Localized escape rate and return times distribution

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Content LOCALIZEDESCAPERATEANDRETURNTIMESDISTRIBUTION by GinPark ADissertationPresentedtothe FACULTYOFTHEUSCGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (MATHEMATICS) August2023 Copyright2023 GinPark Abstract Weshowthatindynamicalsystemswithleft-mixingmeasures,thenumberofvisitstoasuitably chosentargetsetofvanishingmeasureconvergestoacompoundPoissonrandomvariablewhenthe timescaleisinthereciprocalofthevanishingmeasureusinggeneratingfunctionsandconvolution formulae.Weusethisresulttoprovethatallhigherorderlocalizedescaperatesareidenticaltothe firstorderone,whichistheextremalindexataperiodicpoint. iii iv Chapter1 Introduction Webeginbyintroducingsome(butnotall)ofthedefinitionsthatwillappearinthispaper(butnot in the order that they appear), as well as some historical background that led us to this discussion. Thosedefinitionsthatfailtomakeappearanceinthisintroductionwillbeproperlyaddressedasthe needtodosoarises. Let X be a probability space1 with a probability measure . A measurable map2 T on X is said be a transformation on X if its codomain is also X. We say that the transformation T is measure-preserving if for every measurable subsetU ofX, .T 1 U/D .U/. Often we have a transformationT onX given,andifsomeprobabilitymeasure⌫onX satisfiesforeverymeasurable U,⌫.T 1 U/D ⌫.U/,thenwesaythat⌫isaT-invariantmeasure.TheKrylov-Bogoliubovtheorem asserts that ifX is a compact metric space andT is continuous, thenT admits an invariant Borel probabilitymeasure. Forameasure-preservingtransformationT onX,iftheonlymeasurableU suchthatT 1 U D U is either a null set or X up to a null set3, then we say that T is ergodic (with respect to ). AlthoughthisisthedefinitionforT beingergodicwithrespectto,therearemanyotherequivalent characterizations,perhapsthemostusefulofwhichwepresenthere:T is ergodic if and only if for each measurableU 2X having.U/>0, S 1 iD0 T i U DX[Y for some null setY. A point x 2 X is said to be a periodic point if there exists some positive integer m such that 1Infact,anyspaceendowedwithafinitemeasurewilldo. 2T WX !X ismeasurablewithrespectto ifandonlyifforeverymeasurableU ✓X,T 1 U ismeasurable. 3ThismeansthatitisoftheformX[Y whereY isanullset. 1 T m xDx.ThesmallestpositiveintegermforwhichT m xDx istheminimalperiodofx.Apoint x2 X issaidtobe eventually periodicifforsomepositiveintegern,T n x isaperiodicpoint.For apointx2X,its(positivesemi-)orbit istheset O C .x/Dπ T k xWk2Z C º: (1.1) Hencetheperiodicandtheeventuallyperiodicpointsarecharacterizedbyhavingafiniteorbit. SupposeA⇢2 X isameasurable,finitepartitionofX.Thatis,jAj<1, S A2A ADX,every A2A is measurable, and for allA;B 2A,A\B D¿. SupposeT is not invertible. We denote byT j Athej-th pullback of the partitionA: T j ADπ T j AWA2Aº: (1.2) NotethatthisitselfisapartitionofX.ThendenotebyA n itsn-th join n 1 _ jD0 T j AD ´ n 1 \ jD0 A j WA j 2T j A µ : (1.3) IfT isinvertible,thenforthen-thjoin,weshouldconsiderthetwo-sidedintersections: n 1 _ jD1 n T j AD ´ n 1 \ jD1 n A j WA j 2T j A µ : (1.4) In either case, each element ofA n is called an n-cylinder. Observe thatA n is a partition of X still4,andsoforeachx2X,thereexistsauniquen-cylinder,whichwedenotebyA n .x/,suchthat x 2 A n .x/. We then say that a partition is a generating partition if T 1 nD1 A n .x/Dπ xº for each x2X. 4In fact,A n is a finer partition ofX thanA m in the sense that for everyA2A n , there is someB 2A such that A✓B ifn>m. 2 ForasubsetU ofX,the entry/return time function⌧ U isdefinedas ⌧ U .x/D inf j π j 1WT j x2Uº: (1.5) That is, this is the first time that a particular element x enters or returns to U. If we consider the restriction of ⌧ U toU, then it is strictly the return time. The Poincaré recurrence theorem asserts thatifX isaprobabilityspace(ormoregenerallyanymeasurespaceendowedwithafinitemeasure) andT is a measure-preserving transformation, then for almost everyx 2 U, ⌧ U .x/ <1. That is, almosteverypointreturnstoU infinitetime.Itfurtherassertsthatalmostallpointsreturninfinitely often. As mentioned, ⌧ U .x/ counts the first time the elementx enters or returns (depending whether x2U ornot).Wecanthenextendedthisnotionabittodefinethek-th order return time function ⌧ k U inductivelyasfollows:First,⌧ 1 U .x/D ⌧ U .x/,andforeverypositiveintegerk>1, ⌧ k U .x/D ⌧ U ıT ⌧ k1 U .x/ x: (1.6) Namely,⌧ k U .x/istheminimumtimerequiredforx tomakek visitstothesetU. If in addition we assume thatT is ergodic, then we have Kac’s theorem at our disposal, which assertsthat Z U ⌧ U .x/d.x/ D1; (1.7) whose implication is that on average, ⌧ U .x/ D 1=.U/. We shall call the time scale of 1=.U/ Kac’s time scale. Crudely speaking, this is the amount of time one expects to wait to witness a returnpurelyduetoergodicity. From this point on, we shall assume that all transformations are ergodic with respect to the measure with which the transformation is measure-preserving.5 A measure-preserving dynamical 5Infact, we will not haveto explicitly state ergodicityas wewill assumethat ourmeasure isl LisaQuintana show with Pankaj Nashville fuck fuck that fuck that fuck that Nada throwing up Zaza Ashana helpeft- mixing, whose definitionwillappearmomentarily,andthemixingconditionimpliesergodicity. system is the quadruple .X;;M;T/ where .X;;M/ is a measure space, and T is a measure- preserving(withrespectto)transformation.SinceT isatransformation,itfollowsthatT n xexists for every positive integer n. That is, no element x leaves the system, and for that reason, we call the system a closed one. But one can imagine a situation in which we place a (measurable) “hole” U inX, and we no longer keep record of the orbit ofx once its orbit contains a point ofU. That is,weconsidertheelementx asbeing“lost”thefirsttimeitentersU.Wecanthentrytoquantify the rate at which the elements fall into this hole. To be precise, a point x falling into the hole U meansthatthereexistssomepositiveintegerj suchthatT j x2U.NotethatifT isergodic,andif .U/>0,then S 1 jD0 T j U DX uptoanullset,i.e.,givenenoughtime,almosteverythingwill fallintothehole.Henceausefulquantitytodefineisthe escape ratetoU,whichisdefinedby ⇢ U D lim t!1 1 t jlog.⌧ U >t/j; (1.8) ifthelimitexists. Aneasypropertytodeduceisthat,ifU ✓V,then⇢ U  ⇢ V .Thatis,intuitively,thelargerthe hole, the higher the escape rate into that hole. Indeed, observe that if U ✓ V, then .⌧ U >t/ .⌧ V >t/,hencetheinequalityfollows.Butnotethattwosetscanhavethesameescaperateeven whenoneisapropersubsetoftheother:TakeU[T 1 U,andobservethat⇢.U[T 1 U/D ⇢.U/. Notethatbyergodicity,U ¨U [T 1 U. We can then easily generalize this notion to the higher order cases. Namely, we don’t stop tracking the orbit ofx as soon as it falls intoU, but rather, wait until it makes itsk-th fall intoU and then remove it from the system. We then require a dierent escape rate, the k-th order escape ratetoU,whichwedefineasfollows: ⇢ k U D lim t!1 1 t jlog.⌧ k U >t/j; (1.9) again,ifthelimitexists.Asbefore,set⇢ 1 U D ⇢ U . Theescaperates,bethemthefirstorderorthehigherorder,aremeasuredundertheassumption 4 thatthe“targetset”U remainsunchanged,andinparticular,itisofpositivemeasure.Andofcourse, asthedefinitionstands,itisimpossibletodefinetheescaperateforanullsetas π xW ⌧ k U .x/>tº✓ 1 [ iD1 T i U; (1.10) andthelatersetisanullsetbythemeasure-preservingnatureofT. That is, what if we insist on knowing the “local” behavior? Since we cannot deal directly with thecasethatU ispossiblynull,wecaninsteadimagineasequenceofnested,positivemeasuresets converging to the null target set U. That is, we imagine a sequence of sets .U n / 1 nD1 , U nC1 ✓ U n , suchthat 1 \ nD1 U n DU: (1.11) Since each U n has a positive measure, it is possible, if the limit exists, to define ⇢ k U n for each U n . Butifwesimplytakethelimitasntendstowardsinfinity,namely,ifwesimplytake lim n!1 ⇢ k U n D lim n!1 lim t!1 1 t jlog.⌧ k U n >t/j; (1.12) itmightbeameaninglessonesince⇢ k U n mightblowup.Soitmightbenecessarytoproperlyscale theescaperateasnincreases.Weproposethefollowingformulaforthedefinitionofthek-thorder localized escape rate: ⇢ k .x/D lim n!1 ⇢ k U n .U n / D lim n!1 1 .U n / lim t!1 1 t jlog.⌧ k U >t/j; (1.13) where .U n / 1 nD1 , U n 2 .A n /, is any sequence of nested sets such that T 1 nD1 U n Dπ xº. The simplest of such examples would be the sequence .A n .x// 1 nD1 of a generating partition. Part of what Haydn and Yang showed in [14] is that the above limit whenk D 1, under some reasonable assumptions,isindependentofthechoiceof.U n / 1 nD1 . The quantity in (1.13), in general, is a dicult one to compute as it is a double limit. The way inwhichthistechnicaldicultywasovercomein[14]forthefirst-ordercaseisfirstbycomputing thesinglelimit lim n!1 jlog.⌧ 1 U n >s n /j s n .U n / (1.14) with the auxiliary quantity s n , under some mild assumptions on the rate of convergence of s n in relationtothatof.U n /given.Theythengeneralizedthistothedoublelimitcasebytheuseofthe so-called“convolutionlemma.”Inthethirdsection,wecomputethehigherorderlocalizedescape rates under the assumptions that are very similar to those of [14], using the generalization of the convolutionlemmathattheyusedin[14]. Throughoutthispaper,itwillbeassumedthattheunderlyingspaceX isashiftspacewiththe metricd suchthat d..a n / 1 nD1 ;.b n / 1 nD1 /D2 m ; (1.15) where m D inf n π n W a n € b n º. Observe that this is a compact metric space, and hence if T is continuous,wethenhaveaT-invariantmeasurebyKrylov-Bogoliubov.(Infact,if isergodicas well, then this measure is extreme in the sense that there are no twoT-invariant measures 1 and 2 and˛2.0;1/suchthat D ˛ 1 C.1 ˛/ 2 .) 6 Chapter2 TheLimitingDistributionof⇣ t=.U n / U n Let.X;;M;T/beameasure-preservingdynamicalsystem.Definethediscreterandomvariable ⇣ s U WX !Z by ⇣ s U .x/D bsc X jD1 T j U .x/: (2.1) Observethatif⇣ s U isrestrictedtothesetU,itgivesthenumberoftimesthataparticularelementx returnstoU inthetimespanofs.Wecallthiss the cuto time .Thendenoteby1 s l themeasure s l D .⇣ s U Dl/: (2.2) Thatis,itisthemeasureofthesetofelementsthatmake exactlyl entriesintoU withinthecuto timeofs.Evidentlythisquantitytakesonintegervaluesbetween0andbsc. Consider now a sequence of nested measurable sets .U n / 1 nD1 such that .U n / ! 0. Given a measurablefinitepartitionAofthespace,assumethat is left-mixing2 withrespecttoA.That is,foreveryA2.A n /andB 2. S j A j /, j.A \T n k B/ .A/.B/j .A/.k/; (2.3) 1Note that we no longer denote the set U to which x returns, so it must be understood from the context what set U this notation s l is referring to. Most often it will be pretty clear what U is, but if not, it will be given explicitly elsewhere. 2Alternatively, one can prove all of the following propositions assuming that is right -mixing whose definition isasfollows:Underthesameassumptionsoftheleft- mixingcase,j.A \T n k B/ .A/.B/j .B/.k/ . 7 and.k/ &0.Assumefurtherthat isleft-mixingwith decayingatleastpolynomiallywith somepowerp.Thatis,.x/ DO.x p /.Supposenowthats,thecutotime,isgivenasafunction of.U n /.Inparticular,supposethatforsomepositiveconstantt, sDs n D t .U n / : (2.4) That is, the time scale is that of Kac. For each fixed n, therefore fixed s n and U n , ⇣ s n U n is some random variable with the distribution that is the function of the dynamical system, as well as the geometryofU n .Wethenseektofindthelimitingdistributionof⇣ t=.U n / U n asn!1. 2.1 O ˛ k sand k s Some assumptions and preliminary results for answering our question are from [12], which we summarizeinthissection.Lets>0,andlet.U n / 1 nD1 beanestedsequenceofmeasurablesetssuch that T 1 nD1 U n isanullset.Put O ˛ 1 .s;U n /⌘1,andforeveryintegerk>1, O ˛ k .s;U n /D .π ⌧ k 1 U n sº\U n / .U n / : (2.5) Assume that for all s large enough, O ˛ k .s/D lim n!1 O ˛ k .s;U n / exists for each positive integer k. Observe that for each s fixed, O ˛ k .s/ is a decreasing monotone sequence in k as π ⌧ kC1 U n  sº✓ π ⌧ k U n sº,andalsoforeachk fixed,ass!1, O ˛ k .s/%O ˛ k becauseπ ⌧ k U n tº✓π ⌧ k U n sºwhen t s.Henceitfollowsthat O ˛ k O˛ kC1 foreachk.Notethat O ˛ k isafunctionofboththedynamical systemandthegeometryof T 1 nD1 U n andofthesequence.U n / 1 nD1 .3 Similarly,foreachpositiveintegerk,put k .s;U n /D .⇣ s U n Dk/ .⌧ U n s/ ; (2.6) 3These quantities do not necessarily exist for any null set T 1 nD1 U n and a sequence .U n / 1 nD1 “converging” to T 1 nD1 U n ,norisitatrivialaairtocomputeitevenwhentheydoexist.Inalatersection,wecomputethesequantities forsomenicelybehavingsystemsand T 1 nD1 U n s. 8 andput k .s/D lim n!1 k .s;U n /,providedthatthelimitexists.Put˛ k DO ˛ k O˛ kC1 .InTheorem 2of[12],theyprovedthatif P 1 kD1 kO ˛ k <1,then k D ˛ k ˛ kC1 ˛ 1 ; (2.7) andthat(accordingtoRemark7of[12]) lim s!1 lim n!1 .⇣ s U n Dk/ s.U n / D˛ 1 k : (2.8) Therefore,if˛ 1 and k sareknown,andifsDs n !1asn!1,thenwecanwrite .⇣ s n U n Dk/D.1Co.1//˛ 1 s n .U n / k : (2.9) NotethatbyputtingsDs n ,wearecomputingthedoublelimit(2.8)alongaparticularpath.Butit isnotaproblem,asweassumedthatthedoublelimit˛ 1 k existsforeachk. 2.2 TheCompoundPoissonDistribution SupposethatP ⇠ Pois./ isaPoissonrandomvariable,andthatX 1 ;X 2 ;:::arepositive,integer- valued,independent,andidenticallydistributedrandomvariablesthatareallindependentfromP. Therandomvariable W D P X iD1 X i (2.10) iscalledthe compound Poisson random variablewithP andX i s. ObservefromthedefinitionthattheprobabilitymassfunctionofthecompoundPoissonrandom variablegivenin(2.10)isgivenby .W Dk/D k X iD1 .P Di/.S i Dk/De k X jD1 i iä .S i Dk/; (2.11) whereS i D P i jD1 X j .Ifinadditionweknowthat.X j Dk/D k ,thenwecanwrite .S i Dk/D i X jD1 X j Dk ! D [ P i jD1 k j Dk i \ jD1 π X j Dk j º D X P i jD1 k j Dk i Y jD1 k j ; (2.12) wherethelastequalityfollowsfromindependence. Special distributions for the random variables X i s give rise to special distributions for the corresponding compound Poisson random variables. A notable kind is when the X i s are geomet- rically distributed with the parameter ✓. In that case, the distribution ofW is characterized by the probabilitymassfunction .W Dk/De k X jD1 ✓ k j .1 ✓/ j j jä k 1 j 1 ! ; (2.13) and .W D 0/ D e . The random variable with this distribution is called the Pólya-Aeppeli random variableorthe geometric Poisson random variablewithparameters and✓. Without delaying any further, we present the main theorem of this section, but do note that its statementinvolvesalemmathathasnotmadeitsappearanceyetinthispaper.Forthatreason,when wehavegatheredalllemmatatoprovethefollowingtheorem,wewillrestateit. Theorem 1. Lett>0 be fixed. LetA be a finite generating partition of .X;;M;T/, and let be left- mixing with .x/ D O.x p /. Suppose .U n / 1 nD1 is a sequence of nested sets such that U n 2.A n /and.U n /!0.Let.s n / 1 nD1 beasequenceofpositivenumbersdivergingtoinfinityin suchawaythatitispossibletochooseˇ satisfying(2.82)and(2.84).Assumethatforthissequence .s n / 1 nD1 , k s as defined in 2.1 all exist and (2.9) holds. Suppose W is the compound Poisson random variable with the Poisson random variable P ⇠ Pois.˛ 1 t/ and the identically distributed independent random variables X i s that are also independent from P, such that .X i D k/ D k . Then as n!1, ⇣ t=.U n / U n as defined in (2.1) converges in distribution to the random variableW. 10 2.3 TheConvolutionFormula:TheFirstOrderCase ApivotallemmatoproveTheorem1istheso-called“convolutionformula,”thefirstorderversion ofwhichwasprovenin[14].Thestatementofthefirstorderversionisasfollows: Lemma2(H+YLemma4:TheFirstOrderConvolutionFormula). Let.X;;M;T/beameasure- preserving dynamical system, and letA be its finite generating partition. Let0sCt/ .⌧ U >t/.⌧ U >s/j .⌧ U >t Å/.2Å.U/ C2.Å n//: (2.14) For our purpose, we need a generalized version of (2.14).4 Here we present a skeleton proof of the first order case as that of the generalized version which we will present later resembles this skeleton proof very closely. To prove, consider the time gap Œt;t CÅç, and consider the triangle inequalityontheexpression j.⌧ U >sCt/ .π ⌧ U >tº\π ⌧ U ıT tCÅ >s Åº/ C.π ⌧ U >tº\π ⌧ U ıT tCÅ >s Åº/ .⌧ U >t/.⌧ U >s Å/ C.⌧ U >t/.⌧ U >s Å/ .⌧ U >t/.⌧ U >s/j: (2.15) Toboundthefirstquantity(thefirstlineof(2.15)),observethat j.π ⌧ U >tº\π ⌧ U ıT tCÅ >s Åº/ .⌧ U >sCt/j D .π ⌧ U >tº\π ⌧ U ıT t  Åº/  .⌧ U >t/..⌧ U  Å/C.Å n//  .⌧ U >t Å/.Å.U/ C.Å n//: (2.16) This is relatively straightforward in that the setπ ⌧ U >sCtº, the set of elements that make no 4Wewillmakeexplicitwhatthe“generalizedversion”preciselymeansinthefollowingsection. returntoU inthefirstsCt amountoftime,isasubsetofπ ⌧ U >tº\π ⌧ U ıT tCÅ >s Åº,theset of elements that make no return toU in the firstt amount of time, and no returns in the timespan ŒtCÅt;sCtç. The former is the subset of the latter that consists of those elements that make no returntoU inthetimegapŒt;tCÅtç. Theboundonthesecondquantity(thesecondlineof(2.15))isgivenbythemixinginequality (2.3): j.π ⌧ U >tº\π ⌧ U ıT tCÅ >s Åº/ .⌧ U >t/.⌧ U >s Å/j  .⌧ U >t/.Å n/  .⌧ U >t Å/.Å n/; (2.17) andthatonthelastquantity(thelastlineof(2.15))isgivenby j.⌧ U >t/.⌧ U >s Å/ .⌧ U >t/.⌧ U >s/jDj.⌧ U >t/.s Å<⌧ U s/j  .⌧ U >t/Å.U/  .⌧ U >t Å/Å.U/: (2.18) Theupshotofthisformulaisthatwecantreataneventthathappensoveranextendedperiodof time as if it happened over multiple shorter periods in succession, modulo a controllable amount oferror. 2.4 TheConvolutionFormula:TheHigherOrderCase Sowhatwouldbethemostimmediategeneralizationof(2.14)?Itwouldbeverytemptingtowrite j.⌧ k U >sCt/ .⌧ k U >t/.⌧ k U >s/j .⌧ k U >t Å/.2Å.U/ C2.Å n//: (2.19) 12 However, whether this inequality holds or not, it helps little in the way of proving Theorem 1. Instead,itishelpfultoseetheinequality(2.14)inthefollowingway: j.⇣ sCt U D0/ .⇣ t U D0/.⇣ s U D0/j .⌧ 1 U >t Å/.2Å.U/ C2.Å n//: (2.20) Without building any more suspense, we present the following lemma as the appropriate general- izationof(2.14): Lemma 3 (The Generalized Convolution Formula). Let .X;;M;T/ be a measure-preserving dynamical system, and letA be its finite generating partition. Let0t Å/.4Å.U/ C3.Å n//: (2.21) The form of the approximation for .⇣ tCs U D k/ is indeed that of discrete convolution, hence the name. Another notable deviation from (2.14) is the bounding term, in particular, 4Å.U/ C 3.Å n/, as opposed to the original 2Å.U/ C 2.Å n/. As it will turn out, the extra 2Å.U/ C .Å n/ comes from some of the relevant sets not being contained properly into others, therefore we had to consider their symmetric dierences, a consideration that was not requiredinthefirstordercaseasallthecontainmentswereproper. Proof. GiventhetimegapŒt;tCÅç,consider k X jD0 .π ⇣ t U Djº\π ⇣ s Å U ıT tCÅ Dk jº/: (2.22) This is the probability of having exactly j returns in the first t amount of time, and then having exactlyk j returnsinthetimespanŒtCÅ;s Åç.Notethatanythingcanhappeninthetimegap Œt;tCÅç. Put E D k [ jD0 π ⇣ t U Djº\π ⇣ s Å U ıT tCÅ Dk jº: (2.23) WecanpartitionthisE bythenumberofreturnsinthegapŒt;tCÅç.Inparticular,wecanwrite k [ jD0 π ⇣ t U Djº\π ⇣ s Å U ıT tCÅ Dk jºD.E\π ⌧ 1 U ıT t >Åº/[.E\π ⌧ 1 U ıT t  Åº/: (2.24) That is, we partitionE into two sets, one having no returns in the gap, and the other having at least one return in the gap. Observe that the former is a subset ofπ ⇣ tCs U D kº, since if no return occursinadditiontothealreadyexistingk returns,wehaveexactlyk returns.Thus j.⇣ tCs U Dk/ .E/j .π ⇣ tCs U Dkº4E/ D .⇣ tCs U Dk/ .E \π ⌧ U ıT t >Åº/C.E \π ⌧ U ıT t  Åº/ D .π ⇣ tCs U Dkº\.E\π ⌧ U ıT t >Åº/ c /C.E \π ⌧ U ıT t  Åº/ D .π ⇣ tCs U Dkº\.E c [π ⌧ U ıT t >Åº c //C.E \π ⌧ U ıT t  Åº/ D .π ⇣ tCs U Dkº\.E c [π ⌧ U ıT t  Åº//C.E \π ⌧ U ıT t  Åº/  .π ⇣ tCs U Dkº\π ⌧ U ıT t >Åº/C.E \π ⌧ U ıT t  Åº/: (2.25) Thefirstmeasure,.π ⇣ tCs U Dkº\π ⌧ U ıT t  Åº/,canbeapproximatedasfollows: .π ⇣ tCs U Dkº\π ⌧ U ıT t  Åº/ .π ⇣ t Å U kº\π ⌧ U ıT t  Åº/ D k X jD0 .π ⇣ t Å U Djº\π ⌧ U ıT t  Åº/: (2.26) Thatis,ifthetotalnumberofreturnsadduptok,thenitmustbethatduringthefirstt Åamount 14 oftime,nomorethank returnsshouldoccur.Sinceπ ⇣ t Å U D jºisin ..A n / t Å /D.A nCt Å /, andsincefortheeventπ ⌧ U  Åº,wehave π ⌧ U ıT t <ÅºDT t π ⌧ U <ÅºDT n tCÅCn Å π ⌧ U <Åº; (2.27) wecanapproximatethefirstmeasureusingthe-mixingproperty: k X jD0 .π ⇣ t Å U Djº\π ⌧ U ıT t  Åº/ k X jD0 .π ⇣ t Å U Djº\T t π ⌧ U  Åº/  k X jD0 .⇣ t Å U Dj/..⌧ U  Å/C.Å n// D .⌧ kC1 U >t Å/..⌧ U  Å/C.Å n//: (2.28) Toapproximatetherest,wemakeaverycrudeestimatethat EDπ ⇣ t U C⇣ s Å U ıT tCÅ Dkº⇢π ⇣ t Å U kºD k [ jD0 π ⇣ t Å U Djº; (2.29) whichletsusconcludethat .E \π ⌧ U ıT t  Åº/ k X jD0 .π ⇣ t Å U Djº\π ⌧ U ıT t  Åº/  .⌧ kC1 U >t Å/..⌧ U  Å/C.Å n//: (2.30) Wealsohaveasthemiddleerrorterm k X jD0 .π ⇣ t U Djº\π ⇣ s Å U ıT tCÅ Dk jº/ k X jD0 .⇣ t U Dj/.⇣ s Å U Dk j/: (2.31) Observethatforeachsummand .π ⇣ t U Djº\π ⇣ s Å U ıT tCÅ Dk jº/ .⇣ t U Dj/.⇣ s Å U Dk j/; (2.32) sincewehave π ⇣ s Å U ıT tCÅ Dk jºDT t Å π ⇣ s Å U Dk jº; (2.33) combinedwiththefactthatπ ⇣ t U Djº2 ..A n / t /D.A nCt /,weget j.π ⇣ t U Djº\π ⇣ s Å U ıT tCÅ Dk jº/ .⇣ t U Dj/.⇣ s Å U Dk j/j .⇣ t U Dj/.Å n/; (2.34) usingthe-mixingproperty.Hencewhensummedoverj from0tok,weobtain k X jD0 .⇣ t U Dj/.Å n/D .⌧ kC1 U >t/.Å n/ .⌧ kC1 U >t Å/.Å n/: (2.35) Finally,observethatthelastpart ˇ ˇ ˇ ˇ k X jD0 .⇣ t U Dj/.⇣ s Å U Dk j/ k X jD0 .⇣ t U Dj/.⇣ s U Dk j/ ˇ ˇ ˇ ˇ D k X jD0 .⇣ t U Dj/j.⇣ s U Dk j/ .⇣ s Å U Dk j/j (2.36) can be approximated by making the observation similar to that we made for the first error term. Namely,webeginbynoticingthat j.⇣ s U Dk j/ .⇣ s Å U Dk j/j .π ⇣ s U Dk jº4π ⇣ s Å U Dk jº/: (2.37) Observe that the intersection ofπ ⇣ s U D k jº andπ ⇣ s Å U D k jº is precisely the setπ ⇣ s Å U D k jº\π ⌧ U ıT s Å >Åº,therefore π ⇣ s Å U Dk jº\π ⌧ U ıT s Å >Åº⇢π ⇣ s U Dk jº; (2.38) 16 meaningonepartofthesymmetricdierenceis π ⇣ s U Dk jºn.π ⇣ s Å U Dk jº\π ⌧ U ıT s Å >Åº/Dπ ⇣ s U Dk jº\π ⌧ U ıT s Å  Åº: (2.39) HencebymonotonicityandbythefactthatT ismeasure-preserving, .π ⇣ s U Dk jº\π ⌧ U ıT s Å  Åº/ .⌧ U ıT s Å  Å/ D .T Å s π ⌧ U  Åº/ D .⌧ U  Å/: (2.40) Fortheotherpartofthesymmetricdierence,bythesametoken, .π ⇣ s Å U Dk jº\π ⌧ U ıT s Å  Åº/ .⌧ U  Å/: (2.41) Hencethethirderrortermisboundedby k X jD0 .⇣ t U Dj/j.⇣ s U Dk j/ .⇣ s Å U Dk j/j k X jD0 2.⇣ t U Dj/.⌧ U  Å/ D2.⌧ kC1 U >t/.⌧ U  Å/ 2.⌧ kC1 U >t Å/.⌧ U  Å/: (2.42) Combined,thisgivesthefinalerrorestimate ˇ ˇ ˇ ˇ .⇣ tCs U Dk/ k X jD0 .⇣ t U Dj/.⇣ s U Dk j/ ˇ ˇ ˇ ˇ < .⌧ kC1 U >t Å/.4.⌧ U  Å/C3.Å n//: (2.43) To get the formula in (2.21), we use the estimate .⌧ U  Å/ Å.U/, which follows from the observationthat π xW ⌧ U  Åº✓ bÅc [ jD1 T j U; (2.44) hence .⌧ U  Å/ bÅc X jD1 .T j U/DbÅc.U/  Å.U/ (2.45) bythemeasure-preservingnatureofT. ⌅ 2.5 TheLimitingDistributionof⇣ t=.U n / U n :TheGeneratingFunc- tionMethod Forapositiveconstantt,putt=.U n /Dt n .Thatis,weassumethatthecutotimeisinKac’stime scale.Tofindthelimitingdistributionof⇣ t n U n istantamounttofindingthelimit t n k D .⇣ t n U n Dk/ foreverynonnegativeintegerk.Tothatend,wepresentawayofcomputing.⇣ s U Dk/usingthe convolutionformula(2.21)weprovedin2.4. LetU be some fixed measurable set. As before, denote by s k the measure .⇣ s U D k/. Since most of the times it will be evident what U we are referring to (which, later, will be an element of a sequence .U n / 1 nD1 ), we will suppress it and will simply write s k . For s fixed, consider the probabilitygeneratingfunction F s .z/D 1 X kD0 s k z k D 1 X kD0 s k z k : (2.46) Observe that sincej s k j 1 for every k, F s .z/ is an analytic function on any disk of radius less 18 than1centeredattheorigin.ObservefurtherthatF s .0/D s 0 D .⌧ 1 U >s/,andthat 1 lä d l dz l F s .z/ ˇ ˇ ˇ ˇ zD0 D s l : (2.47) Put ⌘.Å/D4Å.U/ C3.Å n/: (2.48) Recallthatbythegeneralizedconvolutionformula(2.21),wehave ˇ ˇ ˇ ˇ tCs k k X jD0 t j s k j ˇ ˇ ˇ ˇ < .⌧ kC1 U >t Å/.4Å.U/ C3.Å n// < ⌘.Å/ k X jD0 t Å j  ⌘.Å/: (2.49) Assume that Å is a function ofs, so ÅD Å.s/, whereupon ⌘ implicitly depends ons. Denote by Q ⌘thecomposition⌘ıÅ,sothat Q ⌘.s/D ⌘.Å.s//.Ifwedefine⇠.s;k;t/tobetheexacterror ⇠.s;k;t/D k X jD0 t j s k j tCs k ; (2.50) thenby(2.49),j⇠.s;k;t/jQ ⌘.s/uniformlyink andt.Wewillapplythistothecasewhent Das andwillput⇠ as k D ⇠.s;k;as/fora2N.Notethatforeverya,wehavet s,sotheconditionthat 01, onjzj<1, the analytic functionF s .z/ satisfies F s .z/ r DF rs .z/C r X kD2 G k .z/F s .z/ r k ; (2.60) whereG n .z/ is as defined in (2.56). Proof. Wehavealreadyshownthecasesforr D2;3.Supposethattheresultholdsforr.Thenby induction, F s .z/ rC1 DF s .z/ r F s .z/DF rs .z/F s .z/C r X kD2 G k .z/F s .z/ rC1 k : (2.61) ThetermF rs .z/F s .z/gives F rs .z/F s .z/D 1 X kD0 rs k z k ! 1 X kD0 s k z k ! D 1 X kD0 z k k X jD0 rs j s k j D 1 X kD0 . .rC1/s k C⇠ rs k /z k DF .rC1/s .z/CG rC1 .z/ (2.62) whichprovestheformulaclaimed. ⌅ Our objective is to take the k-th derivative of (2.60) with respect to the complex variable z, evaluate it atz D 0, and obtain the formula for rs k for large r ands. Let us denote byE r s .z/ the errorfunction E r s .z/D r X kD2 G k .z/F s .z/ r k ; (2.63) sothat(2.60)takestheform F rs .z/DF s .z/ r E r s .z/: (2.64) Evidently (as being the finite sum of functions that are analytic onjzj<1) this error function is analyticonthediskjzj<1,andsowemayconsideritsderivativeofanarbitraryorder.Apotential 22 snaginexecutingthisidea(oftakingthek-thorderderivative)isthatalthoughtheerrortermitself maybehavewell(inthesensethatitsmaximummodulusonjzj<1remainssmall),itsderivatives may not. To show that the derivatives too behave well, we find an upper bound on the modulus of the derivative of the k-th order using the Cauchy Estimation Theorem. Recall that the Cauchy EstimationTheoremstatesthefollowing: Theorem 5 (The Cauchy Estimation Theorem). Suppose f is analytic on a neighborhood of the closed ballB.z 0 ;R/, and suppose that M ´ supπj f.z/jWjz z 0 jDRº<1: (2.65) Then ˇ ˇ ˇ ˇ d k dz k f.z/ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ zDz 0  käM R k : (2.66) Withthistheoreminmind,wepresentthefollowinglemma: Lemma6. LetU be a measurable set, and letE r s .z/ be defined for thatU as in (2.63). Then ˇ ˇ ˇ ˇ 1 kä d k dz k E r s .0/ ˇ ˇ ˇ ˇ DrQ ⌘.s/O.k/; (2.67) therefore ˇ ˇ ˇ ˇ rs k 1 kä d k dz k F s .z/ r ˇ ˇ ˇ ˇ zD0 ˇ ˇ ˇ ˇ DrQ ⌘.s/O.k/: (2.68) Proof. Let0<⇢<1befixed.UponapplyingtheCauchyestimateonthediskjzj ⇢,weobtain that ˇ ˇ ˇ ˇ d k dz k E r s .0/ ˇ ˇ ˇ ˇ  käM ⇢ k ; (2.69) where M D sup zWjzjD⇢ jE r s .z/j: (2.70) WenowlookmorecarefullyattheanalyticfunctionF s .z/.Sincethepowerseriesrepresentation of F s .z/ has only real coecients (that sum to 1), we have jF s .z/j F s .jzj/. Then under the assumptionthatjzj⇢<1, F s .jzj/ s 0 C 1 X kD1 s k jzj k  s 0 C 1 X kD1 s k ⇢ k  s 0 C 1 X kD1 s k ⇢ D .⌧ 1 U >s/C⇢.⌧ 1 U s/ < .⌧ 1 U >s/C.⌧ 1 U s/; (2.71) unless s 0 D1.ButtheninthatcaseF s .z/⌘1,whichconsequentlyimplies jE r s .z/j r X kD2 jG k .z/j Q ⌘.s/.r 1/ 1 ⇢ : (2.72) Sointhiscase,thetotalerrorinthek-thderivativeisboundedby ˇ ˇ ˇ ˇ d k dz k E r s .0/ ˇ ˇ ˇ ˇ  kä Q ⌘.s/.r 1/ .1 ⇢/⇢ k : (2.73) Inallothercases,put D .⌧ U >s/C⇢.⌧ U s/<1.Then jE r s .z/j r X kD2 jG k .z/jjF s .z/j r k  Q ⌘.s/ 1 ⇢ r X kD2 F s .jzj/ r k  Q ⌘.s/ 1 ⇢ r X kD2 r k < Q ⌘.s/.r 1/ .1 ⇢/ : (2.74) 24 Hencethetotalerrorinthek-thderivativeisboundedbythesamelimitingterm: ˇ ˇ ˇ ˇ d k dz k E r s .0/ ˇ ˇ ˇ ˇ  kä Q ⌘.s/.r 1/ .1 ⇢/⇢ k : (2.75) The factor .1 ⇢/ 1 ⇢ k attains the minimum on .0;1/ at ⇢ D k=.1Ck/, hence for that ⇢,we obtain.1 ⇢/ 1 ⇢ k Dk k .1Ck/ 1Ck .Therefore ˇ ˇ ˇ ˇ d k dz k E r s .0/ ˇ ˇ ˇ ˇ Q ⌘.s/.r 1/käk k .1Ck/ 1Ck Q ⌘.s/.r 1/kä.1 Ck/.1Ck 1 / k eQ ⌘.s/.r 1/.kC1/ä; (2.76) whereweusedthefactthat.1Ck 1 / k <e.Hencewededucethat ˇ ˇ ˇ ˇ 1 kä d k dz k E r s .0/ ˇ ˇ ˇ ˇ eQ ⌘.r 1/.s/.kC1/<erQ ⌘.s/.kC1/; (2.77) implying ˇ ˇ ˇ ˇ 1 kä d k dz k E r s .0/ ˇ ˇ ˇ ˇ DrQ ⌘.s/O.k/: (2.78) (2.68)followsfromusing(2.47)onF rs .z/. ⌅ Theresultspresentedin2.5thusfarhavebeenobtainedwithoutinvokinganypropertiesabout the underlying dynamical system. Indeed, the only property of the dynamical system being used, ifwecancallitthat,isthegeneratingfunctiongivenin(2.46)isaprobabilitygeneratingfunction. Lemma4isalgebraicinnature,andLemma6requiresonlyrudimentarycomplexanalysistheorems to prove. In order to apply this general method to our more structured scenarios, we need to prove anotherkeylemmathatwecalltheTwo-SequenceLemma. 2.6 TheTwo-sequenceLemma Thefollowinglemma,theTwo-sequenceLemma,isnamedsobecauseitinvolves“twosequences” thatarecloselyintertwined.Mostimportantly,webeginpayingcloseattentiontothespeedatwhich .U n /decays. Lemma 7 (The Two-sequence Lemma). LetA be a finite generating partition of .X;;M;T/, and let be left- mixing with .x/ D O.x p /. Suppose .U n / 1 nD1 is a sequence of nested sets suchthatU n 2.A n /and.U n /!0.Lett>0befixed.If.s n / 1 nD1 isasequenceofpositivereal numbers such thats n !1, and lim n!1 s 2 n .U n /D0; (2.79) then there exists a sequence.Q s n / 1 nD1 such that F Q s n .z/DF s n .z/; (2.80) and there exists an integer sequence.r n / 1 nD1 such that r n Q s n D t .U n / : (2.81) If in addition for the aforementioned .s n / 1 nD1 there exists some ˇ;c 2 .0;1/ such that for all n suciently large, cs ˇ n s ˇ n n; (2.82) then t=.U n / k D 1 kä d k dz k F s n .z/ r n ˇ ˇ ˇ ˇ zD0 CO.maxπ s ˇ 1 n ;s 1 pˇ n .U n / 1 º/: (2.83) 26 Hence if in additionˇ can be chosen so as to satisfy lim n!1 1 s 1Cpˇ n .U n / D0; (2.84) then lim n!1 t=.U n / k D lim n!1 1 kä d k dz k F s n .z/ r n ˇ ˇ ˇ ˇ zD0 : (2.85) A remark is in order: The hypotheses, especially those that limit the converging speed of s n (namely, (2.79), (2.82), and (2.84)) may seem at first a bit artificial, but they are there to carefully calibrate the converging speed ofs n and the modulus ofˇ. Note that (2.82) governs the minimum rate at which s n blows up: s ˇ n must be faster than linearly. Here, if ˇ 0 satisfied (2.82), then any ˇ>ˇ 0 will also satisfy .2:82/. (2.84) also gives a condition on ˇ, but this time, in relation to p. Thecondition(2.79)putsalimitonthemaximumspeedatwhichs n cangrowinrelationtothatof .U n /. Proof. Let.O s n / 1 nD1 beasequencesuchthat O s n D bs n cCds n e 2 : (2.86) ThenO s n isanintegeranditequalss n ifandonlyifs n isaninteger,andifs n isnotaninteger,then itsdecimalpartis0:5.Moreover,bO s n cDbs n c,hence F O s n .z/DF s n .z/; (2.87) therefore O s n k D s n k : (2.88) Forthissequence.O s n / 1 nD1 ,define.O r n / 1 nD1 by O r n D t O s n .U n / : (2.89) Put  n DO r n bOr n c: (2.90) Forthis n ,put ı n D  n O s n O r n  n : (2.91) Then .O r n  n /.O s n Cı n /DO r n O s n  n O s n CO r n ı n  n ı n D t .U n / : (2.92) Observethatby(2.79),andbytheobservationthatforallnsucientlylarge O s n <2s n , ı n D  n O s n O r n  n  2O s n O r n D 2O s 2 n .U n / t  8s 2 n .U n / t !0: (2.93) SincebO s n C✏cDbO s n cDbs n cforevery✏ <1=2,wehavebO s n Cı n cDbs n c,therefore F O s n Cı n .z/DF s n .z/: (2.94) Moreover,.O r n  n / 1 nD1 D.bO r n c/ 1 nD1 isanintegersequence.IfweputQ s n DO s n Cı n andr n DbO r n c, then r n Q s n D t .U n / ; (2.95) 28 andsobyLemma6, t=.U n / k D r n Q s n k D 1 kä d k dz k F Q s n .z/ r n ˇ ˇ ˇ ˇ zD0 Cr n Q ⌘.Q s n /O.k/ D 1 kä d k dz k F s n .z/ r n ˇ ˇ ˇ ˇ zD0 Cr n Q ⌘.Q s n /O.k/: (2.96) Recallthatwedefinedin(2.48)that ⌘.Å/D4Å.U/ C3.Å n/; (2.97) whereÅisthegaplengthcorrespondingtothetotaltimeintervals.Here,thatroleisservedbyQ s n , soweneedtochooseÅD Å n sothatitsatisfies0<Å n < Q s n =2ifwewishtoapplythegeneralized convolutionformula(2.21).Tothatend,put Å n D Å.Q s n /Ds ˇ n : (2.98) Thentrivially lim n!1 Å n s n Ds ˇ 1 n D0 (2.99) byconstruction,henceforallnsucientlylarge, 0<Å n 1. Suppose .U n / 1 nD1 is a sequence of nested sets such that U n 2.A n / and assume that.U n /DO.n / for some>2. Lett>0 be fixed. Then: 30 1. There exist positive real numbers˛2.0;1=2/ andˇ2.0;1/ such that > 1 ˛ˇ (2.105) and p> ✓ 1 ˛ 1 ◆ 1 ˇ (2.106) simultaneously, and 2. For any˛;ˇ chosen according to (2.105) and (2.106), if we define the sequence.s n / 1 nD1 by s n D .U n / ˛ ; (2.107) and.r n / 1 nD1 as aorded by Lemma 7. Then lim n!1 t=.U n / k D lim n!1 1 kä d k dz k F s n .z/ r n ˇ ˇ ˇ ˇ zD0 : (2.108) Proof. The first part of the lemma is trivial, as we can select˛ˇ arbitrarily close to 1=2 and 1, respectively,tosatisfy(2.105)and(2.106). To prove the second part, it suces to show that the given sequence .s n / 1 nD1 satisfies (2.79), (2.82),and(2.84). The first condition is (2.79) is trivially met, as ˛ < 1=2. To prove that (2.82) holds, observe thatbythehypothesisthat.U n /DO.n /,forallnsucientlylarge,thereexistssomepositive constantc such that .U n /<cn . Then we have .U n / ˛ˇ <c 0 n ˛ˇ forc 0 D c ˛ˇ >0. Hence forallnsucientlylarge, n.U n / ˛ˇ <c 0 n 1 ˛ˇ !0 (2.109) as1 ˛ˇ < 0.Thus1 n.U n / ˛ˇ !1asn!1,meaning .U n / ˛ˇ n> 1 3.U n / ˛ˇ ; (2.110) forallsucientlylarge n.Thisisbecause .U n / ˛ˇ n> 1 3.U n / ˛ˇ () 1 n.U n / ˛ˇ > 1 3 ; (2.111) whichholds. Toverifythatthelastcondition(2.84)ismet,observethat s 1Cpˇ n .U n /D .U n / 1 ˛ p˛ˇ : (2.112) By(2.106),wehave p˛ˇ >1 ˛; (2.113) whichsubsequentlyimpliesthat1 ˛ p˛ˇ <0.As.U n /!0,wehavethedesiredresult. ⌅ 2.7 ProofofTheorem1 WearenowreadytoproveTheorem1whichforthereader’sconveniencewerestate: Theorem9(Theorem1). Lett>0befixed.LetAbeafinitegeneratingpartitionof.X;;M;T/, and let be left- mixing with .x/ D O.x p /. Suppose .U n / 1 nD1 is a sequence of nested sets such thatU n 2 .A n / and .U n /! 0. Let.s n / 1 nD1 be a sequence of positive numbers diverging toinfinityinsuchawaythatitispossibletochooseˇ satisfying(2.82)and(2.84).Assumethatfor this sequence.s n / 1 nD1 , k s as defined in 2.1 all exist and (2.9) holds. Suppose W is the compound Poisson random variable with the Poisson random variable 32 P ⇠ Pois.˛ 1 t/ and the identically distributed independent random variables X i s that are also independent from P, such that .X i D k/ D k . Then as n!1, ⇣ t=.U n / U n as defined in (2.1) converges in distribution to the random variableW. Proof. Let .s n / 1 nD1 be chosen according to the hypothesis, and let .r n / 1 nD1 be a sequence oered byLemma7.WeshallassumethroughouttheproofthatsDs n andr Dr n ,toreduceclutter. BythegeneralizedLeibnizrulefromCalculus, d k dz k F s .z/ r D X P r jD1 k j Dk k j 0 kä Q r jD1 k j ä r Y iD1 F .k i / s .z/; (2.114) whereeachk j 0.By(2.47),uponsubstitutingzD0,weobtain d k dz k F s .z/ r ˇ ˇ ˇ ˇ zD0 D X P r jD1 k j Dk k j 0 kä Q r jD1 k j ä r Y iD1 F .k i / s .0/ D X P r jD1 k j Dk k j 0 kä Q r jD1 k j ä r Y iD1 k i ä s k i Dkä X P r jD1 k j Dk k j 0 r Y iD1 s k i ; (2.115) whereweused.2:47/towriteF .k i / s .0/Dk i ä s k i . To compute, we proceed with the following combinatorial argument: We first choose amongr manyk i sl manythatwillbepositive,ofwhichtherearer choosel manywaystoaccomplish.But since the positive k i s must sum to k, we then require 1  l  k. The remaining r l many k i s willbethen0,hence X P r jD1 k j Dk k j 0 r Y iD1 s k i D k X lD1 r l ! . s 0 / r l X P l jD1 k j Dk k j 1 l Y iD1 s k i D k X lD1 r.r 1/.r lC1/ lä . s 0 / r l X P l jD1 k j Dk k j 1 l Y iD1 s k i : (2.116) Observethat r.r 1/.r lC1/ lä D r l lä l 1 Y kD1 ✓ 1 k r ◆ D.1Co.1// r l lä ; (2.117) therefore(2.116)becomes X P r jD1 k j Dk k j 0 r Y iD1 s k i D k X lD1 r l ! . s 0 / r l X P l jD1 k j Dk k j 1 l Y iD1 s k i D.1Co.1//. s 0 / r k X lD1 r l lä . s 0 / l X P l jD1 k j Dk k j 1 l Y iD1 s k i : (2.118) Asassumed,sinceforeveryk i 1wehave s k i D.1Co.1//˛ 1 k i s.U n /; (2.119) thusfollowsthat l Y iD1 s k i D.1Co.1//˛ l 1 .s.U n // l l Y iD1 k i : (2.120) 34 Andsincer l Dt l =.s.U n // l ,thelastexpressionof(2.116)ultimatelybecomes k X lD1 r l ! . s 0 / r l X P l jD1 k j Dk k j 1 l Y iD1 s k i D.1Co.1//. s 0 / r k X lD1 .˛ 1 t/ l lä X P l jD1 k j Dk k j 1 l Y iD1 k i : (2.121) Since . s 0 / r D.1Co.1//.1 ˛ 1 s.U n // t=s.U n / D.1Co.1//..1 ˛ 1 s.U n // 1=˛ 1 s.U n / / t˛ 1 !e ˛ 1 t ; (2.122) thedesiredresultfollowsfrom(2.115): 1 kä d k dz k F s n .z/ r n ˇ ˇ ˇ ˇ zD0 D t=.U n / k !e ˛ 1 t k X lD1 .˛ 1 t/ l lä X P l jD1 k j Dk k j 1 l Y iD1 k i ; (2.123) meaningthat⇣ t=.U n / U n convergesindistributiontothecompoundPoissonrandomvariableW with P ⇠ Pois.˛ 1 t/,and.X i Dk/D k . ⌅ InChapter3,weputthistheoremtousetofindtheformulaforthehigherorderlocalizedescape rates. 2.8 Example:TheBaker’sMap In this section, we present examples in which we can explicitly compute O ˛ k s and k s, and we find the limiting distribution ⇣ t=U n U n for these examples. Throughout the examples, we will use, as the transformation,theBaker’smaponŒ0;1ç⇥Œ0;1ç,anddierentsequences .U n / 1 nD1 s. ConsidertheBaker’smaponI 2 DŒ0;1ç⇥Œ0;1çgivenby B.x;y/D 8 ˆ < ˆ : .x=2;2y/ ify1=2, ..xC1/=2;2y 1/ ify >1=2. (2.124) EndowI 2 withtherestrictionoftheLebesguemeasure onR 2 .ThenB WI 2 !I 2 isameasure- preserving,invertibletransformationasdetB D1locally.Put D 1 n;m D ≤ .x;y/W n 2 x< nC1 2 ; m 2 y< mC1 2 ≥ ; (2.125) andconsiderthepartition ADπ D n;m W0n;m<1;n;m2Nº: (2.126) That is,A is the set consisting of the four congruent squares of side length1=2. Then the joins of this partition are thedyadic squares. To wit, for someN 2 and0n;m<2 N , if we denoteby D N n;m theset D N n;m D ≤ .x;y/W n 2 N 1 x< nC1 2 N 1 ; m 2 N 1 y< mC1 2 N 1 ≥ ; (2.127) then A N Dπ D N n;m W0n;m<2 N 1 Wn;m2Nº: (2.128) Evidently this is a generating partition, as diamD N n;m ! 0 asN!1 for anyn;m. With respect tothispartition,theLebesguemeasure isleft- mixing: ˇ ˇ ˇ ˇ .A \T n k B/ .A/ .B/ ˇ ˇ ˇ ˇ  .k/: (2.129) Sincethemapisinvertible,itsucestoshowthisresultforan n-cylinderAandanm-cylinderB. 36 Alsoobservethatbyinvertibility, ˇ ˇ ˇ ˇ .A \T n k B/ .A/ .B/ ˇ ˇ ˇ ˇ D ˇ ˇ ˇ ˇ .T n A\T k B/ .T n A/ .B/ ˇ ˇ ˇ ˇ ; (2.130) and T n A is a full vertical strip (meaning its set of y-coordinates is I) having width 2 n 1 . Note that for any m-cylinder B, T k B is by the same token a full horizontal strip (meaning its set of x-coordinatesisI)havingwidth2 m 1 .Moreover,forallk m,wemusthave ˇ ˇ ˇ ˇ .T n A\T k B/ .T n A/ .B/ ˇ ˇ ˇ ˇ D0; (2.131) as the percentage of the set T k B takes in the set T n A, which is .T n A\T k B/=.T n A/, is exactly the same as thatT k B takes inI, which is .B/. This happens so long ask m. When k<m,thenonlyapartial(orevenno)eclipsingofT n A byT k B occurs,therefore ˇ ˇ ˇ ˇ .T n A\T k B/ .T n A/ .B/ ˇ ˇ ˇ ˇ 2.B/ D2 m 1 <2 k 1 : (2.132) Hence putting .k/ D 2 k 1 , we have (2.129) holding, and is decaying faster than any polyno- mial,so.x/ DO.x p /foranyp>0. Observe that .0;0/ and .1;1/ are the only fixed points of B. To identify all periodic points, observe that if we denote by B C and B the maps B W I ⇥Œ0;1=2ç ! Œ0;1=2ç⇥I and B C W I ⇥.1=2;1ç!.1=2;1ç⇥I givenby B .x;y/D 2 6 4 1=2 0 02 3 7 5 2 6 4 x y 3 7 5 ; (2.133) and B C .x;y/D 2 6 4 1=2 1 3 7 5 C 2 6 4 1=2 0 02 3 7 5 2 6 4 x y 3 7 5 ; (2.134) then any periodic point of (not necessarily minimal) period m must be a fixed point of some permutationoflengthmofB C andB .Forinstance,allperiodicpointsofperiod2arefixedpoints ofthemaps:B 2 C ,B 2 ,B C B ,andB B C .Forgeneralm,anym-compositionB m isoftheform B m .x;y/DAC 2 6 4 2 m 0 02 m 3 7 5 2 6 4 x y 3 7 5 ; (2.135) andsincethematrix 2 6 4 1 2 m 0 01 2 m 3 7 5 (2.136) isinvertible,theequationB m .x;y/D.x;y/alwayshasauniquesolution.Thusthereare2 m many periodicpointstotheBaker’smap,arisingfrom2 m manydierentpermutationsof B C andB of lengthm.Observefurtherthatforallm,themapB m islocallycontinuous(ane),thereforeinany neighborhoodofafixedpoint.x;y/ofB m whosey-coordinateisnotequalto1=2,thereexistsan opendiskofradius⇢centeredat.x;y/onwhichthemapiscontinuous.Denoteby.xC˛;yCˇ/ apointinthiscontinuouspatch.Theninthecoordinatescenteredatthefixedpoint.x;y/,onehas 2 6 4 2 m 0 02 m 3 7 5 2 6 4 xC˛ yCˇ 3 7 5 2 6 4 2 m 0 02 m 3 7 5 2 6 4 x y 3 7 5 D 2 6 4 2 m ˛ 2 m ˇ 3 7 5 : (2.137) Recallthedefinitionof O ˛ k : O ˛ k D lim s!1 lim n!1 .π ⌧ k 1 U n sº\U n / .U n / : (2.138) For every cuto time s fixed, by local continuity and by the fact that m is the minimal period (meaning for no0<j <m is B j .x;y/D .x;y/) we can choose N large enough so that for all n>N and0j<m,B j U n arealldisjoint.Thismeansforalln>N,noelementinU n makes its first return toU n until the periodic point does, which happens at timem. Thus we only need to considerthetimesthataremultiplesofm. 38 2.8.1 TheCasewhentheTargetSetsareSquares Suppose now that for a point.x;y/ with minimal periodm, we select as.U n / 1 nD1 the sequence of nestedsquaresoflengthl n thatdecreasesto0,whosecenterofmassisat.x;y/.Then O ˛ 2 D lim s!1 lim n!1 .π ⌧ 1 U n sº\U n / .U n / : (2.139) Let s D qm, and let q be a large natural number. Assuming that all B jm U n are disjoint for all n>N,wehave π ⌧ 1 U n qmº\U n D q [ jD1 U n \B jm U n : (2.140) Bythelocalcontinuityassumption,eachsquareU n underB jm ismappedtotherectanglewhose side lengths are 2 jm l n and 2 jm l n and the center of mass is at .x;y/. Hence the percentage of overlap,.U n \B jm U n /=.U n /,isgivenby .U n \B jm U n / .U n / D2 jm : (2.141) Moreover,if0<j N, returns to U n occur only at the multiplesofm.ThentheboundariesofU n anditsimageB jm U n aregivenasfollows: NotethatB jm islocallygivenby B jm .˛;ˇ/D 2 6 4 2 jm 0 02 jm 3 7 5 2 6 4 ˛ ˇ 3 7 5 ; (2.151) hencetheimageistheellipsewhosesemimajoraxisisalongthey-axisandisoflength2 jm ⇢,and whosesemiminoraxisisalongthex-axisandisoflength2 jm ⇢. Withoutlossofgenerality,assumethat⇢D1.DenotebyF theintersectioninthefirstquadrant betweentheboundaryofthediskandthatoftheimageellipse.Thecirclex 2 Cy 2 D1,underthe map B jm in the local coordinates, is mapped to the ellipse 2 2jm x 2 Cy 2 =2 2jm D 1. Solving this Figure 2.1: The first quadrant view of the boundary ofU n , which is a circle (blue), and that of its imageunderB jm U n ,whichisanellipse(orange). quadraticsystem,weobtainthattheintersectionF occursatthepointwhere 2 6 4 x y 3 7 5 F D 2 6 4 .2 2jm C1/ 1=2 2 jm .2 2jm C1/ 1=2 3 7 5 : (2.152) DenotebyE thepreimageofF underB jm .ThenthecoordinatesofE isgivenby 2 6 4 x y 3 7 5 E D 2 6 4 2 jm 0 02 jm 3 7 5 2 6 4 x y 3 7 5 F D 2 6 4 2 jm .2 2jm C1/ 1=2 .2 2jm C1/ 1=2 3 7 5 : (2.153) Note also that this happens to be the intersection between the unit circle and the image of it under B jm ,whichtheellipse2 2jm x 2 C2 2jm y 2 D1. DenotebyO theorigin.0;0/andbyAthepoint.1;0/.Thentheangle†EOA,whichwewill denoteby,isgivenas D arctan y E x E D arctan2 jm : (2.154) 42 Here,x E andy E representsthex-andthey-coordinateofE,respectively. DenotebyB thepoint.0;1/andbyC thepoint.2 jm ;0/,whichisthex-interceptoftheellipse. From the identity arctanx 1 D ⇡=2 arctanx, we observe that the sectorsBOF is congruent to thesectorEOA.Also,sincethemapisane,thelinesegment OE ismappedtothelinesegment OF,andthecircularsectorEOAismappedtoanellipticalsectorFOC.Butbyapplyingthesame argumentonthepreimage2 2jm x 2 C2 2jm y 2 D 1,andbydenotingbyG thepoint.0;2 jm /,the y-intercept of the preimage ellipse, we find that the elliptical sector GOE is the preimage of the circular sector BOF. Hence the preimage of the region of overlap, whose preimage consists of those points that return to U n , comprises the circular sector EOA and the elliptical sector GOE. Thatis,itistheregionofoverlapbetweenthecircleandtheimageofitunderB jm . To find the area, note that since the map is measure-preserving, the areas of the sectorsFOC andEOA are identical. Since the circular sectionBOF is congruent the circular sectorEOA,we findthepercentageoftheoverlappingregiontobe .U n \B jm U n / .U n / D .U n \B jm U n / .U n / D ⇢ 2 arctan2 jm ⇡⇢ 2 D arctan2 jm ⇡ : (2.155) Notethatasinthesquarecase,wehave U n \B im U n ⇢U n \B jm U n (2.156) when0<j 0,thenthereexistK 2.0;1çandpositiveconstantsc 1 ;c 2 suchthatforallsucientlylarge n,.U j n /c 1 j forallj Knifx isnon-periodic,and .U j n;u /c 2 j forallj K.nCum/ifx isperiodicwithminimalperiodm; (5) If .U n / D O.⇠ n / for some0<⇠<1, then there exist K 2 .0;1ç and positive constants c 1 ;c 2 suchthatforallsucientlylarge n,.U j n /c 1 ⇠ j forallj Knifx isnon-periodic, and.U j n;u /c 2 ⇠ j forallj K.nCum/ifx isperiodicwithminimalperiodm; (6) Ifx isperiodicwithperiodm,thenthereexistsasequence.J n / 1 nD1 withJ n 2.0;1/insuch awaythatnJ n !1asn!1,and k \ jD0 T i j U n ! D.1Co.1// i k =m \ jD0 T jm U n ; (3.5) forallE { D .i 0 ;:::;i k /andallk,where0D i 0 0 and2<< 0 such that for alln suciently large, c 1 n 0 < .U n /<c 2 n ; (3.6) and thatc 3 n ⇣ <nJ n for some⇣2.2=;1ç. Furthermore, p> max ≤ 2; 2 0 ⇣ 1 ≥ I (3.7) (2) There existc 1 ;c 2 >0 and0<⇠ 0 <⇠<1 such that for alln suciently large, c 1 ⇠ 0n < .U n /<c 2 ⇠ n ; (3.8) 48 and thatJ n ⌘J 2.0;1/ is a constant. Furthermore, p> 8log⇠ J log⇠ 0 1: (3.9) Then ⇢.x/D lim n!1 ⇢ U n .U n / D lim n!1 1 .U n / lim t!1 1 t jlog.⌧ 1 U n >t/jD1 ✓: (3.10) Themostimportantquantitythatonemustcomputeinordertoprovethistheoremis.⌧ 1 U n s/, which,inprinciple,canbeexpressedbyemployingtheinclusion-exclusionprinciple: .⌧ 1 U n s/D s [ jD1 T j U n ! D s X jD1 . 1/ jC1 M j : (3.11) Here,wedefine M j D X E {2I j .C E { /; (3.12) where I j .s/D ® E {D.i 1 ;:::;i j /W1i 1 0ands,weput s l DP s l CR s l ; (3.24) whereP s l isthemeasureofthoseelementsthatreturnexactlyl timesallwithinasinglecluster,i.e, P s l D .π xW ⇣ s U n .x/Dl;⌧ l U n .x/ ⌧ 1 U n .x/ nJ n º/; (3.25) andR s l isthemeasureoftherest,i.e, R s l D .π xW ⇣ s U n .x/Dl;⌧ l U n .x/ ⌧ 1 U n .x/>nJ n º/: (3.26) Notethatwiththesenotations,wecanwrite,foranyk>1, .⌧ k U >s/D k 1 X lD0 s l D s 0 C k 1 X lD1 P s l C k 1 X lD1 R s l : (3.27) Notethat(anasymptoticformulafor) s 0 wasfoundin[14]. Withoutanyfurtherdelay,wepresentthemaintheoremofthissection,whichgivestheformula forallhigherorderlocalizedescaperatesundersomemodifiedconditions(modifiedfromthoseof [14]forthefirstordercase): Theorem 12. Let .X;;M;T/ be a measure-preserving ergodic dynamical system. LetA be a finite generating partition of .X;;M;T/, and let be left- mixing with .x/ D O.x p /. Suppose.U n / 1 nD1 isanadaptedneighborhoodsystemofaperiodicpointx withminimalperiodm. Suppose further that for every positive integerl, lim n!1 .U n;l / .U n / D ✓ l ; (3.28) where✓ D ✓.x/ is as defined in (3.1). 52 Suppose for.U n / one of the following conditions holds: (1) There existc 1 ;c 2 ;c 3 >0 and2<< 0 such that for alln suciently large, c 1 n 0 < .U n /<c 2 n ; (3.29) and thatc 3 n ⇣ <nJ n for some⇣2.2=;1ç. Furthermore, p> max ≤ 2; 2 0 ⇣ 1 ≥ I (3.30) (2) There existc 1 ;c 2 >0 and0<⇠ 0 <⇠<1 such that for alln suciently large, c 1 ⇠ 0n < .U n /<c 2 ⇠ n ; (3.31) and thatJ n ⌘J 2.0;1/ is a constant. Furthermore, p> 8log⇠ J log⇠ 0 1: (3.32) Then for every positive integerk, ⇢ k .x/D lim n!1 ⇢ k U n .U n / D lim n!1 1 .U n / lim t!1 1 t jlog.⌧ k U n >t/jD1 ✓: (3.33) That is, all localized escape rates are identical. Note that Theorem 12 is not, strictly speaking, acompletegeneralizationofTheorem11becausethecondition(3.1)hasbeenmodifiedsomewhat to(3.28). It is worth noting that the condition given in (3.28) is indeed satisfied for the cylinders in a shift space endowed with a Bernoulli measure2: A cylinder U n is determined by fixing the first n 2Let E pD.p 1 ;:::;p n /beaprobabilityvector.Thatis, P i p i D1.Let bedefinedasfollows: .U.v 1 ;v 2 ;:::;v k //Dp v 1 p v 2 p v k ; lettersinasequence,andaperiodicpointwithperiodmisasequenceofrepeatedwordsoflength m. Therefore if a periodic point is included in a cylinder U n , then the first n letters fixed must be those of the periodic point. This then forces U n;1 to be the U nCm cylinder formed by additionally fixing them letters following the already determinedn letters ofU n to be the repeated word inx. Therefore, inductively, we obtain that U n;l is the U nClm cylinder whoselm letters following the firstn letters is thel repeats of the word of lengthm ofx. If we denote by 0 the measure of the cylinderU m whosefirstmlettersformtherepeatunitofx,then lim n!1 .U n;1 / .U n / D lim n!1 .U n / 0 .U n / D 0 D✓; (3.34) andmoreover, lim n!1 .U n;l / .U n / D lim n!1 .U n / l 0 .U n / D l 0 D ✓ l : (3.35) Thusthecondition(3.28)ismetforthecylindersinashiftspace. Notethat(3.28)ismetifwehave, lim n!1 .U n;1 / .U n / D lim n!1 .U n \T m U n / .U n / D✓; (3.36) andforeverypositiveintegerq, lim n!1 .U n;qC1 / .U n ;q/ D✓; (3.37) as lim n!1 .U n;l / .U n / D l 1 Y jD0 lim n!1 .U n;jC1 / .U n;j / D ✓ l : (3.38) whereU.v 1 ;v 2 ;:::;v k /isthecylinderwhosefirstk lettersarev 1 ;v 2 ;:::;v k ,respectively. 54 3.3 TheTemperedLimitLemma For us to prove Theorem 12, we need to prove a pivotal lemma that we shall call the Tempered LimitLemma: Lemma 13 (The Tempered Limit Lemma). Let .X;;M;T/ be a measure-preserving ergodic dynamical syste. LetA be a finite generating partition of.X;;M;T/, and let be left- mixing with .x/ D O.x p /. Suppose.U n / 1 nD1 is an adapted neighborhood system of a periodic pointx with minimal periodm. Suppose further that for all positive integerl, lim n!1 .U n;l / .U n / D ✓ l ; (3.39) where✓ D ✓.x/ is as defined in (3.1). 1. (Principal part) Suppose that.s n / 1 nD1 is a sequence of positive numbers such thats n !1. Thenforeverypositiveintegerl and.s n / 1 nD1 ,P s n l asdefinedin(3.24)and(3.25)isgivenby P s n l D.1Co.1//s n .U n /.1 ✓/ 2 ✓ l 1 I (3.40) 2. (Remainderpart) Suppose that either (i)there exists>0such that.U n /DO.n /, and there exists a sequence.s n / 1 nD1 such thats n !1 and lim n!1 s n .nJ n / D0; (3.41) the case which we shall call the polynomial case, or (ii) there exists ⇠ 2 .0;1/ such that .U n /DO.⇠ n /, and there exists a sequence.s n / 1 nD1 such thats n !1 and lim n!1 s n ⇠ nJ n =4 D0; (3.42) the case which we shall call the exponential case. Further suppose that K in the periodic case of Definition 10-(4) and 10-(5) andJ n in Definition 10-(6) satisfy the relationK>J n for all n suciently large. Then for this sequence and every positive integer l>1, R s n l as defined in (3.26) is given by R s n l Do.s n .U n //: (3.43) The name Tempered Limit Lemma is due to the fact that the speed at which s n diverges is “tempered”inthesensethatitbehaveswellinrelationto.U n /. The proof of Lemma 13 is quite lengthy, so we will split the proof into two parts. In Section 3.4,wewillproveLemma13-1,andinSection3.5,wewillproveLemma13-2.InSection3.7,we provethenon-periodicversionofLemma13. 3.4 TheProofofLemma13-1 Webegintheproofwiththefollowinglemmata: Lemma 14. The sets in .A n /, the -algebra generated by n-cylinders, are all of the possible countable unions ofn-cylinders. Proof. We only need to show that the set of all possible unions of n-cylinders is closed under complement.LetU n beann-cylinder.ThenU c n isthesetofsequencesthatavoidthefirstnletters thatdefineU n .ThenU c n ispreciselytheunionofalln-cylindersotherthanU n . ⌅ Lemma 15 (H+Y Lemma 5). Suppose.U n / 1 nD1 is an adapted neighborhood system of a periodic pointx with minimal periodm. For everyu2N, there exists anN u 2N such thatU n ✓ A mu .x/ for alln>N u . Here,A mu .x/ is the uniquemu cylinder that containsx. (The cylinder whose first mn letters are those ofx.) Proof. Suppose an adapted neighborhood system .U n / 1 nD1 of x is given. Consider the function 56 N WA N !Nwhosevalueata€x isgivenby N.a/D infπ nWa62U n º: (3.44) Thisisawell-definedfunctionforeverya€xbecause.U n / 1 nD1 isanadaptedneighborhoodsystem ofx€a.Wefirstshowthatthismapiscontinuous(relativetothesubspacemetrictopologies)on anysetA N nV whereV isacylindercontainingx. Let0<✏<1begiven.WeneedtofindanopenballB a suchthatforallp2B a ,N.p/DN.a/. Obviously, ifa62 U n for alln > N.a/, then for alln > N.a/, none of then-cylinders comprising U n hasthefirstnlettersagreeingwiththoseofa.ThusifwechoosetheopenballA n .a/,wehave forallp2A n .a/,N.p/DN.a/. Let V be fixed. Since V is open, A N nV is closed, and since A N is a compact metric space, A N nV iscompact.ThusbytheextremevaluetheoremN attainsthemaximumonA N nV. Letu2Nbegiven.Thenbytheabove,N attainsthemaximumonthesetA N nA mu .x/.Call thatnumberNN u .Thenifweconsider.U n / 1 nDN u C1 ,noelementa thatisnotinA mu .x/isinany oftheseU n s.Thatis,U n ✓A mu .x/foralln>N u . ⌅ TheupshotofLemma15isthatwecanalwayschooseN largeenoughsothatallN-cylinders inU N havethefirstarbitrarylengthoflettersincommonwithx. Lemma 16. Suppose .U n / 1 nD1 is an adapted neighborhood system of a periodic point x with minimal periodm. If there are two positive integersi;j withi>j such that for alln suciently largeT im y;T jm y 2 U n , thenT lm y 2 U n for everyj  l  i. That is, the periodic returns that occur within a single cluster must be (periodically) consecutive3. Proof. WefirstprovethelemmawhenthesetsU n sarecylinders.LetA N beashiftspacewiththe shift map . Since a cylinder set U n is determined by fixing the first n letters in the sequences, if x 2 U n , the first n letters of U n must agree with those of x. If y is such that T im y;T jm y 2 U n , then y 2 T im U n \T jm U n . But T im U n \T jm U n is a set such that the firstjm 1 letters 3Itmeansthatifmistheminimalperiod,returnsoccurattimeslm;.lC1/m;.lC2/m,andsoon. are arbitrary, and the letters between the positions jm and imC n match those in x. Thus if T im y;T jm y2U n ,itmustalsobethatforeveryj l j,T lm y2U n aswell. Supposenowthegeneralcase.Lety besuchthatforalln>N,T im y;T jm y2U n .IfN u isthe number specified by Lemma 15 for the minimalu satisfyingmu > N, then allU n s withn>N u aresubsetsofA mu .x/.Thatis,thefirstmulettersineverysequenceinthecylindersthatcomprise U n sareidenticaltothoseinx.Thusapplyingtheaboveargumentforthecylindercasewearriveat thedesiredconclusion. ⌅ Proof of Lemma 13-1. To reduce clutter, we put s D s n where .s n / 1 nD1 is any sequence diverging toinfinity. By Lemma 16, we know that the returns within a cluster occur consecutively. If all k returns occur consecutively, then all we need to decide is the head of the cluster, i.e., the time when the first return occurs. Suppose that x is an element that makes l consecutive returns to U n . Then it belongsinoneofthesets q U n;l 1 D l 1 \ jD0 T jm qm U n ; (3.45) which,bytheorder-preservingnatureofT,mustsatisfyforeveryq . q U n;l 1 /D l 1 \ jD0 T jm qm U n ! D l 1 \ jD0 T jm U n ! D .U n;l 1 /: (3.46) Here,q representsthelocationoftheheadofthecluster.Buteach q U n;l 1 containsmorethanjust those elements that make l consecutive returns, namely, those elements that make l consecutive returns starting fromq, and other returns elsewhere. But again, since those additional returns too must occur within a single cluster in a consecutive fashion, to get only those elements that make exactlyl returns,weneedtoconsiderthesets q U n;l 1 n. q U n;l [ q 1 U n;l /D q U n;l 1 n. q U n;l [T m q U n;l /: (3.47) 58 The set q U n;l accounts for those elements that make additional returns after the l consecutive returns starting at q, and the set T m q U n;l , before. Evidently we are assuming here thatq>0. Notethatthesets q U n;l and q 1 U n;l arebothsubsetsof q U n;l 1 ,therefore . q U n;l 1 n. q U n;l [ q 1 U n;l //D . q U n;l 1 / . q U n;l [ q 1 U n;l /: (3.48) Bytheinclusion-exclusionprinciple, . q U n;l [ q 1 U n;l /D . q U n;l /C. q 1 U n;l / . q U n;l \ q 1 U n;l /; (3.49) where q U n;l \ q 1 U n;l D q 1 U n;lC1 DT m q U n;lC1 : (3.50) Thereforeforeveryq>0,wehave . q U n;l 1 n. q U n;l [ q 1 U n;l // D . q U n;l 1 n. q U n;l [T m q U n;l // D . q U n;l 1 / .. q U n;l /C.T m q U n;l / .T m q U n;lC1 // D . q U n;l 1 / . q U n;l / . q U n;l /C. q U n;lC1 / D . q U n;l 1 / 2. q U n;l /C. q U n;lC1 / D .U n;l 1 / 2.U n;l /C.U n;lC1 /: (3.51) ForqD0,wehave . 0 U n;l 1 n 0 U n;l /D .U n;l 1 nU n;l /D .U n;l 1 / .U n;l /: (3.52) Sinceeachset q U n;l 1 n. q U n;l [ q 1 U n;l /isthesetofelementsthatmakeexactlyl returnsstarting attimeq,theyarealldisjoint.Therefore P s l D . 0 U n;l 1 n 0 U n;l /C s l X qD1 . q U n;l 1 n. q U n;l [ q 1 U n;l // D .U n;l 1 / .U n;l /C s l X qD1 ..U n;l 1 / 2.U n;l /C.U n;lC1 // D .U n;l 1 / .U n;l /C.s l/..U n;l 1 / 2.U n;l /C.U n;lC1 // D.s lC1/.U n;l 1 / .2s 2lC1/.U n;l /C.s l/.U n;lC1 /; (3.53) whichgives P s l D.1Co.1//s.U n /.✓ l 1 2✓ l C✓ lC1 /D.1Co.1//s.U n /.1 ✓/ 2 ✓ l 1 ; (3.54) byourassumptionthat.U n;l /=.U n /D.1Co.1//✓ l . ⌅ 3.5 TheProofofLemma13-2 WhatremainsnowistoshowthattheerrorpartR s l remainsnegligible. Proof of Lemma 13-2. Let.s n / 1 nD1 beasequencesatisfyingeithertheconditionsforthepolynomial case (3.41) or the exponential case (3.42). To distinguish the cases at this point is not necessary at this point, however, and all reasoning up to (3.75) is true in either case. As in the principal part case,weusesDs n toreduceclutter. Observe thatl>1 returns4 that take place over multiple clusters are fully determined by the followingparameters:(i)N,thenumberofclusters,whichdoesnotexceedl;(ii)Å i ,thegaplength between the i-th and the .i C1/-th clusters, which must sum to a number no greater than s; (iii) i ,thenumberofreturnsinthei-thcluster,eachofwhichisgreaterthan1,butmustsumtol,and (iv)Å 0 ,thelocationoftheheadofthefirstcluster,whichdoesnotexceeds l. 4Ifl D1,thentherecanonlybeonecluster. 60 Forthej-thcluster5⇤ q j j oflengthm j (meaningthereare j manyperiodicreturns)thatstarts attimeq j , .⇤ q j j /D j 1 \ iD0 T im q j m U n ! D .T q j U n; j 1 / D .U n; j 1 / D.1Co.1//.U n /✓ i 1 : (3.55) Notethatforthefirstcluster,q 1 D Å 0 ,andforalltheotherclusters, q j D j 1 X iD0 Å i C j 1 X iD1 m i : (3.56) Observe that by the measure-preserving nature of T, we can assume that Å 0 D 0, at the expense ofmultiplyingbys l,thenumberofpossibleÅ 0 s.Put ⇤ j D j 1 \ iD0 T im U n DU n; j 1 : (3.57) Then R s l D.s l/ l X ND2 X 1 CC N Dl X 0Å 1 CCÅ N s N \ jD1 ⇤ q j j ! D.s l/ l X ND2 X 1 CC N Dl X 0Å 1 CCÅ N s N \ jD1 T q j ⇤ j ! s l X ND2 X 1 CC N Dl X 0Å 1 CCÅ N s N \ jD1 T q j ⇤ j ! ; (3.58) whereÅ 0 Dq 1 D0. In order to use the -mixing property, we need to ensure that the gaps Å i s between the 5Onemaytakethefollowingexpressionasthedefinitionof⇤ q j j . consecutive clusters are large, which is not necessarily guaranteed. To circumvent that issue, we follow the argument given in [14]. Observe first that⇤ j 2 .A nCm j /. If we put ⌘ n D J n =4 and putforj 2 ⇤ 0 j DT .1 ⌘ n /.nCm j / ⇤ j ; (3.59) then⇤ 0 j 2.A ⌘ n .nC j / /.Alsowehave ⇤ j ✓T .1 ⌘ n /.nCm j / ⇤ 0 j ; (3.60) andthusifweput⇤ 0 1 D⇤ 1 ,then N \ jD1 T q j ⇤ j ✓⇤ 1 \ N \ jD2 T .q j C.1 ⌘ n /.nCm j // ⇤ 0 j : (3.61) Putforeachj 1 j Dq j C.1 ⌘ n /.nCm j /: (3.62) Observethat jC1 j Dq jC1 C.1 ⌘ n /.nCm jC1 / q j C.1 ⌘ n /.nCm j / D Å j Cm j C.1 ⌘ n /.m jC1 m j /; (3.63) thus jC1 j ⌘ n .nCm j /D Å j Cm jC1 ⌘ n .nCm jC1 /: (3.64) Since Å j Cm jC1 isthedistancebetweenthetailsofthej-thclusterandthenext,wemusthave Å j Cm jC1 >nJ n .Thus(asm j <nJ n foreveryj) Å j Cm jC1 ⌘ n .nCm jC1 />nJ n nJ n 4 nJ 2 n 4 > nJ n 2 ; (3.65) 62 andfurthermore, Å j Cm jC1 ⌘ n .nC2m jC1 />nJ n nJ n 4 nJ 2 n 2 > nJ n 4 ; (3.66) bothofwhichdivergetoinfinityasntendstowardsinfinity. Because T j ⇤ 0 j 2 .A j C⌘ n .nCm j / /, if A 2 .A j C⌘ n .nCm j / /, by the -mixing property, foreveryB 2. S j A j /, .A \T jC1 B/ .A/.. jC1 j ⌘ n .nCm jC1 //C.B// D .A/..Å j Cm jC1 ⌘ n .nCm jC1 //C.B//: (3.67) PutÅ j Cm jC1 ⌘ n .nC2m jC1 /Dg jC1 forj 1.Then(3.67)yields .A \T jC1 B/ .A/..g jC1 C⌘ n m jC1 /C.B//: (3.68) Thusapplyingthe-mixingpropertyinductively,weobtainforallN>2, ⇤ 1 \ N \ jD2 T j ⇤ 0 j ! D ..⇤ 1 \T 2 ⇤ 0 2 \\T N1 ⇤ 0 N 1 /\T N ⇤ 0 N /  .⇤ 1 \T 2 ⇤ 0 2 \\T N1 ⇤ 0 N 1 /..g N C⌘ n m N /C.⇤ 0 N //  .⇤ 1 \T 2 ⇤ 0 2 / N Y jD3 ..g j C⌘ n m j /C.⇤ 0 j //  .⇤ 1 / N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j // D.1Co.1//.U n /✓ 1 1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j //: (3.69) HencetheremainderR s l isboundedbythesum(forsomepositiveconstantc) R s l cs.U n / l X ND2 X 1 CC N Dl X Å 1 CCÅ N 0 ✓ 1 1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j //: (3.70) Note that by summing over all possible values Å i such that 0  P N 1 iD1 Å i  s, which is an overestimate,wecaninterchangethesumsover i sandÅ i s.Furthermore,wecaninsteadsumover g j s,whichyieldsanoverestimate X 1 CC N Dl X Å 1 CCÅ N 0 ✓ 1 1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j //  s X g 2 nJ n =4 s X g N nJ n =4 X 1 CC N Dl ✓ 1 1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j //: (3.71) Notealsothat X 1 CC N Dl ✓ 1 1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j //  l X 1 D1 l X N D1 ✓ 1 1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j //  1 1 ✓ l X N D1 l X 2 D1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j //: (3.72) Observethatasthesummationsoverg j stakeplaceafterthesumsover j stakeplace,eachfactor .g j C⌘ n m j /C.⇤ 0 j /intheproducthasdependenceon j only,therefore l X 2 D1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j // D N Y jD3 ..g j C⌘ n m j /C.⇤ 0 j // l X 2 D1 ..g 2 C⌘ n m 2 /C.⇤ 0 2 //  N Y jD3 ..g j C⌘ n m j /C.⇤ 0 j //. 1 .g 2 /Cl.⇤ 0 2 //; (3.73) 64 where 1 .x/isthediscretetailsum P 1 yD0 .xCy/,whichisassumedtobefinite(andsummable oncemore).Continuinginthisfashion,weobtain l X N D1 l X 2 D1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j //D N Y jD2 . 1 .g j /Cl.⇤ 0 j // (3.74) Usingthesametechniqueforthesummationoverg i s,weobtain s X g 2 nJ n =4 s X g N nJ n =4 N Y jD2 . 1 .g j /Cl.⇤ 0 j // N Y jD2 ✓ 2 ✓ nJ n 4 ◆ Csl.⇤ 0 j / ◆ : (3.75) Wenowdistinguishthecasesinordertoestimate.⇤ 0 j /.Beforethat,observethat .⇤ 0 j /D .T .1 ⌘ n /.nCm j / ⇤ j /D .T .1 ⌘ n /.nCm j / U n; j 1 /D .U ⌘ n .nCm j / n; j 1 /: (3.76) BytheassumptionthatK>J n forallnsucientlylarge,forthosesucientlylarge n, nJ n 4 Dn⌘ n <⌘ n .nCm j /<K.nCm j /: (3.77) Suppose first the polynomial case (3.41). Then for all n suciently large, there exists a positive constantc sothat .U n; j 1 / .U ⌘ n .nCm j / n; j 1 /<c ✓ nJ n 4 ◆ Dc 0 .nJ n / ; (3.78) wherec 0 D4 c and inaordedbyDefinition10-(4).Put r n;l D 2 ✓ nJ n 4 ◆ Cc 0 sl.nJ n / : (3.79) Assuming thatn is suciently large so that 2 .nJ n =4/ < 1=2 andc 0 sl.nJ n / < 1=2, we obtain thepenultimateestimateas R s l cs.U n / l X ND2 X 1 CC N Dl X Å 1 CCÅ N 0 ✓ 1 1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j // cs.U n / l X ND2 N Y jD2 r n;l Dcs.U n / l X ND2 r N 1 n;l cs.U n / r n;l .1 r l n;l / 1 r n;l cs.U n / r n;l 1 r n;l C r lC1 n;l 1 r n;l ! cs.U n / r n;l 1=2 c 0 sl.nJ n / C r lC1 n;l 1=2 c 0 sl.nJ n / ! cs.U n / 2 .nJ n =4/ 1=2 c 0 sl.nJ n / C c 0 sl.nJ n =4/ 1=2 c 0 sl.nJ n / C r lC1 n;l 1=2 c 0 sl.nJ n / ! : (3.80) Sinces.nJ n / ! 0, for alln suciently large, we can ensure that c 0 sl.nJ n / < 1=4. Thus we canusetheestimates 2 .nJ n =4/ 1=2 c 0 sl.nJ n /  2 .nJ n =4/ 1=2 1=4 D4 2 ✓ nJ n 4 ◆ DO..nJ n / 2 p /; (3.81) and c 0 sl.nJ n =4/ 1=2 c 0 sl.nJ n /  c 0 sl.nJ n =4/ 1=2 1=4 DO..nJ n / /; (3.82) 66 and r lC1 n;l 1=2 c 0 sl.nJ n /  r lC1 n;l 1=2 1=4 D4 ✓ 2 ✓ nJ n 4 ◆ Cc 0 sl ✓ nJ n 4 ◆ ◆ lC1 <4 2 ✓ nJ n 4 ◆ C4c 0 sl ✓ nJ n 4 ◆ DO.maxπ .nJ n / 2 p ;.nJ n / º/ (3.83) to arrive at the ultimate estimate that bounds the quantity inside of the parentheses of the last line of(3.80): R s l Ds.U n /O.maxπ .nJ n / 2 p ;.nJ n / º/Do.s.U n //; (3.84) wherethefinalequalitycomesfromobservingthatboth.nJ n / 2 p and.nJ n / convergeto0. Assumenowtheexponentialcasesothat(3.42)holds.Thenforallnsucientlylarge, .U n; j 1 / .U ⌘ n .nCm j / n; j 1 /<c 00 ⇠ nJ n =4 ; (3.85) where⇠ isaordedbyDefinition10-(5).Put r n;l D 2 ✓ nJ n 4 ◆ Cc 00 sl⇠ nJ n =4 : (3.86) Observethatwecanstillassurethatforallnsucientlylarge, 2 .nJ n =4/<1=2andc 0 ⇠ nJ n =4 <1=2 sothat R s l cs.U n / l X ND2 X 1 CC N Dl X Å 1 CCÅ N 0 ✓ 1 1 N Y jD2 ..g j C⌘ n m j /C.⇤ 0 j // cs.U n / l X ND2 N Y jD2 r n;l Dcs.U n / l X ND2 r N 1 n;l cs.U n / r n;l .1 r l n;l / 1 r n;l cs.U n / r n;l 1 r n;l C r lC1 n;l 1 r n;l ! cs.U n / r n;l 1=2 c 00 ⇠ nJ n =4 C r lC1 n;l 1=2 c 00 ⇠ nJ n =4 ! cs.U n / 2 .nJ n =4/ 1=2 c 00 ⇠ nJ n =4 C c 00 ⇠ nJ n =4 1=2 c 00 ⇠ nJ n =4 C r lC1 n;l 1=2 c 00 ⇠ nJ n =4 ! : (3.87) Again,asbefore,toarriveatthefirstestimate,weobservethat Thuswecanusetheestimates 2 .nJ n =4/ 1=2 c 00 ⇠ nJ n =4  2 .nJ n =4/ 1=2 1=4 D4 2 ✓ nJ n 4 ◆ DO..nJ n / 2 p /; (3.88) and c 00 ⇠ nJ n =4 1=2 c 00 ⇠ nJ n =4  c 00 ⇠ nJ n =4 1=2 1=4 DO.⇠ nJ n =4 /; (3.89) 68 and r lC1 n;l 1=2 c 00 ⇠ nJ n =4  r lC1 n;l 1=2 1=4 D4 ✓ 2 ✓ nJ n 4 ◆ Cc 00 ⇠ nJ n =4 ◆ lC1 <4 2 ✓ nJ n 4 ◆ C4c 00 ⇠ nJ n =4 DO..nJ n / 2 p / (3.90) toarriveatthesameestimateasbefore. ⌅ 3.6 TheProofoftheMainTheorem We now present the proof of the main theorem, Theorem 12, the precise statement of which we restateforthereader’sconvenience: Theorem17(Theorem12). Let.X;;M;T/beameasure-preservingergodicdynamicalsystem. LetA be a finite generating partition of .X;;M;T/, and let be left- mixing with .x/ D O.x p /. Suppose.U n / 1 nD1 is an adapted neighborhood system of a periodic pointx with minimal periodm. Suppose further that for every positive integerl, lim n!1 .U n;l / .U n / D ✓ l ; (3.91) where✓ D ✓.x/ is as defined in (3.1). Suppose for.U n / one of the following conditions holds: (1) (Polynomial case) There exist c 1 ;c 2 ;c 3 >0 and2<< 0 such that for all n suciently large, c 1 n 0 < .U n /<c 2 n ; (3.92) and thatc 3 n ⇣ <nJ n for some⇣2.2=;1ç. Furthermore, p> max ≤ 2; 2 0 ⇣ 1 ≥ : (3.93) (2) (Exponential case) There existc 1 ;c 2 >0 and0<⇠ 0 <⇠<1 such that for alln suciently large, c 1 ⇠ 0n < .U n /<c 2 ⇠ n ; (3.94) and thatJ n ⌘J 2.0;1/ is a constant, and>1 in Definition 10-(4). Furthermore, p> 8log⇠ 0 J log⇠ 1: (3.95) Then for every positive integerk, ⇢ k .x/D lim n!1 ⇢ k U n .U n / D lim n!1 1 .U n / lim t!1 1 t jlog.⌧ k U n >t/jD1 ✓: (3.96) Proof. Tobegin,putt Drs.Inordertocomputethedoublelimit(3.33),observethatwecanwrite foranypositives, lim n!1 1 .U n / lim t!1 1 t jlog.⌧ k U n >t/jD lim n!1 1 .U n / lim r!1 1 rs jlog.⌧ k U n >rs/j D lim n!1 1 .U n / lim r!1 1 rs ˇ ˇ ˇ ˇ ˇ log k 1 X lD0 rs l ˇ ˇ ˇ ˇ ˇ D lim n!1 1 .U n / lim r!1 1 rs ˇ ˇ ˇ ˇ ˇ log rs 0 C k 1 X lD1 rs l !ˇ ˇ ˇ ˇ ˇ : (3.97) Butnotethatwecanalsowrite log rs 0 C k 1 X lD1 rs l ! D log rs 0 Clog 1C k 1 X lD1 rs l rs 0 ! : (3.98) 70 Observefirstthat lim n!1 1 .U n / lim r!1 1 rs logj rs 0 jD lim n!1 1 .U n / lim r!1 1 rs logj.⌧ 1 U n >rs/jD1 ✓; (3.99) which is the result of Theorem 11. To compute the second limit of (3.98), set s D s n where s n D1=.U n /.Then rs l D r=.U n / l ,andso lim n!1 1 .U n / lim r!1 1 rs log 1C k 1 X lD1 rs l rs 0 ! D lim n!1 lim r!1 1 r log 1C k 1 X lD1 r=.U n / l r=.U n / 0 ! : (3.100) Assumefirstthepolynomialcase.NowitisagoodtimetoinvokeLemma7.Touse,let.w n / 1 nD1 beasequencegivenby w n D.nJ n / ˛ ; (3.101) where0<˛<=2.Thensince.U n /c 2 n forallsucientlylarge n,forsuchnwehave w 2 n .U n /c 2 .nJ n / 2˛ n c 2 n 2˛ !0; (3.102) so the condition (2.79) is met. To show that the additional assumptions (2.82) and (2.84) are also met,wefirstobservethat w ˇ n D.nJ n / ˛ˇ n ˛ˇ⇣ ; (3.103) sowerequirethat˛ˇ⇣ > 1,andfor(2.84),since w 1Cpˇ n .U n / c 1 n ˛⇣.1Cpˇ/ 0 ; (3.104) weneed˛⇣.1Cpˇ/ 0 >0.ObservethatifweselectpD2 0 =⇣ ı forsomeı<1,wethen have ˛⇣.1Cpˇ/ 0 D˛⇣Cp˛⇣ˇ 0 D˛⇣C 2˛ˇ 0 ˛⇣ˇı 0 D˛⇣C 2˛ˇ 0 ˛⇣ˇı 0 D˛⇣.1 ˇı/ 0 ✓ 1 2˛ˇ ◆ : (3.105) Thisquantityispositiveifandonlyif ˛⇣ 0 > 1 .2˛=/ˇ 1 ıˇ : (3.106) Theleft-handsideisstrictlypositive,andwith˛ chosensucientlycloseto =2,itcanbegreater than 1= 0 . On the other hand, for anyı<1 fixed, choosing ˛ suciently close to =2 and ˇ suciently close to 1, it can be arbitrarily small. For example, consider the limit ˇ D e n and ˛De n =2.Thentheright-handsideconvergesto0,andsoforallsucientlylarge n,theright- handsideislessthan1= 0 .Sincethiscanbedoneforanyı<1,wecanselectanyp >2 0 =⇣ 1 and˛andˇsucientlycloseto =2and1,respectively.Then(3.103)and(3.104)arebothsatisfied, sobyLemma7,foreverynonnegativeintegerk, lim n!1 r=.U n / k D 1 kä d k dz k F w n .z/ v n ˇ ˇ ˇ ˇ ˇ zD0 ; (3.107) where.v n / 1 nD1 isacorrespondingsequenceoeredbyLemma7. NowweneedtoinvokeLemma13todecidetheprobabilitygeneratingfunctionF w n .z/.Since weareassumingthepolynomialcase,weonlyneedtoensure(3.41)issatisfied.Andindeed, s n .nJ n / D.nJ n / ˛ !0; (3.108) 72 as˛<=2.HencebyLemma13,foreverypositiveintegerl, w n l D .⇣ w n U n Dl/D.1Co.1//w n .U n /.1 ✓/ 2 ✓ l 1 Co.s n .U n //: (3.109) Hencewecanput6,bysymbolmatching, ˛ 1 D1 ✓; (3.110) and l D.1 ✓/✓ l 1 : (3.111) Henceby(2.123),theprincipalpartof r=.U n / l ,meaningthepartwithouttheadditiveerrorterm, isgivenby r=.U n / l D 1 kä d k dz k F w n .z/ v n ˇ ˇ ˇ ˇ ˇ zD0 D.1Co.1//e .1 ✓/r l X kD1 .1 ✓/ 2k r k kä X P k jD1 l j Dl l j 1 k Y iD1 ✓ l i 1 D.1Co.1//e .1 ✓/r ✓ l l X kD1 .1 ✓/ 2k r k kä X P k jD1 l j Dl l j 1 D.1Co.1//e .1 ✓/r ✓ l l X kD1 .1 ✓/ 2k r k kä l 1 k 1 ! : (3.112) 6Notethatinthefollowingsteps,wearenotcomputing l and˛ 1 .Wearemerelymatchingthesymbolsinorderto usetheformula(2.123) Togetthelastequality,weusedthecombinatorialidentitythat X P k jD1 l j Dl l j 1 D l 1 k 1 ! ; (3.113) which counts the number of positive integer solutions to the equation l 1 CCl k D l, which amountstothenumberofwaystoplacek 1manybarsbetweenl manydots,forwhichthereare l 1spots.Inanycase,howevercomplicateditlooks,allofthequantitiesfollowingtheexponential e .1 ✓/r issomepolynomialinr ofdegreel,whichwewilldenotebyP l .r/. Alsonotethewiththeseassumptionsandourchoiceofw n D.nJ n / ˛ with˛2.0;=2/,allof thenecessaryconditions[14]Lemma3arealsosatisfied,andtherefore w n 0 D.1Co.1//e .1 ✓/w n .U n / D.1Co.1//e .1 ✓/r=v n (3.114) asaresultof[14]Lemma3.ApplyingLemma7againforthecasekD0,weobtain r=.U n / 0 D. w n 0 / v n D.1Co.1//e .1 ✓/r : (3.115) Puttingitalltogether,weobtain k 1 X lD1 r=.U n / l r=.U n / 0 D k 1 X lD1 .1Co.1//e .1 ✓/r P l .r/CO.r/ .1Co.1//e .1 ✓/r CO.r/ (3.116) where we resurrected the dependence of the error term on the parameter t which now goes to infinity.7 Henceasr!1, k 1 X lD1 .1Co.1//e .1 ✓/r P l .r/CO.r/ .1Co.1//e .1 ✓/r CO.r/ DO.Q.R//; (3.117) 7See(2.102)forthefullformula. 74 forsomerationalfunctionQ.R/.Hence lim r!1 1 r log 1C k 1 X lD1 r=.U n / l r=.U n / 0 ! D lim r!1 log.1CO.Q.R/// 1=r D0: (3.118) Thusinthepolynomialcase,thedesiredresultthat⇢ k .x/D1 ✓ follows. Assume now the exponential case. Observe that we only need to show that in this case we can still satisfy the conditions (2.79), (2.82), and (2.84) of Lemma 7, as well as (3.42) of Lemma 13, hencewearriveatthesameconclusion.Fortunately,thiscasecanbecoveredinwholebyCorollary 8, because we may select as any positive integer we desire. The only outstanding property to be checkedis(3.41),whichiseasilymet: s n ⇠ nJ n =4 DO.⇠ nJ n ˛=4 ⇠ nJ n =4 /DO.⇠ .1 ˛/nJ n =4 /; (3.119) whichvanishesas˛ <1=2.Wekeeptherequirementonp inordertosatisfytheconditionsforthe firstordercase. ⌅ 3.7 TheTemperedLimitLemma:TheNon-periodicCase Supposenowwehaveasystemofadaptedneighborhoodsaroundx thatisnon-periodic.Inthecase where k D 1, it was proved in [14] that for if s n is any sequence .s n / 1 nD1 such that s n !1 and s n .U n /!0, .⌧ U n s n / s n .U n / !1 : (3.120) Wegiveananalogueoftheirlemmainthecasewhenk>1: Lemma18. Let.X;;M;T/beameasure-preservingergodicdynamicalsystem.LetAbeafinite generating partition of.X;;M;T/, and let be left- mixing with.x/ DO.x p /. Let.U n / 1 nD1 beasystemofadaptedneighborhoodsaroundx thatisnon-periodic.Thenforany sequence.s n / 1 nD1 such thats n !1 ands n .U n /!0, and for every integerk>1, .⌧ k U n s n / s n .U n / !0 C : (3.121) Proof. Firstobservethatsinceforeverypositiveintegerk andameasurableU n ⌧ k U n .x/ ⌧ kC1 U n .x/; (3.122) itfollowsthat .⌧ kC1 U n s n / .⌧ k U n s n / (3.123) foreverypositiveintegerk.Therefore .⌧ k U n s n / s n .U n /  .⌧ 2 U n s n / s n .U n / ; (3.124) soitsucestoshowthat .⌧ 2 U n s n / s n .U n / !0 C : (3.125) Observethatbythelawoftotalprobability, .⌧ 2 U n s n /D bs n 1c X tD1 .π ⌧ 2 U n s n º\π ⌧ 1 U n Dtº/: (3.126) Forthesetπ ⌧ 2 U n s n º\π ⌧ 1 U n Dtº,wehave π ⌧ 2 U n s n º\π ⌧ 1 U n DtºDT t .π ⌧ 1 U n s n tº\U n /\π ⌧ 1 U n Dtº; (3.127) 76 implyingthat .π ⌧ 2 U n s n º\π ⌧ 1 U n Dtº/D .T t .π ⌧ 1 U n s n tº\U n /\π ⌧ 1 U n Dtº/  .T t .π ⌧ 1 U n s n tº\U n // D .π ⌧ 1 U n s n tº\U n / D .⌧ 1 U n s n t jU n /.U n /: (3.128) Therefore .⌧ 2 U n s n / bs n 1c X tD1 .⌧ 1 U n s n t jU n /.U n /  .U n / bs n 1c X tD1 .⌧ 1 U n s n t jU n /  .U n / bs n 1c X tD1 .⌧ 1 U n s n jU n / s n .U n /.⌧ 1 U n s n jU n /; (3.129) whichgivestheinequality .⌧ 2 U n s n / s n .U n /  .⌧ 1 U n s n jU n /D .π ⌧ 1 U n sº\U n / .U n / : (3.130) By Lemma 1 of [14], ⌧.U n /!1 as x 2 T 1 nD1 U n is a non-periodic point. Here, ⌧.U n / is the periodofU n thatisdefinedasinfπ j 1WT j U n \U n €¿º.Thuswehavetheestimate .π ⌧ 1 U n s n º\U n / .U n / D 1 .U n / bs n c [ jD1 T j U n \U n D 1 .U n / bs n c [ jD⌧.U n / T j U n \U n  1 .U n / bs n c X jD⌧.U n / .T j U n \U n /: (3.131) Forj 2Œ⌧.U n /;b2Kncç,takeU bj=2c n sothatT nb j=2c U n ⇢U bj=2c n .Thentheleft-mixingproperty gives .T j U n \U n / .T .jCnb j=2c/ U bj=2c n \U n / .U n / ✓ .U bj=2c n /C ✓ j 2 ◆◆ : (3.132) Forj 2Œb2KncC1;sç,again,theleft-mixingpropertygives .T j U n \U n / .U n /..U n /C.j n// .U n / ✓ .U n /C ✓ j 2 ◆◆ : (3.133) Therefore 1 .U n / bs n c X jD⌧.U n / .T j U n \U n / b2Knc X jD⌧.U n / .U bj=2c n /C bs n c X jDb2KncC1 .U n /C2 bs n c X jD⌧.U n / ✓ j 2 ◆  b2Knc X jD⌧.U n / .U bj=2c n /Cs n .U n /C2 bs n c X jD⌧.U n / ✓ j 2 ◆ : (3.134) Thefirstsumcanbeapproximatedbyexploitingthecondition(4): b2Knc X jD⌧.U n / .U bj=2c n / bKnc X jDb⌧.U n /=2c j  Z 1 b⌧.U n /=2c .x 1/ dxD .b⌧.U n /=2c 1/ 1 1 : (3.135) Since>1 and ⌧.U n /!1, the first sum goes to zero. Since we assumed thats n .U n /! 0 as well,thesecondtermgoestozeroaswell.Finally,since.x/ DO.x p /withp>2,thetailsum of goestozero. ⌅ 3.8 Example:TheExpandingIntervalMap As a parting exercise, we show that the expanding interval mapT.x/D 2x mod 1 onŒ0;1/ with respect to the restriction of the Lebesgue measure on Œ0;1/ with the initial partition (the Markov 78 partition)Œ0;1=2/;Œ1=2;1/satisfiesallconditionsofTheorem12,sowemayapplyit.Wewillnot belabor the widely known details such as the left -mixing property8, or that the map is ergodic. It is also known that ✓ D 1=2 m for any periodic point of minimal period m. With respect to the initial partition Œ0;1=2/;Œ1=2;1/, the n-cylinders are all clopen intervals Œj2 n ;.j C1/2 n / for 0j<2 n . We will take as your periodic point 0 which has period 1, andU n D Œ0;2 n / as the neighbor- hoods.Inthiscase,.U n /D2 n ,sowearedealingwiththeexponentialcase.Theonlynontrivial propertiestocheckareDefinition10-(5)and10-(6),aswellas(3.28). To show Definition 10-(5), observe that forU n D Œ0;2 n /, ifj  n, thenT n j U n D Œ0;2 j /, andsoitisapproximatedbyasinglej-cylinderU j .Then .U j n /D .U j /2 j (3.136) forallj n,whichshowsthatDefinition10-(5)holdswithK D1and⇠ D1=2. To show 10-(6), observe that the j-th preimage of U n are 2 j many disjoint clopen intervals Œk2 j ;k2 j C2 n j / for 0 k<2 j . For j  n, the only intersection with U n occurs when kD0.Observealsothattheseoverlapsareallnested,sotheintersectionofanynumberpreimages, so long as that number does not exceedn, is the same as the intersection withU n and the smallest one, or that of allU j betweenU n and the smallest one. Here, we may take asJ n any constant less than1. Toshow(3.28),observethat lim n!1 .U n;l / .U n / D lim n!1 2 n .Œ0;2 n /\T l Œ0;2 n // D lim n!1 2 n .Œ0;2 n /\Œ0;2 n l // D2 l : (3.137) Since✓ D1=2,theconditionholds. 8Withrespecttothepartitiongiven, canbetakentobeanexponentiallydecayingfunction. ApplyingTheorem12tothisexample,wefindthat⇢ k .0/D1 ✓ forallk. 80 Bibliography [1] MiguelAbadi,Exponentialapproximationforhittingtimesinmixingprocesses,Mathematical PhysicsElectronicJournal[electroniconly]7(2)(2001). [2] Mark Bolding and Leonid Abramowitsch Bunimovich, Where and when orbits of strongly chaotic systems prefer to go,Nonlinearity32(2019),no.5,1731. [3] Michael Brin and Garrett Stuck, Introduction to dynamical systems, Cambridge University Press,2002. [4] Henk Bruin, Mark Francis Demers, and Mike Todd, Hitting and escaping statistics: Mixing, targets and holes,AdvancesinMathematics328(2018),1263–1298. [5] Leonid Abramowitsch Bunimovich and Alex Yurchenko, Where to place a hole to achieve a maximal escape rate,IsraelJournalofMathematics182(2011),229–252. [6] ConnorDavis,NicolaiHaydn,andFanYang,Escaperateandconditionalescaperatefroma probabilistic point of view,AnnalesHenriPoincaré,22(2021),2195–2225. [7] AndrewFergusonandMarkPollicott, Escape rates for Gibbs measures,ErgodicTheoryand DynamicalSystems32(3)(2012),961–988. [8] Sandro Gallo, Nicolai Haydn, and Sandro Vaienti, Number of visits in arbitrary sets for -mixing dynamics,Annalesdel’InstitutHenriPoincaré,ProbabilitésetStatistiques(2021). [9] Antonio Galves and Bernard Schmitt, Inequalities for hitting times in mixing dynamical systems,Random&ComputationalDynamics(1997). [10] Nicolai Haydn, Entry and return times distribution, Dynamical Systems 28(3) (2013), 333– 353. [11] Nicolai Haydn and Sandro Vaienti, The distribution of return times near periodic orbits, ProbabilityTheoryandRelatedFields144(2009),517–542. [12] ,Limitingentryandreturntimesdistributionforarbitrarynullsets,Communications inMathematicalPhysics378(2020),149–184. [13] Nicolai Haydn and Fan Yang, Entry times distribution for dynamical balls on metric spaces, JournalofStatisticalPhysics167(2017),297–316. [14] , Local escape rate for -mixing dynamical systems, Ergodic Theory and Dynamical Systems40(2020),no.10,2854–2880. 81 [15] CarlangeloLiveraniandVéroniqueMaume-Deschamps,Lasota-Yorkemapswithholes:Con- ditionally invariant probability measure and invariant probability measures on the survivor set,Annalesdel’InstitutHenriPoincaré,ProbabilitésetStatistiques39(2003),385–412. [16] BorisPitskel,PoissonlawforMarkovchains,ErgodicTheoryandDynamicalSystems(1991), 501–513. [17] HenryvandenBedemandNikolaiChernov, Expanding maps of interval with holes,Ergodic TheoryandDynamicalSystems22(2002),637–654. [18] PeterWalters, An introduction to ergodic theory,Springer,1982. 82 Appendix:CombinatorialProofofthePrincipalPartofthe HigherOrderLocalizedEscapeRates In this section, we present a combinatorial argument for finding the formula for the principal part provedin3.4. Beginbydefiningthefunction ⇣ s .x/D s X jD1 T j U .x/; (3.138) which gives the number of times that a particular element x returns to U in the time span of s. Thendenoteby s k themeasure s k D .⇣ s Dk/: (3.139) That is, this is the measure of the set of elements that make exactlyk returns ins amount of time. Thenthequantityweneedtocomputeis .⌧ k U n s/D s X lDk s l : (3.140) Observe that each M k consists of the sum of measures of all k-fold intersections. There every element inπ ⇣ s D kº is counted exactly once. Similarly, every element ofπ ⇣ s D kCtº is counted 83 for kCt k manydistinctC E { ’s,hencewegettheformula M k D k k ! s k C kC1 k ! s kC1 C kC2 k ! s kC2 CC kCj k ! s kCj CC s k ! s s D s X tDk t k ! s t D s X tD0 t k ! s t : (3.141) Our objective is to invert this formula and obtain the formula for each s k in terms of M k . If this formula turns out to be nice enough, we then can use the already known asymptotic formula for M k tocomputeeach s k .Weproposethatthefollowingformuladoesexactlythat: s k D s k X jD0 kCj j ! . 1/ j M kCj D s k X jD0 kCj j ! . 1/ j s X tDkCj t kCj ! s t : (3.142) To verify, we compare the coecients. Upon expanding the right-side, when j D 0, we easily verify that the coecient of s k is 1. For s kCl withl>0 (butkCl  s, sol  s k), observe thatitscoecientis s k X jD0 kCj j ! . 1/ j kCl kCj ! D l X jD0 . 1/ j .kCl/ä .l j/äjäkä D l X jD0 . 1/ j l j ! kCl k ! D0: (3.143) Thuswecanwrite .⌧ k U n s/D s X lDk s l D s X lDk s l X jD0 lCj j ! . 1/ j M lCj D s X lDk s l X jD0 lCj j ! . 1/ j .P lCj CR lCj /; (3.144) whereP lCj andR lCj representtheprincipalandtheremainderparts,respectively,asdescribedin HaydnandYang. Wenowhandleeachpartseparately.Firstly,letusexaminetheprincipalpart.HaydnandYang 84 provedintheirpaperthattheprincipalpartP j canbeexpressas P lCj D.1Co.1//s.U n / ✓ ✓ 1 ✓ ◆ lCj 1 sE lCj ; (3.145) whereE lCj istheerrortermboundedby jE lCj jc.1Co.1//.U n / ✓ ✓ 1 ✓ ◆ n 2m .lCj 1/ Dc.1Co.1//.U n /⇢ lCj 1 : (3.146) Here,⇢D.✓=.1 ✓// n=2m .Wewillhandlethiserrortermlater. TocomputetheprincipalpartofP l ,putr D ✓=.1 ✓/.Then s X lDk s l X jD0 lCj j ! . 1/ j ✓ ✓ 1 ✓ ◆ lCj 1 D s X lDk s l X jD0 lCj j ! . 1/ j r lCj 1 : (3.147) PuttinglCj Dq,weobtain s X lDk s l X jD0 lCj j ! . 1/ j r lCj 1 D s X lDk s X qDl q l ! . 1/ q l r q 1 D 1 r s X lDk . 1/ l s X qD0 q l ! . r/ q : (3.148) Observethat s X qD0 q l ! . r/ q Dr l s X qD0 . 1/ l q.q 1/.q 2/.q lC1/ lä . r/ q l D r l lä d l dr l s X qD0 . r/ q (3.149) Assuming that jrjd<1, the limit of the term-wise dierentiated power series converges uniformlytothederivativeoftheoriginalpowerseries,whichyields lim s!1 r l lä d l dr l s X qD0 . r/ q D r l lä d l dr l 1 X qD0 . r/ q D r l lä d l dr l 1 1Cr D r l lä . 1/ l lä .1Cr/ lC1 D . 1/ l 1Cr ✓ r 1Cr ◆ l : (3.150) Thus 1 r s X lDk . 1/ l s X qD0 q l ! . r/ q D 1 r.1Cr/ 1 X lDk ✓ r 1Cr ◆ l D 1 r ✓ r 1Cr ◆ k D.1 ✓/✓ k 1 : (3.151) ObservethatthisformulaagreeswellwiththepreviouslyknownresultwhenkD1. We now handle the error term of the principal term. The technique is almost identical, except nowinsteadofr,wehave⇢,andthereisnosignalternation: s X lDk s l X jD0 lCj j ! E lCj c.1Co.1//.U n / s X lDk s l X jD0 lCj j ! ⇢ lCj 1 ; (3.152) where⇢D.✓=.1 ✓// n=2m .Applyingthesametechnique,weobservethat s X lDk s l X jD0 lCj j ! ⇢ lCj 1 D 1 ⇢ s X lDk s X qDl q l ! ⇢ q D 1 ⇢ s X lDk s X qD0 q l ! ⇢ q ; (3.153) which,ass goestoinfinity,convergesto 1 ⇢ 1 1 ⇢ 1 X lDk ✓ ⇢ 1 ⇢ ◆ l D 1 .1 ⇢/ k ⇢ k 1 1 2⇢ !0; (3.154) asntendstowardsinfinity. 86

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Creator Park, Gin (author)

Core Title Localized escape rate and return times distribution

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School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Degree Conferral Date 2023-08

Publication Date 07/14/2023

Defense Date 07/13/2023

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Advisor Haydn, Nicolai (committee chair), Lototsky, Sergey (committee member), Zanardi, Paolo (committee member)

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Abstract We show that in dynamical systems with left φ-mixing measures, the number of visits to a suitably chosen target set of vanishing measure converges to a compound Poisson random variable when the time scale is in the reciprocal of the vanishing measure using generating functions and convolution formulae. We use this result to prove that all higher order localized escape rates are identical to the first order one, which is the extremal index at a periodic point.