STATISTICALPROPERTIESOFSOMEALMOSTANOSOVSYSTEMS By XuZhang ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Mathematics{DoctorofPhilosophy 2016 ABSTRACT STATISTICALPROPERTIESOFSOMEALMOSTANOSOVSYSTEMS By XuZhang Weinvestigatethepolynomiallowerandupperboundsfordecayofcorrelationsofaclass oftwo-dimensionalalmostAnosovwithrespecttotheirSinai-Ruelle-Bowen measures(SRBmeasures),wherethealmostAnosovisasystemwhichis hyperboliceverywhereexceptforonepoint.Attheentpoint,theJacobian matrixisanidentitymatrix.Thedegreesoftheboundsaredeterminedbytheexpansion andcontractionratesastheorbitsapproachthetpoint,andcanbeexpressed byusingcotsofthethirdordertermsintheTaylorexpansionsofthe atthetpoints. WediscusstherelationshipbetweentheexistenceofSRBmeasuresandthetia- bilityofsomealmostAnosovnearthetpointsindimensions biggerthanone.TheeigenvalueofJacobianmatrixattheindtpointalongthe one-dimensionalcontractionsubspaceislessthanone,whiletheothereigenvaluesalongthe expansionsubspacesareequaltoone.Asaconsequence,therearettiableal- mostAnosovthatadmitSRBmeasuresintwoorthree-dimensional spaces;thereexistttiablealmostAnosovwithSRBmeasures indimensionsbiggerthanthree.Further,weobtainthepolynomiallowerandupperbounds forthecorrelationfunctionsofthesealmostAnosovmapsthatadmitSRBmeasures. Tomyfamilyandfriendsfortheirloveandsupport. iii ACKNOWLEDGMENTS IwouldliketoexpressmydeepestgratitudetomyadvisorDr.HuyiHu,whoprovides exceptionalguidanceandtremendoussupportduringmygraduatestudy.Withouthisguid- ance,thisthesiswouldnothavebeenpossible.Ifeelveryluckytohavetheopportunityto studyunderthesupervisionofsuchaknowledgableandsmartmathematicianwithgreat personality.Healwaysgenerouslytimetomeetforquestionsanddiscussions,heis veryinspiringandpatientinenhancingmymathematicalunderstanding,andhisviewon mathematicshelpsmetoshapemyownunderstanding.Iamalsoverythankfulforhis encouragementsandsuggestionswhenImeetwithHiselegantpersonalityand persistentpassioninmathematicsandlifewillhavealongtimeonme. IwouldliketothankDr.SheldonNewhouse,Dr.BenjaminSchmidt,Dr.ZhenqiWang, andDr.ZhengfangZhou,forservingasmembersofmydoctoralcommitteeandfortheir invaluablesuggestions.InparticularIwanttothankDr.SheldonNewhouseforhisconsistent supportandgreathelpduringmygraduatestudy. IwouldliketothankDr.CasimAbbas,Dr.Tien-YienLi,Dr.GabrielNagy,Dr. Schenker,Dr.MoxunTang,Dr.XiaodongWang,andDr.YiminXiao,fortheirhelpand greatsupport.AndIamgratefultoMs.LeslieAitcheson,Ms.AltinselBritta,andMs. BarbaraMillerfortheirassistanceinprovidinginformationandhelp. Also,Iwouldliketotakethisopportunitytothankmygraduatefellowsandfriendsfor providingkindhelp,sharingwithmetheirexperienceandknowledge. Finally,Iwouldliketothankmyparentsfortheirendlessloveandalwaysstanding besidesmeovertheyears. iv TABLEOFCONTENTS Chapter1Introduction ............................... 1 1.1Preliminary....................................4 1.2RenewalTheory..................................7 Chapter2PolynomialdecayofcorrelationsforalmostAnosov phisms .................................... 10 2.1Introduction....................................10 2.2Statementofresults................................13 2.3Proofofthemaintheorem............................17 2.3.1Inducetoone-dimensionalmap.....................17 2.3.2Polynomialdecayrates..........................19 2.3.3Polynomialdecayratesfor..............23 2.4Somedistortionestimates............................27 2.5Ratesofconvergenceofthelevelsets......................31 2.6Estimatesofthesizeofelementsof P k;k ...................43 2.7Somelargedeviationestimation.........................51 Chapter3SomestatisticalpropertiesofalmostAnosov withspectralgap ............................. 57 3.1Introduction....................................57 3.2Mainresults....................................60 3.3TheexistenceofSRBmeasures.........................62 3.4Decayofcorrelations...............................76 3.4.1Inducetoone-dimensionalmap.....................77 3.4.2Polynomialdecayrates..........................79 3.4.3Polynomialdecayratesfor..............82 BIBLIOGRAPHY .................................... 86 v Chapter1 Introduction Theresearchofdynamicalsystemsismotivatedbytheproblemsinclassicalphysics, statisticalmechanicsandsoon.Givenaspace X ,adeterministicanddiscretedynamical systemisbyamap T : X ! X ,where X isaRiemannianmanifold,andthemap T preservesaninvariantprobabilitymeasure .Thisdynamicalsystemisdenotedby( X;T; ). Theorbitofaninitialstate x 2 X isdenotedby f x;Tx;:::;T n x;::: g ,whichrepresentsthe longtermbehaviourofthesystem. Indynamicalsystems,thereexistlotsofsimplemaps T withcomplicateddynamics, whichleadtointerestingstoriesaboutchaostheory.Weareconcernedwithergodictheory, whichcanbeseenasaquantitativedescriptionofthedynamicswiththehelpofmeasure theory.Thestatespace X shouldcomewitha ˙ -algebra B ofmeasurablesubsets. Thestatisticalpropertiesofanobservablefunction ˚ on X withrespecttothemap T is alsoaninterestingproblem.Weintroduceasequenceofrandomvariables X n = ˚ T n ,this sequenceofrandomvariablesareidenticallydistributedsincethemeasure isinvariantwith respecttothemap T .Forthestatisticalpropertiesofdynamicalsystems,thereexistlots ofinterestingproblems,forexample,theexistenceofSinai-Bowen-Ruellemeasures(SRB measures),decayofcorrelationsforsomeobservablefunctions,centrallimittheorems,large deviationprinciples,almostsureinvarianceprinciples,andsoon. ThereexistlotsofworkonthestudyoftheSRBmeasures.Givenatrentiable 1 AnosovonacompactconnectedRiemannianmanifoldwiththeRiemannian measure,thereisauniqueinvariantBorelprobabilitymeasurewithrespecttothis morphismsuchthatthemeasurehasabsolutelycontinuousconditionalmeasureonunstable manifolds,themaphaspositiveLyappunovexponents,themetricentropyisequaltothe sumofthepositiveLyapunovexponents,andthemaphasexponentialdecayofcorrelations foroldercontinuousobservablefunctions[42].ForAxiomAattractors,similarresults havebeenobtainedbyBowen,Ruelle,andsoon[4].Pesin,Ledrappier,Youngandothers haveextendedthetheoryonnonuniformlyhyperbolicsets[29,45].ForHenonattractors, BenedicksandYoungshowedthatthereexistSRBmeasuresforcertainparametersandgood statisticalproperties[3].FormoreinformationonSiani-Ruelle-Bowenmeasure,pleaserefer to[47].WewillstudytheexistenceofSRBorSRBmeasuresforaclassofalmost Anosovindimensionsbiggerthanone. Thecorrelationfunctionofasystemisusedtodescribehowfastthestateofthesys- tembecomesuncorrelatedwithitsfuturestatus,andtoestimatethisfunctionisavery interestingproblemindynamicalsystems.Toinvestigatethestatisticalproperties,Young introducedapowerfultool\YoungTower",whichhasbeensuccessfullyappliedtostudy manysystems[45].In[46],Youngappliedthe\couplingmethod"toobtainthepolynomial upperboundsforthecorrelationfunctionsofsomesystems.Later,Sarigintroducedapow- erfulmethod,estimatingtheasymptoticnormsofrenewalsequencesofboundedoperators actingonBanachspaces,andgavethepolynomiallowerboundsforcorrelationfunctions [40].And,GouezelsharpedSarig'sresultsandobtainedbetterestimatesforsomesystems [8]. Forthestudyofthecorrelationfunctionsofthemapsontwo-dimensionalspaces,Liverani andMartensinvestigatedaclassofareapreservingmapsontorus,andobtainedthepoly- 2 nomialupperbounds.In[12],HushowedthatthereexisteitherSRBmeasuresor SRBmeasuresforalmostAnosovomorphismswithnon-degeneracyconditions,where thedecompositionofthetangentspaceofthealmostAnosovsystemsisdiscontinuousatthe tpoints.Itisaninterestingproblemtoinvestigatethestatisticalproperties ofalmostAnosovsystems,sincethiskindofsystemscanbethoughtofasthegeneralization ofthemap x ! x + x 1+ s [13],whichhaspolynomiallowerboundsforcorrelationfunctions. WewillshowthatsomealmostAnosovhaveboththepolynomialupper andlowerbounds. Forthestudyofthelargedeviationprinciples,therearelotsofinterestingresults.Kifer providedamethodtoestablishlargedeviationprinciplesbasedontheexistenceof apressurefunctionalandontheuniquenessofequilibriumstatesforcertaindensesetsof functions[19].Youngstudiedthelargedeviationestimatesforcontinuousmapsofcompact metricspacesandappliedtheseresultsintiablemapsandshiftspaces[43].In[37], theauthorsobtainedtheratefunctionsforcertainmapsbasedonthetheoryofYoung Towerswithexponentialreturntimefunctions.Melbourneinvestigatedthelargedeviation principlesforaclassofnonuniformlyhyperbolicdynamicalsystemswithpolynomialdecay ofcorrelationsandsomemoderatedeviations[26].In[27],MelbourneandNicolstudied thelargedeviationestimatesforalargeclassofnonuniformlyhyperbolicsystems,which areonYoungtowerswithsummabledecayofcorrelations.In[31],Pollicottand Sharpstudiedthelargedeviationbehavioroftheorbitsofintervalmapswithindt points,andobtainedthepolynomialandtheexponentiallevelIestimationresults forfunctions,aswellasthepolynomialandtheexponentiallevelIIestimationresultsfor measures.Wewillstudythelargedeviationestimatesfortwo-dimensionalalmostAnosov andapplytheseresultstothestudyofthedecayofcorrelationsforolder 3 observablefunctions. Therestisorganizedasfollows.InChapter1,someusefulconceptsandresultsare introduced.InChapter2,westudythepolynomiallowerandupperboundsfordecayof correlationsofaclassoftwo-dimensionalalmostAnosovismswithrespectto theirSRBmeasures.Itisdiscoveredthatthedegreesoftheboundscouldbedescribed bytheexpansionandcontractionratesastheorbitsapproachthetpoint, andcanbeexpressedbyusingcotsofthethirdordertermsintheTaylorexpansions oftheatthetpoints.InChapter3,itistoinvestigate therelationshipbetweentheexistenceofSRBmeasuresandthetiabilityofsome almostAnosovnearthetpointsindimensionsbiggerthan one,wherethealmostAnosovisasystemwhichishyperboliceverywhere exceptforonepoint.Asaconsequence,therearettiablealmostAnosov morphismsthatadmitSRBmeasuresintwoorthree-dimensionalspaces;thereexist ttiablealmostAnosovismswithSRBmeasuresindimensionsbigger thanthree.Further,weobtainthepolynomiallowerandupperboundsforthecorrelation functionsofsomealmostAnosovmapsthatadmitSRBmeasures. 1.1Preliminary Inthissection,weintroducesomebasicandusefulproperties. Consideranon-singularmeasurablemap T : X ! X ,where X ismeasurablespace, B is the ˙ algebra, isa ˙ measure.Themeasure iscallednon-singularif ( T 1 ( E ))=0 isequivalentto ( E )=0forany E 2B . 1.1.1. [41]Thetransferoperatorofanon-singularmap( X; B ;T )istheop- 4 erator T : L 1 ( ) ! L 1 ( ),whichisby T f = f T 1 ; where f isthemeasure f ( E )= R E f . Proposition1.1.1. [41]Thereisauniquesolution 2 L 1 ( )totheequation R ˚ = R ( ˚ T ) f foranyfunction ˚ 2 L 1 .Thesolutionis = T f . Proposition1.1.2. [41]Thetransferoperatorisapositiveboundedlinearoperatorwith normone,andsatisfyingthat (1) forany ˚ 2 L 1 and 2 L 1 ,wehave T [( T ) ˚ ]= ( T ˚ ), -almosteverywhere; (2) if T isameasure-preservingmap,thenforany ˚ 2 L 1 ( ),wehave( T ˚ ) T = E ( ˚ j T 1 B ), -almosteverywhere. Foranygivenmap f anditsinvariantprobabilitymeasure ,thecorrelationfunctionfor twoobservablefunctionsandisby Cor n ; f; ):= Z ( f n Z Z where n isapositiveinteger. 1.1.2. Let bean f -invariantBorelprobabilitymeasureandlet H beaclass offunctionson M .Wesaythat( f; )hasexponentialdecayofcorrelationsforfunctionsin H ifthereis0 <˝< 1suchthatforany ; 2H ,thereexists C = C ; suchthat j Cor n ; f; ) j C˝ n : 5 1.1.3. Let bean f -invariantBorelprobabilitymeasureandlet H beaclass offunctionson M .Wesaythat( f; )haspolynomialdecayofcorrelationsforfunctionsin H ifthereis ˝> 0suchthatforany ; 2H ,thereexists C = C ; suchthat j Cor n ; f; ) j Cn ˝ : 1.1.4. Givenameasurablespace X withaprobabilitymeasure andameasur- ablepartition ˘ ,thereexistsafamilyofprobabilitymeasures f ˘ x : x 2 X g ,whichiscalled acanonicalsystemofconditionalmeasuresfor and ˘ [39],satisfyingthat (i) ˘ x ( ˘ ( x ))=1,where ˘ ( x ) 2 ˘ containing x ; (ii) foranymeasurableset B ˆ X ,themap x ! ˘ x ( B )ismeasurable; (iii) ( B )= R X ˘ x ( B ) ( x ) : Let M bea C 1 compactRiemannianmanifoldwithoutboundary.Let betheLebesgue measureon M .Let beaninvariantmeasurewithrespecttoamap f on M ,where f :( M; ) ! ( M; )isa C 1+ measurablemapwithpositiveLyapunovexponentsalmost everywhere,and > 0.ItfollowsfromPesintheory[29]thattheunstablemanifold W u ( x ) existsalmosteverywhereanditisanimmersedsubmanifoldof M .Denoteby u x theRie- mannianmeasureinducedon W u ( x ).Givenameasurablepartition ˘ ,if ˘ ( x ) ˆ W u ( x )and ˘ ( x )containsanopenneighborhoodof x in W u ( x )foralmostevery x withrespecttothe measure ,then ˘ issaidtobesubordinatetounstablemanifolds;further,if ˘ x isabsolutely continuouswithrespectto u x for almosteverywhere x 2 M ,thenthemeasure issaid tohaveabsolutelycontinuousconditionalmeasuresonunstablemanifolds,where ˘ x isa canonicalsystemofconditionalmeasuresfor and ˘ [22]. 6 1.1.5. AninvariantBorelprobabilitymeasure forthemap f on M issaidto beanSRBmeasureif (a) f haspositiveLyapunovexponentsalmosteverywherewithrespecttothemeasure ; (b) hasabsolutelycontinuousconditionalmeasuresonunstablemanifolds. 1.1.6. AniteinvariantBorelprobabilitymeasure forthemap f on M is saidtobeaniteSRBmeasureif (i) thereisaset E ,foranyopenneighborhood V oftheset E ,onehas ( M n V ) < 1 ; (ii) thereturnmap(see3.3.4)ontheset M n V haspositiveLya- punovexponentsalmosteverywherewithrespectto ; (iii) themeasure hasabsolutelycontinuousconditionalmeasuresonunstablemanifolds. 1.2RenewalTheory Inthissection,wetalkabouttheapplicationoftherenewaltheoryindynamicalsystems, whichcouldbeappliedtostudythesystemswiththepolynomialreturntime.Thismethod wasintroducedbySarig[40],andwasextendedbyGouezel[8]. Givenameasurabledynamicalsystem( X; B ;T ),asubset A 2B ,theinducedtrans- formationon A is( A; B\ A; A ;T A ),where B\ A = f B \ A : B 2Bg , A ( E )= ( A \ E ) ( A ) , and T A ( x )= T R A ( x ) ( x ),where R A ( x ):=1 A ( x )inf f n 1: T n ( x ) 2 A g . Proposition1.2.1. [40,Proposition1]Foraconservativenon-singulartransformation ( X; B ;T ), A 2B with0 < ( A ) < 1 .Set T n ˚ :=1 A T n ( ˚ 1 A )and R n ˚ =1 A T n ( ˚ 1 R A = n ). Then,forany z 2 D , T ( z )=( I R ( z )) 1 ; 7 where R ( z )= 1 X n =1 z n R n ;T ( z )= 1 X n =0 z n T n ;T 0 = I;z 2 D : And, T n = n X k =1 R k T n k = n 1 X k =0 T k R n k : Theorem1.2.1. Let T n beboundedlinearoperatorsonaBanachspace L suchthat T ( z )= I + P n 1 z n T n convergesinHom( L ; L )forevry z 2 D , (1) RnewalEquation:forevery z 2 D , T ( z )=( I R ( z )) 1 ,where R ( z )= P n 1 z n R n 2 Hom( L ; L )and P k R n k < 1 . (2) SpectralGap:thespectrumof R (1)consistsofanisolatedsimpleeigenvalueat1and acompactsubsetof D . (3) Aperiodicity:thespectralradiusof R ( z )isstrictlylessthanoneforall z 2 D nf 1 g . Let P betheeigenprojectionof R (1)at1.If P k>n k R k k = O (1 =n forsome > 2and PR 0 (1) P 6 =0,thenforall n T n = 1 P + 1 2 1 X k = n +1 P k + E n ; where isgivenby PR 0 (1) P = , P n = P l>n PR l P ,and E n 2 Hom( L ; L )satisfy k E n k = O (1 =n b c ). Lemma1.2.1 (Sarig,2002;Gouezel,2004) . Let( X; B ;m;T; F )beatopologicallymixing probabilitypreservingMarkovmap,andlog g m F hasa( T F ; F F )locallyoldercontinuous versionforsome F ,where g m F = dm dm T F .Assumethat T F hasthebigimageproperty,i.e., themeasureoftheimagesoftheelementsofthepartitionareboundedawayfrom0(which 8 isalwaystruewhenthenumberofelementin F isIf m [ R F >n ]= O (1 =n )with > 1,thenthereare 2 (0 ; 1)and C> 0suchthatforany 2L and 2 L 1 supported inside F ,onehas Cor n ; T;m ) 1 X k = n +1 m [ R F >k ] Z Z CF ( n ) k k 1 k k L ; where F ( n )=1 =n , if> 2; F ( n )=(log n ) =n 2 ,if =2; F ( n )=1 =n 2 2 ,if2 >> 1. 9 Chapter2 Polynomialdecayofcorrelationsfor almostAnosov 2.1Introduction Thetheoryofdynamicalsystemsplaysanimportantroleintheunderstandingofphysical phenomena,andmanysystemsinphysicsprovidegoodmodelsofdynamicalsystemssuchas thependulumequation,Billiardsystems,Lorentzgas,etc.([17]).Someinterestingphysical systemsarethoughtofasdynamicalsystems,likeanomaloustransport,fractionalkinetics, [20,48].Manyphysicalsystemsexhibitavarietyofmixingproperties.Itiswellknownthat hyperbolicitygivesrisetoexponentialmixingwithrespecttothephysicalmeasures.For systemswithslowerdecayrates,sometphysicalphenomenacouldbeobserved,e.g. stickydomain,intermittency,andsoon([35,48]).Inthisworkwepresentasimplemodelin thecategogyof invertiblesmooth dynamicalsystemsinwhichthesystemshaveintermittent behavior([34,35])andthereforetheratesofmixingcanberegardedaspolynomial. Thesystemsweconsiderare C r , r 4,almostAnosov f ofatwo- dimensionalmanifold M withantedpoint p atwhich Df p =id.Weshow thatundersomenondegeneracyconditions,ifthecotsofthethirdordertermsin theTaylorexpansionsof f at p satisfycertainconditionsthen f haspolynomialdecayof 10 correlations,andthedegreesofthedecayratesaregivenbythecotsofthe xy 2 and y 3 terms. 1 Polynomialdecayforone-dimensionalexpandingmapswithantpoint hasbeenstudiedextensively(seee.g.[23,33,45,13]).Therearesomesystematicways developedtoobtainpolynomialdecayrates.Thetowerstructuresintroducedin[44,45] arewidelyusedthatcanapplyforbothexponentialandsubexponentialdecayrates.The renewmethodsproposedin[40]provideawaytoobtainupperandlowerboundestimates. Forhigher-dimensionalexpandingmapswithanntperiodicpoints,upperbounds estimatesweremadein[33].Recentlybothupperandlowerboundestimateswereobtained in[15]forsomenon-Morkovmaps.Thoughthemethodsinboth[44]and[40]canbeapplied toinvertiblecase,therearefewerresultsinthisdirection.LiveraniandMartensinvestigated aclassofareapreservingmapsontorusandobtainedtheupperboundsforthecorrelation functions[24].Inthisworkweobtainbothupperandlowerboundestimatesofpolynomial decayratesforisms. Ourstrategytoprovetheresultsismoreorlessstandard.Weinducetwo-dimensional almosthyperbolicsystemstoone-dimensionalalmostexpandingsystemsbycollapsingthe stableleavesinaMarkovpartitions,followingtheschemedescribedin[44]inparticular. Thenweuseacorrespondingtheorem,statedin[40](and[8]aswell),fortheinduced systemstoobtainpolynomialdecayrates,inwhichreturnmapsareused.Thelaststep istopasstheratesweobtainedfortheinducedsystemstotheoriginalones. Themostchallengingpartoftheworkistoestimatethesizeofthelevelsets[ ˝>n ], 1 WementionherethatintheTaylorexpansion,theconditions Df p =idmeansthat thelineartermsaretrivial,andhyperbolicityimpliesthatthesecondordertermsmust vanish.Sounderthenondegeneracyconditionsthethirdordertermsdeterminetheergodic propertiesofthesystems. 11 where ˝ isthereturntimewithrespecttotheset M n P ,where P isarectanglewhose interiorcontains p .Notethatrestrictedtotheunstablemanifoldofthetpoint p ,themaphastheform f ( r ) ˇ r + a 0 r 3 .(See(2.2.2)and(2.2.3)with x = r and y =0.) Soifwetakeanypoint z inthethelocalunstablemanifoldof p ,thenthebackwardorbit f n ( z )convergesto p ataspeedproportionalto n 1 = 2 ,thatisunsummable.Fortunately, thesizeofthelevelsets[ ˝>n ]isoforderbetween n 1 and n 1 ,where1 > 1 > 2, becausethestablefoliationis not Lipschitzcontinuousnearthetpoint p ! (See(2.2.4)forthevalueof and ,andProposition2.5.1fortheestimates.)Weobtain suchestimatesbycontrollingtheslopesofthestableleavesatthepointsclosetothelocal stablemanifoldof p . Anotherproblemcomesfromthelaststep,whenweusethedecayratesoftheinduced systemstoobtainthedecayratesoftheoriginalones.Inthisstepweneedtoestimateof thesizesoftherectanglesafter n thiteration.Weuselargedeviationestimationtogetthat mostrectanglesshrinkexponentiallyfast,andprovedirectlythatotherrectanglesshrink fastenough,andthemeasureoftheunionofsuchrectanglesissmall. Itiswellknownthatforalmostexpandingmapsoftheintervalwitht point p =0,if f ( x ) ˇ x + x 1+ s , s 2 (0 ; 1),thentheratesofdecayofcorrelationsareofthe order n (1 =s 1) .Sofasterdecayratesaregivenbystrongerexpansionnearthet point(smaller s ).Inourcase,nearthepoint f ( x;y ) ˇ x (1+ a 2 y 2 ) ;y (1 b 2 y 2 ) , and a 2 = 2 b 2 playstheroleas1 =s inone-dimensionalsystems.Theratesofdecayareroughly oftheorder n ( a 2 = 2 b 2 1) .Thismeansthattheratesofdecayfortwo-dimensionalalmost hyperbolicsystemsaredeterminedbytheofbothcontractionandexpansionwhen orbits approach thetpoint,andfasterdecayratesaregivenbyeitherstronger 12 expansion(larger a 2 )orweakercontraction(smaller b 2 )orboth. 2 Wewouldliketomentionthatbesides[24],therearealsosomeupperboundestimatesfor billiards(see[49]andthereferencestherein).Also,thelowerboundestimatesareannounced in[7]. Therestofthechapterisorganizedasfollows.InSection2.2,weintroducesomerelated andstatetheMainTheorem.InSection2.3,wegivetheproofofthetheorem. Theproofconsistsofthreemajorsteps,whicharecarriedoutinthreesubsections.InSub- section2.3.1,weintroduceaquotientmapbycollapsingthemapalongthestablemanifolds. InSubsection2.3.2,weobtainboththelowerandupperpolynomialboundsfortheinduced systems.InSubsection2.3.3,weobtainthepolynomialboundsforoldercontinuousob- servablesfortheoriginalsystems.Section2.4isfordistortionestimates,mainlyusedin Subsection2.3.1.ThesizeofthelevelsetsareestimatedinSection2.5,wherequantita- tiveanalysisisperformed.Andthedecayratesofthesizeofrectanglesareestimatedin Sections2.6and2.7. 2.2Statementofresults Inthissection,somebasicconceptsandthemainresultsareintroduced. Considera C 1 two-dimensionalcompactRiemannianmanifold M withoutboundary, andtheRiemannianmeasureon M is m .Let 4 ( M )bethesetoffourtimestiable 2.2.1. [[12]1]Amap f 2 4 ( M )iscalledanalmostAnosov ifthereexisttwocontinuousfamiliesofcones x !C u x ; C s x suchthat,except 2 WereferRemark2.2.6forthereasonsthat a 0 and b 0 arenotinvolvedhere. 13 foraset S , (i) Df x C u x C u f ( x ) and Df x C s x C s f ( x ) ; (ii) j Df x v j > j v j forany v 2C u x and j Df x v j < j v j forany v 2C s x . Since S isaset,weonlyneedtoconsiderthat S isaninvariantsetbystudying f n insteadof f forsomenonnegativeinteger n .Assumethat S consistsofasinglepoint p .Apoint p iscalled ent if Df p hasaneigenvalueofmodulus1. Remark2.2.1. (i)ByProposition4.2in[12],thereisaninvariantdecompositionofthe tangentbundleinto TM = E u E s ,thedecompositioniscontinuousexceptatthet point.By2.2.1,awayfromthepointanglebetween E s and E u is boundedawayfromzero. (ii)ItfollowsfromProposition4.4in[12]localunstablemanifoldsexistforall x 2 M . Existenceoflocalstablemanifoldsfollowssimilarly. 2.2.2. [[12]2]AnalmostAnosov f issaidtobenon- degenerate(uptothethirdorder),ifthereexistconstants r 0 > 0and u ; s > 0suchthat forany x 2 B ( S;r 0 ), j Df x v j (1+ u d ( x;S ) 2 ) j v j ; 8 v 2C u x ; j Df x v j (1 s d ( x;S ) 2 ) j v j ; 8 v 2C s x : (2.2.1) Bychoosingasuitablecoordinatesystem,thereisaneighborhood B ( p;r )of p suchthat p =(0 ; 0)and f canbeexpressedas f ( x;y )= x (1+ ˚ ( x;y )) ;y (1 ( x;y )) ; (2.2.2) 14 where( x;y ) 2 R 2 and ˚ ( x;y )= a 0 x 2 + a 1 xy + a 2 y 2 + O ( j ( x;y ) j 3 ) ; ( x;y )= b 0 x 2 + b 1 xy + b 2 y 2 + O ( j ( x;y ) j 3 ) : (2.2.3) Remark2.2.2. By(2.2.1),weknowthat ˚ ( x;y ) ; ( x;y ) > 0forany( x;y ) 2 B ( p;r ) nf p g . Hence,wehave a 0 ;a 2 ;b 0 ;b 2 > 0.Inthispaper,wewillconsiderthecase a 1 = b 1 =0. InLemma7.1of[12],itisinfactprovedthatif f isanalmostAnosov ofatorus M = T 2 ,thenforanyneighborhood U of p ,thereexists 2 (0 ; 1),suchthatthe unstablesubspacesareoldercontinuouswitholderexponent . Byapplyingtherenewaltheorydevelopedby[40]and[8],wecouldobtainthefollowing results: MainTheorem. Let f 2 4 ( M )beatopologicallymixingalmostAnosov thathasanntxedpoint p atwhich(2.2.1){(2.2.3)areSuppose a 0 b 2 a 2 b 0 > 0,4 b 2 0,then f ( P i ) ˙ P j (mod ); (iii) (Localinvertibility)forany P i 2 P with ( P i ) > 0, f : P i ! f ( P i )isinvertiblewith measurableinverse. Bytheassumptionthat f istopologicallymixing,theMarkovmapisirreducible. 2.3.2Polynomialdecayrates Recallthatthetpoint p 2 int P 0 ,andhence, p 2 int P 0 .Denote f M = M n P 0 . Takethereturnmap e f = f ˝ of f withrespectto M n P 0 ,thatis, e f ( x )= f ˝ ( x ) ( x ), where ˝ isthereturntime, ˝ ( x )=min f n> 0: f n ( x ) 2 M n P 0 g .Clearly e f : M n P 0 ! M n P 0 inducesareturnmapfrom f M toitself.Forthesakeofsimplicityofnotation wealsodenoteitby e f . Let T 0 = f [ ˝ = n ]: n =1 ; 2 ;::: g beapartitionintothelevelsets.Thenlet T = T 0 _ P 0 , where P 0 = Pnf P 0 g istheMarkovpartitionof f M .Itisclearthat T isaMarkovpartition of f M . Foranypoint x; y 2 f M ,theseparationtimeisby s ( x; y ):=sup f n 0: e f i ( y ) 2 T ( e f i ( x )) ; 0 i n g : Wemayalsoregard s ( x;y )= s ( x; y )if x 2 x and y 2 y . 19 Let =sup fk Df x j E u x k 1 ; k Df x j E s x k : x 2 M n P 0 g : (2.3.1) Clearly 2 (0 ; 1).Let 2 (0 ; 1)asinLemma2.4.1,andthentake 2 [ ; 1). Foranyfunctionon M ,takeasemi-normby D :=sup x; y 2 f M j x ) y ) j p s ( x; y ) : ThenweconsidertheBanachspace L := f :supp ˆ f M; k k 1 + D < 1g : (2.3.2) andtakethenormin L by k k L = k k 1 + D Itisclearthat L containsolderfunctionswitholderexponent supportedon f M .If 2L ,thenforany x; y with s ( x; y ) n ,wehave j x ) y ) j ( D s ( x; y ) ( D ) n ( D p ) n : Thatis,is locallyoldercontinuous inthesensegivenin[40](seealso[1]). ByLemma3.3.1,weknowthatlog J ( e f ) 2L .Bystandardarguments,itiseasytoknow (e.g.seeLemma2inSubsection3.1in[44])that e f admitsanabsolutelycontinuousinvariant measure e on f M withthedensityfunction e h withrespectto e ,andthedensityfunction log e h 2L andisboundedawayfrom0andy.Byuniquenessweknowthat e istheconditionalmeasurementionedinthelastsubsectionwithrespectto f M . 20 TheJacobianof e f withrespectto e isgivenby J e ( e f )= J ( e f ) e h e f e h : Sincebothlog J ( e f )andlog e h arein L ,sois log J e ( e f ).Hence, log J e ( e f )islocallyolder continuous. NowwearereadytoapplythefollowingtheoremthatisdirectlyderivedfromTheorem2 in[40]. Theorem. Let( M; B ; f; P )beanirreduciblemeasurepreservingMarkovmapwith ( M )= 1,andassumethat log j J e ( e f ) j hasa( e f; T )-locallyoldercontinuousversionfor M .If g.c.d. f ˝ ( x ) ˝ ( y ): x; y 2 M g =1,and [ ˝>n ]= O (1 =n % )with %> 2,thenthereexists C> 0suchthatforany 2L and 2 L 1 withsupp ; supp ˆ f M ,onehas Cor n ; f; ) 1 X k = n +1 [ ˝>k ] Z Z CF % ( n ) k k 1 k k L ; where F % ( n )= O (1 =n % ). WehaveanirreduciblemeasurepreservingMarkovmap( M; B ; f; P )bytheprevious subsection.Byaboveargumentsweknowthat log j J e ( e f ) j hasa( e f; T )-locallyolder continuousversion.Itisclearthat f ˝ ( x ) ˝ ( y ): x; y 2 M g =1byourconstruction.So, whatweneedtodoistoestimate [ ˝>n ],thatis,toestimatetheexponent % . Recallthat P = P 0 istheelementoftheMarkovPartition P with p 2 int P .Denote Q = f 1 P n P .Clearly Q isarectangleandthesetofpoints x 2 M with ˝ ( x ) > 1,where ˝ isthereturntimegivenatthebeginningofthissubsection.Denote Q k =[ ˝ k ]. Clearly Q = Q 2 and Q k +1 ˆ Q k forany k 2.Moreover, Q k arerectanglessuchthatfor 21 any x 2 Q k , W s ( x;Q k )= W s ( x;Q )and W u ( x;Q k ) ˆ W u ( x;Q ). Foranyunstablecurve u 2 W u ( Q ),let u k = u \ Q k .ByProposition2.5.1,weknow thatthereexist D > 0and D > 0suchthat D k 1 m u ( u k ) D k 1 ; where and aregivenintheMainTheorem,and m u istheLebesguemeasurerestricted to u . Denoteby u theconditionalmeasureoftheSRBmeasure on u .Sincethedistortion of f alonganyunstablecurveisuniformlyboundedaboveandbelowawayfrom p (see Lemma2.4.1,alsoProposition7.5in[12]),soisthedensityfunction u dm u .Hence,there exist C ;C > 0suchthat C k 1 u ( u k ) C k 1 : Byintegration,wegetthatsimilarinequalitiesaretruefor k = [ ˝>k ]witht constantcots,thatis,thereexisttwopositiveconstants B ;B > 0suchthat B k 1 ( Q k ) B k 1 : (2.3.3) Itgivesthat 1 X k = n +1 [ ˝>k ]hastheorderbetween n ( 1 1) and n ( 1 1) . By(3.4.5),wecantake % =1 .Since F % ( n )isoforderof n % and %> 1 1,weget thatthereexist A ;A > 0suchthat A n 1 1 Cor n ; f; ) A n 1 1 : (2.3.4) 22 2.3.3Polynomialdecayratesfor Inthissubsection,weestablishpolynomialdecayofcorrelationsforalmostAnosov morphismsusingtheresultsweobtainedinthereducedsystems. Recallthat P isaMarkovpartition,and P = P 0 istheelementof P containing p ,and M 0 = M n P 0 . Weintroduceatypeofolderfunctions: H := : 9 H > 0s.t. j x ) y ) j H j x y j and ˆ M 0 g ; where 2 (max f (1 1 )(3 = 2+ b 0 = (2 a 0 )) 1 ; g ; 1],and 2 (0 ; 1)isspinLemma 7.1of[12],whichisdependentonthemap f andtheelement P 0 . Set P 0 := P and P k;n := W n i = k f i ( P 0 ),and P n = P 0 ;n . Forany ; 2H andforany k> 0,we k by k j B :=inf f x ): x 2 f k ( B ) g forany B 2P 0 ; 2 k ,and k inthesameway. ByLemma2.3.1below,thedirectcalculationgives j Cor n k ; f k ; f; ) Cor n k ; k ; f; ) j Z f k k ) ( f n k ) + Z f k k ) Z (2max j j ) Z j f k k j (2max j j ) C A H k ; (2.3.5) where isspdinLemma2.3.1. For k asabove,let k bethesignedmeasurewhosedensitywithrespectto is k ,andset k := d (( f k ) ( k )) . Let jj bethetotalvariationofasignedmeasure,andnotethat( f k ) ( f k ) )= , 23 where j j ( A )= R A d j j foranyBorelset A ˆ M .ApplyingLemma2.3.1forwecanget Z j k j = j k j ( M )= j ( f k ) ( f k ) ) ( f k ) ( k ) j ( M ) f k k ) j ( M )= Z j f k k j C A H k : Hence,bysimilarcomputationaspreviously,wehave Cor n k ; k ; f; ) Cor n k k ; k ; f; ) Z ( k ( f n k k ) + Z k Z k ) (2max j j ) Z j k j (2max j j ) C A H k : (2.3.6) NowweshowthatCor n k k ; k ; f; )canbeexpressedasfunctionsonlydependenton theunstablemanifolds,whichmeansthatthesefunctionsareconstantalongstablemanifolds oneachelementof P i .Since k isconstantalongstablemanifoldsoneachrectangle P i 2P , wecanregarditasafunctionon M aswell.Alsowehave ˇ ( k )= k ( ˇ )= k ( ), and f ˇ = ˇ f .So, Z ( k ( f n k k = Z ( k ( f n k )) d (( f k ) ( k )) = Z k d (( f n k ) ( f k ) ( k ))= Z k d (( f n ) ( k )) = Z k d ( ˇ ( f n ) ( k ))= Z k d (( f n ) ( k ))= Z k f n k d and, Z k Z k = Z d (( f k ) ( k )) Z k d = Z k d Z k d Itmeans j Cor n k k ; k ; f; ) j = j Cor n k ( k ; k ; f; ) j .Hence,by(3.4.9)and(3.4.10), 24 wehave j Cor n ; f; ) j = j Cor n k ; f k ; f; ) j Cor n k ; f k ; f; ) Cor n k ; k ; f; ) j + j Cor n k ; k ; f; ) Cor n k k ; k ; f; ) j + j Cor n k k ; k ; f; ) j =(2max j j ) C A H k +(2max j j ) C A H k + j Cor n k ( k ; k ; f; ) j : Take k =[ n= 2].Since > 1 1,by(2.3.4),weobtainthatthereexist A> 2 1 1 A and A 0 < 2 1 1 A suchthat A 0 n 1 1 j Cor n ; f; ) j A n 1 1 : ThiscompletesthewholeproofoftheMainTheorem. Lemma2.3.1. Givenany 2 (max f (1 1 )(3 = 2+ b 0 = (2 a 0 )) 1 ; g ; 1],thereexist C A > 0, K> 0and = ( ) > 1 1suchthatforany 2H and k K , Z f k k C A H k : Proof. Recallthatbythe, k j B :=inf f x ): x 2 f k ( B ) g ,where B 2P 0 ; 2 k .So forany x ,thereis y 2P 0 ; 2 k ( x )suchthat f k ( x ) k ( x )= f k ( x ) f k ( y ).Since 2H and f k ( P 0 ; 2 k ( x ))= P k;k ( f k ( x )),wehavethatfor x 2 B with B 2P 0 ; 2 k , j f k ( x ) k ( x ) j = j f k ( x ) f k ( y ) j H j f k ( x ) f k ( y ) j H diam( f k ( B )) = H diam( P k;k ( f k ( x ))) : 25 Itmeans j x ) k ( f k ( x )) j H diam( P k;k ( x )) : (2.3.7) Hence,weneedtoestimatethediameterofthesetsin P k;k . Let 2 (0 ; 0 ),where 0 isgiveninProposition2.7.1.Let S k = f x 2 M n P :diam( P k;k ( x )) e k g : ByRemark2.2.1,thereisauniformlowerboundfortheanglebetween E u x and E s x forall x 2 M n P .Hence,thereexist C ` > 0suchthatforany x 2 S k ,eitherthereexistsanunstable manifold u k ( x u ) ˆP k;k ( x )withthelengthlargerthan C ` e k ,where x u 2P k;k ( x ),or thereexistsastablemanifold s k ( x s ) ˆP k;k ( x )withthelengthlargerthan C ` e k ,where x s 2P k;k ( x ). Intheformercase,bythefact f k ( u k ( x u ))= s 0 ( f k ( x u )),thereis C d > 0and y u 2 u k ( x u )suchthat j Df k y u j E u y u j 0,where J 0 s and J 0 u aregiveninLemma2.4.2.Sowecanget S k ˆ n x 2 M : j Df k x j E u x j 0suchthat ( S k ) C D (log k ) 2(1 1) k 1 1 ; 26 where C D = C D + C 0 D , Let T k begiveninProposition2.6.1.Bythisproposition, ( T k ) C s log k k 1 .Forany x 2 T k ,byPropositions2.6.1and2.6.2,diam( P k;k ( x )) C h k 1 = 2+ 0 ,where 0 = b 0 2 a 0 ,and C h isaconstantlargerthantheconstants C s and C u givenbyProposition2.6.1and2.6.2. Forany x= 2 T k ,diam( P k;k ( x )) C s k 3 = 2+ 0 byProposition2.6.1. Hence,byinvarianceof and(2.3.7),theaboveestimatesgive Z f k k = Z k f k = Z T c k \ S c k j f k k j + Z T c k \ S k j f k k j + Z T k j f k k j H e k + H C s k (3 = 2+ 0 ) C D (log k ) 2(1 1) k 1 1 + H C s k (1 = 2+ 0 ) C s log k k 1 C A H k forsome C A > 0independentofwhere > 3 2 + 0 + 1 1.Bythechoiceof ,we havethat > 1 1. 2.4Somedistortionestimates InthissectionweprovidesomedistortionestimateswhichwereusedinSubsection2.3.1and willbeusedinSection2.6aswell. Lemma2.4.1. Therearepositiveconstants J s ;J u > 0,and 2 (0 ; 1]suchthatforany s 2 W s ( P i ), i =1 ; ;r , x;y 2 s and n 0, log j Df n y j E u y j j Df n x j E u x j J s d s ( x;y ) ;(2.4.1) 27 andforany u 2 W u ( P i ), i =1 ; ;r , x;y 2 u and n 0, log j Df n y j E u y j j Df n x j E u x j J u d u ( x;y ) : (2.4.2) Proof. Denote P = P 0 .BythesamemethodasintheproofofLemma7.4in[12],wecanget thatthereexistsconstant I s > 0suchthatif s ˆ f 1 P n P isa W s -segmentwith f i s ˆ P , i =1 ; n 1,thenforany x;y 2 s , log Df n y j E u y Df n x j E u x I s d u ( x;y ) ; where = isgiveninLemma7.1of[12]. Withthisresultwecangetaproofof(2.4.1)usingthesameideaasintheproofof Proposition7.5in[12],whosedetailscanbefoundinProposition3.1in[16]. Thesecondinequality(2.4.2)canbeobtainedsimilarly. Similarly,wehavethefollowingresult: Lemma2.4.2. Therearetwopositiveconstants J 0 s and J 0 u ,and 2 (0 ; 1]suchthatfor any s 2 W s ( P i ), i =1 ; ;r , x;y 2 s and n 0, log j Df n y j E s y j j Df n x j E s x j J 0 s d s ( x;y ) ; andforany u 2 W u ( P i ), i =1 ; ;r , x;y 2 u and n 0, log j Df n y j E s y j j Df n x j E s x j J 0 u d u ( x;y ) : 28 Lemma2.4.3. (1) Let u ; ^ u i 2 W u ( P i ).Fortheslidingmap ˇ : u ! ^ u i ,onehasthat ˇ = ^ u i . (2) J ( f )( x )= J ( f )( y )forany y 2 s ( x ). Proof. Thestatementsandproofarethesameas(1)and(2)ofLemma1inSubsection3.1 in[44]. Lemma2.4.4. Thereare C> 0, 2 (0 ; 1),and 2 (0 ; 1)suchthatforany u 2 W u ( P i ), i =1 ;:::;r , x;y 2 u , log J ( e f )( x ) J ( e f )( y ) C p s ( x;y ) ; where s ( x;y )isgiveninSubsection2.3.2. Proof. Forany x 2 u \ P i , i 6 =0,onehas J ( e f )( x )= j D e f x j E u x j e u ( e f ( x )) e u ( x ) : Denote ˚ ( x )=log j D e f x j E u x j .Wecanwrite j u ( x ) u ( y ) j k X i =0 [ ˚ ( e f i ( x )) ˚ ( e f i ( y ))] + k X i =0 [ ˚ ( e f i ( b x )) ˚ ( e f i ( b y ))] + 1 X i = k +1 [ ˚ ( e f i ( x )) ˚ ( e f i ( b x ))] + 1 X i = k +1 [ ˚ ( e f i ( y )) ˚ ( e f i ( b y ))] : Wetake k> 0suchthat f k = e f s ( x;y ) = 2 ,where s ( x;y )= s ( x;y )if s ( x;y )isevenand s ( x;y )= s ( x;y )+1,otherwise.Hence, f k ( x ) ;f k (^ x ) ;f k ( y ) ;f k (^ y ) = 2 P ,and(2.4.1)and 29 (2.4.2)canbeappliedtothesumsoftherighthandside.So,wecanget j u ( x ) u ( y ) j J u d u ( f k ( x ) ;f k ( y )) + J u d u ( f k (^ x ) ;f k (^ y )) + J s d s ( f k ( x ) ;f k (^ x )) + J s d s ( f k ( y ) ;f k (^ y )) : Recallthat isin(3.4.2).Wecangetthat d u ( f k ( x ) ;f k ( y )) = d u ( e f s ( x;y ) ( x ) ; e f s ( x;y ) ( y )) d u ( e f s ( x;y ) = 2 ( x ) ; e f s ( x;y ) = 2 ( y )) d u ( e f s ( x;y ) ( x ) ; e f s ( x;y ) ( y )) C d s ( x;y ) = 2 ; where C d isdeterminedbythemaximumradiusofeachelementintheMarkovpartition, weusethefactthat e f s ( x;y ) ( x )and e f s ( x;y ) ( y )areinthesameelementoftheMarkovpar- tition P ,andhence, d u ( e f s ( x;y ) ( x ) ; e f s ( x;y ) ( y )) isuniformlybounded.Similarly,wehave d u ( f k (^ x ) ;f k (^ y )) , J u d s ( f k ( x ) ;f k (^ x )) , J u d s ( f k ( y ) ;f k (^ y )) C 0 s ( x;y ) = 2 ,where C 0 is apositiveconstant.Hence, j u ( x ) u ( y ) j 4 C s ( x;y ) = 2 ; where C isapositiveconstant. Sincelog j D e f x j E u x j log j D e f y j E u y j and u ( e f ( x )) u ( e f ( y ))canbeestimatedinasimilar way,wegettheinequalityweneed. Thiscompetestheproof. 30 2.5Ratesofconvergenceofthelevelsets Inthissection,weproveProposition2.5.1thatisthekeysteptoestimatetheterm [ ˝>n ]. Recallthat Q = Q 2 = f 1 P n P ,and Q i =[ ˝ i ]for i 2. Notethatthemap f hasalocalproductstructure,thatis,thereexistpositiveconstants and suchthatforany x;y 2 M with d ( x;y ) ,[ x;y ]:= W u ( x ) \ W s ( y )contains exactlyonepoint. Takeacoordinatesysteminaneighborhood U of p suchthatthemaphastheformgiven in(2.2.2)and(2.2.3).Hence,the y -axisand x -axisarethestableandunstablemanifoldof p ,respectively.Recallthatweassume a 1 =0= b 1 . Let r> 0besmallsuchthattheballcenteredat p ofradius r iscontainedin U .We alsoassumethat P = P 0 issmallenoughsuchthat P , f ( P ),and f 1 ( P )arecontainedin theball. Proposition2.5.1. Suppose ; 2 (0 ; 1) < 2 a 2 b 2 a 2 2 + a 2 b 2 + b 2 2 < 2 b 2 a 2 < .Then thereexist D , D > 0suchthatforanyunstablecurve u 2 W u ( Q ),forany k> 0,we have D k 1 m u ( u k ) D k 1 : where u k = u \ Q k and m u istheLebesguemeasurerestrictedto u . Proof. Let u 2 W u ( Q )beanunstablecurvein Q .Denote q = u \ W s " ( p ). Forany z =( x;y ) 2 u ,denote z 1 =( x 1 ;y 1 )= f ( z ),and z =( x; y )=[ z;fz ]= W u ( z ) \ W s ( fz ).Sinceboth z 1 and z areinthesamestablecurve, z 2 Q k ifandonly if z 2 Q k 1 .Soif z isanendpointof u k ,then z isanendpointof u k 1 .Inorderto estimatethelengthof u k ,weestimatetheratio m u ( u k 1 ) =m u ( u k ).Thisisequivalent toestimate x=x . 31 Denoteby v s z arealnumberor 1 suchthat( v s z ; 1)isatangentvectorof W s r ( z ).Take thefunction^ ˆ on[0 ;r ]asinProposition2.5.2.ByLemmas2.5.2and2.5.4below,weknow thatif z = z 0 istlycloseto q ,then a 2 b 2 +^ ˆ ( y 0 ) (1 x 0 ) x 0 y 0 v s z 0 a 2 b 2 +^ ˆ ( y 0 ) (1 x 0 ) x 0 y 0 : Withtheestimatesfor v s z ,wecangetbyLemmas2.5.3and2.5.5thatthereexist E ;E > 0 suchthat x 0 + E x 1+ 0 x 0 x 0 + E x 1+ 0 : Ifwedenote s k = m u ( u k ),theinequalitiesmean s k + E s 1+ k s k 1 s k + E s 1+ k : forall k tlylarge.Hence,itfollows(e.g.seeLemma3.1in[14])thatthereexist D , D > 0suchthatforall k> 0, D k 1 s k D k 1 : Thisiswhatweneed. ToobtainLemmas2.5.2and2.5.4,weconsider v s z ,where z isnearthe y -axis.Assume that v s z hastheform v s z = ˆ x y ; where ˆ = ˆ ( x;y ). 32 Since( v s z ; 1)isinthestableconeat z ,withoutlossofgenerality,assumethat 1 v s z 1 ; 8 z 2 B ( p;r ) : (2.5.1) Let ˆ baafunctionon U .Set z 1 := f ( z )and ˆ 1 := ˆ ( z 1 ). ˆ ( x;y ):=( ˆ ˆ 1 )(1+ ˚ )(1 )+ ˆ 1 y (1+ ˚ ) y y (1 ) ˚ y ˆ 1 ˆx (1+ ˚ ) x + ˆx (1 ) ˚ x ; where ˚ = ˚ ( x;y )and = ( x;y ).Weneedthefollowingfacts. Lemma2.5.1 ([12]Lemma8.3) . If v s z ˆ ( z ) x y and0 ˆ ( x;y ),then v s z 1 ˆ ( z 1 ) x 1 y 1 . Theresultalsoholdsifall\ 6 "arereplacedby\ > ". Togetmorepreciseformof ˆ ,weneedthefollowingresults. Proposition2.5.2 ([12]Proposition8.4) . ThereexistsaLipschitzfunction^ ˆ on[0 ;r ]with ^ ˆ (0)=0satisfyingthefollowingtwoequations: a 2 b 2 +^ ˆ (0 ;y )=(^ ˆ ( y ) ^ ˆ ( y (0) 1 ))(1+ ˚ )(1 ) + a 2 b 2 +^ ˆ ( y (0) 1 ) y (1+ ˚ ) y y (1 ) ˚ y =0 ; and b 2 log(1+ ˚ )+ a 2 log(1 ) b 2 Z y y (0) 1 ^ ˆ ( t ) t dt =0 ; (2.5.2) where ˚ = ˚ (0 ;y ), = (0 ;y ),and y (0) 1 = y (1 (0 ;y )). Theupperboundestimateshavebeenprovedin[12].Westatethecorrespondinglemmas hereforcompletion,whichareLemmas9.1and9.2in[12] 33 Lemma2.5.2. Suppose a 2 > 2 b 2 ,0 << 1,and a 0 b 2 a 2 b 0 > 0.Thenforanypoint q =(0 ;y q )with y q > 0small,thereexists > 0suchthatforany z 0 =( x 0 ;y 0 ) 2 W u ( q ) with x 0 > 0, v s z 0 a 2 b 2 +^ ˆ ( y 0 ) (1 x 0 ) x 0 y 0 : Lemma2.5.3. Let z 0 =( x 0 ;y 0 )with x 0 > 0.Ifforall z =( x;y )inthestablecurvethat joins z 0 and z 1 , v s z a 2 b 2 +^ ˆ ( y ) (1 x ) x y ; then x 0 x 0 + E x 1+ 0 ; where E isapositiveconstantdependenton y 0 . Thefollowinglemmaisthekeysteptogetthelowerboundestimatesfor x 0 =x 0 . Lemma2.5.4. Givenany ; 2 (0 ; 1)with < 2 a 2 b 2 a 2 2 + a 2 b 2 + b 2 2 < 2 b 2 a 2 < .Thenforany point q =(0 ;y q )with y q > 0small,thereexists "> 0suchthatforany z 0 =( x 0 ;y 0 ) 2 W u " ( q ) with x 0 > 0small, v s z 0 a 2 b 2 +^ ˆ ( y 0 ) (1 x 0 ) x 0 y 0 : (2.5.3) Proof. Foreach z 0 =( x 0 ;y 0 ) 2 W u r ( q ), z i =( x i ;y i )= f i ( z 0 ), c 0 :=0 ;c i := A 1 x 0 y 2 0 Q i 1 j =0 1 0 y j y (0 ;y j ) 8 i 1 ; where A 1 = a 2 2 b 2 (2 b 2 a 2 )and 0 isspdinLemma2.5.6.Itisevidentthat c i +1 c i = c i +1 0 y i y (0 ;y i ) ; 8 i> 0 : (2.5.4) 34 Set ˆ i := ˆ ( z i )= a 2 b 2 +^ ˆ ( y i ) (1 x i ) ;i 0 ; (2.5.5) and ~ ˆ i := ˆ i c i ;i 0 : (2.5.6) Forany z i =( x i ;y i ),set ~ ˆ i ( x i ;y i ):=(~ ˆ i ~ ˆ i +1 )(1+ ˚ i )(1 i ) +~ ˆ i +1 y i (1+ ˚ i ) y ( x i ;y i ) y i (1 i ) ˚ y ( x i ;y i ) ~ ˆ i ~ ˆ i +1 x i (1+ ˚ i ) x ( x i ;y i )+~ ˆ i x i (1 i ) ˚ x ( x i ;y i ) ; where ˚ i = ˚ ( z i )= ˚ ( x i ;y i ), i = ( z i )= ( x i ;y i ). Bycontradiction,supposethat(2.5.3)isincorrect.Itistoshowthatfor y q > 0small enough,thereis "> 0suchthatforany z 0 =( x 0 ;y 0 ) 2 W u " ( q )with q =(0 ;y q ), x 0 ;y 0 > 0, v s z i ~ ˆ i x i y i and0 ~ ˆ i ( x i ;y i ) ; this,togetherwithLemma2.5.1,yieldsthat v s z i +1 ~ ˆ i +1 x i +1 y i +1 : ByLemma2.5.6below,wecantake "> 0smallenoughsuchthat c n 0 > 1+max f a 2 =b 2 + ^ ˆ ( y i ): y 2 [0 ;r ] g andhence,~ ˆ n 0 < 1forsome n 0 = n ( z 0 ).Since c i increaseswith i ,it followsthat~ ˆ i < 1forany i n 0 .Notethat x i isincreasingand y i isdecreasingwhen theorbitundertheiterationof f isintheneighborhoodoftheorigin.Thenthereexists 35 n 1 n 0 suchthat v s z n 1 > ~ ˆ n 1 x n 1 y n 1 > 1.Thiscontradicts(2.5.1). Now,wewillshowthatforall i 0with x i 0.By(2.5.11),wehave R ~ ˆ ( x i ;y i ) 8 > > > < > > > : = O ( x 2 i + x i y 2 i )if~ ˆ i 1; < 0if~ ˆ i < 0 : For i =0, c 0 =0and c 1 = a 2 2 b 2 (2 b 2 a 2 ) x 0 y 2 0 bytheof c i .Hence, ~ ˆ 0 ( x 0 ;y 0 )= a 2 2 b 2 (2 b 2 a 2 ) x 0 y 2 0 c 1 y i y (0 ;y i )+ O ( x 2 0 + x 0 y 3 0 + x 0 y 3 0 ) < 0 ; sinceweassume x 0 issmallcomparedwith y 0 . For0 0tlylarge. If i n 0 ,then~ ˆ i < 0.Hence, R ~ ˆ ( x i ;y i ) < 0.Thenby(2.5.18), ~ ˆ i ( x i ;y i )= a 2 2 b 2 (2 b 2 a 2 ) x i y 2 i c i +1 (1 0 ) y i y (0 ;y i ) R ~ ˆ ( x i ;y i ) j + O ( x 2 i y i + x i y 2 i )+ x i O ( x 2 i + y 3 i )) < 0 : Thiscompletestheproof. Lemma2.5.5. Let z 0 =( x 0 ;y 0 )with x 0 ;y 0 > 0.Ifforall z =( x;y )inthestablecurve thatjoins z 0 and z 1 , v s z a 2 b 2 +^ ˆ ( y 0 ) (1 x 0 ) x 0 y 0 ; (2.5.19) then x 0 x 0 + E x 1+ 0 ; where E isapositiveconstantdependenton y 0 . Proof. Since( v s z ; 1)formsatangentlineofthestablemanifold W s r ( z ),(2.5.19)gives dx dy a 2 b 2 +^ ˆ ( y ) (1 x ) x y ; whichimpliesthat dx x (1 x ) + a 2 b 2 +^ ˆ ( y ) dy y 0 : Integratingthefunctionfrom z 1 =( x 1 ;y 1 )to z 0 =( x 0 ; y 0 ),wehave log x 0 x 1 1 log 1 x 0 1 x 1 + a 2 b 2 log y 0 y 1 + Z y 0 y 1 ^ ˆ ( y ) y dy 0 : 39 Inthefollowingdiscussions,weomitthesubscript0.Theaboveinequalitygives x x 1 1 x 1 x 1 1 y 1 y a 2 b 2 exp Z y y 1 ^ ˆ ( y ) y dy : This,togetherwith x 1 = x (1+ ˚ ( x;y ))and y 1 = y (1 ( x;y )),yieldsthat x x (1+ ˚ ( x;y ))(1 ( x;y )) a 2 b 2 1 x 1 x 1 1 y y a 2 b 2 exp Z y y (1 ( x;y )) ^ ˆ ( y ) y dy : By(2.5.7), ˚ ( x;y )= ˚ (0 ;y )+ O ( x 2 + xy 2 )and ( x;y )= (0 ;y )+ O ( x 2 + xy 2 ).Hence, Z y (1 (0 ;y )) y (1 ( x;y )) ^ ˆ ( y ) y dy = O ( x ),wherewetreat y asaconstant.By(2.5.2),onehas (1+ ˚ ( x;y ))(1 ( x;y )) a 2 b 2 exp Z y y (1 ( x;y )) ^ ˆ ( y ) y dy =(1+ O ( x 2 ))exp Z y y ^ ˆ ( y ) y dy : Since z =( x; y )and z =( x;y )areinthesamelocalunstablemanifold,onehasthat j y y j N ( x x ) N ( x 1 x )= Nx˚; where N isapositiveconstant.So, y y a 2 b 2 1+ Nx˚ y a 2 b 2 =1+ O ( x )andexp Z y y ^ ˆ ( y ) y dy =1+ O ( x ) : Nowweget x x 1 x 1 x 1 1 1+ O ( x ) : 40 Usingthefacts x 1 = x (1+ ˚ ) = x + x ˚ + x O ( ˚ 2 )and x< x ,wehave 1 x 1 x 1 =1+ x 1 x 1 x 1 =1+ x + x ˚ x + x O ( ˚ 2 ) 1 x 1 1+ x ˚ + x O ( ˚ 2 ) 1 x 1 : Therefore, x x 1+ E x ; where E isapositiveconstantdependenton y 0 . Thiscompletestheproof. Lemma2.5.6. Suppose ; 2 (0 ; 1) < 2 a 2 b 2 a 2 2 + a 2 b 2 + b 2 2 < 2 b 2 a 2 < .Thenthere exist 0 2 (0 ; 1)and > a 2 b 2 a 2 2 + b 2 2 suchthatforanypositiveconstants K and N ,apoint q =(0 ;y q )with y q > 0small,thereis "> 0suchthatforany z 0 =( x 0 ;y 0 ) 2 W u " ( q )with x 0 > 0,thefollowinginequalitiesholdsimultaneouslyforsomepositiveinteger n = n ( z 0 ): x 0 y 2 0 n Y j =0 1 0 y j y (0 ;y j ) 1 N;Kx n 1+ a 2 b 2 a 2 2 + b 2 2 such that = 2 a 2 b 2 ( a 2 2 + b 2 2 ) .Take a 2 b 2 a 2 2 + b 2 2 << 1andthentake 0 > 0suchthat 1 > 0 > max n 2 ; 2 ( 1) + 2 o : Clearlywehave 2 < 2 a 2 b 2 2( a 2 2 + a 2 b 2 + b 2 2 ) 2 a 2 b 2 = a 2 b 2 a 2 2 + b 2 2 < . Bythechoicesof 0 and ,wecouldassumethat K islargeenoughsuchthatif Kx y , 41 then 1 0 y y (0 ;y ) 1 1 2 b 2 y 2 (1 ) 2 2 (2.5.20) and (1+ ˚ ) (1 ) 2 1 ; (2.5.21) where 1 and 2 satisfy max n 2 ; 2 ( 1) + 2 o < 2 < 1 < 0 : (2.5.22) Hence,forany z 0 =( x 0 ;y 0 )with Kx 0 y 1+ n +1 . Since0 0.Hence,if z 0 istlycloseto q ,then y n +1 42 canbearbitrarilysmallandtherighthandsideoftheinequalitycanbearbitrarilylarge. Thislemmaisthusproved. 2.6Estimatesofthesizeofelementsof P k;k Recallthat P isaMarkovpartition.Denote P k;n = _ n i = k f i ( P )and P n = P 0 ;n .Denoteby P k;n ( x )theelementof P k;n thatcontains x . Also,denoteby s n ( x )theconnectedstablecurvesthatcontains x andiscontainedin P n ( x ),andby u n ( x )theconnectedunstablecurvesthatcontains x andiscontainedin P n; 0 ( x ). Recallthat m s istheLebesguemeasurerestrictedtostablecurves.Recallalsothat Q = Q 2 = f 1 P n P ,and Q k =[ ˝ k ], k 2,areintroducedinSubsection2.3.2. Denote R k =[ ˝ = k ]= Q k n Q k +1 for k 2.Thenwedenote Q + k = f ˝ ( Q k )and R + k = f ˝ ( R k )= f k ( R k ),where f ˝ isthereturnmapof f withrespectto M 0 = M n P 0 . Clearly Q k = [ 1 i = k R i and Q + k = [ 1 i = k R + i . Proposition2.6.1. Thereexist K s > 0and C s > 0suchthatforany k K s ,wecan aset T k withthefollowingproperties: (i) ( T k ) C s log k k 1 ; (ii) m s ( s k ( x )) C s k 1 = 2+ 0 forany x 2 T k ; (iii) m s ( s k ( x )) C s k 3 = 2+ 0 forany x= 2 T k [ P , where 0 = b 0 = 2 a 0 . 43 Proof. Take K s 2 K 1 ,where K 1 isgiveninCorollary2.6.1. Recallthat isin(3.4.2).Foreach k> 0,take ` = ` k = j log k log k .Thenfor any j ` k , j < 1 k . T k = ` [ i =0 ( f ˝ ) i ( Q + b k= 2 c ) ; where ˝ isthereturntimewithrespectto M n P .By(3.4.5), ( Q b k= 2 c ) 2 1 B k 1 for some B > 0.Since ispreservedunderthemap f ˝ ,wecanget ( T k ) 2 1 B k 1 ` C 0 log k k 1 forsome C 0 > 0.Hence,wegetpart(i)if C s C 0 . Forany x 2 M ,denote x k := f k ( x ).If x k 2 P ,wee ˝ ( x k )=min f i> 0: f i ( x k ) 2 M n P g ,thetimetheorbitof x k enter M n P . Wenowproveaclaimstrongerthantherequirementsin(ii)and(iii):Forany x= 2 P , theinequalityin(ii)holdsforany x 2 T k with x k 2 P and ˝ ( x k ) >k= 2;andthatin(iii) holdsotherwise. If x k = 2 P ,thenbyCorollary2.6.2(i), m s ( s k ( x )) C 2 k 3 = 2+ 0 . If x k 2 P and ˝ ( x k ) k= 2,thenwehave f ˝ ( x k ) = 2 P and k ˝ ( x k ) max f K 1 ;k= 2 g . UsingCorollary2.6.2(i)with f ˝ ( x k ) ( x k )and x = f k ˝ ( x k ) ( f ˝ ( x k ) ( x k ))weget m s ( s k ( x )) C 2 ( k ˝ ( x k ) 3 = 2+ 0 2 3 = 2+ 0 C 2 k 3 = 2+ 0 : If x k 2 P , ˝ ( x k ) >k= 2and x= 2 T k ,thenwehave s ˝ ( x k ) ( f ˝ ( x k ) ( x k )) ˆ Q + b k= 2 c .By 44 Corollary2.6.2(ii)wehave m s ( s ˝ ( x k ) ( f ˝ ( x k ) ( x k ))) C 2 b k= 2 c 1 = 2+ 0 2 1 = 2+ 0 C 2 k 1 = 2+ 0 .Onthe otherhand, x= 2 T k implies k ˝ ( x k ) ˝ ( f ˝ ( x k ))+ ˝ (( f ˝ ) 2 ( x k ))+ + ˝ (( f ˝ ) ` ( x k )).Hence k Df k ˝ ( x k ) y j E s y k ` 1 k forany y 2 s ˝ ( x k ) ( f ˝ ( x k ) ( x k ))bythechoiceof ` .Notethat f k ˝ ( x k ) s ˝ ( x k ) ( f ˝ ( x k ) ( x k )) = ˝ s k ( x ).Weget m s ( s k ( x )) 1 k m s s ˝ ( x k ) ( f ˝ ( x k ) ( x k )) 1 k 2 1 = 2+ 0 C 2 k 1 = 2+ 0 = 2 1 = 2+ 0 C 2 k 3 = 2+ 0 : Ontheotherhand,if x k 2 P , ˝ ( x k ) >k= 2and x 2 T k ,thenwecanonlyget m s ( s k ( x )) m s s ˝ ( x k ) ( f ˝ ( x k ) ( x k )) C 2 b k= 2 c 1 = 2+ 0 2 1 = 2+ 0 C 2 k 1 = 2+ 0 : Nowwegetwhatweclaimedifwetake C s =2 1 = 2+ 0 C 2 . Proposition2.6.2. Thereexist K u > 0and C u > 0suchthatforany k K u , m u ( u k ( x )) C u k 1 forany x= 2 P . Proof. TheproofissimilartothatforProposition2.6.1byusingtheestimatesgivenin Proposition2.5.1for u k 2 W u ( Q k ),insteadofCorollary2.6.2for s k 2 W s ( Q + k ). ToproveLemma2.6.3below,weneedthefollowingfacts. Lemma2.6.1 ([14]Lemmas3.1and3.2) . If t n 1 t n + Ct 1+ % n + O ( t 1+ % 0 n ) 8 n> 0 ; (2.6.1) 45 where % 0 >% ,thenforalllarge n , t n 1 ( %C ( n + k )) 1 =% + O 1 ( n + k ) 0 ; (2.6.2) forsome 0 > 1 =% and k 2 Z . Moreover,if(2.6.2)holdsandforall n> 0, r ( t n ) 1 C 0 t % n + O ( t 1+ % 0 n ) ; where C 0 > 0,thenthereexists D> 0suchthatforall k 0 >k , n + k 0 k Y i = k 0 k r ( t i ) D k n + k C 0 =%C : Theresultsremaintrueifweinterchange\ "and\ ".Therefore,if(2.6.1)becomesan equality,thensodoes(2.6.2). Lemma2.6.2 ([12]Propositions2.6and2.8) . Forany "> 0,thereexistsaconstant 0 0suchthatforany x 2 Q with n = ˝ ( x ), k Df n x j E s x k C 1 n 3 = 2+ 0 ,where 0 = b 0 = 2 a 0 . Proof. Choose u ; s > 0small.Thentakesectors S u = f z 2 U : j \ ( z;E u p ) j u g and S s = f z 2 U : j \ ( z;E s p ) j s g ,where \ ( z;E u p )istheanglebetweenthevectorfrom p to z andtheline E u p .Thenlet S c = P n ( S s [S s ). If N 0 > 0islargeenough,thenforany x 2 Q N 0 ,theorbitof x passesthrough S s , S c ,and S u consecutivelybeforeitleaves P .Notethatif x 2 R n ˆ Q N 0 ,then n = n x = ˝ ( x ) N 0 . Wetake n s , n c , n u > 0suchthat n s =max f j> 0: f i ( x ) 2S s ; 8 1 i j g , n c =max f j> 0: f n s + i ( x ) 2S c ; 8 1 i j g ,and n u = n x n s n c .Thatis, x;f ( x ) ;:::;f n s ( x ) 2S s , f n s +1 ( x ) ;:::;f n s + n c ( x ) 2S c ,and f n s + n c +1 ( x ) ;:::;f n x ( x ) 2S u . Notethat(2.2.3)impliesthat f hastheform f ( r )= r (1 b 2 r 2 + O ( r 3 ))restrictedto W s " ( p ),and Df hastheform Df j E s =1 3 b 2 r 2 + O ( r 3 )restrictedto E s x for x =(0 ;r ) 2 W s " ( p ).Hence,byLemma2.6.1,foranypoint^ x 2 W s " ( p ) \ Q , j f n (^ x ) jˇ 1 p 2 b 2 n and k Df n ^ x j E s ^ x k˘ ^ d s p n 3 forsomeconstant ^ d s > 0,where a k ˇ b k meanslim k !1 a k b k =1,and a k ˘ b k means a k =b k isboundedawayfrom0andity.Sincethepointsin S s areclose to W s " ( p ),wecangetthatthereexist c s >c 0 s > 0and d s >d 0 s > 0suchthat c 0 s p n s j f n s ( x ) j c s p n s and d 0 s p ( n s ) 3 k Df n s x j E s x k d s p ( n s ) 3 : (2.6.3) Nowweconsiderthepartoftheorbitin S c .Take z 2S s suchthat f k ( z ) 2S u \ Q + N 0 withsome k> 0. k s and k c inawaysimilarwiththatof n s and n c asabove, thatis, k s isthelargestpositiveintegersuchthat f 1 ( z ) ;:::;f k s ( z ) 2S s ,and k c isthe 47 largestpositiveintegersuchthat f k s +1 ( z ) ;:::;f k s + k c ( z ) 2S c .ConsiderLemma2.6.2with " small.If N 0 istlylarge,thenfor x 2 Q N 0 , j f n s ( x ) j = t j f k s ( z ) j issmall.Hence, byLemma2.6.2,for i =0 ; 1 ;:::;n c , (1 " ) k c j f n s ( x ) jj f n s + i ( x ) j (1+ " ) k c j f n s ( x ) j and n c ˘ k c t 2 = k c j f k s ( z ) j 2 j f n s ( x ) j 2 : So,thereexist c n >c 0 n > 0and c c >c 0 c > 0suchthatfor i =0 ; 1 ;:::;n c , c 0 n j f n s ( x ) j 2 n c c n j f n s ( x ) j 2 ; and c 0 c j f n s ( x ) jj f n s + i ( x ) j c c j f n s ( x ) j : (2.6.4) Notethat(2.2.2)and(2.2.3)implythatthereexist c>c 0 > 0suchthat1 c j y j 2 k Df y j E s y k 1 c 0 j y j 2 forany y with j y j small.Hence,bytaking y = f n s + i ( x ), i = 0 ; 1 ;:::;n c ,weobtainthatthereexist0 c 0 u > 0and d u >d 0 u > 0suchthat c 0 u p n u f n s + n c ( x ) j c u p n u ; d 0 u ( n u ) b 0 = 2 a 0 Df n u f n s + n c ( x ) j E s f n s + n c ( x ) k d u ( n u ) b 0 = 2 a 0 : (2.6.6) 48 Bythesecondinequalityof(2.6.4), j f n s + n c ( x ) j˘j f n s ( x ) j .Hence,by(2.6.3),(2.6.4), and(2.6.6),all n s , n c and n u areroughlyproportional.Since n s + n c + n u = n = n x ,we knowthatthereexist ˆ s ;ˆ u 2 (0 ; 1)suchthat n s ˆ s n and n u ˆ u n .Soby(2.6.3),(2.6.5) and(2.6.6),weget k Df n x j E s x k C 1 n 3 = 2+ b 0 = 2 a 0 forsome C 1 > 0. Theproofiscompleted. Corollary2.6.1. Thereexists K 1 > 0suchthatforany n>K 1 ,if x;f n ( x ) = 2 P ,then k Df n x j E s x k C 1 n 3 = 2+ 0 ,where C 1 and 0 areasinLemma2.6.3. Proof. Take K 0 1 > 0suchthat C 1 k 3 = 2+ 0 C 1 n 3 = 2+ 0 C 1 2( k + n ) 3 = 2+ 0 ,whenever k;n K 0 1 . Let S = S K 0 1 = f f i ( x ) 2 P : x 2 Q K 0 1 ;i =1 ;:::;n x 1 g ,where n x = ˝ ( x ).Since f is uniformlyhyperbolicon M n S ,thereexists ˆ = ˆ S 2 (0 ; 1)suchthat k Df z j E s x k ˆ forany x 2 M n S .Take K 00 1 > 0suchthatforany n K 00 1 , ˆ n C 1 (2 n ) 3 = 2+ 0 . Take K 1 =max f 2 K 0 1 ; 2 K 00 1 g .For x;f n x= 2 P with n K 1 ,wedenote I = f i 2 (1 ;n ): f i ( x ) = 2 S g ,andlet k x bethecardinalityof I .If k x n= 2 >K 00 1 ,then k Df n x j E s x k Y i 2 I k Df f i ( x ) j E s f i ( x ) k ˆ k x C 1 (2 k x ) 3 = 2+ 0 C 1 n 3 = 2+ 0 : If k x n= 2,thenwemayassumethattheorbit f x;:::;f n 1 ( x ) g passesthrough Q K 0 1 ` times.Let k 1 n= 2. Thiscompletestheproof. Recallthat Q n , R n , Q + n , R + n and s n ( x )aregivenatthebeginningofthissection.Also, wehave Q + n 2P n . Corollary2.6.2. Thereexists C 2 > 0suchthatforany k> 0, (i) m s s k ( f k ( x )) C 2 k 3 = 2+ 0 if x;f k ( x ) = 2 P ; (ii) m s s k ( x ) C 2 k 1 = 2+ 0 if x 2 Q + k . Proof. (i)Notethat f n ( s 0 ( x ))= s k ( f k ( x )).ByCorollary2.6.1,anddistortionestimates giveninLemma2.4.2,wecangetthat m s s k ( f k ( x )) C 0 1 k 3 = 2+ 0 m s s 0 ( x ) forsome C 0 1 > 0.Thenweusethefactthat m s s 0 ( x ) areboundedaboveforall x 2 M . (ii)Notethatfor y 2 R i , f i ( y ) 2 R + i and f i ( s 0 ( y i ))= s i ( f i ( y i )).Byusingthe sameargumentsasabove,andusingLemma2.6.3toreplaceCorollary2.6.1,wecanget m s s i ( f i ( y )) C 2 i 3 = 2+ 0 forall y 2 R i .Sinceforany x 2 Q + k , s k ( x )istheunionofthe stablecurves s i ( z i ), z i 2 R + i \ s k ( x ), i = k;k +1 ;::: ,wegetthat m s ( s k ( x )) 1 X i = k C 2 i 3 = 2+ 0 . Nowwecanincrease C 2 togettheresultofpart(ii). 50 2.7Somelargedeviationestimation Inthissection,westudythelargedeviationestimatesfortheobservablefunction 2L withrespecttothequotientmap f .Weadoptthediscussionsusedin[32]. Recallthat( f; M )istheone-dimensionalsysteminducedfrom( f;M ),and( ~ f; f M )isthe returnmapsof f withrespectto f M = M n P 0 . Lemma2.7.1. Let0 << 1 2 .Givenany > 0,foranyfunction 2L satisfying j R d j ,onehasthat n x 2 M : n 1 X i =0 f i ( x )) Z d > o = O ((log n ) 2( 1 1) n ( 1 1) ) : (2.7.1) ThetransferoperatoroftheMarkovmap f isasfollows: T x )= X f y = x g ( y y ) ; where g = d f and 2 L 1 ( M ).Since isinvariantwithrespecttothequotient map f , g issaidtobethe g -functionof . thefollowingoperators: T n :=1 Q T n 1 Q ) ;R n :=1 Q T 1 [ R Q = n ] ) : ByProposition1of[40],onehastherenewalequation: T ( z )=( I R ( z )) 1 ;z 2 D ; 51 where D istheunitdiskinthecomplexplane,and R ( z )= 1 X n =1 z n R n ;T ( z )= I + 1 X n =1 z n T n ;z 2 D : ProofofLemma2.7.1. Forconvenience,set:= R d . Itfollowsfrom(2.3.4)andthefactthat isaninvariantmeasureof f that Z f k d = Z f k Z d Z d d = Z f k d Z f k d Z d = j Cor n ; f; ) j C k 1 1 : Bytherenewaltheory,Theorem1in[40]orTheorem1.1in[8], T n = 1 r Pr + 1 r 2 1 X k = n +1 P k + E n ; where Pr istheeigenprojectionof R (1)at1, r isgivenby Pr R 0 (1) Pr = r Pr , P n = P l>n Pr R l Pr , E n 2 Hom( L ; L ).ByusingLemma6.5in[8]and(3.4.5),wehavethat k R n k = O ( 1 n ).So, wehave k E n k = o (1 =n 1 1 ). Bythefactthat Pr = R Q d (seetheproofofTheorem2in[40]), R d =0,and Theorem1.2in[8],onehas Z kT n k d = Z k T n k d = O 1 n 1 1 : Next,itistoapplythemethodoftheproofofProposition2.3in[32]toprove(2.7.1). ByProposition1.2in[41]andthefactthat f ismeasurepreservingwithrespecttothe measure , E j f k B )=( T k f k foranypositiveinteger k and 2 L 1 ( M ).Bydirect 52 computation, n x 2 M : n 1 X i =0 f i ( x )) > o 1 ( ) 2 # Z n 1 X i =0 f i ( x )) 2 # d ( x ) Cn # ( ) 2 # k k 2 # +240 n X k =1 k 1 = 2 k E f k j B ) k 2 # 2 # = Cn # ( ) 2 # k k 2 # +240 n X k =1 k 1 = 2 k E j f k B ) k 2 # 2 # = Cn # ( ) 2 # k k 2 # +240 n X k =1 k 1 = 2 kT k k 2 # 2 # Cn # ( ) 2 # k k 2 # +240 k k (2 # 1) = (2 # ) 1 n X k =1 k 1 = 2 Z jT k j d 1 2 # 2 # C n # 2 # k k 2 # +240 k k (2 # 1) = (2 # ) 1 n X k =1 1 k 2 # ; where # = 1 1 > 1andCorollary1from[28]isusedinthesecondinequality.Thisshows (2.7.1). FinallyweshowapropositionwhichisusedinSubsection2.3.3. Proposition2.7.1. Thereexists 0 > 0suchthatforany0 << 0 , E;E 0 > 0,wecan C D , C 0 D > 0respectivelyand N d > 0satisfying n x 2 M : j Df n x j E u x j suchthat Ee e forall n>N d . Nowletusprove(2.7.2). FortheMarkovpartition P = f P 0 ;P 1 ; ;P r g and^ u i 2 W u ( P i ),0 i r , considerthefollowingfunction ( x )= 8 > > < > > : 0if x 2 P 0 ; log j Df ˇ ( x ) j E u ˇ ( x ) j if x 62 P 0 ; where ˇ istheslidingmapinSubsection2.3.1.Clearly isconstantalongthestable manifoldsin P i ,0 i r .Itcanberegardedasanelementin L aswell.Itisevidentthat R d > 0. Since f isuniformlyhyperbolicon M n P ,thereexisttwopositiveconstants C u and C 0 u suchthat C u log j Df x j E u x j C 0 u 8 x 2 M n P: Hence,ifwelet C L = C u C 0 u and C 0 L = C 0 u C u ,then C L log j Df x j E u x j log j Df ˇ ( x ) j E u ˇ ( x ) j C 0 L 8 x 2 P i ;i 6 =0 : So, log j Df n x j E u x j = n 1 X i =0 log j Df f i ( x ) j E u f i ( x ) j n 1 X i =0 1 M n P 0 log j Df f i ( x ) j E u f i ( x ) j C L n 1 X i =0 ( f i ( x )) ; 54 where 1 M n P 0 istheindicatorfunction.Hence, n x 2 M : 1 n log j Df n x j E u x j < o ˆ n x 2 M : 1 n n 1 X i =0 ( f i ( x )) < C L o (2.7.4) forany > 0. Take 0 = C L R ,andlet0 << 0 .Set := R =C L .Clearly > 0.Recall thatwementionedthat canberegardedasfunctionsin L .SobyLemma2.7.1,onehas that n x 2 M : n 1 X i =0 ( f i ( x )) Z d > o = O ((log n ) 2( 1 1) n ( 1 1) ) ; andtherefore, n x 2 M : 1 n n 1 X i =0 ( f i ( x )) < Z d o = O ((log n ) 2( 1 1) n ( 1 1) ) : (2.7.5) By(2.7.4)and(2.7.5),andthefactthat isthequotientmeasureof ,wehavethat n x 2 M : j Df n x j E u x j 0.Thisis(2.7.2). 55 Toget(2.7.3),weintroducethefollowingfunction ( x )= 8 > > < > > : 0if x 2 P 0 ; log j Df ˇ ( x ) j E s ˇ ( x ) j if x 62 P 0 : Hence isconstantalongthestablemanifoldsandcanberegardedasafunctionin L .Itis alsoobviousthat R d > 0.Byusingsimilarmethodsasabove,wecanobtain n x 2 M : j Df n x j E s x j >e o C 0 D (log n ) 2( 1 1) n 1 1 forsome C 0 D > 0.Notethat E s isone-dimensional.So j Df n f n ( x ) j E s f n ( x ) j e .Since isaninvariantmeasure,weget(2.7.3). 56 Chapter3 Somestatisticalpropertiesofalmost Anosovwithspectral gap 3.1Introduction Theexistenceofaninvariantmeasureofamapisabasicprobleminergodictheory[25]. Insmoothergodictheory,oneimportantresultofSinaiisthatattiableAnosov onacompactconnectedRiemannianmanifoldhasaninvariantmeasure, whichhasabsolutelycontinuousconditionalmeasuresonunstablemanifolds[42].Thiswas generalizedtoAxiom-AsystemsbyBowenandRuelle[4].BasedonSinai,Ruelle,and Bowen'swork,akindofinvariantmeasureswithabsolutelycontinuousconditionalmeasures onunstablemanifoldsiscalledSRBmeasures.ThemapswithSRBmeasureshavesome gooddynamicalpropertiesinphysics[5].FormoreinformationonSRBmeasures,please refertothesurvey[47].LotsofresultsabouttheexistenceofSRBmeasureshavebeen obtainedformanysystems,forexample,non-uniformlyhyperbolicsystemsbyPesin[2], singularsystemsbyKatoketal.[18],thebilliardsystemsandsoon[6,21]. Insmoothergodictheoryandphysics,someinterestingsystemsaregeneratedbyfunction- 57 sofhightiability.Forexample,theLorenzsystem,Logisticmap,Henonmap,andso on[38].And,thetiabilityofthemapsctsthedynamics.Forexample,thereexists aone-dimensionalmap T :[0 ; 1] ! [0 ; 1],whichispiecewisettiableexpanding andthederivativeatonepointisequaltoone,but T cannotadmitaabsolutely continuousinvariantmeasure[30];HuandYoungobtainedthatsometwtiable almostAnosovontwo-dimensionalspacesadmitSRBmea- sures[16];HuobtainedsomeresultsontheexistenceofSRBmeasuresandSRB measuresforalmostAnosovsystems[12]. Foramixingdynamicalsystem,thecorrelationfunctionprovidesuswiththequantitative descriptionabouthowfastthestateofthesystembecomesuncorrelatedwithitsfuture status.TheSRBmeasuresplayanimportantroleinthestudyofthecorrelationfunctions. Thetransferoperatorwithsomefunctionspacesisapowerfultoolinthestudyofthedecay rateofcorrelationfunctions.Forinstance,theideaoftheconstructionof\YoungTower" hasbeensuccessfullyappliedtothestudyofmanysystemswithexponentialdecayrates,like Henonmap,piecewisehyperbolicsystems,scatteringbilliardsandsoon[45].Theestimation ofthepolynomialupperboundsforthecorrelationfunctionsofsomesystemsisobtainedby the\couplingmethod"[46].Later,theestimationofthepolynomiallowerboundsforthe correlationfunctionsofsomemapsisstudiedbythe\renewaltheory"[40],whichissharped byGouezel'sresults[8]. Thereexistmanyinterestingresultsabouttheestimationofthecorrelationfunctionsof themapsontwo-dimensionalmanifolds.Forinstance,theworkofBenedicksandYoungon HenonmapprovedtheexistenceofSRBmeasures,exponentialdecayofcorrelationsand soon[3],thestudyofLiveraniandMartensonaclassofareapreservingmapsontorus gavetheupperboundsforthecorrelationfunctions[24].In[9],theupperboundsforthe 58 correlationfunctionshavebeenobtainedforsomesystemswithonecenterunstabledirection Manneville-Pomeau-likemapbyHatomoto. Inthischapter,weprovidesomealmostAnosovmapsonspaceswithdimensions nolessthantwo,sincetherearefewexamplesinhigherdimensions,thesemapscouldbe regardedasthegeneralizationofManneville-Pomeau-likemapsinhigherdimensions.And, westudytheexistenceofSRBorSRBmeasuresforthistypeofmaps.Weobtainthat thetiabilityofthemapsneartheentpointsandthedimensionofthe spacestheexistenceofSRBmeasures(SeeTheorem3.2.1).Asaconsequence,there arettiablealmostAnosovthatadmitSRBmeasuresin spaceswithdimensionsequaltotwoorthree,whichisageneralizationoftheresultsof[16]; thereexistttiablealmostAnosovthatadmitSRBmeasuresin spaceswithdimensionsbiggerthanthree.Further,weapplytherenewaltheorytoinvestigate thepolynomiallowerandupperboundsformapsthatadmitSRBmeasures(seeTheorem 3.2.2). Therestisorganizedasfollows.InSection3.2,somebasicandthemainresults areintroduced.InSection3.3,itistostudytheexistenceofSRBorSRBmeasures inspaceswithdimensionsbiggerthanorequaltotwo.InSection3.4,thepolynomial lowerandupperboundsareobtainedbyusingtherenewaltheory.Thissectionconsistsof threeparts.InSubsection3.4.1,aquotientmapbycollapsingthemapalongthestable manifoldsisintroduced.InSubsection3.4.2,boththelowerandupperpolynomialbounds forthedecayrateofthecorrelationfunctionsareobtainedbyusingtherenewaltheory, wheretheobservablefunctionsareonthequotientmanifold.InSubsection3.4.3, thepolynomialboundsforolderobservablefunctionsfortheoriginalismsare obtained. 59 3.2Mainresults Inthissection,themainresultsareintroduced. Assume M isa C 1 compactRiemannianmanifoldwithoutboundary,andthedimension of M is m 2.Let f beanedon M satisfyingthefollowingproperties: (1) if m 3,themap f isttiableon M ,andhasapoint p ; (1) 0 if m =2,themap f is C 1+ on M ,andhasapoint p ,where > 0; (2) themap f istopologicallymixing,andtopologicallyconjugatewithanAnosov morphism; (3) thereexistaconstant0 < s < 1andacontinuousfunction u with u ( x ) 8 > < > : =1at x = p > 1elsewhere ; andthereisadecompositionofthetangentspace T x M : T x M = E u x E s x ; suchthat j Df x j E s x j s ;m ( Df x j E u x ) u ( x ) ;Df p j E u p = id; s L 0 < 1 ; where m ( Df x j E u x )=inf v 2 E u x ;v 6 =0 j Df x v j j v j ;L 0 =sup x 2 M;v 2 E u x ;v 6 =0 j Df x v j j v j ; 60 (4) thedimensionof E u x and E s x is m 1and1,respectively; (5) thereisacoordinatesystemonasmallneighborhood U of p suchthatthemap f can bewrittenasfollows: f ( x 1 ;x 2 ;:::;x m 1 ;x m ) =((1+ j ( x 1 ;:::;x m 1 ) j + ˆx 2 m ) x 1 ;:::; (1+ j ( x 1 ;:::;x m 1 ) j + ˆx 2 m ) x m 1 ; s x m ) (3.2.1) where ˆ isanonzeroconstantand j ( x 1 ;:::;x m 1 ) j = q P m 1 i =0 x 2 i . Remark3.2.1. The ˆx 2 m termcouldbereplacedbysomegeneraltiablefunction ( x m )with (0)=0. Theorem3.2.1. Forthe f satisfyingtheaboveassumptionsand m 2,if > 0andmin f 2 ;m 2 g 0,min f m 2 ; 2 g 0,thereis C L > 0suchthatforany( W 1 ;W 2 ;H )with d s ( x;H ( x )) < forany x 2 W 1 ,theLipschitz constantislessthan C L .Further,if m 3,thentheholonomymapistiable. Proof. First,itistostudythecase m 3.ByAssumptions(1)and(3),themap f istwice- tiableand s L 0 < 1.OnecouldapplythemethodusedintheproofofTheorem6.3 in[10]toobtainthatthestablefoliationis C 1 .Hence,thestablemanifold W s isLipschitz. Second,itistoconsiderthecase m =2.ByAssumption(1) 0 , f is C 1+ .So,the argumentsfor m 3donotwork.Onecouldapplytheargumentsusedintheproofof Proposition2.5in[16]toobtaintheLipschitzpropertyofthestablemanifold W s . Thiscompletestheproof. Lemma3.3.1. [12,Lemma8.1]Let f a k g 1 k =1 beasequenceofpositivenumbers, C and betwopositiveconstants. (i) If a k 1 a k + Ca 1+ k forany k 1,thenthereexist D> 0and k 0 1suchthat a k D ( k k 0 ) 1 fortlylarge k . 65 (ii) If a k 1 a k + Ca 1+ k forany k 1,thenthereexist D 0 > 0and k 0 0 1suchthat a k D 0 ( k + k 0 0 ) 1 fortlylarge k . Lemma3.3.2. Let h :[ 1 ; 1] ! R beamap,whichcanbewrittenas h ( x )= x (1+ x + o ( x ))for x inasmallneighborhood I of0,where > 0.Forany a 0 2 (0 ; 1] \ I , set a k := h k ( a 0 ), k 1.Givenanypositiveinteger m 2,if0 < 0, D 1 j Df k y j E u y j j Df k z j E u z j D: (3.3.1) Proof. Firstofall,wewillstudythecase m =3,theargumentsfor m =3belowalsowork for m> 3. Assumethat P istlysmallsuchthat P [ f ( P ) ˆ U anddiam( P [ f ( P )) < , where U isintroducedinAssumption(5)and isspinCorollary3.3.1.So,the map H introducedin3.3.3iswellwithrespecttothelocalunstable manifoldscontainedin U .ByAssumption(5),onedoesnotneedtoconsiderthecurvature ontheunstablemanifold ˆ U .Let d u ( y;z )denotethemetriconwithrespecttothe Riemannianmetricrestrictedtotheunstablemanifold.Let d ( y;z )bethedistance bytheEuclideanmetric.ByAssumption(5),onecouldassumethat d ( y;z )= d u ( y;z ), where y and z areinacommonunstablemanifoldcontainedin U . 66 Onecouldassumethatthereisapositiveconstant suchthat P =[ W s ( p ) ;W u ( p )],it isalsoreasonableto ˝ =inf w 2 @ s P \ W u ( p;P ) f d u ( w;f ( w )):theminimallengthcurvecontainedin W u ( p ) joining w and f ( w ) g ; (3.3.2) where @ s P = f w 2 P : w 62 int W u ( w;P ) g and @ s ( f ( P ))issimilarly. Now,itistoinvestigatethedistortionestimationalonganyunstablesubmanifold.Con- sider ˆ (( f ( P ) n P ) \ W u ( x ))forsome x 2 f ( P ) n P ,and f i ˆ P for1 i k 1, thenforany y;z 2 log j Df k y j E u y j j Df k z j E u z j E 1 d u ( y;z ) ˝ ; (3.3.3) where E 1 isapositiveconstantdeterminedlater. By(3.2.1),thelocalunstablemanifoldforthepointin U iscontainedinsomehorizon plane,wherethehorizonplanecouldberepresentedby f ( x 1 ;x 2 ;x 3 ): x 3 isequaltosomeconstant g : So,assumethat ˆf ( x 1 ;x 2 ;x 3 ): x 3 = E 2 g ,where E 2 isarealnumber.Set i := f i 0 i k .Hence, i ˆ A i := f ( x 1 ;x 2 ;x 3 ): x 3 = i s E 2 g ; 0 i k: Next,letusintroduceafunction ˚ ( w )= j Df 1 w j E u w j ,where j Df 1 w j E u w j =det( Df 1 w j E u w ). Now,itistostudytheanalyticexpressionof ˚ ( w )on U .By(3.2.1),set r := x 2 1 + x 2 2 , 67 themap f restrictedtotheunstablemanifoldcanbewrittenas ((1+ r 2 ) x 1 + 1 ( x 1 ;x 2 ;x 3 ) ; (1+ r 2 ) x 2 + 2 ( x 1 ;x 2 ;x 3 )) ; where 1 ( x 1 ;x 2 ;x 3 )= ˆx 2 3 x 1 , 2 ( x 1 ;x 2 ;x 3 )= ˆx 2 3 x 2 .Bydirectcalculation,theJacobian matrixof ˚ ( w )withrespectto x 1 and x 2 is 0 B @ 1+ r 2 + r 2 1 x 2 1 + @ 1 @x 1 r 2 1 x 1 x 2 + @ 1 @x 2 r 2 1 x 1 x 2 + @ 2 @x 1 1+ r 2 + r 2 1 x 2 2 + @ 2 @x 2 1 C A : Hence,thedeterminantis 1+ r 2 + r 2 1 x 2 1 + @ 1 @x 1 1+ r 2 + r 2 1 x 2 2 + @ 2 @x 2 r 2 1 x 1 x 2 + @ 1 @x 2 r 2 1 x 1 x 2 + @ 2 @x 1 =1+(2+ ) r 2 +(1+ ) r + @ 1 @x 1 (1+ r 2 + r 2 1 x 2 2 )+ @ 2 @x 2 (1+ r 2 + r 2 1 x 2 1 ) + @ 1 @x 1 @ 2 @x 2 r 2 1 x 1 x 2 @ 2 @x 1 r 2 1 x 1 x 2 @ 1 @x 2 @ 1 @x 2 @ 2 @x 1 =1+(2+ ) r 2 +(1+ ) r + ˆx 2 3 (2+2 r 2 + r 2 )+ ˆ 2 x 4 3 : (3.3.4) Hence,for x 3 ,thelevelcurvesforthefunction ˚ ( w )arecirclescontainedinsome horizonplane.This,togetherwith(3.2.1)and(3.3.4),yieldsthattheimageoflevelcurves under f arealsolevelcurves. Denoteby y i := f i ( y )and z i := f i ( z ), i 0.Let O 1 betheplanecontainingthe x 3 -axisandthepoint y , O 2 betheplanecontainingthe x 3 -axisandthepoint z .By(3.2.1), onehasthat y i 2 O 1 and z i 2 O 2 ,0 i k .Denoteby l z i thelevelcurvescontainedin 68 A i .Theset l z i \ O 1 hastwopoints,take z i 2 l z i \ O 1 ,whichisclosertothepoint y i .Let S i bethelinesegmentintheplane A i joiningthepoints z i and y i .By(3.2.1),onehasthat S i +1 = f 1 ( S i ),andthereisalinesegment 0 suchthat S 0 ˆ 0 andtheendpointsof 0 aretwopoints w and w 0 ,where w 2 @ s P and w 0 2 @ s f ( P ),and @ s P isspin(3.3.2). Set i := f i 0 ), i 0.So,len 0 ) C 0 ˝ and S i ˆ i ,where C 0 isapositiveconstant determinedbyProposition3.3.2. ByTheorem9.2in[36]and f isttiableinAssumption(1),onehasthat j Df 1 y i j E u y i jj Df 1 z i j E u z i j = Z 1 0 D ( j Df 1 ( z i + t ( y i z i )) j E u ( z i + t ( y i z i )) j )( y i z i ) dt C 1 j D ( j Df 1 y i j E u y i j ) jj y i z i j C 2 d ( y i ;z i ) ; (3.3.5) where C 1 and C 2 aretwopositiveconstants,thesearederivedbythefactthat j Df 1 j E u j isuniformlycontinuousandrentiable,since f isttiableonthecompact manifold M ,andthepoint z i fallsintoauniformlysmallneighborhoodofthepoint y i for tlylarge i .So,for j k ,byAssumption(1),onehas log j Df j y j E u y j j Df j z j E u z j log j 1 Y i =0 1+ j Df 1 y i j E u y i jj Df 1 z i j E u z i j j Df 1 z i j E u z i j C 3 j 1 X i =0 j Df 1 y i j E u y i jj Df 1 z i j E u z i j = C 3 j 1 X i =0 j Df 1 y i j E u y i jj Df 1 z i j E u z i j C 3 C 2 j 1 X i =0 j y i z i j = C 3 C 2 j 1 X i =0 d u ( y i ;z i ) ; (3.3.6) where C 3 > 0isaconstantdependenton f . Since i isalinesegment,forany w 2 i ,let j Df 1 j i ( w ) j = j Df 1 ( w ) ~v w j ,where ~v w isatangentvectorofthecurve i atthepoint w withunitlength,and j Df 1 ( w ) ~v w j isthe lengthofthevector Df 1 ( w ) ~v w . 69 Itfollowsfromthettiabilityof f andTheorem9.2in[36]that j Df 1 j i ( y i ) jj Df 1 j i ( z i ) j = Z 1 0 D ( j Df 1 j i ( z i + t ( y i z i )) j )( y i z i ) dt C 4 j D ( j Df 1 j i ( y i ) j ) jj y i z i j C 5 d ( y i ;z i ) C 5 i ) ; (3.3.7) where C 4 and C 5 aretwopositiveconstants.Hence,for j k ,onehas log j Df j j 0 ( y ) j j Df j j 0 ( z ) j log j 1 Y i =0 1+ j Df 1 j i ( y i ) jj Df 1 j i ( z i ) j j Df 1 j i ( z i ) j C 6 j 1 X i =0 j Df 1 j i ( y i ) jj Df 1 j i ( z i ) j C 6 C 5 j 1 X i =0 j y i z i j = C 6 C 5 j 1 X i =0 d u ( y i ;z i ) ; (3.3.8) where C 6 > 0isaconstantdependenton f .Thus,onehas d u ( y j ;z j ) j ) C 7 d u ( y;z ) 0 ) ; 8 j k; (3.3.9) where C 7 isapositiveconstantdependenton f . Let ^ i betheimageof i underthemap H : i ! W u ( p ),0 i k .Since ^ i 'sare pairwisedisjointandProposition3.3.2,onehasthat j 1 X i =1 i ) j 1 X i =1 C L length( ^ i ) C L diam( W u ( p;P )) : (3.3.10) 70 Hence,itfollowsfrom(3.3.6){(3.3.10)that log j Df j y j E u y j j Df j z j E u z j j 1 X i =0 C 3 C 2 d u ( y i ;z i ) C 3 C 2 C 7 C L diam( W u ( p;P )) d u ( y;z ) 0 ) E 1 d u ( y;z ) ˝ E 1 d u ( y;z ) ˝ ; where E 1 = C 3 C 2 C 7 C L diam( W u ( p;P )) =C 0 .Thisv(3.3.3). Now,itistoshow(3.3.1). BythepropertiesoftheMarkovpartition,thereisaconstant > 0suchthatthe diameterofislessthan andifin i ) \ ( f ( P ) n P ) 6 = ; ,then i ˆ f ( P ) n P .Suppose thatthenumberoftheorbitsof y and z comesbackto P is s 0 ,andthereexistpositive integers k i and l i ,1 i s 0 ,suchthat P \ j 6 = ; ; 8 j 2 [ 1 i s 0 (( k i ;k i + l i ) \ Z ) ; and P \ j = ; ; 8 j 62 [ 1 i s 0 (( k i ;k i + l i ) \ Z ) ; where j = f j So, log j Df k y j E u y j j Df k z j E u z j = s 0 X i =1 log j Df l i y k i j E u y k i j j Df l i z k i j E u z k i j + s 0 X i =0 k i +1 1 X j = k i + l i log j Df 1 y j j E u y j j j Df 1 z j j E u z j j : Thepartcanbeestimatedby(3.3.3),andthesecondpartisoutsideof P ,whichisa geometricsequencebyAssumption(3),where f isuniformlyhyperbolicoutsideof P .So, (3.3.1)holds. 71 Finally,itistostudythecase m =2and0 << 1.Forthecasethat m =2and 1, onecouldapplysimilarargumentsfor m =3asabove. Supposethat P [ f ( P ) ˆ U anddiam( P [ f ( P )) < .Fixany0 < ,itistoverify thatifishomermorphictoanintervalsuchthat ˆ (( f ( P ) n P ) \ W u ( x ))forsome x 2 f ( P ) n P , ,and f i ˆ P for1 i k 1,thenforany y;z 2 log j Df k z j E u z j j Df k y j E u y j D 00 d u ( y;z ) # ; (3.3.11) where D 00 isapositiveconstantand # = 1+ . ByProposition3.3.2,ittostudythedistortionestimatesalongtheunstable manifoldofthetpoint p ,thatis,itisenoughtostudy f : W u ( p;P ) ! W u ( p ). Itisevidentthat f isinjectivewhenitisrestrictedto W u ( p;P )and f 1 ( W u ( p;P )) ˆ W u ( p;P ).Suppose f ( x 1 )= x 1 + x 1+ 1 + ˚ ( x 1 )for x 1 > 0,when f isrestrictedtotheunstable manifold,where ˚ ( x 1 )isthehigherorderterm.InAssumption(5),thereisnohigherorder term,thereasonweaddthishigherordertermisthatwetheargumentsherealsowork forthemapwiththishigherorderterm.Inotherwords,if m =2and0 << 1,wecould generalizeAssumption(5).So,assumethat ˆ f ( W u ( p;P )) n W u ( p;P ).Hence,onehas thatforany y;z 2 with d ( y;z ) j y j = 2, d ( f ( y ) ;f ( z )) (1+ C 0 1 j y j ) d ( y;z ) ; log det Df ( y ) det Df ( z ) C 2 j y j 1 d ( y;z ) ; where C 0 1 and C 2 aretwopositiveconstants.Forany y;z 2 set y i := f i ( y )and 72 z i := f i ( z ).Bydirectcalculation,onehasthat d ( y i ;z i ) 1 # d ( y i ;y i +1 ) 1 # D 1 j y i + y +1 i y i j 1 # = D 1 j y i j (1+ )(1 # ) = D 1 j y i j ; where D 1 isapositiveconstant.ItfollowsfromLemma3.3in[14]that(3.3.11)holds. Byusing(3.3.11)andthesameargumentinthecase m 3asabove,onecanshow (3.3.1)holdsfor m =2and0 << 1. Thiscompletestheproof. Proposition3.3.4. Theunstablemanifold W u ( p )andthestablemanifold W s ( p )aredense in M ,respectively. Proof. Itistoshowthat W u ( p )isdense.Similarargumentsalsoworkfor W s ( p ). Takeanyrectangle X withint X 6 = ; .Takeastrictlysmallerrectangle ^ X ˆ int X .It followsAssumption(2)thatthereis k> 0suchthat f k ( ^ X ) \ P 6 = ; .So,if k istly large,then f k ( X ) s -crosses P .Hence, f k ( W u ( p;P )) \ X 6 = ; .Thiscompletestheproof. 3.3.4. Givenanysubset E ˆ M ,thereturnmapisgivenby g = f ˝ ( x ) ( x ): M n E ! M n E ,where ˝ ( x )=min f i> 0: f i ( x ) 2 M n E g isthereturntimefunction withrespecttotheset E . Lemma3.3.3. ThereexistsanergodicinvariantBorelprobabilitymeasure g forthemap g ,whichhasabsolutelycontinuousconditionalmeasuresontheunstablemanifoldsof f . Proof. Firstofall,itistoshowtheexistenceofaninvariantmeasure,whichhasabsolutely continuousconditionalmeasuresonunstablemanifolds. Supposethat P =[ W u ( p ) ;W s ( p )],where isasmallpositiveconstant.Denote ^ P := f ( P ) n P .Ifthedimensionof M isbiggerthantwo,then ^ P isconnected.Set Q := W u ( x; ^ P ). 73 Denoteby Q theLebesguemeasureon Q ,and( g k Q )( E )= Q ( g k ( E )).Takealimitof thesequence 1 k P k 1 i =0 g i ( Q )intheweakstartopology,denotedby g .Itisevidentthat g isinvariant. Now,itistoshowthat g hasabsolutelycontinuousconditionalmeasuresonunstable manifolds. Foranysmallrectangle K in M n P ,eachcomponentof g i ( Q )isadisjointunionof W u leaves,whicharecontainedinsomeelementfromtheMarkovpartition,andifanycomponent of g i ( Q )intersects K ,thenit u -crosses K byAssumption(2)andthediscussionsusedin theproofofProposition3.3.4.Let ˆ i bethedensityof g i ( Q )withrespecttotheLebesgue measureon g i ( Q ),where Q istheRiemannianmeasure inducedon Q .Itfollowsfrom Proposition3.3.3thatforany x;y inthesamecomponentof g i ( Q ) \ K , D 1 ˆ i ( x ) ˆ i ( y ) D; where D isindependentof i .Itisevidentthatsimilarestimatesonthelimitdensitiescould beobtained. Finally,theergodicityof g withrespectto g canbederivedbythediscussionsinthe proofofLemma5.3in[16],wheretheapplicationofLemma5.1intheproofofLemma5.3 of[16]isreplacedbyAssumption(2). Thiscompletestheproof. Denote S := f 1 P n P ,where P = P 0 istheelementoftheMarkovpartition P containing p .Withoutlossofgenerality,assumethat P =[ W u ( p ) ;W s ( p )].Itisevidentthat S consists ofpoints x 2 M with ˝ ( x ) > 1,where ˝ istherstreturntimefunctionin 74 3.3.4.Set S ( k ) :=[ ˝ k ].So,onehasthat S = S (2) and S ( k +1) ˆ S ( k ) forany k 2. Further,onehasthat W s ( x;S ( k ) )= W s ( x;S )and W u ( x;S ( k ) ) ˆ W u ( x;S )forany x 2 S ( k ) . Foranyunstableleaf u 2 W u ( S ),denoteby u k = u \ S ( k ) .ByAssumption(5)and Lemma3.3.1,onehasthatthereexist D 0 > 0and D > 0suchthat D 0 k m 1 u ( u k ) D k m 1 ; (3.3.12) where u istheLebesguemeasurerestrictedto u . Finally,itistoshowTheorem3.2.1. Proof. Set R i := f x 2 M n P : R ( x )= i g .Denote := 1 X i =1 i 1 X j =0 f j ( g j R i ) ; where g isthemeasurespinLemma3.3.3.So, hasabsolutelycontinuousconditional measuresontheunstablemanifoldsbyLemma3.3.3. Let e S ( i ) betheprojectionof S ( i ) onto W u ( p;P )along W s .ByProposition3.3.2,one has ( S ( i ) ) ˇ ( e S ( i ) ) ; where ( e S ( i ) )isthevolumeortheLebesguemeasureof e S ( i ) restrictedto W u ( p;P ).This, togetherwiththefactthat f i ( S ( i ) )arepairwisedisjointsubsetsof P , isinvariant,(3.3.12), andLemma3.3.2,yieldsthat ( P ) ˇ 1 X i =1 ( f i ( S ( i ) ))= 1 X i =1 ( S ( i ) ) ˇ 1 X i =1 ( e S ( i ) ) ˇ 1 X i =1 i m 1 ; 75 whichisconvergentwhenever0 < 0forany P i ; P j 2 P ; (iii) (Localinvertibility)themap f : P i ! f ( P i )isinvertiblewithmeasurableinversefor any P i 2 P with ( P i ) > 0. ItfollowsfromAssumption(2)thatthisMarkovmapisirreducible. 78 3.4.2Polynomialdecayrates Inthissubsection,thelowerandupperboundsforthedecayratesoftheobservablefunctions fortheinducedsystem( f; M )isinvestigatedbyapplyingtherenewaltheory. Set f M := M n P .Recallthat g = f ˝ isthereturnmapon M n P and P = P 0 .It isevidentthat g on M n P inducesareturnmapfrom f M toitself,denotedby e f .It followsfrom p 2 int P that p 2 int P . Let P 0 = Pnf P 0 g betheMarkovpartitionof f M .Set T := T 0 _ P 0 ,where T 0 = f T k = [ ˝ = k ]: k =1 ; 2 ; g isapartitionintosetswiththesamereturntime. Theseparationtimeisgivenby s ( x; y ):=sup f k 0: e f i ( y ) 2 T ( e f i ( x )) ; 0 i k g ; 8 x; y 2 f M: Forany x 2 x and y 2 y ,itisalsoreasonabletoset s ( x;y ):= s ( x; y ). ItfollowsfromAssumption(3)thatthemap f isuniformlyhyperbolicoutsideofany neighborhoodofthepoint p .Onecould :=sup fk Df x j E u x k 1 ; k Df 1 x j E s x k 1 : x 2 M n P g ; (3.4.2) where k Df x j E u x k =sup v 2 E u x ;v 6 =0 j Df x v j j v j and k Df 1 x j E s x k =sup v 2 E s x ;v 6 =0 j Df 1 x v j j v j .Itisevi- dentthat 2 (0 ; 1). Next,itistointroduceaBanachspaceon M : L := f : ˆ f M; k k L := k k 1 + D < 1g ; (3.4.3) 79 where kk L isthenorm, D isasemi-normgivenby D :=sup x; y 2 f M j x ) y ) j s ( x; y ) ; > 0,min f 2 ;m 2 g 0,min f 2 ;m 2 g 0such thatforany 2L and 2 L 1 with ˆ f M ,onehas Cor n ; f; ) 1 X k = n +1 [ ˝>k ] Z Z CF % ( n ) k k 1 k k L ; (3.4.4) where 1 X k = n +1 [ ˝>k ]hasorder n ( % 1) , F % ( n )= O (1 =n 2 % 2 ),and % = m 1 . Proof. ItfollowsfromthediscussionsintheprevioussubsectionthattheMarkovmap ( M; B ; f; P )isirreduciblemeasurepreserving. Next,itistoapplyTheorem6.3in[8]toshow(3.4.4). First,itistoprovethat e f hasbigimageproperty.Thiscouldbederivedbythe oftheMarkovpartition P andthediscussionsaboutthethebigimagepropertyinSection 6.2in[8]. Second,itistoverifythatlog J ( e f )islocallyoldercontinuous,whichisintroducedin [40](seealso[1]). 80 ItfollowsfromProposition3.3.3thatlog J ( e f ) 2L .Byapplyingsimilararguments usedinLemma2inSubsection3.1in[45],onehasthat e f admitsanabsolutelycontinuous invariantmeasure e on f M withthedensityfunction e h withrespectto e ˛ ,andthedensity functionlog e h 2L andisboundedawayfrom0andy.Byuniquenessweknow that e istheconditionalmeasurementionedinthelastsubsectionwithrespectto f M . TheJacobianof e f withrespectto e isasfollows J e ( e f )= J ( e f ) e h e f e h : Bythefactthatlog J ( e f )andlog e h arein L ,onehasthat log J e ( e f )isalsoin L ,yielding that log J e ( e f )islocallyoldercontinuous. Now,itistoprovethatgreatestcommondivisorof f ˝ ( x ) ˝ ( y ): x; y 2 M g isone,and [ ˝>k ]= O (1 =k % ). Itfollowsfromourconstructionthatthegreatestcommondivisorof f ˝ ( x ) ˝ ( y ): x; y 2 M g isone.So,oneonlyneedstoestimate [ ˝>k ].Let u 2 W u ( S )beanyunstableleaf. Denoteby u theconditionalmeasureoftheSRBmeasure whenitisrestrictedto u .By Proposition3.3.3,thedistortionof f alonganyunstableleafisuniformlybounded.Similar conclusionsalsoworkforthedensityfunction u u . Hence,by(3.3.12),thereexist C 0 1 ;C 1 > 0suchthat C 1 n % u ( u n ) C 0 1 n % : BydirectintegrationandProposition3.3.2,onehasthatsimilarinequalitiesalsohold for [ ˝>n ]withtpositiveconstantcots,thatis,thereexisttwopositive 81 constants B 0 1 and B 1 suchthat B 0 1 n % [ ˝>n ] B 1 n % : (3.4.5) Itgivesthat 1 X k = n +1 [ ˝>k ]hastheorder n ( % 1) . ItfollowsfromTheorem6.3in[8]thatthestatementofthislemmaiscorrect. Theproofiscompleted. 3.4.3Polynomialdecayratesfor Inthissubsection,itistoestablishthepolynomialdecayratesofcorrelationfunctionfor thealmostAnosovsmsbyusingtheresultsinprevioussubsections. First,itistointroduceatypeofolderfunctions: H := : 9 H > 0s.t. j x ) y ) j H j x y j and ˆ M n P g ; where m 2, > 0,min f m 2 ; 2 g 0,min f m 2 ; 2 g m 1 1= % 1. 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