 964 y Fq(#)831 947 y Fk(0)853 964 y Fo(2F)g Fl(\()p Fq(#)p Fl(\))1057 860 y Ft(V)-7 b(al\()p Fr(#)1258 826 y Fo(0)1282 860 y Ft(\))1314 765 y Fp(\014)1314 815 y(\014)1314 865 y(\014)1365 860 y Fn(\024)23 b Fr(q)1493 826 y Fl(2)p Fq(N)1579 834 y Fi(1)1611 826 y Fl(\()p Fq(#)p Fl(\))1707 860 y Fn(j)p Fr(\020)6 b Fn(j)1795 826 y Fo(\000)p Fl(2)p Fq(N)1933 834 y Fi(0)1965 826 y Fl(\()p Fq(#)p Fl(\))2061 860 y Fr(q)2101 826 y Fl(2)p Fq(s)2170 860 y Fr(D)2241 826 y Fq(s)2239 881 y Fl(0)2277 860 y Fr(C)2342 826 y Fq(k)2336 881 y Fl(1)2383 860 y Fr(D)2454 826 y Fq(k)2452 881 y Fl(1)2495 860 y Ft(\()p Fr(\016)s Ft(\))p Fr(k)2645 826 y Fl(4)2682 860 y Fn(j)p Fr(\020)g Fn(j)2770 826 y Fl(2)p Fq(s)2839 860 y Fr(e)2878 826 y Fo(\000)p Fl(\()p Fq(\030)r Fo(\000)p Fq(\016)r Fl(\))p Fo(j)p Fq(\027)t Fo(j)3279 860 y Ft(\(4.6\))450 1106 y Fj(wher)l(e)30 b Fr(D)753 1118 y Fl(0)820 1106 y Fj(and)g Fr(D)1050 1118 y Fl(1)1087 1106 y Ft(\()p Fr(\016)s Ft(\))h Fj(ar)l(e)f(the)g(same)g(c)l(onstants) f(as)h(in)g(lemma)g(5.)450 1315 y(Pr)l(o)l(of.)43 b Ft(Note)26 b(that)h(this)f(is)g(p)r(ossible)g(since)f(the)i(mec)n(hanisms)e (generating)g(the)h(cancellations)450 1454 y(for)g(eac)n(h)g(resonance) f Fr(V)1188 1466 y Fq(i)1216 1454 y Ft(,)i Fr(i)c Ft(=)g(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(s)p Ft(,)27 b(do)f(not)h(in)n(terfere)f(with) h(eac)n(h)f(other.)36 b(Note)27 b(also)f(that)450 1594 y(in)36 b(our)f(case,)i(as)e(w)n(ell)g(as)g(in)h([1],)i(no)d(com)n (binatorial)f(problems)h(arise)g(due)g(to)h(resonances)450 1733 y(con)n(tained)d(inside)g(eac)n(h)f(other,)i(as)f(there)g(are)f (only)h(t)n(w)n(o)g(scales:)47 b(that)33 b(is,)i(scales)d(are)g(used) 450 1873 y(just)25 b(to)f(accoun)n(t)f(for)h(singular)f(\(div)n(ergen)n (t\))h(and)g(regular)f(propagators.)32 b(So)24 b(w)n(e)g(ha)n(v)n(e)f (just)i(to)450 2012 y(use)g(lemma)h(5)f(for)g(eac)n(h)g(resonance)e Fr(V)1658 2024 y Fq(i)1686 2012 y Ft(:)36 b(see)26 b([1)o(])g(for)f (more)f(details,)i(as)f(the)h(pro)r(of)f(is)g(iden)n(tical)450 2152 y(in)41 b(that)f(case.)74 b(A)41 b(minor)e(observ)-5 b(ation)39 b(is)h(that,)k(if)d Fx(V)h Ft(denotes)e(the)g(set)h(of)f (resonances,)450 2291 y Fx(V)25 b Ft(=)d Fn(f)p Fr(V)724 2303 y Fl(1)761 2291 y Fr(;)14 b(:)g(:)g(:)28 b(;)14 b(V)1008 2303 y Fq(s)1044 2291 y Fn(g)p Ft(,)27 b(then)1330 2401 y Fp(X)1312 2579 y Fq(V)14 b Fo(2)p Fd(V)1481 2480 y Ft(\()q Fr(k)1557 2492 y Fq(V)1633 2480 y Fn(\000)k Ft(1)g(+)g Fr(m)1932 2492 y Fq(V)1989 2480 y Ft(\))h(+)2134 2401 y Fp(X)2123 2583 y Fq(u)7 b(=)-41 b Fo(2)p Fd(V)2278 2480 y Fr(m)2351 2492 y Fq(u)2418 2480 y Ft(=)22 b Fr(k)s(;)705 b Ft(\(4.7\))450 2715 y(where)27 b Fr(u)32 b(=)-51 b Fn(2)23 b Fx(V)29 b Ft(means)e Fr(u)32 b(=)-51 b Fn(2)23 b Fr(V)1391 2727 y Fq(j)1454 2715 y Fn(8)p Fr(j)28 b Ft(=)22 b(1)p Fr(;)14 b(:)g(:)g(:)27 b(;)14 b(s)p Ft(.)p 3389 2715 4 57 v 3393 2662 50 4 v 3393 2715 V 3443 2715 4 57 v 550 2904 a(Next,)32 b(w)n(e)f(estimate)h Fn(j)p Fr(u)1314 2874 y Fl(\()p Fq(k)q Fl(\))1406 2904 y Ft(\()p Fr(\013;)14 b(!)s Ft(\))p Fn(j)p Ft(.)49 b(Let)32 b Fr(N)1939 2874 y Fq(R)1930 2925 y Fl(0)1993 2904 y Ft(\()p Fr(#)p Ft(\))e(=)f Fr(N)2297 2916 y Fl(0)2334 2904 y Ft(\()p Fr(#)p Ft(\))22 b Fn(\000)e Fr(N)2630 2874 y Fo(\003)2621 2925 y Fl(0)2668 2904 y Ft(\()p Fr(#)p Ft(\))32 b(b)r(e)g(the)g(n)n(um) n(b)r(er)f(of)450 3044 y(resonan)n(t)24 b(lines)h(in)h(a)f(giv)n(en)g (tree)g Fr(#)h Ft(\(as)f(eac)n(h)f(resonance)g(has)h Fj(one)32 b Ft(exiting)25 b(line,)h(the)g(n)n(um)n(b)r(er)450 3183 y(of)i(resonances)e(is)i(the)h(same)e(as)h Fr(N)1572 3153 y Fq(R)1563 3204 y Fl(0)1626 3183 y Ft(\()p Fr(#)p Ft(\),)h(of)f(course\).)38 b(By)28 b(the)g(corollary)e(to)i(lemma)g(5,) g(with)450 3322 y Fr(s)23 b Ft(=)g Fr(N)676 3292 y Fo(\003)667 3343 y Fl(0)714 3322 y Ft(\()p Fr(#)p Ft(\),)28 b(w)n(e)f(obtain)619 3489 y(1)p 543 3527 193 4 v 543 3603 a Fr(k)586 3618 y Fo(F)6 b Fl(\()p Fq(#)p Fl(\))746 3450 y Fp(\014)746 3500 y(\014)746 3550 y(\014)840 3467 y(X)773 3649 y Fq(#)813 3632 y Fk(0)836 3649 y Fo(2F)g Fl(\()p Fq(#)p Fl(\))1040 3546 y Ft(V)-7 b(al\()p Fr(#)1241 3511 y Fo(0)1265 3546 y Ft(\))1297 3450 y Fp(\014)1297 3500 y(\014)1297 3550 y(\014)1348 3546 y Fn(\024)803 3813 y(\024)22 b 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b(al\()p Fr(#)2651 4475 y Fo(0)2675 4509 y Ft(\))p Fr(;)450 4750 y Ft(from)27 b(whic)n(h)h(it)g(easily)f(follo)n(ws)f(that)1278 4938 y Fn(j)p Fr(u)1349 4904 y Fl(\()p Fq(k)q Fl(\))1349 4958 y Fq(\027)1441 4938 y Fn(j)d(\024)g Fr(B)1642 4904 y Fq(k)1638 4958 y Fl(0)1683 4938 y Fr(C)1748 4904 y Fq(k)1742 4958 y Fl(1)1789 4938 y Fr(D)1860 4904 y Fq(k)1858 4958 y Fl(1)1901 4938 y Ft(\()p Fr(\016)s Ft(\))p Fn(j)p Fr(\020)6 b Fn(j)2093 4904 y Fo(\000)p Fl(2)p Fo(b)p Fq(k)q(=r)2311 4879 y Fk(\003)2347 4904 y Fo(c)2382 4938 y Fr(e)2421 4904 y Fo(\000)p Fq(\030)r Fo(j)p Fq(\027)t Fo(j)2585 4938 y Fr(;)671 b Ft(\(4.9\))450 5126 y(for)29 b(some)h(constan)n(t)f Fr(B)1190 5138 y Fl(0)1227 5126 y Ft(,)i(as)e(the)i(n)n(um)n(b)r(er)e(of)h(trees)g(of)g(order)e Fr(k)33 b Ft(is)d(b)r(ounded)h(b)n(y)e(a)h(constan)n(t)450 5266 y(to)d(the)h(p)r(o)n(w)n(er)f Fr(k)s Ft(.)p eop %%Page: 15 15 15 14 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)880 b(15)430 241 y Ft(So)33 b(from)f(\(4.9\))h(one)g(deduces)g (that,)i(for)e(an)n(y)f Fr(\016)k(>)c Ft(0,)i Fr(u)2251 211 y Fl(\()p Fq(k)q Fl(\))2343 241 y Ft(\()p Fr(\013;)14 b(!)s Ft(\))34 b(is)f(analytic)g(in)g Fr(\013)h Ft(in)f(a)330 380 y(strip)27 b(of)h(width)g Fr(\030)23 b Fn(\000)18 b Fr(\016)31 b Ft(around)26 b(the)i(real)f(axis)g(and)g(satis\014es)g (the)h(b)r(ound)1043 569 y Fn(k)p Fr(u)1133 534 y Fl(\()p Fq(k)q Fl(\))1224 569 y Ft(\()p Fn(\001)p Fr(;)14 b(!)s Ft(\))p Fn(k)1445 581 y Fq(\030)r Fo(\000)p Fq(\016)1589 569 y Fn(\024)22 b Fr(B)1739 581 y Fl(1)1777 569 y Ft(\()p Fr(\016)s Ft(\))p Fr(C)1946 534 y Fq(k)1940 589 y Fl(1)1987 569 y Fr(D)2058 534 y Fq(k)2056 589 y Fl(1)2099 569 y Ft(\()p Fr(\016)s Ft(\))p Fn(j)p Fr(\020)6 b Fn(j)2291 534 y Fo(\000)p Fl(2)p Fo(b)p Fq(k)q(=r)2509 509 y Fk(\003)2546 534 y Fo(c)2580 569 y Fr(;)515 b Ft(\(4.10\))330 757 y(for)32 b(some)g(constan)n(t)g Fr(B)1078 769 y Fl(1)1115 757 y Ft(\()p Fr(\016)s Ft(\),)i(div)n(erging)d(as)h Fr(\016)1788 727 y Fo(\000)p Fl(1)1910 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y(b)r(e)f(ac)n(hiev)n(ed)e(in)i(sect.)42 b(6,)30 b(where)f(it)h(will)g (b)r(e)g(sho)n(wn)e(that)i(the)g(limit)g(\(1.18\))f(is)g(indep)r(enden) n(t)330 1594 y(on)d(the)g(non-tangen)n(tial)e(path)i(c)n(hosen)f(in)h (the)h(complex)e Fr(!)k Ft(plane)d(and)f(where)h(this)g(limit)g(can)330 1733 y(b)r(e)j(explicitly)g(computed)h(\(at)f(least)f(in)i(principle\)) f(as)f(solution)g(of)h(a)g(di\013eren)n(tial)g(equation,)330 1873 y(pro)n(ving)d(that)i(it)g(is)f(actually)g(a)h(function)g(of)f Fr(")1820 1843 y Fq(r)1853 1817 y Fk(\003)1919 1873 y Ft(and)h Fr(q)s(\013)p Ft(.)1102 2082 y(5.)46 b Fs(Renormaliza)-6 b(tion)31 b(of)h(resonances)430 2271 y Ft(F)-7 b(or)27 b(eac)n(h)f(resonance)g Fr(V)47 b Ft(in)28 b Fr(#)f Ft(and)h(for)f(eac) n(h)g(no)r(de)g Fr(u)c Fn(2)g Fr(V)c Ft(,)28 b(write)1396 2459 y Fr(\027)1437 2471 y Fq(`)1465 2479 y Fg(u)1531 2459 y Ft(=)23 b Fr(\027)1665 2425 y Fl(0)1660 2480 y Fq(`)1688 2488 y Fg(u)1750 2459 y Ft(+)1915 2381 y Fp(X)1833 2563 y Fq(`)1861 2546 y Fk(0)1883 2563 y 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5352 y Ft(\))g(to)f(a)g(no)r(de)h(in)g Fr(V)1871 5364 y Fl(1)1908 5352 y Ft(\()p Fr(`)1975 5364 y Fq(u)2019 5352 y Ft(\).)p eop %%Page: 16 16 16 15 bop 450 42 a Fl(16)702 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f (GUIDO)g(GENTILE)550 241 y Ft(T)-7 b(o)31 b(a)n(v)n(oid)f(confusion,)j (as)e(in)h(sect.)49 b(3,)32 b(in)g(the)g(follo)n(wing)f(discussion,)h (for)f(an)n(y)g(tree)g Fr(#)3341 211 y Fo(0)3395 241 y Fn(2)450 380 y(F)510 392 y Fq(V)567 380 y Ft(\()p Fr(#)p Ft(\))k(w)n(e)e(use)g(the)h(partial)f(ordering)f(relations)g(de\014ned) i(in)g Fr(#)g Ft(\(whic)n(h)g(of)f(course)f(can)i(b)r(e)450 520 y(di\013eren)n(t)28 b(from)f(those)g(of)h Fr(#)1335 490 y Fo(0)1358 520 y Ft(\).)550 659 y(It)21 b(is)f(easy)g(to)g(reason) g(as)g(in)g(sect.)35 b(3)20 b(to)h(conclude)f(that)h(the)g(action)f(of) h(the)g(transformations)450 799 y(of)f(the)g(group)f Fn(P)958 811 y Fq(V)1035 799 y Ft(do)r(es)h(not)f(mo)r(dify)i(the)f (com)n(binatorial)e(factors)h(\(4.2\))g(if)i(eac)n(h)e(tree)g(obtained) 450 938 y(through)28 b(a)g(transformation)f(is)i(coun)n(ted)f(once)h (and)f(only)g(once:)39 b(simply)29 b(rep)r(eat)f(iterativ)n(ely)450 1078 y(the)g(same)f(argumen)n(t)g(as)g(in)h(sect.)38 b(3,)27 b(b)n(y)h(noting)g(that)g(eac)n(h)f(transformation)f(in)i Fn(P)3127 1090 y Fq(V)3212 1078 y Ft(can)g(b)r(e)450 1217 y(obtained)h(b)n(y)g(com)n(bining)g(a)g(\014nite)h(n)n(um)n(b)r (er)f(of)h(elemen)n(tary)e(op)r(erations)g(consisting)h(just)h(in)450 1357 y(shifting)d(a)g(single)f(line.)37 b(So)26 b(in)h(the)g(follo)n (wing)f(analysis)g(the)h(com)n(binatorial)e(factors)g(pla)n(y)h(no)450 1496 y(r^)-42 b(ole)27 b(as)g(they)g(are)g(in)n(v)-5 b(arian)n(t)26 b(under)i(the)g(action)f(of)h(the)g(group)e Fn(P)2546 1508 y Fq(V)2603 1496 y Ft(.)550 1636 y(Then)h(for)g(an)n(y)g (line)h Fr(`)23 b Fn(2)g Fr(L)p Ft(\()p Fr(V)c Ft(\),)28 b(write)1398 1824 y Fr(!)s(\027)1494 1836 y Fq(`)1549 1824 y Ft(=)1646 1768 y Fr(p)p 1646 1805 42 4 v 1647 1881 a(q)1698 1824 y(\027)1739 1836 y Fq(`)1789 1824 y Ft(+)18 b Fr(\026)1922 1836 y Fq(`)1954 1824 y Fr(;)180 b(\026)2207 1836 y Fq(`)2262 1824 y Fn(\021)23 b Fr(\020)6 b(\027)2433 1836 y Fq(`)2465 1824 y Fr(;)791 b Ft(\(5.5\))450 2016 y(and,)32 b(for)f(an)n(y)g(tree)g Fr(#)1150 1986 y Fo(0)1203 2016 y Fn(2)f(F)1348 2028 y Fq(V)1405 2016 y Ft(\()p Fr(#)p Ft(\),)j(consider)e(the)h(resonance)d(factor)i (\(4.3\))g(as)g(a)g(function)h(of)450 2156 y(the)c(quan)n(tities)f Fr(\026)1026 2168 y Fq(`)1054 2176 y Fi(1)1091 2156 y Fr(;)14 b(:)g(:)g(:)27 b(;)14 b(\026)1339 2168 y Fq(`)1367 2176 y Fg(m)1418 2190 y(V)1476 2156 y Ft(:)1369 2344 y Fn(V)1420 2356 y Fq(V)1478 2344 y Ft(\()p Fr(#)1559 2310 y Fo(0)1582 2344 y Ft(\))24 b Fn(\021)e(V)1776 2356 y Fq(V)1834 2344 y Ft(\()p Fr(#)1915 2310 y Fo(0)1939 2344 y Ft(;)14 b Fr(\026)2026 2356 y Fq(`)2054 2364 y Fi(1)2090 2344 y Fr(;)g(:)g(:)g(:)27 b(;)14 b(\026)2338 2356 y Fq(`)2366 2364 y Fg(m)2417 2378 y(V)2475 2344 y Ft(\))p Fr(:)749 b Ft(\(5.6\))450 2533 y(As)28 b(in)g([1)o(])g(w)n(e) f(can)g(write)533 2756 y 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y(0)1651 3282 y Ft(d)p Fr(t)c Ft(\(1)k Fn(\000)g Fr(t)p Ft(\))2105 3226 y Fr(@)2154 3195 y Fl(2)p 1989 3263 319 4 v 1989 3339 a Fr(@)5 b(\026)2088 3351 y Fq(`)2116 3359 y Fg(i)2146 3339 y Fr(@)g(\026)2245 3351 y Fq(`)2273 3359 y Fg(j)2317 3282 y Fn(V)2368 3294 y Fq(V)2426 3282 y Ft(\()p Fr(#)2507 3247 y Fo(0)2530 3282 y Ft(;)14 b Fr(t\026)2647 3294 y Fq(`)2675 3302 y Fi(1)2712 3282 y Fr(;)g(:)g(:)g(:)27 b(;)14 b(t\026)2990 3294 y Fq(`)3018 3302 y Fg(m)3069 3316 y(V)3127 3282 y Ft(\))p Fr(;)97 b Ft(\(5.7\))450 3519 y(with)28 b(the)g(same)f(notations)g(as)g(in)h([1)o(])g(to)g (denote)f(the)h(deriv)-5 b(ativ)n(es.)550 3658 y(No)n(w)27 b(w)n(e)g(pass)g(to)g(the)h(pro)r(of)f(of)h(lemma)f(5.)450 3847 y Fj(Pr)l(o)l(of.)43 b Ft(W)-7 b(e)28 b(can)f(reason)f(as)h(in)h (sect.)37 b(3)27 b(to)h(deduce)f(that)1373 3957 y Fp(X)1284 4139 y Fq(#)1324 4122 y Fk(0)1347 4139 y Fo(2F)1440 4147 y Fg(V)1489 4139 y Fl(\()p Fq(#)p Fl(\))1595 4036 y Fn(S)1645 4048 y Fq(V)1703 4036 y Ft(\()p Fr(#)1784 4001 y 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5027 y Ft(\))1742 4910 y Fp(\024)1852 4971 y Fr(@)p 1796 5008 162 4 v 1796 5084 a(@)5 b(\026)1895 5096 y Fq(`)1923 5104 y Fg(j)1968 5027 y Fn(V)2019 5039 y Fq(V)2076 5027 y Ft(\()p Fr(#)2157 4993 y Fo(0)2181 5027 y Ft(;)14 b(0)p Fr(;)g(:)g(:)g(:)27 b(;)14 b Ft(0\))2532 4910 y Fp(\025)2598 5027 y Ft(=)23 b(0)551 b(\(5.9\))450 5268 y(for)27 b(an)n(y)g Fr(j)h Ft(=)23 b(1)p Fr(;)14 b(:)g(:)g(:)26 b(;)14 b(m)1196 5280 y Fq(V)1254 5268 y Ft(.)37 b(First)27 b(note)h(that)p eop %%Page: 17 17 17 16 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)880 b(17)479 215 y Fr(@)p 423 252 162 4 v 423 329 a(@)5 b(\026)522 341 y Fq(`)550 349 y Fg(j)595 272 y Fn(V)646 284 y Fq(V)703 272 y Ft(\()p Fr(#)784 237 y Fo(0)808 272 y Ft(;)14 b(0)p Fr(;)g(:)g(:)g(:)27 b(;)14 b Ft(0\))22 b(=)1005 546 y(=)1096 467 y Fp(X)1092 646 y Fq(`)p Fo(2)p Fq(V)1218 404 y Fp(\(")1400 490 y Fr(@)p 1344 527 V 1344 603 a(@)5 b(\026)1443 615 y Fq(`)1471 623 y Fg(j)1515 546 y Fr(g)s Ft(\()p Fr(\027)1636 512 y Fl(0)1631 566 y Fq(`)1674 546 y Ft(\))1706 404 y Fp(#)q(")1829 467 y(Y)1803 653 y Fq(`)1831 628 y Fk(0)1853 653 y Fo(2)p Fq(V)t(;)1817 721 y(`)1845 696 y Fk(0)1867 721 y Fo(6)p Fl(=)p Fq(`)1974 546 y Fr(g)s Ft(\()p Fr(\027)2090 558 y Fq(`)2118 542 y Fk(0)2145 546 y Ft(\))2177 404 y Fp(#")2289 467 y(Y)2274 645 y Fq(u)p Fo(2)p Fq(V)2528 490 y Fr(f)2569 502 y Fq(\027)2602 510 y Fg(u)2645 490 y Fr(\027)2691 460 y Fq(m)2750 468 y Fg(u)2686 510 y Fq(u)p 2435 527 452 4 v 2435 603 a Fr(s)2474 615 y Fq(u;)p Fl(1)2570 603 y Ft(!)14 b Fr(:)g(:)g(:)g(s)2757 615 y Fq(u;j)2843 623 y Fg(u)2896 404 y Fp(#)q(\))3012 546 y Fr(:)83 b Ft(\(5.10\))430 860 y(Then)20 b(tak)n(e)f(the)h(term)g(in)g(\(5.10\))f (in)h(whic)n(h)g(the)g(deriv)-5 b(ativ)n(e)19 b(acts)g(on)h(the)g(line) g Fr(`)j Fn(\021)f Fr(`)3020 872 y Fq(u)3083 860 y Ft(exiting)330 1000 y(the)30 b(no)r(de)g Fr(u)d Fn(2)g Fr(V)19 b Ft(.)44 b(Consider)29 b(together)g(all)h(the)g(trees)f Fr(#)2161 970 y Fo(0)2212 1000 y Fn(2)e(F)2354 1012 y Fq(V)2411 1000 y 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Ft(obtained)g(in)h(this)g(w)n(a)n(y)e(has)h(a)g (v)-5 b(alue)32 b(giv)n(en)e(b)n(y)i(a)f(common)g(factor)f({)h(call)330 2584 y(it)d(\002\()p Fr(#)559 2554 y Fo(0)582 2584 y Ft(\))23 b Fn(\021)g Ft(\002\()p Fr(#)p Ft(\))28 b({)f(times)h(a)f (factor)f(whic)n(h,)i(on)f(the)h(con)n(trary)-7 b(,)25 b(dep)r(ends)j(on)g(the)f(particular)330 2724 y(transformations)f(whic) n(h)h(ha)n(v)n(e)g(b)r(een)h(p)r(erformed)f({)g(call)g(it)h(\012\()p Fr(#)2396 2694 y Fo(0)2420 2724 y Ft(\))g({.)430 2863 y(The)g(factor)f(\012\()p Fr(#)980 2833 y Fo(0)1004 2863 y Ft(\))i(con)n(tains)e(the)i(deriv)n(ed)e(propagator)e(in)k(\(5.10\))e (times)i(a)f(factor)f(whic)n(h)330 3003 y(is)i(giv)n(en)f(as)h(follo)n (ws.)40 b(Either)29 b(the)h(op)r(eration)e(1)h(pro)r(duces)f(a)h (factor)f Fr(\027)2593 3015 y Fq(w)2641 3023 y Fi(0)2678 3003 y Ft(,)h Fr(w)2789 3015 y Fl(0)2853 3003 y Fn(2)d Fr(V)2982 3015 y Fl(2)3019 3003 y Ft(\()p Fr(`)3086 3015 y Fq(u)3130 3003 y Ft(\),)k(the)330 3142 y(op)r(eration)36 b(2)g(pro)r(duces)g(a)g(factor)g Fr(\027)1513 3154 y Fq(w)1561 3162 y Fg(j)1596 3142 y Ft(,)j Fr(w)1717 3154 y Fq(j)1791 3142 y Fn(2)f Fr(V)1932 3154 y Fl(1)1970 3142 y Ft(\()p Fr(`)2037 3154 y Fq(u)2080 3142 y Ft(\))f(and)g(the)g (op)r(eration)e(3)i(pro)r(duces)e(a)330 3282 y(factor)28 b Fr(\027)610 3294 y Fq(w)658 3302 y Fg(i)688 3282 y Ft(,)h Fr(w)799 3294 y Fq(i)852 3282 y Fn(2)c Fr(V)48 b Ft(for)28 b(eac)n(h)g(en)n(tering)f(line)i Fr(`)1858 3294 y Fq(i)1914 3282 y Ft(\(case)f(1\),)h(or)f(the)h(op)r(eration)e(1) i(pro)r(duces)e(a)330 3421 y(factor)d Fr(\027)606 3433 y Fq(w)654 3441 y Fi(0)690 3421 y Ft(,)i Fr(w)798 3433 y Fl(0)858 3421 y Fn(2)e Fr(V)985 3433 y Fl(1)1023 3421 y Ft(\()p Fr(`)1090 3433 y Fq(u)1133 3421 y Ft(\),)h(the)g(op)r (eration)f(2)g(pro)r(duces)g(a)g(factor)g Fr(\027)2476 3433 y Fq(w)2524 3441 y Fg(j)2559 3421 y Ft(,)h Fr(w)2666 3433 y Fq(j)2725 3421 y Fn(2)e Fr(V)2851 3433 y Fl(2)2889 3421 y Ft(\()p Fr(`)2956 3433 y Fq(u)2999 3421 y Ft(\))i(and)g(the)330 3561 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Fr(\027)2365 5009 y Fq(m)2424 5017 y Fg(V)2475 5009 y Fo(\000)p Fl(1)2360 5071 y Fq(V)2564 4954 y Fp(h)2669 4990 y Fr(@)p 2613 5027 V 2613 5103 a(@)5 b(\026)2712 5115 y Fq(`)2740 5123 y Fg(j)2784 5046 y Fr(g)s Ft(\()p Fr(\027)2905 5012 y Fl(0)2900 5067 y Fq(`)2928 5075 y Fg(j)2963 5046 y Ft(\))2995 4954 y Fp(i)3118 5046 y Ft(\(5.13\))330 5251 y(for)38 b(case)g(2.)71 b(W)-7 b(e)39 b(used)g(that)g(in)g(the)h (second)e(sum)h(\(5.13\))f(the)h(deriv)n(ed)g(propagator)d(has)330 5391 y(c)n(hanged)29 b(its)h(sign:)41 b(the)30 b(propagator)d(is)i (indeed)i(ev)n(en)e(\(so)g(that)h(its)g(deriv)-5 b(ativ)n(e)29 b(is)h(o)r(dd\))g(and)p eop %%Page: 18 18 18 17 bop 450 42 a Fl(18)702 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f (GUIDO)g(GENTILE)450 241 y Ft(the)h(arro)n(w)e(sup)r(erimp)r(osed)i(to) g(the)g(line)g Fr(`)1799 253 y Fq(u)1872 241 y Ft(has)f(a)h(di\013eren) n(t)g(direction)g(in)g(the)g(sets)g(of)g(trees)450 380 y(corresp)r(onding)c(to)h(the)h(t)n(w)n(o)f(sums)g(ab)r(o)n(v)n(e.)550 520 y(Then)g(the)h(sum)g(of)34 b(\(5.12\))27 b(and)g(\(5.13\))g(giv)n (es)g(zero,)f(hence)i(implies)g(\(5.9\))o(.)550 659 y(In)38 b(order)f(to)g(conclude)h(the)h(pro)r(of)e(of)h(lemma)g(5,)i(it)f(is)f (enough)f(to)h(use)g(the)g(follo)n(wing)450 799 y(prop)r(erties.)550 938 y(By)32 b(using)i(\(1.3\),)g(\(2.5\))f(and)g(the)h(fact)f(that)g Fr(x)2058 908 y Fq(m)2154 938 y Fn(\024)f Fr(m)p Ft(!)p Fr(\032)2390 908 y Fo(\000)p Fq(m)2504 938 y Fr(e)2543 908 y Fq(\032x)2653 938 y Ft(for)g(all)h Fr(x;)14 b(\032)32 b Fn(2)h Fh(R)3207 950 y Fl(+)3268 938 y Ft(,)h(one)450 1078 y(\014nds)1058 1174 y Fp(\020)1123 1187 y(Y)1108 1365 y Fq(u)p Fo(2)p Fq(V)1259 1266 y Fn(j)p Fr(f)1323 1278 y Fq(\027)1356 1286 y Fg(u)1399 1266 y Fn(j)1446 1210 y(j)p Fr(\027)1510 1222 y Fq(u)1554 1210 y Fn(j)1577 1180 y Fq(m)1636 1188 y Fg(u)p 1446 1247 233 4 v 1493 1323 a Fr(m)1566 1335 y Fq(u)1609 1323 y Ft(!)1689 1174 y Fp(\021)1762 1266 y Fn(\024)22 b Fr(C)1914 1232 y Fq(k)1908 1286 y Fl(1)1955 1266 y Fr(D)2026 1229 y Fq(k)2061 1237 y Fg(V)2112 1229 y Fo(\000)p Fl(1+)p Fq(m)2307 1237 y Fg(V)2024 1288 y Fl(1)2361 1266 y Ft(\()p Fr(\016)s Ft(\))p Fr(e)2504 1232 y Fo(\000)p Fl(\()p Fq(\030)r Fo(\000)p Fq(\016)r Fl(\))p Fo(j)p Fq(\027)t Fo(j)2805 1266 y Fr(;)410 b Ft(\(5.14\))450 1497 y(for)27 b(an)n(y)g Fr(\030)g(>)c(\016)j(>)c Ft(0)28 b(and)f(some)g(constan)n(t)g Fr(D)1878 1509 y Fl(1)1915 1497 y Ft(\()p Fr(\016)s Ft(\))h(suc)n(h)f(that)1537 1686 y(0)c Fr(<)54 b Ft(lim)1689 1744 y Fq(\016)r Fo(!)p Fl(0)1820 1727 y Fi(+)1882 1686 y Fr(D)1951 1698 y Fl(1)1988 1686 y Ft(\()p Fr(\016)s Ft(\))14 b Fr(\016)26 b(<)d Fn(1)p Fr(:)450 1874 y Ft(F)-7 b(urthermore)30 b(the)i(second)f(deriv) -5 b(ativ)n(e)31 b(of)g(a)g(propagator)e(on)i(scale)g(1)g(is)g (trivially)g(b)r(ounded)450 2013 y(b)n(y)25 b Fr(C)6 b(D)697 2025 y Fl(0)735 2013 y Fr(q)775 1983 y Fo(\000)p Fl(4)890 2013 y Ft(\()p Fr(C)32 b Ft(is)25 b(the)i(same)e(b)r(ound)h (as)f(in)h(\(2.6\))o(\),)h(so)e(that)h(the)g(second)f(line)h(in)g 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Fl(1)1921 2940 y Fp(X)1920 3118 y Fq(\027)t Fo(2)p Fm(Z)2055 3018 y Fr(")2094 2984 y Fq(k)2134 3018 y Fr(e)2173 2984 y Fq(i\027)t(\013)2309 3007 y Ft(\026)2309 3018 y(\026)-47 b Fr(u)2351 2984 y Fl(\()p Fq(k)q Fl(\))2351 3039 y Fq(\027)2444 3018 y Fr(:)812 b Ft(\(6.1\))550 3257 y(As)19 b(w)n(e)h(shall)f(sho)n(w)f (no)n(w,)j(only)e(the)h(trees)f(without)h(resonances)e(can)h(con)n (tribute)g(to)3191 3246 y(\026)3191 3257 y(\026)-47 b Fr(u)o Ft(\()p Fr(\013;)14 b(")p Ft(\).)450 3466 y Fx(Lemma)30 b(6.)40 b Fj(If)30 b Fr(#)24 b Fn(2)f(T)1187 3478 y Fq(\027;k)1311 3466 y Fj(and)30 b Fr(N)1548 3436 y Fq(R)1539 3487 y Fl(0)1603 3466 y Ft(\()p Fr(#)p Ft(\))23 b Fn(6)p Ft(=)g(0)p Fj(,)30 b(then)f Fr(N)2184 3436 y Fo(\003)2175 3487 y Fl(0)2222 3466 y Ft(\()p Fr(#)p Ft(\))24 b Fr(<)f(k)s Ft(\()p Fr(#)p Ft(\))p Fr(=r)2687 3436 y Fo(\003)2725 3466 y Fj(.)450 3675 y(Pr)l(o)l(of.)43 b Ft(The)25 b(pro)r(of)g(is)g(b) n(y)f(induction)i(on)f(the)g(order)f(of)h(the)g(tree.)36 b(Of)25 b(course)f(for)g Fr(k)i Ft(=)d(1)h(there)450 3815 y(can)j(b)r(e)h(no)g(resonance,)d(so)i(that)h(the)g(b)r(ound)g(is) g(trivially)f(satis\014ed.)550 3954 y(If)33 b(the)h(ro)r(ot)e(line)i (of)f Fr(#)g Ft(is)g(either)g(on)g(scale)f(1)h(or)f(on)h(scale)g(0)g (and)g(resonan)n(t,)g(then)g(\(4.1\))450 4094 y(sho)n(ws)26 b(that)i Fr(N)944 4064 y Fo(\003)935 4114 y Fl(0)982 4094 y Ft(\()p Fr(#)p Ft(\))c Fr(<)f(k)s Ft(\()p Fr(#)p Ft(\))p Fr(=r)1447 4064 y Fo(\003)1486 4094 y Ft(.)550 4233 y(If)28 b(the)g(ro)r(ot)e(line)i(of)g Fr(#)g Ft(is)f(on)g(scale)g (0)g(and)h(nonresonan)n(t,)e(then)1500 4444 y Fr(N)1576 4409 y Fo(\003)1567 4464 y Fl(0)1614 4444 y Ft(\()p Fr(#)p Ft(\))d(=)g(1)18 b(+)2011 4340 y Fq(m)1981 4365 y Fp(X)1983 4541 y Fq(j)s Fl(=1)2115 4444 y Fr(N)2191 4409 y Fo(\003)2182 4464 y Fl(0)2229 4444 y Ft(\()p Fr(#)2310 4456 y Fq(j)2345 4444 y Ft(\))p Fr(;)450 4676 y Ft(where)38 b Fr(#)750 4688 y Fl(1)787 4676 y Fr(;)14 b(:)g(:)g(:)28 b(;)14 b(#)1035 4688 y Fq(m)1136 4676 y Ft(are)38 b(the)h(subtrees)f(en)n (tering)g(the)h(cluster)f Fr(T)12 b Ft(:)58 b(b)n(y)38 b(lemma)g(4)g(one)h(has)450 4815 y Fr(k)s Ft(\()p Fr(#)577 4827 y Fl(1)614 4815 y Ft(\))12 b(+)g Fr(:)i(:)g(:)f Ft(+)f Fr(k)s Ft(\()p Fr(#)1049 4827 y Fq(m)1112 4815 y Ft(\))23 b Fn(\024)g Fr(k)15 b Fn(\000)d Fr(r)1429 4785 y Fo(\003)1468 4815 y Ft(.)36 b(Then)25 b(w)n(e)f(can)g(iterate)g (the)h(pro)r(cedure)e(b)n(y)i(considering)e(the)450 4955 y(trees)31 b Fr(#)702 4967 y Fl(1)739 4955 y Fr(;)14 b(:)g(:)g(:)28 b(;)14 b(#)987 4967 y Fq(m)1050 4955 y Ft(.)48 b(If)32 b(at)f(least)g(one)g(of)g(them,)i(sa)n(y)d Fr(#)2214 4967 y Fl(\026)-46 b Fq(m)2264 4955 y Ft(,)33 b(with)e(1)e Fn(\024)45 b Ft(\026)-58 b Fr(m)29 b Fn(\024)g Fr(m)p Ft(,)j(has)f(the)h(ro)r(ot)450 5094 y(line)26 b(either)g(on)f(scale)g(1)g(or)g(on)h(scale)e(0)i(and)f(resonan)n(t,)g (one)g(can)h(reason)e(as)h(ab)r(o)n(v)n(e)f(to)i(deduce)450 5234 y(that)34 b Fr(N)712 5204 y Fo(\003)703 5254 y Fl(0)750 5234 y Ft(\()p Fr(#)844 5246 y Fl(\026)-46 b Fq(m)895 5234 y Ft(\))34 b Fr(<)f(k)s Ft(\()p Fr(#)1199 5246 y Fl(\026)-46 b Fq(m)1249 5234 y Ft(\))p Fr(=r)1362 5204 y Fo(\003)1435 5234 y Ft(and)34 b(the)g(statemen)n(t)g(is)g(pro)n(v)n (en.)54 b(Otherwise)33 b(one)h(con)n(tin)n(ues)450 5373 y(un)n(til)29 b(a)f(line)h Fr(`)g Ft(on)f(scale)g(0)h(and)f(resonan)n (t)f(is)i(reac)n(hed)e(\(as)i Fr(N)2424 5343 y Fq(R)2415 5394 y Fl(0)2478 5373 y Ft(\()p Fr(#)p Ft(\))d Fn(6)p Ft(=)e(0)k(suc)n(h)h(a)f(line)h(exists\).)p eop %%Page: 19 19 19 18 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)880 b(19)330 241 y Ft(F)-7 b(or)33 b(the)h(subtree)f(ha)n(ving)f Fr(`)h Ft(as)g(ro)r(ot)g(line)g(one)g(can)g(reason)f(as)h(ab)r(o)n(v)n (e)f(and)h(the)h(statemen)n(t)330 380 y(follo)n(ws)27 b(again.)p 3269 380 4 57 v 3273 328 50 4 v 3273 380 V 3323 380 4 57 v 430 570 a(Then)g(the)h(follo)n(wing)f(result)g(can)g(b) r(e)h(pro)n(v)n(en.)330 779 y Fx(Lemma)i(7.)40 b Fj(The)31 b(function)1286 768 y Ft(\026)1286 779 y(\026)-47 b Fr(u)o Ft(\()p Fr(\013;)14 b(")p Ft(\))31 b Fj(admits)f(the)g(tr)l(e)l(e)f(r)l (epr)l(esentation)886 1041 y Ft(\026)886 1052 y(\026)-47 b Fr(u)928 1017 y Fl(\()p Fq(k)q Fl(\))928 1072 y Fq(\027)1044 1052 y Ft(=)1132 857 y Fp(8)1132 931 y(>)1132 956 y(<)1132 1106 y(>)1132 1131 y(:)1206 918 y(P)1293 939 y Fo(0)1293 1006 y Fq(#)p Fo(2T)1416 1015 y Fg(\027;k)1519 981 y Ft(V)-7 b(al)1639 944 y Fo(0)1662 981 y Ft(\()p Fr(#)p Ft(\))p Fr(;)85 b Fj(if)31 b Fr(\027)d Fn(2)c Fr(q)s Fh(Z)12 b Fn(n)18 b(f)p Ft(0)p Fn(g)p Fr(;)c(k)24 b Fn(2)f Fr(r)2633 951 y Fo(\003)2672 981 y Fh(N)t Fr(;)1206 1148 y Ft(0)635 b Fj(otherwise,)3159 1052 y Ft(\(6.2\))330 1336 y Fj(wher)l(e)569 1274 y Fp(P)656 1295 y Fo(0)713 1336 y Fj(me)l(ans)34 b(that)g(only)g(tr)l(e)l(es)g Fr(#)g Fj(with)g Fr(N)1872 1306 y Fq(R)1863 1357 y Fl(0)1926 1336 y Ft(\()p Fr(#)p Ft(\))e(=)e(0)j Fj(and)h Fr(N)2482 1306 y Fo(\003)2473 1357 y Fl(0)2520 1336 y Ft(\()p Fr(#)p Ft(\))e(=)e Fr(k)s(=r)2887 1306 y Fo(\003)2959 1336 y Fj(have)35 b(to)f(b)l(e)330 1476 y(summe)l(d)28 b(over)g(and)h Ft(V)-7 b(al)1110 1439 y Fo(0)1133 1476 y Ft(\()p Fr(#)p Ft(\))29 b Fj(di\013ers)g(fr)l(om)f Ft(V)-7 b(al\()p Fr(#)p Ft(\))29 b Fj(as,)g(for)g Fr(\027)k Fj(multiple)c(of)g Fr(q)s Fj(,)g Fr(\015)5 b Ft(\()p Fr(\027)g Ft(\))28 b Fj(has)h(to)f(b)l(e)330 1615 y(r)l(eplac)l(e)l(d)j(with)f Ft(\(2)p Fr(\031)s(i\027)5 b Ft(\))1056 1585 y Fl(2)1094 1615 y Fj(.)330 1825 y(Pr)l(o)l(of.)43 b Ft(As)21 b Fr(\015)5 b Ft(\()p Fr(\027)g Ft(\))24 b(=)f Fr(O)r Ft(\()p Fr(\020)1115 1795 y Fl(2)1153 1825 y Ft(\))f(for)f Fr(\027)28 b Fn(2)23 b Fr(q)s Fh(Z)15 b Ft(and)21 b Fr(\015)5 b Ft(\()p Fr(\027)g Ft(\))24 b(=)e Fr(O)r Ft(\(1\))g(otherwise)f(\(see)g(\(2.6\))o(\),)i(b) n(y)e(lemma)330 1964 y(4,)27 b(lemma)h(6)f(the)h(cancellation)f(mec)n (hanisms,)f(one)i(has)f(that)1415 2152 y(lim)1406 2207 y Fq(\020)t Fo(!)p Fl(0)1553 2152 y Fn(j)p Ft(V)-7 b(al\()p Fr(#)p Ft(\))p Fn(j)29 b Fr(\020)1903 2118 y Fl(2)p Fq(k)q(=r)2039 2093 y Fk(\003)2102 2152 y Ft(=)22 b(0)p Fr(;)330 2353 y Ft(whenev)n(er)40 b Fr(N)784 2323 y Fq(R)775 2374 y Fl(0)838 2353 y Ft(\()p Fr(#)p Ft(\))46 b Fn(\025)e Ft(1,)g(so)c(that)h (only)g(trees)f(without)h(resonances)e(are)h(not)h(v)-5 b(anishing)330 2493 y(for)36 b Fr(\020)43 b Fn(!)38 b Ft(0;)i(this)c(means)g(that)g(one)g(m)n(ust)g(ha)n(v)n(e)f Fr(N)2034 2505 y Fl(0)2071 2493 y Ft(\()p Fr(#)p Ft(\))j(=)f Fr(N)2400 2463 y Fo(\003)2391 2513 y Fl(0)2438 2493 y Ft(\()p Fr(#)p Ft(\).)63 b(Moreo)n(v)n(er)34 b Fr(N)3078 2505 y Fl(0)3114 2493 y Ft(\()p Fr(#)p Ft(\))k Fn(\024)330 2565 y Fp(\004)369 2632 y Fr(k)s Ft(\()p Fr(#)p Ft(\))p Fr(=r)609 2602 y Fo(\003)648 2565 y Fp(\005)715 2632 y Ft(and)1571 2772 y(lim)1562 2826 y Fq(\020)t Fo(!)p Fl(0)1709 2676 y Fp(\014)1709 2726 y(\014)1709 2776 y(\014)1742 2761 y Ft(\026)1742 2772 y(\026)-47 b Fr(u)1784 2737 y Fl(\()p Fq(k)q Fl(\))1784 2792 y Fq(\027)1877 2676 y Fp(\014)1877 2726 y(\014)1877 2776 y(\014)1928 2772 y Fr(<)22 b Fn(1)330 2940 y Ft(for)27 b(an)n(y)g Fr(k)f Fn(\025)c Ft(1)28 b(and)f(for)g(an)n(y)g Fr(\027)h Fn(2)c Fh(Z)o Ft(.)31 b(Then)d(if)g Fr(k)e Fn(2)d Fr(r)2027 2910 y Fo(\003)2066 2940 y Fh(N)38 b Ft(and)27 b Fr(N)2382 2952 y Fl(0)2419 2940 y Ft(\()p Fr(#)p Ft(\))d(=)f Fr(k)s(=r)2771 2910 y Fo(\003)2836 2940 y Ft(one)28 b(has)1580 3129 y(lim)1571 3183 y Fq(\020)t Fo(!)p Fl(0)1718 3033 y Fp(\014)1718 3083 y(\014)1718 3133 y(\014)1751 3118 y Ft(\026)1751 3129 y(\026)-47 b Fr(u)1793 3094 y Fl(\()p Fq(k)q Fl(\))1793 3149 y Fq(\027)1886 3033 y Fp(\014)1886 3083 y(\014)1886 3133 y(\014)1937 3129 y Fn(\025)22 b Ft(0)p Fr(;)330 3317 y Ft(while)1619 3457 y(lim)1610 3511 y Fq(\020)t Fo(!)p Fl(0)1763 3446 y Ft(\026)1763 3457 y(\026)-48 b Fr(u)1805 3422 y Fl(\()p Fq(k)q Fl(\))1805 3477 y Fq(\027)1921 3457 y Ft(=)22 b(0)330 3638 y(otherwise.)38 b(So)29 b(rescaling)d Fr(")f Fn(!)f Fr("\020)1445 3608 y Fl(2)p Fq(=r)1545 3583 y Fk(\003)1612 3638 y Ft(and)29 b(taking)e(the)i(limit)g Fr(\020)i Fn(!)25 b Ft(0,)j(one)g(\014nds)h(that)f(only)330 3777 y(the)20 b(trees)e(with)i Fr(k)26 b Fn(2)d Fr(r)1022 3747 y Fo(\003)1061 3777 y Fh(Z)12 b Ft(and)19 b Fr(N)1355 3789 y Fl(0)1392 3777 y Ft(\()p Fr(#)p Ft(\))24 b(=)f Fr(k)s(=r)1744 3747 y Fo(\003)1801 3777 y Ft(can)c(ha)n(v)n(e)f(a)h (non)n(v)-5 b(anishing)18 b(v)-5 b(alue)19 b(con)n(tributing)330 3917 y(to)438 3906 y(\026)438 3917 y(\026)-47 b Fr(u)481 3874 y Fl(\()p Fq(k)q Fl(\))481 3926 y Fq(\027)573 3917 y Ft(.)42 b(Finally)-7 b(,)30 b(for)e(suc)n(h)h(trees)g(to)g(ha)n(v)n (e)f(really)g(a)h(non)n(v)-5 b(anishing)28 b(con)n(tribution,)h(one)f (can)330 4056 y(easily)k(see,)j(b)n(y)e(using)g(that)g Fr(N)1334 4026 y Fo(\003)1325 4077 y Fl(0)1372 4056 y Ft(\()p Fr(#)p Ft(\))g(=)f Fr(k)s(=r)k Ft(and)d Fr(N)2019 4026 y Fq(R)2010 4077 y Fl(0)2073 4056 y Ft(\()p Fr(#)p Ft(\))g(=)f(0,)i(the)g(scale)f(of)g(the)g(ro)r(ot)g(line)330 4196 y(m)n(ust)i(b)r(e)h(0)f(\(otherwise)f Fr(N)1226 4165 y Fo(\003)1217 4216 y Fl(0)1264 4196 y Ft(\()p Fr(#)p Ft(\))i Fr(<)g(k)s(=r)1641 4165 y Fo(\003)1714 4196 y Ft(b)n(y)g(\(4.1\)\),)h(so)e(that)g(the)h(momen)n(tum)f Fr(\027)41 b Ft(\015o)n(wing)330 4335 y(through)27 b(the)h(ro)r(ot)f (line)g(has)h(to)f(b)r(e)h(a)f(m)n(ultiple)h(of)g Fr(q)s Ft(.)p 3269 4335 V 3273 4282 50 4 v 3273 4335 V 3323 4335 4 57 v 430 4524 a(No)n(w)f(w)n(e)g(come)g(bac)n(k)g(to)g(\(1.15\)) g(and)h(pro)n(v)n(e)e(the)i(follo)n(wing)e(trivial)h(prop)r(erties.)330 4734 y Fx(Lemma)j(8.)40 b Fj(Ther)l(e)33 b(exists)f(a)g(solution)g(to) 37 b Ft(\(1.15\))31 b Fj(with)i Fr(r)2216 4746 y Fl(0)2281 4734 y Ft(=)26 b(2)32 b Fj(if)g(and)h(only)f(if)h(ther)l(e)f(exist)330 4873 y Fr(\027)371 4885 y Fq(i)399 4873 y Fr(;)14 b(\027)477 4885 y Fq(j)535 4873 y Fn(2)23 b(A)p Ft(\()p Fr(f)9 b Ft(\))p Fj(,)27 b(with)e Fr(\027)1061 4885 y Fq(i)1112 4873 y Fr(<)e(\027)1241 4885 y Fq(j)1276 4873 y Fj(,)j(such)f(that)g Fr(\027)1717 4885 y Fq(i)1768 4873 y Ft(=)d Fr(c)1891 4885 y Fq(i)1927 4873 y Ft(+)8 b Fr(l)2025 4885 y Fq(i)2052 4873 y Fr(q)28 b Fj(and)e Fr(\027)2315 4885 y Fq(j)2373 4873 y Ft(=)c Fr(c)2496 4885 y Fq(j)2539 4873 y Ft(+)8 b Fr(l)2637 4885 y Fq(j)2672 4873 y Fr(q)s Fj(,)26 b(with)g Fr(c)2975 4885 y Fq(i)3010 4873 y Ft(+)8 b Fr(c)3119 4885 y Fq(j)3177 4873 y Ft(=)23 b Fr(q)s Fj(.)330 5082 y(Pr)l(o)l(of.)43 b Ft(If)37 b(\(1.15\))30 b(has)g(a)g(solution)g(with) h Fr(r)1694 5094 y Fl(0)1760 5082 y Ft(=)c(2)j(then)h(there)g(exist)f Fr(i)d Fn(6)p Ft(=)h Fr(j)35 b Ft(suc)n(h)30 b(that)h(either)330 5222 y Fr(r)367 5234 y Fq(i)416 5222 y Ft(+)21 b Fr(r)539 5234 y Fq(j)604 5222 y Ft(=)29 b(2)i(or)g Fr(r)914 5234 y Fq(i)963 5222 y Ft(+)21 b Fr(r)1088 5192 y Fo(0)1086 5243 y Fq(j)1151 5222 y Ft(=)29 b(2.)48 b(The)32 b(\014rst)f(relation)g (giv)n(es)f Fr(\027)2265 5234 y Fq(i)2314 5222 y Ft(+)21 b Fr(\027)2441 5234 y Fq(j)2505 5222 y Ft(=)30 b Fr(R)q(q)k Ft(for)d(some)g Fr(R)f(>)f Ft(0,)330 5361 y(while)34 b(the)h(second)f(one)g(w)n(ould)f(giv)n(e)h Fr(\027)1604 5373 y Fq(i)1654 5361 y Fn(\000)23 b Fr(\027)1783 5373 y Fq(j)1852 5361 y Ft(=)33 b Fr(R)q(q)s Ft(,)j(so)e(that)g(it)h(has)f (to)g(b)r(e)h(discarded,)g(if)330 5501 y Fn(j)p Fr(I)389 5513 y Fq(c)423 5501 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)24 b Ft(=)e(1)i(for)f(an)n(y)h Fr(c)f Fn(2)g Fr(A)1213 5516 y Fq(p=q)1318 5501 y Ft(\()p Fr(f)9 b Ft(\),)25 b(b)n(y)f(lemma)g(1.)35 b(W)-7 b(riting)25 b Fr(\027)2299 5513 y Fq(i)2349 5501 y Ft(=)e Fr(c)2473 5513 y Fq(i)2512 5501 y Ft(+)11 b Fr(l)2613 5513 y Fq(i)2640 5501 y Fr(q)27 b Ft(and)d Fr(\027)2903 5513 y Fq(j)2961 5501 y Ft(=)f Fr(c)3085 5513 y Fq(j)3131 5501 y Ft(+)11 b Fr(l)3232 5513 y Fq(j)3267 5501 y Fr(q)s Ft(,)p eop %%Page: 20 20 20 19 bop 450 42 a Fl(20)702 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f (GUIDO)g(GENTILE)450 241 y Ft(with)g Fr(l)665 253 y Fq(i)692 241 y Fr(;)14 b(l)754 253 y Fq(j)813 241 y Fn(2)24 b Fh(Z)953 253 y Fl(+)1031 241 y Ft(and)k Fr(c)1229 253 y Fq(i)1256 241 y Fr(;)14 b(c)1329 253 y Fq(j)1388 241 y Fr(<)23 b(q)s Ft(,)29 b(one)f(has)f Fr(\027)1910 253 y Fq(i)1957 241 y Ft(+)18 b Fr(\027)2081 253 y Fq(j)2140 241 y 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3443 1078 4 57 v 450 1287 a Fj(R)l(emark)39 b Ft(5)p Fj(.)i Ft(Note)33 b(that)h(when)f Fr(q)44 b(=)-51 b Fn(2)32 b Fr(A)1710 1302 y Fq(p=q)1816 1287 y Ft(\()p Fr(f)9 b Ft(\))33 b(and)g Fn(j)p Fr(I)2191 1299 y Fl(\026)-35 b Fq(c)2223 1287 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)32 b(\025)g Ft(2)h(for)f(some)j(\026)-44 b Fr(c)p Ft(,)34 b(then)g(w)n(e)e(can)450 1426 y(also)e(consider)g(the)i(Diophan)n(tine)f(problem)g(\(1.15\))f (and)h(it)h(is)f(easy)g(to)g(see)f(that)i(in)f(general)450 1566 y(there)e(can)f(b)r(e)h(solutions)f(to)h(\(1.15\))f(with)h Fr(r)1850 1578 y Fl(0)1913 1566 y Ft(=)c(2.)40 b(This)29 b(do)r(es)f(not)h(mo)r(dify)g(the)h(v)-5 b(alue)28 b(of)h Fr(r)3411 1536 y Fo(\003)450 1705 y Ft(in)k(\(1.16\))o(,)g(but,)i(as)c (w)n(e)h(shall)g(see)g(b)r(elo)n(w,)i(it)e(is)h(a)f(case)f(whic)n(h)h (has)g(to)g(b)r(e)h(explicitly)g(tak)n(en)450 1845 y(in)n(to)27 b(accoun)n(t)g(if)h(one)f(w)n(an)n(ts)g(to)h(construct)f(all)g(p)r (ossible)g(clusters)g(with)h(t)n(w)n(o)f(no)r(des.)550 2054 y(Giv)n(en)g(an)n(y)g(tree)g Fr(#)p Ft(,)h(to)f(an)n(y)g(no)r(de)h Fr(u)22 b Fn(2)i Fr(#)k Ft(w)n(e)f(asso)r(ciate)f(the)i(lab)r(els)1700 2242 y Fr(\033)1747 2254 y Fq(u)1814 2242 y Ft(=)23 b Fn(\006)p Ft(1)p Fr(;)1206 b Ft(\(6.3a\))1700 2417 y Fr(c)1736 2429 y Fq(u)1803 2417 y Fn(2)23 b Fr(A)1943 2432 y Fq(p=q)2048 2417 y Ft(\()p Fr(f)9 b Ft(\))p Fr(;)1048 b Ft(\(6.3b\))1700 2591 y Fr(l)1725 2603 y Fq(u)1792 2591 y Fn(2)23 b Fr(B)1933 2603 y Fq(c)1963 2611 y Fg(u)2006 2591 y Ft(\()p Fr(f)9 b Ft(\))p Fr(;)1099 b Ft(\(6.3c\))450 2779 y(where)23 b Fr(A)748 2794 y Fq(p=q)853 2779 y Ft(\()p Fr(f)9 b Ft(\))23 b(and)g Fr(B)1210 2791 y Fq(c)1244 2779 y Ft(\()p Fr(f)9 b Ft(\))23 b(are)f(de\014ned)i(in)f(\(1.12\))g (and)g(\(1.13\))o(,)h(resp)r(ectiv)n(ely;)g(then)f(for)g(an)n(y)450 2919 y(no)r(de)28 b Fr(u)22 b Fn(2)i Fr(#)j Ft(one)h(has)f(that)h(its)f (mo)r(de)h(lab)r(el)g Fr(\027)1938 2931 y Fq(u)2009 2919 y Ft(can)f(b)r(e)h(expressed)e(as)1598 3107 y Fr(\027)1639 3119 y Fq(u)1705 3107 y 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(um)n(b)r(er)g(of)g(no)r(des\))f(suc)n(h)450 4090 y(that)28 b(V)-7 b(al)750 4053 y Fo(0)773 4090 y Ft(\()p Fr(#)p Ft(\))24 b Fn(6)p Ft(=)e(0)28 b(ha)n(v)n(e)e(order)g Fr(k)s Ft(\()p Fr(#)p Ft(\))e(=)f Fr(r)1785 4060 y Fo(\003)1851 4090 y Ft(and)28 b(they)f(are)g(obtained)g(in)h(the)g(follo)n(wing)f(w) n(a)n(y)-7 b(.)573 4247 y Fn(\017)41 b Ft(If)36 b Fr(r)786 4217 y Fo(\003)861 4247 y Ft(=)g(1,)h(\014x)f Fr(\033)j Ft(=)d Fn(\006)p Ft(1)f(and)g Fr(l)j Fn(2)f Fr(B)1906 4259 y Fq(q)1943 4247 y Ft(\()p Fr(f)9 b Ft(\):)53 b(the)35 b(only)h(line)f Fr(`)h Fn(2)h Fr(#)f Ft(has)f(momen)n(tum)656 4387 y Fr(\027)697 4399 y Fq(`)752 4387 y Ft(=)22 b Fr(\033)s Ft(\()p Fr(q)h Ft(+)18 b Fr(l)r(q)s Ft(\).)573 4526 y Fn(\017)41 b Ft(If)27 b Fr(r)777 4496 y Fo(\003)839 4526 y Ft(=)22 b(2,)27 b(\014x)g Fr(\033)f Ft(=)d Fn(\006)p Ft(1.)36 b(F)-7 b(or)26 b(an)n(y)g Fr(c)d Fn(2)g Fr(C)6 b Ft(\()p Fr(f)j Ft(\),)28 b(\014x)e Fr(l)2274 4538 y Fl(1)2311 4526 y Fr(;)14 b(l)2373 4538 y Fl(2)2433 4526 y Fn(2)24 b Fr(B)2575 4538 y Fq(c)2609 4526 y Ft(\()p Fr(f)9 b Ft(\))26 b(with)i Fr(\033)s Ft(\()p Fr(l)3045 4538 y Fl(1)3099 4526 y Fn(\000)16 b Fr(l)3205 4538 y Fl(2)3242 4526 y Ft(\))24 b Fr(>)e Ft(0:)656 4666 y(then)33 b(the)h(t)n(w)n(o)e(no)r(des)h Fr(u)1448 4678 y Fl(1)1518 4666 y Ft(and)g Fr(u)1733 4678 y Fl(2)1803 4666 y Ft(of)g Fr(#)g Ft(ha)n(v)n(e)f(mo)r(de)i(lab)r(els)f Fr(\027)2692 4678 y Fl(1)2762 4666 y Ft(and)g Fn(\000)p Fr(\027)3035 4678 y Fl(2)3105 4666 y Ft(suc)n(h)g(that)656 4805 y Fr(\027)697 4817 y Fq(i)751 4805 y Ft(=)26 b Fr(c)20 b Ft(+)f Fr(l)1007 4817 y Fq(i)1034 4805 y Fr(q)33 b Ft(for)c Fr(i)d Ft(=)g(1)p Fr(;)14 b Ft(2.)42 b(Otherwise)29 b(\014x)h Fn(f)p Fr(c)2158 4817 y Fl(1)2194 4805 y Fr(;)14 b(c)2267 4817 y Fl(2)2304 4805 y Fn(g)30 b Ft(suc)n(h)f(that)h Fr(c)2783 4817 y Fl(1)2840 4805 y Ft(+)19 b Fr(c)2960 4817 y Fl(2)3024 4805 y Ft(=)26 b Fr(q)33 b Ft(and)c(set)656 4945 y Fr(\027)697 4957 y Fq(i)748 4945 y Ft(=)24 b Fr(c)873 4957 y Fq(i)919 4945 y Ft(+)19 b 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Ft(,)24 b(with)g Fr(\027)2500 5236 y Fq(j)2558 5224 y Fn(2)f(A)p Ft(\()p Fr(f)9 b Ft(\))24 b(for)f Fr(j)28 b Ft(=)22 b(1)p Fr(;)14 b(:)g(:)g(:)27 b(;)14 b(r)3389 5236 y Fl(0)3427 5224 y Ft(,)656 5363 y(suc)n(h)24 b(that)h Fr(\027)1058 5375 y Fl(1)1108 5363 y Ft(+)13 b Fr(:)h(:)g(:)d Ft(+)i Fr(\027)1413 5375 y Fq(r)1444 5383 y Fi(0)1503 5363 y Ft(=)23 b Fr(R)q(q)s Ft(,)i(with)g Fr(R)f(>)e Ft(0:)35 b(the)25 b(ro)r(ot)f(line)h Fr(`)f Ft(of)h Fr(#)g Ft(has)f(momen)n(tum)656 5503 y Fr(\027)697 5515 y Fq(`)752 5503 y Ft(=)e Fr(R)q(q)s Ft(.)p eop %%Page: 21 21 21 20 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)880 b(21)330 241 y Fj(R)l(emark)39 b Ft(6)p Fj(.)i Ft(Note)35 b(that,)h(in)e(the)h(case)f Fr(r)1649 211 y Fo(\003)1722 241 y Fn(\025)g Ft(3,)h Fn(8)p Fr(c)f Fn(2)g Fr(A)2189 256 y Fq(p=q)2294 241 y Ft(\()p Fr(f)9 b Ft(\))35 b(one)f(has)g Fn(j)p Fr(I)2816 253 y Fq(c)2850 241 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)34 b Ft(=)g(1)g(and)330 380 y Fn(j)p Fr(B)416 392 y Fq(c)450 380 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)30 b Ft(=)f(1,)j(that)g(is)g(there)f(is)g(only)h(one)f (in)n(teger)f Fr(l)i Fn(2)e Fr(B)2209 392 y Fq(c)2242 380 y Ft(\()p Fr(f)9 b Ft(\))32 b(and)g(one)f(in)n(teger)g Fr(\027)k Fn(2)30 b(A)p Ft(\()p Fr(f)9 b Ft(\))330 520 y(suc)n(h)27 b(that)h Fr(\027)h Ft(=)22 b Fr(c)d Ft(+)f Fr(l)r(q)s Ft(.)430 729 y(By)35 b(using)g(the)h(ab)r(o)n(v)n(e)e (results)h(and)g(reasoning)f(as)g(in)i([1],)h(sect.)61 b(6,)37 b(w)n(e)e(can)g(pro)n(v)n(e)f(the)330 868 y(follo)n(wing)27 b(result,)g(analogous)e(to)j(lemma)f(5)g(of)h([1)o(].)330 1078 y Fx(Lemma)i(10.)40 b Fj(The)c(function)1343 1067 y Ft(\026)1343 1078 y(\026)-47 b Fr(u)p Ft(\()p Fr(\013;)14 b(")p Ft(\))35 b Fj(de\014ne)l(d)f(in)41 b Ft(\(1.18\))34 b Fj(satis\014es)h(the)f(di\013er)l(ential)i(e)l(qua-)330 1217 y(tion:)518 1349 y Ft(d)564 1319 y Fl(2)606 1338 y Ft(\026)606 1349 y(\026)-47 b Fr(u)p Ft(\()p Fr(\013;)14 b(")p Ft(\))p 518 1386 325 4 v 611 1462 a(d)q Fr(\013)711 1438 y Fl(2)875 1405 y Ft(=)23 b Fr(")1044 1327 y Fp(X)1016 1502 y Fq(\033)r Fl(=)p Fo(\006)p Fl(1)1266 1327 y Fp(X)1206 1508 y Fq(l)p Fo(2)p Fq(B)1322 1516 y Fg(q)1355 1508 y Fl(\()p Fq(f)7 b Fl(\))1460 1405 y Fn(C)1509 1362 y Fq(\033)r Fl(\()p Fq(l)p Fl(+1\))p Fq(q)1504 1434 y(p=q)1743 1405 y Ft(\()p Fr(f)i Ft(\))27 b(exp)14 b([)p Fr(i)p Ft(\()p Fr(\033)s Ft(\()p Fr(l)21 b Ft(+)d(1\))p Fr(q)e Ft(\()q Fr(\013)i Ft(+)2640 1394 y(\026)2640 1405 y(\026)-48 b Fr(u)p Ft(\()p Fr(\013;)14 b(")p Ft(\)\))q(])g Fr(;)191 b Ft(\(6.6\))330 1648 y Fj(for)31 b Fr(r)502 1618 y Fo(\003)564 1648 y Ft(=)22 b(1)p Fj(,)423 1824 y Ft(d)469 1794 y Fl(2)512 1813 y Ft(\026)512 1824 y(\026)-48 b Fr(u)p Ft(\()p Fr(\013;)14 b(")p Ft(\))p 423 1861 V 517 1937 a(d)p Fr(\013)616 1913 y Fl(2)781 1880 y Ft(=)22 b Fr(")907 1846 y Fl(2)986 1801 y Fp(X)958 1977 y Fq(\033)r Fl(=)p Fo(\006)p Fl(1)1134 1788 y Fp(h)1222 1801 y(X)1174 1983 y Fq(c)p Fo(2)p Fq(C)t Fl(\()p Fq(f)7 b Fl(\))1559 1801 y Fp(X)1437 1983 y Fq(l)1458 1991 y Fi(1)1490 1983 y Fq(;l)1531 1991 y Fi(2)1564 1983 y Fo(2)p Fq(B)1659 1991 y Fg(c)1690 1983 y Fl(\()p Fq(f)g Fl(\))p Fq(;)1404 2049 y(\026)p Fo(\021)p Fq(\033)r Fl(\()p Fq(l)1583 2057 y Fi(1)1617 2049 y Fo(\000)p Fq(l)1690 2057 y Fi(2)1722 2049 y Fl(\))p Fq(>)p Fl(0)1847 1880 y Fn(C)1896 1840 y Fq(\026q)1891 1908 y(p=q)1995 1880 y Ft(\()p Fr(f)i Ft(\))28 b(exp)14 b([)p Fr(i\026q)i Ft(\()q Fr(\013)i Ft(+)2626 1869 y(\026)2626 1880 y(\026)-48 b Fr(u)p Ft(\()p Fr(\013;)14 b(")p Ft(\)\))q(])k(+)349 2211 y(+)594 2132 y Fp(X)432 2314 y Fq(c)462 2322 y Fi(1)494 2314 y Fq(;c)544 2322 y Fi(2)576 2314 y Fo(2)p Fq(A)671 2325 y Fg(p=q)765 2314 y Fl(\()p Fq(f)7 b Fl(\))p Fq(;)525 2380 y(c)555 2388 y Fi(1)587 2380 y Fl(+)p Fq(c)668 2388 y Fi(2)700 2380 y Fl(=)p Fq(q)991 2132 y Fp(X)889 2314 y Fq(l)910 2322 y Fi(1)943 2314 y Fo(2)p Fq(B)1038 2322 y Fg(c)1065 2334 y Fi(1)1101 2314 y Fl(\()p Fq(f)g Fl(\))p Fq(;)899 2387 y(l)920 2395 y Fi(2)953 2387 y Fo(2)p Fq(B)1048 2395 y Fg(c)1075 2407 y Fi(2)1111 2387 y Fl(\()p Fq(f)g Fl(\))1226 2211 y Fn(C)1275 2168 y Fq(\033)r Fl(\()p Fq(l)1362 2176 y Fi(1)1394 2168 y Fl(+)p Fq(l)1466 2176 y Fi(2)1498 2168 y Fl(+1\))p Fq(q)1270 2239 y(p=q)1645 2211 y Ft(\()p Fr(f)i Ft(\))28 b(exp)14 b([)p Fr(i\033)s Ft(\()p Fr(l)2087 2223 y Fl(1)2142 2211 y Ft(+)k Fr(l)2250 2223 y Fl(2)2306 2211 y Ft(+)g(1\))p Fr(q)f Ft(\()p Fr(\013)i Ft(+)2709 2200 y(\026)2709 2211 y(\026)-47 b Fr(u)o Ft(\()p Fr(\013;)14 b(")p Ft(\)\))q(])3000 2119 y Fp(i)3039 2211 y Fr(;)97 b Ft(\(6.7\))330 2531 y Fj(for)31 b Fr(r)502 2501 y Fo(\003)564 2531 y Ft(=)22 b(2)29 b Fj(and)443 2672 y Ft(d)489 2641 y Fl(2)532 2661 y Ft(\026)532 2672 y(\026)-47 b Fr(u)o Ft(\()p Fr(\013;)14 b(")p Ft(\))p 443 2709 V 537 2785 a(d)p Fr(\013)636 2761 y Fl(2)801 2728 y Ft(=)22 b Fr(")927 2693 y Fq(r)958 2701 y Fi(0)1037 2649 y Fp(X)1008 2825 y Fq(\033)r Fl(=)p Fo(\006)p Fl(1)1206 2649 y Fp(X)1199 2827 y Fq(R>)p Fl(0)1529 2649 y Fp(X)1348 2831 y Fq(\027)1381 2839 y Fi(1)1413 2831 y Fq(;:::)11 b(;\027)1557 2839 y Fg(r)1585 2851 y Fi(0)1621 2831 y Fo(2A)p Fl(\()p Fq(f)c Fl(\))p Fq(;)1360 2900 y(\027)1393 2908 y Fi(1)1425 2900 y Fl(+)p Fq(:::)o Fl(+)p Fq(\027)1619 2908 y Fg(r)1647 2920 y Fi(0)1684 2900 y Fl(=)p Fq(Rq)1843 2728 y Fn(C)1892 2688 y Fq(\033)r(Rq)1887 2756 y(p=q)2020 2728 y Ft(\()p Fr(f)i Ft(\))27 b(exp)14 b([)p Fr(i\033)s(R)q(q)j Ft(\()p Fr(\013)i Ft(+)2714 2717 y(\026)2714 2728 y(\026)-47 b Fr(u)o Ft(\()p Fr(\013;)14 b(")p Ft(\)\))q(])g Fr(;)117 b Ft(\(6.8\))330 3048 y Fj(for)31 b Fr(r)502 3018 y Fo(\003)564 3048 y Fn(\025)22 b Ft(3)p Fj(,)30 b(with)g(b)l(oundary)h(c)l (onditions)1688 3037 y Ft(\026)1688 3048 y(\026)-48 b Fr(u)p Ft(\(0)p Fr(;)14 b(")p Ft(\))23 b(=)2028 3037 y(\026)2028 3048 y(\026)-48 b Fr(u)p Ft(\(2)p Fr(\031)s(;)14 b(")p Ft(\))23 b(=)g(0)29 b Fj(and)1368 3236 y Fn(C)1417 3202 y Fq(\027)1412 3259 y(p=q)1517 3236 y Ft(\()p Fr(f)9 b Ft(\))23 b(=)1803 3158 y Fp(X)1742 3336 y Fq(#)p Fo(2T)1865 3348 y Fg(r)1894 3336 y Fk(\003)1928 3348 y Fg(;\027)1998 3236 y Ft(V)-7 b(al)2118 3199 y Fo(0)2141 3236 y Ft(\()p Fr(#)p Ft(\))p Fr(;)882 b Ft(\(6.9\))330 3484 y Fj(wher)l(e)30 b Fr(r)603 3453 y Fo(\003)665 3484 y Ft(=)23 b Fr(r)792 3453 y Fo(\003)831 3484 y Ft(\()p Fr(f)9 b Ft(\))30 b Fj(is)g(de\014ne)l(d)g(in)36 b Ft(\(1.16\))o Fj(.)430 3693 y Ft(W)-7 b(e)28 b(can)f(de\014ne)400 4189 y Fn(N)12 b Ft(\()p Fr(f)d Ft(\))23 b(=)705 3770 y Fp(8)705 3845 y(>)705 3869 y(>)705 3894 y(>)705 3919 y(>)705 3944 y(>)705 3969 y(>)705 3994 y(>)705 4019 y(>)705 4044 y(>)705 4069 y(>)705 4094 y(<)705 4243 y(>)705 4268 y(>)705 4293 y(>)705 4318 y(>)705 4343 y(>)705 4368 y(>)705 4392 y(>)705 4417 y(>)705 4442 y(>)705 4467 y(>)705 4492 y(:)779 3867 y Fn(f)p Fr(\027)k Fn(2)d Fh(N)t Fn(j)p Fr(\027)34 b Fn(2)24 b Fr(I)1235 3879 y Fq(q)1272 3867 y Ft(\()p Fr(f)9 b Ft(\))p Fn(g)p Fr(;)1059 b Ft(if)28 b Fr(r)2625 3837 y Fo(\003)2687 3867 y Ft(=)22 b(1)p Fr(;)779 4034 y Fn(f)p Fr(\027)27 b Ft(=)c Fr(\027)1018 4046 y Fl(1)1074 4034 y Ft(+)18 b Fr(\027)1198 4046 y Fl(2)1235 4034 y Fn(j)p Ft(\()p Fr(\027)1331 4046 y Fl(1)1369 4034 y Fr(;)c(\027)1447 4046 y Fl(2)1484 4034 y Ft(\))23 b Fn(2)h Fh(N)1672 4004 y Fl(2)972 4202 y Ft(and)k(either)f Fr(\027)1410 4214 y Fl(1)1466 4202 y Ft(+)18 b Fr(\027)1590 4214 y Fl(2)1650 4202 y Ft(=)23 b Fr(q)31 b Ft(or)26 b Fr(\027)1948 4214 y Fl(1)2004 4202 y Fn(\000)18 b Fr(\027)2128 4214 y Fl(2)2188 4202 y Fn(2)24 b Fr(q)s Fh(Z)o Fn(g)p Fr(;)77 b Ft(if)28 b Fr(r)2625 4172 y Fo(\003)2687 4202 y Ft(=)22 b(2)p Fr(;)779 4369 y Fn(f)p Fr(\027)27 b Ft(=)c Fr(\027)1018 4381 y Fl(1)1074 4369 y Ft(+)18 b Fr(:)c(:)g(:)k Ft(+)g Fr(\027)1396 4381 y Fq(r)1427 4389 y Fi(0)1464 4369 y Fn(j)p Ft(\()p Fr(\027)1560 4381 y Fl(1)1597 4369 y Fr(;)c(:)g(:)g(:)28 b(;)14 b(\027)1837 4381 y Fq(r)1868 4389 y Fi(0)1904 4369 y Ft(\))23 b Fn(2)h Fh(Z)2099 4339 y Fq(r)2130 4347 y Fi(0)972 4537 y Ft(and)k Fr(\027)1175 4549 y Fl(1)1230 4537 y Ft(+)18 b Fr(:)c(:)g(:)19 b Ft(+)f Fr(\027)1553 4549 y Fq(r)1584 4557 y Fi(0)1643 4537 y Ft(=)23 b Fr(R)q(q)30 b Ft(with)e Fr(R)c(>)f Ft(0)p Fn(g)p Fr(;)177 b Ft(if)28 b Fr(r)2625 4506 y Fo(\003)2687 4537 y Ft(=)22 b Fr(r)2811 4549 y Fl(0)2872 4537 y Fn(\025)h Ft(3)p Fr(;)3118 4189 y Ft(\(6.10\))330 4702 y(so)h(that)i(the)f(three)g(di\013eren)n(tial)g (equations)f(app)r(earing)g(in)i(the)f(statemen)n(t)g(of)g(lemma)g(10)f (can)330 4841 y(b)r(e)k(all)f(expressed)g(as)767 4982 y(d)813 4952 y Fl(2)855 4971 y Ft(\026)855 4982 y(\026)-47 b Fr(u)o Ft(\()p Fr(\013;)14 b(")p Ft(\))p 767 5019 V 860 5095 a(d)q Fr(\013)960 5071 y Fl(2)1124 5038 y Ft(=)23 b Fr(")1251 5004 y Fq(r)1284 4979 y Fk(\003)1364 4959 y Fp(X)1336 5135 y Fq(\033)r Fl(=)p Fo(\006)p Fl(1)1584 4959 y Fp(X)1526 5141 y Fq(\027)t Fo(2N)9 b Fl(\()p Fq(f)e Fl(\))1776 5038 y Fn(C)1825 5004 y Fq(\033)r(\027)1820 5061 y(p=q)1924 5038 y Ft(\()p Fr(f)i Ft(\))28 b(exp)14 b([)p Fr(i\033)s(\027)19 b Ft(\()p Fr(\013)g Ft(+)2562 5027 y(\026)2562 5038 y(\026)-48 b Fr(u)p Ft(\()p Fr(\013;)14 b(")p Ft(\)\))q(])g Fr(;)228 b Ft(\(6.11\))330 5279 y(where)27 b Fr(\027)h Fn(2)c Fr(q)s Fh(N)t Ft(.)p eop %%Page: 22 22 22 21 bop 450 42 a Fl(22)702 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f (GUIDO)g(GENTILE)450 241 y Fj(R)l(emark)39 b Ft(7)p Fj(.)i Ft(F)-7 b(or)23 b Fr(f)9 b Ft(\()p Fr(x)p Ft(\))24 b(=)f(sin)14 b Fr(x)24 b Ft(\(standard)f(map\),)i(\(6.11\))e(reduces)h(to)f(Eq.)35 b(\(6.13\))24 b(of)g([1)o(])g(and)450 380 y(\(6.9\))34 b(reduces)g(to)g(Eq.)57 b(\(6.14\))34 b(of)g([1].)58 b(So)34 b(lemma)g(10)g(extends)g(lemma)h(5)f(of)g([1])h(to)f(more)450 520 y(general)26 b(p)r(erturbations.)450 729 y Fj(R)l(emark)39 b Ft(8)p Fj(.)i Ft(In)28 b(the)g(case)f Fr(r)1337 699 y Fo(\003)1399 729 y Ft(=)22 b(1,)28 b(one)f(immediately)g(sees)g(that) h Fr(C)2615 699 y Fq(\027)2609 756 y(p=q)2738 729 y Ft(=)22 b(\(2)p Fr(\031)s(i\027)5 b Ft(\))3056 699 y Fo(\000)p Fl(1)3146 729 y Fr(f)3187 741 y Fq(\027)3228 729 y Ft(.)550 938 y(Note)27 b(that,)h(b)n(y)g(the)f(b)r(ounds)h(of)g(sect.)37 b(4,)27 b(one)g(has)1286 1031 y Fp(\014)1286 1081 y(\014)1286 1131 y(\014)1314 1126 y Fn(C)1363 1092 y Fq(\033)r(\027)1358 1149 y(p=q)1462 1126 y Ft(\()p Fr(f)9 b Ft(\))1576 1031 y Fp(\014)1576 1081 y(\014)1576 1131 y(\014)1627 1126 y Fn(\024)23 b Fr(B)1782 1092 y Fq(r)1815 1067 y Fk(\003)1778 1147 y Fl(0)1854 1126 y Fr(C)1919 1092 y Fq(r)1952 1067 y Fk(\003)1913 1147 y Fl(1)1990 1126 y Fr(D)2061 1092 y Fq(r)2094 1067 y Fk(\003)2059 1147 y Fl(1)2133 1126 y Ft(\()p Fr(\016)s Ft(\))14 b Fr(e)2290 1092 y Fo(\000)p Fl(\()p Fq(\030)r Fo(\000)p Fq(\016)r Fl(\))p Fo(j)p Fq(\027)t Fo(j)2591 1126 y Fr(;)450 1315 y Ft(for)27 b(an)n(y)g Fr(\030)g(>)c(\016)j(>)c Ft(0)28 b(\(see)f(\(1.3\)\),)h (assuring)e(summabilit)n(y)h(on)h Fr(\027)g Fn(2)23 b(N)12 b Ft(\()p Fr(f)d Ft(\))29 b(in)e(\(6.11\).)550 1454 y(Eq.)36 b(\(6.11\))27 b(is)g(easily)g(reducible)g(to)h(an)f(ordinary)f (di\013eren)n(tial)h(equation)g(with)i(separable)450 1594 y(v)-5 b(ariables)35 b(\(just)h(tak)n(e)g Fr(\030)41 b Ft(=)36 b Fr(\013)24 b Ft(+)1545 1583 y(\026)1545 1594 y(\026)-47 b Fr(u)p Ft(\()p Fr(\013;)14 b(")p Ft(\))36 b(to)g(reduce)f(it)i(to)e(the)i(Hamilton)f(equations)f(of)g(a)450 1733 y(one-dimensional)26 b(system\);)i(moreo)n(v)n(er)d(if)j Fr(f)9 b Ft(\()p Fr(x)p Ft(\))28 b(is)g(a)f(trigonometric)f(p)r (olynomial)h(then)1337 1843 y Fp(X)1279 2025 y Fq(\027)t Fo(2N)9 b Fl(\()p Fq(f)e Fl(\))1529 1921 y Fr(C)1594 1887 y Fq(\033)r(\027)1588 1944 y(p=q)1693 1921 y Ft(\()p Fr(f)i Ft(\))28 b(exp)13 b([)p Fr(i\033)s(\027)19 b Ft(\()q Fr(\013)g Ft(+)2330 1910 y(\026)2330 1921 y(\026)-47 b Fr(u)o Ft(\()p Fr(\013;)14 b(")p Ft(\)\))q(])450 2167 y(is)20 b(a)g(trigonometric)e(p)r(olynomial,)j(so)e(that)i(-)f(b)n(y)f (a)h(classical)f(c)n(hange)f(of)i(v)-5 b(ariable,)21 b(see)f Fj(e.g.)k Ft([17)o(],)450 2306 y(p.)35 b(72)20 b(-)g(the)h(solution)f(of)27 b(\(6.11\))20 b(is)h(the)g(in)n(v)n(erse)e (of)h(an)h(h)n(yp)r(erelliptic)g(in)n(tegral)e(\(whic)n(h)i(reduces)450 2446 y(to)27 b(an)g(elliptic)h(function)g(in)f(the)h(case)e(of)i(the)f (standard)g(map)g(explicitly)g(considered)f(in)i([1]\);)450 2585 y(see)f([18)o(],)h([19)o(])g(for)f(an)g(in)n(tro)r(duction)h(to)f (the)h(theory)f(of)g(h)n(yp)r(erelliptic)h(functions.)1637 2794 y(7.)45 b Fs(Conclusions)550 2984 y Ft(Generically)28 b(the)i(radius)f(of)h(con)n(v)n(ergence)d(of)i(the)h(KAM)g(in)n(v)-5 b(arian)n(t)28 b(curv)n(es)h(for)g(the)h(gen-)450 3123 y(eralized)f(standard)g(maps)h(of)g(the)h(form)f(\(1.1\))g(scales)f(as) g Fr(\032)p Ft(\()p Fr(!)s Ft(\))f(=)f Fr(O)r Ft(\()p Fr(\020)2728 3093 y Fl(2)2766 3123 y Ft(\),)32 b(for)d Fr(!)h Ft(=)d Fr(p=q)c Ft(+)c Fr(\020)6 b Ft(,)450 3263 y(b)r(ecause)27 b(for)g(generic)g(analytic)g(p)r(erturbations)g Fr(f)9 b Ft(\()p Fr(x)p Ft(\))28 b(one)f(has)g Fr(q)f Fn(2)e Fr(A)2695 3278 y Fq(p=q)2800 3263 y Ft(\()p Fr(f)9 b Ft(\).)550 3402 y(The)33 b(situation)g(is)g(quite)h(di\013eren)n(t)f (if)h(trigonometric)e(p)r(olynomials)g(are)g(explicitly)i(con-)450 3542 y(sidered.)69 b(In)39 b(suc)n(h)f(a)h(case)e(there)i(is)f(strong)f (dep)r(endence)j(of)e(the)h(scaling)f(factor)f(on)i(the)450 3681 y(rational)34 b(rotation)f(n)n(um)n(b)r(er)i Fr(p=q)i Ft(\(more)d(precisely)g(on)h Fr(q)s Ft(\);)k(compare)33 b(with)j(the)f(n)n(umerical)450 3821 y(results)27 b(in)h([5)o(].)550 3960 y(Note)h(that)h(prop)r(osition)f(1)h(yields)f Fr(r)1719 3930 y Fo(\003)1785 3960 y Fn(\024)d Fr(q)s Ft(,)k(so)f(that)h(the)h(b) r(est)f(b)r(eha)n(viour)e(one)i(can)f(hop)r(e)450 4099 y(to)e(get)h(for)f(the)h(radius)f(of)g(con)n(v)n(ergence)e(is)j(of)f (the)h(form)g Fr(O)r Ft(\()p Fr(\020)2420 4069 y Fl(2)p Fq(=q)2525 4099 y Ft(\).)550 4239 y(Consider)36 b(the)i(set)f(of)h (analytic)e(functions)i Fr(f)46 b Ft(with)38 b(\(arbitrarily\))e (pre\014xed)h(norm,)i(sa)n(y)450 4378 y Fn(k)p Fr(f)9 b Fn(k)584 4390 y Fl(0)643 4378 y Ft(=)23 b(1:)36 b(w)n(e)27 b(shall)g(denote)h(it)g(b)n(y)f Fn(S)1664 4390 y Fl(1)1702 4378 y Ft(,)h(with)1344 4567 y Fn(S)1394 4579 y Fl(1)1455 4567 y Ft(=)23 b Fn(f)p Fr(f)31 b Fn(2)23 b(B)1790 4579 y Fq(\030)1822 4563 y Fk(0)1872 4567 y Fn(8)p Fr(\030)1959 4532 y Fo(0)2004 4567 y Fr(<)g(\030)t Fn(jk)p Fr(f)9 b Fn(k)2289 4579 y Fl(0)2348 4567 y Ft(=)23 b(1)p Fn(g)p Fr(;)736 b Ft(\(7.1\))450 4755 y(using)27 b(the)h(notations)f(in)n(tro) r(duced)g(in)h(sect.)37 b(1.)g(De\014ne)1107 4943 y Fr(R)q Ft(\()p Fr(!)s Ft(\))23 b(=)51 b(inf)1401 4998 y Fq(f)7 b Fo(2S)1526 5006 y Fi(1)1571 4943 y Fr(\032)1614 4955 y Fq(f)1657 4943 y Ft(\()p Fr(!)s Ft(\))p Fr(;)180 b(R)2042 4955 y Fq(r)2079 4943 y Ft(\()p Fr(!)s Ft(\))24 b(=)93 b(inf)2342 4998 y Fq(f)7 b Fo(2S)2467 5006 y Fi(1)2499 4998 y Fq(;)2309 5062 y(r)2342 5037 y Fk(\003)2377 5062 y Fl(\()p Fq(f)g Fl(\)=)p Fq(r)2565 4943 y Fr(\032)2608 4955 y Fq(f)2651 4943 y Ft(\()p Fr(!)s Ft(\))p Fr(:)486 b Ft(\(7.2\))550 5199 y(The)27 b(theorem)g(in)h(sect.)37 b(1)27 b(sho)n(ws)g(that)1263 5388 y Fr(R)1326 5400 y Fq(r)1363 5388 y Ft(\()p Fr(!)s Ft(\))c Fn(\025)g Fr(C)1652 5400 y Fq(r)1689 5388 y Fn(j)p Fr(\020)6 b Fn(j)1777 5353 y Fl(2)p Fq(=r)1881 5388 y Fr(;)180 b(R)q Ft(\()p Fr(!)s Ft(\))23 b Fn(\025)g Fr(C)2437 5400 y Fl(0)2474 5388 y Fn(j)p Fr(\020)6 b Fn(j)2562 5353 y Fl(2)2600 5388 y Fr(;)656 b Ft(\(7.3\))p eop %%Page: 23 23 23 22 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)880 b(23)330 241 y Ft(for)27 b(some)g(constan)n(ts)g Fr(C)1092 253 y Fq(r)1156 241 y Ft(and)h Fr(C)1377 253 y Fl(0)1414 241 y Ft(.)430 380 y(On)36 b(the)h(other)f(hand)g(if)h Fr(r)1295 350 y Fo(\003)1372 380 y Ft(=)g Fr(q)j Ft(one)c(has)g(at)g (least)g(one)g(function)h Fr(f)46 b Fn(2)39 b(S)2908 392 y Fl(1)2982 380 y Ft(suc)n(h)d(that)330 520 y Fr(\032)373 532 y Fq(f)416 520 y Ft(\()p Fr(!)s Ft(\))23 b Fn(\024)g Fr(C)6 b Fn(j)p Fr(\020)g Fn(j)799 490 y Fl(2)p Fq(=q)931 520 y Ft(for)27 b(some)g(constan)n(t)g Fr(C)6 b Ft(:)37 b(just)29 b(tak)n(e)d Fr(f)9 b Ft(\()p Fr(x)p Ft(\))24 b(=)f(sin)14 b Fr(x)p Ft(,)28 b(see)f([1)o(].)430 659 y(More)38 b(generally)-7 b(,)40 b(for)e(an)n(y)h Fr(r)44 b Fn(2)f(f)p Ft(1)p Fr(;)14 b(:)g(:)g(:)26 b(;)14 b(q)s Fn(g)p Ft(,)42 b(one)c(can)h(consider)f(a)g(function)i Fr(f)47 b Ft(whic)n(h)330 799 y(is)36 b(a)f(trigonometric)f(p)r (olynomial)i(of)f(degree)g Fr(N)1898 811 y Fq(f)1978 799 y Fr(<)h(q)j Ft(and)c(with)i Fr(r)2561 769 y Fo(\003)2636 799 y Ft(=)g Fr(r)r Ft(:)54 b(the)36 b(radius)f(of)330 938 y(con)n(v)n(ergence)20 b(of)j(the)g(solution)1326 927 y(\026)1326 938 y(\026)-47 b Fr(u)22 b Ft(to)g(the)h(equation)f (\(1.18\))g(is)g(non)n(v)-5 b(anishing,)23 b(as)f(this)h(solution)330 1078 y(m)n(ust)41 b(dep)r(end)g(analytically)f(on)g Fr(")h Ft(in)g(a)f(neigh)n(b)r(orho)r(o)r(d)g(of)h(the)g(origin,)i(and)d (cannot)h(b)r(e)330 1217 y(in\014nite,)21 b(b)n(y)e(the)g(prop)r (erties)e(of)i(the)g(h)n(yp)r(erelliptic)f(functions.)35 b(As)18 b(the)h(radius)f(of)h(con)n(v)n(ergence)330 1357 y Fr(\032)373 1369 y Fq(f)416 1357 y Ft(\()p Fr(!)s Ft(\))33 b(is)h(related)e(to)h(the)h(radius)e(of)h(con)n(v)n(ergence)e(of)2122 1346 y(\026)2122 1357 y(\026)-48 b Fr(u)33 b Ft(b)n(y)g(the)h(scaling)e (la)n(w)h Fr(")f Fn(!)g Fr("\020)3168 1326 y Fl(2)p Fq(=r)3268 1301 y Fk(\003)3307 1357 y Ft(,)330 1496 y(this)e(means)f(that)h(there) f(exists)g(a)h(constan)n(t)e Fr(C)1844 1508 y Fq(f)1917 1496 y Ft(suc)n(h)i(that)g Fr(\032)2332 1508 y Fq(f)2375 1496 y Ft(\()p Fr(!)s Ft(\))c Fn(\024)g Fr(C)2670 1508 y Fq(f)2713 1496 y Fn(j)p Fr(\020)6 b Fn(j)2801 1466 y Fl(2)p Fq(=r)2901 1441 y Fk(\003)2970 1496 y Ft(for)29 b(suc)n(h)g(a)330 1636 y(function)f Fr(f)9 b Ft(.)430 1775 y(So)27 b(w)n(e)g(can)g(conclude)h(that)1014 1963 y Fr(R)1077 1975 y Fq(r)1114 1963 y Ft(\()p Fr(!)s Ft(\))23 b(=)g Fr(D)1413 1975 y Fq(r)1450 1963 y Ft(\()p Fr(!)s Ft(\))p Fn(j)p Fr(\020)6 b Fn(j)1657 1929 y Fl(2)p Fq(=r)1761 1963 y Fr(;)180 b(R)q Ft(\()p Fr(!)s Ft(\))23 b(=)g Fr(D)2327 1975 y Fl(0)2364 1963 y Ft(\()p Fr(!)s Ft(\))p Fn(j)p Fr(\020)6 b Fn(j)2571 1929 y Fl(2)2609 1963 y Fr(;)527 b Ft(\(7.4\))330 2152 y(where)27 b Fr(D)639 2164 y Fq(r)676 2152 y Ft(\()p Fr(!)s Ft(\))g(and)h Fr(D)1053 2164 y Fl(0)1090 2152 y Ft(\()p Fr(!)s Ft(\))g(are)e(b)r(ounded)i(functions)g (suc)n(h)g(that)472 2340 y(0)22 b Fr(<)h(D)693 2352 y Fq(r)r Fl(1)786 2340 y Fr(<)f(D)942 2352 y Fq(r)979 2340 y Ft(\()p Fr(!)s Ft(\))h Fr(<)g(D)1278 2352 y Fq(r)r Fl(2)1370 2340 y Fr(<)g Fn(1)p Fr(;)180 b Ft(0)23 b Fr(<)f(D)1965 2352 y Fl(01)2058 2340 y Fr(<)h(D)2215 2352 y Fl(0)2252 2340 y Ft(\()p Fr(!)s Ft(\))g Fr(<)g(D)2551 2352 y Fl(02)2644 2340 y Fr(<)g Fn(1)p Fr(;)321 b Ft(\(7.5\))330 2528 y(for)27 b(some)g(constan)n(ts)g Fr(D)1102 2540 y Fq(r)r Fl(1)1171 2528 y Fr(;)14 b(D)1277 2540 y Fq(r)r Fl(2)1347 2528 y Fr(;)g(D)1453 2540 y Fl(01)1523 2528 y Fr(;)g(D)1629 2540 y Fl(02)1699 2528 y Ft(,)27 b(indep)r(enden)n(t)i(on)e Fr(!)s Ft(.)430 2668 y(Note)j(that)h(the)h(b)r(eha)n(viour)d Fr(\032)1394 2680 y Fq(f)1437 2668 y Ft(\()p Fr(!)s Ft(\))g(=)f Fr(C)1737 2680 y Fq(f)1780 2668 y Ft(\()p Fr(!)s Ft(\))p Fn(j)p Fr(\020)6 b Fn(j)1987 2638 y Fl(2)p Fq(=q)2092 2668 y Ft(,)31 b(with)g Fr(C)2397 2680 y Fq(f)2441 2668 y Ft(\()p Fr(!)s Ft(\))g(uniformily)g(b)r(ounded,)330 2807 y(holding)d(for)g(the)g(standard)f(map,)i Fr(f)9 b Ft(\()p Fr(x)p Ft(\))25 b(=)e(sin)14 b Fr(x)p Ft(,)29 b(is)f(related)g(to)g(the)g(in)n(terp)r(olation)g(through)330 2947 y(the)j(Bryuno)e(function)i(of)f(the)g(radius)g(of)g(con)n(v)n (ergence)e Fr(\032)2200 2959 y Fq(f)2242 2947 y Ft(\()p Fr(!)s Ft(\))j(for)f(real)f Fr(!)k Ft(whic)n(h)d(has)g(b)r(een)330 3086 y(pro)n(v)n(en)23 b(in)h([2];)i(the)e(relation)g(can)g(b)r(e)h (exactly)e(form)n(ulated)h(b)n(y)g(using)g([20)o(],)i(where)e(the)g (exten-)330 3226 y(sion)j(to)h(the)g(complex)f(plane)g(of)h(the)g (Bryuno)e(function)i(is)g(discussed.)430 3365 y(Therefore)35 b(no)i(univ)n(ersal)f(dep)r(endence)h(of)g(the)h(radius)e(of)h(con)n(v) n(ergence)e(on)h Fr(p=q)k Ft(is)d(ex-)330 3505 y(p)r(ected)30 b(in)h(general)d(for)h(the)i(in)n(v)-5 b(arian)n(t)28 b(curv)n(es)h(of)h(maps)f(of)h(the)g(forms)f(\(1.1\):)41 b(therefore)29 b(no)330 3644 y(in)n(terp)r(olation)24 b(through)g(the)i(Bryuno)d(function)j(\(or)e(p)r(ossible)h (generalizations)e(of)i(it\))g(can)g(b)r(e)330 3783 y(attempted)33 b(generically)e(for)h Fr(\032)1329 3795 y Fq(f)1372 3783 y Ft(\()p Fr(!)s Ft(\))h(or)f Fr(R)q Ft(\()p Fr(!)s Ft(\),)i(in)f(the)g (case)f(of)g(real)g(rotation)f(n)n(um)n(b)r(ers,)j(as)330 3923 y(done)c(in)h([2])f(in)h(the)g(case)f(of)g(the)h(standard)e(map.) 46 b(Compare)29 b(also)h(the)h(results)f(holding)g(for)330 4062 y(the)e(Siegel)f(problem,)g([21)o(],)h(for)f(whic)n(h)h(the)g (situation)f(is)h(quite)g(di\013eren)n(t.)430 4202 y(On)37 b(the)g(con)n(trary)e(if)j(w)n(e)f(restrict)f(ourselv)n(es)f(to)i(the)h (space)e Fn(f)p Fr(f)48 b Fn(2)39 b(S)2717 4214 y Fl(1)2754 4202 y Fn(j)p Fr(r)2816 4172 y Fo(\003)2855 4202 y Ft(\()p Fr(f)9 b Ft(\))40 b(=)e Fr(r)r Fn(g)p Ft(,)i(at)330 4341 y(least)26 b(for)g Fr(r)688 4311 y Fo(\003)750 4341 y Ft(=)d Fr(q)29 b Ft(the)e(in)n(terp)r(olation)f(through)f(the)i(Bryuno) f(function)h(holds)f(for)g Fr(R)3018 4353 y Fq(q)3054 4341 y Ft(\()p Fr(!)s Ft(\);)h(see)330 4481 y(ab)r(o)n(v)n(e.)37 b(Then)28 b(one)g(could)g(conjecture)f(that)h(some)g(p)r(ossible)g (generalization)e(of)i(the)g(Bryuno)330 4620 y(function)g(in)n(terp)r (olates)f Fr(R)1171 4632 y Fq(r)1208 4620 y Ft(\()p Fr(!)s Ft(\),)h(but)g(of)f(course)g(one)g(needs)g(to)h(w)n(ork)e(it)i(out.)430 4760 y(As)23 b(a)g(\014nal)g(commen)n(t,)h(w)n(e)f(note)h(that,)g (instead)f(of)h(generalized)e(standard)g(map)h(lik)n(e)g(\(1.1\),)330 4899 y(one)36 b(can)g(study)g(also)g(generalized)e(semistandard)i (maps,)i(b)n(y)e(considering)f(p)r(erturbations)330 5039 y Fr(f)9 b Ft(\()p Fr(x)p Ft(\))28 b(of)g(the)g(form)f(\(1.2\),)g(with) i(the)f(sum)f Fj(only)36 b Ft(o)n(v)n(er)26 b(the)i(in)n(tegers)e Fr(\027)i Fn(\025)23 b Ft(1,)k Fj(i.e.)p Ft(,)1496 5227 y Fr(f)9 b Ft(\()p Fr(x)p Ft(\))24 b(=)1769 5148 y Fp(X)1768 5326 y Fq(\027)t Fo(\025)p Fl(1)1904 5227 y Fr(f)1945 5239 y Fq(\027)1986 5227 y Fr(e)2025 5193 y Fq(i\027)t(x)2127 5227 y Fr(;)1009 b Ft(\(7.6\))p eop %%Page: 24 24 24 23 bop 450 42 a Fl(24)702 b(ALBER)-5 b(TO)29 b(BERRETTI)h(AND)f (GUIDO)g(GENTILE)450 241 y Ft(where)h(the)h(co)r(e\016cien)n(ts)f Fr(f)1303 253 y Fq(\027)1374 241 y Ft(satisfy)i(\(1.3\))o(.)46 b(If)31 b Fr(f)2004 253 y Fl(1)2069 241 y Ft(=)c(1)j(and)h Fr(f)2439 253 y Fq(\027)2507 241 y Ft(=)d(0)i Fn(8)p Fr(\027)i Fn(\025)c Ft(2,)j(\(7.6\))f(reduces)450 380 y(to)d(the)h(semistandard)f(map.)550 520 y(Note)35 b(that)h(in)g(suc)n (h)f(a)g(case)g(the)h(analysis)e(p)r(erformed)h(in)h(this)g(pap)r(er)f (could)g(b)r(e)h(easily)450 659 y(adapted)d(\(and)g(less)f(w)n(ork)g(w) n(ould)h(b)r(e)g(required,)h(as)e(there)h(w)n(ould)g(not)g(b)r(e)g(the) h(problem)e(of)450 799 y(resonances:)38 b(all)28 b(clusters)h(are)f (nonresonan)n(t)f(for)h(p)r(erturbations)h(of)g(the)g(form)g(\(7.6\))o (\))h(and)e(a)450 938 y(result)f(similar)g(to)h(the)g(theorem)f(in)h (sect.)36 b(1)28 b(w)n(ould)f(immediately)g(follo)n(w.)550 1078 y(Ho)n(w)n(ev)n(er)c(the)i(absence)g(of)g(negativ)n(e)f(F)-7 b(ourier)24 b(lab)r(els)h(induces)g(a)g(condition)g(whic)n(h)g(can)g(b) r(e)450 1217 y(di\013eren)n(t)31 b(to)h(\(1.15\))e(to)h(determine)h (the)f(scaling)g(of)g(the)h(radius)e(of)h(con)n(v)n(ergence,)f(as)h(it) g(can)450 1357 y(b)r(e)36 b(easily)e(c)n(hec)n(k)n(ed.)59 b(F)-7 b(or)35 b(instance)g(it)h(is)f(straigh)n(tforw)n(ard)d(to)k(see) f(that,)i(ev)n(en)e(if)h(there)f(is)450 1496 y(a)e Fr(c)f Fn(2)h Fr(A)743 1511 y Fq(p=q)848 1496 y Ft(\()p Fr(f)9 b Ft(\))33 b(suc)n(h)g(that)g Fr(c)f Fn(6)p Ft(=)g Fr(q)37 b Ft(and)c Fn(j)p Fr(I)1838 1508 y Fq(c)1872 1496 y Ft(\()p Fr(f)9 b Ft(\))p Fn(j)32 b(\025)g Ft(2)h(\(the)h(same)e(sym)n(b)r(ols)h (in)n(tro)r(duced)g(for)450 1636 y(the)27 b(maps)g(\(1.1\))f(can)h(b)r (e)g(used)f(with)i(the)f(same)f(meaning\),)h(it)g(is)g(no)f(more)g(p)r (ossible)h(to)f(form)450 1775 y(a)33 b(\(nonresonan)n(t\))g(cluster)g (with)i(t)n(w)n(o)e(no)r(des)g Fr(u)1985 1787 y Fl(1)2056 1775 y Ft(and)g Fr(u)2271 1787 y Fl(2)2342 1775 y Ft(suc)n(h)h(that)g Fr(\027)2763 1787 y Fq(u)2802 1795 y Fi(1)2861 1775 y Fn(\000)22 b Fr(\027)2989 1787 y Fq(u)3028 1795 y Fi(2)3098 1775 y Fn(2)34 b Fr(q)s Fh(Z)27 b Ft(and)450 1914 y Fr(\027)491 1926 y Fq(u)530 1934 y Fi(1)585 1914 y Ft(+)18 b Fr(\027)709 1926 y Fq(u)748 1934 y Fi(2)808 1914 y Fn(2)24 b Fr(q)s Fh(Z)o Ft(,)e(simply)27 b(b)r(ecause)h(all)f(mo)r(de)h(lab)r(els)f(are) g(strictly)g(p)r(ositiv)n(e.)550 2054 y(This)d(means)g(that)h(the)g (scaling)e(b)r(eha)n(viour)g(of)i(the)f(radii)g(of)h(con)n(v)n(ergence) d(for)h(generalized)450 2193 y(standard)d(and)h(semistandard)f(maps)h (ha)n(ving)f(the)i(same)f(F)-7 b(ourier)20 b(lab)r(els)h(\()p Fj(i.e.)p Ft(,)j(admitting)e(the)450 2333 y(same)i(sequences)1039 2312 y(~)1029 2333 y Fr(I)1065 2345 y Fq(m)1128 2333 y Ft(\()p Fr(f)9 b Ft(\)\))26 b(can)e(b)r(e)h(completely)f(di\013eren)n (t:)36 b(this)25 b(is)f(a)h(feature)f(quite)h(new)g(with)450 2472 y(resp)r(ect)31 b(to)g(the)h(case)e(of)h(the)h(standard)e(and)h (semistandard)g(maps,)h(for)e(whic)n(h)h(the)h(scaling)450 2612 y(b)r(eha)n(viour)d(is)h(the)h(same)f(\(and)g(whose)g(radii)f(of)i (con)n(v)n(ergence,)d(for)i(real)f(rotation)h(n)n(um)n(b)r(ers,)450 2751 y(admit)e(the)g(same)f(in)n(terp)r(olation)g(in)g(terms)h(of)f (the)h(Bryuno)f(function;)h(see)f([2])h(and)f([7]\).)450 3127 y Fx(Ac)m(k)m(o)m(wlegdmen)m(ts.)36 b Ft(W)-7 b(e)27 b(thank)h(M.)g(Kon)n(tsevic)n(h)d(and)j(S.)f(Marmi)g(for)g(useful)h (discussions.)450 3267 y(One)f(of)h(us)f(\(GG\))i(thanks)e(IHES)h(for)f (hospitalit)n(y)g(while)h(part)f(of)g(this)h(w)n(ork)e(w)n(as)h(done.) 1708 3712 y Fs(References)485 3878 y Fw([1])35 b(Berretti,)20 b(A.)f(and)i(Gen)n(tile,)g(G.,)43 b Fc(Sc)l(aling)22 b(Pr)l(op)l(erties)h(for)g(the)e(R)l(adius)j(of)e(Conver)l(genc)l(e)g (of)g(a)h(Lindste)l(dt)595 3994 y(Series:)32 b(the)26 b(Standar)l(d)h(Map)p Fw(,)d(J.)g(Math.)f(Pures.)g(Appl.)g(\(9\))i Fb(78)e Fw(\(1999\),)i(no.)f(2,)f(159{176)485 4110 y([2])35 b(Berretti,)23 b(A.)g(and)i(Gen)n(tile,)f(G.,)48 b Fc(Bryuno)26 b(F)-5 b(unction)26 b(and)h(the)e(Standar)l(d)j(Map)p Fw(,)23 b(Preprin)n(t)h(\(1998\).)485 4227 y([3])35 b(Berretti,)h(A.,)f (Celletti,)h(A.,)g(Chierc)n(hia,)g(L.)d(and)h(F)-6 b(alcolini,)36 b(C.,)72 b Fc(Natur)l(al)37 b(Boundaries)f(for)f(A)n(r)l(e)l(a-)595 4343 y(Pr)l(eserving)26 b(Twist)f(Maps)p Fw(,)f(J.)g(Stat.)g(Ph)n(ys.)g Fb(66)e Fw(\(1992\),)k(no.)d(5{6,)h(1613{1630)485 4459 y([4])35 b(F)-6 b(alcolini,)23 b(C.)f(and)i(de)g(la)f(Lla)n(v)n(e,)g (R.,)48 b Fc(Numeric)l(al)26 b(Calculation)h(of)e(Domains)h(of)g(A)n (nalyticity)e(for)h(Per-)595 4575 y(turb)l(ation)g(The)l(ories)g(in)f (pr)l(esenc)l(e)g(of)h(Smal)t(l)g(Divisors)p Fw(,)d(J.)g(Stat.)g(Ph)n (ys.)g Fb(67)g Fw(\(1992\),)i(no.)e(3{4,)g(645{666.)485 4692 y([5])35 b(Berretti,)28 b(A.)e(and)i(Marmi,)d(S.,)56 b Fc(Sc)l(aling,)30 b(Perturb)l(ative)f(R)l(enormalization)i(and)e(A)n (nalyticity)f(for)h(the)595 4808 y(Standar)l(d)f(Map)e(and)g(Some)g (Gener)l(alizations)p Fw(,)f(Chaos)f(Solitons)g(F)-6 b(ractals)24 b Fb(5)f Fw(\(1995\),)i(no.)f(2,)g(257{269)485 4924 y([6])35 b(Berretti,)18 b(A.,)f Fc(A)n(nalyticity,)i(Sc)l(aling)g (and)h(R)l(enormalization)h(for)f(Some)f(Complex)h(A)n(nalytic)f (Dynamic)l(al)595 5040 y(Systems)p Fw(,)k(Planet.)h(Space)h(Sci.)e Fb(46)g Fw(\(1998\),)i(1487{????)485 5157 y([7])35 b(Da)n(vie,)28 b(A.M.,)e Fc(The)j(critic)l(al)g(function)g(for)h(the)e(semistandar)l (d)j(map)p Fw(,)e(Nonlinearit)n(y)f Fb(7)f Fw(\(1994\),)j(no.)d(1,)595 5273 y(219{229)485 5389 y([8])35 b(Garey)-6 b(,)26 b(M.)f(R.,)g(and)g (Johnson,)i(D.)e(S.,)g Fc(Computers)j(and)g(Intr)l(actability:)36 b(A)27 b(Guide)h(to)f(the)g(The)l(ory)h(of)595 5505 y(NP-Completeness)p Fw(,)c(F)-6 b(reeman,)23 b(San)h(F)-6 b(rancisco,)24 b(1979)p eop %%Page: 25 25 25 24 bop 1276 42 a Fl(GENERALIZED)29 b(ST)-5 b(AND)n(ARD)29 b(MAPS)880 b(25)365 241 y Fw([9])35 b(Gen)n(tile,)23 b(G.)g(and)g(Mastropietro,)g(V.,)f Fc(Metho)l(ds)j(for)g(the)g(A)n (nalysis)g(of)g(the)f(Lindste)l(dt)h(Series)g(for)g(KAM)475 357 y(tori)j(and)h(R)l(enormalizability)g(in)e(Classic)l(al)j(Me)l (chanics.)e(A)g(R)l(eview)f(with)i(Some)f(Applic)l(ations)p Fw(,)g(Rev.)475 473 y(Math.)c(Ph)n(ys.)f Fb(8)g Fw(\(1996\),)j(no.)d (3,)h(393{444)330 589 y([10])35 b(Galla)n(v)n(otti,)24 b(G.,)g Fc(Twistless)h(KAM)g(tori)p Fw(,)f(Comm.)d(Math.)i(Ph)n(ys.)h Fb(164)e Fw(\(1994\),)j(no.)f(1,)f(145{156)330 706 y([11])35 b(Galla)n(v)n(otti,)i(G.,)f Fc(Twistless)f(KAM)f(tori,)k(quasi\015at)d (homo)l(clinic)i(interse)l(ctions,)f(and)g(other)g(c)l(anc)l(el-)475 822 y(lations)d(in)g(the)f(p)l(erturb)l(ation)i(series)f(of)f(c)l (ertain)h(c)l(ompletely)g(inte)l(gr)l(able)g(Hamiltonian)h(systems.)e (A)475 938 y(r)l(eview)p Fw(,)23 b(Rev.)h(Math.)g(Ph)n(ys.)f Fb(6)g Fw(\(1994\),)i(no.)f(3,)f(343{411)330 1054 y([12])35 b(Gen)n(tile,)23 b(G.)e(and)i(Mastropietro,)f(V.,)f Fc(KAM)i(the)l(or)l (em)i(r)l(evisite)l(d)p Fw(,)d(Ph)n(ys.)f(D)h Fb(90)f Fw(\(1996\),)i(no.)f(3,)g(225{234)330 1171 y([13])35 b(Eliasson,)28 b(L.)f(H.,)g Fc(A)n(bsolutely)i(Conver)l(gent)g(Series)g (Exp)l(ansions)j(for)d(Quasi-p)l(erio)l(dic)h(Motions)p Fw(,)f(Uni-)475 1287 y(v)n(ersit)n(y)d(of)f(Sto)r(c)n(kholm)g(preprin)n (t)h(\(1988\),)h(and)f(Math.)f(Ph)n(ys.)g(Electron.)h(J.)f Fb(2)g Fw(\(1996\),)i(pap)r(er)f(4,)f(1{33,)475 1403 y Fa()330 1519 y Fw([14])35 b(P)n(oincar)n(\023)-33 b(e,)23 b(H.,)e Fc(L)l(es)j(m)n(\023)-35 b(etho)l(des)27 b(nouvel)t(les)d(de)h(la)g(m)n (\023)-35 b(ec)l(anique)25 b(c)n(\023)-35 b(eleste)p Fw(,)22 b(Gauthier-Villars,)f(P)n(aris,)g(V)-6 b(ol.)22 b(I,)475 1636 y(1892,)i(V)-6 b(ol.)23 b(I)r(I,)h(1893,)g(V)-6 b(ol.)23 b(I)r(I)r(I,)i(1899)330 1752 y([15])35 b(Moser,)c(J.,)g Fc(On)h(invariant)g(curves)f(of)h(ar)l(e)l(a)h(pr)l(eserving)f (mappings)h(of)f(the)g(annulus)p Fw(,)h(Nac)n(hr.)d(Ak)l(ad.)475 1868 y(Wiss.)23 b(G\177)-35 b(ottingen,)25 b(Math.)f(Ph)n(ys.)f(Kl.)g (I)r(I)h Fb(1962)e Fw(\(1962\),)k(1{20)330 1984 y([16])35 b(Chierc)n(hia,)22 b(L.,)f(and)i(F)-6 b(alcolini,)21 b(C.,)g Fc(A)j(Dir)l(e)l(ct)f(Pr)l(o)l(of)j(of)e(a)g(The)l(or)l(em)i (by)d(Kolmo)l(gor)l(ov)j(in)e(Hamiltonian)475 2100 y(Systems)p Fw(,)f(Ann.)g(Scuola)i(Norm.)c(Sup.)j(Pisa)f(Cl.)g(Sci.)g(\(4\))i Fb(21)e Fw(\(1994\),)i(no.)f(4,)f(541{593)330 2217 y([17])35 b(Whittak)n(er,)i(E.)32 b(T.,)j Fc(A)f(T)-5 b(r)l(e)l(atise)35 b(on)f(the)h(A)n(nalytic)l(al)g(Dynamics)f(of)h(Particles)f(and)i(R)n (igid)e(Bo)l(dies)p Fw(,)475 2333 y(Cam)n(bridge)23 b(Univ)n(ersit)n(y) g(Press,)g(Cam)n(bridge,)f(1947)330 2449 y([18])35 b(Bak)n(er,)d(H.)e (F.,)i Fc(A)n(b)l(elian)g(functions)p Fw(,)g(reprin)n(t)f(of)f(the)i (1897)f(original,)h(Cam)n(bridge)d(Univ)n(ersit)n(y)i(Press,)475 2565 y(Cam)n(bridge)23 b(\(1995\))330 2682 y([19])35 b(Siegel,)22 b(C.)f(L.,)h Fc(T)-5 b(opics)24 b(in)g(Complex)h(F)-5 b(unction)24 b(The)l(ory)p Fw(,)e(John)h(Wiley)f(&)f(Sons,)h(New)g(Y)-6 b(ork,)22 b(V)-6 b(ol.)21 b(I.)h(El-)475 2798 y(liptic)d(F)-6 b(unctions)21 b(and)f(Uniformization)e(Theory)-6 b(,)20 b(1969)g(V)-6 b(ol.)19 b(I)r(I.)h(Automorphic)f(functions)h(and)g(Ab)r (elian)475 2914 y(In)n(tegrals,)j(1971)h(V)-6 b(ol.)22 b(I)r(I)r(I.)i(Ab)r(elian)e(F)-6 b(unctions)24 b(and)g(Mo)r(dular)e(F) -6 b(unctions)24 b(of)f(Sev)n(eral)g(V)-6 b(ariables,)22 b(1973)330 3030 y([20])35 b(Marmi,)21 b(S.,)i(Moussa,)g(P)-6 b(.)23 b(and)i(Y)-6 b(o)r(ccoz,)24 b(J.-C.,)e Fc(Complex)27 b(Brjuno)f(function)p Fw(,)d(Preprin)n(t)h(\(1999\))330 3147 y([21])35 b(Y)-6 b(o)r(ccoz,)20 b(J.-C.,)d Fc(Th)n(\023)-35 b(eor)n(\022)g(eme)22 b(de)f(Sie)l(gel,)f(Nombr)l(es)h(de)f(Bruno)h(et) e(Polin^)-36 b(omes)21 b(Quadr)l(atiques)p Fw(,)f(Ast)n(\023)-33 b(erisque)475 3263 y Fb(231)23 b Fw(\(1995\),)i(3{88)430 3717 y Fv(Alber)l(to)19 b(Berretti,)i(Dip)l(ar)l(timento)d(di)i(Ma)l (tema)l(tica,)f(I)r(I)g(Universit)2471 3711 y(\022)2469 3717 y(a)h(di)f(R)n(oma)g(\(Tor)h(Ver)o(ga)l(t)l(a\),)330 3833 y(Via)26 b(della)g(Ricer)o(ca)g(Scientifica,)g(00133)f(R)n(oma,)h (It)l(al)l(y)f(and)h(Istituto)h(Nazionale)f(di)g(Fisica)h(Nuc-)330 3949 y(leare,)e(Sez.)33 b(Tor)26 b(Ver)o(ga)l(t)l(a)430 4066 y Fc(E-mail)g(addr)l(ess)5 b Fw(:)33 b Fa(berretti@roma2.infn.it) 430 4263 y Fv(Guido)18 b(Gentile,)i(Dip)l(ar)l(timento)e(di)h(Ma)l (tema)l(tica,)f(Universit)2274 4257 y(\022)2272 4263 y(a)h(di)g(R)n(oma)f(3,)i(Lar)o(go)e(S.)g(Leonardo)330 4379 y(Murialdo)25 b(1,)g(00146)f(R)n(oma,)h(It)l(al)l(y)430 4496 y Fc(E-mail)h(addr)l(ess)5 b Fw(:)33 b Fa (gentile@matrm3.mat.uniro)q(ma3)q(.it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9910080857181--