This is a multi-part message in MIME format. ---------------0303040821683 Content-Type: text/plain; name="03-87.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-87.keywords" Liapunov Exponents, Decay of Correlations ---------------0303040821683 Content-Type: application/postscript; name="final.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="final.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: final.dvi %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Italic Times-Roman %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D 600 -Z -f final %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.03.04:0945 %%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: 8r.enc % @@psencodingfile@{ % author = "S. 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Fw(j)p Fv(S)1843 899 y FE(k)1892 872 y Fw(j)17 b Fl(&)2057 804 y Fv(C)p 2042 849 108 4 v 2042 943 a(\025)2100 909 y FE(k)2100 972 y FF(1)2186 872 y Fv(;)1455 b Fu(\()p Fx(2)p Fv(:)p Fx(2)p Fu(\))94 1198 y Fx(for)34 b(some)f(constant)g Fv(C)7 b Fx(.)62 b(While)34 b(this)e(property)h(is)h(in)f(a)h(w)o(ay)f (true)h(for)g FG(short)e(times)h Fx(\(small)g Fv(k)s Fx(\))h(because)94 1317 y(generally)39 b(there)g(is)f(a)h(component)e (of)i(the)g(observ)n(ables)e(which)i(\223feels\224)g(the)f(f)o(ast)h (rate)g(of)g(the)f(lar)n(gest)94 1437 y(Liapuno)o(v)19 b(multiplier)l(,)g(the)i(purpose)e(of)i(this)e(paper)i(is)e(to)h(sho)n (w)f(that)h(\(2.2\))g Ft(does)h(not)g(hold)f Fx(asymptotically)94 1556 y(in)25 b(general.)316 1676 y(First)41 b(of)h(all,)j(closer)c (scrutin)o(y)f(of)i(the)f(separation)g(ar)n(gument)g(gi)n(v)o(en)f(abo) o(v)o(e)h(indicates)f(that)h(the)94 1795 y(e)o(xpected)23 b(beha)n(vior)g(of)g Fv(S)1007 1822 y FE(k)1079 1795 y Fx(should)f(be)i(dictated)e(not)h(by)f(the)h(lar)n(gest)g(Liapuno)o (v)f(multiplier)l(,)f(b)n(ut)i(rather)g(by)94 1915 y(the)i FG(smallest)f(abo)o(ve)g Fx(1:)1722 2089 y Fw(j)p Fv(S)1811 2116 y FE(k)1860 2089 y Fw(j)17 b Fl(&)2057 2022 y Fv(C)p 2010 2066 172 4 v 2010 2157 a(\025)2068 2129 y FE(k)2068 2184 y FF(min)2219 2089 y Fv(;)1422 b Fu(\()p Fx(2)p Fv(:)p Fx(3)p Fu(\))94 2383 y Fx(where)337 2340 y FF(5)1413 2529 y Fv(\025)1471 2556 y FF(min)1629 2529 y Fu(=)44 b Fx(min)o Fw(f)p Fv(\025)2013 2556 y FE(i)2099 2529 y Fx(:)52 b Fv(\025)2237 2556 y FE(i)2298 2529 y Fv(>)29 b Fx(1)p Fw(g)24 b Fv(:)1113 b Fu(\()p Fx(2)p Fv(:)p Fx(4)p Fu(\))94 2745 y Fx(W)-8 b(e)20 b(will)e(see)h(that)g(\(2.3\))f (holds)g(for)h(certain)h(special)e(e)o(xamples,)h(b)n(ut)f(for)i(a)f (general)g(map)g(the)f(Equation)g(\(2.3\))94 2865 y Ft(cannot)25 b(be)g(an)f(equality)g Fx(for)g(generic)g(observ)n(ables)f(in)g Fw(C)2134 2822 y FF(1)2174 2865 y Fx(,)h(e)n(v)o(en)f(if)h(we)g(a)n(v)n (oid)f(the)h(resonances.)35 b(Namely)-6 b(,)94 2985 y(we)26 b(e)o(xpect)e(for)h(maps)f Fv(f)36 b Fx(with)24 b(non\255constant)f (deri)n(v)n(ati)n(v)o(e)g(and)h(for)h(observ)n(ables)f(in)h Fw(C)3099 2942 y FF(1)3164 2985 y Fx(an)g(inequality)1734 3298 y Fw(j)p Fv(S)1823 3325 y FE(k)1872 3298 y Fw(j)17 b Fl(&)2057 3231 y Fv(C)p 2022 3275 149 4 v 2022 3367 a(\025)2080 3338 y FE(k)2080 3394 y FF(ess)2207 3298 y Fv(;)1434 b Fu(\()p Fx(2)p Fv(:)p Fx(5)p Fu(\))94 3642 y Fx(with)22 b(1)28 b Fv(<)g(\025)535 3669 y FF(ess)653 3642 y Fv(<)g(\025)816 3669 y FF(min)930 3642 y Fx(:)34 b Ft(In)23 b(general,)g(the)h(decorr)n(elation)g(is)e(slo)o(wer)g(than) i Fv(C)t(=\025)3006 3599 y FE(k)3006 3669 y FF(min)3120 3642 y Fx(.)35 b(Furthermore,)23 b Fv(\025)3784 3669 y FF(ess)94 3761 y Fx(is)30 b(a)f(much)g(stronger)g(barrier)h(to)f (decay)h(than)f(the)h(resonances:)45 b(Only)29 b(a)g(v)o(ery)h(radical) f(restriction)g(of)g(the)94 3881 y(observ)n(ables)20 b(\(to)h(a)g(subspace)g(of)g Fw(C)1323 3838 y FF(1)1384 3881 y Fx(with)f FG(in\002nite)g Fx(codimension\))f(will)h(in)g (general)h(lead)g(to)g(a)g(f)o(aster)g(decay)-6 b(.)316 4001 y(The)25 b(purpose)f(of)h(this)f(paper)h(is)f(to)h(clarify)g(the)f (issues)g(related)h(to)g(these)f(questions.)p 94 5091 1200 4 v 241 5159 a FD(5)316 5202 y FC(W)-6 b(e)22 b(do)g(not)g (consider)g(systems)h(with)f(Liapuno)o(v)f(multipliers)i(equal)g(to)f (1,)h(where)f(it)g(is)g(kno)n(wn)g(that)g(the)h(decorrelation)g(rate)94 5291 y(may)e(not)e(e)n(v)o(en)i(be)f(e)o(xponential.)p eop %%Page: 4 4 4 3 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1196 b Ft(4)94 194 y Fy(3.)30 b(The)g(Essential)f(Decorr)n(elation)i(Radius) 94 383 y Fx(In)26 b(this)e(section)g(we)h(de\002ne)h(the)e(essential)h (decorrelation)f(radius)h Fv(\045)2475 410 y FF(ess)2565 383 y Fx(.)36 b(The)25 b(essential)f(decorrelation)h(rate)94 503 y Fv(\025)152 530 y FF(ess)268 503 y Fx(is)f(then)h(de\002ned)g(by) 1680 623 y Fv(\025)1738 649 y FF(ess)1873 623 y Fu(=)44 b Fx(1)p Fv(=\045)2146 649 y FF(ess)2261 623 y Fv(;)94 803 y Fx(so)25 b(that)f(the)h(correlation)f(functions)g Fv(S)1442 829 y FE(k)1516 803 y Fx(will)g(basically)g(decay)h(lik)o(e)g Fv(\025)2564 760 y Fq(\000)p FE(k)2564 829 y FF(ess)2703 803 y Fu(=)j Fv(\045)2860 760 y FE(k)2860 829 y FF(ess)2950 803 y Fx(.)316 922 y(The)c(de\002nition)f(of)h Fv(\045)1060 949 y FF(ess)1173 922 y Fx(depends)g(on)g(tw)o(o)f(Banach)2118 879 y FF(6)2183 922 y Fx(spaces)h Fv(X)31 b Fx(and)24 b Fv(Y)e Fx(,)i(with)f Fv(X)31 b Fx(a)25 b(subspace)e(of)h(the)94 1042 y(dual)30 b(of)g Fv(Y)22 b Fx(,)31 b(The)f(reader)h(should)e (think)g(of)h Fv(X)38 b Fx(and)30 b Fv(Y)52 b Fx(as)30 b(the)g(Banach)g(space)h(of)f Fw(C)3074 999 y FF(1)3144 1042 y Fx(functions)f(with)g(the)94 1161 y(norm)d Fw(k)p Fv(h)p Fw(k)j Fu(=)h Fx(sup)763 1188 y FE(x)829 1161 y Fw(j)p Fv(h)p Fu(\()p Fv(x)p Fu(\))p Fw(j)22 b Fu(+)i Fx(sup)1338 1188 y FE(x)1405 1161 y Fw(j)p Fv(D)s(h)p Fu(\()p Fv(x)p Fu(\))p Fw(j)p 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FE(M)2450 2096 y Fj(0)2477 2116 y Fq(nf)p FF(0)p Fq(g)2663 1958 y Fw(jh)p Fv(y)t(;)17 b(U)2907 1916 y FE(n)2959 1958 y Fv(x)p Fw(ij)p 2663 2003 420 4 v 2718 2094 a(k)p Fv(x)p Fw(kk)p Fv(y)t Fw(k)3094 1855 y Fr(!)3173 1894 y FF(1)p FE(=n)3347 2026 y Fv(:)294 b Fu(\()p Fx(3)p Fv(:)p Fx(1)p Fu(\))94 2489 y Ft(Remark.)28 b Fx(The)e(reason)h(we)f(w)o(ant)g(the)h (space)f Fv(X)34 b Fx(to)26 b(be)h(in)l(v)n(ariant)e(under)i Fv(U)37 b Fx(is)26 b(to)g(mak)o(e)g(connection)g(later)94 2609 y(on)33 b(with)g(the)g(spectral)g(radius.)60 b(This)33 b(will)f(force)i(us)f(to)f(use)h(spaces)h Fv(X)40 b Fx(whose)33 b(de\002nitions)f(are)i(a)f(little)94 2728 y(in)l(v)n(olv)o(ed.)38 b(Although)24 b(such)i(a)g(problem)f(does)h(not)f(seem)h(to)f(appear)i (in)e(the)h(de\002nition)f(of)h(the)f(correlation)94 2848 y(function,)f(it)h(is)f(hidden)g(in)h(the)f(duality)g(relation)g (between)h(the)g(tw)o(o)f(observ)n(ables.)316 2968 y(The)i(idea)g(of)h (De\002nition)e(3.1)h(is)g(to)g(peal\255out)g(the)g(v)n(arious)f 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Fw(1)c Fk(or)50 b Fu(\()p Fv(U)33 b Fw(\000)23 b Fv(\025)p Ft(1)p Fu(\))p Fv(X)56 b Fk(is)25 b(not)f(closed)g Fw(g)h Fv(;)143 b Fu(\()p Fx(3)p Fv(:)p Fx(2)p Fu(\))94 4074 y Fk(and)25 b(the)g(point\255essential)e(spectral)i(radius)f(by) 405 4289 y Fv(\033)462 4316 y FF(p)p Fq(\000)p FF(ess)650 4289 y Fu(\()p Fv(X)r(;)17 b(U)11 b Fu(\))42 b(=)j Fx(sup)p Fw(fj)p Fv(\025)p Fw(j)51 b Fx(:)25 b Fv(\025)j Fw(2)g Ft(C)d Fk(is)f(an)h(accumulation)f(point)g(of)g(eigen)l(v)n(alues)1508 4456 y(or)h(an)g(eigen)l(v)n(alue)f(of)h(in\002nite)f(multiplicity)m Fw(g)h Fv(:)3669 4373 y Fu(\()p Fx(3)p Fv(:)p Fx(3)p Fu(\))94 4838 y Ft(Theor)n(em)i(3.3.)35 b Fk(Let)25 b Fv(U)36 b Fk(be)25 b(a)g(continuous)e(linear)i(operator)g(on)f Fv(X)8 b Fk(.)35 b(If)26 b Fv(X)35 b Fw(\032)28 b Fv(Y)2881 4795 y Fq(\003)2927 4838 y Fk(,)d(then)1343 5056 y Fv(\045)1395 5083 y FF(ess)1484 5056 y Fu(\()p Fv(X)r(;)17 b(Y)6 b(;)17 b(U)11 b Fu(\))42 b Fw(\025)j Fv(\033)2101 5083 y FF(p)p Fq(\000)p FF(ess)2288 5056 y Fu(\()p Fv(X)r(;)17 b(U)11 b Fu(\))23 b Fv(:)1043 b Fu(\()p Fx(3)p Fv(:)p Fx(4)p Fu(\))p 94 5182 1200 4 v 241 5248 a FD(6)316 5291 y FC(Hilbert)20 b(spaces)h(are)f(not)g(adequate)h(since)f(we)g(w)o(ork)f(with)h (functions)f(in)h FA(C)2420 5248 y FD(1)2455 5291 y FC(.)p eop %%Page: 5 5 5 4 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1196 b Ft(5)94 432 y(Remark.)42 b Fx(It)e(w)o(ould)g(be)h(much)f(nicer)h(if) f(we)h(kne)n(w)f(that)g Fv(\045)2290 459 y FF(ess)2380 432 y Fu(\()p Fv(X)r(;)17 b(Y)6 b(;)17 b(U)11 b Fu(\))63 b Fw(\025)k Fv(\033)3040 459 y FF(ess)3130 432 y Fu(\()p Fv(X)r(;)17 b(U)11 b Fu(\))p Fx(.)81 b(Some)41 b(of)94 552 y(the)25 b(dif)n(\002culties)e(of)i(this)e(paper)i(w)o(ould)f (disappear)l(,)g(and)h(the)f(considerations)g(of)g(Section)h(8)f(and)h (Section)f(9)94 671 y(w)o(ould)19 b(immediately)f(gi)n(v)o(e)g(the)h (inequality)f(\(2.5\).)34 b(Ne)n(v)o(ertheless,)19 b(Theorem)g(3.3)g (is)g(still)f(some)n(what)g(useful)94 791 y(because)28 b(information)f(on)g Fv(\033)1120 818 y FF(p)p Fq(\000)p FF(ess)1335 791 y Fx(is)g(relati)n(v)o(ely)f(easy)i(to)f(get)g(at.)45 b(One)27 b(might)f(be)i(tempted)f(to)g(conjecture)94 911 y(that)h Fv(\045)324 937 y FF(ess)414 911 y Fu(\()p Fv(X)r(;)17 b(Y)6 b(;)17 b(U)11 b Fu(\))30 b Fw(\025)j Fv(\033)1007 937 y FF(ess)1097 911 y Fu(\()p Fv(X)r(;)17 b(U)11 b Fu(\))p Fx(.)46 b(Ho)n(we)n(v)o(er)l(,)28 b(we)g(found)g(no)g (proof,)i(since)e(we)h(do)f(not)g(kno)n(w)f(those)h Fv(\025)94 1030 y Fx(for)e(which)f Fv(U)34 b Fw(\000)23 b Fv(\025)j Fx(has)f(closed)g(range.)39 b(On)25 b(the)g(other)h(hand,)f(for)h (those)f Fv(U)37 b Fx(and)25 b Fv(X)33 b Fx(we)26 b(will)f(consider)l (,)g(we)94 1150 y(shall)20 b(\002nd)h Fv(\033)536 1177 y FF(p)p Fq(\000)p FF(ess)723 1150 y Fu(\()p Fv(X)r(;)c(U)11 b Fu(\))27 b(=)h Fv(\033)1199 1177 y FF(ess)1289 1150 y Fu(\()p Fv(X)r(;)17 b(U)11 b Fu(\))p Fx(,)20 b(so)g(that)g(in)g(the)h 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FC(Note)f(that)g(if)g(all)g(the)h(slopes)f(are)g(positi)n(v)o (e,)g(we)g(are)g(really)h(talking)f(about)g(an)g Fm(`)p FC(\255fold)f(map)h(of)f(the)i(circle)g(to)e(itself.)p eop %%Page: 16 16 16 15 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(16)94 194 y Fx(abo)o(v)o(e)37 b(has)h(no)g(ef)n(fect,)j(since)d (the)f(boundary)h(terms)f(in)g([4,)k(proof)d(of)g(Lemma)f(5])h(do)g (not)f(contrib)n(ute.)94 314 y(Therefore,)26 b(one)f(\002nds)94 505 y Ft(Theor)n(em)32 b(8.2.)49 b Fk(The)30 b(essential)e(point)h (spectral)g(radius)g Fv(\033)2170 532 y FF(p)p Fq(\000)p FF(ess)2357 505 y Fk(for)h(maps)f(with)g Ft(constant)h(deri)o(v)o(ati)o (v)o(e)g(in)94 625 y(each)c(piece)g(of)f(the)g(Mark)o(o)o(v)g (partition)g Fv(\033)1643 652 y FF(p)p Fq(\000)p FF(ess)1855 625 y Fk(is)g(gi)n(v)o(en)e(by)767 897 y Fv(\033)824 924 y FF(p)p Fq(\000)p FF(ess)1011 897 y Fu(\()p Fw(C)1108 854 y FF(1)1148 897 y Fv(;)17 b Fc(L)p Fu(\))43 b(=)i 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Fv(d\026)p Fu(\()p Fv(x)p Fu(\))17 b Fw(j)p Fu(\()p Fv(f)2700 1762 y FE(n)2752 1805 y Fu(\))2791 1762 y Fq(0)2819 1805 y Fu(\()p Fv(x)p Fu(\))p Fw(j)2982 1762 y Fq(\000)p FF(1)3083 1664 y Fr(\023)3198 1805 y Fv(:)443 b Fu(\()p Fx(8)p Fv(:)p Fx(5)p Fu(\))94 2052 y Fx(This)40 b(means)f(that)g(for)i(such)e(maps)g(the)h(decay)g(of)g (correlations)g(is)f(indeed)g(related)h(to)g(the)g(Liapuno)o(v)94 2171 y(multiplier)l(,)30 b(and)g(as)g(we)g(shall)f(see)h(belo)n(w)f(in) h(Theorem)f(8.4,)i(equality)e(only)g(holds)g(if)h(all)g(the)f(slopes)g (are)94 2291 y(the)c(same)g(\(in)f(modulus\).)94 2460 y Ft(Pr)n(oof)43 b(of)f(Theor)n(em)i(8.2.)e Fx(W)-8 b(e)42 b(start)g(with)f(a)i(setting)e(which)h(is)f(some)n(what)g(more)h (general)g(than)g(the)94 2580 y(assumptions)28 b(of)h(Theorem)g(8.2.)49 b(W)-8 b(e)30 b(consider)f(a)h(map)f Fv(f)40 b Fx(of)29 b(the)g(unit)g(interv)n(al)f(which)h(is)g(piece)n(wise)g Fw(C)3834 2537 y FF(2)94 2700 y Fx(e)o(xpanding)d(and)g(Mark)o(o)o(v)-6 b(,)25 b(namely)h(there)h(is)e(a)i(\002nite)g(partition)e Fw(A)h Fx(of)g(the)g(interv)n(al)g(by)g(subinterv)n(als)e(such)94 2819 y(that)30 b(on)f(each)i(atom)e Fv(f)40 b Fx(is)29 b(monotone)g(and)g Fw(C)1690 2776 y FF(2)1760 2819 y Fx(on)h(the)g(closure)f(and)h(such)f(that)h(the)f(image)h(of)f(each)i (atom)94 2939 y(is)h(the)g(union)g(of)g(atoms)g(\(modulo)f(closure\).) 58 b(W)-8 b(e)33 b(also)f(assume)f(that)h(there)h(is)f(an)g(inte)o(ger) g Fv(k)j Fx(for)e(which)94 3058 y Fw(j)p Fv(f)182 3015 y FE(k)231 3010 y Fq(0)259 3058 y Fw(j)28 b Fv(>)h(\020)36 b(>)29 b Fx(1,)d(and)f Fv(f)36 b Fx(is)25 b(topologically)f(mixing.)36 b(Under)26 b(these)f(assumptions)f(it)h(is)g(well)g(kno)n(wn)f(that)i Fv(f)94 3178 y Fx(has)i(a)h(unique)e(absolutely)g(continuous)f(in)l(v)n (ariant)h(probability)f(measure)i Fv(d\026)k Fu(=)h Fv(h)f(dx)c Fx(which)f(is)h(er)n(godic)94 3297 y(with)e(e)o(xponential)f(decay)i (of)f(correlations)g(\(see)g([2])h(and)f(references)i(therein\).)40 b(It)26 b(is)g(also)g(easy)h(to)f(v)o(erify)94 3417 y(that)f Fv(h)g Fx(is)f Fw(C)500 3374 y FF(1)566 3417 y Fx(on)g(each)h(atom)g (of)g Fw(A)f Fx(\(with)g Fw(C)1627 3374 y FF(1)1692 3417 y Fx(e)o(xtension)g(to)g(the)h(closure\).)316 3536 y(W)-8 b(e)22 b(will)e(consider)h(the)g(decay)h(of)f(correlations)g(in)f(the)h (space)h Fv(X)29 b Fx(of)21 b(functions)f(which)h(are)h Fw(C)3554 3494 y FF(1)3615 3536 y Fx(e)o(xcept)94 3656 y(maybe)34 b(on)g(the)g(boundary)f(of)h(the)g(atoms)f(of)h Fw(A)p Fx(.)63 b(F)o(or)34 b Fv(Y)22 b Fx(,)37 b(we)d(use)g(the)g (space)g(of)g Fw(C)3140 3613 y FF(1)3215 3656 y Fx(functions)e(whose)94 3775 y(inte)o(gral)23 b(o)o(v)o(er)h(each)g(atom)f(of)h Fw(A)g Fx(is)g(equal)f(to)h(zero.)36 b(This)23 b(insures)h(that)f(if)h Fv(g)31 b Fw(2)d Fv(Y)22 b Fx(,)i(we)g(can)h(\002nd)f(a)g(function)94 3895 y Fv(v)32 b Fw(2)c(C)326 3852 y FF(2)391 3895 y Fx(such)d(that)f Fv(v)825 3852 y Fq(0)880 3895 y Fu(=)29 b Fv(g)f Fx(and)c Fv(v)29 b Fx(v)n(anishes)24 b(on)g(the)h(boundary)f (of)h(the)g(atoms)f(of)g Fw(A)p Fx(.)316 4015 y(W)-8 b(e)25 b(will)f(denote)h(by)f Fw(A)1146 4041 y FE(n)1225 4015 y Fx(the)g(partition)1734 3940 y Fr(W)1817 3972 y FE(n)1817 4044 y FF(0)1888 4015 y Fv(f)1948 3972 y Fq(\000)p FE(j)2052 4015 y Fw(A)p Fx(.)35 b(If)25 b Fv(u)g Fx(and)g Fv(v)k Fx(are)c Fw(C)2815 3972 y FF(1)2880 4015 y Fx(functions,)f(we)h(ha)n(v)o(e)423 4132 y Fr(Z)539 4267 y Fv(u)d Fw(\001)g Fv(v)720 4225 y Fq(0)770 4267 y Fw(\016)g Fv(f)902 4225 y FE(n)956 4267 y Fv(d\026)27 b Fu(=)1200 4132 y Fr(Z)1342 4200 y Fv(uh)p 1329 4245 142 4 v 1329 4336 a(f)1389 4307 y FE(n)t Fq(0)1504 4267 y Fw(\001)22 b Fv(v)1606 4225 y Fq(0)1656 4267 y Fw(\016)g Fv(f)1788 4225 y FE(n)1842 4267 y Fv(f)1902 4225 y FE(n)5 b Fq(0)1983 4267 y Fv(dx)28 b Fu(=)2296 4173 y Fr(X)2225 4386 y FE(I)5 b Fq(2A)2383 4404 y Fn(n)p Fj(\000)p Fo(1)2528 4132 y Fr(Z)2583 4358 y FE(I)2671 4200 y Fv(uh)p 2657 4245 V 2657 4336 a(f)2717 4307 y FE(n)g Fq(0)2833 4267 y Fw(\001)22 b Fv(v)2935 4225 y Fq(0)2985 4267 y Fw(\016)g Fv(f)3117 4225 y FE(n)3193 4267 y Fw(\001)g Fv(f)3303 4225 y FE(n)t Fq(0)3384 4267 y Fv(dx)j(;)94 4575 y Fx(and)g(inte)o (grating)e(by)i(parts)f(we)i(get)354 4691 y Fr(Z)470 4827 y Fv(u)d Fw(\001)e Fv(v)651 4784 y Fq(0)701 4827 y Fw(\016)h Fv(f)833 4784 y FE(n)887 4827 y Fv(d\026)44 b Fu(=)h Fw(\000)1330 4732 y Fr(X)1259 4946 y FE(I)5 b Fq(2A)1417 4964 y Fn(n)p Fj(\000)p Fo(1)1562 4691 y Fr(Z)1617 4918 y FE(I)1679 4686 y Fr(\022)1778 4760 y Fv(uh)p 1765 4804 V 1765 4895 a(f)1825 4866 y FE(n)t Fq(0)1918 4686 y Fr(\023)1992 4725 y Fq(0)2036 4827 y Fv(v)26 b Fw(\016)c Fv(f)2242 4784 y FE(n)2296 4827 y Fv(dx)635 5176 y Fu(+)800 5081 y Fr(X)729 5295 y FE(I)5 b Fq(2A)887 5313 y Fn(n)p Fj(\000)p Fo(1)1027 5108 y Fv(u)p Fu(\()p Fv(b)1166 5065 y Fq(\000)1166 5137 y FE(I)1233 5108 y Fu(\))p Fv(h)p Fu(\()p Fv(b)1411 5065 y Fq(\000)1411 5137 y FE(I)1478 5108 y Fu(\))p Fv(v)t Fu(\()p Fv(f)1668 5065 y FE(n)1721 5108 y Fu(\()p Fv(b)1803 5065 y Fq(\000)1803 5137 y FE(I)1870 5108 y Fu(\)\))p 1027 5153 920 4 v 1323 5251 a Fv(f)1383 5222 y FE(n)g Fq(0)1465 5251 y Fu(\()p Fv(b)1547 5208 y Fq(\000)1547 5280 y FE(I)1613 5251 y Fu(\))1982 5176 y Fw(\000)2152 5081 y Fr(X)2081 5295 y FE(I)g Fq(2A)2239 5313 y Fn(n)p Fj(\000)p Fo(1)2396 5108 y Fv(u)p Fu(\()p Fv(a)2545 5065 y Fp(+)2545 5137 y FE(I)2610 5108 y Fu(\))p Fv(h)p Fu(\()p Fv(a)2798 5065 y Fp(+)2798 5137 y FE(I)2864 5108 y Fu(\))p Fv(v)t Fu(\()p Fv(f)3054 5065 y FE(n)3107 5108 y Fu(\()p Fv(a)3199 5065 y Fp(+)3199 5137 y FE(I)3265 5108 y Fu(\)\))p 2396 5153 947 4 v 2700 5251 a Fv(f)2760 5222 y FE(n)g Fq(0)2842 5251 y Fu(\()p Fv(a)2934 5208 y Fp(+)2934 5280 y FE(I)2999 5251 y Fu(\))3382 5176 y Fv(;)3669 5032 y Fu(\()p Fx(8)p Fv(:)p Fx(6)p Fu(\))p eop %%Page: 17 17 17 16 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(17)94 194 y Fx(where)26 b(the)e(boundary)h(points)e Fv(a)1232 221 y FE(I)1302 194 y Fx(and)i Fv(b)1514 221 y FE(I)1585 194 y Fx(are)g(de\002ned)g(for)g Fv(I)36 b Fw(2)28 b(A)2444 221 y FE(n)p Fq(\000)p FF(1)2620 194 y Fx(by)p 1716 347 52 4 v 1716 427 a Fv(I)35 b Fu(=)28 b Fx([)p Fv(a)1986 453 y FE(I)2032 427 y Fv(;)17 b(b)2120 453 y FE(I)2164 427 y Fx(])28 b Fv(:)94 645 y Fx(Note)h(that)f(the)g (tw)o(o)g(sequences)h Fu(\()p Fv(a)1347 672 y FE(I)1392 645 y Fu(\))f Fx(and)h Fu(\()p Fv(b)1714 672 y FE(I)1759 645 y Fu(\))g Fx(are)g(identical)f(e)o(xcept)g(for)h(the)f(\002rst)h (and)f(last)g(terms,)h(and)94 765 y(the)o(y)23 b(are)i(gi)n(v)o(en)d (by)h(all)h(the)f(preimages)h(of)g(order)g(up)f(to)g Fv(n)d Fw(\000)g Fx(1)k(of)f(the)h(boundaries)f(of)h(the)f(atoms)g(of)h Fw(A)p Fx(.)35 b(In)94 884 y(particular)l(,)25 b(for)g(each)h Fv(I)35 b Fw(2)28 b(A)1124 911 y FE(n)1178 884 y Fx(,)d Fv(f)1288 842 y FE(n)1341 884 y Fu(\()p Fv(a)1433 911 y FE(I)1478 884 y Fu(\))g Fx(and)g Fv(f)1771 842 y FE(n)1825 884 y Fu(\()p Fv(b)1907 911 y FE(I)1952 884 y Fu(\))f Fx(belong)g(to)h Fv(@)6 b Fw(A)p Fx(.)316 1004 y(W)-8 b(e)31 b(no)n(w)e(use)h(the)g(assumption)f(of)h(Theorem)g(8.2,)h (namely)f(that)g Fv(f)2724 961 y Fq(0)2781 1004 y Fx(is)g(constant)g (on)g(the)g(atoms)f(of)94 1124 y Fw(A)p Fx(.)40 b(This)25 b(implies)g(that)g Fv(f)997 1081 y FE(n)5 b Fq(0)1105 1124 y Fx(is)26 b(constant)f(on)h(the)g(atoms)f(of)h Fw(A)2283 1150 y FE(n)p Fq(\000)p FF(1)2434 1124 y Fx(.)40 b(Therefore,)27 b(the)f(\002rst)g(term)g(of)g(\(8.6\))h(is)94 1243 y(gi)n(v)o(en)d(by)288 1523 y Fw(\000)453 1428 y Fr(X)382 1642 y FE(I)5 b Fq(2A)540 1660 y Fn(n)p Fj(\000)p Fo(1)685 1387 y Fr(Z)740 1614 y FE(I)803 1383 y Fr(\022)902 1456 y Fv(uh)p 888 1500 142 4 v 888 1591 a(f)948 1563 y FE(n)g Fq(0)1042 1383 y Fr(\023)1115 1421 y Fq(0)1159 1523 y Fv(v)26 b Fw(\016)c Fv(f)1365 1480 y FE(n)1419 1523 y Fv(dx)44 b Fu(=)h Fw(\000)1788 1387 y Fr(Z)1916 1456 y Fu(\()p Fv(uh)p Fu(\))2108 1413 y Fq(0)p 1916 1500 220 4 v 1955 1591 a Fv(f)2015 1563 y FE(n)5 b Fq(0)2176 1523 y Fv(v)26 b Fw(\016)c Fv(f)2382 1480 y FE(n)2435 1523 y Fv(dx)44 b Fu(=)h Fw(\000)2804 1387 y Fr(Z)2921 1523 y Fw(L)2990 1480 y FE(n)3060 1383 y Fr(\022)3145 1456 y Fu(\()p Fv(uh)p Fu(\))3337 1413 y Fq(0)p 3145 1500 V 3184 1591 a Fv(f)3244 1563 y FE(n)5 b Fq(0)3377 1383 y Fr(\023)3467 1523 y Fv(v)t(dx)25 b(;)94 1857 y Fx(where)i Fw(L)f Fx(is)g(the)g(Perron\255Frobenius)h(operator)f (associated)g(to)f Fv(f)11 b Fx(.)40 b(Note)26 b(that)g(when)g Fv(f)37 b Fx(is)25 b(not)h(constant)f(on)94 1976 y(each)g(atom)e(of)h Fw(A)p Fx(,)f(another)h(term)f(appears)h(in)l(v)n(olving)e(the)i(deri)n (v)n(ati)n(v)o(e)d(of)j Fv(f)2742 1933 y FE(n)t Fq(0)2823 1976 y Fx(.)36 b(This)23 b(term)g(corresponds)h(to)94 2096 y(a)29 b(compact)f(operator)g(and)g(did)f(not)h(interv)o(ene)f(in) h(the)f(computation)g(of)h(the)g(essential)f(spectral)h(radius)g(in)94 2215 y([4].)37 b(It)24 b(is)h(not)f(clear)h(ho)n(w)f(such)h(a)g(term)g (w)o(ould)f(in\003uence)h(the)f(present)h(computation.)316 2335 y(T)-8 b(o)24 b(complete)h(the)f(proof)h(of)g(Theorem)f(8.2)h(one) g(\002rst)g(applies)f(Theorem)g(3.3)h(to)f(the)h(operator)1642 2610 y Fv(U)11 b Fu(\()p Fv(g)t Fu(\))27 b(=)h Fw(L)2069 2470 y Fr(\022)2172 2543 y Fv(g)p 2153 2587 88 4 v 2153 2679 a(f)2213 2650 y Fq(0)2253 2470 y Fr(\023)94 2904 y Fx(in)35 b(the)h(space)f Fv(X)712 2861 y Fq(0)775 2904 y Fx(of)h(functions)e(which)h(are)h(piece)n(wise)f Fw(C)2220 2861 y FF(0)2296 2904 y Fx(e)o(xcept)g(possibly)e(at)j(the)f(boundary)g (of)g(the)94 3023 y(atoms)27 b(of)g Fw(A)p Fx(,)g(and)g Fv(Y)853 2981 y Fq(0)908 3023 y Fx(the)f(space)i(of)f Fw(C)1473 2981 y FF(2)1540 3023 y Fx(functions)f(v)n(anishing)f(on)h Fv(@)6 b Fw(A)p Fx(.)42 b(One)27 b(then)f(applies)h(Lemma)f(5)h(of)94 3143 y([4])c(to)f(conclude)h(that)f(each)h(point)e(in)h(the)h(open)f (disk)f(of)i(the)f(essential)g(spectrum)g(is)g(an)g(eigen)l(v)n(alue.) 35 b(\(This)94 3263 y(Lemma)27 b(has)g(only)f(been)i(pro)o(v)o(en)e (for)h(full)f(Mark)o(o)o(v)h(maps)f(b)n(ut)h(the)g(proof)f(easily)h(e)o (xtends)f(to)h(the)g(general)94 3382 y(Mark)o(o)o(v)22 b(case.\))36 b(Note)22 b(that)g(since)h Fv(h)28 b Fw(6)p Fu(=)g Fx(0,)23 b(multiplication)d(by)i Fv(h)h Fx(is)f(a)h(bounded)f (in)l(v)o(ertible)f(operator)i(in)f Fv(X)8 b Fx(.)94 3502 y(This)25 b(pro)o(vides)e(the)i(desired)f(lo)n(wer)h(bound)f(if)g (there)h(is)g(only)f(the)g(\002rst)h(term)g(in)f(equation)g(\(8.6\).) 316 3621 y(It)j(remains)f(to)h(sho)n(w)e(that)i(the)g(last)f(tw)o(o)g (terms)h(are)g(equal)g(to)g(zero,)h(b)n(ut)e(this)g(follo)n(ws)f(at)i (once)g(from)94 3741 y(the)e(requirement)f Fv(v)t Fu(\()p Fv(@)6 b Fw(A)p Fu(\))26 b(=)j Fx(0.)94 3916 y Ft(Remark.)d Fx(The)f(r)-5 b(.h.s.)24 b(of)h(\(8.5\))f(is)h(a)g(special)f(v)n(alue)h (of)f(the)h(function)1111 4182 y Fv(F)14 b Fu(\()p Fv(\014)5 b Fu(\))44 b(=)83 b Fx(lim)1494 4242 y FE(n)p Fq(!1)1736 4115 y Fx(1)p 1731 4159 60 4 v 1731 4250 a Fv(n)1819 4182 y Fx(log)1963 4046 y Fr(Z)2079 4182 y Fv(d\026)p Fu(\()p Fv(x)p Fu(\))17 b Fw(j)p Fu(\()p Fv(f)2470 4139 y FE(n)2522 4182 y Fu(\))2561 4139 y Fq(0)2589 4182 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Fv(f)53 b Fu(=)42 b Fv(T)1130 3001 y FE(\025)1183 2974 y Fu(\011)p Fx(.)67 b(So)35 b(we)h(conclude)f(that)f(if)h(v)n(ar)p Fu(\()p Fv(u)p Fu(\))42 b(=)g Fx(0)35 b(the)g(required)g Fv(h)h Fx(e)o(xists,)g(and)94 3093 y(furthermore,)25 b(computing)e(\(8.8\))i(in)f(the)h(coordinate)g(system)e(de\002ned)i (by)g Fu(\011)p Fx(,)g(we)g(see)g(that)g Fv(\025)3379 3120 y FF(ess)3497 3093 y Fu(=)j Fv(\025)p Fx(.)316 3213 y(If)18 b(v)n(ar)p Fu(\()p Fv(u)p Fu(\))27 b Fv(>)h Fx(0,)18 b(then,)h(since)e Fv(F)31 b Fx(of)17 b(\(8.7\))h(is)e(a)i(con)l(v)o(e)o (x)e(function)h(and)g Fv(F)2759 3170 y Fq(00)2810 3213 y Fu(\()p Fx(0)p Fu(\))27 b(=)h Fx(v)n(ar)p Fu(\()p Fx(log)15 b Fw(j)p Fv(f)3465 3170 y Fq(0)3492 3213 y Fw(j)5 b(\000)g Fx(log)17 b Fv(\025)p Fu(\))p Fx(,)94 3332 y(we)26 b(see)f(that)f Fv(\025)620 3359 y FF(ess)738 3332 y Fv(<)k(\025)p Fx(.)316 3452 y(Finally)-6 b(,)24 b(if)i Fv(\025)787 3479 y FF(ess)906 3452 y Fv(<)j(\025)d Fx(then)f(clearly)h Fv(f)36 b Fx(cannot)25 b(be)h(conjugated)e(to)h(a)h(function)e(with)h(constant)g(slope,)94 3571 y(because)h(in)e(that)g(case)i(we)f(w)o(ould)f(ha)n(v)o(e)g Fv(\025)k Fu(=)g Fv(\025)1780 3598 y FF(ess)1896 3571 y Fx(from)c(\(8.8\).)316 3691 y(This)g(completes)g(the)g(proof)h(of)g (Theorem)g(8.4)f(\(and)h(also)f(of)h(Corollary)g(8.5\).)p 3812 3637 63 2 v 3812 3695 2 59 v 3872 3695 V 3812 3697 63 2 v 94 4155 a Fy(9.)30 b(Expanding)h(Maps)f(of)f(Smooth)h(Manif)m (olds)94 4344 y Fx(The)37 b(results)f(of)h([4])g(ha)n(v)o(e)g(been)g(e) o(xtended)f(to)h(the)f(multi\255dimensional)e FG(e)n(xpanding)i Fx(case)h(in)g(the)f(w)o(ork)94 4463 y(of)g(Gundlach)f(and)g(Latushkin) f([6].)68 b(Simplifying)34 b(their)h(statement)f(for)i(our)f(purpose,)i (the)o(y)e(sho)n(w)f(the)94 4583 y(follo)n(wing)94 4775 y Ft(Theor)n(em)28 b(9.1.)39 b Fk(The)26 b(Perron\255Frobenius)g (operator)g(for)g(a)g Fw(C)2227 4732 y FF(2)2293 4775 y Fk(e)o(xpanding)f(map)g Fv(\036)i Fk(of)f(a)g(smooth)e(manifold)94 4894 y Fw(M)p Fk(,)h(when)g(acting)f(on)h(the)f(space)h(of)g Fw(C)1456 4851 y FF(1)1521 4894 y Fk(functions,)f(has)h(an)g(essential) f(spectral)g(radius)h(gi)n(v)o(en)e(by)627 5207 y Fv(\033)684 5234 y FF(ess)818 5207 y Fu(=)45 b Fx(e)o(xp)1099 5036 y Fr( )1207 5207 y Fx(sup)1178 5292 y FE(\027)5 b Fq(2)p FF(Er)o(g)1392 5066 y Fr(\022)1465 5207 y Fv(h)1522 5234 y FE(\027)1594 5207 y Fu(+)1694 5071 y Fr(Z)1749 5297 y Fq(M)1815 5207 y Fv(d\027)h Fu(\()p Fv(x)p Fu(\))33 b Fx(log)2217 5126 y Fr(\000)2263 5207 y Fw(j)17 b Fx(det)f(D)p Fv(\036)p Fu(\()p Fv(x)p Fu(\))p Fw(j)2740 5164 y Fq(\000)p FF(1)2841 5126 y Fr(\001)2909 5207 y Fw(\000)23 b Fv(\037)3071 5234 y FE(\027)3120 5066 y Fr(\023)3194 5036 y(!)3314 5207 y Fv(;)327 b Fu(\()p Fx(9)p Fv(:)p Fx(1)p Fu(\))p eop %%Page: 20 20 20 19 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(20)94 194 y Fk(where)29 b(the)f(sup)f(is)h(o)o(v)o(er)f(all)h(er)n (godic)g(measures)g(of)g(the)g(system,)g 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b(case,)g(one)g(tak)o(es)g Fv(\036)j Fu(=)g Fv(f)11 b Fx(.)35 b(F)o(or)25 b(an)g(in)l(v)n(ariant)e(er)n(godic)i(measure)g Fv(\027)32 b Fx(the)24 b(inte)o(gral)g(in)g(\(9.1\))h(equals)1502 1378 y Fv(\025)1560 1405 y FE(\027)1654 1378 y Fu(=)45 b Fw(\000)1870 1243 y Fr(Z)1986 1378 y Fv(d\027)40 b Fx(log)16 b Fw(j)p Fv(f)2359 1335 y Fq(0)2386 1378 y Fw(j)25 b Fv(:)94 1622 y Fx(The)g(unique)f(Liapuno)o(v)g(e)o(xponent)f (of)i Fv(D)s(f)36 b Fx(for)25 b(the)g(in)l(v)n(ariant)e(er)n(godic)i (measure)g Fv(\027)31 b Fx(is)1564 1868 y Fv(\037)1626 1894 y FE(\027)1720 1868 y Fu(=)1842 1732 y Fr(Z)1958 1868 y Fx(log)16 b Fw(j)p Fv(f)2190 1825 y Fq(0)2217 1868 y Fw(j)p Fv(d\027)31 b(:)94 2109 y Fx(From)25 b(\(9.1\))g(we)g (conclude)g(the)f(that)989 2390 y Fv(\033)1046 2416 y FF(ess)1136 2309 y Fr(\000)1181 2390 y Fw(C)1239 2347 y FF(1)1279 2309 y Fr(\001)1369 2390 y Fu(=)45 b Fx(e)o(xp)1650 2219 y Fr( )1759 2390 y Fx(sup)1729 2475 y FE(\027)5 b Fq(2)p FF(Er)o(g)1927 2390 y Fw(f)p Fv(h)2034 2416 y 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5159 a FD(11)316 5202 y FC(This)19 b(is)h(obtained)g(from)f (Eq.\(1.2\))f(in)h([6],)g(where)g(the)i(authors)e(allo)n(w)h(a)g(coc)o (ycle)h(deri)n(v)o(ed)e(from)g(a)h(b)n(undle)g(automorphism)f(in)94 5291 y(place)j(of)d Fm(D)r(\036)p FC(.)p eop %%Page: 21 21 21 20 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(21)94 194 y Fx(T)-8 b(o)32 b(get)g(a)h(lo)n(wer)f(bound)f(on)h (the)g(essential)f(spectral)h(radius,)i(we)e(can)h(plug)e(in)h(a)g (particular)g(measure)h(in)94 314 y(e)o(xpression)24 b(\(9.1\).)36 b(Using)24 b(the)g(SRB)j(measure)d Fv(\026)p Fx(,)h(we)g(get)1696 528 y Fv(\033)1753 555 y FF(ess)1887 528 y Fw(\025)45 b Fv(e)2055 486 y Fq(\000)p FE(\037)2167 503 y Fn(\026)2245 528 y Fv(:)94 743 y Fx(Thus,)25 b(we)g(get)f(in)h (this)f(case)h(the)g(follo)n(wing)d(corollary)j(from)g(Theorem)f(9.1:) 94 935 y Ft(Cor)n(ollary)17 b(9.2.)33 b Fk(The)17 b(essential)g 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2251 5568 l 2274 5586 l 2298 5604 l 2323 5623 l 2348 5644 l 2375 5666 l 2403 5689 l 2432 5713 l 2462 5739 l 2492 5764 l 2523 5791 l 2553 5817 l 2583 5843 l 2611 5867 l 2638 5891 l 2662 5912 l 2683 5930 l 2701 5946 l 2716 5959 l 2727 5969 l 2736 5977 l 2741 5981 l 2744 5984 l 2745 5985 l gs col0 s gr % Polyline n 6435 3195 m 6525 3195 l 6930 3420 l 7020 3420 l 6975 3465 l 7200 3645 l 7155 3690 l gs col0 s gr % Polyline n 2700 6120 m 2790 6125 l %2880 6120 l %2880 6120 l %2880 6030 l 2900 6055 l 2950 6055 l 2910 6055 l 3015 5985 l 3015 5895 l gs col0 s gr % Polyline gs 410 300 translate n 6975 1035 m 7290 1215 l 7290 2025 l 7425 2150 l 7290 2250 l 7290 3195 l 7155 3285 l 6975 3380 l gs col0 s gr gr $F2psEnd rs %%EndDocument @endspecial 2640 2687 a Fv(W)2748 2644 y FE(u)1469 4577 y Fw(W)1575 4534 y FE(s)1091 4463 y Fv(W)1199 4420 y FE(s)3510 3008 y Fw(W)3616 2965 y FE(u)3132 3367 y Fw(W)3238 3324 y FE(u)3285 3342 y Fo(min)2716 3632 y Fv(w)i Fu(=)d Fv(W)3031 3589 y FE(u)3078 3607 y 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Fk(for)g(which)941 3113 y FE(k)r Fq(\000)p FF(1)940 3143 y Fr(X)945 3356 y FE(j)t Fp(=)p FF(0)1100 3237 y Fv(\021)1153 3157 y Fr(\000)1199 3237 y Fv(f)1259 3195 y Fq(\000)p FE(j)1363 3237 y Fu(\()p Fv(x)p Fu(\))1498 3157 y Fr(\001)1587 3237 y Fw(\025)1710 3113 y FE(k)r Fq(\000)p FF(1)1709 3143 y Fr(X)1714 3356 y FE(j)t Fp(=)p FF(0)1870 3237 y Fv(')1935 3157 y Fr(\000)1980 3237 y Fv(f)2040 3195 y Fq(\000)p FE(j)2144 3237 y Fu(\()p Fv(x)p Fu(\))2279 3157 y Fr(\001)2346 3237 y Fu(+)e Fv(k)s(")2547 3264 y Fq(\003)2615 3237 y Fu(+)g Fx(log)16 b Fv(C)2930 3264 y Fq(\003)3001 3237 y Fv(;)590 b Fu(\()p Fx(10)p Fv(:)p Fx(7)p Fu(\))94 3559 y Fk(uniformly)24 b(for)h(suf)n(\002ciently)f(lar) n(ge)h Fv(k)j Fk(and)d(for)g Ft(all)f Fv(x)h Fk(in)f Fw(M)p Fk(.)316 3734 y Fx(This)42 b(assumption)f(implies)h Fv(\013)p Fu(\()l Fv(~)-46 b(s)53 b Fu(\))g(=)g Fv(\014)5 b Fu(\()1889 3716 y Fv(~)1895 3734 y(t)54 b Fu(\))f(=)h Fx(0,)47 b(because)d(the)f(Liapuno)o(v)e(multipliers)g(are)94 3854 y(dif)n(ferent.)48 b(In)29 b(other)g(w)o(ords,)h(the)e(e)o (xpansion)g(rates)h Fv(\021)k Fx(and)c Fv(')g Fx(are)h(allo)n(wed)e(to) h(\003uctuate,)h(b)n(ut)e(there)h(must)94 3973 y(remain)35 b(a)h(\223gap\224)f Fv(")799 4000 y Fq(\003)880 3973 y Fx(between)h(them)e(e)n(v)o(erywhere,)j(and)f(at)f(lar)n(ge)g(times.) 66 b(It)35 b(w)o(ould)f(be)i(interesting)e(to)94 4093 y(understand)25 b(to)h(which)f(e)o(xtent)g(\(10.7\))g(could)h(be)f (replaced)i(by)e(a)h(condition)e(on)i(the)f(Liapuno)o(v)g(e)o(xponents) 94 4212 y(alone.)43 b(A)27 b(stronger)g(statement)f(than)h(\(10.7\))g (is)f(to)h(assume)g Fv(\021)t Fu(\()p Fv(x)p Fu(\))j Fv(>)h(')p Fu(\()p Fv(x)p Fu(\))c Fx(for)g(all)g Fv(x)p Fx(.)43 b(This)26 b(is)h(in)g(f)o(act)g(the)94 4332 y(\223b)n(unching) 21 b(condition,)-7 b(\224)20 b(since)h(from)g(the)g(continuity)f(of)h Fv(')g Fx(and)g Fv(\021)26 b Fx(and)21 b(the)g(compactness)f(of)h(the)g (manifold)94 4451 y(it)k(follo)n(ws)e(that)h Fv(\021)t(=')k(>)g Fx(1)22 b Fu(+)h Fv(")28 b(>)g Fx(1,)d(uniformly)e(in)i 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FE(n)1977 5199 y Fp(+)2131 5172 y Fx(for)37 b(the)g(unstable)f(directions,)j(and)e(then)g Fv(")3706 5199 y Fq(\003)3797 5172 y Fu(=)94 5291 y Fx(log)16 b Fv(\025)296 5318 y Fp(+)385 5291 y Fw(\000)22 b Fx(log)16 b Fv(\025)686 5318 y Fq(\000)754 5291 y Fx(.)36 b(So)25 b(for)g(hyperbolicity)-6 b(,)22 b(the)j(strong)f(form)g(of)h(\(10.7\))g (is)f(being)g(required.)p eop %%Page: 24 24 24 23 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(24)316 194 y Fx(The)31 b(follo)n(wing)e(result)h(is)g(formulated)g (as)h(a)g(conjecture,)i(since)d(the)h(ar)n(guments)f(to)n(w)o(ard)g (its)g(proof)94 314 y(are)c(only)e(sk)o(etched.)94 505 y Ft(Conjectur)n(e)41 b(10.3.)78 b Fk(Consider)38 b(a)h(dynamical)f (system)g(which)g(is)h(uniformly)e(hyperbolic)h(and)g(has)h(an)94 625 y(SRB)45 b(measure)e Fv(\026)h Fk(whose)e(Liapuno)o(v)g (multipliers)f(are)j(all)f(dif)n(ferent)g(from)g(1.)91 b(Assume)42 b(it)g(satis\002es)94 744 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FE(u)2074 2324 y FF(loc)2164 2297 y Fu(\()p Fv(x)p Fu(\))p Fx(,)g(and)g(which)g(does)g(not)g(v)n (anish)f(on)h(the)g(hull)94 2417 y(of)f Fv(A)278 2444 y FF(0)318 2417 y Fx(.)38 b(This)25 b(is)h(possible)e(because)i(these)g (manifolds)e(are)i Fw(C)2230 2374 y FF(1)p Fp(+)p FE(\016)2396 2417 y Fx(in)g Fv(x)p Fx(.)38 b(W)-8 b(e)26 b(also)f(assume)h(that)f (this)g(v)o(ector)94 2536 y(\002eld)h(has)e(zero)i(di)n(v)o(er)n(gence) e(in)g(the)h(unstable)f(directions.)316 2656 y(W)-8 b(e)33 b(further)g(choose)g(a)g(function)g Fv(v)k Fx(which)32 b(is)h Fw(C)2055 2613 y FF(2)2128 2656 y Fx(in)f(the)h(hull)f(\(in)h Fw(M)p Fx(\))g(of)g Fv(A)3102 2683 y FF(0)3142 2656 y Fx(,)i(v)n(anishing)c(on)h(the)94 2775 y(stable)25 b(boundaries)f(of)h Fv(A)997 2802 y FF(0)1036 2775 y Fx(.)316 2895 y(The)g(observ)n(ables)e Fv(g)1036 2922 y FF(2)1101 2895 y Fx(in)h Fv(X)32 b Fx(are)26 b(de\002ned)f(by)g(the)f(equation)1624 3149 y Fv(g)1672 3176 y FF(2)1755 3149 y Fu(=)1893 3082 y Fx(1)p 1889 3126 58 4 v 1889 3218 a Fv(h)1958 3149 y(dv)t Fu(\()p Fv(V)2159 3176 y FE(A)2219 3194 y Fo(0)2254 3149 y Fu(\))g Fv(;)94 3380 y Fx(where)31 b Fv(h)f Fx(is)g(de\002ned)g(in)g(\(10.3\),) h(and)f(is)f(e)o(xtended)h(to)f(a)h(positi)n(v)o(e)e(function)h(on)h (the)g(hull)f(of)h Fv(A)3479 3407 y FF(0)3518 3380 y Fx(.)52 b(By)30 b(our)94 3500 y(abo)o(v)o(e)k(assumptions,)g Fv(g)966 3526 y FF(2)1039 3500 y Fx(is)g Fw(C)1198 3457 y FF(1)1272 3500 y Fx(in)g(the)f(unstable)h(directions)f(and)h Fv(\016)t Fx(\255H)2672 3499 y(\250)2664 3500 y(older)g(continuous)e (in)i(the)g(stable)94 3619 y(directions.)62 b(As)33 b(we)h(v)n(ary)f Fv(A)1152 3646 y FF(0)1226 3619 y Fx(and)g Fv(v)38 b Fx(o)o(v)o(er)33 b(all)g(possible)g(choices,)i(we)f(obtain)f(a)h(set)f Fv(X)3288 3646 y FF(0)3362 3619 y Fx(of)g(functions.)94 3739 y(Since)j Fv(v)46 b Fw(2)c(C)611 3696 y FF(2)651 3739 y Fx(,)c(the)d(map)g Fv(v)46 b Fw(7!)41 b Fv(dv)t Fu(\()p Fv(V)1514 3766 y FE(A)1574 3783 y Fo(0)1608 3739 y Fu(\))35 b 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b(belo)n(w)e(for)g(a)i(precise)f(description.)211 5248 y FD(15)316 5291 y FC(F)o(or)f(more)h(subtle)g(aspects)h(of)f(the) g(conjugation,)g(see)h(Theorem)e(8.4.)p eop %%Page: 25 25 25 24 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(25)316 194 y Fx(Finally)-6 b(,)24 b(the)g(operator)h Fv(U)36 b Fx(is)24 b(the)h(K)m(oopman)e(operator)i(of)g(the)g(map)f Fv(f)11 b Fx(,)24 b(that)h(is)1529 327 y Fr(\000)1575 407 y Fv(U)11 b(g)1706 327 y Fr(\001)1750 407 y Fu(\()p Fv(x)p Fu(\))44 b(=)h Fv(g)2103 327 y Fr(\000)2147 407 y Fv(f)11 b Fu(\()p Fv(x)p Fu(\))2342 327 y Fr(\001)2412 407 y Fv(:)1179 b Fu(\()p Fx(10)p Fv(:)p Fx(9)p Fu(\))316 621 y Fx(No)n(w)25 b(that)h(the)g(spaces)g(are)h(in)f(place,)h(we)f (can)h(w)o(ork)f(on)g(\(10.2\).)39 b(T)-8 b(ak)o(e)27 b Fv(g)2890 648 y FF(1)2956 621 y Fx(and)f Fv(g)3174 648 y FF(2)3239 621 y Fx(in)g(a)h(piece)f Fv(A)3725 648 y FF(0)3791 621 y Fx(of)94 740 y(the)h(partition.)40 b(W)-8 b(e)27 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b(Assumption)e(10.2)h(implies)f(a)i(uniform)f (bound)g(for)h(\(10.11\):)420 2903 y Fv(dg)520 2930 y FF(1)559 2935 y FE(f)606 2915 y Fj(\000)p Fn(k)703 2935 y Fp(\()p FE(x)p Fp(\))816 2823 y Fr(\000)861 2903 y Fv(D)943 2930 y FE(x)994 2903 y Fv(f)1054 2861 y Fq(\000)p FE(k)1165 2903 y Fu(\()p Fv(V)d Fu(\))1323 2823 y Fr(\001)1412 2903 y Fu(=)45 b Fv(dg)1634 2930 y FF(1)1673 2903 y Fu(\()1706 2885 y Fv(~)1712 2903 y(t)28 b Fu(\))1815 2930 y FE(f)1862 2910 y Fj(\000)p Fn(k)1959 2930 y Fp(\()p FE(x)p Fp(\))2093 2903 y Fw(\001)22 b Fv(\013)p Fu(\()p Fv(V)g Fu(\))2365 2930 y FE(x)2437 2903 y Fw(\001)g Fv(e)2533 2846 y Fq(\000)2609 2778 y Fr(P)2714 2803 y Fn(k)q Fj(\000)p Fo(1)2714 2883 y Fn(j)s Fb(=)p Fo(0)2850 2846 y FE(')2902 2765 y Fr(\000)2948 2846 y FE(f)2995 2803 y Fj(\000)p Fn(j)3086 2846 y Fp(\()p FE(x)p Fp(\))3193 2765 y Fr(\001)1481 3135 y Fu(+)h Fc(O)1651 3054 y Fr(\000)1697 3135 y Fv(e)1743 3077 y Fq(\000)p FE(")1843 3095 y Fj(\003)1885 3077 y FE(k)r Fq(\000)1991 3010 y Fr(P)2097 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y FF(1)1776 4739 y Fv(e)1822 4696 y Fq(\000)p Fp(\006)1941 4714 y Fn(k)2026 4739 y Fu(~)g Fv(g)2071 4766 y FF(2)2132 4739 y Fw(\016)22 b Fv(f)2264 4696 y FE(k)2335 4739 y Fu(+)h Fc(O)p Fu(\()p Fv(e)2590 4696 y Fq(\000)p FE(")2690 4714 y Fj(\003)2737 4739 y Fv(Z)2805 4766 y FE(k)2854 4739 y Fu(\))h Fv(;)625 b Fu(\()p Fx(10)o Fv(:)p Fx(14)p Fu(\))94 4995 y Fx(where)957 5189 y Fu(\006)1029 5216 y FE(k)1078 5189 y Fu(\()p Fv(x)p Fu(\))44 b(=)1380 5065 y FE(k)r Fq(\000)p FF(1)1378 5095 y Fr(X)1384 5308 y FE(j)t Fp(=)p FF(0)1539 5189 y Fv(')1604 5146 y FE(u)1656 5109 y Fr(\000)1702 5189 y Fv(f)1762 5146 y FE(j)1803 5189 y Fu(\()p Fv(x)p Fu(\))1938 5109 y Fr(\001)2008 5189 y Fv(;)219 b Fu(~)-53 b Fv(g)2300 5216 y FF(1)2339 5189 y Fu(\()p Fv(x)p Fu(\))44 b(=)h Fv(dg)2740 5216 y FF(1)2779 5189 y Fu(\()2812 5171 y Fv(~)2818 5189 y(t)16 b Fu(\))2909 5216 y FE(x)2984 5189 y Fv(:)p eop %%Page: 26 26 26 25 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(26)316 194 y Fx(W)-8 b(e)34 b(no)n(w)e(use)h(the)g(thermodynamic)f (formalism)g([3].)61 b(De\002ne)34 b(the)f(H)2841 193 y(\250)2833 194 y(older)g(continuous)f(function)94 314 y Fv(')159 271 y FE(u)237 314 y Fx(by)24 b Fv(')426 271 y FE(u)506 314 y Fu(=)k Fw(\000)17 b Fx(log)f(det)g Fv(J)1052 271 y FE(u)1129 314 y Fx(where)25 b Fv(J)1462 271 y FE(u)1539 314 y Fx(is)f(the)h(Jacobian)f(matrix)g(in)g(the)h(unstable)f(b)n (undle.)3336 271 y FF(16)3445 314 y Fx(Recall)i(that)94 433 y(under)d(our)f(assumptions)f(there)i(is)f(a)h(homeomorphism)d Fv(\031)26 b Fx(conjugating)20 b(the)j(dynamical)e(system)h Fv(f)33 b Fx(on)22 b(the)94 553 y(attractor)j(to)g(a)g(subshift)f Fw(S)33 b Fx(of)24 b(\002nite)h(type.)36 b(The)25 b(SRB)h(measure)f Fv(\026)g Fx(is)g(then)f(transformed)h(to)f(a)h(Gibbs)f(state)94 672 y Fv(\015)32 b Fx(with)26 b(the)h(H)611 671 y(\250)603 672 y(older)f(continuous)g(potential)f Fv(')1739 629 y FE(u)1814 672 y Fw(\016)e Fv(\031)t Fx(.)41 b(W)-8 b(e)28 b(get)e(for)h(the)g(\002rst)g(term)f(on)g(the)h(right)f(hand)h (side)94 792 y(of)e(\(10.14\):)1069 955 y Fv(D)1151 982 y FE(k)1244 955 y Fu(=)45 b Fw(\000)1460 820 y Fr(Z)1577 955 y Fv(d\015)36 b Fu(~)-54 b Fv(g)1761 982 y FF(1)1823 955 y Fw(\016)22 b Fv(\031)31 b(e)2029 912 y Fq(\000)p Fp(\006)2148 930 y Fn(k)2192 912 y Fq(\016)p FE(\031)2325 955 y Fw(\001)42 b Fu(~)-53 b Fv(g)2440 982 y FF(2)2501 955 y Fw(\016)22 b Fv(\031)k Fw(\016)c(S)2795 912 y FE(k)2872 955 y Fv(:)670 b Fu(\()p Fx(10)o Fv(:)p Fx(15)p Fu(\))94 1177 y Fx(De\002ne)26 b(an)f(operator)g Fw(T)50 b Fx(by)1534 1316 y Fw(T)26 b Fv( )48 b Fu(=)c Fv(e)1894 1273 y Fq(\000)p FE(')2008 1230 y Fn(u)2056 1273 y Fq(\016)p FE(\031)2172 1289 y Fu(~)2151 1316 y Fv( )26 b Fw(\016)c(S)33 b Fv(;)94 1493 y Fx(where)26 b Fv( )i Fx(is)d(a)g(function)f(on)g(the)h(shift)f (space.)36 b(Then)25 b(\(10.15\))f(becomes)1163 1753 y Fv(D)1245 1780 y FE(k)1339 1753 y Fu(=)44 b Fw(\000)1554 1617 y Fr(Z)1671 1753 y Fv(d\015)1807 1672 y Fr(\000)1853 1753 y Fw(T)1933 1710 y Fq(\003)1978 1672 y Fr(\001)2024 1710 y FE(k)2073 1753 y Fu(\()s(~)-53 b Fv(g)2160 1780 y FF(1)2221 1753 y Fw(\016)22 b Fv(\031)t Fu(\))49 b Fw(\001)42 b Fu(~)-53 b Fv(g)2557 1780 y FF(2)2618 1753 y Fw(\016)22 b Fv(\031)31 b(:)764 b Fu(\()p Fx(10)o Fv(:)p Fx(16)p Fu(\))94 2031 y Fx(W)-8 b(e)32 b(no)n(w)f(apply)g(Theorem)g (3.3)g(with)g Fv(X)1557 1988 y Fq(0)1621 2031 y Fu(=)38 b Fv(Y)1816 1988 y Fq(0)1880 2031 y Fu(=)g Fw(C)2053 1988 y FF(0)2125 2031 y Fx(and)31 b Fv(U)2379 1988 y Fq(0)2444 2031 y Fu(=)37 b Fw(T)2638 1988 y Fq(\003)2715 2031 y Fx(and)31 b(we)h(\002nd)g Fv(\045)3277 2058 y FF(ess)3366 2031 y Fu(\()p Fv(X)r(;)17 b(Y)6 b(;)17 b(U)11 b Fu(\))35 b(=)94 2151 y Fv(\045)146 2178 y FF(ess)236 2151 y Fu(\()p Fv(X)366 2108 y Fq(0)393 2151 y Fv(;)17 b(Y)518 2108 y Fq(0)545 2151 y Fv(;)g(U)669 2108 y Fq(0)697 2151 y Fu(\))27 b Fw(\025)h Fv(\033)925 2178 y FF(p)p Fq(\000)p FF(ess)1112 2151 y Fu(\()p Fv(X)1242 2108 y Fq(0)1269 2151 y Fv(;)17 b(U)1393 2108 y Fq(0)1421 2151 y Fu(\))p Fx(.)34 b(It)23 b(remains)f(to)g(gi)n(v)o(e)f(a)i(lo)n(wer)f (bound)g(on)g Fv(\033)2996 2178 y FF(p)p Fq(\000)p FF(ess)3183 2151 y Fu(\()p Fv(X)3313 2108 y Fq(0)3340 2151 y Fv(;)17 b(U)3464 2108 y Fq(0)3491 2151 y Fu(\))23 b Fx(in)f(terms)94 2270 y(of)i(the)f(pressure.)35 b(Using)22 b(a)i(well\255kno)n(wn)e(de)n (vice)g([2,)i(Lemma)f(1.3],)g(we)g(can)h(conjugate)e Fw(T)49 b Fx(to)23 b(an)g(operator)94 2390 y Fw(T)148 2417 y Fp(+)240 2390 y Fx(de\002ned)i(by)1509 2545 y Fw(T)1563 2572 y Fp(+)1630 2545 y Fv( )47 b Fu(=)e Fv(e)1910 2499 y Fq(\000)p FE(')2024 2456 y Fn(u)2024 2526 y Fb(+)2081 2499 y Fq(\016)p FE(\031)2197 2519 y Fu(~)2176 2545 y Fv( )26 b Fw(\016)c(S)33 b Fv(;)94 2723 y Fx(where)26 b Fv(')428 2680 y FE(u)428 2749 y Fp(+)516 2723 y Fw(\016)c Fv(\031)28 b Fx(depends)d(only)f(on)g(the)h(future,)g FG(i.e)o(.)p Fx(,)f Fv(')2021 2680 y FE(u)2021 2749 y Fp(+)2112 2723 y Fx(is)h(constant)e(on)i(the)g(stable)f(\(local\))h (lea)n(v)o(es.)35 b(Note)94 2842 y(no)n(w)24 b(that)h(when)g Fw(T)783 2799 y Fq(\003)758 2869 y Fp(+)854 2842 y Fx(acts)g(on)f(a)h (function)f Fv( )1649 2869 y Fp(+)1740 2842 y Fx(which)h(depends)f (only)g(on)h(the)f FG(futur)l(e)p Fx(,)g(it)h(is)f(gi)n(v)o(en)f(by) 1306 3114 y Fw(T)1386 3072 y Fq(\003)1360 3141 y Fp(+)1432 3114 y Fv( )1497 3141 y Fp(+)1607 3114 y Fu(=)1779 3047 y Fx(1)p 1741 3091 126 4 v 1741 3183 a Fv(\036)1800 3210 y Fp(+)1878 3114 y Fc(L)1949 3034 y Fr(\000)1995 3114 y Fv(\036)2054 3141 y Fp(+)2121 3114 y Fv(e)2167 3068 y Fq(\000)p FE(')2281 3026 y Fn(u)2281 3095 y Fb(+)2338 3068 y Fq(\016)p FE(\031)2433 3114 y Fv( )2498 3141 y Fp(+)2564 3034 y Fr(\001)2634 3114 y Fv(;)908 b Fu(\()p Fx(10)o Fv(:)p Fx(17)p Fu(\))94 3391 y Fx(where)26 b Fc(L)f Fx(is)f(the)h(Perron\255Frobenius)g(operator)g(and)g Fv(\036)2004 3417 y Fp(+)2095 3391 y Fx(satis\002es)1714 3603 y Fc(L)p Fv(\036)1844 3630 y Fp(+)1955 3603 y Fu(=)45 b Fv(\036)2136 3630 y Fp(+)2227 3603 y Fv(:)94 3815 y Fx(The)25 b(eigen)l(v)n(alue)f(abo)o(v)o(e)f(is)h(1)h(because)g(we)g (are)g(dealing)f(with)g(an)g(SRB)i(measure,)f(and)f(the)h(eigen)l(v)o (ector)f(is)94 3935 y(unique.)35 b(W)-8 b(e)24 b(no)n(w)f(see)h(that)f Fv(\033)1154 3962 y FF(p)p Fq(\000)p FF(ess)1341 3935 y Fu(\()p Fv(X)1471 3892 y Fq(0)1498 3935 y Fv(;)17 b(U)1622 3892 y Fq(0)1649 3935 y Fu(\))24 b Fx(is)f(bounded)f(belo)n(w)h(by)g (the)h(essential)f(point)f(spectral)i(radius)94 4054 y(of)h Fc(L)17 b Fx(e)o(xp)p Fu(\()p Fw(\000)p Fv(')614 4011 y FE(u)614 4081 y Fp(+)702 4054 y Fw(\016)22 b Fv(\031)t Fu(\))p Fx(.)35 b(One)25 b(no)n(w)f(introduces)g(the)h(pressure)827 4335 y Fv(P)14 b Fu(\()p Fv(')1009 4292 y FE(u)1083 4335 y Fw(\000)23 b Fv(h)p Fu(\))44 b Fw(\021)80 b Fx(lim)1445 4400 y FE(k)r Fq(!1)1679 4267 y Fx(1)p 1676 4312 56 4 v 1676 4403 a Fv(k)1760 4335 y Fx(log)1904 4199 y Fr(Z)1959 4425 y Fq(M)2074 4335 y Fv(d\026)p Fu(\()p Fv(x)p Fu(\))17 b Fv(e)2384 4277 y Fq(\000)2459 4209 y Fr(P)2565 4234 y Fn(k)q Fj(\000)p Fo(1)2565 4314 y Fn(j)s Fb(=)p Fo(0)2701 4277 y FE(h)2748 4196 y Fr(\000)2793 4277 y FE(f)2840 4234 y Fj(\000)p Fn(j)2931 4277 y Fp(\()p FE(x)p Fp(\))3038 4196 y Fr(\001)3114 4335 y Fv(;)94 4603 y Fx(where)33 b Fv(P)14 b Fu(\()p Fv(')552 4560 y FE(u)603 4603 y Fu(\))37 b(=)h Fx(0,)33 b(because)f(we)g(are)g(dealing)f(with)g(an)h(SRB)h (measure.)56 b(Then,)33 b(it)e(is)g(kno)n(wn)g([4],)i([2,)94 4722 y(Theorem)h(1.5.7])f(that)g(e)n(v)o(ery)g(point)g(in)g(the)g(open) h(disk)f(of)g(radius)h(e)o(xp)2657 4642 y Fr(\000)2703 4722 y Fv(P)14 b Fu(\()p Fx(2)p Fv(')2935 4679 y FE(u)2986 4722 y Fu(\))3025 4642 y Fr(\001)3104 4722 y Fx(is)33 b(an)h(eigen)l(v)n(alue)f(of)94 4852 y Fc(L)17 b Fx(e)o(xp)p Fu(\()p Fw(\000)p Fv(')506 4809 y FE(u)506 4879 y Fp(+)594 4852 y Fw(\016)22 b Fv(\031)t Fu(\))p Fx(.)33 b(Since)16 b(e)o(xp)1204 4771 y Fr(\000)1249 4852 y Fv(P)e Fu(\()p Fx(2)p Fv(')1481 4809 y FE(u)1533 4852 y Fu(\))1572 4771 y Fr(\001)1645 4852 y Fv(>)28 b Fx(1)p Fv(=\025)1908 4879 y FF(min)2022 4852 y Fx(,)18 b(the)e(desired)g(inequality)g(follo) n(ws.)30 b(This)16 b(completes)94 4971 y(the)25 b(sk)o(etch)g(of)f(the) h(proof)g(of)g(Conjecture)g(10.3.)p 3812 4917 63 2 v 3812 4976 2 59 v 3872 4976 V 3812 4978 63 2 v 94 5182 1200 4 v 211 5248 a FD(16)316 5291 y FC(In)19 b(principle,)h(for)f(the) i(case)g(of)e(2)h(positi)n(v)o(e)g(Liapuno)o(v)f(e)o(xponents,)h Fm(')2269 5248 y Fh(u)2338 5291 y FC(can)h(be)f(computed)g(from)f Fm(')p FC(,)h Fm(\021)s FC(,)3241 5276 y Fm(~)3253 5291 y(t)p FC(,)12 b Fm(~)-34 b(s)p FC(,)20 b Fm(\013)p FC(,)g(and)g Fm(\014)t FC(.)p eop %%Page: 27 27 27 26 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(27)94 194 y(10.1.)25 b(Suf\002cient)i(Conditions)94 383 y Fx(W)-8 b(e)23 b(ne)o(xt)d(address)i(the)f(question)g(of)g(suf)n (\002cient)h(conditions)d(for)j(Assumption)e(10.1)h(and)g(Assumption)f (10.2)94 503 y(to)26 b(hold.)37 b(A)26 b(typical)e(such)i(condition)e (is)h(the)g(b)n(unching)g(condition)f(from)h([9,)h(p.)38 b(602],)25 b(or)l(,)h(the)g(concept)f(of)94 623 y(domination)e(de)n(v)o (eloped)h(in)g([7].)36 b(Consider)25 b(a)g(point)f Fv(x)h Fx(and)f(write)h Fv(D)s(f)2581 649 y FE(x)2656 623 y Fx(in)f(matrix)g(form)1506 907 y Fv(D)s(f)1640 934 y FE(x)1734 907 y Fu(=)1856 766 y Fr(\022)1946 847 y Fv(A)2021 874 y FE(x)2212 847 y Fx(0)1984 966 y(0)137 b Fv(D)2253 993 y FE(x)2320 766 y Fr(\023)2435 907 y Fv(;)94 1181 y Fx(with)25 b(the)f(blocks)g(corresponding)g(to)g(unstable)g(and)h (stable)g(subspaces,)f(respecti)n(v)o(ely)-6 b(.)34 b(W)-8 b(e)25 b(de\002ne)1247 1404 y Fv(\025)1305 1430 y FE(x)1400 1404 y Fu(=)45 b Fw(k)p Fv(D)1654 1430 y FE(x)1704 1404 y Fw(k)25 b Fv(;)116 b(\026)1983 1430 y FE(x)2077 1404 y Fu(=)2199 1323 y Fr(\000)2244 1404 y Fw(k)p Fv(A)2369 1361 y Fq(\000)p FF(1)2369 1430 y FE(x)2471 1404 y Fw(k)2521 1323 y Fr(\001)2567 1361 y Fq(\000)p FF(1)2694 1404 y Fv(:)94 1641 y Fx(Let)25 b Fv(\027)301 1668 y FE(x)376 1641 y Fx(be)g(the)g(in)l(v)o(erse)f(of)h(the)g(Lipschitz)e(constant)h (for)i Fv(f)2158 1598 y Fq(\000)p FF(1)2259 1641 y Fx(:)903 1923 y Fv(\027)952 1950 y FE(x)1047 1923 y Fu(=)1334 1855 y Fx(1)p 1181 1900 358 4 v 1181 1991 a Fv(L)p Fu(\()p Fv(f)1348 1962 y Fq(\000)p FF(1)1449 1991 y Fu(\))1488 2018 y FE(x)1575 1923 y Fv(;)115 b(L)p Fu(\()p Fv(g)t Fu(\))1916 1950 y FE(x)2009 1923 y Fw(\021)125 b Fx(sup)2131 2013 y Fq(j)p FE(x)p Fq(\000)p FE(y)s Fq(j)p FE(<")2457 1855 y Fw(j)p Fv(g)t Fu(\()p Fv(x)p Fu(\))20 b Fw(\000)j Fv(g)t Fu(\()p Fv(y)t Fu(\))p Fw(j)p 2457 1900 544 4 v 2586 1991 a(j)p Fv(x)e Fw(\000)i Fv(y)t Fw(j)3038 1923 y Fv(;)94 2220 y Fx(with)i Fv(")j(>)g Fx(0)d(some)f(small)g(constant.) 316 2340 y(One)h(de\002nes)g(the)g(b)n(unching)e(constant)h(by)1348 2611 y Fv(B)1429 2568 y FE(u)1481 2611 y Fu(\()p Fv(f)11 b Fu(\))43 b(=)i Fx(inf)1819 2671 y FE(x)1928 2544 y Fx(log)16 b Fv(\026)2132 2571 y FE(x)2204 2544 y Fw(\000)23 b Fx(log)16 b Fv(\025)2506 2571 y FE(x)p 1928 2588 629 4 v 2120 2679 a Fx(log)g Fv(\027)2313 2706 y FE(x)2593 2611 y Fv(:)94 2981 y Ft(Theor)n(em)29 b(10.5.)40 b Fk(Let)26 b Fv(f)37 b Fk(be)27 b(a)g Fw(C)1242 2938 y FF(3)1309 2981 y Fk(map)f(of)g(the)h(manifold)e Fw(M)h Fk(which)g(gi)n(v)o(es)f (rise)i(to)f(an)g(Axiom)g(A)g(system)94 3100 y(whose)19 b(unstable)e(manifolds)h Fv(W)1244 3058 y FE(u)1314 3100 y Fk(are)i Fw(C)1513 3058 y FF(2)1571 3100 y Fk(and)f(the)f(stable)g (ones,)i Fv(W)2461 3058 y FE(s)2503 3100 y Fk(,)g(are)g Fw(C)2747 3058 y FF(1)2787 3100 y Fk(.)33 b(Assume)18 b(that)g(the)h(Liapuno)o(v)94 3220 y(multipliers)k(of)i Fv(f)35 b Fk(satisfy)25 b(the)f(follo)n(wing)f(conditions:)111 3340 y(1\))50 b(The)25 b(smallest)e(Liapuno)o(v)h(multiplier)e(abo)o(v) o(e)i(1)h(is)f Fv(\025)2082 3366 y FF(min)2221 3340 y Fk(and)h(the)f(corresponding)g(dimension)f(is)i(1.)111 3459 y(2\))50 b(There)25 b(are)h(no)e(Liapuno)o(v)g(multipliers)e (equal)j(to)f(1.)111 3579 y(3\))50 b(The)25 b(\(multidimensional)c(v)o (ersion)j(of\))h(inequality)f(\(10.7\))g(holds.)111 3698 y(4\))50 b(The)25 b(b)n(unching)e(constant)h(satis\002es)1790 3818 y Fv(B)1871 3775 y FE(u)1923 3818 y Fu(\()p Fv(f)11 b Fu(\))43 b Fv(>)i Fx(1)24 b Fv(;)1242 b Fu(\()p Fx(10)o Fv(:)p Fx(18)p Fu(\))244 4000 y Fk(and)25 b(for)g(the)f(in)l(v)o(erse)h (map)1739 4119 y Fv(B)1820 4076 y FE(u)1872 4119 y Fu(\()p Fv(f)1971 4076 y Fq(\000)p FF(1)2072 4119 y Fu(\))44 b Fv(>)h Fx(1)24 b Fv(:)1191 b Fu(\()p Fx(10)o Fv(:)p Fx(19)p Fu(\))94 4301 y Fk(Then)25 b(Assumption)e(10.1)h(and)h (Assumption)d(10.2)i(hold.)94 4533 y Ft(Pr)n(oof)o(.)j Fx(The)h(proofs)f(of)g(all)g(assertions)g(e)o(xcept)g(for)g(the)h (smoothness)d(of)j Fv(h)f Fx(can)h(be)g(found)f(in)g([9,)h(Chapter)94 4652 y(19,)d(p607].)316 4772 y(So)i(it)g(remains)g(to)g(pro)o(v)o(e)f (the)h(dif)n(ferentiability)e(of)j Fv(h)p Fx(.)43 b(W)-8 b(e)28 b(recall)f(that)g(using)f(a)i(base)f(point)f Fv(x)3656 4799 y FE(W)3774 4772 y Fx(on)94 4891 y(the)c(leaf)g Fv(W)14 b Fx(,)22 b(we)g(ha)n(v)o(e)f(for)h(the)g(density)e(of)i(the)f (SRB)i(measure)f(on)f(the)h(unstable)e(manifold)h Fv(W)3419 4849 y FE(u)3471 4891 y Fu(\()p Fv(x)p Fu(\))g Fx(of)h(an)o(y)94 5011 y Fv(x)28 b Fw(2)g Fv(W)14 b Fx(:)1215 5189 y Fv(h)p Fu(\()p Fv(x)p Fu(\))28 b(=)1567 5065 y Fq(1)1543 5095 y Fr(Y)1540 5308 y FE(j)t Fp(=)p FF(0)1690 5189 y Fv(e)1736 5144 y FE(')1788 5101 y Fn(u)1836 5063 y Fr(\000)1881 5144 y FE(f)1928 5101 y Fj(\000)p Fn(j)2019 5144 y Fp(\()p FE(x)p Fp(\))2126 5063 y Fr(\001)2172 5144 y Fq(\000)p FE(')2286 5101 y Fn(u)2334 5063 y Fr(\000)2380 5144 y FE(f)2427 5101 y Fj(\000)p Fn(j)2518 5144 y Fp(\()p FE(x)2594 5162 y Fn(W)2671 5144 y Fp(\))2702 5063 y Fr(\001)p eop %%Page: 28 28 28 27 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(28)94 194 y Fx(where)32 b Fv(x)426 221 y FE(W)547 194 y Fx(is)e(a)h(reference)i(point)c(chosen)i(once)g(and)g(for)g(all)f (on)g Fv(W)2573 151 y FE(u)2625 194 y Fu(\()p Fv(x)p Fu(\))p Fx(.)54 b(When)30 b(v)n(arying)g Fv(x)h Fx(along)f(an)94 314 y(unstable)d(leaf,)h(the)f(reference)i(point)d(does)h(not)g (change.)43 b(Each)27 b(term)g(in)g(the)g(abo)o(v)o(e)g(product)f(is)h (dif)n(feren\255)94 433 y(tiable)f(in)f(the)g(unstable)g(direction,)g (and)h(the)f(re)o(gularity)g(properties)g(of)h Fv(h)g Fx(follo)n(w)e(easily)h(by)h(checking)f(the)94 553 y(con)l(v)o(er)n (gence)i(of)g(the)f(series.)42 b(The)26 b(H)1430 552 y(\250)1422 553 y(older)h(continuity)d(of)j Fv(h)g Fx(follo)n(ws)e(by)i (standard)f(ar)n(guments)g(from)g(the)94 672 y(H)174 671 y(\250)166 672 y(older)f(continuity)e(of)i(the)g(stable)f (foliation.)p 3812 618 63 2 v 3812 677 2 59 v 3872 677 V 3812 679 63 2 v 94 938 a Ft(Remark.)29 b Fx(In)g(the)f(case)h(of)g (sk)o(e)n(w)e(products)h(of)h(Baladi)f(maps,)h(it)f(is)g(easy)g(to)g(v) o(erify)g(that)g(the)g(local)h(stable)94 1058 y(manifolds)h(are)h(v)o (ertical)e(se)o(gments.)51 b(Because)32 b(of)e(the)h(local)f (\003atness)g(of)h(the)f(in)l(v)n(ariant)f(measure)i(of)f(the)94 1177 y(one)d(dimensional)d(system,)i(it)f(follo)n(ws)g(from)h(the)h(e)o (xplicit)d(e)o(xpression)h(of)i(the)f(map)g(that)g(the)g(dif)n (ferential)94 1297 y(is)i(diagonal.)43 b(This)27 b(implies)g(that)g (the)g(\002eld)h(of)g(unstable)f(directions)g(is)g(horizontal)g(\(as)h (well)f(as)h(the)f(local)94 1416 y(unstable)38 b(manifolds\).)77 b(This)38 b(implies)f(that)h Fu(\002)1813 1443 y FE(w)1917 1416 y Fx(is)g(constant)g(on)g(the)h(local)g(unstable)e(manifold.)77 b(By)94 1536 y(changing)25 b(if)h(necessary)f(the)h(transv)o(erse)f (measure,)g(we)h(can)g(assume)f(that)g Fu(\002)2819 1563 y FE(w)2912 1536 y Fu(=)k Fx(1.)38 b(Therefore,)26 b(if)f Fv(v)30 b Fx(is)25 b(a)94 1656 y Fw(C)152 1613 y FF(2)210 1656 y Fx(function)16 b(with)g(compact)h(support)f(contained)h(in)f(an) i(atom)e(of)h(the)g(Mark)o(o)o(v)f(partition,)h(the)g(observ)n(able)g Fv(g)3835 1682 y FF(2)94 1775 y Fx(de\002ned)23 b(as)g(abo)o(v)o(e)f (is)g Fw(C)921 1732 y FF(1)983 1775 y Fx(and)h(we)g(can)g(apply)f(the)g (abo)o(v)o(e)g(technique.)34 b(Note)22 b(that)h(in)f(this)f(case)i (Assumption)94 1895 y(10.1)i(is)f(violated)g(since)h(the)f(map)h(is)f (area)i(preserving.)94 2220 y Ft(10.2.)f(An)g(Example)94 2410 y Fx(W)-8 b(e)39 b(construct)d(an)i(e)o(xample)f(where)h(all)f (the)h(abo)o(v)o(e)e(assumptions)g(are)i(satis\002ed.)74 b(This)37 b(e)o(xample)g(is)g(a)94 2529 y(generalization)26 b(of)h(the)f(solenoid)f(and)i(can)g(also)f(be)g(vie)n(wed)g(as)h(a)f (sk)o(e)n(w)g(product.)40 b(First)27 b(of)f(all,)h(let)f Fv(p)k(>)g Fx(6)94 2649 y(be)25 b(an)g(odd)g(inte)o(ger)-5 b(.)34 b(Let)25 b Fv(`)f Fx(be)h(that)g(solution)e(of)i(the)f(equation) 1275 2838 y Fv(`)1317 2796 y FF(2)1357 2758 y Fr(\000)1402 2838 y Fx(1)e Fu(+)h Fx(cos)p Fu(\()p Fx(2)p Fv(\031)t(=p)p Fu(\))1996 2758 y Fr(\001)2062 2838 y Fw(\000)g Fx(4)p Fv(`)e Fu(+)i Fx(2)44 b Fu(=)h Fx(0)24 b Fv(;)94 3028 y Fx(which)35 b(is)f(less)h(than)f(one.)66 b(Let)34 b Fv(r)45 b Fu(=)d Fx(1)27 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Fv(x;)g(y)t(;)g(z)t Fu(\)\))40 b(=)45 b(\()p Fv(p#;)17 b(q)t(';)g(r)s(x)j Fu(+)j Fv(`)17 b Fx(cos)e Fv(#)i Fx(cos)f Fv(';)h(r)s(y)25 b Fu(+)e Fv(`)17 b Fx(sin)e Fv(#)i Fx(cos)e Fv(';)i(r)s(z)26 b Fu(+)d Fv(`)17 b Fx(sin)e Fv(')p Fu(\))25 b Fv(;)94 3885 y Fx(where)h(the)e(angles)g(are)i (modulo)d(2)p Fv(\031)28 b Fx(and)c(we)h(use)g(Cartesian)g(coordinates) f(on)g Ft(B)2941 3843 y FF(3)2981 3885 y Fx(.)36 b(It)25 b(is)f(left)g(to)h(the)f(reader)94 4005 y(to)c(v)o(erify)g(that)f (because)i(of)f(our)g(choice)g(of)g Fv(`)f Fx(and)h Fv(r)j Fx(the)d(map)g(is)f(injecti)n(v)o(e.)33 b(It)20 b(is)f(ob)o(viously)f (a)i(sk)o(e)n(w)g(product)94 4125 y(abo)o(v)o(e)30 b(the)h(map)f(of)h (the)g(torus)f Fu(\()p Fv(#;)17 b(')p Fu(\))35 b Fw(7!)h Fu(\()p Fv(p#;)17 b(q)t(')p Fu(\))30 b Fx(which)g(is)h(er)n(godic)f (and)h(mixing)e(for)j(the)e(Lebesgue)94 4244 y(measure.)94 4412 y Ft(Remark.)e Fx(More)g(balls)e(can)i(be)f(pack)o(ed)h(and)f (also)g(balls)g(with)f(lar)n(ger)i(radius)f(using)f(a)i(Peano)g(surf)o (ace)g(for)94 4532 y(the)d(position)e(of)i(the)g(centers)g(instead)f (of)h(the)f(sphere)h(of)g(radius)g Fv(`)f Fx(as)h(abo)o(v)o(e.)316 4652 y(The)g(dif)n(ferential)f(of)h Fv(f)35 b Fx(is)24 b(gi)n(v)o(en)g(by)1347 5067 y Fv(D)s(f)39 b Fu(=)1625 4777 y Fr(0)1625 4953 y(B)1625 5012 y(B)1625 5072 y(B)1625 5136 y(@)1747 4828 y Fv(p)135 b Fx(0)117 b(0)100 b(0)f(0)1747 4948 y(0)136 b Fv(q)122 b Fx(0)100 b(0)f(0)1729 5067 y Fw(X)114 b(X)h Fv(r)104 b Fx(0)99 b(0)1729 5187 y Fw(X)114 b(X)g Fx(0)101 b Fv(r)i Fx(0)1729 5306 y Fw(X)114 b(X)g Fx(0)100 b(0)g Fv(r)2465 4777 y Fr(1)2465 4953 y(C)2465 5012 y(C)2465 5072 y(C)2465 5136 y(A)2593 5067 y Fv(;)p eop %%Page: 29 29 29 28 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(29)94 194 y Fx(and)849 519 y Fv(D)s(f)994 476 y Fq(\000)p FF(1)1123 519 y Fu(=)1228 229 y Fr(0)1228 404 y(B)1228 464 y(B)1228 524 y(B)1228 587 y(@)1451 280 y Fx(1)p Fv(=p)387 b Fx(0)317 b(0)196 b(0)h(0)1501 399 y(0)388 b(1)p Fv(=q)272 b Fx(0)196 b(0)h(0)1332 519 y Fw(\000X)15 b Fv(=)p Fu(\()p Fv(r)s(p)p Fu(\))99 b Fw(\000X)15 b Fv(=)p Fu(\()p Fv(r)s(q)t Fu(\))99 b Fx(1)p Fv(=r)150 b Fx(0)197 b(0)1332 638 y Fw(\000X)15 b Fv(=)p Fu(\()p Fv(r)s(p)p Fu(\))99 b Fw(\000X)15 b Fv(=)p Fu(\()p Fv(r)s(q)t Fu(\))148 b Fx(0)g(1)p Fv(=r)i Fx(0)1332 758 y Fw(\000X)15 b Fv(=)p Fu(\()p Fv(r)s(p)p Fu(\))99 b Fw(\000X)15 b Fv(=)p Fu(\()p Fv(r)s(q)t Fu(\))148 b Fx(0)196 b(0)149 b(1)p Fv(=r)2964 229 y Fr(1)2964 404 y(C)2964 464 y(C)2964 524 y(C)2964 587 y(A)3092 519 y Fv(;)94 897 y Fx(where)26 b Fw(X)39 b Fx(denotes)25 b(v)n(arious)e(quantities)h(of)h(order)g (one.)316 1017 y(W)-8 b(e)19 b(no)n(w)e(v)o(erify)h(the)h(b)n(unching)e (conditions.)32 b(First)19 b(of)f(all,)i(the)e(stable)g(b)n(undle)g(is) g(ob)o(viously)e(obtained)94 1136 y(by)22 b(setting)f(the)g(\002rst)h (tw)o(o)f(components)g(of)h(a)g(tangent)f(v)o(ector)g(equal)h(to)f (zero.)36 b(Therefore,)23 b Fv(\025)28 b Fu(=)g Fv(r)s Fx(,)22 b(and)g(also)94 1256 y(the)j(stable)f(manifold)g(of)h(a)g (point)f Fu(\()p Fv(#;)17 b(';)g(x;)g(y)t(;)g(z)t Fu(\))22 b Fx(is)i(the)h(set)f(of)h(points)f(with)g(the)g(same)h(angles)f Fv(#)h Fx(and)g Fv(')p Fx(.)316 1375 y(The)g(unstable)g(b)n(undle)f(is) h(not)g(so)g(tri)n(vial.)35 b(As)25 b(in)g([9],)h(the)f(unstable)f(b)n (undle)h(is)g(obtained)f(as)i(a)f(graph)94 1495 y(abo)o(v)o(e)32 b(the)g(space)h(of)g(v)o(ectors)e(whose)i(last)e(tw)o(o)h(coordinates)g (are)h(equal)g(to)f(zero.)59 b(In)33 b(other)f(w)o(ords,)i(for)94 1614 y(e)n(v)o(ery)23 b(point)f Fv(P)42 b Fw(2)28 b Ft(M)p Fx(,)23 b(there)h(is)e(a)i(linear)f(operator)g Fv(L)1958 1641 y FE(P)2048 1614 y Fx(from)g Ft(R)2337 1571 y FF(2)2400 1614 y Fx(to)f Ft(R)2572 1571 y Fa(3)2635 1614 y Fx(such)h(that)g(the)g (unstable)f(subspace)94 1734 y(at)j Fv(P)39 b Fx(is)24 b(the)h(set)1328 1853 y Fv(E)1408 1811 y FE(u)1459 1853 y Fu(\()p Fv(P)14 b Fu(\))27 b(=)1747 1773 y Fr(\010)1805 1853 y Fu(\()p Fv(z)t(;)17 b(L)2007 1880 y FE(P)2073 1853 y Fv(z)t Fu(\))2179 1769 y Fr(\014)2179 1829 y(\014)2229 1853 y Fv(z)33 b Fw(2)28 b Ft(R)2474 1811 y FF(2)2513 1773 y Fr(\011)2613 1853 y Fv(;)94 2009 y Fx(with)d(the)h(canonical)g (identi\002cations.)37 b(From)25 b(the)h(equation)f(satis\002ed)g(by)h Fv(L)2785 2036 y FE(P)2877 2009 y Fx(\(see)g([9]\))g(it)f(follo)n(ws)g (easily)94 2129 y(that)1501 2248 y(sup)1479 2334 y FE(P)11 b Fq(2)p Fa(M)1677 2248 y Fw(k)p Fv(L)1795 2275 y FE(P)1861 2248 y Fw(k)44 b(\024)h(O)s Fu(\()p Fx(1)p Fu(\))p Fv(q)2335 2205 y Fq(\000)p FF(1)2461 2248 y Fv(:)94 2485 y Fx(It)30 b(then)f(follo)n(ws)f(that)h Fv(\026)951 2443 y Fq(\000)p FF(1)1087 2485 y Fu(=)34 b Fv(q)1246 2443 y Fq(\000)p FF(1)1348 2485 y Fu(\()p Fx(1)25 b Fu(+)g Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(q)1822 2443 y Fq(\000)p FF(1)1923 2485 y Fu(\))p Fx(.)49 b(Finally)29 b Fv(\027)2403 2443 y Fq(\000)p FF(1)2535 2485 y Fx(is)g(at)h(most)e(the)h(sup)g(norm)g(of)g Fv(D)s(f)3772 2443 y Fq(\000)p FF(1)94 2605 y Fx(and)c(we)g(get)g Fv(\027)606 2562 y Fq(\000)p FF(1)736 2605 y Fw(\024)j Fv(r)889 2562 y Fq(\000)p FF(1)1013 2605 y Fu(+)23 b Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(r)1371 2562 y Fq(\000)p FF(1)1472 2605 y Fv(q)1520 2562 y Fq(\000)p FF(1)1622 2605 y Fx(.)36 b(Recalling)25 b(that)f Fv(r)31 b Fw(\031)d Fv(\031)t(q)2554 2562 y Fq(\000)p FF(1)2680 2605 y Fx(for)e(lar)n(ge)f Fv(q)t Fx(,)g(we)g(get)631 2769 y Fv(\025\026)749 2726 y Fq(\000)p FF(1)852 2769 y Fv(\027)907 2726 y Fq(\000)p FF(2)1054 2769 y Fw(\024)44 b Fv(r)s(q)1271 2726 y Fq(\000)p FF(1)1373 2769 y Fv(r)1421 2726 y Fq(\000)p FF(2)1523 2769 y Fu(\()p Fx(1)22 b Fu(+)g Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(q)1991 2726 y Fq(\000)p FF(1)2093 2769 y Fu(\))44 b Fw(\024)h Fv(\031)2359 2726 y Fq(\000)p FF(1)2460 2769 y Fu(\()p Fx(1)22 b Fu(+)g Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(q)2928 2726 y Fq(\000)p FF(1)3030 2769 y Fu(\))44 b Fv(<)h Fx(1)25 b Fv(;)94 2932 y Fx(for)37 b Fv(q)k Fx(lar)n(ge)c(enough,)i(namely)d(the)g(unstable)g(b)n(undle)f(is)h(e)n (v)o(en)g Fw(C)2468 2890 y FF(2)2545 2932 y Fx(\(we)h(only)f(require)g Fw(C)3321 2890 y FF(1)p Fp(+)p FE(\013)3511 2932 y Fx(for)h(some)94 3052 y Fv(\013)28 b(>)g Fx(0\).)316 3172 y(The)d(stable)f(b)n(undle)g (is)h(ob)o(viously)d(in\002nitely)i(re)o(gular)g(b)n(ut)h(we)g(can)g (check)g(the)g(b)n(unching)f(condition)94 3291 y(for)i(the)e(in)l(v)o (erse.)35 b(W)-8 b(e)26 b(obtain)e Fv(\025)j Fu(=)h Fv(q)1393 3248 y Fq(\000)p FF(1)1518 3291 y Fu(+)22 b Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(r)1875 3248 y Fq(\000)p FF(1)1977 3291 y Fv(q)2025 3248 y Fq(\000)p FF(2)2127 3291 y Fx(,)j Fv(\026)2237 3248 y Fq(\000)p FF(1)2366 3291 y Fu(=)k Fv(r)s Fx(,)24 b Fv(\027)2624 3248 y Fq(\000)p FF(1)2754 3291 y Fu(=)29 b Fv(p)22 b Fu(+)g Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fx(.)36 b(W)-8 b(e)25 b(get)691 3455 y Fv(\025\026)809 3412 y Fq(\000)p FF(1)912 3455 y Fv(\027)967 3412 y Fq(\000)p FE(\013)1114 3455 y Fw(\024)j Fv(q)1267 3412 y Fq(\000)p FF(1)1369 3455 y Fv(r)s(p)1467 3412 y FE(\013)1524 3455 y Fu(\()p Fx(1)22 b Fu(+)g Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(q)1992 3412 y Fq(\000)p FF(1)2094 3455 y Fu(\))44 b(=)h Fv(\031)t(q)2408 3412 y Fq(\000)p FF(2)2509 3455 y Fv(p)2559 3412 y FE(\013)2616 3455 y Fu(\()p Fx(1)22 b Fu(+)g Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(q)3084 3412 y Fq(\000)p FF(1)3186 3455 y Fu(\))i Fv(;)94 3619 y Fx(and)f(this)e(is) h(smaller)f(than)h(one)h(for)f Fv(q)27 b Fx(lar)n(ge)22 b(enough)g(if)g Fv(\013)28 b(<)g Fx(2)22 b(and)g Fv(p)h Fx(is)f(not)f(much)h(lar)n(ger)h(than)f Fv(q)t Fx(.)35 b(In)22 b(other)94 3738 y(w)o(ords,)j(we)g(can)g(construct)f(e)o (xamples)g(with)g(the)h(stable)f(b)n(undle)g Fw(C)2439 3695 y FF(1)p Fp(+)p FE(\033)2614 3738 y Fx(for)h(an)o(y)f(0)k Fv(<)g(\033)j(<)d Fx(1.)316 3858 y(Finally)h(we)i(ha)n(v)o(e)e(to)h (check)g(the)g(condition)f(inf)21 b Fv(\021)t(=')35 b(>)g Fx(1.)52 b(In)30 b(the)f(abo)o(v)o(e)g(e)o(xample)h(this)f(is)g(made)94 3977 y(simpler)35 b(by)g(the)h(observ)n(ation)e(that)h(the)g(set)h(of)f (tangent)g(v)o(ectors)g(with)g(\002rst)h(coordinate)f(equal)g(to)h (zero)94 4097 y(is)j(co)o(v)n(ariant.)76 b(The)38 b(same)h(is)f(true)h (for)g(the)f(set)h(of)f(v)o(ectors)g(with)g(second)h(coordinate)f (equal)g(to)h(zero.)94 4216 y(Therefore,)26 b(the)f(tw)o(o)f(in)l(v)n (ariant)g(b)n(undles)f(are)j(graphs.)36 b(The)24 b(lar)n(gest)h(one)g (is)f(a)h(set)g(of)g(v)o(ectors)1139 4299 y Fr(\010\000)1243 4380 y Fv(s;)17 b Fx(0)p Fv(;)g(u)1487 4407 y FF(1)1525 4380 y Fu(\()p Fv(P)d Fu(\))p Fv(s;)j(v)1821 4407 y FF(1)1859 4380 y Fu(\()p Fv(P)d Fu(\))p Fv(s;)j(w)2178 4407 y FF(1)2217 4380 y Fu(\()p Fv(P)d Fu(\))p Fv(s)2420 4299 y Fr(\001)2481 4296 y(\014)2481 4355 y(\014)2531 4380 y Fv(s)27 b Fw(2)h Ft(R)2771 4299 y Fr(\011)94 4544 y Fx(and)d(the)g(lo)n(west)e(one)1105 4583 y Fr(\010\000)1208 4664 y Fx(0)p Fv(;)17 b(s;)g(u)1452 4690 y FF(2)1490 4664 y Fu(\()p Fv(P)d Fu(\))p Fv(s;)j(v)1786 4690 y FF(2)1825 4664 y Fu(\()p Fv(P)d Fu(\))p Fv(s;)j(w)2144 4690 y FF(2)2182 4664 y Fu(\()p Fv(P)d Fu(\))p Fv(s)2385 4583 y Fr(\001)2446 4579 y(\014)2446 4639 y(\014)2496 4664 y Fv(s)28 b Fw(2)g Ft(R)2737 4583 y Fr(\011)2836 4664 y Fv(:)94 4819 y Fx(The)23 b(six)f(functions)f Fv(u)861 4846 y FE(i)894 4819 y Fx(,)i Fv(v)990 4846 y FE(i)1023 4819 y Fx(,)g Fv(w)1142 4846 y FE(i)1198 4819 y Fx(satisfy)f(the)g (usual)g(coherence)h(equations,)f(and)g(it)g(follo)n(ws)f(easily)h (that)g(the)o(y)94 4939 y(are)k(all)f(uniformly)e(bounded)h(by)h Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(q)1540 4896 y Fq(\000)p FF(1)1641 4939 y Fx(.)36 b(It)25 b(follo)n(ws)e(easily)h(that)1164 5103 y Fv(\021)32 b Fw(\025)c Fv(p)22 b Fu(+)h Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(pq)1830 5060 y Fq(\000)p FF(1)1981 5103 y Fx(and)50 b Fv(')28 b Fw(\024)g Fv(q)e Fu(+)d Fw(O)s Fu(\()p Fx(1)p Fu(\))h Fv(;)94 5266 y Fx(and)h(our)g(condition)e (is)i(satis\002ed)f(if)h Fv(p=q)32 b(>)c Fx(1)22 b Fu(+)g Fw(O)s Fu(\()p Fx(1)p Fu(\))p Fv(q)2061 5224 y Fq(\000)p FF(1)2188 5266 y Fx(and)i Fv(q)29 b Fx(is)c(lar)n(ge)g(enough.)p eop %%Page: 30 30 30 29 bop 94 -45 a Fs(Liapuno)n(v)38 b(Mul)-7 b(tipliers)37 b(and)h(Deca)-7 b(y)37 b(of)i(Correla)-7 b(tions)1146 b Ft(30)94 194 y Fy(A)m(ppendix)94 383 y Fx(W)-8 b(e)30 b(gi)n(v)o(e)d(here)i(the)g(proof)f(of)h(Lemma)f(3.4.)47 b(The)28 b(l.h.s.)g(of)h(\(3.5\))g(is)f Fv(\045)2569 410 y FF(ess)2658 383 y Fx(.)48 b(The)28 b(r)-5 b(.h.s.)28 b(will)g(be)h(called)38 b Fu(\026)-60 b Fv(\045)3759 410 y FF(ess)3849 383 y Fx(.)94 503 y(W)-8 b(e)26 b(ha)n(v)o(e)e(ob)o (viously)32 b Fu(\026)-60 b Fv(\045)927 530 y FF(ess)1045 503 y Fw(\025)28 b Fv(\045)1202 530 y FF(ess)1292 503 y Fx(.)35 b(T)-8 b(o)25 b(pro)o(v)o(e)f(the)g(con)l(v)o(erse)h (inequality)-6 b(,)22 b(let)j Fv(")j(>)g Fx(0.)36 b(From)24 b(the)h(de\002nition)94 623 y(of)g Fv(\045)254 649 y FF(ess)369 623 y Fx(we)g(can)g(\002nd)g(tw)o(o)g(subspaces)f Fv(M)35 b Fx(and)25 b Fv(M)1859 580 y Fq(0)1911 623 y Fx(of)g(\002nite)g(codimension)e(such)i(that)972 996 y(lim)17 b(sup)1012 1077 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b(are)h(the)f(closures)h(of)f Fv(M)48 b Fx(and)38 b Fv(M)3126 2590 y Fq(0)3191 2633 y Fx(are)g(also)g(of)f(\002nite)94 2753 y(codimension.)e(Moreo)o(v)o(er)l(,)23 b(for)i(each)h Fv(n)p Fx(,)e(we)h(can)h(\002nd)f Fv(x)i Fw(2)2258 2728 y Fu(\026)2221 2753 y Fv(M)36 b Fx(and)24 b Fv(y)31 b Fw(2)2734 2728 y Fu(\026)2697 2753 y Fv(M)2805 2710 y Fq(0)2832 2753 y Fx(,)25 b(both)f(of)h(norm)f(1)h(such)f(that)455 2874 y Fr(\014)455 2934 y(\014)515 2959 y Fv(<)29 b(y)t(;)17 b(U)798 2916 y FE(n)850 2959 y Fv(x)28 b(>)1040 2874 y Fr(\014)1040 2934 y(\014)p 455 3010 619 4 v 610 3101 a Fw(k)p Fv(y)t Fw(kk)p Fv(x)p Fw(k)1113 3033 y(\025)74 b Fx(sup)1243 3129 y FE(x)p Fq(2)1370 3111 y Fp(\026)1342 3129 y FE(M)1230 3216 y(y)s Fq(2)1355 3199 y Fp(\026)1327 3216 y FE(M)1410 3174 y Fj(0)1478 2874 y Fr(\014)1478 2934 y(\014)1539 2959 y Fv(<)28 b(y)t(;)17 b(U)1821 2916 y FE(n)1874 2959 y Fv(x)27 b(>)2063 2874 y Fr(\014)2063 2934 y(\014)p 1478 3010 V 1633 3101 a Fw(k)p Fv(y)t Fw(kk)p Fv(x)p Fw(k)2131 3033 y(\000)2230 2952 y Fr(\000\000)2322 3033 y Fv(\045)2374 3060 y FF(ess)2485 3033 y Fu(+)c Fv(")2631 2952 y Fr(\001)2677 2990 y FE(n)2754 3033 y Fw(\000)2853 2952 y Fr(\000)2899 3033 y Fv(\045)2951 3060 y FF(ess)3063 3033 y Fu(+)f Fv("=)p Fx(2)3308 2952 y Fr(\001)3354 2990 y FE(n)3408 2952 y Fr(\001)3498 3033 y Fv(:)94 3424 y Fx(If)k Fv(x)38 b(=)-61 b Fw(2)29 b Fv(M)35 b Fx(or)25 b Fv(y)42 b(=)-61 b Fw(2)28 b Fv(M)887 3381 y Fq(0)940 3424 y Fx(or)c(both,)g(we)i(can)f(\002nd)g(tw)o(o)f (sequences)h Fu(\()p Fv(x)2457 3451 y FE(j)2499 3424 y Fu(\))i Fw(\032)h Fv(M)36 b Fx(and)24 b Fu(\()p Fv(y)3059 3451 y FE(j)3101 3424 y Fu(\))j Fw(\032)i Fv(M)3381 3381 y Fq(0)3433 3424 y Fx(con)l(v)o(er)n(ging)94 3543 y(to)c Fv(x)g Fx(and)g Fv(y)j Fx(respecti)n(v)o(ely)-6 b(.)34 b(Therefore,)601 3673 y Fr(\014)601 3733 y(\014)662 3757 y Fv(<)28 b(y)t(;)17 b(U)944 3715 y FE(n)997 3757 y Fv(x)28 b(>)1187 3673 y Fr(\014)1187 3733 y(\014)p 601 3808 V 757 3900 a Fw(k)p Fv(y)t Fw(kk)p Fv(x)p Fw(k)1260 3831 y 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FC([1])39 b(V)-10 b(.)19 b(Baladi.)i(Unpublished)e(\(1989\).)134 563 y([2])39 b(V)-10 b(.)29 b(Baladi.)i Fz(P)-6 b(ositive)31 b(tr)o(ansfer)f(oper)o(ator)o(s)g(and)g(decay)g(of)g(corr)m(elations)p FC(,)h(v)n(olume)g(16)e(of)h Fz(Advanced)g(Series)h(in)f(Nonlinear)267 652 y(Dynamics)20 b FC(\(Ri)n(v)o(er)f(Edge,)h(NJ:)f(W)-6 b(orld)20 b(Scienti\002c)i(Publishing)e(Co.)f(Inc.,)h(2000\).)134 742 y([3])39 b(R.)18 b(Bo)n(wen.)f(Equilibrium)f(States)j(and)f(the)f (Er)o(godic)g(Theory)f(of)h(Anoso)o(v)f(Dif)n(feomorphisms.)g Fz(Spring)o(er)h(Lect.)i(Notes)e(in)h(Math.)267 832 y FB(470)i FC(\(1975\),)e(Berlin\226Heidelber)o(g\226Ne)n(w)h(Y)-9 b(ork.)134 921 y([4])39 b(P)-9 b(.)17 b(Collet)g(and)f(S.)g(Isola.)g (On)g(the)g(essential)i(spectrum)e(of)f(the)i(transfer)e(operator)h (for)f(e)o(xpanding)h(Mark)o(o)o(v)f(maps.)i Fz(Comm.)e(Math.)267 1011 y(Phys.)20 b FB(139)f FC(\(1991\),)g(551\226557.)134 1101 y([5])39 b(D.)16 b(E.)g(Edmunds)f(and)h(W)-7 b(.)16 b(D.)g(Ev)n(ans.)g 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FC(\(1984\),)267 2356 y(181\226193.)94 2446 y([12])39 b(P)-9 b(.)27 b(Mattila.)g Fz(Geometry)g(of)g(sets)f(and)g(measur)m(es)h(in)f(Euclidean)h(spaces)p FC(,)g(v)n(olume)g(44)f(of)f Fz(Cambridg)o(e)i(Studies)f(in)h(Advanced) 267 2535 y(Mathematics)21 b FC(\(Cambridge:)f(Cambridge)g(Uni)n(v)o (ersity)f(Press,)h(1995\).)f(Fractals)i(and)f(recti\002ability)-5 b(.)94 2625 y([13])39 b(J.)17 b(Rousseau\255Egele.)h(Un)f(th)1067 2624 y(\264)1063 2625 y(eor)1169 2624 y(\036)1165 2625 y(eme)h(de)f(la)h(limite)h(locale)f(pour)f(une)g(classe)i(de)f (transformations)e(dilatantes)j(et)f(monotones)f(par)267 2715 y(morceaux.)j Fz(Ann.)f(Pr)l(obab)m(.)h FB(11)f FC(\(1983\),)g(772\226788.)94 2804 y([14])39 b(D.)20 b(Ruelle.)g Fz(Thermodynamic)g(Formalism)f FC(\(Addison)g(W)-6 b(esle)o(y)h(,)20 b(Reading)g(MA,)g(1978\).)94 2894 y([15])39 b(D.)20 b(Ruelle.)g(Resonances)i(of)d(chaotic)i(dynamical)g(systems.)f Fz(Phys.)g(Re)o(v)-6 b(.)20 b(Lett.)h FB(56)f FC(\(1986\),)e (405\226407.)94 2984 y([16])39 b(D.)20 b(Ruelle.)g(Resonances)i(for)d (Axiom)g FB(A)g FC(\003o)n(ws.)h Fz(J)n(.)g(Dif)o(fer)m(ential)h(Geom.) e FB(25)g FC(\(1987\),)g(99\226116.)94 3073 y([17])39 b(L.\255S.)19 b(Y)-9 b(oung.)19 b(Recurrence)i(times)g(and)f(rates)g (of)g(mixing.)f Fz(Isr)o(ael.)h(J)n(.)g(Math.)g FB(110)g FC(\(1999\),)e(153\226188.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303040821683--