Content-Type: multipart/mixed; boundary="-------------0303210648513" This is a multi-part message in MIME format. ---------------0303210648513 Content-Type: text/plain; name="03-128.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-128.keywords" reccurence,Poincare return time,substitution,subshifts ---------------0303210648513 Content-Type: application/postscript; name="FEIGEN.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="FEIGEN.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: FEIGEN.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\Local\TeX\bin\win32\dvips.exe FEIGEN.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.02.19:1514 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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23 w Fq(R)q Fs(\()p Fq(y)s Fs(\).)36 b(Moreo)n(v)n(er)21 b(if)j Fq(\026)g Fs(is)g(an)f(in)n(v)-5 b(arian)n(t)23 b(measure)g(and)h(\(\006)p Fq(;)14 b(\033)n(;)g(\026)p Fs(\))25 b(is)f(ergo)r(dic,)456 3699 y(then)39 b(b)r(oth)g Fq(R)p 863 3712 V 39 w Fs(and)p 1138 3633 V 39 w Fq(R)g Fs(are)f Fq(\026)p Fs(-almost)f(ev)n(erywhere)g (constan)n(t,)k(so)d(there)g(exist)g(constan)n(ts)456 3799 y(0)22 b Fp(\024)h Fv(r)647 3811 y Fl(0)707 3799 y Fp(\024)g Fv(r)834 3811 y Fl(1)895 3799 y Fp(\024)f(1)p Fs(,)28 b(suc)n(h)f(that)h Fq(R)p 1483 3812 V 1 w Fs(\()p Fq(y)s Fs(\))23 b(=)g Fv(r)1805 3811 y Fl(0)1870 3799 y Fs(a.e.)37 b(and)p 2193 3732 V 27 w Fq(R)q Fs(\()p Fq(y)s Fs(\))23 b(=)g Fv(r)2515 3811 y Fl(1)2580 3799 y Fs(a.e.)555 3899 y(Saussol)35 b(et)i(al.)61 b([9])36 b(sho)n(w)f(that)i(if)f(\(\006)p Fq(;)14 b(\033)n(;)g(\026)p Fs(\))38 b(is)e(ergo)r(dic)f(with)h(p)r(ositiv)n(e)g(en)n(trop)n(y)-7 b(,)37 b(then)456 3998 y Fq(R)p 456 4011 V Fs(\()p Fq(y)s Fs(\))g Fp(\025)g 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b(another)h(generalization)f(of)h(Cassaigne)f(et)h (al)h([2)o(].)37 b(W)-7 b(e)28 b(sho)n(w)f(that)g(in)h(substitutiv)n(e) 456 4712 y(subshifts)35 b Fv(r)847 4724 y Fl(0)920 4712 y Fs(is)h(the)g(minim)n(um)g(of)f(the)h(range)e Fq(R)p 2029 4725 64 4 v 1 w Fs(\(\006\))i(while)f Fv(r)2516 4724 y Fl(1)2589 4712 y Fs(is)h(the)f(maxim)n(um)h(of)f(the)456 4814 y(range)p 686 4747 V 30 w Fq(R)q Fs(\(\006\).)51 b(Moreo)n(v)n(er)29 b(w)n(e)j(describ)r(e)g(an)f(algorithm)g(whic)n(h)h (computes)g Fv(r)2927 4826 y Fl(0)2997 4814 y Fs(and)g Fv(r)3202 4826 y Fl(1)3271 4814 y Fs(for)g(a)456 4914 y(giv)n(en)26 b(substitution.)p 456 5041 499 4 v 555 5133 a Fu(F)-6 b(acult)n(y)31 b(of)f(Mathematics)g(and)g(Ph)n(ysics,)h (Charles)e(Univ)n(ersit)n(y)h(in)f(Prague,)j(Malostransk)n(\023)-33 b(e)30 b(n\023)-35 b(am.)48 b(25,)456 5216 y(CZ-11800)24 b(Praha)g(1,)f(Czec)n(hia.)1933 5315 y Fl(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fl(2)1233 b(PETR)23 b(K)1981 236 y(\027)1981 251 y(URKA)1699 450 y Fs(2.)41 b Fr(Subshifts)555 600 y Fs(F)-7 b(or)28 b(an)h(alphab)r(et)f Fq(A)h Fs(denote)g(b)n(y)f Fq(A)1704 569 y Fm(\003)1772 600 y Fs(the)h(set)f(of)h(\014nite)g(w)n (ords)e(and)i(b)n(y)f Fq(A)2935 569 y Fn(N)3011 600 y Fs(the)h(space)f(of)456 699 y(one-sided)h(in\014nite)i(w)n(ords)e(with) i(the)f(pro)r(duct)h(top)r(ology)-7 b(.)43 b(Denote)31 b(b)n(y)f Fp(j)p Fq(u)p Fp(j)g Fs(the)g(length)h(of)f(a)456 799 y(w)n(ord)d Fq(u)e Fp(2)g Fq(A)877 769 y Fm(\003)944 799 y Fs(and)k(b)n(y)f Fp(j)p Fq(u)p Fp(j)1317 811 y Fo(a)1386 799 y Fs(the)h(n)n(um)n(b)r(er)f(of)h(o)r(ccurrences)e(of)i (a)f(letter)h Fq(a)g Fs(in)g Fq(u)p Fs(.)40 b(The)28 b(empt)n(y)456 898 y(w)n(ord)g(is)h(denoted)g(b)n(y)g Fq(\025)g Fs(and)g Fq(A)1482 868 y Fl(+)1563 898 y Fs(=)c Fq(A)1715 868 y Fm(\003)1773 898 y Fp(n)19 b(f)p Fq(\025)p Fp(g)29 b Fs(is)g(the)g(set)g(of)g(nonempt)n(y)g(w)n(ords.)40 b(W)-7 b(e)30 b(write)456 998 y Fq(v)c Fp(v)d Fq(u)p Fs(,)k(if)h Fq(v)e Fs(=)d Fq(u)986 1013 y Fl([)p Fo(i;j)s Fl(\))1131 998 y Fs(=)g 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b(cylinder)f(set)h(of)g(a)g(w)n (ord)456 1405 y Fq(u)22 b Fp(2)i(L)p Fs(\(\006\))k(is)g([)p Fq(u)p Fs(])22 b(=)h Fp(f)p Fq(x)g Fp(2)g Fs(\006)g(:)g Fq(x)1468 1420 y Fl([0)p Fo(;)p Fm(j)p Fo(u)p Fm(j)p Fl(\))1672 1405 y Fs(=)g Fq(u)p Fp(g)p Fs(.)555 1505 y(Giv)n(en)28 b Fq(y)e Fp(2)e Fs(\006)k(w)n(e)g(de\014ne)g(the)g (sequence)g(of)g(free)f(p)r(ositions)h Fq(s)23 b Fs(=)h(\()p Fq(s)2707 1517 y Fo(k)2748 1505 y Fs(\))2780 1517 y Fo(k)q Fm(\025)p Fl(0)2934 1505 y Fs(in)k Fq(y)j Fs(b)n(y)c(induc-)456 1605 y(tion.)36 b(Set)26 b Fq(s)838 1617 y Fl(0)898 1605 y Fs(=)d(0)i(and)g(if)h Fq(s)1325 1617 y Fo(k)q Fm(\000)p Fl(1)1476 1605 y Fs(has)f(b)r(een)h(already)e(de\014ned,)i(then)g Fq(s)2640 1617 y Fo(k)2704 1605 y Fq(>)c(s)2830 1617 y Fo(k)q Fm(\000)p Fl(1)2982 1605 y Fs(is)j(the)h(largest)456 1704 y(in)n(teger,)g(suc)n(h)i(that)f(for)h(all)f Fq(n)p Fs(,)1295 1842 y Fq(s)1334 1854 y Fo(k)q Fm(\000)p Fl(1)1483 1842 y Fq(<)22 b(n)h Fp(\024)g Fq(s)1770 1854 y Fo(k)1857 1842 y Fp(\))46 b Fs([)p 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2261 3926 245 4 v 2336 4002 a Fq(\013)2389 3978 y Fo(k)2539 3945 y Fs(=)22 b Fq(\027)2667 3957 y Fo(a)456 4124 y Fs(If)k Fq(#)h Fs(is)f(a)g(primitiv)n(e)g (substitution,)h(then)g(there)f(exists)g(a)g Fq(#)p Fs(-p)r(erio)r(dic) g(p)r(oin)n(t)h Fq(x)c Fp(2)h Fq(A)3117 4094 y Fn(N)3190 4124 y Fs(and)i(w)n(e)456 4224 y(assume)f(that)h Fq(x)g Fs(is)f(not)h Fq(\033)s Fs(-p)r(erio)r(dic.)36 b(By)26 b(passing)e(to)i(a)f(p)r(o)n(w)n(er)f(of)i Fq(#)p Fs(,)g(w)n(e)g(can)f (assume)g(that)h Fq(x)456 4324 y Fs(is)i(a)g(\014xed)h(p)r(oin)n(t,)g Fq(#)p Fs(\()p Fq(x)p Fs(\))c(=)f Fq(x)29 b Fs(and)g(moreo)n(v)n(er,)d (the)j(lo)n(w)n(er)e(norm)h Fp(j)p Fq(#)p Fp(j)c Fs(=)h(min)p Fp(fj)p Fq(#)p Fs(\()p Fq(a)p Fs(\))p Fp(j)g Fs(:)f Fq(a)h Fp(2)g Fq(A)p Fp(g)456 4423 y Fs(is)i(at)h(least)f(2.)36 b(The)28 b(corresp)r(onding)e(subshift)i(is)f(the)h(orbit)f(closure) 1074 4573 y(\006)1134 4585 y Fo(#)1202 4573 y Fs(=)p 1289 4493 183 4 v 22 w Fi(O)q Fs(\()p Fq(x)p Fs(\))d(=)f Fp(f)p Fq(y)i Fp(2)e Fq(A)1831 4538 y Fn(N)1901 4573 y Fs(:)g Fp(8)p Fq(n;)14 b Fp(9)p Fq(k)s(;)g(y)2251 4588 y Fl([0)p Fo(;n)p Fl(\))2415 4573 y Fs(=)23 b Fq(x)2550 4588 y Fl([)p Fo(k)q(;k)q Fl(+)p Fo(n)p Fl(\))2784 4573 y Fp(g)456 4711 y Fs(and)g(do)r(es)h(not)g(dep)r(end)g(on)g(the)g(c)n (hoice)f(of)h(the)h(\014xed)e(p)r(oin)n(t)i Fq(x)p Fs(.)36 b(The)24 b(subshift)g(\006)3001 4723 y Fo(#)3069 4711 y Fs(is)g(minimal)456 4811 y(and)j(uniquely)h(ergo)r(dic.)36 b(In)27 b(particular,)g(for)g(ev)n(ery)f Fq(y)g Fp(2)d Fs(\006)2336 4823 y Fo(#)2380 4811 y Fs(,)1411 4949 y(lim)1382 4999 y Fo(n)p Fm(!1)1569 4949 y Fs(#)p Fp(f)p Fq(i)g(<)f(n)h Fs(:)g Fq(y)1979 4961 y Fo(i)2030 4949 y Fs(=)f Fq(a)p Fp(g)p Fq(=n)g Fs(=)h Fq(\026)2455 4961 y Fo(a)2495 4949 y Fq(:)456 5116 y Fs(W)-7 b(e)23 b(use)h(the)g(same)e(sym)n(b)r(ol)h Fq(\026)h Fs(for)f(the)h(measure)e Fq(\026)p Fs(\()p Fq(W)12 b Fs(\))24 b(of)g(a)f(set)g Fq(W)35 b Fp(\022)23 b Fs(\006)2781 5128 y Fo(#)2825 5116 y Fs(.)36 b(The)23 b(complexit)n(y)456 5216 y(function)30 b Fq(P)12 b Fs(\()p Fq(n)p Fs(\))26 b(=)g(#)p Fp(L)1205 5185 y Fo(n)1251 5216 y Fs(\(\006)1343 5228 y Fo(#)1387 5216 y Fs(\))h(=)e(#)p Fp(f)p Fq(u)h Fp(2)g(L)p Fs(\(\006)1951 5228 y Fo(#)1996 5216 y Fs(\))h(:)f Fp(j)p Fq(u)p Fp(j)g Fs(=)g Fq(n)p Fp(g)i Fs(is)i(sublinear,)f(i.e.,)h(there)g(exist)p eop %%Page: 3 3 3 2 bop 1038 251 a Fl(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)h(SUBSTITUTIVE)h(SUBSHIFTS)548 b(3)456 450 y Fs(0)32 b Fq(<)h(a)g(<)f(b)i Fs(suc)n(h)f(that)h Fq(an)f Fp(\024)f Fq(P)12 b Fs(\()p Fq(n)p Fs(\))34 b Fp(\024)e Fq(bn)h Fs(for)g(eac)n(h)g Fq(n)p Fs(.)55 b(The)34 b(return)f(times)h(of)f (cylinders)456 550 y(are)f(sublinear)g(to)r(o.)53 b(If)34 b Fq(u)d Fp(2)i(L)1468 520 y Fo(n)1513 550 y Fs(\(\006)1605 562 y Fo(#)1650 550 y Fs(\),)i(then)e Fq(an)f Fp(\024)g Fq(\034)9 b Fs(\([)p Fq(u)p Fs(]\))33 b Fp(\024)f Fq(bn)p Fs(.)53 b(W)-7 b(e)33 b(sho)n(w)f(no)n(w)h(that)g(in)456 649 y(substitutiv)n(e)28 b(subshifts)g Fv(r)1293 661 y Fl(0)1353 649 y Fq(<)23 b Fv(r)1480 661 y Fl(1)1517 649 y Fs(.)456 797 y Fv(Prop)s(osition)35 b(1.)44 b Fh(If)35 b Fs(\006)f Fh(is)g(a)h(substitutive)e(subshift,)k(then)d(ther)l(e)g (exists)g Fq(y)g Fp(2)d Fs(\006)j Fh(such)g(that)p 456 830 64 4 v 456 897 a Fq(R)p Fs(\()p Fq(y)s Fs(\))24 b Fq(>)e(R)p 738 910 V 1 w Fs(\()p Fq(y)s Fs(\))p Fh(.)456 1045 y(Pr)l(o)l(of.)43 b Fs(Let)23 b(0)g Fq(<)g(a)g(<)f(b)h Fs(b)r(e)h(constan)n(ts)f(whic)n(h)g(satisfy)h Fq(an)e Fp(\024)h Fq(P)12 b Fs(\()p Fq(n)p Fs(\))23 b Fp(\024)g Fq(bn)g Fs(and)g Fq(an)g Fp(\024)g Fq(\034)9 b Fs(\([)p Fq(u)p Fs(]\))24 b Fp(\024)456 1145 y Fq(bn)f Fs(for)h(eac)n(h)g Fq(u)e Fp(2)i(L)1079 1115 y Fo(n)1124 1145 y Fs(\(\006)1216 1157 y Fo(#)1261 1145 y Fs(\).)36 b(Fix)24 b(a)g(real)f(n)n(um)n(b)r (er)i(0)d Fq(<)h(c)g(<)f Fs(1)i(and)g(assume)g(that)h(for)e(all)h Fq(y)i Fp(2)d Fs(\006)456 1245 y(and)k(for)g(all)g Fq(k)s Fs(,)h Fq(s)995 1257 y Fo(k)q Fl(+1)1143 1245 y Fp(\024)23 b Fs(\()p Fq(c)18 b Fs(+)g(1\))p Fq(s)1513 1257 y Fo(k)1554 1245 y Fs(.)37 b(Then)27 b Fq(s)1869 1257 y Fo(k)1933 1245 y Fp(\024)c Fs(\()p Fq(c)c Fs(+)f(1\))2265 1214 y Fo(k)q Fm(\000)p Fl(1)2418 1245 y Fs(and)1421 1383 y(2)1463 1349 y Fo(k)1526 1383 y Fp(\024)23 b Fq(P)12 b Fs(\()p Fq(s)1750 1395 y Fo(k)1790 1383 y Fs(\))24 b Fp(\024)e Fq(bs)2008 1395 y Fo(k)2072 1383 y Fp(\024)h Fq(b)p Fs(\()p Fq(c)18 b Fs(+)g(1\))2439 1349 y Fo(k)456 1518 y Fs(and)33 b(this)i(is)e(a)h(con)n(tradiction.)54 b(Th)n(us)34 b(there)g(exists)f Fq(y)j Fp(2)e Fs(\006)g(and)g(an)g (increasing)e(sequence)456 1617 y Fq(k)499 1629 y Fl(1)559 1617 y Fq(<)23 b(k)690 1629 y Fl(2)750 1617 y Fq(<)g Fp(\001)14 b(\001)g(\001)g Fs(,)27 b(suc)n(h)h(that)f Fq(s)1405 1629 y Fo(k)1440 1637 y Fk(i)1467 1629 y Fl(+1)1574 1617 y Fp(\000)18 b Fq(s)1696 1629 y Fo(k)1731 1637 y Fk(i)1784 1617 y Fp(\025)23 b Fq(cs)1947 1629 y Fo(k)1982 1637 y Fk(i)2013 1617 y Fs(.)37 b(It)28 b(follo)n(ws)1318 1718 y Fq(\034)1354 1730 y Fo(k)1389 1738 y Fk(i)1416 1730 y Fl(+1)p 1318 1755 187 4 v 1359 1831 a Fq(s)1398 1843 y Fo(k)1433 1851 y Fk(i)1533 1774 y Fp(\000)1627 1718 y Fq(\034)1663 1730 y Fo(k)1698 1738 y Fk(i)1725 1730 y Fl(+1)p 1626 1755 189 4 v 1626 1831 a Fq(s)1665 1843 y Fo(k)1700 1851 y Fk(i)1726 1843 y Fl(+1)1847 1774 y Fp(\025)1945 1718 y Fq(\034)1981 1730 y Fo(k)2016 1738 y Fk(i)2043 1730 y Fl(+1)2149 1718 y Fp(\001)19 b Fq(c)f Fp(\001)h Fq(s)2326 1730 y Fo(k)2361 1738 y Fk(i)p 1945 1755 447 4 v 2021 1831 a Fq(s)2060 1843 y Fo(k)2095 1851 y Fk(i)2126 1831 y Fq(s)2165 1843 y Fo(k)2200 1851 y Fk(i)2227 1843 y Fl(+1)2424 1774 y Fp(\025)k Fq(ac)456 1968 y Fs(so)p 558 1902 64 4 v 27 w Fq(R)p Fs(\()p Fq(y)s Fs(\))c Fp(\000)f Fq(R)p 831 1981 V 1 w Fs(\()p Fq(y)s Fs(\))23 b Fp(\025)g Fq(ac)p Fs(.)2163 b Fg(\003)555 2118 y Fs(W)-7 b(e)28 b(shall)f(use)h(frequen)n(tly)f(the)h(follo)n (wing)f("deco)r(ding")f(theorem.)456 2266 y Fv(Theorem)32 b(2)i Fs(\(Moss)n(\023)-39 b(e)28 b([8)o(]\))p Fv(.)43 b Fh(L)l(et)31 b Fq(#)g Fh(b)l(e)h(a)g(primitive)h(substitution)d(with) i(an)g(ap)l(erio)l(dic)i(\014xe)l(d)456 2366 y(p)l(oint)i Fq(x)p Fh(.)57 b(De\014ne)35 b(a)h(function)g Fq(h)e Fs(:)g Ff(N)43 b Fp(!)34 b Ff(N)46 b Fh(by)36 b Fq(h)p Fs(\()p Fq(n)p Fs(\))e(=)g Fp(j)p Fq(#)p Fs(\()p Fq(x)2485 2381 y Fl([0)p Fo(;n)p Fl(\))2629 2366 y Fp(j)p Fh(.)56 b(Then)37 b(ther)l(e)f(exists)f(a)456 2466 y(c)l(ontext)g(length)i Fq(m)e(>)g Fs(0)h Fh(such)h(that)g(for)g(every)g Fq(u)e Fp(2)h(L)p Fs(\(\006)2345 2478 y Fo(#)2390 2466 y Fs(\))h Fh(of)g(length)g(at)g(le)l(ast)f Fs(2)p Fq(m)g Fh(ther)l(e)456 2565 y(exist)i Fq(i;)14 b(j)45 b Fp(2)40 b Ff(N)49 b Fh(with)40 b Fs(0)f Fp(\024)h Fq(i)f Fp(\024)h Fq(m)p Fh(,)h Fp(j)p Fq(u)p Fp(j)25 b(\000)g Fq(m)40 b Fp(\024)f Fq(j)45 b Fp(\024)40 b(j)p Fq(u)p Fp(j)e Fh(and)i(unique)e(wor)l(d)i Fq(v)j Fp(2)e(L)p Fs(\(\006\))456 2665 y Fh(with)e Fq(u)693 2680 y Fl([)p Fo(i;j)s Fl(\))856 2665 y Fs(=)g Fq(#)p Fs(\()p Fq(v)s Fs(\))p Fh(,)k(such)c(that)g(whenever)h Fq(x)1979 2680 y Fl([)p Fo(n;n)p Fl(+)p Fm(j)p Fo(u)p Fm(j)p Fl(\))2300 2665 y Fs(=)f Fq(u)g 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Fq(u)p Fs(])1643 3133 y Fo(n)1711 3121 y Fs(=)23 b Fp(f)p Fq(y)j Fp(2)d Fs(\002)2051 3133 y Fo(#)2119 3121 y Fs(:)g Fq(y)2206 3136 y Fl([)p Fo(n;n)p Fl(+)p Fm(j)p Fo(u)p Fm(j)p Fl(\))2510 3121 y Fs(=)f Fq(u)p Fp(g)p Fs(.)37 b(The)28 b(cylinder)f(of)h(the)456 3221 y(empt)n(y)d(w)n(ord)g(is)g (the)h(full)g(space)f([)p Fq(\025)p Fs(])e(=)g([)p Fq(\025)p Fs(])1795 3233 y Fl(0)1856 3221 y Fs(=)g(\002)2009 3233 y Fo(#)2052 3221 y Fs(.)37 b(W)-7 b(e)25 b(extend)h(the)g(substitution) g(to)f(a)h(map)456 3321 y Fq(#)d Fs(:)g Fq(A)636 3291 y Fn(Z)702 3321 y Fp(!)g Fq(A)870 3291 y Fn(Z)941 3321 y Fs(b)n(y)923 3455 y Fq(#)p Fs(\()p Fq(:)14 b(:)g(:)g(u)1163 3467 y Fm(\000)p Fl(2)1252 3455 y Fq(u)1300 3467 y Fm(\000)p Fl(1)1389 3455 y Fq(:u)1460 3467 y Fl(0)1496 3455 y Fq(u)1544 3467 y Fl(1)1595 3455 y Fq(:)g(:)g(:)p Fs(\))23 b(=)g Fq(:)14 b(:)g(:)g(#)p Fs(\()p Fq(u)2075 3467 y Fm(\000)p Fl(2)2164 3455 y Fs(\))p Fq(#)p Fs(\()p Fq(u)2325 3467 y Fm(\000)p Fl(1)2414 3455 y Fs(\))p Fq(:#)p Fs(\()p Fq(u)2598 3467 y Fl(0)2635 3455 y Fs(\))p Fq(#)p Fs(\()p Fq(u)2796 3467 y Fl(1)2834 3455 y Fs(\))g Fq(:)g(:)g(:)456 3590 y Fs(where)27 b(the)h(dot)g(is)f(placed)h(immediately)g(b)r(efore) f(the)h(zero)f(co)r(ordinate.)36 b(As)28 b(a)f(consequence)456 3689 y(of)g(Theorem)g(2)g(w)n(e)g(ha)n(v)n(e)456 3837 y Fv(Prop)s(osition)j(3.)461 3923 y Fh(1.)43 b Fq(#)p Fs(\(\002)717 3935 y Fo(#)761 3923 y Fs(\))23 b Fp(\022)g Fs(\002)969 3935 y Fo(#)1013 3923 y Fh(.)461 4014 y(2.)43 b Fq(#)23 b Fs(:)g(\002)754 4026 y Fo(#)821 4014 y Fp(!)g Fs(\002)992 4026 y Fo(#)1066 4014 y Fh(is)30 b(one-to-one)f(and)i(op)l (en.)461 4105 y(3.)43 b(If)30 b Fq(u)23 b Fp(2)g(L)p Fs(\(\006)956 4117 y Fo(#)1001 4105 y Fs(\))p Fh(,)30 b(then)g Fq(#)p Fs(\([)p Fq(u)p Fs(])1448 4117 y Fl(0)1485 4105 y Fs(\))23 b(=)g([)p Fq(#)p Fs(\()p Fq(u)p Fs(\)])1835 4117 y Fl(0)1902 4105 y Fh(in)30 b Fs(\002)2069 4117 y Fo(#)2113 4105 y Fh(.)461 4197 y(4.)43 b(F)-6 b(or)27 b(every)i Fq(y)c Fp(2)f Fs(\002)1151 4209 y Fo(#)1222 4197 y Fh(ther)l(e)j(exists)g(a)h(unique)f Fq(z)f Fp(2)e Fs(\002)2195 4209 y Fo(#)2266 4197 y Fh(and)k(unique)f Fq(i)22 b(<)h Fp(j)p Fq(#)p Fs(\()p Fq(z)2972 4209 y Fl(0)3009 4197 y Fs(\))p Fp(j)p Fh(,)29 b(such)e(that)574 4296 y Fq(y)e Fs(=)e Fq(\033)778 4266 y Fo(i)806 4296 y Fs(\()p Fq(#)p Fs(\()p Fq(z)t Fs(\)\))p Fh(.)456 4444 y Fv(De\014nition)30 b(4.)41 b Fh(F)-6 b(or)30 b(a)g(clop)l(en)h (\(close)l(d)g(and)f(op)l(en\))g(set)g Fq(W)35 b Fp(\022)22 b Fs(\002)2560 4456 y Fo(#)2604 4444 y Fh(,)30 b(we)g(set)652 4583 y Fq(l)r Fs(\()p Fq(W)12 b Fs(\))83 b(=)g(max)o Fp(f)p Fq(l)24 b Fp(\024)f Fs(0)g(:)g Fp(8)p Fq(y)i Fp(2)e Fq(W)n(;)14 b Fp(8)p Fq(z)25 b Fp(2)f Fq(A)2064 4548 y Fn(Z)2107 4583 y Fq(;)14 b Fs(\()p Fq(z)2215 4598 y Fl([)p Fo(l;)p Fm(1)p Fl(\))2394 4583 y Fs(=)23 b Fq(y)2523 4598 y Fl([)p Fo(l;)p Fm(1)p Fl(\))2725 4583 y Fp(\))46 b Fq(z)26 b Fp(2)d Fq(W)12 b Fs(\))p Fp(g)637 4707 y Fq(p)p Fs(\()p Fq(W)g Fs(\))83 b(=)g(min)p Fp(f)p Fq(n)23 b Fp(\024)f Fs(0)h(:)g Fp(8)p Fq(y)s(;)14 b(z)25 b Fp(2)e Fq(W)n(;)14 b(y)1939 4722 y Fl([)p Fo(n;)p Fl(0\))2105 4707 y Fs(=)23 b Fq(z)2232 4722 y Fl([)p Fo(n;)p Fl(0\))2374 4707 y Fp(g)639 4832 y Fq(q)s Fs(\()p Fq(W)12 b Fs(\))83 b(=)g(max)o Fp(f)p Fq(n)23 b Fp(\025)f Fs(0)h(:)g Fp(8)p Fq(y)s(;)14 b(z)25 b Fp(2)f Fq(W)n(;)14 b(y)1956 4847 y Fl([0)p Fo(;n)p Fl(\))2121 4832 y Fs(=)23 b Fq(z)2248 4847 y Fl([0)p Fo(;n)p Fl(\))2390 4832 y Fp(g)639 4970 y Fq(r)r Fs(\()p Fq(W)12 b Fs(\))84 b(=)f(min)p Fp(f)p Fq(l)24 b Fp(\025)f Fs(0)f(:)i Fp(8)p Fq(y)g Fp(2)g Fq(W)n(;)14 b Fp(8)p Fq(z)25 b Fp(2)f Fq(A)2048 4935 y Fn(Z)2091 4970 y Fq(;)14 b Fs(\()p Fq(z)2199 4985 y Fl(\()p Fm(\0001)p Fo(;l)p Fl(\))2437 4970 y Fs(=)22 b Fq(y)2565 4985 y Fl(\()p Fm(\0001)p Fo(;l)p Fl(\))2826 4970 y Fp(\))47 b Fq(z)26 b Fp(2)d Fq(W)12 b Fs(\))p Fp(g)456 5116 y Fh(Denote)35 b(by)h Fp(j)p Fq(W)12 b Fp(j)34 b Fs(=)g Fq(r)r Fs(\()p Fq(W)12 b Fs(\))24 b Fp(\000)e Fq(l)r Fs(\()p Fq(W)12 b Fs(\))36 b Fh(the)g(length)g(of)g Fq(W)48 b Fh(and)36 b(by)g Fq(c)p Fs(\()p Fq(W)12 b Fs(\))35 b Fp(2)f Fq(A)2931 5086 y Fo(q)r Fl(\()p Fo(W)9 b Fl(\))p Fm(\000)p Fo(p)p Fl(\()p Fo(W)g Fl(\))3336 5116 y Fh(the)456 5216 y(c)l(ommon)30 b(c)l(entr)l(al)f(p)l(art)h(of)h Fq(W)12 b Fh(,)30 b(such)g(that)f(for)i(al)t(l)g Fq(y)25 b Fp(2)f Fq(W)12 b Fh(,)30 b Fq(y)2414 5231 y Fl([)p Fo(p)p Fl(\()p Fo(W)9 b Fl(\))p Fo(;q)r Fl(\()p Fo(W)g Fl(\)\))2819 5216 y Fs(=)22 b Fq(c)p Fs(\()p Fq(W)12 b Fs(\))p Fh(.)p eop %%Page: 4 4 4 3 bop 456 251 a Fl(4)1233 b(PETR)23 b(K)1981 236 y(\027)1981 251 y(URKA)555 450 y Fs(Then)34 b Fq(l)r Fs(\()p Fq(W)12 b Fs(\))33 b Fp(\024)g Fq(p)p Fs(\()p Fq(W)12 b Fs(\))33 b Fp(\024)g Fq(q)s Fs(\()p Fq(W)12 b Fs(\))33 b Fp(\024)g Fq(r)r Fs(\()p Fq(W)12 b Fs(\))35 b(and)e Fq(W)46 b Fs(is)34 b(a)f(union)g(of)h(cylinders)f(of)g(length)456 550 y Fp(j)p Fq(W)12 b Fp(j)28 b Fs(p)r(ositioned)g(at)h Fq(l)r Fs(\()p Fq(W)12 b Fs(\).)39 b(All)29 b(these)f(cylinders)g(coincide)g (at)h([)p Fq(p)p Fs(\()p Fq(W)12 b Fs(\))p Fq(;)i(q)s Fs(\()p Fq(W)e Fs(\)\).)40 b(F)-7 b(or)28 b(the)h(full)456 649 y(set)e Fq(W)35 b Fs(=)23 b([)p Fq(\025)p Fs(])28 b(w)n(e)f(ha)n(v)n(e)g Fq(l)r Fs(\()p Fq(W)12 b Fs(\))23 b(=)f Fq(p)p Fs(\()p Fq(W)12 b Fs(\))24 b(=)e Fq(q)s Fs(\()p Fq(W)12 b Fs(\))24 b(=)f Fq(r)r Fs(\()p Fq(W)12 b Fs(\))24 b(=)f(0.)1359 1303 y Fd(\010)1442 1262 y(\010)1513 1227 y(\010)1359 1109 y(H)1442 1150 y(H)1513 1185 y(H)p 1596 1187 709 4 v 1832 1256 a Fq(c)p Fs(\()p Fq(W)12 b Fs(\))2304 1227 y Fd(H)2387 1268 y(H)2458 1303 y(H)2304 1185 y(\010)2387 1144 y(\010)2458 1109 y(\010)2422 1168 y(H)2458 1185 y(H)1265 854 y Fq(l)r Fs(\()p Fq(W)g Fs(\))p 1358 1020 4 119 v 1359 1020 a Fd(?)1501 854 y Fq(p)p Fs(\()p Fq(W)g Fs(\))p 1594 1020 V 1596 1020 a Fd(?)1808 854 y Fs(0)p 1830 1020 V 1832 1020 a Fd(?)2210 854 y Fq(q)s Fs(\()p Fq(W)g Fs(\))p 2303 1020 V 2304 1020 a Fd(?)2446 854 y Fq(r)r Fs(\()p Fq(W)g Fs(\))p 2539 1020 V 2541 1020 a Fd(?)1509 1502 y Fr(Figure)32 b(1.)41 b Fs(A)28 b(clop)r(en)f(set)555 1687 y(If)33 b Fq(W)42 b Fp(\022)30 b Fs(\002)923 1699 y Fo(#)999 1687 y Fs(is)i(a)f(clop)r (en)h(set,)i(then)e Fq(#)p Fs(\()p Fq(W)12 b Fs(\))33 b(is)f(a)f(clop)r(en)h(set)g(to)r(o.)50 b(W)-7 b(e)32 b(in)n(v)n(estigate)f(the)456 1787 y(prop)r(erties)26 b(of)i(the)g(iterates)f Fq(#)1429 1757 y Fo(k)1470 1787 y Fs(\()p Fq(W)12 b Fs(\).)456 1931 y Fv(Prop)s(osition)29 b(5.)40 b Fh(Ther)l(e)30 b(exists)f(an)g(algorithm)i(which,)g(given)f (a)f(clop)l(en)h(set)f Fq(W)12 b Fh(,)29 b(c)l(omputes)456 2030 y(the)h(limit)1411 2130 y Fq(\037)p Fs(\()p Fq(W)12 b Fs(\))23 b(=)50 b(lim)1728 2184 y Fo(k)q Fm(!1)1911 2130 y Fq(q)s Fs(\()p Fq(#)2032 2095 y Fo(k)2073 2130 y Fs(\()p Fq(W)12 b Fs(\)\))19 b Fp(\001)g Fq(\013)2373 2095 y Fm(\000)p Fo(k)2466 2130 y Fq(:)456 2307 y Fh(Pr)l(o)l(of.)43 b Fs(Let)35 b Fq(f)45 b Fs(:)36 b Fq(A)g Fp(!)g Fq(A)f Fs(b)r(e)h(a)f(\014nite)h(dynamical)e(system)i(giv)n(en)e(b)n(y)h Fq(f)9 b Fs(\()p Fq(a)p Fs(\))36 b(=)f Fq(#)p Fs(\()p Fq(a)p Fs(\))3237 2319 y Fl(0)3311 2307 y Fs(and)456 2406 y(set)g Fq(A)655 2418 y Fl(0)727 2406 y Fs(=)g Fp(f)p Fq(a)g Fp(2)h Fq(A)f Fs(:)h([)p Fq(a)p Fs(])23 b Fp(\\)h Fq(W)47 b Fp(6)p Fs(=)35 b Fp(;g)p Fs(.)58 b(If)36 b(for)e(all)h Fq(k)j Fp(\025)d Fs(0)f Fq(f)2432 2376 y Fo(k)2473 2406 y Fs(\()p Fq(A)2567 2418 y Fl(0)2605 2406 y Fs(\))h(con)n(tains)f(at)h (least)f(t)n(w)n(o)456 2506 y(elemen)n(ts,)28 b(then)h Fq(q)s Fs(\()p Fq(#)1130 2476 y Fo(k)1171 2506 y Fs(\()p Fq(W)12 b Fs(\)\))25 b(=)f(0)k(and)g Fq(\037)p Fs(\()p Fq(W)12 b Fs(\))24 b(=)g(0.)39 b(Assume)28 b(that)g(for)g(some)g Fq(j)h(>)24 b Fs(0,)k Fq(f)3246 2476 y Fo(j)3280 2506 y Fs(\()p Fq(A)3374 2518 y Fl(0)3412 2506 y Fs(\))456 2606 y(is)j(a)f(singleton,)i(so)e Fq(q)s Fs(\()p Fq(#)1220 2576 y Fo(j)1255 2606 y Fs(\()p Fq(W)12 b Fs(\)\))30 b Fq(>)f Fs(0.)46 b(Let)32 b Fq(v)1869 2618 y Fo(k)1939 2606 y Fs(=)c Fq(#)2081 2576 y Fo(k)2122 2606 y Fs(\()p Fq(W)12 b Fs(\))2276 2622 y Fl([0)p Fo(;q)r Fl(\()p Fo(#)2446 2606 y Fk(k)2483 2622 y Fl(\()p Fo(W)d Fl(\)\)\))2662 2606 y Fs(,)32 b(so)f Fq(q)s Fs(\()p Fq(#)2944 2576 y Fo(k)2985 2606 y Fs(\()p Fq(W)12 b Fs(\)\))30 b(=)e Fp(j)p Fq(v)3357 2618 y Fo(k)3398 2606 y Fp(j)p Fs(.)456 2708 y(Since)f Fp(j)p Fq(#)p Fp(j)d(\025)e Fs(2,)28 b Fp(j)p Fq(v)1034 2720 y Fo(k)q Fl(+1)1159 2708 y Fp(j)23 b(\025)g Fs(2)p Fp(j)p Fq(v)1398 2720 y Fo(k)1438 2708 y Fp(j)28 b Fs(and)f Fp(j)p Fq(v)1713 2720 y Fo(k)1755 2708 y Fp(j)g Fs(tend)h(to)g(in\014nit)n(y)-7 b(.)37 b(Set)1039 2892 y Fq(m)1112 2904 y Fl(1)1173 2892 y Fs(=)1260 2775 y Fj(\030)1319 2836 y Fs(2)p Fq(m)p 1319 2873 115 4 v 1329 2949 a Fp(j)p Fq(#)p Fp(j)1443 2775 y Fj(\031)1505 2892 y Fq(;)97 b(m)1698 2904 y Fl(2)1758 2892 y Fs(=)1846 2775 y Fj(\030)1987 2836 y Fq(m)p 1904 2873 239 4 v 1904 2949 a Fp(j)p Fq(#)p Fp(j)19 b(\000)f Fs(1)2153 2775 y Fj(\031)2215 2892 y Fq(;)97 b(q)2372 2904 y Fo(j)2430 2892 y Fs(=)22 b Fq(q)s Fs(\()p Fq(#)2638 2858 y Fo(j)2674 2892 y Fs(\()p Fq(W)12 b Fs(\)\))456 3072 y(where)28 b Fq(m)g Fs(is)h(the)g(con)n(text)g(length)f(from)h(Theorem)f(2.)39 b(Let)29 b Fq(j)2416 3084 y Fl(0)2479 3072 y Fp(\025)24 b Fs(0)29 b(b)r(e)g(the)g(\014rst)f(in)n(teger)g(for)456 3171 y(whic)n(h)f Fq(q)730 3183 y Fo(j)757 3191 y Fc(0)817 3171 y Fp(\025)c Fq(m)978 3183 y Fl(1)1015 3171 y Fs(.)37 b(F)-7 b(or)27 b Fq(j)h Fp(\025)23 b Fq(j)1408 3183 y Fl(0)1473 3171 y Fs(set)1326 3311 y Fq(V)1374 3323 y Fo(j)1433 3311 y Fs(=)f Fp(f)p Fq(y)1603 3326 y Fl([)p Fo(q)1652 3334 y Fk(j)1683 3326 y Fm(\000)p Fo(m)1794 3334 y Fc(1)1826 3326 y Fo(;q)1876 3334 y Fk(j)1907 3326 y Fl(+)p Fo(m)2017 3334 y Fc(2)2049 3326 y Fl(\))2102 3311 y Fs(:)i Fq(y)h Fp(2)f Fq(#)2343 3277 y Fo(j)2378 3311 y Fs(\()p Fq(W)12 b Fs(\))p Fp(g)456 3454 y Fs(By)27 b(Theorem)g(2,)g(for)g(ev)n(ery)f Fq(y)g Fp(2)d Fq(#)1569 3424 y Fo(j)1605 3454 y Fs(\()p Fq(W)12 b Fs(\))28 b(w)n(e)f(ha)n(v)n (e)f Fq(q)2137 3466 y Fo(j)s Fl(+1)2280 3454 y Fp(\024)c(j)p Fq(#)p Fs(\()p Fq(y)2512 3469 y Fl([0)p Fo(;q)2614 3477 y Fk(j)2645 3469 y Fl(\))2675 3454 y Fs(\))p Fp(j)d Fs(+)f Fq(m)28 b Fs(and)f(therefore)690 3591 y Fp(j)p Fq(#)p Fs(\()p Fq(y)835 3606 y Fl([0)p Fo(;q)937 3614 y Fk(j)968 3606 y Fl(+)p Fo(m)1078 3614 y Fc(2)1110 3606 y Fl(\))1141 3591 y Fs(\))p Fp(j)18 b(\000)g Fq(q)1334 3603 y Fo(j)s Fl(+1)1477 3591 y Fp(\025)k(j)p Fq(#)p Fs(\()p Fq(y)1709 3606 y Fl([)p Fo(q)1758 3614 y Fk(j)1790 3606 y Fo(;q)1840 3614 y Fk(j)1871 3606 y Fl(+)p Fo(m)1981 3614 y Fc(2)2013 3606 y Fl(\))2043 3591 y Fs(\))p Fp(j)d(\000)f Fq(m)23 b Fp(\025)f Fq(m)2456 3603 y Fl(2)2512 3591 y Fp(\001)d(j)p Fq(#)p Fp(j)f(\000)g Fq(m)23 b Fp(\025)g Fq(m)3007 3603 y Fl(2)456 3728 y Fs(Th)n(us)36 b Fq(#)p Fs(\()p Fq(y)s Fs(\))832 3743 y Fl([)p Fo(q)881 3751 y Fk(j)r Fc(+1)983 3743 y Fm(\000)p Fo(m)1094 3751 y Fc(1)1126 3743 y Fo(;q)1176 3751 y Fk(j)r Fc(+1)1278 3743 y Fl(+)p Fo(m)1388 3751 y Fc(2)1421 3743 y Fl(\))1487 3728 y Fs(is)h(a)f(sub)n(w)n(ord)e(of)j Fq(#)p Fs(\()p Fq(y)2220 3743 y Fl([)p Fo(q)2269 3751 y Fk(j)2300 3743 y Fm(\000)p Fo(m)2411 3751 y Fc(1)2443 3743 y Fo(;q)2493 3751 y Fk(j)2525 3743 y Fl(+)p Fo(m)2635 3751 y Fc(2)2667 3743 y Fl(\))2697 3728 y Fs(\))g(and)f Fq(V)2984 3740 y Fo(j)s Fl(+1)3140 3728 y Fs(is)g(deter-)456 3830 y(mined)c(b)n(y)h Fq(V)878 3842 y Fo(j)913 3830 y Fs(.)51 b(Since)33 b Fq(V)1257 3842 y Fo(j)1325 3830 y Fs(are)e(\014nite)i(\(and)f(b)r(ounded\),)j(there)d(exist)g Fq(j)2736 3842 y Fl(0)2805 3830 y Fp(\024)e Fq(j)36 b(<)31 b(j)26 b Fs(+)c Fq(r)35 b Fs(suc)n(h)456 3929 y(that)27 b Fq(V)683 3941 y Fo(j)s Fl(+)p Fo(r)r Fl(+)p Fo(i)900 3929 y Fs(=)c Fq(V)1036 3941 y Fo(j)s Fl(+)p Fo(i)1173 3929 y Fs(for)k(all)h Fq(i)22 b Fp(\025)h Fs(0.)36 b(There)27 b(exist)h Fq(b;)14 b(c)22 b Fp(2)i(L)p Fs(\(\006)2452 3941 y Fo(#)2497 3929 y Fs(\))k(suc)n(h)f(that)1033 4069 y Fq(y)f Fp(2)e Fq(#)1228 4035 y Fo(j)1263 4069 y Fs(\()p Fq(W)12 b Fs(\))83 b Fp(\))g Fq(y)1707 4084 y Fl([0)p Fo(;q)1809 4092 y Fk(j)1840 4084 y Fl(\))1893 4069 y Fs(=)23 b Fq(b)950 4206 y(y)j Fp(2)d Fq(#)1144 4172 y Fo(j)s Fl(+)p Fo(r)1263 4206 y Fs(\()p Fq(W)12 b Fs(\))83 b Fp(\))g Fq(y)1707 4221 y Fl([0)p Fo(;q)1809 4229 y Fk(j)r Fc(+)p Fk(r)1912 4221 y Fl(\))1965 4206 y Fs(=)23 b Fq(#)2102 4172 y Fo(r)2139 4206 y Fs(\()p Fq(b)p Fs(\))p Fq(c)929 4349 y(y)i Fp(2)f Fq(#)1123 4314 y Fo(j)s Fl(+)p Fo(lr)1263 4349 y Fs(\()p Fq(W)12 b Fs(\))83 b Fp(\))g Fq(y)1707 4364 y Fl([0)p Fo(;q)1809 4373 y Fk(j)r Fc(+)p Fk(lr)1932 4364 y Fl(\))1985 4349 y Fs(=)23 b Fq(#)2122 4314 y Fo(lr)2180 4349 y Fs(\()p Fq(b)p Fs(\))p Fq(#)2329 4314 y Fl(\()p Fo(l)p Fm(\000)p Fl(1\))p Fo(r)2524 4349 y Fs(\()p Fq(c)p Fs(\))14 b Fp(\001)g(\001)g(\001)g Fq(#)2798 4314 y Fo(r)2835 4349 y Fs(\()p Fq(c)p Fs(\))p Fq(c)456 4483 y Fs(It)28 b(follo)n(ws)902 4656 y Fq(\037)p Fs(\()p Fq(W)12 b Fs(\))83 b(=)102 b(lim)1339 4710 y Fo(l)p Fm(!1)1516 4600 y Fp(j)p Fq(#)1588 4570 y Fo(lr)1646 4600 y Fs(\()p Fq(b)p Fs(\))p Fq(#)1795 4570 y Fl(\()p Fo(l)p Fm(\000)p Fl(1\))p Fo(r)1991 4600 y Fs(\()p Fq(c)p Fs(\))14 b Fp(\001)g(\001)g (\001)g Fq(#)2265 4570 y Fo(r)2302 4600 y Fs(\()p Fq(c)p Fs(\))p Fq(c)p Fp(j)p 1516 4637 945 4 v 1892 4713 a Fq(\013)1945 4689 y Fo(j)s Fl(+)p Fo(lr)1191 4839 y Fs(=)83 b Fq(\013)1392 4805 y Fm(\000)p Fo(j)1505 4760 y Fj(X)1493 4942 y Fo(i<)p Fm(j)p Fo(b)p Fm(j)1651 4839 y Fq(\027)1692 4851 y Fo(b)1721 4859 y Fk(i)1770 4839 y Fs(+)1853 4772 y Fj(\000)1891 4839 y Fq(\013)1944 4805 y Fm(\000)p Fo(j)s Fm(\000)p Fo(r)2134 4839 y Fs(+)18 b Fq(\013)2270 4805 y Fm(\000)p Fo(j)s Fm(\000)p Fl(2)p Fo(r)2493 4839 y Fs(+)g Fp(\001)c(\001)g(\001) 2687 4772 y Fj(\001)2751 4760 y(X)2739 4942 y Fo(i<)p Fm(j)p Fo(c)p Fm(j)2897 4839 y Fq(\027)2938 4851 y Fo(c)2968 4859 y Fk(i)1191 5123 y Fs(=)83 b Fq(\013)1392 5089 y Fm(\000)p Fo(j)1505 5044 y Fj(X)1493 5226 y Fo(i<)p Fm(j)p Fo(b)p Fm(j)1651 5123 y Fq(\027)1692 5135 y Fo(b)1721 5143 y Fk(i)1770 5123 y Fs(+)1909 5067 y Fq(\013)1962 5037 y Fm(\000)p Fo(j)p 1863 5104 234 4 v 1863 5180 a Fq(\013)1916 5156 y Fo(r)1971 5180 y Fp(\000)18 b Fs(1)2132 5044 y Fj(X)2120 5226 y Fo(i<)p Fm(j)p Fo(c)p Fm(j)2278 5123 y Fq(\027)2319 5135 y Fo(c)2349 5143 y Fk(i)p eop %%Page: 5 5 5 4 bop 1038 251 a Fl(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)h(SUBSTITUTIVE)h(SUBSHIFTS)548 b(5)3380 450 y Fg(\003)456 600 y Fv(Prop)s(osition)28 b(6.)40 b Fh(Ther)l(e)29 b(exists)f(an)h (algorithm)h(which,)h(given)e(a)g(clop)l(en)g(set)f Fq(W)12 b Fh(,)29 b(yields)h(the)456 699 y(limit)1336 799 y Fq(\045)p Fs(\()p Fq(W)12 b Fs(\))24 b(=)49 b(lim)1645 853 y Fo(k)q Fm(!1)1827 799 y Fq(\034)9 b Fs(\()p Fq(#)1953 765 y Fo(k)1995 799 y Fs(\()p Fq(W)j Fs(\)\))20 b Fp(\001)e Fq(\013)2295 765 y Fm(\000)p Fo(k)2411 799 y Fq(>)23 b Fs(0)p Fq(:)456 979 y Fh(Pr)l(o)l(of.)43 b Fs(Set)28 b Fq(b)22 b Fs(=)h Fq(r)r Fs(\()p Fq(W)12 b Fs(\))20 b Fp(\000)e Fq(l)r Fs(\()p Fq(W)12 b Fs(\).)37 b(Let)27 b Fq(U)37 b Fs(b)r(e)28 b(the)g(set)f(of)h(all)f(w)n(ords)g Fq(u)22 b Fp(2)i(L)p Fs(\()p Fq(x)p Fs(\))k(suc)n(h)g(that)1278 1114 y([)p Fq(u)1349 1129 y Fl([0)p Fo(;b)p Fl(\))1480 1114 y Fs(])1503 1129 y Fo(l)p Fl(\()p Fo(W)9 b Fl(\))1674 1114 y Fp(\022)23 b Fq(W)n(;)97 b Fs([)p Fq(u)2029 1129 y Fl([)p Fo(a;a)p Fl(+)p Fo(b)p Fl(\))2249 1114 y Fs(])2272 1129 y Fo(l)p Fl(\()p Fo(W)9 b Fl(\))2444 1114 y Fp(\022)23 b Fq(W)456 1249 y Fs(for)35 b(some)g Fq(a)i(>)g Fs(0)e(\(Figure)h(2\).) 61 b(Let)36 b Fq(a)1729 1261 y Fo(u)1809 1249 y Fs(=)h Fq(a)f Fs(b)r(e)g(the)g(least)g(in)n(teger)f(with)h(this)g(prop)r(ert)n (y)-7 b(,)456 1362 y(so)35 b Fp(j)p Fq(u)p Fp(j)h Fs(=)g Fq(a)841 1374 y Fo(u)908 1362 y Fs(+)24 b Fq(b)p Fs(.)61 b(Assume)36 b(that)g Fq(k)j Fp(\025)d Fs(0)g(and)f(let)h Fq(w)j Fp(2)e Fq(#)2418 1332 y Fo(k)2459 1362 y Fs(\()p Fq(W)12 b Fs(\))25 b Fp(\\)f Fq(\033)2767 1332 y Fm(\000)p Fo(\034)7 b Fl(\()p Fo(#)2923 1307 y Fk(k)2959 1332 y Fl(\()p Fo(W)i Fl(\)\))3112 1362 y Fs(\()p Fq(#)3193 1332 y Fo(k)3235 1362 y Fs(\()p Fq(W)j Fs(\)\).)456 1475 y(There)30 b(exist)g Fq(z)t(;)14 b(v)30 b Fp(2)f Fq(W)42 b Fs(suc)n(h)31 b(that)g Fq(w)f Fs(=)e Fq(#)1857 1445 y Fo(k)1898 1475 y Fs(\()p Fq(z)t Fs(\),)j Fq(\033)2109 1445 y Fo(\034)7 b Fl(\()p Fo(#)2213 1420 y Fk(k)2249 1445 y Fl(\()p Fo(W)i Fl(\)\))2402 1475 y Fs(\()p Fq(w)r Fs(\))30 b(=)d Fq(#)2698 1445 y Fo(k)2739 1475 y Fs(\()p Fq(v)s Fs(\).)47 b(By)30 b(Theorem)g(2)456 1574 y(there)h(exists)f Fq(a)f(>)g Fs(0)i(with)h Fq(z)g Fs(=)d Fq(\033)1552 1544 y Fo(a)1592 1574 y Fs(\()p Fq(v)s Fs(\))j(Then)g Fq(u)c Fs(=)h Fq(z)2161 1589 y Fl([)p Fo(l)p Fl(\()p Fo(W)9 b Fl(\))p Fo(;l)p Fl(\()p Fo(W)g Fl(\)+)p Fo(a)p Fl(+)p Fo(b)p Fl(\))2715 1574 y Fp(2)29 b Fq(U)40 b Fs(and)31 b Fq(a)3105 1586 y Fo(u)3177 1574 y Fs(=)e Fq(a)p Fs(,)j(so)456 1683 y Fq(\034)9 b Fs(\()p Fq(#)582 1653 y Fo(k)624 1683 y Fs(\()p Fq(W)j Fs(\)\))23 b(=)g Fp(j)p Fq(#)993 1653 y Fo(k)1034 1683 y Fs(\()p Fq(u)1114 1698 y Fl([0)p Fo(;a)1222 1706 y Fk(u)1261 1698 y Fl(\))1291 1683 y Fs(\))p Fp(j)p Fs(.)37 b(F)-7 b(or)27 b(ev)n(ery)f Fq(u)d Fp(2)g Fq(U)37 b Fs(there)27 b(exists)g(a)h(limit)1246 1843 y Fq(t)1276 1855 y Fo(u)1343 1843 y Fs(=)49 b(lim)1430 1897 y Fo(k)q Fm(!1)1613 1843 y Fp(j)p Fq(#)1685 1808 y Fo(k)1726 1843 y Fs(\()p Fq(u)1806 1858 y Fl([0)p Fo(;a)1914 1866 y Fk(u)1953 1858 y Fl(\))1983 1843 y Fs(\))p Fp(j)19 b(\001)f Fq(\013)2151 1808 y Fm(\000)p Fo(k)2268 1843 y Fs(=)2371 1764 y Fj(X)2355 1941 y Fo(i)f Fs(0.)774 b Fg(\003)934 3008 y Fq(l)r Fs(\()p Fq(W)12 b Fs(\))p 1003 2890 4 119 v 1005 2855 a Fd(6)1643 3008 y Fq(r)r Fs(\()p Fq(W)g Fs(\))p 1712 2890 V 1714 2855 a Fd(6)2115 3008 y Fq(l)r Fs(\()p Fq(W)g Fs(\))p 2185 2890 V 2186 2855 a Fd(6)2824 3008 y Fq(r)r Fs(\()p Fq(W)g Fs(\))p 2893 2890 V 2895 2855 a Fd(6)p 1005 2726 709 4 v 2186 2726 V 1005 2537 1890 4 v 1336 2701 a Fq(W)1103 b(W)1903 2512 y(u)982 2300 y Fs(0)p 1003 2489 4 119 v 1005 2489 a Fd(?)1690 2300 y Fq(b)p 1712 2489 V 1714 2489 a Fd(?)2139 2300 y Fq(a)2183 2312 y Fo(u)p 2185 2489 V 2186 2489 a Fd(?)2777 2300 y Fq(a)2821 2312 y Fo(u)2883 2300 y Fs(+)18 b Fq(b)p 2893 2489 V 2895 2489 a Fd(?)1499 3207 y Fr(Figure)32 b(2.)41 b Fs(Return)28 b(time)456 3429 y Fv(De\014nition)34 b(7.)42 b Fh(We)32 b(say)h(that)f(a)h(clop)l(en)g(set)f Fq(W)40 b Fp(\022)27 b Fs(\002)2235 3441 y Fo(#)2311 3429 y Fh(is)32 b(de)l(c)l(o)l(dable,)k (if)d(for)g(some)f Fq(i)c Fp(2)g Ff(Z)o Fh(,)456 3529 y Fq(\033)506 3499 y Fm(\000)p Fo(i)586 3529 y Fs(\()p Fq(W)12 b Fs(\))29 b Fp(\022)f Fq(#)p Fs(\(\002)1008 3541 y Fo(#)1053 3529 y Fs(\))p Fh(.)48 b(If)33 b Fq(i)c Fp(\025)f Fs(0)33 b Fh(is)g(the)g(le)l(ast)g(inte)l(ger)g(with)h(this)f (pr)l(op)l(erty,)i(we)e(write,)i(by)e(an)456 3628 y(abuse)d(of)g (notation,)1002 3763 y Fq(#)1051 3729 y Fm(\000)p Fl(1)1140 3763 y Fs(\()p Fq(W)12 b Fs(\))24 b(=)e Fq(#)1454 3729 y Fm(\000)p Fl(1)1544 3763 y Fs(\()p Fq(\033)1626 3729 y Fm(\000)p Fo(i)1706 3763 y Fs(\()p Fq(W)12 b Fs(\)\))24 b(=)e Fp(f)p Fq(z)k Fp(2)e Fs(\002)2254 3775 y Fo(#)2320 3763 y Fs(:)g Fq(\033)2417 3729 y Fo(i)2445 3763 y Fs(\()p Fq(#)p Fs(\()p Fq(z)t Fs(\)\))f Fp(2)h Fq(W)12 b Fp(g)456 3897 y Fh(We)29 b(say)g(that)g(a)h(clop)l(en)g(set)e Fq(W)35 b Fp(\022)23 b Fs(\002)1640 3909 y Fo(#)1713 3897 y Fh(is)29 b(short,)h(if)g(b)l(oth)f Fq(p)p Fs(\()p Fq(W)12 b Fs(\))17 b Fp(\000)g Fq(l)r Fs(\()p Fq(W)12 b Fs(\))29 b Fh(and)g Fq(r)r Fs(\()p Fq(W)12 b Fs(\))18 b Fp(\000)f Fq(q)s Fs(\()p Fq(W)12 b Fs(\))456 3997 y Fh(ar)l(e)30 b(less)g(than)g Fs(\()p Fq(m)18 b Fs(+)g(1\))p Fp(j)p Fq(#)p Fp(j)p Fq(=)p Fs(\()p Fp(j)p Fq(#)p Fp(j)h(\000)f Fs(1\))p Fh(,)30 b(wher)l(e)g Fq(m)g Fh(is)g(the)g(c)l(ontext)e(length) i(fr)l(om)h(The)l(or)l(em)f(2.)555 4144 y Fs(If)e Fq(W)40 b Fs(is)27 b(deco)r(dable,)h(then)g(clearly)e Fq(#)p Fs(\()p Fq(#)1832 4114 y Fm(\000)p Fl(1)1922 4144 y Fs(\()p Fq(W)12 b Fs(\)\))24 b(=)e Fq(\033)2269 4114 y Fm(\000)p Fo(i)2349 4144 y Fs(\()p Fq(W)12 b Fs(\).)456 4291 y Fv(Prop)s(osition)30 b(8.)40 b Fh(If)30 b Fq(W)42 b Fh(is)30 b(a)g(clop)l(en)g(set)f(with)h Fp(j)p Fq(c)p Fs(\()p Fq(W)12 b Fs(\))p Fp(j)24 b Fs(=)f Fq(q)s Fs(\()p Fq(W)12 b Fs(\))18 b Fp(\000)g Fq(p)p Fs(\()p Fq(W)12 b Fs(\))23 b Fp(\025)g Fs(2)p Fq(m)p Fh(,)29 b(wher)l(e)i Fq(m)456 4391 y Fh(is)f(the)g(c)l(ontext)e(length,)j(then)e Fq(W)42 b Fh(is)30 b(de)l(c)l(o)l(dable,)i(and)974 4568 y Fq(r)r Fs(\()p Fq(#)1094 4534 y Fm(\000)p Fl(1)1184 4568 y Fs(\()p Fq(W)12 b Fs(\)\))19 b Fp(\000)g Fq(q)s Fs(\()p Fq(#)1594 4534 y Fm(\000)p Fl(1)1683 4568 y Fs(\()p Fq(W)12 b Fs(\)\))84 b Fp(\024)2110 4512 y Fq(r)r Fs(\()p Fq(W)12 b Fs(\))20 b Fp(\000)e Fq(q)s Fs(\()p Fq(W)12 b Fs(\))19 b(+)f Fq(m)p 2110 4549 665 4 v 2395 4625 a Fp(j)p Fq(#)p Fp(j)2804 4568 y Fs(+)g(1)972 4798 y Fq(q)s Fs(\()p Fq(#)1093 4763 y Fm(\000)p Fl(1)1183 4798 y Fs(\()p Fq(W)12 b Fs(\)\))19 b Fp(\000)f Fq(p)p Fs(\()p Fq(#)1594 4763 y Fm(\000)p Fl(1)1683 4798 y Fs(\()p Fq(W)12 b Fs(\)\))84 b Fp(\024)2110 4742 y Fq(q)s Fs(\()p Fq(W)12 b Fs(\))19 b Fp(\000)f Fq(p)p Fs(\()p Fq(W)12 b Fs(\))p 2110 4779 493 4 v 2309 4855 a Fp(j)p Fq(#)p Fp(j)2631 4798 y Fs(+)18 b(1)986 5027 y Fq(p)p Fs(\()p Fq(#)1109 4993 y Fm(\000)p Fl(1)1198 5027 y Fs(\()p Fq(W)12 b Fs(\)\))19 b Fp(\000)f Fq(l)r Fs(\()p Fq(#)1594 4993 y Fm(\000)p Fl(1)1683 5027 y Fs(\()p Fq(W)12 b Fs(\)\))84 b Fp(\024)2110 4971 y Fq(p)p Fs(\()p Fq(W)12 b Fs(\))19 b Fp(\000)f Fq(l)r Fs(\()p Fq(W)12 b Fs(\))19 b(+)f Fq(m)p 2110 5008 654 4 v 2389 5084 a Fp(j)p Fq(#)p Fp(j)2792 5027 y Fs(+)g(1)456 5216 y Fh(If)30 b Fq(W)41 b Fh(is)31 b(also)f(short,)h(then)e(so)h(is)h Fq(#)1587 5185 y Fm(\000)p Fl(1)1676 5216 y Fs(\()p Fq(W)12 b Fs(\))p Fh(.)p eop %%Page: 6 6 6 5 bop 456 251 a Fl(6)1233 b(PETR)23 b(K)1981 236 y(\027)1981 251 y(URKA)456 450 y Fh(Pr)l(o)l(of.)43 b Fs(By)24 b(Theorem)f(2)h (there)h(exist)f Fq(i;)14 b(j)29 b Fs(suc)n(h)24 b(that)g Fq(p)p Fs(\()p Fq(W)12 b Fs(\))24 b Fp(\024)e Fq(i)h Fp(\024)g Fq(p)p Fs(\()p Fq(W)12 b Fs(\))g(+)g Fq(m)p Fs(,)25 b Fq(q)s Fs(\()p Fq(W)12 b Fs(\))g Fp(\000)g Fq(m)23 b Fp(\024)456 550 y Fq(j)34 b Fp(\024)28 b Fq(q)s Fs(\()p Fq(W)12 b Fs(\))32 b(and)f(unique)g Fq(v)k Fs(suc)n(h)c(that)g (for)g(eac)n(h)f Fq(y)i Fp(2)d Fq(W)12 b Fs(,)32 b Fq(y)2394 565 y Fl([)p Fo(i;j)s Fl(\))2546 550 y Fs(=)d Fq(#)p Fs(\()p Fq(v)s Fs(\).)48 b(Moreo)n(v)n(er,)30 b(there)456 656 y(exists)d Fq(z)f Fp(2)e Fs(\002)894 668 y Fo(#)965 656 y Fs(with)k Fq(#)p Fs(\()p Fq(z)t Fs(\))23 b(=)g Fq(\033)1471 626 y Fo(i)1499 656 y Fs(\()p Fq(y)s Fs(\))28 b(and)f Fq(z)g Fp(2)c Fs([)p Fq(v)s Fs(])2029 668 y Fl(0)2067 656 y Fs(,)k(so)g Fq(W)40 b Fs(is)27 b(deco)r(dable.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)729 843 y Fq(r)r Fs(\()p Fq(#)849 808 y Fm(\000)p Fl(1)939 843 y Fs(\()p Fq(W)12 b Fs(\)\))19 b Fp(\000)f Fq(q)s Fs(\()p Fq(#)1348 808 y Fm(\000)p Fl(1)1438 843 y Fs(\()p Fq(W)12 b Fs(\)\))24 b Fp(\024)1745 786 y Fq(r)r Fs(\()p Fq(W)12 b Fs(\))20 b Fp(\000)e Fq(j)p 1745 823 335 4 v 1865 899 a Fp(j)p Fq(#)p Fp(j)2108 843 y Fs(+)g(1)23 b Fp(\024)2354 786 y Fq(r)r Fs(\()p Fq(W)12 b Fs(\))19 b Fp(\000)f Fq(q)s Fs(\()p Fq(W)12 b Fs(\))19 b(+)f Fq(m)p 2354 823 665 4 v 2638 899 a Fp(j)p Fq(#)p Fp(j)3047 843 y Fs(+)g(1)456 1035 y(Similarly)23 b(w)n(e)g(obtain)g (the)h(inequalit)n(y)f(for)g Fq(p)p Fs(\()p Fq(#)1943 1005 y Fm(\000)p Fl(1)2032 1035 y Fs(\()p Fq(W)12 b Fs(\)\))e Fp(\000)g Fq(l)r Fs(\()p Fq(#)2411 1005 y Fm(\000)p Fl(1)2501 1035 y Fs(\()p Fq(W)i Fs(\)\),)25 b(while)f(the)g(inequalit)n(y)456 1135 y(for)j Fq(q)s Fs(\()p Fq(#)704 1105 y Fm(\000)p Fl(1)793 1135 y Fs(\()p Fq(W)12 b Fs(\)\))19 b Fp(\000)g Fq(p)p Fs(\()p Fq(#)1205 1105 y Fm(\000)p Fl(1)1294 1135 y Fs(\()p Fq(W)12 b Fs(\)\))28 b(is)g(ob)n(vious.)35 b(If)28 b(W)g(is)g(short,)f(then)878 1362 y Fq(r)r Fs(\()p Fq(#)998 1328 y Fm(\000)p Fl(1)1088 1362 y Fs(\()p Fq(W)12 b Fs(\)\))19 b Fp(\000)f Fq(q)s Fs(\()p Fq(#)1497 1328 y Fm(\000)p Fl(1)1587 1362 y Fs(\()p Fq(W)12 b Fs(\)\))24 b Fp(\024)1904 1246 y Fl(\()p Fo(m)p Fl(+1\))p Fm(j)p Fo(#)p Fm(j)p 1904 1267 275 4 v 1959 1315 a(j)p Fo(#)p Fm(j\000)p Fl(1)2207 1286 y Fs(+)18 b Fq(m)p 1894 1343 469 4 v 2081 1419 a Fp(j)p Fq(#)p Fp(j)2391 1362 y Fs(+)g(1)23 b Fp(\024)2637 1306 y Fs(\()p Fq(m)18 b Fs(+)g(1\))p Fp(j)p Fq(#)p Fp(j)p 2637 1343 376 4 v 2705 1419 a(j)p Fq(#)p Fp(j)h(\000)f Fs(1)456 1555 y(so)27 b Fq(#)607 1524 y Fm(\000)p Fl(1)696 1555 y Fs(\()p Fq(W)12 b Fs(\))28 b(is)f(short)g(to)r(o.)2065 b Fg(\003)456 1710 y Fv(De\014nition)40 b(9.)46 b Fh(L)l(et)36 b Fq(V)56 b Fp(\032)36 b Fq(W)50 b Fh(b)l(e)37 b(clop)l(en)h(sets.)61 b(We)38 b(say)g(that)f Fq(V)56 b Fh(is)38 b(a)f(maximal)h(clop)l(en)456 1809 y(subset)31 b(of)h Fq(W)12 b Fh(,)33 b(if)f Fq(\037)p Fs(\()p Fq(V)19 b Fs(\))27 b Fq(>)f(\037)p Fs(\()p Fq(W)12 b Fs(\))32 b Fh(and)g(ther)l(e)g(is)g(no)g(clop)l(en)g(set)g Fq(U)40 b Fh(with)32 b Fq(V)46 b Fp(\032)26 b Fq(U)35 b Fp(\032)26 b Fq(W)44 b Fh(and)456 1909 y Fq(\037)p Fs(\()p Fq(U)9 b Fs(\))23 b Fq(>)g(\037)p Fs(\()p Fq(W)12 b Fs(\))p Fh(.)456 2061 y Fv(Lemma)29 b(1.)41 b Fh(L)l(et)29 b Fq(U;)14 b(V)48 b Fh(b)l(e)30 b(maximal)h(clop)l(en)f(subsets)f(of)i Fq(W)12 b Fh(.)38 b(If)31 b Fq(U)c Fp(\\)19 b Fq(V)42 b Fp(6)p Fs(=)22 b Fp(;)p Fh(,)30 b(then)f Fq(U)j Fs(=)23 b Fq(V)c Fh(.)456 2212 y(Pr)l(o)l(of.)43 b Fs(Assume)e(that)h Fq(w)48 b Fp(2)f Fq(U)36 b Fp(\\)28 b Fq(V)60 b Fs(and)41 b(set)h Fq(c)k Fs(=)f(min)p Fp(f)p Fq(\037)p Fs(\()p Fq(U)9 b Fs(\))p Fq(;)14 b(\037)p Fs(\()p Fq(V)19 b Fs(\))p Fp(g)46 b Fq(>)f(\037)p Fs(\()p Fq(W)12 b Fs(\).)79 b(F)-7 b(or)456 2312 y Fq(c)492 2324 y Fo(k)555 2312 y Fs(=)23 b(min)p Fp(f)p Fq(q)s Fs(\()p Fq(#)944 2282 y Fo(k)985 2312 y Fs(\()p Fq(U)9 b Fs(\)\))p Fq(;)14 b(q)s Fs(\()p Fq(#)1305 2282 y Fo(k)1347 2312 y Fs(\()p Fq(V)19 b Fs(\)\))p Fp(g)28 b Fs(w)n(e)f(ha)n(v)n(e)g(lim)2009 2324 y Fo(k)q Fm(!1)2196 2312 y Fq(c)2232 2324 y Fo(k)2273 2312 y Fq(\013)2326 2282 y Fm(\000)p Fo(k)2442 2312 y Fs(=)c Fq(c)p Fs(.)36 b(If)28 b Fq(u;)14 b(v)26 b Fp(2)e Fq(U)j Fp([)19 b Fq(V)f Fs(,)28 b(then)1280 2455 y Fq(#)1329 2420 y Fo(k)1370 2455 y Fs(\()p Fq(u)p Fs(\))1482 2470 y Fl([0)p Fo(;c)1584 2479 y Fk(k)1619 2470 y Fl(\))1673 2455 y Fs(=)22 b Fq(#)1809 2420 y Fo(k)1850 2455 y Fs(\()p Fq(w)r Fs(\))1975 2470 y Fl([0)p Fo(;c)2077 2479 y Fk(k)2114 2470 y Fl(\))2167 2455 y Fs(=)h Fq(#)2304 2420 y Fo(k)2345 2455 y Fs(\()p Fq(v)s Fs(\))2452 2470 y Fl([0)p Fo(;c)2554 2479 y Fk(k)2590 2470 y Fl(\))456 2602 y Fs(so)k Fq(q)s Fs(\()p Fq(#)679 2572 y Fo(k)721 2602 y Fs(\()p Fq(U)h Fp([)19 b Fq(V)g Fs(\)\))25 b Fp(\025)g Fq(c)1194 2614 y Fo(k)1263 2602 y Fs(and)j Fq(\037)p Fs(\()p Fq(U)g Fp([)20 b Fq(V)f Fs(\))25 b Fp(\025)f Fq(c)h(>)f(\037)p Fs(\()p Fq(W)12 b Fs(\).)40 b(Since)29 b Fq(U;)14 b(V)47 b Fs(are)27 b(maximal,)i(w)n(e)f(get)456 2702 y Fq(U)j Fs(=)23 b Fq(U)k Fp([)19 b Fq(V)42 b Fs(=)23 b Fq(V)18 b Fs(.)2323 b Fg(\003)555 2857 y Fs(W)-7 b(e)28 b(construct)f(no)n(w)g(a)g (\014nite)h(graph)f(asso)r(ciated)f(to)h(a)g(substitution.)38 b(Denote)27 b(b)n(y)h Fp(W)34 b Fs(the)456 2956 y(set)29 b(of)f(all)h(clop)r(en)g(sets)g Fq(W)37 b Fp(\022)25 b Fs(\002)1495 2968 y Fo(#)1567 2956 y Fs(whic)n(h)k(are)f(short)g(and) h(not)g(deco)r(dable.)41 b(By)28 b(Prop)r(osition)456 3056 y(8,)h Fp(W)35 b Fs(is)29 b(\014nite.)42 b(W)-7 b(e)29 b(sa)n(y)f(that)h(a)g(pair)f Fq(e)d Fs(=)g(\()p Fq(W)1978 3068 y Fl(0)2016 3056 y Fq(;)14 b(W)e Fs(\))29 b(is)g(an)f(edge,)h(if)h Fq(W)2775 3068 y Fl(0)2838 3056 y Fp(2)25 b(W)36 b Fs(and)29 b Fq(W)41 b Fs(is)29 b(a)456 3156 y(maximal)20 b(clop)r(en)g(subset)h(of)g Fq(W)1455 3168 y Fl(0)1492 3156 y Fs(.)35 b(Denote)21 b(b)n(y)f Fp(E)28 b Fs(the)21 b(set)g(of)g(edges.)34 b(W)-7 b(e)21 b(ha)n(v)n(e)e(the)i(source)f(and)456 3255 y(target)29 b(maps)h Fq(s;)14 b(t)27 b Fs(:)h Fp(E)34 b(!)28 b(W)37 b Fs(de\014ned)30 b(as)g(follo)n(ws.)44 b(If)30 b Fq(e)e Fs(=)f(\()p Fq(W)2475 3267 y Fl(0)2513 3255 y Fq(;)14 b(W)e Fs(\))27 b Fp(2)h(E)37 b Fs(is)31 b(an)f(edge,)g(then)456 3358 y Fq(s)p Fs(\()p Fq(e)p Fs(\))23 b(=)f Fq(W)786 3370 y Fl(0)824 3358 y Fs(.)35 b(Its)23 b(target)e(is)i Fq(t)p Fs(\()p Fq(e)p Fs(\))g(=)g Fq(W)1640 3370 y Fl(1)1700 3358 y Fs(=)g Fq(#)1837 3328 y Fm(\000)p Fo(L)p Fl(\()p Fo(e)p Fl(\))2022 3358 y Fs(\()p Fq(W)12 b Fs(\),)24 b(where)e Fq(L)p Fs(\()p Fq(e)p Fs(\))h Fp(\025)f Fs(0)g(is)h(the)f (least)g(in)n(teger)456 3458 y(suc)n(h)i(that)g Fq(W)894 3470 y Fl(1)957 3458 y Fs(is)g(not)g(deco)r(dable.)36 b(Prop)r(osition)22 b(8)i(implies)h(that)g Fq(W)2641 3470 y Fl(1)2703 3458 y Fs(is)f(short,)h(so)e Fq(W)3193 3470 y Fl(1)3254 3458 y Fp(2)h(W)7 b Fs(.)456 3557 y(The)25 b(o\013set)h(of)f(an)h(edge)f Fq(e)e Fs(=)f(\()p Fq(W)1494 3569 y Fl(0)1532 3557 y Fq(;)14 b(W)e Fs(\))26 b(is)f Fq(\037)p Fs(\()p Fq(e)p Fs(\))f(=)e Fq(\037)p Fs(\()p Fq(W)12 b Fs(\))j Fp(\000)f Fq(\037)p Fs(\()p Fq(W)2526 3569 y Fl(0)2564 3557 y Fs(\))23 b Fq(>)g Fs(0)i(and)h(its)g (probabilit)n(y)456 3657 y(is)i Fq(P)12 b Fs(\()p Fq(e)p Fs(\))24 b(=)f Fq(\026)p Fs(\()p Fq(W)12 b Fs(\))p Fq(=\026)p Fs(\()p Fq(W)1226 3669 y Fl(0)1264 3657 y Fs(\).)39 b(Let)29 b Fp(G)1557 3669 y Fl(0)1618 3657 y Fs(=)24 b(\()p Fp(W)1821 3669 y Fl(0)1859 3657 y Fq(;)14 b Fp(E)1940 3669 y Fl(0)1977 3657 y Fq(;)g(s;)g(t)p Fs(\))28 b(b)r(e)g(the)h(subgraph)e(of)h Fp(G)i Fs(=)23 b(\()p Fp(W)7 b Fq(;)14 b Fp(E)7 b Fq(;)14 b(s;)g(t)p Fs(\))456 3757 y(of)36 b(those)f(v)n(ertices)g(whic)n(h)h (are)f(reac)n(hable)g(from)g(the)i(initial)f(v)n(ertex)f([)p Fq(\025)p Fs(])j(=)f(\002)3025 3769 y Fo(#)3069 3757 y Fs(.)62 b(Giv)n(en)36 b(a)456 3856 y(v)n(ertex)23 b Fq(W)35 b Fp(2)24 b(W)978 3868 y Fl(0)1040 3856 y Fs(the)h(outgoing)f (edges)g(determine)h(a)f(clop)r(en)h(partition)g(of)g Fq(W)36 b Fs(and)25 b(the)g(sum)456 3956 y(of)i(their)h(probabilities)e (is)i(1.)456 4108 y Fv(Lemma)h(2.)41 b Fh(F)-6 b(or)30 b(every)g(me)l(asur)l(able)h(set)e Fq(W)35 b Fp(\022)23 b Fs(\002)2103 4120 y Fo(#)2176 4108 y Fh(we)30 b(have)1460 4289 y Fq(\026)p Fs(\()p Fq(#)p Fs(\()p Fq(W)12 b Fs(\)\))25 b(=)2062 4233 y Fq(\026)p Fs(\()p Fq(W)12 b Fs(\))p 1899 4270 531 4 v 1899 4284 a Fj(P)1987 4371 y Fo(a)p Fm(2)p Fo(A)2136 4346 y Fq(\026)2186 4358 y Fo(a)2226 4346 y Fp(j)p Fq(#)p Fs(\()p Fq(a)p Fs(\))p Fp(j)456 4498 y Fh(Pr)l(o)l(of.)43 b Fs(F)-7 b(or)29 b Fq(y)g Fp(2)e Fs(\002)1084 4510 y Fo(#)1158 4498 y Fs(and)i Fq(n)e(>)f Fs(0)j(set)h Fq(k)1735 4510 y Fo(n)1807 4498 y Fs(=)c Fp(j)p Fq(#)p Fs(\()p Fq(y)2043 4513 y Fl([0)p Fo(;n)p Fl(\))2186 4498 y Fs(\))p Fp(j)p Fs(.)44 b(If)30 b Fq(u)c Fp(2)h(L)p Fs(\(\002)2703 4510 y Fo(#)2747 4498 y Fs(\),)k(then)f Fq(#)p Fs(\()p Fq(u)p Fs(\))g(o)r(ccurs)456 4598 y(in)d Fq(#)p Fs(\()p Fq(y)674 4613 y Fl([)p Fo(m;k)807 4621 y Fk(n)848 4613 y Fm(\000)p Fo(m)p Fl(\))989 4598 y Fs(\))h(only)f(at)h (p)r(ositions)f Fp(j)p Fq(#)p Fs(\()p Fq(y)1827 4613 y Fl([0)p Fo(;j)s Fl(\))1959 4598 y Fs(\))p Fp(j)p Fs(,)h(suc)n(h)g (that)g Fq(y)2474 4613 y Fl([)p Fo(j;j)s Fl(+)p Fm(j)p Fo(u)p Fm(j)p Fl(\))2753 4598 y Fs(=)22 b Fq(u)p Fs(.)37 b(It)28 b(follo)n(ws)778 4799 y Fq(\026)p Fs(\()p Fq(#)p Fs(\([)p Fq(u)p Fs(])1035 4811 y Fl(0)1073 4799 y Fs(\))83 b(=)111 b(lim)1335 4849 y Fo(n)p Fm(!1)1533 4737 y Fs(#)p Fp(f)p Fq(i)22 b(<)h(k)1826 4749 y Fo(n)1894 4737 y Fs(:)g Fq(#)p Fs(\()p Fq(y)s Fs(\))2097 4752 y Fl([0)p Fo(;)p Fm(j)p Fo(#)p Fl(\()p Fo(u)p Fl(\))p Fm(j)p Fl(\))2393 4737 y Fs(=)g Fq(#)p Fs(\()p Fq(u)p Fs(\))p Fp(g)p 1533 4780 1151 4 v 2064 4856 a Fq(k)2107 4868 y Fo(n)1188 5026 y Fs(=)111 b(lim)1335 5076 y Fo(n)p Fm(!1)1533 4964 y Fs(#)p Fp(f)p Fq(i)22 b(<)h(n)g Fs(:)g Fq(y)1943 4979 y Fl([0)p Fo(;u)p Fl(\))2106 4964 y Fs(=)g Fq(u)p Fp(g)p 1533 5007 751 4 v 1883 5083 a Fq(n)2311 5026 y Fp(\001)2382 4970 y Fq(n)p 2363 5007 89 4 v 2363 5083 a(k)2406 5095 y Fo(n)2484 5026 y Fs(=)2724 4970 y Fq(\026)p Fs(\([)p Fq(u)p Fs(])2900 4982 y Fl(0)2937 4970 y Fs(\))p 2582 5007 531 4 v 2582 5021 a Fj(P)2670 5108 y Fo(a)p Fm(2)p Fo(A)2818 5083 y Fq(\026)2868 5095 y Fo(a)2908 5083 y Fp(j)p Fq(#)p Fs(\()p Fq(a)p Fs(\))p Fp(j)3380 5216 y Fg(\003)p eop %%Page: 7 7 7 6 bop 1038 251 a Fl(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)h(SUBSTITUTIVE)h(SUBSHIFTS)548 b(7)456 450 y Fv(Prop)s(osition)25 b(10.)37 b Fh(F)-6 b(or)27 b(every)g Fq(y)e Fp(2)f Fs(\006)1700 462 y Fo(#)1770 450 y Fh(ther)l(e)i(exists)g(a)h(p)l(ath)g Fs(\()p Fq(e)2514 462 y Fo(k)2577 450 y Fs(:)d Fq(W)2702 462 y Fo(k)2766 450 y Fp(!)f Fq(W)2950 462 y Fo(k)q Fl(+1)3075 450 y Fs(\))3107 462 y Fo(k)q Fm(\025)p Fl(0)3260 450 y Fh(in)j Fp(G)3407 462 y Fl(0)456 550 y Fh(fr)l(om)32 b(the)h(initial)g(vertex)f Fq(W)1367 562 y Fl(0)1432 550 y Fs(=)c([)p Fq(\025)p Fs(])k Fh(and)h(inte)l(gers)f Fs(\()p Fq(l)2183 562 y Fo(k)2224 550 y Fs(\))2256 562 y Fo(k)q Fm(\025)p Fl(0)2415 550 y Fh(such)g(that)g Fq(l)2803 562 y Fo(k)q Fl(+1)2948 550 y Fp(\000)20 b Fq(l)3058 562 y Fo(k)3126 550 y Fs(=)27 b Fq(L)p Fs(\()p Fq(e)3346 562 y Fo(k)3387 550 y Fs(\))p Fh(,)456 652 y(and)36 b Fq(W)701 664 y Fo(k)776 652 y Fs(=)e Fq(#)924 622 y Fm(\000)p Fo(l)997 631 y Fk(k)1038 652 y Fs(\([)p Fq(y)1134 667 y Fl([0)p Fo(;s)1237 676 y Fk(k)1273 667 y Fl(\))1303 652 y Fs(]\))p Fh(.)58 b(Conversely)37 b(any)f(in\014nite)g(p)l(ath)g (in)g Fp(G)2671 664 y Fl(0)2745 652 y Fh(which)h(starts)f(in)g Fq(W)3407 664 y Fl(0)456 751 y Fh(yields)31 b(a)f(unique)f(p)l(oint)h Fq(y)c Fp(2)e Fs(\006)1442 763 y Fo(#)1515 751 y Fh(with)31 b(this)f(pr)l(op)l(erty.)39 b(Mor)l(e)l(over,)1397 895 y Fq(\026)p Fs(\([)p Fq(y)1543 910 y Fl([0)p Fo(;s)1646 919 y Fk(k)1683 910 y Fl(\))1713 895 y Fs(]\))23 b(=)g Fq(P)12 b Fs(\()p Fq(e)2015 907 y Fl(0)2052 895 y Fs(\))i Fp(\001)g(\001)g(\001)g Fq(P)e Fs(\()p Fq(e)2345 907 y Fo(k)q Fm(\000)p Fl(1)2470 895 y Fs(\))456 1053 y Fh(Pr)l(o)l(of.)43 b Fs(F)-7 b(or)22 b(a)g(\014xed)h Fq(k)j Fs(set)d Fq(U)1371 1065 y Fo(n)1439 1053 y Fs(=)f Fq(#)1575 1023 y Fm(\000)p Fo(n)1672 1053 y Fs(\()p Fq(y)1745 1068 y Fl([0)p Fo(;s)1848 1077 y Fk(k)1885 1068 y Fl(\))1915 1053 y Fs(\)\))h Fp(2)h(W)7 b Fs(,)24 b(where)e(0)g Fp(\024)h Fq(n)g Fp(\024)g Fq(l)2790 1065 y Fo(k)2853 1053 y Fs(and)g Fq(l)3035 1065 y Fo(k)3098 1053 y Fp(\025)g Fs(0)f(is)h(the)456 1153 y(\014rst)g(in)n(teger)g(for) h(whic)n(h)g Fq(U)1309 1165 y Fo(l)1330 1174 y Fk(k)1394 1153 y Fs(is)g(not)g(deco)r(dable.)35 b(Then)24 b Fq(c)p Fs(\()p Fq(U)2372 1165 y Fo(l)2393 1174 y Fk(k)2434 1153 y Fs(\))f Fq(<)f Fs(2)p Fq(m)i Fs(and)g(b)n(y)f(induction)i(w)n(e)456 1253 y(get)19 b(that)g Fq(U)814 1265 y Fo(l)835 1274 y Fk(k)894 1253 y Fs(is)g(short.)34 b(Th)n(us)19 b Fq(W)1492 1265 y Fo(k)1556 1253 y Fs(=)k Fq(U)1701 1265 y Fo(l)1722 1274 y Fk(k)1785 1253 y Fp(2)g(W)7 b Fs(.)34 b(Set)20 b Fq(V)2192 1265 y Fo(k)2256 1253 y Fs(=)j Fq(#)2393 1223 y Fm(\000)p Fo(l)2466 1232 y Fk(k)2506 1253 y Fs(\()p Fq(y)2579 1268 y Fl([0)p Fo(;s)2682 1277 y Fk(k)q Fc(+1)2789 1268 y Fl(\))2820 1253 y Fs(\).)34 b(Since)19 b([)p Fq(y)3181 1268 y Fl([0)p Fo(;s)3284 1277 y Fk(k)q Fc(+1)3391 1268 y Fl(\))3421 1253 y Fs(])456 1355 y(is)25 b(a)g(maximal)f(clop)r(en)h (subset)h(of)f([)p Fq(y)1607 1370 y Fl([0)p Fo(;s)1710 1379 y Fk(k)1746 1370 y Fl(\))1776 1355 y Fs(],)h Fq(e)d Fs(=)f(\()p Fq(W)2107 1367 y Fo(k)2149 1355 y Fq(;)14 b(V)2234 1367 y Fo(k)2275 1355 y Fs(\))26 b(is)f(an)g(edge)f(and)h(for) g Fq(W)3075 1367 y Fo(k)q Fl(+1)3224 1355 y Fs(=)d Fq(t)p Fs(\()p Fq(e)p Fs(\))456 1463 y(\(target\))27 b(w)n(e)g(get)g(that)h Fq(y)1246 1478 y Fl([0)p Fo(;s)1349 1487 y Fk(k)q Fc(+1)1456 1478 y Fl(\))1509 1463 y Fs(=)23 b Fq(#)1646 1433 y Fo(l)1667 1442 y Fk(k)q Fc(+1)1778 1463 y Fs(\()p Fq(W)1888 1475 y Fo(k)q Fl(+1)2014 1463 y Fs(\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)f Fq(\026)p Fs(\([)p Fq(y)2587 1478 y Fl([0)p Fo(;s)2690 1486 y Fc(0)2722 1478 y Fl(\))2752 1463 y Fs(]\))d(=)e Fq(\026)p Fs(\([)p Fq(\025)p Fs(]\))i(=)f(1)k(and)1050 1609 y Fq(\026)p Fs(\([)p Fq(y)1196 1624 y Fl([0)p Fo(;s)1299 1633 y Fk(k)q Fc(+1)1407 1624 y Fl(\))1437 1609 y Fs(])p 1050 1653 410 4 v 1086 1729 a Fq(\026)p Fs(\([)p Fq(y)1232 1744 y Fl([0)p Fo(;s)1335 1753 y Fk(k)1371 1744 y Fl(\))1401 1729 y Fs(])1493 1672 y(=)1605 1616 y Fq(\026)p Fs(\()p Fq(#)1736 1586 y Fo(l)1757 1595 y Fk(k)1798 1616 y Fs(\()p Fq(V)1878 1628 y Fo(k)1920 1616 y Fs(\)\))p 1590 1653 409 4 v 1590 1729 a Fq(\026)p Fs(\()p Fq(#)1721 1705 y Fo(l)1742 1714 y Fk(k)1783 1729 y Fs(\()p Fq(W)1893 1741 y Fo(k)1935 1729 y Fs(\)\))2032 1672 y(=)2145 1616 y Fq(\026)p Fs(\()p Fq(V)2275 1628 y Fo(k)2317 1616 y Fs(\))p 2130 1653 234 4 v 2130 1729 a Fq(\026)p Fs(\()p Fq(W)2290 1741 y Fo(k)2332 1729 y Fs(\))2397 1672 y(=)c Fq(P)12 b Fs(\()p Fq(e)2621 1684 y Fo(k)2661 1672 y Fs(\))3380 1867 y Fg(\003)456 2028 y Fv(Prop)s(osition)30 b(11.)40 b Fh(F)-6 b(or)30 b(an)g(e)l(dge)h Fq(e)22 b Fs(=)h(\()p Fq(W)1851 2040 y Fl(0)1889 2028 y Fq(;)14 b(W)e Fs(\))23 b(:)g Fq(W)2195 2040 y Fl(0)2256 2028 y Fp(!)g Fq(W)2440 2040 y Fl(1)2507 2028 y Fh(c)l(onsider)31 b(a)f(line)l(ar)h(function) 1329 2219 y Fq(f)1370 2231 y Fo(e)1406 2219 y Fs(\()p Fq(z)t Fs(\))23 b(=)f Fq(a)1667 2231 y Fo(e)1703 2219 y Fq(z)g Fs(+)c Fq(b)1883 2231 y Fo(e)1941 2219 y Fs(=)2039 2163 y Fq(\045)p Fs(\()p Fq(W)2192 2175 y Fl(0)2229 2163 y Fs(\))p Fq(z)k Fs(+)c Fq(\037)p Fs(\()p Fq(e)p Fs(\))p 2039 2200 523 4 v 2095 2278 a Fq(\045)p Fs(\()p Fq(W)2248 2290 y Fl(1)2286 2278 y Fs(\))p Fq(\013)2371 2254 y Fo(L)p Fl(\()p Fo(e)p Fl(\))456 2408 y Fh(Given)28 b Fq(y)e Fp(2)d Fs(\006)901 2420 y Fo(#)946 2408 y Fh(,)28 b(let)g Fq(l)1139 2420 y Fo(k)1208 2408 y Fh(b)l(e)g(the)g(se)l(quenc)l(e)g(fr) l(om)h(Pr)l(op)l(osition)g(10)g(and)g(let)f Fq(k)2845 2420 y Fo(i)2900 2408 y Fh(b)l(e)h(the)f(se)l(quenc)l(e)456 2508 y(of)i(times)g(whose)h(tr)l(ansitions)f(p)l(ass)g(thr)l(ough)g Fq(e)p Fh(,)g(i.e.,)h Fq(W)2248 2520 y Fo(k)2283 2528 y Fk(i)2338 2508 y Fs(=)22 b Fq(W)2503 2520 y Fl(0)2571 2508 y Fh(and)30 b Fq(W)2810 2520 y Fo(k)2845 2528 y Fk(i)2872 2520 y Fl(+1)2983 2508 y Fs(=)23 b Fq(W)3149 2520 y Fl(1)3186 2508 y Fh(.)39 b(Then)1486 2701 y Fs(lim)1466 2753 y Fo(i)p Fm(!1)1645 2644 y Fq(s)1684 2656 y Fo(k)1719 2664 y Fk(i)1746 2656 y Fl(+1)p 1645 2682 189 4 v 1646 2758 a Fq(\034)1682 2770 y Fo(k)1717 2778 y Fk(i)1744 2770 y Fl(+1)1862 2701 y Fp(\000)18 b Fq(f)1986 2713 y Fo(e)2035 2584 y Fj(\022)2107 2644 y Fq(s)2146 2656 y Fo(k)2181 2664 y Fk(i)p 2107 2682 105 4 v 2108 2758 a Fq(\034)2144 2770 y Fo(k)2179 2778 y Fk(i)2221 2584 y Fj(\023)2305 2701 y Fs(=)23 b(0)456 2889 y Fh(The)k(c)l(o)l(e\016c)l (ents)f Fq(a)1056 2901 y Fo(e)1118 2889 y Fh(and)g Fq(b)1311 2901 y Fo(e)1373 2889 y Fh(satisfy)h Fq(a)1672 2901 y Fo(e)1731 2889 y Fp(\024)22 b Fs(1)k Fh(and)h Fq(b)2080 2901 y Fo(e)2138 2889 y Fq(>)c Fs(0)p Fh(.)37 b(Mor)l(e)l(over,)29 b(the)d(pr)l(o)l(duct)g(of)h(slop)l(es)456 2989 y Fq(a)500 3001 y Fo(e)565 2989 y Fh(along)j(a)h(cycle)f(of)h(the)f(gr)l(aph)h(is) f(strictly)g(smal)t(ler)h(than)f Fs(1)p Fh(.)456 3143 y(Pr)l(o)l(of.)43 b Fs(Since)28 b Fq(\034)969 3155 y Fo(k)1004 3163 y Fk(i)1058 3143 y Fs(=)22 b Fq(\034)9 b Fs(\([)p Fq(y)1286 3158 y Fl([0)p Fo(;s)1389 3167 y Fk(k)1420 3180 y(i)1452 3158 y Fl(\))1482 3143 y Fs(]\))23 b(=)g Fq(\034)9 b Fs(\()p Fq(#)1774 3112 y Fo(l)1795 3121 y Fk(k)1826 3134 y(i)1862 3143 y Fs(\()p Fq(W)1972 3155 y Fl(0)2010 3143 y Fs(\)\),)28 b(and)599 3360 y(lim)579 3412 y Fo(i)p Fm(!1)759 3304 y Fq(s)798 3316 y Fo(k)833 3324 y Fk(i)859 3316 y Fl(+1)966 3304 y Fp(\000)18 b Fq(s)1088 3316 y Fo(k)1123 3324 y Fk(i)p 759 3341 395 4 v 896 3417 a Fq(\013)949 3393 y Fo(k)984 3401 y Fk(i)1186 3360 y Fs(=)43 b(lim)1274 3412 y Fo(i)p Fm(!1)1454 3304 y Fq(q)s Fs(\()p Fq(#)1575 3272 y Fo(l)1596 3281 y Fk(k)1627 3294 y(i)1662 3304 y Fs(\()p Fq(W)12 b Fs(\)\))19 b Fp(\000)f Fq(q)s Fs(\()p Fq(#)2071 3272 y Fo(l)2092 3281 y Fk(k)2123 3294 y(i)2159 3304 y Fs(\()p Fq(W)2269 3316 y Fl(0)2307 3304 y Fs(\)\))p 1454 3341 918 4 v 1853 3417 a Fq(\013)1906 3393 y Fo(k)1941 3401 y Fk(i)2404 3360 y Fs(=)23 b Fq(\037)p Fs(\()p Fq(W)12 b Fs(\))19 b Fp(\000)f Fq(\037)p Fs(\()p Fq(W)2962 3372 y Fl(0)3000 3360 y Fs(\))23 b(=)g Fq(\037)p Fs(\()p Fq(e)p Fs(\))p Fq(;)456 3523 y Fs(w)n(e)k(get)667 3655 y Fq(s)706 3667 y Fo(k)741 3675 y Fk(i)768 3667 y Fl(+1)p 667 3692 189 4 v 668 3768 a Fq(\034)704 3780 y Fo(k)739 3788 y Fk(i)766 3780 y Fl(+1)884 3712 y Fp(\000)18 b Fq(f)1008 3724 y Fo(e)1058 3594 y Fj(\022)1129 3655 y Fq(s)1168 3667 y Fo(k)1203 3675 y Fk(i)p 1129 3692 105 4 v 1130 3768 a Fq(\034)1166 3780 y Fo(k)1201 3788 y Fk(i)1243 3594 y Fj(\023)750 3951 y Fs(=)916 3895 y Fq(s)955 3907 y Fo(k)990 3915 y Fk(i)1017 3907 y Fl(+1)1124 3895 y Fp(\000)g Fq(s)1246 3907 y Fo(k)1281 3915 y Fk(i)p 916 3932 395 4 v 920 4014 a Fq(\013)973 3982 y Fo(l)994 3991 y Fk(k)1025 4004 y(i)1079 4014 y Fp(\001)h Fq(\013)1174 3990 y Fo(L)p Fl(\()p Fo(e)p Fl(\))1340 3951 y Fp(\001)1391 3895 y Fq(\013)1444 3864 y Fo(l)1465 3873 y Fk(k)1496 3886 y(i)1523 3873 y Fc(+1)p 1391 3932 212 4 v 1404 4008 a Fq(\034)1440 4020 y Fo(k)1475 4028 y Fk(i)1502 4020 y Fl(+1)1631 3951 y Fs(+)1724 3895 y Fq(s)1763 3907 y Fo(k)1798 3915 y Fk(i)p 1724 3932 105 4 v 1725 4008 a Fq(\034)1761 4020 y Fo(k)1796 4028 y Fk(i)1852 3834 y Fj(\022)1965 3895 y Fq(\034)2001 3907 y Fo(k)2036 3915 y Fk(i)p 1923 3932 187 4 v 1923 4008 a Fq(\034)1959 4020 y Fo(k)1994 4028 y Fk(i)2021 4020 y Fl(+1)2138 3951 y Fp(\000)2324 3895 y Fq(\045)p Fs(\()p Fq(W)2477 3907 y Fl(0)2515 3895 y Fs(\))p 2231 3932 410 4 v 2231 4010 a Fq(\045)p Fs(\()p Fq(W)2384 4022 y Fl(1)2422 4010 y Fs(\))p Fq(\013)2507 3986 y Fo(L)p Fl(\()p Fo(e)p Fl(\))2651 3834 y Fj(\023)2730 3951 y Fp(\000)2950 3895 y Fq(\037)p Fs(\()p Fq(e)p Fs(\))p 2823 3932 V 2823 4010 a Fq(\045)p Fs(\()p Fq(W)2976 4022 y Fl(1)3014 4010 y Fs(\))p Fq(\013)3099 3986 y Fo(L)p Fl(\()p Fo(e)p Fl(\))740 4182 y Fp(!)1045 4126 y Fq(\037)p Fs(\()p Fq(e)p Fs(\))p 916 4163 413 4 v 916 4241 a Fq(\034)9 b Fs(\()p Fq(W)1071 4253 y Fl(1)1110 4241 y Fs(\))p Fq(\013)1195 4217 y Fo(L)p Fl(\()p Fo(e)p Fl(\))1357 4182 y Fs(+)1450 4126 y Fq(s)1489 4138 y Fo(k)1524 4146 y Fk(i)p 1450 4163 105 4 v 1451 4239 a Fq(\034)1487 4251 y Fo(k)1522 4259 y Fk(i)1583 4182 y Fp(\001)19 b Fs(0)f Fp(\000)1906 4126 y Fq(\037)p Fs(\()p Fq(e)p Fs(\))p 1778 4163 413 4 v 1778 4241 a Fq(\034)9 b Fs(\()p Fq(W)1933 4253 y Fl(1)1971 4241 y Fs(\))p Fq(\013)2056 4217 y Fo(L)p Fl(\()p Fo(e)p Fl(\))2223 4182 y Fs(=)23 b(0)456 4370 y(Since)k Fq(W)35 b Fp(\032)23 b Fq(W)951 4382 y Fl(0)989 4370 y Fs(,)1048 4512 y Fq(\034)9 b Fs(\()p Fq(#)1174 4481 y Fo(k)1216 4512 y Fs(\()p Fq(W)1326 4524 y Fl(0)1364 4512 y Fs(\)\))p 1048 4549 381 4 v 1191 4625 a Fq(\013)1244 4601 y Fo(k)1462 4568 y Fp(\024)1559 4512 y Fq(\034)g Fs(\()p Fq(#)1685 4481 y Fo(k)1727 4512 y Fs(\()p Fq(W)j Fs(\)\))p 1559 4549 355 4 v 1690 4625 a Fq(\013)1743 4601 y Fo(k)1947 4568 y Fs(=)2045 4512 y Fq(\034)d Fs(\()p Fq(#)2171 4481 y Fo(k)q Fl(+)p Fo(L)p Fl(\()p Fo(e)p Fl(\))2393 4512 y Fs(\()p Fq(W)2503 4524 y Fl(1)2541 4512 y Fs(\)\))p 2045 4549 561 4 v 2188 4626 a Fq(\013)2241 4602 y Fo(k)q Fl(+)p Fo(L)p Fl(\()p Fo(e)p Fl(\))2634 4568 y Fp(\001)18 b Fq(\013)2728 4533 y Fo(L)p Fl(\()p Fo(e)p Fl(\))456 4752 y Fs(so)31 b Fq(\045)p Fs(\()p Fq(W)715 4764 y Fl(0)753 4752 y Fs(\))g Fp(\024)g Fq(\045)p Fs(\()p Fq(W)1065 4764 y Fl(1)1103 4752 y Fs(\))p Fq(\013)1188 4722 y Fo(L)p Fl(\()p Fo(e)p Fl(\))1321 4752 y Fs(.)52 b(If)32 b Fq(e)f Fs(=)f Fq(e)1687 4764 y Fl(0)1724 4752 y Fq(;)14 b(:)g(:)g(:)g(;)g(e)1948 4764 y Fo(k)q Fm(\000)p Fl(1)2104 4752 y Fs(:)31 b Fq(W)2236 4764 y Fl(0)2305 4752 y Fp(!)g Fq(W)2497 4764 y Fl(1)2565 4752 y Fp(!)g(\001)14 b(\001)g(\001)31 b(!)g Fq(W)2999 4764 y Fo(k)3071 4752 y Fs(=)g Fq(W)3245 4764 y Fl(0)3315 4752 y Fs(is)h(a)456 4854 y(cycle)27 b(in)h Fp(G)5 b Fs(,)28 b(then)g Fq(a)1096 4866 y Fo(e)1154 4854 y Fs(=)23 b Fq(a)1286 4866 y Fo(e)1317 4874 y Fc(0)1368 4854 y Fp(\001)14 b(\001)g(\001)g Fq(a)1523 4866 y Fo(e)1554 4875 y Fk(k)q Fe(\000)p Fc(1)1691 4854 y Fs(=)22 b Fq(\013)1831 4824 y Fm(\000)p Fo(L)p Fl(\()p Fo(e)1986 4832 y Fc(0)2019 4824 y Fl(\))p Fm(\000\001\001\001)o(\000)p Fo(L)p Fl(\()p Fo(e)2311 4833 y Fk(k)q Fe(\000)p Fc(1)2420 4824 y Fl(\))2474 4854 y Fq(<)g Fs(1.)754 b Fg(\003)555 5016 y Fs(F)-7 b(or)33 b(the)h(sequence)f Fq(s)1248 5028 y Fo(k)1289 5016 y Fq(=\034)1367 5028 y Fo(k)q Fl(+1)1525 5016 y Fs(w)n(e)h(consider)e(the)i(graph)f Fp(G)2423 5028 y Fl(2)2494 5016 y Fs(whose)g(v)n(ertices)f(are)h Fp(E)3240 5028 y Fl(0)3311 5016 y Fs(and)456 5116 y(whose)i(edges)g (are)g Fp(E)1130 5128 y Fl(2)1204 5116 y Fs(=)i Fp(f)p Fs(\()p Fq(d;)14 b(e)p Fs(\))36 b Fp(2)i(E)1711 5086 y Fl(2)1785 5116 y Fs(:)f Fq(t)p Fs(\()p Fq(d)p Fs(\))h(=)f Fq(s)p Fs(\()p Fq(e)p Fs(\))p Fp(g)p Fs(.)61 b(The)36 b(source)f(and)h(target)f(maps)456 5216 y(and)30 b(probabilities)g(are) f Fq(s)p Fs(\()p Fq(d;)14 b(e)p Fs(\))29 b(=)e Fq(d)p Fs(,)32 b Fq(t)p Fs(\()p Fq(d;)14 b(e)p Fs(\))28 b(=)g Fq(e)p Fs(,)j Fq(P)12 b Fs(\()p Fq(d;)i(e)p Fs(\))28 b(=)f Fq(P)12 b Fs(\()p Fq(e)p Fs(\).)46 b(The)31 b(paths)f(in)h Fp(G)3265 5228 y Fl(0)3333 5216 y Fs(are)p eop %%Page: 8 8 8 7 bop 456 251 a Fl(8)1233 b(PETR)23 b(K)1981 236 y(\027)1981 251 y(URKA)456 450 y Fs(in)35 b(one-to-one)e(corresp)r(ondence)g(with)i (those)g(paths)g(in)g Fp(G)2368 462 y Fl(2)2441 450 y Fs(whose)f(initial)h(v)n(ertex)f Fq(e)h Fp(2)g(E)3407 462 y Fl(0)456 550 y Fs(satis\014es)26 b Fq(s)p Fs(\()p Fq(e)p Fs(\))e(=)e Fq(\025)28 b Fs(in)g Fp(G)1236 562 y Fl(0)1274 550 y Fs(.)456 726 y Fv(Prop)s(osition)i(12.)40 b Fh(F)-6 b(or)30 b(a)g(p)l(air)h(of)g(e)l(dges)f Fq(W)1925 738 y Fl(0)2010 679 y Fo(d)1986 726 y Fp(!)23 b Fq(W)2170 738 y Fl(1)2256 679 y Fo(e)2231 726 y Fp(!)g Fq(W)2415 738 y Fl(2)2482 726 y Fh(c)l(onsider)31 b(a)f(line)l(ar)h(function)1392 918 y Fq(g)1432 930 y Fo(de)1502 918 y Fs(\()p Fq(z)t Fs(\))23 b(=)1730 862 y Fq(\045)p Fs(\()p Fq(W)1883 874 y Fl(1)1921 862 y Fs(\))p Fq(z)e Fs(+)e Fq(\037)p Fs(\()p Fq(d)p Fs(\))p Fq(\013)2309 832 y Fm(\000)p Fo(L)p Fl(\()p Fo(d)p Fl(\))p 1730 899 769 4 v 1909 977 a Fq(\045)p Fs(\()p Fq(W)2062 989 y Fl(2)2100 977 y Fs(\))p Fq(\013)2185 953 y Fo(L)p Fl(\()p Fo(e)p Fl(\))456 1101 y Fh(Given)28 b Fq(y)e Fp(2)d Fs(\006)901 1113 y Fo(#)946 1101 y Fh(,)28 b(let)g Fq(l)1139 1113 y Fo(k)1208 1101 y Fh(b)l(e)g(the)g(se)l(quenc)l (e)g(fr)l(om)h(Pr)l(op)l(osition)g(10)g(and)g(let)f Fq(k)2845 1113 y Fo(i)2900 1101 y Fh(b)l(e)h(the)f(se)l(quenc)l(e)456 1201 y(of)39 b(times)f(whose)h(tr)l(ansitions)f(p)l(ass)h(thr)l(ough)g Fq(d;)14 b(e)p Fh(,)40 b(i.e.,)i Fq(W)2400 1213 y Fo(k)2435 1221 y Fk(i)2505 1201 y Fs(=)c Fq(W)2686 1213 y Fl(0)2724 1201 y Fh(,)i Fq(W)2867 1213 y Fo(k)2902 1221 y Fk(i)2929 1213 y Fl(+1)3056 1201 y Fs(=)e Fq(W)3237 1213 y Fl(1)3313 1201 y Fh(and)456 1301 y Fq(W)534 1313 y Fo(k)569 1321 y Fk(i)596 1313 y Fl(+2)707 1301 y Fs(=)23 b Fq(W)873 1313 y Fl(2)910 1301 y Fh(.)39 b(Then)1428 1453 y Fs(lim)1408 1506 y Fo(i)p Fm(!1)1588 1397 y Fq(s)1627 1409 y Fo(k)1662 1417 y Fk(i)1688 1409 y Fl(+1)p 1588 1434 189 4 v 1589 1510 a Fq(\034)1625 1522 y Fo(k)1660 1530 y Fk(i)1687 1522 y Fl(+2)1805 1453 y Fp(\000)18 b Fq(g)1928 1465 y Fo(de)2012 1336 y Fj(\022)2123 1397 y Fq(s)2162 1409 y Fo(k)2197 1417 y Fk(i)p 2083 1434 187 4 v 2083 1510 a Fq(\034)2119 1522 y Fo(k)2154 1530 y Fk(i)2180 1522 y Fl(+1)2279 1336 y Fj(\023)2363 1453 y Fs(=)k(0)456 1650 y Fh(Pr)l(o)l(of.)43 b Fs(W)-7 b(e)28 b(ha)n(v)n(e)466 1776 y Fq(s)505 1788 y Fo(k)540 1796 y Fk(i)566 1788 y Fl(+1)p 466 1813 189 4 v 467 1889 a Fq(\034)503 1901 y Fo(k)538 1909 y Fk(i)565 1901 y Fl(+2)683 1832 y Fp(\000)18 b Fq(g)806 1844 y Fo(de)889 1715 y Fj(\022)1001 1776 y Fq(s)1040 1788 y Fo(k)1075 1796 y Fk(i)p 960 1813 187 4 v 960 1889 a Fq(\034)996 1901 y Fo(k)1031 1909 y Fk(i)1058 1901 y Fl(+1)1157 1715 y Fj(\023)548 2072 y Fs(=)802 2016 y Fq(s)841 2028 y Fo(k)876 2036 y Fk(i)903 2028 y Fl(+1)1009 2016 y Fp(\000)g Fq(s)1131 2028 y Fo(k)1166 2036 y Fk(i)p 715 2053 570 4 v 715 2135 a Fq(\013)768 2103 y Fo(l)789 2112 y Fk(k)820 2125 y(i)874 2135 y Fp(\001)g Fq(\013)968 2111 y Fo(L)p Fl(\()p Fo(d)p Fl(\)+)p Fo(L)p Fl(\()p Fo(e)p Fl(\))1313 2072 y Fp(\001)1364 2016 y Fq(\013)1417 1985 y Fo(l)1438 1994 y Fk(k)1469 2007 y(i)1497 1994 y Fc(+2)p 1364 2053 212 4 v 1377 2129 a Fq(\034)1413 2141 y Fo(k)1448 2149 y Fk(i)1475 2141 y Fl(+2)1604 2072 y Fs(+)1738 2016 y Fq(s)1777 2028 y Fo(k)1812 2036 y Fk(i)p 1697 2053 187 4 v 1697 2129 a Fq(\034)1733 2141 y Fo(k)1768 2149 y Fk(i)1795 2141 y Fl(+1)1907 1955 y Fj(\022)1978 2016 y Fq(\034)2014 2028 y Fo(k)2049 2036 y Fk(i)2076 2028 y Fl(+1)p 1978 2053 V 1978 2129 a Fq(\034)2014 2141 y Fo(k)2049 2149 y Fk(i)2076 2141 y Fl(+2)2193 2072 y Fp(\000)2379 2016 y Fq(\045)p Fs(\()p Fq(W)2532 2028 y Fl(1)2570 2016 y Fs(\))p 2286 2053 410 4 v 2286 2131 a Fq(\045)p Fs(\()p Fq(W)2439 2143 y Fl(2)2477 2131 y Fs(\))p Fq(\013)2562 2107 y Fo(L)p Fl(\()p Fo(e)p Fl(\))2705 1955 y Fj(\023)2785 2072 y Fp(\000)3094 2016 y Fq(\037)p Fs(\()p Fq(d)p Fs(\))p 2878 2053 593 4 v 2878 2131 a Fq(\045)p Fs(\()p Fq(W)3031 2143 y Fl(2)3069 2131 y Fs(\))p Fq(\013)3154 2107 y Fo(L)p Fl(\()p Fo(d)p Fl(\)+)p Fo(L)p Fl(\()p Fo(e)p Fl(\))539 2303 y Fp(!)931 2247 y Fq(\037)p Fs(\()p Fq(d)p Fs(\))p 715 2284 V 715 2361 a Fq(\045)p Fs(\()p Fq(W)868 2373 y Fl(2)905 2361 y Fs(\))p Fq(\013)990 2337 y Fo(L)p Fl(\()p Fo(d)p Fl(\)+)p Fo(L)p Fl(\()p Fo(e)p Fl(\))1336 2303 y Fs(+)1469 2247 y Fq(s)1508 2259 y Fo(k)1543 2267 y Fk(i)p 1429 2284 187 4 v 1429 2360 a Fq(\034)1465 2372 y Fo(k)1500 2380 y Fk(i)1527 2372 y Fl(+1)1643 2303 y Fp(\001)h Fs(0)f Fp(\000)2054 2247 y Fq(\037)p Fs(\()p Fq(d)p Fs(\))p 1838 2284 593 4 v 1838 2361 a Fq(\045)p Fs(\()p Fq(W)1991 2373 y Fl(2)2028 2361 y Fs(\))p Fq(\013)2113 2337 y Fo(L)p Fl(\()p Fo(d)p Fl(\)+)p Fo(L)p Fl(\()p Fo(e)p Fl(\))2463 2303 y Fs(=)23 b(0)3380 2485 y Fg(\003)456 2638 y Fv(Theorem)32 b(13.)42 b Fh(L)l(et)30 b Fq(#)c Fs(:)g Fq(A)h Fp(!)f Fq(A)1556 2608 y Fl(+)1642 2638 y Fh(b)l(e)32 b(a)f(primitive)i(substitution)e (with)h(an)f(ap)l(erio)l(dic)j(\014xe)l(d)456 2738 y(p)l(oint)c Fq(x)23 b Fp(2)h Fq(A)877 2708 y Fn(N)923 2738 y Fh(.)39 b(Set)1309 2841 y Fv(r)1348 2853 y Fl(0)1409 2841 y Fs(=)22 b(min)15 b Fq(R)p 1649 2854 64 4 v Fs(\(\006)1804 2853 y Fo(#)1849 2841 y Fs(\))p Fq(;)99 b Fv(r)2042 2853 y Fl(1)2102 2841 y Fs(=)23 b(max)p 2358 2774 V 13 w Fq(R)q Fs(\(\006)2514 2853 y Fo(#)2559 2841 y Fs(\))456 2964 y Fh(Then)30 b Fs(0)23 b Fq(<)f Fv(r)863 2976 y Fl(0)924 2964 y Fq(<)g Fv(r)1050 2976 y Fl(1)1111 2964 y Fq(<)h Fp(1)p Fh(,)30 b Fq(R)p 1337 2977 V 1 w Fs(\()p Fq(y)s Fs(\))23 b(=)g Fv(r)1659 2976 y Fl(0)1726 2964 y Fh(a.e.,)32 b(and)p 2074 2897 V 30 w Fq(R)p Fs(\()p Fq(y)s Fs(\))24 b(=)e Fv(r)2395 2976 y Fl(1)2463 2964 y Fh(a.e.)456 3115 y(Pr)l(o)l(of.)43 b Fs(Sa)n(y)25 b(that)i Fq(C)j Fp(\022)22 b(W)1308 3127 y Fl(0)1372 3115 y Fs(is)k(a)g(\014nal)h(irreducible)f (comp)r(onen)n(t)g(of)g Fp(G)2681 3127 y Fl(0)2719 3115 y Fs(,)h(if)g(for)f(ev)n(ery)f Fq(W)35 b Fp(2)23 b Fq(C)456 3214 y Fs(and)36 b Fq(W)716 3184 y Fm(0)776 3214 y Fp(2)i(W)951 3226 y Fl(0)1025 3214 y Fs(w)n(e)d(ha)n(v)n(e)g Fq(W)1445 3184 y Fm(0)1506 3214 y Fp(2)j Fq(C)k Fs(i\013)37 b(there)f(exists)g(a) g(path)g(from)g Fq(W)48 b Fs(to)36 b Fq(W)3078 3184 y Fm(0)3101 3214 y Fs(.)63 b(Denote)456 3314 y(b)n(y)30 b Fq(C)633 3326 y Fl(1)671 3314 y Fq(;)14 b(:)g(:)g(:)f(;)h(C)914 3326 y Fo(p)984 3314 y Fs(the)31 b(\014nal)f(irreducible)g(comp)r(onen) n(ts)g(of)h Fp(G)2338 3326 y Fl(0)2376 3314 y Fs(.)46 b(The)30 b(set)h Fq(Y)2799 3326 y Fo(i)2855 3314 y Fp(\022)d Fs(\006)3008 3326 y Fo(#)3083 3314 y Fs(of)i(those)h Fq(y)456 3414 y Fs(whic)n(h)i(ultimately)g(attain)g Fq(C)1412 3426 y Fo(i)1474 3414 y Fs(is)g(op)r(en,)h(has)f(p)r(ositiv)n(e)g (measure,)g(and)g Fq(Y)51 b Fs(=)32 b Fq(Y)3024 3426 y Fl(1)3084 3414 y Fp([)22 b(\001)14 b(\001)g(\001)22 b([)h Fq(Y)3406 3426 y Fo(p)456 3513 y Fs(has)34 b(measure)h(1.)59 b(Sa)n(y)35 b(that)g(a)g(path)h Fq(e)f Fs(=)h Fq(e)1912 3525 y Fl(0)1948 3513 y Fq(;)14 b(:)g(:)g(:)g(;)g(e)2172 3525 y Fo(j)s Fm(\000)p Fl(1)2292 3513 y Fq(;)g(e)2368 3525 y Fo(j)2402 3513 y Fq(;)g(:)g(:)g(:)g(;)g(e)2626 3525 y Fo(k)q Fm(\000)p Fl(1)2786 3513 y Fs(in)36 b Fq(C)2950 3525 y Fo(i)3013 3513 y Fs(is)g(simple,)h(if)456 3613 y Fq(e)495 3625 y Fl(0)532 3613 y Fq(;)14 b(:)g(:)g(:)f(;)h(e)755 3625 y Fo(j)s Fm(\000)p Fl(1)905 3613 y Fs(is)31 b(a)f(cycle,)h(i.e.,)h Fq(t)p Fs(\()p Fq(e)1558 3625 y Fo(j)s Fm(\000)p Fl(1)1678 3613 y Fs(\))d(=)e Fq(s)p Fs(\()p Fq(e)1941 3625 y Fl(0)1978 3613 y Fs(\),)32 b Fq(e)2104 3625 y Fl(0)2141 3613 y Fq(;)14 b(:)g(:)g(:)g(;)g(e)2365 3625 y Fo(j)s Fm(\000)p Fl(1)2515 3613 y Fs(are)30 b(pairwise)f(distinct,)k(and)456 3713 y Fq(e)495 3725 y Fo(j)529 3713 y Fq(;)14 b(:)g(:)g(:)g(;)g(e)753 3725 y Fo(k)q Fm(\000)p Fl(1)907 3713 y Fs(are)27 b(pairwise)g (distinct.)39 b(The)29 b(comp)r(osition)e Fq(f)2381 3725 y Fo(e)2412 3733 y Fk(j)r Fe(\000)p Fc(1)2535 3713 y Fp(\001)14 b(\001)g(\001)f Fq(f)2686 3725 y Fo(e)2717 3733 y Fc(0)2782 3713 y Fs(has)28 b(a)g(unique)g(\014xed)456 3812 y(p)r(oin)n(t)i Fq(z)k Fs(and)c(w)n(e)g(set)g Fq(z)1208 3824 y Fo(e)1271 3812 y Fs(=)d Fq(f)1404 3824 y Fo(e)1435 3833 y Fk(k)q Fe(\000)p Fc(1)1562 3812 y Fp(\001)14 b(\001)g(\001)g Fq(f)1714 3824 y Fo(e)1745 3832 y Fk(j)1780 3812 y Fs(\()p Fq(z)t Fs(\).)45 b(The)31 b(set)f(of)g(simple)h(paths)f(is)g(\014nite.) 45 b(Denote)456 3912 y(b)n(y)33 b Fq(c)613 3924 y Fo(i)673 3912 y Fq(>)f Fs(0)g(the)i(minim)n(um)g(of)f(all)g(1)p Fq(=z)1714 3924 y Fo(e)1782 3912 y Fs(o)n(v)n(er)e(all)i(simple)h (paths)f(in)g Fq(C)2744 3924 y Fo(i)2772 3912 y Fs(.)54 b(Then)33 b(for)g(almost)456 4011 y(all)28 b Fq(y)h Fp(2)d Fq(Y)771 4023 y Fo(i)799 4011 y Fs(,)j Fq(R)p 851 4024 V 1 w Fs(\()p Fq(y)s Fs(\))d(=)f Fq(c)1175 4023 y Fo(i)1202 4011 y Fs(.)42 b(Consider)28 b(no)n(w)h(t)n(w)n(o)f(di\013eren)n(t)h (\014nal)g(irreducible)g(comp)r(onen)n(ts)f Fq(C)3393 4023 y Fo(i)3421 4011 y Fs(,)456 4111 y Fq(C)515 4123 y Fo(j)550 4111 y Fs(.)61 b(Since)36 b Fq(Y)907 4123 y Fo(i)935 4111 y Fs(,)h Fq(Y)1043 4123 y Fo(j)1114 4111 y Fs(are)e(op)r(en)h(and)f(\(\006)1731 4123 y Fo(#)1776 4111 y Fq(;)14 b(\033)s Fs(\))36 b(is)g(minimal,)i(there)d(exists)g Fq(k)40 b(>)c Fs(0)f(suc)n(h)g(that)456 4214 y Fq(Y)504 4226 y Fo(ij)585 4214 y Fs(=)23 b Fq(Y)721 4226 y Fo(i)758 4214 y Fp(\\)9 b Fq(\033)872 4183 y Fm(\000)p Fo(k)966 4214 y Fs(\()p Fq(Y)1046 4226 y Fo(j)1082 4214 y Fs(\))23 b(is)g(nonempt)n(y)g(and)g(has)f(p)r(ositiv)n(e)h(measure.)34 b(F)-7 b(or)23 b(almost)f(all)h Fq(y)j Fp(2)d Fq(Y)3268 4226 y Fo(ij)3350 4214 y Fs(w)n(e)456 4316 y(ha)n(v)n(e)c Fq(R)p 640 4329 V Fs(\()p Fq(y)s Fs(\))24 b(=)e Fq(c)958 4328 y Fo(i)1006 4316 y Fs(and)e Fq(c)1196 4328 y Fo(i)1247 4316 y Fp(\025)j Fq(R)p 1335 4329 V Fs(\()p Fq(\033)1480 4286 y Fo(k)1522 4316 y Fs(\()p Fq(y)s Fs(\)\))g Fp(\025)g Fq(c)1809 4328 y Fo(j)1844 4316 y Fs(.)34 b(Th)n(us)20 b(all)g Fq(c)2248 4328 y Fo(i)2296 4316 y Fs(are)f(equal)h Fq(c)2675 4328 y Fl(1)2735 4316 y Fs(=)j Fp(\001)14 b(\001)g(\001)23 b Fs(=)g Fq(c)3067 4328 y Fo(p)3128 4316 y Fs(=)g Fv(r)3255 4328 y Fl(0)3315 4316 y Fq(>)g Fs(0)456 4416 y(and)k(for)h(allmost)f (all)h Fq(y)e Fp(2)e Fs(\006)1358 4428 y Fo(#)1430 4416 y Fs(w)n(e)k(ha)n(v)n(e)f Fq(R)p 1745 4429 V Fs(\()p Fq(y)s Fs(\))d(=)g Fv(r)2068 4428 y Fl(0)2105 4416 y Fs(.)38 b(If)29 b Fq(y)d Fp(2)e Fs(\006)2456 4428 y Fo(#)2519 4416 y Fp(n)18 b Fq(Y)h Fs(,)28 b(then)g(for)g(some)f Fq(k)g Fp(\025)c Fs(0,)456 4515 y Fq(\033)506 4485 y Fo(k)547 4515 y Fs(\()p Fq(y)s Fs(\))g Fp(2)h Fq(Y)18 b Fs(,)28 b(so)f Fq(R)p 976 4528 V 1 w Fs(\()p Fq(y)s Fs(\))c Fp(\025)g Fq(R)p 1259 4528 V Fs(\()p Fq(\033)1404 4485 y Fo(k)1446 4515 y Fs(\()p Fq(y)s Fs(\)\))g Fp(\025)g Fv(r)1736 4527 y Fl(0)1774 4515 y Fs(,)k(so)g Fv(r)1965 4527 y Fl(0)2026 4515 y Fs(=)c(min)14 b Fq(R)p 2266 4528 V Fs(\(\006)2421 4527 y Fo(#)2466 4515 y Fs(\).)555 4615 y(Similarly)33 b(denote)g(b)n(y)g Fq(D)1377 4627 y Fl(1)1414 4615 y Fq(;)14 b(:)g(:)g(:)f(D)1630 4627 y Fo(p)1701 4615 y Fs(all)33 b(\014nal)g(irreducible)g(comp)r(onen)n(ts)f(of)h Fp(G)3040 4627 y Fl(2)3078 4615 y Fs(,)h Fq(Y)3183 4627 y Fo(i)3243 4615 y Fp(\022)e Fs(\006)3400 4627 y Fo(#)456 4715 y Fs(the)23 b(set)g(of)g(those)g(p)r(oin)n(ts)g(whic)n(h)g (ultimately)g(attain)g Fq(D)2203 4727 y Fo(i)2231 4715 y Fs(.)35 b(if)24 b Fq(e)e Fs(=)h Fq(e)2549 4727 y Fl(0)2600 4715 y Fq(:)14 b(:)g(:)f(;)h(e)2786 4727 y Fo(j)s Fm(\000)p Fl(1)2906 4715 y Fq(;)g(e)2982 4727 y Fo(j)3016 4715 y Fq(;)g(:)g(:)g(:)g(;)g(e)3240 4727 y Fo(k)q Fm(\000)p Fl(1)3389 4715 y Fs(is)456 4814 y(a)22 b(simple)h(path)g(in)g Fp(G)1104 4826 y Fl(2)1141 4814 y Fs(,)h(then)f(the)h(comp)r(osition)e Fq(g)2010 4826 y Fo(e)2041 4834 y Fk(j)r Fe(\000)p Fc(1)2163 4814 y Fp(\001)14 b(\001)g(\001)g Fq(g)2314 4826 y Fo(e)2345 4834 y Fc(0)2404 4814 y Fs(has)22 b(a)h(single)f(\014xed)h(p)r(oin)n(t) g Fq(z)j Fs(and)456 4914 y(w)n(e)31 b(set)i Fq(z)756 4926 y Fo(e)821 4914 y Fs(=)e Fq(g)957 4926 y Fo(e)988 4935 y Fk(k)q Fe(\000)p Fc(1)1115 4914 y Fp(\001)14 b(\001)g(\001)g Fq(g)1266 4926 y Fo(e)1297 4934 y Fk(j)1332 4914 y Fs(\()p Fq(z)t Fs(\).)51 b(Since)32 b(all)g(co)r(e\016cien)n(ts)g(of)g(all)g (functions)g Fq(g)2900 4926 y Fo(e)2931 4934 y Fk(j)2999 4914 y Fs(are)f(p)r(ositiv)n(e,)456 5014 y(w)n(e)d(ha)n(v)n(e)f Fq(z)810 5026 y Fo(e)870 5014 y Fq(>)d Fs(0.)39 b(Denote)29 b(b)n(y)f Fq(d)1508 5026 y Fo(i)1561 5014 y Fq(<)c Fp(1)k Fs(the)h(maxim)n(um)g(of)f(all)g(1)p Fq(=z)2628 5026 y Fo(e)2691 5014 y Fs(o)n(v)n(er)f(all)h(simple)h(paths)456 5116 y(in)35 b Fq(D)629 5128 y Fo(i)656 5116 y Fs(.)58 b(Then)35 b(for)f(almost)g(all)h Fq(y)i Fp(2)e Fq(Y)1709 5128 y Fo(i)1737 5116 y Fs(,)p 1797 5049 V 37 w Fq(R)p Fs(\()p Fq(y)s Fs(\))h(=)e Fq(d)2146 5128 y Fo(i)2174 5116 y Fs(.)58 b(Consider)34 b(no)n(w)g(t)n(w)n(o)g(di\013eren)n(t)h (\014nal)456 5216 y(irreducible)23 b(comp)r(onen)n(ts)g Fq(D)1385 5228 y Fo(i)1413 5216 y Fs(,)i Fq(D)1530 5228 y Fo(j)1564 5216 y Fs(.)36 b(Since)24 b Fq(Y)1884 5228 y Fo(i)1912 5216 y Fs(,)h Fq(Y)2008 5228 y Fo(j)2067 5216 y Fs(are)e(op)r(en)h(and)g(\(\006)2649 5228 y Fo(#)2693 5216 y Fq(;)14 b(\033)s Fs(\))25 b(is)f(minimal,)h(there)p eop %%Page: 9 9 9 8 bop 1038 251 a Fl(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)h(SUBSTITUTIVE)h(SUBSHIFTS)548 b(9)456 450 y Fs(exists)30 b Fq(k)h(>)d Fs(0)i(suc)n(h)h(that)g Fq(Y)1349 462 y Fo(ij)1435 450 y Fs(=)d Fq(Y)1576 462 y Fo(i)1625 450 y Fp(\\)21 b Fq(\033)1751 420 y Fm(\000)p Fo(k)1844 450 y Fs(\()p Fq(Y)1924 462 y Fo(j)1959 450 y Fs(\))31 b(is)g(nonempt)n(y)f (and)h(has)f(p)r(ositiv)n(e)g(measure.)456 553 y(The)k(set)g Fq(\033)819 523 y Fo(k)860 553 y Fs(\()p Fq(Y)940 565 y Fo(ij)999 553 y Fs(\))g Fp(\022)f Fq(Y)1211 565 y Fo(j)1281 553 y 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2010 689 a Fq(\013)g Fp(\000)g Fs(1)2239 632 y(=)23 b Fq(\013;)97 b Fv(r)2539 644 y Fl(1)2600 632 y Fs(=)22 b Fq(\013)456 824 y Fv(Corollary)33 b(16.)1469 902 y Fs(1)p 1444 939 91 4 v 1444 1015 a Fq(\013)1497 991 y Fl(2)1568 958 y Fp(\024)23 b Fq(R)p 1656 971 64 4 v Fs(\()p Fq(x)p Fs(\))h Fp(\024)1958 902 y Fs(1)p 1952 939 54 4 v 1952 1015 a Fq(\013)2038 958 y Fp(\024)p 2126 891 64 4 v 23 w Fq(R)q Fs(\()p Fq(x)p Fs(\))g Fp(\024)e Fq(\013)456 1114 y Fh(with)30 b Fq(R)p 636 1127 V 1 w Fs(\()p Fq(x)p Fs(\))24 b(=)e Fq(\013)975 1084 y Fm(\000)p Fl(2)1065 1114 y Fh(,)p 1120 1047 V 30 w Fq(R)q Fs(\()p Fq(x)p Fs(\))i(=)e Fq(\013)31 b Fh(almost)f(everywher)l(e.)456 1263 y Fv(Ac)m(kno)m(wledgemen)m(t.)35 b Fs(A)26 b(part)f(of)h(this)g (researc)n(h)e(has)h(b)r(een)i(done)e(during)h(m)n(y)f(visit)h(at)g (the)456 1363 y(Cen)n(tre)34 b(de)h(Ph)n(ysique)e(Th)n(\023)-39 b(eorique)33 b(in)i(Lumin)n(y)-7 b(,)37 b(Marseille.)57 b(I)35 b(thank)f(Sandro)g(V)-7 b(aien)n(ti)35 b(for)456 1462 y(suggesting)26 b(the)i(problematics.)1708 1632 y Fr(References)491 1765 y Fu([1])35 b(V.)29 b(Afraimo)n(vic)n(h,)g(J.) g(R.)g(Chazottes,)k(B.Saussol:)42 b(P)n(oin)n(t)n(wise)30 b(dimensions)f(for)g(P)n(oincar)n(\023)-33 b(e)30 b(recurrence)601 1848 y(asso)r(ciated)k(with)g(maps)e(and)i(sp)r(ecial)f(\015o)n(ws.)g (Discrete)g(and)h(Con)n(tin)n(uous)g(Dynamical)f(Systems,)i(to)601 1931 y(app)r(ear)491 2014 y([2])g(J.)23 b(Cassaigne,)h(P)-6 b(.)23 b(Hub)r(ert,)h(and)g(S.)f(V)-6 b(aien)n(ti,)24 b(priv)l(ate)g(comm)n(unication.)491 2097 y([3])35 b(F.Durand,)27 b(B.)f(Host,)i(and)f(C.)g(Sk)l(au:)38 b(Substitutional)29 b(dynamical)d(systems,)h(Bratelli)f(diagrams)g(and)601 2180 y(dimensions)c(groups.)i Fb(Er)l(go)l(d.)j(Th.)f Fu(&)f Fb(Dynam.)h(Sys.)d Fa(19)p Fu(,)f(953-993)j(\(1999\))491 2263 y([4])35 b(M.)27 b(Hirata,)i(B.Saussol,)f(and)h(S.)f(V)-6 b(aien)n(ti:)41 b(Statistics)29 b(of)f(return)g(times:)39 b(A)28 b(general)h(framew)n(ork)e(and)601 2346 y(new)d(applications,)g Fb(Comm.)i(Math.)g(Phys.)e Fa(206)p Fu(:)30 b(3-55)24 b(\(1999\))491 2429 y([5])35 b(M.)22 b(Kupsa:)32 b(Lo)r(cal)24 b(return)g(rates)g(in)f(Sturmian)g(subshifts,)f(submitted.)491 2512 y([6])35 b(P)-6 b(.)23 b(K)-7 b(\027)-46 b(urk)l(a:)31 b(T)-6 b(op)r(ological)24 b(and)h(Sym)n(b)r(olic)d(Dynamics,)h(to)h(b)r (e)g(published)g(b)n(y)g(SMF.)491 2595 y([7])35 b(B.)20 b(Moss)n(\023)-33 b(e:)29 b(Puissances)21 b(de)g(mots)f(et)i (reconaissabilit)n(\023)-33 b(e)21 b(des)g(p)r(oin)n(ts)g(\014xes)g (d'une)g(substitution.)h Fb(The)l(or)l(et.)601 2678 y(Comput.)k(Sci.)d Fa(99)p Fu(:)31 b(327-334)24 b(\(1992\))491 2761 y([8])35 b(B.)27 b(Moss)n(\023)-33 b(e:)41 b(Reconnaissabilit)n(\023)-33 b(e)29 b(des)g(substitutions)g(et)g(complexit)n(\023)-33 b(e)29 b(des)g(suites)f(automatiques.)h Fb(Bul)t(l.)601 2844 y(So)l(c.)d(Math.)f(F)-5 b(r)l(anc)l(e)25 b Fa(124)p Fu(,)d(329-346)j(\(1996\))491 2927 y([9])35 b(B.Saussol,)21 b(S.T)-6 b(roubletzk)n(o)n(y)g(,)23 b(and)g(S.V)-6 b(aien)n(ti:)30 b(Recurrence,)23 b(dimensions,)d(and)i(Ly)n(apuno)n(v)i(exp)r(onen)n (ts,)601 3010 y Fb(J.)h(Stat.)h(Phys.)e Fa(106)p Fu(,)e(3/4,)i(623-634) g(\(2002\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303210648513--