2HUYIHUmeasuresoffullHausdor dimensioninalmostexpandingsystemsThedensityfunctionsusuallyapproachtoin nityasxapproachestotheindi erent xedpointpWeshowthattheincreaseratesareboundedbypolynomialsCons ID: 851944
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1 EQUILIBRIUMSTATESWITHPOLYNOMIALDECAYOFCO
EQUILIBRIUMSTATESWITHPOLYNOMIALDECAYOFCORRELATIONSHuyiHuAbstract.Thepurposeofthispaperistostudystatisticalpropertiesofsomeal-mostexpandingdynamicalsystems.Examplesofsuchsystemsincludepiecewiseexpandingmapsontheunitinterval,expandingmapsonCantorsets,andsomepiecewiseexpandingmapsontheunitcubes,allofwhichcontainanindi erent xedpoint.WesupposethesystemshaveMarkovpartition.Sowecanworkonsymbolicdynamicalsystems.WeproveexistenceofabsolutelycontinuousinvariantmeasureswithrespecttoeitherLebesguemeasuresorconformalmeasures,andshowthatthesemeasuresareweakGibbsstatesandequilibriumstates.Further,byusingtheprojectivemetricweobtainthatunderiterationofthePerron-Frobeniusoperators,functionsconvergetotheequilibriumsinpolynomialspeed,andthereforethesystemshavepolynomialdecayofcorrelations.0.IntroductionThemotivationofthepaperistounderstandstatisticalpropertiesforalmostex-pandingdynamicalsystemsthatallowabsolutelycontinuousprobabilityinvariantmeasureswithrespecttoLebesgueorconformalmeasures.Wesaythatapiecewisesmoothsystemisalmostexpanding,ifitisexpandingeverywhereexceptata nitenumberofperiodicorbits.Thesystemscanbede nedonintervals,higherdimen-sionalmanifolds,ornowheredensesubsets(fractals).Inthispaperweonlyconsiderthecasethatthesystemscontainoneindi erent xedpointp.Systemswithmoreindi erentperiodicorbitscanbetreatedsimilarly.InSection2wediscussthreetypesofsystemsasexamples:pi
2 ecewiseexpandingmapsontheunitinterval,ex
ecewiseexpandingmapsontheunitinterval,ex-pandingmapsonnowheredenseclosesubsets,andsomepiecewiseexpandingmapsontheunitcubes.Allofthemhaveanindi erent xedpoint.WeassumethatthesystemshaveMarkovpartitions.Sothesystemscanberepresentedbyone-sidedsubshiftof nitetype,andwecanworkonpotentials'.TheresultsinthispaperareinfactforaclassofpotentialswhichdonotsatisfyingHolderconditionswiththeusualmetriconsymbolicspace.However,wetakeaspecialmetricunderwhichthepotentialsbecomeLipschitzfunctions.Thisdoesnota ecttheresults,sincetheyareindependentofmetric.We rstproveexistenceofinvariantprobabilitymeasuresofcertainpoten-tialfunctions,whichcorrespondtoabsolutelycontinuousinvariantmeasuresorThisworkwassupportedbyNSFundergrantsDMS-9970646.TypesetbyAMS-TEX1 2HUYIHUmeasuresoffullHausdor dimensioninalmostexpandingsystems.Thedensityfunctionsusuallyapproachtoin nity,asxapproachestotheindi erent xedpointp.Weshowthattheincreaseratesareboundedbypolynomials.Consequently,theupperboundsofmeasuresofn-cylindersalsodecreasepolynomiallyasnincreases.Somealmostexpandingsystemsonlyadmit- niteabsolutelycontinuousmeasure.Wearenotgoingtoconsiderthecaseinthispaper.Second,westudysomeergodicpropertiesofthemeasuressuchasexactness,Gibbsstatesandequilibriumstates.Themeasuresareexact,andthereforeareergodicandmixing.However,arenotGibbsstates.Infact,foranygivenn,expnP+Sn'(x) canbearbitrarilycloseto1asxcl
3 osetop,ifP=0and'(0)=0.Thereforewecannote
osetop,ifP=0and'(0)=0.Thereforewecannotexpectuniformboundsfortheratiobetweenthisquantityandmeasuresofn-cylinders.Ourresultsshowthatareunique\weak"Gibbsstatesinthesensethattheboundsaregivenbysubexponentialfunctions.Usingthispropertywecanalsoshowthatthemeasuresareequilibriumstatesofthepotentialfunctions.Anotherimportantpartofthepaperistoprovethepowerlaw:theratesofconvergenceoftestfunctionsundertheiterationofthePerron-Frobeniusoperatorsarepolynomial.Theseratesgivethespeedofconvergenceofsmoothmeasurestendingtotheirequilibriums,andthespeedofdecayofcorrelations.Thepowerlawisaphenomenonthathasonlyprovedforalmostexpandingmaps,thoughitisbelievedtohappeninmanydi erentkindsofsystems.Ourproofshowsthatdegreeofthedecayratedependontherateofchangeofthepotentialfunctionsnearthe xedpointp.Asaconsequence,theCentralLimitTheoremholdsforsuchsystemsiftheorderofdecayratesishigherthanO(n 1 ),�0.ThemaintoolweuseinthispaperisthePerron-FrobeniusoperatorsL'.Forex-istenceofabsolutelycontinuousinvariantmeasure,weobtainthedensityfunctionshas xedpointsofL'insomefunctionalspaces.hisusuallyunboundedneartheindi erent xedpoint.WeusethefactL'h=htoestimatetheincreasingratesofhasxnearp.ThisestimatesallowustounderstandnearphowvaluesoftestfunctionsgchangeunderiterationsoftheoperatorL'.ThisbehaviordeterminestheconvergentrateoffLngg,sincethemotionnearpisslowerthananywhereelseinthedynamical
4 systems.Ourproofforpolynomialdecayratesc
systems.Ourproofforpolynomialdecayratescarriesoverthisobservation.Tocontrolthedistancebetweentestfunctionsandtheirequilibriums,weusetheprojective(Hilbert)metric,atechniqueoriginallyduetoG.Birkho ([Bi])andthenFerreroandSchmitt([FS])forintroducingintodynamicalsystems(seealso[Li1]).Ourprooffordecayofcorrelationgivesagoodgeometricdescriptionfortestfunctionsconvergingtoitslimit.Considerasystemwithdecayraten ( 1).Weshowsthatatthestepn,awayfromp,thedi erencebetweenthetestfunctionanditsequilibriumisoforderO(n )inshape,whichcanbemeasuredbytheprojectivemetric(PropositionC,and(An)intheproofoftheproposition),whilethedi erenceisoforderO(n ( 1))ifmeasuredinC0norm(TheoremD).Ontheotherhand,nearp,functionschangeslowlyunderthePerron-Frobeniusoperators.Forcompletionourstatementsincludesomeknownresults.Thoughwewillgiverelatedreferencesaswestatethetheorems,wewouldliketomentionsomeofthem POLYNOMIALDECAYOFCORRELATIONS3here.Existenceofabsolutelycontinuousinvariantmeasuresforpiecewiseexpandingmapsontheunitintervalwithindi erent xedpointswasstudiedtwodecadesago.(Seeforexample,[Pi],[T].)ForexpandingsystemsonCantorsetswithindi erent xedpoints,someergodicpropertiesincludingconditionsforexistenceofmeasuresoffullHausdor dimensionandequilibriums,aswellasrigidproperties,wasstudiedbyUrbanski([U]).Polynomialdecayofcorrelationswas rstprovedbyM.Mori([M])andbyLambert,SiboniandVaienti([LaSV]
5 )forpiecewiselinearmapswithindi erent xe
)forpiecewiselinearmapswithindi erent xedpoints(Takahashimodel).ForsymbolicdynamicalsystemsitwasstudiedbyFisherandLopesforlocallyconstantpotentials([FL]),andbyPollicottforsummablepotentials([P],seealso[BFG]).Fornonlinearmapsontheinterval,polynomialdecaywasobtainedbyYoung([Y]),Liverani,SaussolandVaienti(LiSV]),PollicottandYuri([PY1]),andtheauthor([H])recently.Amongwhichestimatesin[Y]and[H]aresharp,however,theapproachesfordecayofcorrelationsaredi erent.ThemaintoolisprobabilisticcouplinginYoung'swork([Y]),andisHilbertmetricinours.Ontheotherhand,theargumentsin[H]areessentiallyforlowerboundestimates,andtheupperboundsareobtainedasabyproduct.Thispaperisorganizedasfollows.TheassumptionsandgeneralresultsarestatedinSection1.InSection2weapplytheseresultstopiecewisesmoothal-mostexpandingmaps.TheoremA,mainlydealingwithexistenceandpropertiesofinvariantmeasures,isprovedinSection3.Section4isforsomepreliminaryresultsconcerningpropertiesofthePerron-FrobeniusoperatorL .TheoremBisconcerningergodicproperties:exactness,weakGibbsandequilibriumstates,oftheinvariantmeasures,andisprovedinSections5.ThemainsteptowardspolynomialdecayofcorrelationsistoprovePropositionC,whichiscarriedoverinSection6and7.InthelastsectionweproveTheoremD,EandF,whichareconcerningrateofconvergenceoftestfunctions,decayofcorrelationsandtheCentralLimitTheoremrespectively.1.Assumptions,StatementsofResultsandNotati
6 onsLet+=Q1f0;1;;r 1g,and:+!+bethe
onsLet+=Q1f0;1;;r 1g,and:+!+betheleftshift.IfAisanrrmatrixof0'sand1's,let+=fx2+:Axixi+1=18i0g.Itiswellknownthat+A=+A.Weassumethatistopologicallymixing.WealsoassumeA00=1.So,0=000isa xedpointof.Wesaythatwisann-wordifw=w0w1wn 1andAwiwi+1=180in 2.Theworduwisthewordufollowedbythewordw.Givenann-wordw=w0w1wn 1,wede neRw=fx2+:xi=wi;80in 1g:Thissetiscalledann-cylinder,orsimplyacylinder.Letbethepartitionof+AintofRs:s=0;1r 1g,andn=_ 1__n 1.Foranyn-wordw,wehaveRw2n.Wesimplywritew2ninstead.Fork0,wedenoteOk=R0k,Pk=OknOk+1,andQk=+AnOk+1.Inotherwords,Ok,PkandQkaresetsofthepointsthatstartwithatleast,exact,andatmostkzerosrespectively.Also,wedenoteP0k=Pk[Pk+1. 4HUYIHUTake2(0;1)and �0.LetK0bethelargestnumberksuchthatk(k+1) (1+ ).De neametricon+inductivelybythefollowingrules.i)d(x;y)=1ifx06=y0.ii)d(x;y)=d(x;y)ifx0=y0andx;y2QK0.iii)d(x;y)=(k+1) ( +1)d(kx;ky)ifx2Pk,y2P0k,k�K0.iv)d(x;y)=k+l 1Pi=kd(x(i);x(i+1)),ifx2Pkandy2Pk+l,kK0,wherex(k)=x,x(k+l)=y,andx(i)2Pifori=k;;k+l.Withthismetric,theleftshift:+A!+Aisuniformlyexpandingwitharate 1onQK0.TheexpandingrateonPkconvergesto1ifk!1.Bythemetricwecanseethatifx=0k~x2Pk,y=0k~y2P0ksuchthat~x06=~y0,thend(x;y)=maxfk;(k+1) ( +1)g.Hence,thereexistsC 1suchthatdiamOkC k 8k0;(1:1)wherethediameterofasetSisde nedbydiamS=supfd(x;y):x;y2Sg.Weassumethatthepotentialfunc
7 tion'satis esthefollowing.AssumptionA.(I
tion'satis esthefollowing.AssumptionA.(I)'isacontinuousfunctionon+A;(II)9C'�0suchthatj'(x) '(y)jC'maxfK 10;k 1gd(x;y)8x2Pk;y2P0k;(III)9 �1,K1�0suchthatforallkK1,'(0) '(x) +1k+1 C(k+1)1+8x2Pkforsomeconstants�0andC�0independentofkandx.Wemayassumeminf ;1gsincewecanalwaysreduce.Remark1.1.Itiseasytoseethat'satis estheHoldercondition.However,ifwetakeametricbetweenx=fxigandy=fyigas~d(x;y)=1Xi=0signjxi yij2i,then'maynotbeaHolderfunction.Remark1.2.Ifwedenotevark(')=maxfj'(x) '(y)j:xi=yi;8i=0;;k 1g,thenbyAssumptionA(II)vark(')Ck 1forsomeC�0.Sothepotential'doesnothavesummablevariationssince1Pk=0vark(')isnotnecessarysummable.LetC0(S)denotethesetofcontinuousrealfunctionsonthesetS.De nethePerron-FrobeniusOperatorL'fromC0 +nf0gtoitselfbyL'g(x)=Xy2 1xe'(y)g(y): POLYNOMIALDECAYOFCORRELATIONS5DenoteDk=maxfk;K0g.ForJ0,wede neHJ=fg2C0 +nf0g:g�0;g(y)g(x)eJD kd(x;y)8x2Pk;y2P0k;8k0g:WewillseethatL'HJHJifJislargeenough.Forafunctiong2C0 +nf0g,x=0^x2O1,wedenote^g(x)=Xs6=0e'(s^x)g(s^x):(1:2)DenotebyM(+)thesetofBorelprobabilitymeasureson+A.TheoremA.(Existenceandpropertiesofdensityfunctions.)Suppose'satis esAssumptionA(I)-(III).Thenthereisameasure2M(+A),whichispositiveonnonemptyopensets,aconstante'(0),andafunctionh2HJ'forsomeJ'�0suchthatL=,L'h=h,(h)=1,andlimsupn!1 nh(0nx)^h(0)8x6=0:(1:3)M
8 oreover,(g)=(hg)de nesa-invariantprob
oreover,(g)=(hg)de nesa-invariantprobabilitymeasuresatisfyingPnCn ;OnCn ( 1)8n0(1:4)forsomeC�0.Remark1.3.IfwealsoassumeinadditiontoAssumptionA(III)that'(0) '(x) +1k+1 C(k+1)1+8x2Pkforsomeconstants�0andC�0independentofkandx,thenwehaveaninequalityoppositeto(1.4)forthelowerlimit,andthereforethelimitexistsandisequalto^h(0).(See[H]forone-dimensionalcase.)InthiscaseonecannotexpectthatthemeasureobtainedasaboveisaGibbsmeasure.Thisisnotonlybecausecylindersnear0havelargermeasure,butalsobecausevariationsofthepotentialfunction'arenotsummable.Thelatterimpliesunboundeddistortions,i.e.ratiosbetweeneSn(x)andeSn(y)canbeunboundedforxandyinasamen-cylinderasn!1.De nition1.1.ABorelprobabilityinvariantmeasure='iscalledaweakGibbsmeasureofafunction',if(f0g)=0andifthereisarealnumberP,andafunctionp(x;n)suchthatforeveryx1p(x;n)Rx0x1xn 1expf nP+Sn'(x)gp(x;n);(1:5)wherep(x;n)satis esthefollowing:a)limn!11nlogp(x;n)=0for-a.e.x2+;b)limk!11klogsupx2Qk\ n+1Q0p(x;n)=0. 6HUYIHURemark1.5.Partb)impliesthatwecan nduniformboundsforcylinderswhicharepreimagesofRu,u=1;;r 1,andareawayfrom0,eventhoughwecannotdoitforallcylinders.Remark1.6.ThenotionweakGibbsmeasurewasalsousedbyM.Yuri([Yu1],[Yu2])forsomenonuniformexpandingmapsaboutthesametimeasthispaperwaswriting.Ourde nitionisslightlymoregeneral.Thisisbecauseinherassumption,potentialfunctionssatis
9 fyweakboundedvariation,whichfailsinourse
fyweakboundedvariation,whichfailsinoursetting.Inthecaseweareinterested,thepotentials'areusuallyequalto0atthe xedpoint0,sinceeigenvaluesofDfareequalto1atindi erent xedpoints.Recallthatasystem(;)isanexactendomorphism,if1Tn=0 nBconsistsonlysetsofmeasure0and1for.Itiswellknownthatif(;)isanexactendomor-phism,thenitismixing.(See[R].Alsosee[PY2]forrecentreference.)Aninvariantmeasureisanequilibriumstateof'ifP(';)=h()+Z'd,whereP(';)isthetopologicalpressure,andh()isthemeasuretheoreticen-tropy.TheoremB.(Ergodicpropertiesoftheinvariantmeasures.)Suppose'satis esAssumptionA(I)-(III).Then='hasthefollowingproperties:i)(;)isanexactendomorphism;ii)istheuniqueweakGibbsmeasureof'withconstantP=P(';)=log;iii)isanequilibriumstateof'.Remark1.7.Weshouldnotethatequilibriumstatesfor'isnotunique,sincetheDiracmeasure0isanobviousequilibriumstate.Formanysmoothdynamicalsystems,(1.4)iseasiertocheck.AssumptionA(III0).ThereisC�0suchthatforalln0,OnCn ( 1):ForPropositionCandTheoremD-Fbelow,wealwaysassumethateither(1)'satis esAssumptionA(I)-(III),or(2)'satis esAssumptionA(I),(II)and='satis esAssumptionA(III0).De ne (x)='(x)+logh(x) logh(x) logforx2+nf0g,whereweregardh(0)=1iflimx!0h(x)=1andthereforelogh(0) logh(0)=0.Thende neL :C0(+)!C0(+A)byL g=1hL'(hg),or,equivalentlyL g(x)=Xy2 1(x)e (y)g(y)=1h(x)Xy2 1(x)e'(y)h(y)g(y):However,L HJ6HJforanyJ�
10 0;0ingeneral.(SeeLemma4.6.i)forreasons.)
0;0ingeneral.(SeeLemma4.6.i)forreasons.)ForJ�0,wedenoteG+J=HJ\C0(+) POLYNOMIALDECAYOFCORRELATIONS7PropositionC.ThereexistsJ�0suchthatforanyk�0,wecan ndA�0suchthatforallg2G+Jwith(g)=1,jg(x) 1j1=38x2+andforallx;y2Qk,Lng(y)Lng(x)1+A(n+1) 8n0:ThispropositionisthemainsteptoestimatetheratesofconvergenceofLngandtheratesofdecayofcorrelations.ItsaysthatonQk,thedi erencebetweenLngandaconstantfunctionisoforderO(n ).DenoteG=g2C0 +:9L�0s.t.jg(y) g(x)jLd(x;y)8x2Pk;y2P0k8k0 :Clearly,ifg2G,thenforanyJ�0,therearerealnumbersc1;c2suchthatc1g+c22G+J,(c1g+c2)=1andjc1g(x)+c2 1j1=38x2+.TheoremD.(Rateofconvergence.)Letk�0begiven.Foranyg2G,thereexistsC�0suchthati)jLng(x) (g)jC(n+1) 18x2Qk,n0;ii)ZjLng(x) (g)jd(x)C(n+1) 18n0.Denotebykgk1theL1normofafunctiong.TheoremE.(Decayofcorrelations.)Foranyg2G,thereexistsC�0suchthatforany~g2L1(+;) Z ~gngd Z~gdZgd C(n+1) 1k~gk18n0:WesaythattheCentralLimitTheoremholdsforafunctiongif1pnn 1Pi=0giconvergesinlawtoanormaldistribution.TheoremF.(CentralLimitTheorem.)TheCentralLimitTheoremholdsforallfunctioninGif �2.2.Almostexpandingmaps:ApplicationsInthissectionweconsiderpiecewiseexpandingmapswithanindi erent xedpoint.DenoteI=[0;1].Letf:Im!Rmbeamap.AclosedsubsetImiscalledinvariantiff=. 8HUYIHUAssumptionB(I).fj:!istopologicallymixing.AsubsetR
11 isproperifR=intR.A nitecoverfR0;;Rr
isproperifR=intR.A nitecoverfR0;;Rr 1gofbypropersetsisaMarkovpartitionifi)intRi\intRj=;fori6=j;ii)f(intRi)isaunionofsomeRj.AssumptionB(II).fj:!hasaMarkovpartitionintosubsetsfRigr 1i=0.AssumptionB(III).fispiecewisesmooth.Moreprecisely,foreachi,fjintRiisaC2mapfromintRitoitsimage,anditcanbeC1extendedtoRi.Apiecewisesmoothmapfiscalleduniformlyexpandingonasubset,ifthereis�1suchthatforallx2,v2Rm,kDfx(v)kkvk:A xedpointpoffiscalledindi erentifDfphasaneigenvalueontheunitcircleinC.De nition2.1.Letf:Im!RmandImbeaclosedinvariantsubset.Themapfissaidtobeexpandingonwithanindi erent xedpointpifi)p2andpisanindi erent xedpointoff;ii)fisuniformlyexpandingawayfromp,thatis,foranyopenneighborhoodUofp,fisuniformlyexpandingonnU.Iffisexpandingonaninvariantsubset,withorwithoutindi erent xedpoints,andhasaMarkovpartition,thenthereisamap:+!givenby(fxig)=1Tj=0f iRxj,wherefxig2+A,suchthatf=.Assumethatfhasanindi erent xedpointp,andfissmoothinaneighborhoodofp.ThenwecanalwayschooseaMarkovpartitionfRigr 1i=0suchthatp2intR0.Hence,(0)=p.Notethattheprojection:+!inducesamap^:C0()!C0(+A)by^g=g8g2C0().WedenoteHJ()=g2C0(nfpg):^g2HJ ;G()=g2C0():^g2G :Forthesakeofnotationalsimplicity,weidentifythesetsPkwithPk,QkwithQketc.Sometimeswealsosimplywritefunctions^gasg.Assumefurtherthatnearp,thereisalocalcoordinatesystemsuchthatfx=x 1+jxj\r+(x);(2
12 :1)wherej(x)j;jxjkD(x)k;jxj2kD2(x)k=O
:1)wherej(x)j;jxjkD(x)k;jxj2kD2(x)k=O(jxj1+\r0)(2:2)forsome\r0�\rasxnearp. POLYNOMIALDECAYOFCORRELATIONS9Lemma2.1.Supposef(x)satis es(2.1)and(2.2).Thenthereis01,integersk0k00suchthatforalllargek,1(k+k0)\r1 1(k+k0)jxj\r1(k+k00)\r1+1(k+k00)8x2Pk:(2:3)Proof.WemayassumethatIsothatwecandropthenormsignjj.Weclaimthatiffxx+x1+\r+0x1+\r0forsome0�0,thenthereis01suchthatforalllargen,x\r1n\r1 1nimplies fx\r1(n 1)\r1 1(n 1):(2:4)Thisimpliesthe rstinequalityof(2.3).Infact,foranylargekwecanalways ndk0suchthatx\r1(k+k0)\r1 1(k+k0)8x2Pk+k0.Thenweuseinduction.Denote\rn=\r 1 n 1.Bythecondition, fx\rx\r 1+x\r+0x\r0\r1n\rn1+1n\rn+0(n\rn)\r0=\r\r:Toprovethelemmaweonlyneedshowthat1n\rn1+1n\rn+0(n\rn)\r0=\r\r1(n 1)\rn 1;or,equivalently,n 1n1+1n\r 1n1+\r+0(n\rn)\r0=\r\r\rn\rn 1=1 (n 1) 1 n :(2:5)Takeminf1;\r0=\r 1g.Then(n\rn) (\r0=\r)isofhigherorder.Itiseasytoseelimn!1n1+n 1n 1+1n\r 1n1+\r\r 1=limn!1n1+ \rn1+\r \r 32n2\r= 1;andlimn!1n1+1 (n 1) 1 n 1=limn!1n1 (1 n 1) 1 n = :Soweknowthatasn!1,theleftsidein(2.5)islike1 n (1+)andtherightsideislike1 n (1+).Since1,therightsideislargerforalllargen.Thesecondinequalityin(2.3)canbeprovedsimilarly.2.1Mapsontheunitinterval.LetfbeapiecewisesmoothexpandingmapfromtheunitintervalIontoitselfwithanindi erent xedpointp=0.Denotebyf0thederivativeoff. 10HUYIH
13 UTheoremG.Supposef:I!Isatis esAssumption
UTheoremG.Supposef:I!Isatis esAssumptionB(I)-(III)with=I,andisexpandingwithanindi erent xedpoint0,andnear0hasthefrom(2.1)with0\r1.Thenthefollowingholds:(a)fadmitsanabsolutelycontinuousinvariantprobabilitymeasurewhosedensityfunctionh2HJ(I)forsomeJ�0andsatis eslimsupx!0x\rh(x)Xx2f 1(0)nf0gh(x)f0(x):(2:6)(b)isaweakGibbsstateandanequilibriumstateof'(x)= logf0(x).(c)Foranyg2G(I),thereisaconstantC�0suchthatforany~g2L1(I;), Z ~gngd Z~gdZgd C(n+1) 1k~gk18n0:(2:7)(d)TheCentralLimitTheoremholdsforallg2G(I)if\r12Remark2.1.Therequirementonconditions(2.1)canbeslightlyrelaxed.Forexample,thesameproofcangothroughifweassumex+ax1+\rfxx+bx1+\rforsome0ab.Remark2.2.The rstpartof(a),concerningexistenceofabsolutelycontinuousinvariantmeasure,iswellknown(seee.g.[Pi],[T]).Forthesecondpart,asmen-tionedinRemark1.3,wecanprovethatthelimitexistsandtheequalityholds.(See[H],TheoremA).Formapswith- niteabsolutelycontinuousinvariantmeasures,similarresultswereobtainedbyP.ColletandP.Ferrero(see[CF1]and[CF2]).Remark2.3.ThesameestimatesasinPart(c)wereobtainedbyYoung[Y]andtheauthor[H]withdi erentapproaches.Liverani,SaussolandVaienti(LiSV])obtainedO n ( 1)(logn) andPollicottandYuri([PY1])obtainedO n ( 1 )8�0astheupperboundsofdecayofcorrelations.ProofofTheoremG.Take = =\r 1.Toprovethetheoremweonlyneedshowthat'satis esAssumptionA(I)-(III).Clearly,'iscontin
14 uous.By(2.1)and(2.2)wehavethatthereexist
uous.By(2.1)and(2.2)wehavethatthereexist1;2�0suchthatf0(x)1+(1+\r)x\r 1x\r0;f00(x)\r(1+\r)x 1+\r+2x 1+\r0Cx 1+\rforsomeC\r(1+\r).Letkbealargeintegerandx2Pk.Bythe rstinequalityin(2.3),wehavex\r(k+k0) 1\r 1andthereforef00(x)C (k+k0)\r (1 \r)C\r 1(k+k0) 1C'k 1(2:8) POLYNOMIALDECAYOFCORRELATIONS11forsomeC'C(2\r) 1�0ifkk0.Also,notethatf0(x)isboundedbelowby1+(1+\r)1(k+k0)\r1 1(k+k0) 1 k+k0 \r01 1(k+k0) \r0:Since \r01+,wecan ndCC0�0suchthatf0(x)1+1+\r(k+k0)\r C0(k+k0)1+1+ +1k Ck1+:(2:9)Hence,AssumptionA(II)and(III)followfromthede nitionof'andthefacts�log(1+s)s s2=2.NowweapplyTheoremA,B,EandFtogettheresults.2.2.Cantorsets.Letf:I!R+beapiecewisesmoothexpandingmapwithanindi erent xedpointp=0,andnear0,(2.1)holds.Denote=fx2I:fnx2I8n0g.Clearly,02.LettbetheHausdor dimensionof,i.e.t=dimH().WesayameasureonisameasureoffullHausdor dimensionifdimH()=dimH(),wheredimH()=inffdimH(0):0;(0)=1g.TheoremH.Supposefj:!satis esAssumptionB(I)-(III),andisexpandingwithanindi erent xedpoint0,andnear0hasthefrom(2.1)with\r 1�2t 1 1.Thenthefollowingholds:(a)fadmitsaconformalmeasureandaninvariantmeasure,bothareprobabilitymeasuresoffullHausdor dimension,suchthatwiththedensityfunctionh=d=d2HJ()forsomeJ�0andsatisfying(2.6).(b)isaweakGibbsstateandanequilibriumstateof'(x)= tlogf0(x).(c)Forany
15 g2G(),thereisaconstantC�0suchtha
g2G(),thereisaconstantC�0suchthat(2.7)holdsforany~g2L1(;).(d)TheCentralLimitTheoremholdsforallg2G()if\r 1�3t 1 1.Remark2.4.ThepartsregardingexistenceofprobabilitymeasuresoffullHausdor dimensionandequilibriumstateinthetheoremwereobtainedbyM.Urbanski([U])inasimilarsetting.Healsoprovedthat =t 1+\r 1 11isanecessaryconditionformeasuretobe nite.ProofofTheoremH.Take =\r 1and =t 1+\r 1 1.AssumptionA(I)isclear.AssumptionA(II)followsfrom(2.8)andthede ni-tionof'.ForAssumptionA(III)wecanseeby(2.9)that '(x)tlog1+1+\r 1k Ck1+=t(1+\r 1)k tCk1++O1k2 12HUYIHUforsomeC�0.Since\r 1�2t 1 1,wehave =t(1+\r 1) 1�1.SotheresultfollowsfromTheoremA,B,EandFifwecanprovethatisameasureoffullHausdor dimension.ItisclearthatP(^';)=P(';f)andh^()=h(f),where^(g)=(^g).ByTheoremAandB(ii)wehaveP(';f)=log'(0)=0:Sinceisanequilibriumstate,wehave0P(';f)=h(f)+Z'd=h(f) tZlogf0d:Thatis,th(f)Rlogf0d:Notethatisergodic,therightsideoftheequalityisequaltodimH().WegettdimH().ThisiswhatwewantsincewealwayshavetdimH().2.3.Mapsonhigherdimensionalspaces.Letfbeapiecewisesmoothexpandingsmoothmapfromthem-dimensionalcubeImontoitselfwithanindi erent xedpointp=0.Also,near0,fisoftheform(2.1)and(2.2).Inotherwords,f(x)isthemapx!x(1+jx\rj)plusahigherorderperturbation.DenotebydetDf(x)thedeterminantofDfatx.TheoremI.Supposef:Im!Imsatis esAssump
16 tionB(I)-(III)andisexpand-ingwithanindi
tionB(I)-(III)andisexpand-ingwithanindi erent xedpoint0,andnear0hasthefrom(2.1)with0\rm.Thenthefollowingholds:(a)fadmitsanabsolutelycontinuousinvariantprobabilitymeasurewhosedensityfunctionh2HJ(Im)forsomeJ�0andsatis es(2.6).(b)isaweakGibbsstateandanequilibriumstateof'(x)= logjdetDf(x)j.(c)Foranyg2G(Im),thereisaconstantC�0suchthat(2.7)holdsforany~g2L1(Im;).(d)TheCentralLimitTheoremholdsforallg2G(Im)if\rm=2.Remark2.5.PollicottandYuriobtainedadi erentexampleofalmostexpandingsystemonasubsetoftheplane,theinhomogeneousDiophantineapproximationtransformation,whichhaspolynomialdecayofcorrelations([PY1]).Remark2.6.WecanalsodiscussthecasethatisafractalinIm.Supposethatnearthe xedpointthesethastheformfcs:c2 ;s2Sg,where [0;a],0a1,andSSm 1arebothclosedsubsetwithHausdor dimensionst andtSrespectively.Take =\r 1tS+(1+\r 1)t 1.Thenwecangetresultssimilartopart(a),(b)and(c)if �1andtopart(d)if �2.ProofofTheoremI.Take =\r 1and =m\r 1. POLYNOMIALDECAYOFCORRELATIONS13Sincethemapx!x(1+jx\rj)hasthedeterminant 1+jxj\rm 1 1+(1+\r)jxj\r,wehavejdetDf(x)j= 1+jxj\rm 1 1+(1+\r)jxj\r+O(jxj\r0)=1+(m+\r)jxj\r+O(jxj\r0)ifweassume\r02\r.So'(x)= logh1+(m+\r)jxj\r+O(jxj\r0)i:Clearly,AssumptionA(I)follows.NotethatbyLemma2.1jxj\r(k+k0) 1\r 1ifx2Pk.WecanseethatthenormofthederivativeofdetDf(x)isboundedby(m+\r)jxj 1+\r+O(jxj 1+\r0)C(m+\r)\r 1(k+k0) 1C'k 1fo
17 rsomeC�1andC'(2\r) 1C(m+\r)if\r
rsomeC�1andC'(2\r) 1C(m+\r)if\r1.If\r1,thenweusejxj\r(k+k00) 1\r 1togetasimilarestimate.TheseimplyAssumptionA(II).AlsobyLemma2.1forthelowerboundofjxj,wehavethatforallx2Pk,logjdetDf(x)j=logh 1+(m+\r)jxj\r+O(jxj\r0)i(m+\r)(k+k0)\r C0(k+k0)1+\r 1m+1k Ck1+forsomeCC00.HenceAssumptionA(III)followswith =\r 1m.ByTheoremA,B,EandFwegettheresultsofthetheorem.3.TheOperatorL':ProofofTheoremAProposition3.1.ThereisJ'�0suchthatforallJJ',thefollowinghold.i)Forx2Pk,y2P0k, '(0x) '(0y) +JD k+1d(0x;0y)JD kd(x;y);andifs6=0,thenforx;y2+, '(sx) '(sy) +JK 0d(sx;sy)JK 0d(x;y):ii)Forx2Pk,y2P0k,andforw=w0w1wn 1withwx2Pm,wy2P0m, Sn'(wx) Sn'(wy) +JD md(wx;wy)JD kd(x;y);andifwn 16=0,thenforx;y2+, Sn'(wx) Sn'(wy) +JD md(wx;wy)JK 0d(x;y): 14HUYIHUProof.i)FirstweassumekK0andx2Pk,y2P0k.ByAssumptionA(II), '(0x) '(0y) +J(k+1) d(0x;0y)C'(k+1) 1d(0x;0y)+J(k+1) d(0x;0y)C'k(k+1)2+Jkk+1k d(x;y)Jk d(x;y);ifJ'C'.IfkK0ors6=0,thenwetakeJ'�0suchthatC'K 0+J'K 0J'K 0.Thereforewecangettheresultsinasimilarway.ii)Itcanbeobtainedfromi)byinduction.Recallthat^gisde nedin(1.2).Corollary3.2.LetJJ'andx=0~x;y=0~y2O2.Ifg(s~y)g(s~x)eJK 0d(s~x;s~y)8s6=0,then^g(y)^g(x)eJK 0d(x;y):Proof.ItfollowsfromProposition3.1.i)andthefactthat^g(y)^g(x)maxs6=0ne'(s~y)g(s~y)e'(s~x)g(s~x)omaxs6=0ne'(s~y) '(s~x)eJK 0d(s~x;s~y)o:ProofofTheoremA.Existenceandpropertiesofa
18 ndaregivenbyLemma3.3.FixJJ'.LetB=C0 +
ndaregivenbyLemma3.3.FixJJ'.LetB=C0 +nf0gbutwiththenormkgk=supx2+nf0g(k(x)+1) 2g(x) ;(3:1)wherek(x)=kifx2Pk.ItiseasytocheckthatBisaBanachspace.Lemma3.5belowimpliesthatL':B!Biscontinuous.TakeH=fg2HJ:(g)=1;g(x) k(x)+1H8x2Pk;kK2g:whereHandK2areaconstantstobedeterminedintheproofofLemma3.7.Hisnotemptysinceitcontainsaconstantfunctiong(x)=1.Clearly,Hisaconvexset.ByLemma3.6,Hiscompact.ByLemma3.7, 1L'HH.BytheSchauder-Tychono FixedPointTheorem(seee.g.[DS]), 1L'hasa xedpointh2H.Sowehaveh2HJ,8JJ',andL'h=h.Toprovethatis-invariant,wecancheckdirectlythatL' h(g)=g(L'h),then(g)= h(g)= 1L' h[g]= ( 1L'h)g=(hg)=(g).(Seee.g.[B]formoredetails.)Theinequalities(1.3)isgivenbyLemma3.8.Use(1.3)andLemma3.4wecangetPnC0n forsomeC0�0.SoOn1Pk=nC0k Cn ( 1)forsomeCC0.Thesegive(1.4). POLYNOMIALDECAYOFCORRELATIONS15Lemma3.3.Thereisarealnumbere'(0),andameasure2M(+),whichispositiveonnonemptyopensets,suchthatL=.Proof.Themap!L(L')(1)isacontinuousmapfromM(+)toitself.SinceM(+A)iscompactinweaktopology,bytheSchauder-Tychono FixedPointTheoremthemaphasa xedpoint.Soifwetake=(L)(1),thenL'=.ToproveU�0foranyopensetU,itisenoughtoshowthatforanywordu,Ru�0.Sinceistopologicallymixing,nRu=+forsomen�0.Henceforanyx2+A,thereisann-wordvsuchthatvx2Ru.WehaveLn Ru(x)=Xw2neSn'(wx) Ru(wx)eSn'(vx) Ru(v
19 x)e nk'k�0;SoRu=1n(L)n( Ru)
x)e nk'k�0;SoRu=1n(L)n( Ru)1n(Ln Ru)1n(e nk'k)=1ne nk'k�0:Nowweprovee'(0).Supposee'(0).Since'isacontinuousfunction,thereisk�0suchthat 1e'(x)�18x2Ok.Notethatify2Pi+1,theny2Pi.Wegetthatforik,Pi+1= 1L' Pi+1=Z 1Xy2 1xe'(y) Pi+1(y)d(x)=Z 1e'(0x) Pi(x)d(x)�Z Pi(x)d(x)=Pi:(3:2)Hence,wehavePi�Pk8i�k,contradicting nitenessof.Lemma3.4.ThereexistsC�0suchthatforallk�0,PkCk ( +1).Proof.i)Usingthesameargumentasthatin(3.2),andthenusingAssumption(III),wehavePk+1=Z 1e'(0x) Pk(x)d(x)e ( +1)=(k+1)Pk1 1k+1+12(k+1)2 +1Pk=kk+1 +11+12k(k+1) +1Pk;fork�K0.Inductively,Pk+jkk+j +1k+j 1Yi=k1+12i(i+1) +1PkCkk+j +1PkforsomeC�0sincetheproductconvergesasj!1.Itmeansthatk +1Pkisboundedabove.SoPkisoforderk ( +1). 16HUYIHULemma3.5.L'isaboundedlinearoperator.Proof.Takeg(x)=(k(x)+1)2,wherek(x)=kifx2Pk.Clearlygthemaximalelementintheunitballwithrespecttothenormin(3.1).SinceL'isapositiveoperator,weonlyneedprovethatk(x) 2L'g(x)isbounded.Notethatk(0x)=k(x)+1.Also,k(sx)=0ifs6=0.Sowehave k(x)+1 2L'g(x)= k(x)+1 2e'(0x) k(0x)+12+Xs6=0e'(sx) k(sx)+12;whichisboundedby2ek'k+rek'k,whereristhenumberofdi erentsymbolsusedin+.Lemma3.6.ThesetHiscompact.Proof.Foranyx2Pk,wehaveg(y)g(x)e JD kd(x;y)g(x)e JD k=k +18y2Pkbecauseg2HJ.Hence,1ZPkg(y)d(y)g(x)e JD k=k +
20 1Pk.Thatis,g(x)eJD k=k +1(Pk) 1.Ifx
1Pk.Thatis,g(x)eJD k=k +1(Pk) 1.Ifx2OK2,then k(x)+1 2g(x) k(x)+1 1HH.SoHisboundedabove.SinceHisequicontinuousoutsideOkforanylargek,itisequicontinuous.NowweonlyneedshowthatHisaclosedsubsetinB.ItisenoughtoprovethatforanyconvergentsequencefgfigginH,Zlimi!1gfigd=1.ThisistruebytheLebesgueDominatedConvergenceTheorem.Infact,wecantakethedominantfunctionG(x)=eJD k=k +1(Pk) 1asx2Pk,kK2andG(x)= k(x)+1Hasx2OK2.Clearly,gGforallg2H,andGisintegrablebecauseZOK1Gd=1Xk=K1(k+1)HPk1Xk=K1CHk+1k +11:Lemma3.7. 1L'HH.Proof.Takeg2H.Weprove 1L'g2H.Itisclearthat 1L'g�0and( 1L'g)= 1L(g)=(g)=1.Takex2Pk,y2P0k.Then 1Lg(y) 1Lg(x)=Psy2+e'(sy)g(sy)Psy2+e'(sx)g(sx)maxsx;sy2+Ane'(sy)g(sy)e'(sx)g(sx)o:Notethatg(sy)g(sx)eJK 0d(sx;sy)ifs6=0andg(0y)g(0x)eJD k+1d(0x;0y).SobyPropo-sition3.1,therightsideofaboveinequalityislessthanorequaltoeJD kd(x;y). POLYNOMIALDECAYOFCORRELATIONS17Itremainstoshowthatforx2Pk,kK1, 1L'g(x)(k+1)H.Since �1,wecantakeK2maxfK0;K1glargeenoughsuchthatforallkK2,k+2k=1+2ke( +1)=(k+1) C=(k+1)1+:Hence,byAssumptionA(III)andLemma3.3, 1e'(0x)= 1e'(0)e'(0x) '(0)e ( +1)=(k+2)+C=(k+1)1+kk+2:Sinceg(0x)(k+2)H,wehave1(k+1)L'g(x)=1k+1 1e'(0x)g(0x)+1(k+1)Xs6=0e'(sx)g(sx)1k+1kk+2(k+2)H+1(k+1)Xs6=0e'(sx)g(sx)kk+1H+1(k+1)Xs6=0e'(sx)g(sx):Notethatg(sx)fors6=0isuniformlybounded.IfHislargeenough,thenthesecondtermisless
21 thanorequaltoH=(k+1).Hence, 1L'g(x)(k+
thanorequaltoH=(k+1).Hence, 1L'g(x)(k+1)H.Lemma3.8.ThereisCh�0suchthatforallx2P0,foralllargen, nh(0nx)^h(0)1+Chn:Proof.Recallthatweassumeminf1; gafterAssumptionA(III)isstated.TakeCh�0suchthatforalllargen,( 1)n 2(n+1)2Chn (1+ )2(n+1)2 n+ +2(n+1)22 J0C n C(n 2 )(n+1)2+�0;whereJ0=J'K 0.Firstweclaimthatifthereisx2P0suchthatforChn , nh(0nx)^h(1+(3:3)holdsforsomelargen,then n+1h(0n+1x)^h(0)1++1n+1:(3:4) 18HUYIHUByLemma3.3andAssumptionA(III),e '(0n+1x)=e '(0)e'(0) '(0n+1x)1+ +1n+1 C(n+1)1+:(3:5)SinceL'h=h,wehavee'(0n+1x)h(0n+1x)+^h(0n+1x)=h(0nx),where^hisde nedin(1.2).Thatis,h(0n+1x)=e '(0n+1x) h(0nx) ^h(0n+1x):(3:6)Notethath2HJ'.ByCorollary3.2and(0.1),ifnislargeenough,then^h(0n+1x)^h(0)eJ0d(0nx;0)^h(0)eJ0C =n ^h(0)(1+2J0C =n ):(3:7)Soby(3.3),ifnislargeenough,then1^h(h(0nx) ^h(0n+1x)n 1+ 1+2J0C n =n +n 2J0C n n 2 :(3:8)By(3.6)andthenby(3.5)and(3.8),wehave n+1h(0n+1x)^h( n+1e '(0n+1x)1^h( h(0nx) ^h(0n+1x) n+1n+ +2n+1hn +n 2J0C n i n+1C(n+1)1+n 2 =(n+ +2)(n )(n+1)2+(n+ +2)n(n+1)2 n+ +2(n+1)22 J0C n C(n 2 )(n+1)2+Notethat(n+ +2)(n )(n+1)2=1 (1+ )2(n+1)2;(n+ +2)n(n+1)2=+1n+1+( 1)n 2(n+1)2;AlsonotethatChn .BythechoiceofCh,weget(3.4).Theclaimistrue.Usingthisclaimwecangettheresultofthelemma.Otherwisewehavean�Chn 0suchthat(3.3)holdsforsomelargen0.
22 Thenusingtheclaimrepeatedly,weget n0+kh(
Thenusingtheclaimrepeatedly,weget n0+kh(0n0+kx)^h(0)1+n0+k 1Yi=n0 1+1i+18k0:Sincethesecondtermgoestoin nityask!1,itcontradictsthefactthath(0nx)(n+1)Hforalln�0. POLYNOMIALDECAYOFCORRELATIONS194.TheOperatorL :SomelemmasLemma4.1.i)L c=cforanyconstantfunctionc.ii)(L g)=(g)foranyintegrablefunctiong.iii)(jL gj)(jgj)foranyfunctionginL1(+;).Proof.i)SinceL'h=h,wegetL c=1hL'(ch)=chL'h=c.ii)Thisisbecause(L g)= h1hL'(hg)= 1L'(hg)= hg=(g).iii)Itiseasytocheckthatforanyx,jL g(x)jL jg(x)j.Hence,byPartii),(jL gj)(L jgj)=(jgj).Lemma4.2.Letwbeann-word,n�0.Thenforanywordu,ZRueSn (wx)d(x)=Rwu:Proof.Notethatn:Rwu!Ruisone-to-one,and Rwu(wx)= Ru(x).Hence Ln Rwu(x)=eSn (wx) Rwu(wx)=eSn (wx) Ru(x).ByLemma4.1.ii),ZRueSn (wx)d(x)=ZeSn (wx) Ru(x)d(x)= Ln Rwu=RwuLemma4.3.i)e (0x)=1 1h(x)^h(0x)asx2+.ii)9B�0suchthatforalln,e (0x)1 n+Bn1+,asx2Pn.iii)9C �0suchthatforallk;n,eSn (0nx)C kk+n asx2Pk,and(k+n) Pk+nC k Pk.Proof.i)SinceL h(x)=h(x),wehavee'(0x)h(0x)+^h(0x)=h(x).Sobythede nitionof ,e (0x)=e'(0x)h(0x)h(x)=1 1h(x)^h(0x).ii)ByLemma3.8andasimilarmethodfor(3.7),wehavethatforalllargen,1h(x)^h(0x) n^h(0)1+Chn 1^h(0)1 2C J'n n Bn1+;ifBislargeenough.NowtheresultfollowsfromParti).iii)ByPartii),foralllargek,ifx2Pk,thene (0x)1 k+Bk1+=k k1+Bk(k )1 1k+1 1+Bk(k ): 20HUYIHUTakin
23 gproduct,wegeteSn (0nx)kk+n k+n 1Yi=
gproduct,wegeteSn (0nx)kk+n k+n 1Yi=k1+Bi(i )kk+n 1Yi=k1+Bi(i ):Sincetheproductconvergence,itgivesthe rstinequality.ByLemma4.2,Pn+k=RPkeSn (0nx)d.Thenthesecondinequalityfollowsfromthe rstone.Forx=0~x2O1,denoteg(x)=Ps6=0e (s~x)g(s~x)Ps6=0e (s~x)=Ps6=0e'(s~x)h(s~x)g(s~x)Ps6=0e'(s~x)h(s~x):(4:1)Thatis,g(x)istheaverageofg(s~x),s6=0,withweightse'(s~x).ByLemma4.1.i),Ps6=0e (sx)=1 e (0x).SowehaveL g(x)=e (0x)g(0x)+(1 e (0x))g(0x):(4:2)Lemma4.4.Foranyx2O1,Lng(x)=g(0nx)eSn (0nx)+nXj=1gn j(0jx) 1 e (0jx)eSj 1 (0j 1x):Proof.Itcanbeobtainedby(4.2)andinduction.Lemma4.5.LetkK0.Foralln1,eSn (0ny)eSn (0nx)e2J'k d(x;y)8x;y2Qk:Proof.Notethatbythede nitionof ,Sn (0ny) Sn (0nx)Sn'(0ny) Sn'(0nx)+logh(0ny)h(0nx)+logh(x)h(y):Firstweconsiderthecasethatx2Pjandy2P0j,0jk.Sinceh2HJ',logh(0ny)h(0nx)J'D j+nd(0nx;0ny),andlogh(x)h(y)J'D jd(x;y).ByProposition3.1.ii),wehave Sn (0ny) Sn (0nx) 2J'D jd(x;y).SoeSn (0ny)eSn (0nx)e2J'D jd(x;y)eSn (0nx)e2J'k d(x;y):Ingeneral,weassumex2Pjandy2Pl,0jlk.Wecantakex(i)2Pi,jilsuchthatx(j)=xandx(l)=y.Thenaboveresultsholdforanypairx(i)andx(i+1).Nowweusethefactthatd(x;y)=l 1Pi=jd(x(i);x(i+1)). POLYNOMIALDECAYOFCORRELATIONS21Lemma4.6.LetJ0.i)Ifgsatis esg(y)g(x)eJ(j+m) d(x;y)8x;y2Qj+mwithmK0,,thenLjg(y)Ljg(x)e(2J'+Jmj+m)m d(x;y)8x;y2Qm:ii)Ifgsatis esg(sy)g(sx)eJd(sx;sy)forallsymbolss6=0,theng(0y)g(0x)e(2
24 J'K 0+J)d(x;y)8x;y2+:Proof.i)Notethatf
J'K 0+J)d(x;y)8x;y2+:Proof.i)Notethatforallj-wordw,d(wx;wy) mj+m +1d(x;y).So2J'm d(x;y)+J(j+m) d(wx;wy)2J'+Jmj+mm d(x;y):ByLemma4.5,theratioLjg(y)=Ljg(x)isboundedbymaxw2jneSj (wy)g(wy)eSj (wx)g(wx)omaxw2jne2J'm d(x;y)+J(j+m) d(wx;wy)oe(2J'+Jmj+m)m d(x;y);ii)Clearly,e'(sy)h(sy)e'(sx)h(sx)eJ'K 0d(sx;sy).Also,2J'K 0d(sx;sy)+Jd(sx;sy)(2J'K 0+J)d(x;y):By(4.1)andthesamemethodasabove,wegettheresult.Lemma4.7.Letg2G+J.Thenforanyx2+nf0g,limn!1Lng(x)=(g);andtheconvergenceisuniformonanyclosedsubsetof+nf0g.Proof.WemayassumeJ3J'.ItiseasytoseebyLemma4.6.i)that8n2k,Lng(y)Lng(x)eJk d(x;y)8x;y2Qk:(4:3)Hence,restrictedtoQk,fLng(x)gareuniformlyboundedandequicontinuous.Sothereisasubsequencefnigsuchthatlimi!1Lni g(x)=g(x)forallx2Qkandthereforeforallx2+nf0g.Nowweprovethatgisaconstantfunction. 22HUYIHUForthispurpose,itisenoughtoshowthat g (g) =0.Notethat Lng (g) isdecreasingbyLemma4.1.iii).Soc=limn!1 Lng (g) exists.Sincegiscontinuousandmaxx2+ L g maxx2+ g ,weobtainthatc= Ljg (g) foralllimitpointsgofLng andforallj0.However,ifc6=0,thenthereisx2+nf0g,andl-wordsuandvsuchthatg(ux)�(g)andg(vx)(g).Bythede nitionofL andLemma4.1.i),wehave Llg (g) g (g) ;acontradiction.Sinceonanyclosedsubsetin+nf0g,Lngsatisfy(4.3)ifnislargeenough,andthereforeareequicontinuous.Henceuniformityfollowsfromthefactthatlimn!1 L
25 ng (g) =0.Lemma4.8.Letubeanyword.Foran
ng (g) =0.Lemma4.8.Letubeanyword.Foranyx2+nf0gliml!1Xw2l;RwRueSl (wx)=Ru;andthatconvergenceisuniformonanyclosedsubsetof+nf0g.Proof.Firstweconsiderthecasethatuisanm-wordwithu6=0m.Thereisn�0suchthatn(Ru)=+becauseistopologicallymixing.Notethatforanyn-wordw, Ru(wx)1or0,dependingwhetherRwRuornot.ByLemma4.5,Ln Ru2G+2J'.SincePw2l;RwRueSl (wx)=Ll Ru(x),wegetLl Ru(x)! Ru=Ruuniformlyonanyclosedsubsetof+nf0gbyLemma4.7.Forthecaseu=0m,wenotethatbyLemma4.1.i),Xu2mXw2l;RwRueSl (wx)=Xw2leSl (wx)=1;andthenusethefactsthatthereareonly nitenumberofm-words,andtheconvergenceisuniformforallotherm-words.5.WeakGibbsandEquilibriumStates:ProofofTheoremBProofofTheoremB.ProofofParti)isthesameasthatin[W1],Corollary12.ByLemma5.1and5.2,wehavep(x;n)=B'maxfh(x);h(x) 1gifxn 16=0andp(x;n)=q(k+n)maxfh(x);h(x) 1gotherwise.Sincen 1x2Rxn 1,p(x;n)satis esb)inDe nition1.1.Notethatkinq(k+n)isafunctionde nedbyk(nx)=kifnx2Pk.SincePktendto0inapolynomialrateask!1,logk(x)isintegrable.Hencelimn!11nlogk(nx)=0for-a.e.x2+.Thisprovesthatp(x;n)satis esa).Hence,isaweakGibbsmeasure.UniquenessfollowsfromLemma5.4,andP=log=P(';)isprovedinLemma5.3.ByLemma5.3,isanequilibriumstate. POLYNOMIALDECAYOFCORRELATIONS23Lemma5.1.ThereexistsaconstantB'�0suchthatforanyn-wordw=w0w1wn 1withwn 16=0,B 1'h(wx)Rwexpf nlog+Sn'(wx)gB'h(wx):Proof.ByLemma4.2andthede nitionof ,Rw=Z
26 eSn (wz)d(z)=1nZeSn'(wz)h(wz)d(z):(5:
eSn (wz)d(z)=1nZeSn'(wz)h(wz)d(z):(5:1)Notethate J'C K 0eSn'(wz)h(wz)=eSn'(wx)h(wx)eJ'C K 0byProposition3.1and(1.1).WehaveRw1nZeJ'K 0eSn'(wx)h(wx)d(z)1neJ'C K 0eSn'(wx)h(wx):Ontheotherhand,ifwx2+,thenRw1nZRx0e J'K 0eSn'(wx)h(wx)d(z)Rx01ne J'C K 0eSn'(wx)h(wx):SowecantakeB'eJ'K 0minf(Ru) 1:u=0;1;;r 1g.Lemma5.2.Thereexistsapolynomialqofonevariablewithpositivecoecientssuchthatforx2Pk,n-wordw,h(wx)q(k+n)Rwexpf nlog+Sn'(wx)gq(k+n)h(wx):Proof.WeonlyproveforkK0.ThecasekK0canbediscussedsimilarly.First,weconsiderthecasew=0n.Assumez2Pj.Itfollowswz2Pj+n.Also,wx2Pk+n.Ifjk,thenwetakex(i)2Pifori=j+n;;k+nsuchthatx(j+n)=wzandx(k+n)=wx.ThereforeeSn'(wz)h(wz)eSn'(wx)h(wx)k+n 1Yi=j+neJ'D id(x(i);x(i+1))=eJ'k+n 1Pi=j+nD ii +1C1(k+n)J'(5:2)forsomeC1�0.SinceQk1,wegetZQkeSn'(wz)h(wz)d(z)C1(k+n)J'eSn'(wx)h(wx):(5:3)Ifj�k,thenbyLemma4.2andthede nitionof ,1nZOkeSn'(0nz)h(0nz)d(z)=ZOkeSn (0nz)d(z)=Ok+n:(5:4) 24HUYIHUItiseasytoseebyLemma4.3.iii)thatOk+n=1Xi=k+nPiC 1Xi=k+nk+ni Pk+nC2(k+n)Pk+n:(5:5)forsomeC2�0independentofnandk.Also,byLemma4.2andProposition3.1,Pk+nZPkeSn (wz)d(z)= nZPkeSn'(wz)h(wz)d(z) nCeJ'=keSn'(wx)h(wx)Pk nC3eSn'(wx)h(wx):(5:6)forsomeC;C3�0independentofnandk.Notethat+=Qk[Ok+1.By(5.1),(5.3),and(5.4)-(5.6),Rw nZQkeSn'(uz)h(uz)d(z)+ nZOkeSn'(uz)h(uz)d(z) C1(k+n)J'+C2C3(k+
27 n) neSn'(wx)h(wx):(5:7)Byasimilarmetho
n) neSn'(wx)h(wx):(5:7)Byasimilarmethodfor(5.3),wealsohaveRwZQkeSn'(wz)h(wz)d(z)Q0C 11(k+n) J' neSn'(wx)h(wx):WegetupperandlowerboundforRwasw=0n.Forthecasew6=0n,wemayassumew=wn10n2,wheren=n1+n2andthelastsymbolofwn1isnotequalto0.NotethatSn'(wz)=Sn1'(wz)Sn2'(0n2z):ByProposition3.1,eSn1'(wz)h(wz)h(0n2z)isboundedbye2J'K 0d(0n2x;0n2z)eSn1'(wx)h(wx)h(0n2x)e2J' K 0eSn1'(wx)h(wx)h(0n2x):Byasimilarmethodasthatforthesecondinequalityin(5.7),wecangetZeSn2'(0n2z)h(0n2z)d(z) C1(k+n2)J'+C2C3(k+n2)eSn2'(0n2x)h(0n2x):Henceby(5.1),Rwisboundedaboveby ne2J' K 0eSn1'(wx)h(wx)h(0n2x) C1(k+n2)J'+C2C3(k+n2)eSn2'(0n2x)h(0n2x)e2J' K 0 C1(k+n2)J'+C2C3(k+n2) neSn'(wx)h(wx):Similarly,wecangetthatRwisboundedbelowbye 2J' K 0Q0C 11(k+n) J' neSn'(wx)h(wx):Sowecantakeq(k+n)=e2J' K 0maxC1(k+n)J'+C2C3(k+n);C1(k+n)J'=Q0 : POLYNOMIALDECAYOFCORRELATIONS25Lemma5.3.AweakGibbsmeasureofacontinuousfunction'isanequilibriumstateofthefunction,andtheconstantPinthede nitionofweakGibbsmeasureisequaltothetopologicalpressureof'.Proof.Notethatlimn!11nlogp(x;n)=0for-a.e.x2+.Sinceisergodic,byShannon-McMillan-BreimanTheorem,(1.5)impliesthatfor-a.e.x2+A,h()= limn!11nlogRx0xn 1=P limn!11nSn'(x):Integratingwithrespecttox,andusingBirkho ErgodicTheorem,wegeth()=P Z'd(x):Sobyvariationprinciple,wehaveP(';)h()+R'd(x)=P.ToproveP=P(';),weonlyneedshowthatforany�0
28 ,P+P(';).By[W2]Theorem9.6,P(';)=lims![,P+P(';).By[W2]Theorem9.6,P(';)=lims](851944/p-p-by-w2-theorem9-6-p-limsupn-11nlogxw2.jpg ",P+P(';).By[W2]Theorem9.6,P(';)=lims")
,P+P(';).By[W2]Theorem9.6,P(';)=limsupn!11nlogXw2ninfwx2RweSn'(wx):(5:8)Foreachn-wordw,take~xwith~x06=0andw~x2Rw.Hencew~x2Qn+1.ByDe nition1.1.b),p(w~x;n+1)e(n+1)ifnislargeenough.Soby(1.5),infwx2RweSn'(wx)eSn+1'(w~x)ek'ke(n+1)e(n+1)PRw~x0ek'k:SinceRw~x0RwandPw2nRw=1,wegetP(';)P+by(5.8).Lemma5.4.UnderthesuppositioninLemma5.3,anyweakGibbsstate0of'coincideswith.Proof.Assume0satis es(1.5)withconstantP0andfunctionp0(x;n).ByLemma5.3,weknowthatP0=P(;')=P.Letw=w0wn 1beann-wordwithRwQkforsomekn.Ifwn 16=0,thenbyDe nition1.1.b),p0(wx;n)B0kforsomeB0k�0.Soby(1.5),0RwB0ke nP+Sn'(wx)B0kBkRw:(5:9)Ifwn 1=0,thenwecanalways ndasequenceofwordsfu(i)g1=1whoselastsymbolsarenonzerosuchthatRw=fw0g[1Si=1Rwu(i).Notethatf0g=0impliesfw0g=0.By(5.9)weknow0Rw=1Xi=10Rwu(i)B0kBk1Xi=1Rwu(i)=B0kBkRw: 26HUYIHUSincethisistrueforallcylindersinQk,bytakinglimitweknowthat0(E)B0kB 1k(E)foranyBorelsetEQk.Itimpliesthat0isabsolutelycontinuouswithrespecttoonQkandthereforeon+.BytheRadon-NykodymTheoremwehave0=forsomeintegrablefunction.Sinceboth0andare-invariant,isa-invariantfunction.Sinceismixing,andthereforeisergodic,isconstant-almosteverywhere.Hence=1-a.e.and0=.6.Hilbertmetrics:ProveofPropositionCDenotebyC0(S)thesetofcontinuousfunctionsonametricspaceS.ForanyJ�0,de neaconeCJ(S)=ng2C0(S):g(x)�08x2S;g(y)g(x)eJd(x;y)8x;y2S
29 o:Thenwetakeaprojectivemetric(Hilbertmet
o:Thenwetakeaprojectivemetric(Hilbertmetric)oftwofunctionsg;~g2CJ(S)by(g;~g)=loga(g;~g)b(g;~g);wherea(g;~g)=inffa:a~g g2CJ(S)g;b(g;~g)=supfb:g b~g2CJ(S)g:Inparticular,if~g1,thenwedenote(g)=(g;1)=log(ag=bg),whereag=infna:a g(y)a g(x)eJd(x;y)o;bg=supnb:g(y) bg(x) beJd(x;y)o:(6:1)Notethata g(y)a g(x)eJ0d(x;y)ifandonlyifaeJd(x;y)g(x) g(y)eJd(x;y) 1.Wealsohaveag=supx;y2SeJd(x;y)g(x) g(y)eJd(x;y) 1;bg=infx;y2SeJd(x;y)g(x) g(y)eJd(x;y) 1:(6:2)Lemma6.1.Letg2CJ(S)begiven.SupposeS1;S2Swiththepropertiesthat8x2S1,y2S2,9z2S1\S2suchthatd(x;y)=d(x;z)+d(y;z).Then(gjS1[S2)(gjS1)+(gjS2).Proof.Let(gjSj)=logajb 1j,j=1;2.Takea=maxfa1;a2gandb=minfb1;b2g.Clearly,ifx;y2S1orS2,thena g(y) a g(x)eJd(x;y)andg(y) b g(x) beJd(x;y).Ifx2S1andy2S2,thenwetakez2S1\S2withd(x;y)=d(x;z)+d(y;z)andthereforea g(y)a g(x)=a g(y)a g(z)a g(z)a g(x);g(y) bg(x) b=g(y) bg(z) bg(z) bg(x) beJd(x;y):Sowehave(gjS1[S2)logab 1.Nowweusethefactaba1b1a2b2. POLYNOMIALDECAYOFCORRELATIONS27Lemma6.2.i)Forx;y2S,g(y)g(x)1+e(gjS) 1 1 e Jd(x;y)1+ e(gjS) 1Jd(x;y):ii)Ifg(y)g(x)eJ0d(x;y)forallx;y2S,then(gjS)logsupx;y2Sg(y)g(x)+logJ+J0J J0J0diamS+2J0J J0:Proof.i)By(6.2),8x;y2S,eJd(x;y)g(y) g(x)a eJd(x;y) 1:Sinceg(x)�b,eJd(x;y)g(y)g(x) 1ab eJ0d(x;y) 1;oreJd(x;y)g(y)g(x) 1ab 1 eJd(x;y) 1e(gjS) 1 eJd(x;y) 1:ii)Usingtheassumptionandthefactthattheexponentialfunctionisco
30 nvex,wehaveeJd(x;y)g(x) g(y)eJd(x;y) 1g
nvex,wehaveeJd(x;y)g(x) g(y)eJd(x;y) 1g(x)eJd(x;y) e J0d(x;y)eJd(x;y) 1g(x)J+J0JandeJd(x;y)g(x) g(y)eJd(x;y) 1g(x)eJd(x;y) eJ0d(x;y)eJd(x;y) 1g(x)J J0J:Hence(gjS)=logagbglogsupx;y2Sg(y)g(x)+logJ+J0J J0:Thisistheresult.Lemma6.3.LetA:C0(S1)!C0(S2)beapositivelinearoperatorthatpreservesconstantfunctions(i.e.Ag�0wheneverg�0andAc=c).Suppose(g)=logagbgforsomeg2C0(S1),whereagandbgarede nedasin(6.1).Thene(Ag) 1 tanh2 e(g) 1; 28HUYIHUwhere=maxf(ag Ag);(Ag bg)g.Proof.Denote(ag Ag)=logaag Agbag Ag.By(6.1),aag Ag=infna:a (ag Ag(y))a (ag Ag(x))eJd(x;y)o=ag supnag a:Ag(y) (ag a)Ag(x) (ag a)eJd(x;y)o=ag bAg:Similarly,bag Ag=ag aAg;aAg bg=aAg bg;bAg bg=bAg bg:Bythede nitionof(ag Ag)and(Ag bg),wehaveag bAgag aAg=aag Agbag Age(ag Ag)e;aAg bgbAg bg=aAg bgbAg bge(Ag bg)e:SoaAg bAge 1e+1 ag bg:SincebAg bg=bAg bg0,wegetaAgbAg 1e 1e+1 agbg 1= tanh2 agbg 1:Lemma6.4.Let0c1anda�0.i)Ifg(y)g(x)1+a,then1+c(g(y) 1)1+c(g(x) 1)1+acg(x)1+c(g(x) 1).ii)If1+c(g(y) 1)1+c g(x) 11+a,theng(y)g(x)1+a1+c(g(x) 1)cg(x).Proof.Usethefactsthatg(y)g(x)1+aisequivalenttog(y) g(x)ag(x)andthat 1+c(g(y) 1) 1+c(g(x) 1)=c g(y) g(x).ProofofPropositionC.RecallthatC isde nedin(0.1)andJ'isgivenbyProposition3.1.WealsoassumeC J'20.Takek�0suchthat(6.11)and(6.12)aresatis edandsuchthatkmax14K0;20J'; 1425C2 J2'1= K0 :(6:3)LetJ=3J'k .
31 POLYNOMIALDECAYOFCORRELATIONS29Takem&#x
POLYNOMIALDECAYOFCORRELATIONS29Takem�0suchthateC J(eJC =m 1)19:(6:4)Take �0suchthat eC J+119:(6:5)Takelmaxf14k;mgsuchthatforallm-wordu, Xw2l;RwRueSl (wx) Ru Ru:(6:6)ThisispossiblebecauseofLemma4.8.TakeA0suchthatC J'K 0A3C (k+l) + :(6:7)TakeN0suchthat(N+1) log(25=24)AandsuchthatforallnN,25261(n+1) 1(n+l+1) (6:8)and(7.6)inthenextsectionaresatis ed.The rstrequirementimpliese 125248A(N+1) :(6:9)Denotegn=Lng,andQk;l=Pk[[Pk+l.Wealsoassumethefollowing:Foralln=0;1;;N+l 1,(An)(gnjQk+l)A(n+1) ,(Bn)(gnjQk;l)2:5C J'K 0k A(n+1) ,(Cn)(gnjQk;l)25C2 J2'K 0k A(n+1) ,(Dn)(gnjQk)1 25C2 J2'K 0k A(n+1) ;andforalln=0;1;;N 1,(B0n)gn(y)gn(x)1+8C J2'K 0A(n+1) 8x;y2Qk;l.ThisispossiblebecausebyLemma6.4.i)andLemma6.4.i)wecanalwaysreplacegby1+c(g 1)forsomesmallcindependentofg.WeprovethepropositionbyshowingthatforallnN+l,(An),(Bn),(Cn)and(Dn)hold.ByLemma6.2.i),(An)implytheresultsoftheproposition.ByLemma6.1andthefactQk\Qk;l=Pk,(Cn)and(Dn)imply(An)foreveryn0.Sobyinduction,weonlyneedprovethefollowingclaimstocompletetheproof. 30HUYIHUClaim1.ForallnN,(An)imply(Bn+k).Claim2.ForallnN+l,(B0i)and(Bj)imply(Cn),wherei=0;1;;N 1andj=N;;n 1,.Claim3.ForallnN,(An)imply(Dn+l).ProofofClaim1.Suppose(gnjQk+l)=loganbnA(n+1) .Wehavean gn(y)an gn(x);gn(y) bngn(x) bneJd(x;y)=e3J'k d(x;y)8x;y2Qk:UsingLemma4.6.i)withJ=3J',m=K0andj=
32 k14K0,wegetan gn+k(y)an gn+k(x);gn+k(y)
k14K0,wegetan gn+k(y)an gn+k(x);gn+k(y) bngn+k(x) bne2:2J'K 0d(x;y)8x;y2Q0:(6:10)ThenweuseLemma4.6.ii)withJ=2:2J'K 0togetan gn+k(y)an gn+k(x);gn+k(y) bngn+k(x) bne4:2J'K 0d(x;y)8x;y2O1:NotethatJ=3J'k anddiam(Qk;l)C k .Wecanchooseksuchthat8:43J'k 4:2J'K 06C 10k :(6:11)ThisispossiblesincewehaveassumedC J'20.ByLemma6.2.ii),(an gn+kjQk;l);(gn+kjQk;l bn)4:8C J'K 0k :By(6.10)weknowthatan gn+k;gn+k bn2CJ(P0).Hence(gn+kjP0)loganbnA(n+1) .ClearlyLemma6.3canbeappliedinthiscase.Usingthefacttanhand(6.9),weget(gn+kjQk;l)124:8C J'K 0k 2524(gn+kjP0)2:5C J'K 0k A(n+1) :ProofofClaim2.ByLemma6.2.i)and(6.9),forallNjn 1,(Bj)implygj(y)gj(x)1+8C J'K 0A3k (j+1) Jd(x;y)=1+8C J2'K 0A(j+1) d(x;y)8x;y2Qk;l: POLYNOMIALDECAYOFCORRELATIONS31Hence,theinequalityistrueforallj=0;1;;n 1becauseoftheassumption(B0i),i=0;1;;N 1.NowweuseProposition7.1,andnotethat8C J2'K 0AiscorrespondingtoAinthisproposition,andthengetgn(y)gn(x)1+24C J2'K 0A(n+1) d(x;y)8x;y2Qk;l:NotethatA(n+1) log(25=24)1andC J'20.Wecanchooseksuchthat48C J2'3J'k 24C J2'C2 J2'k :(6:12)ThenbyLemma6.2.ii)weget(Cn).ProofofClaim3.Assume(gnjQl+k)=loga=b.Thena gn;gn b2CJ(Ql+k).Soa gn(y)a gn(x);gn(y) bgn(x) beJd(x;y)=e3J'k d(x;y)8x;y2Ql+kByProposition7.4,a gn+l(y)a gn+l(x);gn+l(y) bgn+l(x) b28x;y2Qk:Also,byusingLemma4.6.i)withJ=3J',m=kandj=l,andthenusingthefactl14k,wehavea gn+l(y)a gn
33 +l(x);gn+l(y) bgn+l(x) be2:2J'k d(x;y)8
+l(x);gn+l(y) bgn+l(x) be2:2J'k d(x;y)8x;y2Qk:Hence,byLemma6.2.ii),(a gn+ljQk);(gn+ljQk b)log2+log3J'k +2:2J'k 3J'k 2:2J'k =log13:ThenbyLemma6.3,e(gn+ljQk) 1 tanhlog132 e(gnjQk+l) 1=67 e(gnjQk+l) 1:Hence,by(6.9),(6.8),and(6.3),(gn+ljQk)e(gn+ljQk) 167 e(gnjQk+l) 12528(gnjQk+l)2528A(n+1) 2628A(n+l+1) A(n+l+1) 1 25C2 J2'K 0k : 32HUYIHU7.EstimatesofgnonQk;landQk:TwosupplementarypropositionsProposition7.1.Supposeg2G+Jwith(g)=1andjg(x) 1j1=38x2+,andgj(y)gj(x)1+A(j+1) d(x;y)8x;y2O1(7:1)forall0jn 1,nN.Thengn(y)gn(x)1+3A(n+1) d(x;y)8x;y2Qk;l:Proof.Sincegn(x)2=3,itisenoughtoprovethatgn(y) gn(x)2A(n+1) d(x;y)8x;y2Qk;l:ByLemma4.4,wehavethatforx;y2Qk;l,gn(y) gn(x)=eSn (0ny)g(0ny) eSn (0nx)g(0nx)+nXj=1gn j(0jy) gn j(0jx) 1 e (0jx)eSj 1 (0j 1x)+nXj=1gn j(0jy)h 1 e (0jy)eSj 1 (0j 1y) 1 e (0jx)eSj 1 (0j 1x)i=(I)+(II)+(III):(7:2)If(III)0.ThereforebyLemma7.2and7.3,gn(y) gn(x)16+32A(n+1) d(x;y)=5A3(n+1) d(x;y):Nowweconsiderthecase(III)0.Wereplacegby2 gandthengetanequalitysimilarto(7.2).ItisclearthatestimatesinLemma7.2and7.3stillholdfor2 gbecause 2 g(y) 2 g(x)=g(x) g(y).Sotheupperboundsfor(I)and(II)remainthesame.Now(III)becomesnXj=1 2 gn j(0jy)h 1 e (0jy)eSj 1 (0j 1y) 1 e (0jx)eSj 1 (0j 1x)inXj=12h eSj 1 (0j 1y) eSj (0jy) eSj 1 (0j 1x) eSj (0jx)i2 eSn (0nx) eSn (0ny)2A6(n+k) d(x;y);whereweuseLemma7.2withg=1inthe
34 laststep.Hence, 2 gn(y) 2 gn(x)16+3
laststep.Hence, 2 gn(y) 2 gn(x)16+32+26A(n+1) d(x;y)=2A(n+1) d(x;y):Thisistheresultsincetheleftsideisequaltogn(x) gn(y). POLYNOMIALDECAYOFCORRELATIONS33Lemma7.2.UndertheconditionsofProposition7.1,forallx;y2Qk;l,eSn (0ny)g(0ny) eSn (0nx)g(0nx)A6(n+1) d(x;y):Proof.Firstweconsiderthecasethatx2Pi,y2P0iforsomekik+l.By(6.3),thechoiceofkimpliesthat2J'i d(x;y)2J'=i2J'=klog(3=2).SousingLemma4.5.i)withJ=J',m=iandj=n,wegeteSn (0ny)g(0ny)eSn (0nx)g(0nx)e2J'i d(x;y)1+3J'i d(x;y):ByLemma4.3.iii),eSn (0nx)g(0nx)C in+i 434C 3k+ln+k+l :Recall(6.7)andthefactthatwehavereplaced8C'J2'K 0AbyA.ItfollowsthateSn (0ny)g(0ny) eSn (0nx)g(0nx)eSn (0nx)g(0nx)3J'i d(x;y)4C 3k+ln+k+l 3J'i d(x;y)A6(n+1) d(x;y):Ingeneral,forx;y2Qk;l,wetakex(0);x(1);;x(j)suchthatx(0)=x,x(j)=y,andx(i)2Pi+i0forsomei0.Foreachpairx(i),x(i+1),weuseaboveresult.Sinced(x;y)=j 1Pi=0d(x(i);x(i+1))andeSn (0ny)g(0ny) eSn (0nx)g(0nx)=j 1Pi=0eSn (0nx(i+1))g(0nx(i+1)) eSn (0nx(i))g(0nx(i)),wegettheresult.Lemma7.3.UndertheconditionsofProposition7.1,forallx;y2Qk;l,nXj=1gn j(0jy) gn j(0jx) 1 e (0jx)eSj 1 (0j 1x)3Ad(x;y)2(n+1) :(7:3)Proof.Notethatd(0jx;0jy)k+lj+k+l +1d(x;y)foranyx;y2Qk;l.By(7.1),gn j(0j 1y) gn j(0j 1x)gn j(0j 1x)A(n j+1) d(0jx;0jy)gn j(0j 1x)A(n j+1) k+lj+k+l +1d(x;y):(7:4)Letn1=maxfi:( +1+ )i( +1)(n+1) (k+l)g:(7:5) 34HUYIHUWritethesumin(7.3)asnXj=1=n1Xj=1
35 +nXj=n1+1=(IIa)+(IIb):Forallj=1;;n1,
+nXj=n1+1=(IIa)+(IIb):Forallj=1;;n1,(n+1) (k+l) +1(n j+1) (j+k+l) +1;because(j+k+l) +1(n j+1) increasesas0jn1.Henceby(7.4)andLemma4.4,(IIa)n1Xj=1gn j(0j 1x)Ad(x;y)(n+1) 1 e (0jx)eSj 1 (0j 1x)Ad(x;y)(n+1) gn(x)4Ad(x;y)3(n+1) :WeassumethatNislargeenoughsuchthatforallnN8C k+ln+k+l +1k+ln1+k+l 1(n+1) ;(7:6)wheren1isde nedin(7.5)andC isgivenbyLemma4.3.iii).Itispossiblesinceby(7.5),n1nisroughlyproportionalto +1 +1+ .Notethatforallj=n1+1;;n,(n+k+l) +1(n j+1) (j+k+l) +1;(7:7)becausetherightsideisdecreasingwithjifn1jn.AlsobyLemma4.3.iii),nXj=n1+1 1 e (0jx)eSj 1 (0j 1x)eSn1 (0n1x)C k+ln1+k+l :(7:8)Henceby(7.4),(7.7)andby(7.8),andthenby(7.6),(IIb)4A3k+ln+k+l +1d(x;y)nXn1+1 1 e (0jx)eSj 1 (0j 1x)4A3k+ln+k+l +1C k+ln1+k+l d(x;y)A6(n+1) d(x;y):Nowwehave(IIa)+(IIb)43+16A(n+1) d(x;y)=3A2(n+1) d(x;y): POLYNOMIALDECAYOFCORRELATIONS35Proposition7.4.Foranyg�0suchthatg(y)g(x)eJd(x;y)8x;y2Qk+l,Llg(y)2Llg(x)8x;y2Qk:Proof.Bytheconditionand(1.1),wehaveg(x)(g)eC Jforanyx2Qk+l.Wemayassumethattheinequalityholdsforanyx2+sincethevaluesofLlgonQkonlydependonthevaluesofgonQk+l.Forawordw,denoteIw(g)=1RwZRw g (g)d:Letubeanm-word.By(1.1)and(6.4),forx;y2Ru,g(y) g(x)g(x) eJd(x;y) 1(g)eC J eC J=m 119(g):Sincel�m,itimpliesthatforanyl-wordw, g(wx) (g) Iw(g) 19(g);(7:9)andifRwRu,then Iu(g) Iw(g) 19(g):(7:10)Recallth
36 attheconditionalexpectationofg (g)corre
attheconditionalexpectationofg (g)correspondingtolisE g (g)jl(x)=1l(x)Zl(x) g(z) (g)d(z)=Xw2lIw(g) Rw(x);wherel(x)istheelementoflcontainingx.HenceLlE g (g)jl(x)=Xw2leSl (wx)Xw02lIw0(g) Rw0(wx)=Xw2leSl (wx)Iw(g):Foranyx2Qk,wehave Llg(x) (g) Llg(x) LlE(gjl)(x) + LlE(gjl)(x) (g) :(7:11)ByLemma4.1.i)and(7.9),the rsttermisboundedby Xw2leSk (wx)g(wx) (g) Iw(g) 19(g)Xw2leSk (wx)=19(g): 36HUYIHUToestimatethesecondterm,wewriteLlE g (g) l(x)=Xw2leSl (wx)Iw(g)=Xu2mXw2lRwRueSl (wx)Iw(g)=Xu2mXw2lRwRueSl (wx)Iu(g)+Xu2mXw2lRwRueSl (wx) Iw(g) Iu(g):(7:12)Notethat Iu(g) maxxfjg(x)jg+(g) eC J+1(g):Denote (u)=1RuXRwRueSl (wx) 1.By(6.6),j (u)j foranym-wordu.Soby(6.5),the rstsumin(7.12)isboundedby XuIu(g)XwRueSl (wx) = XumIu(g) 1+ (u)Ru = XumIu(g)Ru+Xum (u)Iu(g)Ru 0+ eC J+1(g)19(g):By(7.10),thesecondsumin(7.12)isalsoboundedby19(g).Henceby(7.11), Llg(x) (g) (g)=3.Thatis,23 Llg(x)(g) 438x2Qk;whichimpliestheresultoftheproposition.8.Convergencerate:ProofofTheoremD,EandFProofofTheoremD.Denotegn=Lng.Byconsideringc1g+c2wemayassumeg2G+J,(g)=1andjg(x) 1j1=38x2+sothatPropositionCcanbeapplied.Takekk1suchthatQk2=3,wherek1isasinSublemma8.2.TakeCmaxf6A;4Ckg 1k0gwhereAisgivenbyPropositionC,Cisgivenby(1.4),andkk0istheC0norm.Letn1�0satisfySublemma8.2andsatisfythatforanynn1,C(n+1)6A.WemayincreaseC
37 ifnecessarysuchthatforallj=0;1;;n1,jg
ifnecessarysuchthatforallj=0;1;;n1,jgj(x) 1jC(j+1) 18x2Qk:(8:2) POLYNOMIALDECAYOFCORRELATIONS37Toprovethetheorem,weonlyneedshowthatforalln�n1,(En) gn(x) 1 C(n+1) 1foranyx2Qk;(Fn)ZOk+1 gn(x) 1 dC2(n+1) 1.Clearly,(En)and(8.2)givetheresultsofParti).Sincetheintegralofjgn(x) 1jonQkisboundedbyC(n+1) ( 1)Qk,andQk[Ok+1=+,wehavePartii)ifwereplaceC(Qk+1=2)byC.ByLemma8.1,(Ej),j=0;1;;n 1,imply(Fn).AndbyLemma8.3andPropositionC,(Fn)imply(En).Thentheresultsfollowfrominduction.Lemma8.1.Letnn1.Ifforj=0;1;;n 1,jgj(x) 1jC(j+1) 18x2Qk;(8:3)thenZOk+1 gn(x) 1 dC2(n+1) 1:Proof.ByLemma4.2,(1.4),andthechoiceofC,ZOk+1jgn(x) 1jeSn (0nx)dkg 1kOk+n+1Ckg 1k(n+k+1) 1C4(n+1) 1:Ontheotherhand,(8.3)andthede nitionofgimplythatforj=0;1;;n 1,jgj(x) 1jC(j+1) 18x2O1:Also,byLemma4.2,ZOk+1 1 e (0jx)eSj 1 (0j 1x)d(x)=Ok+j Ok+j+1=Pk+j:Hence,by(1.4)andSublemma8.2,ZOk+1nXj=1 gn j(0j 1x) 1 1 e (0jx)eSj 1 (0j 1x)d(x)nXj=1C(n j+1) 1Pk+jnXj=1C(n j+1) 1C(k+j) C4(n+1) 1NowtheresultfollowsfromLemma4.4. 38HUYIHUSublemma8.2.Therearek1;n1�0suchthatifkk1,thennXj=11(n j+1) 11(j+k) 14C(n+1) 18nn1:Proof.NotethatnXj=11(n j+1) 11(j+k) 1(n2+1) 1[n=2]Xj=11(j+k) +1(n2+k) nXj=[n=2]+11(n j+1) 12 1(n+1) 11Xj=11(j+k) +2 (n+1) 1[n=2]+1Xj=11j 1(n+1)=(I)+(II):Theseriesin(I)isoforderk ( 1),andthesumin(II)isofordern ( 1).Since �
38 0;1,theresultfollows.Lemma8.3.Suppose(
0;1,theresultfollows.Lemma8.3.Suppose(g)=1,gn(y)gn(x)1+A(n+1) 8x;y2Qk;(8:4)andZOk+1jgn(x) 1jdC2(n+1) 1:Thenjgn(x) 1jC(n+1) 18x2Qk:Proof.Since(g)=1, ZQk gn 1d = ZOk+1 gn 1d ZOk+1 gn 1 dC2(n+1) 1:By(8.4),foranyx;y2Qk,gn(y) 1gn(x) 1+gn(x)A(n+1) :IntegratingwithrespecttoxoverQk,andusingthefact6AC(n+1),weget gn(y) 1 Qk ZQk gn(x) 1d +(gn)A(n+1) C2(n+1) 1+A(n+1) =2C3(n+1) 1: POLYNOMIALDECAYOFCORRELATIONS39SinceQk2=3,wegettheresult.ProofofTheoremE.Notethatbythede nitionofL ,foranyfunctions~gandgde nedon+,Ln (~g)g=~g L g.So,wehave (~g)g= L (~g)g= ~gL g:(8:5)Hence, (~gn)g (~g)(g) = ~gLng ~g(g) = ~g[Lng (g)] (j~gj) Lng (g) :NowtheresultfollowsfromTheoremD.ToproveTheoremF,we rstrecallageneralresultfortheCentralLimitTheo-remfrom[Li2].Theorem.Let(X;)beaprobabilityspace,andletT:X!Xbeanoninvertibleergodicmeasure-preservingtransformation.Letg2L1(X;)besuchthat(g)=0.Assumei)1Pn=1 R(gTn)gd 1,ii)1Pn=1^Tn(g)isabsolutelyconvergenta.e.ThentheCentralLimitTheoremholdsforg.Inthestatement,^Tdenotesthedualoftheoperator^T:L2(X;)!L2(X;)de nedby^T(g)=gT.ProofofTheoremF.Weonlyneedchecktheconditionsinabovetheorem.Clearly,conditioni)followsfromTheoremEwith~g=g.NotethatL isthedualoperatorof^becauseof(8.5)andthefact^~g=~g.Thenconditionii)followsfromTheoremD.i).References[B]R.Bowen,Equilibriumstatesa
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