graph directed markov systems (mauldin r d,urbanski m)

《graph directed markov systems (mauldin r d,urbanski m)》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

Thispageintentionallyleftblank CAMBRIDGETRACTSINMATHEMATICSGeneralEditorsB.BOLLOBAS,W.FULTON,A.KATOK,F.KIRWAN,P.SARNAK148GraphDirectedMarkovSystems R.DanielMauldinandMariuszUrba´nskiGraphDirectedMarkovSystemsGeometryanddynamicsoflimitsets Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPauloCambridgeUniversityPressTheEdinburghBuilding,Cambridge,UnitedKingdomPublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYorkwww.cambridge.orgInformationonthistitle:www.cambridge.org/9780521825382©CambridgeUniversityPress2003Thisbookisincopyright.Subjecttostatutoryexceptionandtotheprovisionofrelevantcollectivelicensingagreements,noreproductionofanypartmaytakeplacewithoutthewrittenpermissionofCambridgeUniversityPress.Firstpublishedinprintformat2003-isbn-13978-0-511-07091-4eBook(EBL)-isbn-100-511-07091-8eBook(EBL)-isbn-13978-0-521-82538-2hardback-isbn-100-521-82538-5hardbackCambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofsforexternalorthird-partyinternetwebsitesreferredtointhisbook,anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. ContentsIntroductionpagevii1Preliminaries12SymbolicDynamics42.1Topologicalpressureandvariationalprinciples52.2Gibbsstates,equilibriumstatesandpotentials122.3Perron–Frobeniusoperator262.4Ionescu-TulceaandMarinescuinequality312.5Stochasticlaws402.6AnalyticpropertiesofpressureandthePerron–Frobeniusoperator432.7TheexistenceofeigenmeasuresoftheConjugatePerron–FrobeniusoperatorandofGibbsstates483H¨olderFamiliesofFunctionsandF-ConformalMeasures543.1SummableH¨olderfamilies543.2F-conformalmeasures574ConformalGraphDirectedMarkovSystems624.1SomepropertiesofconformalmapsinRdwithd≥2624.2Conformalmeasures;Hausdorffandboxdimensions714.3Stronglyregular,hereditarilyregularandirregularsystems874.4Dimensionsofmeasures904.5Hausdorff,packingandLebesguemeasures944.6Porosityoflimitsets1034.7Theassociatediteratedfunctionsystem1074.8Refinedgeometry,F-conformalmeasuresversusHausdorffmeasures1094.9Multifractalanalysis123v viContents5ExamplesofGDMSs1365.1ExamplesofGDMSsinotherfieldsofmathematics1365.2Exampleswithspecialgeometricfeatures1396ConformalIteratedFunctionSystems144dµ6.1TheRadon-Nikodymderivativeρ=144dm6.2RateofapproximationoftheHausdorffdimensionbyfinitesubsystems1536.3Uniformperfectness1566.4Geometricrigidity1606.5Refinedgeometricrigidity1657DynamicalRigidityofCIFSs1717.1Generalresults1717.2One-dimensionalsystems1767.3Two-dimensionalsystems1817.4Rigidityindimensiond≥31958ParabolicIteratedFunctionSystems2098.1Preliminaries2098.2Topologicalpressureandassociatedparameters2128.3Perron–Frobeniusoperator,semiconformalmeasuresandHausdorffdimension2188.4Theassociatedhyperbolicsystem.Conformalandinvariantmeasures2228.5Examples2349ParabolicSystems:HausdorffandPackingMeasures2389.1Preliminaries2389.2TheCased≥32409.3Theplanecase,d=22469.4Proofsofthemaintheorems255Appendix1Ergodictheory262Appendix2Geometricmeasuretheory264GlossaryofNotation269Bibliography272Index280 IntroductionThegeometricanddynamictheoryofthelimitsetgeneratedbytheit-erationoffinitelymanysimilaritymapssatisfyingtheopensetconditionhasbeenwelldevelopedforsometimenow.Overthepastseveralyears,theauthorshaveinturndevelopedatechnicallymorecomplicatedgeo-metricanddynamictheoryofthelimitsetgeneratedbytheiterationofinfinitelymanyuniformlycontractingconformalmaps,a(hyperbolic)conformaliteratedfunctionsystem.Thistheoryallowsonetoanalyzemanymorelimitsets,forexamplesetsofcontinuedfractionswithre-strictedentries.Werecallandextendthistheoryinthelaterchapters.Themainfocusofthisbookistheexplorationofthegeometricanddy-namicpropertiesofafarreachinggeneralizationofaconformaliteratedfunctionsystemcalledagraphdirectedMarkovsystem(GDMS).Thesesystemsareveryrobustinthattheyapplytomanysettingsthatdonotfitintotheschemeofconformaliteratedsystems.Whilethebasictheoryislaidouthereandwetouchonmanynaturalquestionsarisinginitscontext,weemphasizethattherearemanyissuesandcurrentresearchtopicswhichwedonotcover:forexamples,thedetailedanalysisofthestructureofharmonicmeasuresoflimitsetsprovidedin[UZd],theex-aminationofthedoublingpropertyofconformalmeasuresperformedin[MU6],theextensivestudyofgeneralizedpolynomiallikemappings(see[U7]and[SU]),themultifractalanalysisofgeometricallyfiniteKleiniangroups(see[KS]),andtheconnectiontoquantizationdimensionfromengineering(see[LM]and[GL]).Therearemanyresearchproblemsinthisactiveareathatremainunsolved.Ourbookisorganizedasfollows.IntheveryshortfirstchapterwedescribethebasicsettingforGDMSs.Essentially,oneiteratesafamilyofuniformlycontractingmapswhichareindexedbythedirectededgesinamultigraphwhichmayhaveinfinitelymanyverticesandinfinitelyvii viiiIntroductionmanyedges.(Thisincludesthefinitegraphdirectedsystemsofsimil-aritiesintroducedin[MW2,EM]andexpoundedbyEdgarin[E].)Onegeneratespointsinthelimitsetbyperforminganinfinitedirectedwalkthroughthegraph.Thisleadstoanaturalmapfromthecodingspaceorspaceofinfinitewalksthroughthegraphtothepointsofthelimitset.Chapter2formsaself-containedunitandcanbereadindependentlyoftherestofthebook.Herewedevelopthesymbolicdynamicsandther-modynamicformalismforsubshiftsoffinitetypewithinfinitelymanysymbols.Thus,wearegivenanincidencematrixAandweconsiderthespaceEofallinfinitesequencesofsymbolssuchthatAhasvalue1onallpairsofconsecutivetermsandtheshiftmaponE.Thisformalismincludestheactiononthecodingspacedescribedinthefirstchapter.Oneofourmaingoalsistodevelopthetheoryandpropertiesofvari-ousmeasuresandfunctionsonthesymbolspace.Toafunctionfonthesymbolspaceisassociateditstopologicalpressurewithrespecttotheshiftmap.Ofcourse,thisisastandardthingtodo,butcaremustbetakenbecausewehaveinfinitelymanysymbolsandthespaceEisnotcompact.Inthefirstsectionwedeterminesomeconditionsunderwhichthetopologicalpressuremaybeapproximatedbythemoreusualpressureswhenthesystemisrestrictedtosubsystemsonfinitelymanysymbols.Next,wedetermineconditionsunderwhichthereexistsaninvariantGibbsstateforthefunctionsf.WepresentsomeresultsofourselvesandofSarigwhichstatethatforareasonableclassoffunc-tionsfwhichwecallacceptable,thereisaninvariantGibbsstateforfifandonlyifthematrixAisfinitelyirreducible.Wealsodeterminesomeconditionsunderwhichfhasauniqueinvariantequilibriumstate.InthethirdsectionwedevelopthepropertiesofthetransferorPerron–Frobeniusoperatorassociatedtof.Inordertofullyanalyzeoursystem,weprovidesomeresultsfromfunctionalanalysisinthefourthsection.Weproveanexponentialdecayofcorrelations,acentrallimittheoremandageneralizedlawoftheiteratedlogarithminsection5.Insection6,wevarythefunctionfwithacomplexparametert.Weshowthatvariousoperatorsarethenholomorphic,whichimpliesunderappropri-ateassumptionsrealanalyticityofthepressurefunction.Insection7,weshowthatforcertainfunctionsf,theassociatedconjugatePerron–FrobeniusoperatorhasaBorelprobabilitymeasureasaneigenmeasure.ThisallowsustoconcludethatifAisfinitelyprimitivethenthereisaGibbsstateforf. IntroductionixInChapter3,byusingthetoolsdevelopedinChapter2,weputthethermodynamicformalismforinfinitesubshiftsoffinitetypeintothecontextofGDMSs.WebeginwithF,aH¨olderfamilyofweightfunctionsassociatedtotheGDMS.ByusinganassociatedtopologicalpressureandPerron–Frobeniusoperator,wedetermineconditionsunderwhichthereisanF-conformalmeasure.ThisverygeneraldefinitionofF-conformalmeasurenotonlygeneralizestheusualnotionofconformalmeasure,butformsabasictoolforlateruseinobtainingthegeometricpropertiesofthelimitsets.InChapter4wedealindetailwithgeometricandfractalpropertiesofconformalGDMSs.WebeginbyprovingvariouskindsofdistortionpropertiesofconformalmapsinIRdwithd≥2.Wethendealinthischapterwiththevariousbasicnotionsofdimensionforthelimitset:Hausdorff,upperandlowerMinkowskiorbox(ball)countingdimensionandpackingdimensionandthecorrespondingHausdorffandpackingmeasures.WedealwiththeHausdorffandpackingdimensionofvariousnaturalmeasuressupportedonthelimitsetandsomegeometricprop-ertiesofthelimitset,e.g.porosity.Finally,weobtainthemultifractalanalysisofvariousconformalmeasuressupportedonthelimitset.Weemphasizethatitisinthischapterthatwemusttransfermanyres-ultsfromtheabstractcodingspacetothelimitset.Asapointmayhavemorethanonecode,thisleadstoseveraldelicateissuesingeomet-ricmeasuretheory.Therefore,therolesofdistortionpropertiesofoursystemofmapsandthegeometricpropertiesofourseedsets,e.g.arelativelyreasonableboundary,“theconecondition,”playacrucialroleinobtainingthenecessaryestimatesofouranalysis.Chapter5isdevotedtovariousillustrativeexamplesincludingKlein-iangroupsofSchottkytype,expandingrepellersandanumberofone-dimensionalsystemswithprescribedgeometricfeatures.InChapter6westarttopresentthespecialcaseofaGDMS,acon-formaliteratedfunctionsystem(CIFS).Westudythereal-analyticex-tensionoftheRadon-Nikodymderivativeofaninvariantmeasurewithrespecttoaconformalmeasure,theclassicalexamplebeingGauss’meas-urefortheshiftmaponcontinuedfractions.WeestimatetherateatwhichtheHausdorffdimensionofthelimitsetsgeneratedbythefinitesubsystemsofaCIFSapproximatesthedimensionofthelimitset.Wedetermineconditionsunderwhichthelimitsetisuniformlyperfect.InSection4ofthischapterwebeginthediscussionofgeometricrigiditybydealingwiththelimitssetofaCIFSwhoseclosureisconnected.Inessencetherigidityinthissectionmeansthatincased≥3eitherthe xIntroductionHausdorffdimensionofthelimitsetislargerthan1orelseincased≥3thelimitsetisasubsetofcircleorlineandincased=2itisasubsetofananalyticJordancurve.Inthenextsectionweimproveonthisrigiditybyshowingthatifd≥3thenessentiallyeithertheHausdorffdimensionofthelimitsetJexceedsthetopologicaldimensionkoftheclosureofJorelsetheclosureofJisapropercompactsubsetofeitherageometricsphereoranaffinesubspaceofdimensionk.InChapter7wedealwithdynamicalrigiditystemmingfromtheworkofSullivan(see[Su3])onconformalexpandingrepellersinthecomplexplane.Weaskthefundamentalquestionwhentwotopologicallycon-jugateinfiniteiteratedfunctionsystemsareconjugateinasmootherfashion.Theansweristhatsuchconjugacyextendstoaconformalcon-jugacyonsomeneighborhoodsoflimitsetsifandonlyifitisLipschitzcontinuous.Thisturnsouttoequivalentlymeanthatthisconjugacyexchangesmeasureclassesofappropriateconformalmeasuresorthatthemultipliersofcorrespondingfixedpointsofallcompositionsofourgeneratorscoincide.InChapter8westudyPIFS,paraboliciteratedfunctionsystems.Thesearesystemsthatarealmostconformalexceptthatweallowfinitelymanyofthemapstohaveaparabolicorneutralfixedpointinsteadofbeinguniformlycontracting.AprimeexampleofsuchasystemisgivenbythesystemofthreeconformalmapsintheplanewhoselimitsetistheresidualsetinApollonianpacking.Weanalyzethesesystemsbyshowinghowtoassociateaconformaliteratedfunctionsystemtotheparabolicsystemandhowthepropertiesoftheparaboliclimitsetandmeasuresmaybederivedfromthisassociatedconformalsystem.Itisinterestingthattheparabolicsystemmayconsistoffinitelymanymaps,buttheassociatedconformalsystemisinfinite.Bymovingfromthefinitetotheinfinite,theanalysisbecomeseasier.InChapter9weprovideadetailedquantativeanalysisofthedynam-icalbehaviorofparabolicmaps(indimensiond≥2)aroundparabolicpointsandweapplyittoprovideacompletecharacterizationofcon-formalmeasuresoffiniteparabolicsystemsintermsofHausdorffandpackingmeasures.Thissimultaneouslyprovidestheanswertotheques-tionaboutnecessaryandsufficientconditionsforthesetwogeometricmeasurestobefiniteandpositive.InfirstsectionoftheAppendixwecollectsomebasicconceptsandtheoremsfromergodictheoryandinthesecondsectioncontainsacom-pressedexpositionofsometopicsfromgeometricmeasuretheorywhichareofinteresthere. IntroductionxiWehavealsoprovidedtwoindexes,oneforterminologyandtheotherforspecialsymbolicnotation,andsomereferences.WethankCambridgeUniversityPressfortheenormoushelpprovidedinbringingthisprojecttofruition.WethankLarryLindsayforhiscorrectionstopartsofthemanuscript.Ofcourse,webearresponsibilityforallerrorsandomissionsandaskforegivenessofallwhomwehaveoverlookedinourcredits.Finally,wewishtothanktheNationalScienceFoundationforitssupportforourresearchduringthepreparationofthisbook. 1PreliminariesGraphdirectedMarkovsystemsarebaseduponadirectedmultigraphandanassociatedincidencematrix,(V,E,i,t,A).Themultigraphcon-sistsofafinitesetVofverticesandacountable(eitherfiniteorinfinite)setofdirectededgesEandtwofunctionsi,t:E→V.Foreachedgee,i(e)istheinitialvertexoftheedgeeandt(e)istheterminalvertexofe.Theedgegoesfromi(e)tot(e).Also,afunctionA:E×E→{0,1}isgiven,calledanincidencematrix.ThematrixAisanedgeincidencematrix.Itdetermineswhichedgesmayfollowagivenedge.So,thematrixhasthepropertythatifAuv=1,thent(u)=i(v).Wewillcon-siderfiniteandinfinitewalksthroughthevertexsetconsistentwiththeincidencematrix.Thus,wedefinethesetofinfiniteadmissiblewordsE∞={ω∈E∞:A=1foralli≥1},Aωiωi+1byEnwedenotethesetofallsubwordsofE∞oflengthn≥1,andAAbyE∗wedenotethesetofallfinitesubwordsofE∞.WewilldropAAthesubscriptAwhenthematrixisclearfromcontext.Wewillconsidertheleftshiftmapσ:E∞→E∞definedbydroppingthefirstentryofω.Sometimeswealsoconsiderthisshiftasdefinedonwordsoffinitelength.Givenω∈E∗by|ω|wedenotethelengthofthewordω,i.e./theuniquensuchthatω∈En.Ifω∈E∞andn≥1,thenω|n=ω1...ωn.AGraphDirectedMarkovSystem(GDMS)consistsofadirectedmulti-graphandincidencematrixtogetherwithasetofnon-emptycompactmetricspaces{Xv}v∈V,anumbers,00,wedefineametricd,onI∞,bysettingd(ω,τ)=e−α|ω∧τ|.TheseααmetricsareallequivalentandinducethesametopologyandBorelsets.Afunctionisuniformlycontinuouswithrespecttooneofthesemetricsifandonlyifitisuniformlycontinuouswithrespecttoall.Also,afunctionisH¨olderwithrespecttooneofthesemetricsifandonlyifitisH¨olderwithrespecttoall;ofcoursetheH¨olderorderdependsonthemetric.Ifnometricisspecificallymentioned,wetakeittobed1.Inthischapterwepresentvariousaspectsofthethermodynamicform-alismofacontinuouspotentialonashiftspacegeneratedbyacountablealphabet.Ourapproachhasbeendevelopedinseveralpapersculminat-ingin[MU3]andstemsfromthatofRuelle[Ru]andBowen[B1],cf.also[Wa]and[PU].Thecaseofacountableshifthasalsobeenconsidered4 2.1Topologicalpressureandvariationalprinciples5(seee.g.[Gu],[GS],[PP],[Sar]and[Zar]).Ourdefinitionoftopologicalpressureismoretraditionalthanthatproposedintheseworks.Itfitsbetterwithourgeometricapplicationsandneedsnocompactificationsoftheshiftspace.InparticularweareabletoconstructGibbsstatesandequilibriumstatesofunboundedpotentials.2.1TopologicalpressureandvariationalprinciplesTheincidencematrixAissaidtobeirreducibleifforalli,j∈Ithereexistsapathω∈E∗suchthatω=iandω=j.Thisisequivalent1|ω|tosayingthatthedirectedgraphisstronglyconnected:foranytwoelementsa,bofIthereisafinitepathstartingataandendingatb.Thisisalsoequivalenttosayingthattheleftshiftmap,σ,onE∞=I∞Aistopologicallymixing:foranytwonon-emptyopensubsetsU,VofI∞Athereisanon-negativeintegernsuchthatσn(U)∩V=∅.WesayAisprimitiveifthereexistsp≥1suchthatalltheentriesofAparepositive,orinotherwords,foralli,j∈Ithereexistsapathω∈Epsuchthatω1=iandω|ω|=j.ThematrixAissaidtobefinitelyirreducibleifthereexistsafinitesetΛ⊂E∗suchthatforalli,j∈Ithereexistsapathω∈Λforwhichiωj∈E∗.Wenotethefollowingfact.IfAisirreducible,thenAisfinitelyirreducibleifandonlyifthereisafinitesetoflettersFsuchthatforeverya∈I,therearep,q∈FsuchthatAap=Aqa=1.Finally,AissaidtobefinitelyprimitiveifthereexistsafinitesetΛ⊂E∗consistingofwordsofthesamelengthsuchthatforalli,j∈Ithereexistsapathω∈Λforwhichiωj∈E∗.Noticethatafinitelyirreduciblematrixdoesnothavetobeprimitivenorconversely.NoticealsothatthesetΛ(associatedeitherwithafinitelyirreducibleorfinitelyprimitivematrix)canbetakentobeemptyprovidedE∞consistsofallinfinitewordsfromI.GivenasetF⊂I,weput∞∞EF={ω∈E:ωi∈Fforalli≥1}.Asequence{a}∞consistingofrealnumbersissaidtobesubadditivenn=1ifan+m≤an+anforallm,n≥1.Forthesakeofcompletnessweprovidetheproofofthefollowingwell-knownelementaryfact.Lemma2.1.1Ifasequence{a}∞issubadditive,thenlimann=1n→∞nexistsandisequaltoinfnan/n.Thelimitcouldbe−∞,butifthean’sareboundedbelow,thenthelimitisnonnegative. 6SymbolicDynamicsProof.Fixm≥1.Eachn≥1canbeexpressedasn=km+iwith0≤iPF(f).Bythedefinitionofpressurethereexistsnt≥1suchthatforeveryn≥nt
t−PF(f)logexpsup(Snf|[ω∩F])0.Bytheacceptabilityoff,thereexistsl≥1suchthat|f(ω)−f(τ)|<",ifω|l=τ|landM=osc(f)<∞.Now,fixk≥l.Bysubadditivity,1logZ(f)≥P(f).NoticethatthereexistsafinitesetkkF⊂Isuchthat1logZk(F,f)>P(f)−".(2.2)kWemayassumethatFcontainsΛ.Putk
−1f=f◦σj.j=0Now,foreveryelementτ=τ,τ,...,τ∈Fk∩Ek×···×Fk∩Ek12n(nfactors)onecanchooseelementsα1,α2,...,αn−1∈Λsuchthatτ=τατα...τατ∈E∗.Noticethatthefunctionτ→τisat1122n−1n−1n 2.1Topologicalpressureandvariationalprinciples9mostqn−1-to-1(infactun−1-to-1,whereuisthenumberoflengthsofwordscomposingΛ.Thenforeveryn≥1,kn+
p(n−1)qn−1Z(F,f)ii=kn

|τ|≥expsupf◦σjτ∈(Fk∩Ek)n[τ∩F]j=0

|τ|≥expinff◦σj[τ]τ∈(Fk∩Ek)nj=0

n≥expinff|[τi]+T(n−1)τ∈(Fk∩Ek)ni=1

n=exp(T(n−1))expinff|[τi]τ∈(Fk∩Ek)ni=1

n≥exp(T(n−1))exp(supf|[τi]−(k−l)"−Ml)τ∈(Fk∩Ek)ni=1

n=exp(T(n−1)−(k−l)"n−Mln)expsupf|[τi]τ∈(Fk∩Ek)ni=1n
=e−Texpn(T−(k−l)"−Ml)exp(supf|).[τ]τ∈(Fk∩Ek)Hence,thereexistskn≤in≤(k+p)nsuchthat1Z(F,f)≥e−Texpn(T−(k−l)"−Ml−logq)Z(F,f)ninkpnandtherefore,using(2.2),weobtain1−|T|l"Ml+logpPF(f)=limlogZin(F,f)≥−"+−n→∞inkk+pk+P(f)−2"≥P(f)−7"providedthatkislargeenough.Thus,letting"0,thetheoremfollows.ThecaseP(f)=∞canbetreatedsimilarly.
Wesayaσ-invariantBorelprobabilitymeasure˜µonE∞isfinitelysup-portedprovidedthereexistsafinitesetF⊂Isuchthat˜µ(E∞)=1.FThewell-knownvariationalprincipleforfinitelysupportedmeasures(see 10SymbolicDynamics[B1],[Ru],comp.[Wa]and[PU])tellsusthatforeveryfinitesetF⊂IPF(f)=sup{hµ˜(σ)+fdµ˜},wherethesupremumistakenoverallσ-invariantergodicBorelprob-abilitymeasures˜µwith˜µ(F∞)=1andh(σ)istheentropyofµ˜withµ˜respecttoσ.ApplyingTheorem2.1.5,wethereforeobtainthefollowing.Theorem2.1.6(1stvariationalprinciple)IfAisfinitelyirreducibleandiff:E∞→IRisacceptable,thenP(f)=sup{(h(σ)+fdµ˜},µ˜wherethesupremumistakenoverallσ-invariantergodicBorelprobab-ilitymeasuresµ˜whicharefinitelysupported.Forn≥1,letαnbethestandardpartitionofE∞intocylindersoflengthn:αn={[ω]:|ω|=n}.Ifn=1,wewritealsoαforα1.IfβisacountablemeasurablepartitionofE∞and˜µisaprobabilitymeasure,thentheentropyofµ˜withrespecttothepartitionβisHµ˜(β)=−B∈βµ˜(B)log˜µ(B).Ournexttheoremisthefollowing.Theorem2.1.7(2ndvariationalprinciple)Iff:E∞→IRisacon-tinuousfunctionandµ˜isaσ-invariantBorelprobabilitymeasureonE∞suchthatfdµ>˜−∞,thenhµ˜(σ)+fdµ˜≤P(f).Inaddition,ifP(f)<∞,thenthereexistsq≥1suchthatH(αq)<∞.µ˜Proof.IfP(f)=+∞,thereisnothingtoprove.So,supposethatP(f)<∞.Thenthereexistsq≥1suchthatZn(f)<∞foreveryn≥q.Also,foreveryn≥1,wehave
µ˜([ω])sup(Snf|[ω])≥Snfdµ˜=nfdµ>˜−∞.|ω|=n 2.1Topologicalpressureandvariationalprinciples11Therefore,usingtheconcavityofthefunctionh(x)=−xlogx,weobtainforeveryn≥q,H(αn)+Sfdµ˜µ˜n
≤µ˜([ω])(supSnf|[ω]−log˜µ([ω]))|ω|=n
=Z(f)Z(f)−1esupSnf|[ω]hµ˜([ω])e−supSnf|[ω]nn|ω|=n
≤Z(f)hZ(f)−1esupSnf|[ω]µ˜([ω])e−supSnf|[ω]nn|ω|=n=Z(f)h(Z(f)−1)nn
=logexpsupSnf|[ω]=logZn(f).|ω|=nTherefore,H(αn)≤logZ(f)+n(−f)dµ<˜∞foreveryn≥q,andµ˜nsinceinadditionαqisagenerator,weobtain1nhµ˜(σ)+fdµ˜≤liminfHµ˜(α)+Snfdµ˜)n→∞n1≤limlogZn(f)=P(f).n→∞n
AsanimmediateconsequenceofTheorem2.1.6andTheorem2.1.7,wefindthefollowing.Theorem2.1.8(3rdvariationalprinciple)Supposetheincidencemat-rixAisfinitelyirreducible.Iff:E∞→IRisacceptable,thenP(f)=sup{hµ˜(σ)+fdµ˜},wherethesupremumistakenoverallσ-invariantergodicBorelprobab-ilitymeasuresµ˜suchthatfdµ>˜−∞.Weendthissectionwiththefollowingusefultechnicalfact.Proposition2.1.9Iftheincidencematrixisfinitelyirreducibleandthefunctionfisacceptable,thenP(f)<∞ifandonlyifZ1(f)<∞.Proof.LetΛ⊂E∗beasetofwordswhichwitnessesthefiniteir-reducibilityoftheincidencematrixA.Lets=max{|α|:α∈Λ}and 12SymbolicDynamicslet|α
|−1M=mininff◦σj:α∈Λ.[α]j=0Forn≥1andω∈In,letω=ωαωα...ωαω,whereall1122n−1n−1nα,...,αareappropriatelytakenfromΛsothatω∈E∗.Thus,1n−1n≤|ω|≤n+s(n−1).SimilarlyasintheproofofTheorem2.1.5,thefunctionω→ωisatmostqn−1-to-1,whereq=#Λ.Sincefisacceptable,wethereforegetn+
s(n−1)n+
s(n−1)

i−1qn−1Z(f)=qn−1expsupf◦σjii=ni=ni[ω]j=0ω∈E
|ω
|−1≥expsupf◦σjω∈In[ω]j=0

n≥expinff|[ωj]+M(n−1)ω∈Inj=1

n≥eM(n−1)expsupf|−osc(f)n[ωj]ω∈Inj=1n
=exp−M+(M−osc(f))nexp(supf|[e]e∈In=exp−M+(M−osc(f))nZ1(f).FromthisitfollowsthatifP(f)<∞,thenalsoZ1(f)<∞.TheoppositeimplicationisobvioussinceZ(f)≤Z(f)n.
n12.2Gibbsstates,equilibriumstatesandpotentialsIff:E∞→IRisacontinuousfunction,thenaBorelprobabilitymeas-ure˜monE∞iscalledaGibbsstateforf(cf.[B1],[HMU],[PU],[Ru],[Wa]and[U1])ifthereexistconstantsQ≥1andPm˜suchthatforeveryω∈E∗andeveryτ∈[ω]−1m˜([ω])Q≤≤Q.(2.3)expS|ω|f(τ)−Pm˜|ω|Ifadditionally˜misshift-invariant,itisthencalledaninvariantGibbsstate. 2.2Gibbsstates,equilibriumstatesandpotentials13Remark2.2.1NoticethatthesumS|ω|f(τ)in(2.3)canbereplacedbysup(S|ω|f|[ω])orbyinf(S|ω|f|[ω]).Also,noticethatifm˜isaGibbsstateandifµ˜andm˜areboundedlyequivalent(meaningthereissomeK≥1suchthatK−1≤µ˜([ω])/m˜([ω])≤Kforallω∈E∗),thenµ˜isaGibbsstateforthepotentialf.Wewillusethesefactsfromtimetotime.Westartwiththefollowing.Proposition2.2.2(a)ForeveryGibbsstatem˜,Pm˜=P(f).(b)AnytwoGibbsstatesforthefunctionfareequivalent,withRadon-Nikodymderivativesboundedawayfromzeroandinfin-ity.Proof.Weshallfirstprove(a).Towardsthisendfixn≥1and,usingRemark2.2.1,sum(2.3)overallwordsω∈En.Sincem˜([ω])=|ω|=n1,wethereforeget

Q−1e−Pm˜nexpsupSf|≤1≤Qe−Pm˜nexpsupSf|.n[ω]n[ω]|ω|=n|ω|=nApplyinglogarithmstoallthreetermsofthisformula,dividingallthetermsbynandtakingthelimitasn→∞,weobtain−Pm˜+P(f)≤0≤−Pm˜+P(f),whichmeansthatPm˜=P(f).Theproofofitem(a)isthuscomplete.Inordertoprovepart(b)supposethatmandνaretwoGibbsstatesofthefunctionf.Noticenowthatpart(a)impliestheexistenceofaconstantT≥1suchthat−1ν([ω])T≤≤Tm([ω])forallwordsω∈E∗.StraightforwardreasoninggivesnowthatνandmareequivalentandT−1≤dν≤T.
dmAsanimmediateconsequenceof(2.3)andRemark2.2.1wegetthefollowing.Proposition2.2.3Anyuniformlycontinuousfunctionf:E∞→IRthathasaGibbsstateisacceptable.Forω∈E∗andn≥1,letEω={τ∈En:A=1}andEω={τ∈E∗:A=1}.nτnω1∗τ|τ|ω1 14SymbolicDynamicsWeshallprovethefollowingresultconcerninguniquenessandsomestochasticpropertiesofGibbsstates.StrongerstochasticpropertieswillbeprovedinSection2.5Theorem2.2.4IfanacceptablefunctionfhasaGibbsstateandtheincidencematrixAisfinitelyirreducible,thenfhasauniqueinvariantGibbsstate.TheinvariantGibbsstateisergodic.Moreover,ifAisfinitelyprimitive,theinvariantGibbsstateiscompletelyergodic.Proof.Let˜mbeaGibbsstateforf.Fixingω∈E∗,using(2.3),Remark2.2.1andProposition2.2.2(a)wegetforeveryn≥1m˜(σ−n([ω]))

=m˜([τω])≤Qexpsup(Sn+|ω|f|[τω])−P(f)(n+|ω|)τ∈Eωτ∈Eωnn
≤Qexpsup(Snf|[τ])−P(f)nexpsup(S|ω|f|[ω])−P(f)|ω|τ∈Eωn
≤QQm˜([τ])Qm˜([ω])≤Q3m˜([ω]).τ∈Eωn(2.4)LetthefinitesetofwordsΛwitnessthefiniteirreducibilityoftheincid-encematrixAandletpbethemaximallengthofawordinΛ.Sincefisacceptable,T=min{inf(Sα|f|[α])−P(f)|α|:α∈Λ}>−∞.Foreachτ,ω∈E∗,letα=α(τ,ω)∈Λbesuchthatταω∈E∗.Then,wehaveforallω∈E∗andallnn
+pn
+p

m˜(σ−i([ω]))=m˜([τω])≥m˜([τα(τ,ω)ω])i=ni=nτ∈Eωτ∈Eni
≥Q−1expinf(Sf|)|τ|+|α(τ,ω)|+|ω|[ταω]τ∈En−P(f)(|τ|+|α(τ,ω)|+|ω|)(2.5)
≥Q−1expinf(Sf|)−P(f)(n)n[τ]τ∈En+inf(S|α(τ,ω)||f|[α(τ,ω)])−P(f)|α(τ,ω)|)+inf(S|ω|f|[ω])−P(f)|ω| 2.2Gibbsstates,equilibriumstatesandpotentials15
≥Q−1eTexpinf(Sf|)−P(f)|ω||ω|[ω]τ∈En×expinf(Snf|[τ])−P(f)(n)
≥Q−2eTm˜([ω])expinf(Sf|)−P(f)(n)n[τ]τ∈En
−2T−1≥Qem˜([ω])Qm˜([τ])τ∈En|=Q−3eTm˜([ω]).LetLbeaBanachlimitdefinedontheBanachspaceofallboundedsequencesofrealnumbers.Itisnotdifficulttocheckthattheformulaµ˜(A)=L(˜m(σ−n(A)))definesafinite,non-zero,invariant,finitelyn≥0additivemeasureonBorelsetsofE∞satisfyingQ−3eT3m˜(A)≤µ˜(A)≤Qm˜(A).(2.6)pSince˜misacountablyadditivemeasure,wededucethat˜µisalsocount-ablyadditive.Letusprovetheergodicityof˜µor,equivalently,of˜m.Letω∈En.Foreachτ∈E∗,wefind:n
+p−im˜(σ([τ])∩[ω])≥m˜([ωα(ω,τ)τ])(2.7)i=n−3T≥Qem˜([τ])˜m([ω]).TakenowanarbitraryBorelsetA⊂E∞.Fix">0.Sincethenestedfamilyofsets{[τ]:τ∈E∗}generatestheBorelσ-algebraonE∞,foreveryn≥0andeveryω∈EnwecanfindasubfamilyZofE∗consistingofmutuallyincomparablewordssuchthatA⊂{[τ]:τ∈Z}andforn≤i≤n+p,
−i−im˜(σ([τ])∩[ω])≤m˜([ω]∩σ(A))+"/p.τ∈ZThen,using(2.7)wegetn
+pn
+p
"+m˜([ω]∩σ−i(A))+"≥m˜([ω]∩σ−i(τ))i=ni=nτ∈Z
−3T(2.8)≥Qem˜([τ])˜m([ω])τ∈Z≥Q−3eTm˜(A)˜m([ω]). 16SymbolicDynamicsHence,letting"0,wegetn
+pm˜([ω]∩σ−i(A))≥Q−3eTm˜(A)˜m([ω]).i=nFromthisinequalitywefindn
+pn
+pm˜σ−i(E∞B)∩[ω]=m˜[ω]σ−i(B)∩[ω]i=ni=nn
+p=m˜([ω])−m˜σ−i(B)∩[ω]i=n≤[p+1−Q−3eTm˜(B)]˜m([ω]).Thus,foreveryBorelsetB⊂E∞,foreveryn≥0,andforeveryω∈Enwehaven
+pm˜(σ−i(B)∩[ω])≤(p+1−Q−3eT(1−m˜(B)))˜m([ω]).(2.9)i=nInordertoconcludetheproofoftheergodicityofσ,supposethatσ−1(B)=Bwith01sosmallthatγη<1andchooseasubfamilyRofE∗consistingofmutuallyincom-parablewordsandsuchthatB⊂{[ω]:ω∈R}and˜m{[ω]:ω∈R}≤ηm˜(B).Then˜m(B)=m˜(B∩[ω])≤γm˜([ω])=ω∈Rω∈Rγm˜{[ω]:ω∈R}≤γηm˜(B)0.Sincethenestedfamilyofsets{[τ]:τ∈E∗}generatestheBorelσ-algebraonE∞,for 2.2Gibbsstates,equilibriumstatesandpotentials17everyn≥0andeveryω∈EnwecanfindasubfamilyZofE∗consistingofmutuallyincomparablewordssuchthatA⊂{[τ]:τ∈Z}and
−(n+qr)−(n+qr)m˜(σ([τ])∩[ω])≤m˜[ω]∩σ(A)+".τ∈ZThen,using(2.10)weget
−(n+qr)−3rT"+˜m[ω]∩σ(A)≥Qem˜([τ])˜m([ω])τ∈Z≥Q−3erTm˜(A)˜m([ω]).Hence,letting"0,wegetm˜[ω]∩σ−(n+qr)(A)≥Q˜(r)˜m(A)˜m([ω]),whereQ˜(r)=Q−3exp(rT).NotethatitfollowsfromthislastinequalitythatQ˜=Q˜(r)≤1.Also,fromthisinequalitywefind˜mσ−(n+qr)(E∞B)∩[ω]=˜m[ω]σ−(n+qr)(B)∩[ω]=˜m([ω])−m˜σ−(n+qr)(B)∩[ω]≤(1−Q˜m˜(B))˜m([ω]).Thus,foreveryBorelsetB⊂E∞,foreveryn≥0,andforeveryω∈Inwehavem˜(σ−(n+qr)(B)∩[ω])≤1−Q˜(1−m˜(B))m˜([ω]).(2.11)Inordertoconcludetheproofofthecompleteergodicityofσsupposethatσ−r(B)=Bwith01sosmallthatγη<1andchooseasubfamilyRof(Er)∗consistingofmutuallyincomparablewordsandsuchthatB⊂{[ω]:ω∈R}andm˜{[ω]:ω∈R}≤ηm˜(B).Then˜m(B)≤m˜(B∩[ω])≤ω∈Rγm˜([ω])=γm˜{[ω]:ω∈R}≤γηm˜(B)k0expsupf|[aj]≤KwhereKcomesfromlemma2.2.5. 2.2Gibbsstates,equilibriumstatesandpotentials19LetF={ai:i≤k0}.Now,itfollowsfromtheprecedinglemmathateveryletterisfollowedbysomeelementofFandeveryletterisprecededbysomeelementofF.SinceAisirreducible,itfollowsthatAisfinitelyirreducible.
Recallthatforω,τ∈I∞,wedefinedω∧τ∈I∞∪I∗tobethelongestinitialblockcommontobothωandτ.Wesaythatafunctionf:E∞→IRisH¨oldercontinuouswithanexponentα>0ifVα(f):=sup{Vα,n(f)}<∞,n≥1whereV(f)=sup{|f(ω)−f(τ)|eα(n−1):ω,τ∈E∞and|ω∧τ|≥n}.α,nNotethatifgisH¨oldercontinuousoforderαandθ,ψ∈E∞,thenV(g)d(θ,ψ)=V(g)e−α|θ∧ψ|≥e−α|g(θ)−g(ψ)|.αααAlso,notethateachH¨oldercontinuousfunctionisacceptable.Wesaythattwofunctionsf,g:E∞→IRarecohomologousinaclassHifthereexistsafunctionu:E∞→IRintheclassHsuchthatg−f=u−u◦σ.WeshallnowprovidealistofnecessaryandsufficientconditionsfortwoH¨oldercontinuousfunctionstohavethesameinvariantGibbsstates.TheproofisanalogoustotheproofofTheorem1.28in[B1](seealso[HMU]).Theorem2.2.7Supposethatf,g:E∞→IRaretwoH¨oldercontinu-ousfunctionsthathaveinvariantGibbsstatesµ˜fandµ˜grespectively.SupposealsothattheincidencematrixAisfinitelyirreducible.Thenthefollowingconditionsareequivalent:(1)µ˜f=˜µg.(2)ThereexistsaconstantRsuchthatforeachn≥1,ifσn(ω)=ω,thenSnf(ω)−Sng(ω)=nR.(3)Thedifferenceg−fiscohomologoustoaconstantRintheclassofboundedH¨oldercontinuousfunctions.(4)Thedifferenceg−fiscohomologoustoaconstantintheclassofboundedcontinuousfunctions. 20SymbolicDynamics(5)ThereexistconstantsSandTsuchthatforeveryω∈E∞andeveryn≥1|Snf(ω)−Sng(ω)−Sn|≤T.IftheseconditionsaresatisfiedthenR=S=P(f)−P(g).Proof.(1)⇒(2).Itfollowsfrom(2.3)that−2expSkf(ω)−P(f)k)2Q≤≤QexpSkg(ω)−P(g)k)foreveryω∈E∞andeveryk≥1.Supposethatσn(ω)=ω.Thenforeveryk=ln,l≥1,andeveryh,Slnh(ω)=lSnh(ω)andso−22Q≤expl[(Snf(ω)−Sng(ω))−(P(f)−P(g))n]≤Q.Hence,thereexistsaconstantT≥0suchthatforallpositiveintegersl,l|Snf(ω)−Sng(ω)−(P(f)−P(g))n|≤T.Therefore,lettingl∞,weconcludethatSnf(ω)−Sng(ω)=(P(f)−P(g))n.Thus,puttingR=P(f)−P(g)completestheproofoftheimplication(1)⇒(2).(2)⇒(3).Defineη=f−g−R.SincetheincidencematrixAisirreducible,thereexistsapointτ∈E∞transitivefortheshiftmapσ:E∞→E∞.PutΓ={σk(τ):k≥1}anddefinethefunctionu:Γ→IRbysettingk
−1kju(σ(τ))=η(σ(τ)).j=0Notethatthefunctionuiswelldefinedsincethepointsσk(τ),k≥1,aremutuallydistinct.TakingtheminimumofexponentswemayassumethatbothfunctionsfandgareH¨oldercontinuouswiththesameorderβ.LetΛwitnessthefiniteirreducibilityoftheincidencematrixA.Let|Λ|=sup{|α|:α∈Λ}andS=sup{|S|α|η|:α∈Λ}.Fixk≥1,somepointα∈Λsuchthatτατ∈E∗,andconsidertheperiodicpointk1 2.2Gibbsstates,equilibriumstatesandpotentials21ω=(τ|α)∞.Thenbyourassumptionkk
−1|u(σk(τ))|=η(σj(τ))−f(σj(ω))−g(σj(ω))+Rkj=0|α|−1
+g(σk+jω)−f(σk+jω)+R|α|j=0k
−1=(f(σj(τ))−f(σj(ω)))−(g(σj(τ))j=0−g(σj(ω)))−Sη(σk(ω)|α|(2.12)k
−1k
−1jjjj≤|f(σ(τ))−f(σ(ω))|+|g(σ(τ))−g(σ(ω))|j=0j=0+|Sη(σk(ω))||α|k
−1k
−1≤V(f)e−β(k−j)+V(g)e−β(k−j)+Sββj=0j=0e−β≤(Vβ(f)+Vβ(g))+S<∞.1−e−βAssumenowσk(τ)|=σl(τ)|forsomekinf(−f|[i])˜µf([i])≥Qinf(−f|[i])expinff|[i]−P(f)i∈Ii∈I
=Q−1e−P(f)inf(−f|)expinff|.[i][i]i∈I(b)⇒(c).Assumethati∈Iinf(−f|[i])expinf(f|[i])<∞.WeshallshowthatHµ˜f(α)<∞.Bydefinition,

Hµ˜f(α)=−µ˜f([i])log˜µf([i])≤−µ˜f([i])(inf(f|[i])−P(f)−logQ).i∈Ii∈ISincei∈Iµ˜f([i])(P(f)+logQ)<∞,itsufficestoshowthati∈I−µ˜f([i])inf(f|[i])<∞.Andindeed,

−µ˜f([i])inf(f|[i])=µ˜f([i])sup(−f|[i])i∈Ii∈I
≤µ˜f([i])inf(−f|[i])+osc(f).i∈ISincei∈Iµ˜f([i])osc(f)=osc(f),itisenoughtoshowthat
µ˜f([i])inf(−f|[i])<∞.i∈ISince˜µfisaprobabilitymeasure,limi→∞µ˜f([i])=0.Therefore,itfollowsfrom(2.3)thatlimi→∞sup(f|[i])−P(f)=−∞.Thus,forallisufficientlylarge,sayi≥k,sup(f|[i])<0.Hence,foralli≥k,inf(−f|[i])=−sup(f|[i])>0.So,using(2.3)again,weget

µ˜f([i])inf(−f|[i])≤Qexpinf(f|[i])−P(f)inf(−f|[i])i≥ki≥k
−P(f)=Qeexpinf(f|[i])inf(−f|[i])i≥kwhichisfiniteduetoourassumption.Finally,wefindHµ˜f(α)<∞.(c)⇒(a).SupposethatHµ˜f(α)<∞.Weneedtoshowthat−fdµ˜f<∞.Wehave
∞>Hµ˜f(α)=−µ˜f([i])logµ˜f([i])i∈I
≥−µ˜f([i])inf(f|[i])−P(f)+logQ.i∈I 24SymbolicDynamicsHence,i∈I−µ˜f([i])inf(f|[i])<∞andtherefore

−fdµ˜f=−fdµ˜f≤sup(−f|[i])˜µf([i])[i]i∈Ii∈I
=−inf(f|[i])˜µf([i])<∞.i∈I
Thenexttheoremshowsthattheassumption−fdµ˜f<∞issufficientfortheappropriateGibbsstatetobeauniqueequilibriumstate.Theorem2.2.9SupposethattheincidencematrixAisfinitelyirredu-cible.Supposethatf:E∞→IRisaH¨oldercontinuousfunctionwithaGibbsstateandthat−fdµ˜f<∞,whereµ˜fistheuniqueinvari-antGibbsstateforthepotentialf(seeTheorem2.2.4).Thenµ˜fistheuniqueequilibriumstateforthepotentialf.Proof.Inordertoshowthat˜µfisanequilibriumstateofthepotentialfconsiderα={[i]:i∈I},thepartitionofE∞intoinitialcylindersoflength1.ByLemma2.2.8,Hµ˜f(α)<∞.ApplyingtheShannon–McMillan–Breimantheorem,Birkhoff’sergodictheorem,and(2.3),wegetfor˜µ-a.e.ω∈E∞f1hµ˜f(σ)≥hµ˜f(σ,α)=lim−log˜µf([ω|n])n→∞n1≥lim−logQ+Snf(ω)−P(f)nn→∞n−1=limSnf(ω)+P(f)=−fdµ˜f+P(f)n→∞nwhich,inviewofTheorem2.1.7,impliesthat˜µfisanequilibriumstateforthepotentialf.Toproveuniquenessoftheequilibriumstatewesupposethat˜νisanequilibriumstateforthepotentialf:E∞→IRand˜ν=˜µ.Applyingftheergodicdecompositiontheorem,wemayassumethat˜νisergodic.Then,using(2.3),wehaveforeveryn≥10=n(h(σ)+(f−P(f))dν˜)≤H(αn)+(Sf−P(f)n)dν˜ν˜ν˜n
1=−ν˜([ω])log˜ν([ω])−(Snf−P(f)n)dν˜ν˜([ω])[ω]|ω|=n 2.2Gibbsstates,equilibriumstatesandpotentials25
≤−ν˜([ω])(log˜ν([ω])−(Snf(τω)−P(f)n))|ω|=nforasuitableτω∈[ω]
=−ν˜([ω])(log[˜ν([ω])exp(P(f)n−Snf(τω))])|ω|=n
−1≤−ν˜([ω])log[˜ν([ω])(µf([ω])Q)]|ω|=n
ν˜([ω])=logQ−ν˜([ω])log.µ˜f([ω])|ω|=nTherefore,inordertoconcludetheproof,itsufficestoshowthat
ν˜([ω])lim−ν˜([ω])log=−∞.n→∞µ˜f([ω])|ω|=nSincebothmeasures˜νand˜µfareergodicand˜ν=˜µf,themeasures˜νand˜µfmustbemutuallysingular.Inparticular,∞ν˜([ω|n])limν˜ω∈E:≤S=0n→∞µ˜f([ω|n])foreveryS>0.Foreveryj∈Zandeveryn≥1,set∞−jν˜([ω|n])−j+1Fn,j=ω∈E:e≤0andisassumedtosatisfy
exp(sup(f|[e]))<∞.(2.16)e∈IFunctionsfsatisfyingthisconditionwillbecalledinthesequelsum-mable.WenotethatiffhasaGibbsstate,thenfissummable.ThisrequirementallowsustodefinethePerron–FrobeniusoperatorLf:C(E∞)→C(E∞),actingonthespaceofboundedcontinuousfunc-bbtionsC(E∞)providedwith||·||,theuniformnorm,asfollows:b0
Lf(g)(ω)=exp(f(eω))g(eω).e∈I:Aeω=11Then||Lf||0≤e∈Iexp(sup(f|[e]))<∞andforeveryn≥1,
nLf(g)(ω)=expSnf(τω)g(τω).τ∈En:Aτnω1=1TheconjugateoperatorL∗actingonthespaceC∗(E∞)hasthefollowingfbform:L∗(µ)(g)=µ(L(g))=L(g)dµ.fffFromnowonthroughoutthissectionwealsoassume˜misaprobabilitymeasuredefinedontheBorelsubsetsofE∞whichisaneigenmeasureoftheconjugateoperatorL∗:C∗(E∞)→C∗(E∞).Thecorrespondingfbbeigenvalueisdenotedbyλ.SinceLfisapositiveoperator,λ≥0.ObviouslyL∗n(˜m)=λnm˜.Theintegralversionofthisequalitytakesfonthefollowingform
nexpSnf(τω)g(τω)dm˜(ω)=λgdm,˜(2.17)τ∈En:Aτnω1=1foreveryfunctiong∈C(E∞).InfactthisequalityextendstothespacebofallboundedBorelfunctionsonE∞.Inparticular,takingω∈E∗,sayω∈En,aBorelsetB⊂E∞suchthatA=1,foreveryτ∈B,andωnτ1g=11ωB,weobtainfrom(2.17)
λnm˜(ωB)=expSf(τρ)11(τρ)dm˜(ρ)nωBτ∈En:Aτnρ1=1=expSnf(ωρ)dm˜(ρ)(2.18){ρ∈B:Aωnρ1=1}=expSnf(ωρ)dm˜(ρ)B 28SymbolicDynamicsRemark2.3.2Notethatif(2.18)holds,thenbyrepresentingaBorelsetB⊂E∞asaunion∞:A=ω∈En[ωBω],whereBω={α∈Eωnα11andωα∈B},astraightforwardcalculationbasedon(2.18)demon-stratesthat(2.17)issatisfiedforthecharacteristicfunction11BofthesetB.Next,itfollowsfromstandardapproximationargumentsthat(2.17)issatisfiedforallm˜-integrablefunctionsg.Finally,wenotethatm˜isaneigenmeasureoftheconjugateoperatorL∗ifandonlyifformulaf(2.18)issatisfied.Theorem2.3.3Iftheincidencematrixisfinitelyirreducible,thentheeigenmeasurem˜isaGibbsstateforf.Inaddition,λ=eP(f).Proof.Itimmediatelyfollowsfrom(2.18)andLemma2.3.1thatforeveryω∈E∗andeveryτ∈[ω]m˜([ω])≤λ−nT(f)expSf(τ)=T(f)expSf(τ)−nlogλ,(2.19)nnwheren=|ω|.Ontheotherhand,letΛbeaminimalsetwhichwitnessesthefiniteirreducibilityofA.Foreveryα∈Λ,letE={τ∈E∞:ωατ∈E∞}.αBythedefinitionofΛ,E=E∞.Hence,thereexistsγ∈Λsuchα∈Λαthat˜m(E)≥(#Λ)−1.Writingp=|γ|wethereforehaveγ−(n+p)m˜([ω])≥m˜([ωγ])=λexpSn+pf(ωγρ)dm˜(ρ)ρ∈E∞:Aγpρ1=1=λ−(n+p)expSf(ωγρ)expSf(γρ)dm˜(ρ)npρ∈E∞:Aγpρ1=1≥λ−nexpmin{inf(Sf|):α∈Λ}−plogλ|α|[α]×expSnf(ωγρ)dm˜(ρ)ρ∈E∞:Aγpρ1=1=Cλ−nexpSf(ωγρ)dm˜(ρ)nEγ≥CT(f)−1λ−nm˜(E)expSf(τ)γn−1−1≥CT(f)(#Λ)expSnf(τ)−nlogλ,(2.20)whereC=expmin{inf(S|α|f|[α]):α∈Λ}−plogλ.Thus˜misaGibbsstateforf.Theequalityλ=eP(f)followsnowimmediatelyfromProposition2.2.2.
2.3Perron–Frobeniusoperator29InmostofourworkconcerningthePerron–Frobeniusoperatoranditsapplicationswewilldealwiththefollowingnormalizedversionofit.L=e−P(f)L.0fThefirstresultconcerningthenormalizedPerron–Frobeniusoperatoristhefollowing.Theorem2.3.4Ifafunctionf:E∞→IRhasaGibbsstatewithratioboundingconstantQ,thenforeveryn≥1andeveryω∈InnL0(11)(ω)≤Q.Thus,forallg∈C(E∞),||Ln(g)||≤Q||g||.b000Proof.LetνbeaGibbsmeasureforf.InviewofLemma2.3.1andthedefinitionofGibbsstatesweget
Ln(11)(ω)=expSf(τ)−P(f)n0nτ∈σ−n(ω)
≤Qν([τ|])≤Qν(σ−n([ω]))≤Q.nτ∈σ−n(ω)
WeemphasizethatinTheorem2.3.4weassumedonlytheexistenceofaGibbsstateandnotaneigenmeasureoftheconjugatePerron–Frobeniusoperator.Weshallnowprovethefollowing.Theorem2.3.5SupposeΛisafinitesetofwordswitnessingthefiniteirreducibilityoftheincidencematrix.LetMbethemaximallengthofawordinΛ.ThenthereexistsaconstantR>0suchthatn
+MjL0(11)(ω)≥R,j=nforalln≥1andallω∈E∞.Proof.Itfollowsfrom(2.17)andTheorem2.3.3thatLk(11)dm˜=10forallk≥1.So,foreveryk≥1,thereexistsγ(k)∈E∞suchthatLk(11)(γ(k))≥1.SinceΛisfiniteandfisH¨oldercontinuous,wehave0N=min{expinf(S|α|f|[α])−|α|P(f):α∈Λ}>0. 30SymbolicDynamicsLetn≥1andletτ∈E∞.Thus,n
+Mn
+M
jL0(11)(τ)=expSjf(ωτ)−P(f)j.j=nj=nω∈Ej:Aω=1jτ1Foreachω∈Enchooseα(ω,τ)∈Λsuchthatωα(ω,τ)τisadmissible.Thus,n
+M
jL0(11)(τ)≥expSn+|α(ω,τ)|f(ωα(ω,τ)τ)−P(f)(n+|α(ω,τ)|)j=nω∈En
≥expSnf(ωα(ω,τ)τ)−P(f)nω∈En×expS|α(ω,τ)|f(α(ω,τ)τ)−P(f)|α(ω,τ)|
≥NexpSnf(ωα(ω,τ)τ)−P(f)n.ω∈EnNotingthatifAωnγ(n)1=1thenexpSf(ωα(ω,τ)τ)−P(f)n≥(Tf)−1expSf(ωγ(n))−P(f)n,nnwehaven
+M
Lj(11)(τ)≥N(Tf)−1expSf(ωγ(n))−P(f)n0nj=nω∈En:Aωnγ(n)1=1−1n−1≥N(Tf)L0(11)(γ(n))≥NT(f).
Remark:ifintheorem2.3.5weassumeAisfinitelyprimitivethenthereisaconstantR>0suchthatLn(11)(ω)≥Rforallnandω.0Theorem2.3.6Iftheincidencematrixisfinitelyirreducible,thentheconjugateoperatorL∗fixesatmostoneBorelprobabilitymeasure.0Proof.Supposethat˜mand˜m1aresuchtwofixedpoints.InviewofProposition2.2.2(b)andTheorem2.3.3,themeasures˜mand˜m1areequivalent.ConsidertheRadon-Nikodymderivitiveρ=dm˜1.Tempor-dm˜arilyfixω∈E∗,sayω∈En. 2.4Ionescu-TulceaandMarinescuinequality31Itthenfollowsfrom(2.18)andTheorem2.3.3thatm˜([ω])=expSnf(ωτ)−P(f)ndm˜(τ)τ∈E∞:Aωnτ1=1=expSn−1f(σ(ωτ))−P(f)(n−1)τ∈E∞:Aωnτ1=1×expf(ωτ)−P(f)dm˜(τ)=expSn−1f(σ(ωτ))−P(f)(n−1)τ∈E∞:A(σ(ω))n−1τ1=1×expf(ωτ)−P(f)dm˜(τ).Thus,infexpf|[ω]−P(f)m˜([σω])≤m˜([ω])≤supexpf|[ω]−P(f)m˜([σω]).Sincef:E∞→IRisH¨oldercontinuous,wethereforeconcludethatforeveryω∈E∞m˜[ω|n]lim=expf(ω)−P(f)(2.21)n→∞m˜[σ(ω)|n−1]andthesameformulaistruewith˜mreplacedby˜m1.UsingThe-orem2.3.3andTheorem2.2.4,thereexistsasetofpointsω∈E∞with˜mmeasure1forwhichtheRadon-Nikodymderivativesρ(ω)andρ(σ(ω))arebothdefined.Letω∈E∞besuchapoint.Thenfrom(2.21)anditsversionfor˜m1weobtainm˜1[ω|n]ρ(ω)=limn→∞m˜[ω|n]m˜1[ω|n]m˜1[σ(ω)|n−1]m˜[σ(ω)|n−1]=lim··n→∞m˜1[σ(ω)|n−1]m˜[σ(ω)|n−1]m˜[ω|n]=expf(ω)−P(f)ρ(σ(ω))expP(f)−f(ω)=ρ(σ(ω)).ButaccordingtoTheorem2.2.4,σ:E∞→E∞isergodicwithrespecttoaσ-invariantmeasureequivalentwith˜m.Weconcludethatρism˜-almosteverywhereconstant.Since˜m1and˜marebothprobabilitymeasures,˜m1=˜m.
2.4Ionescu-TulceaandMarinescuinequalityInthissectionweproveinourcontextthefamousTulcea-IonescuandMarinescuinequality(seeLemma2.4.1).Itisouraimtoobtaina 32SymbolicDynamicsspectralpictureandtherateofconvergenceoftheiteratesofthePerron–Frobeniusoperatorand,inthenextsection,someimportantstochasticlaws.LetH={g:E∞→CI:gisboundedandcontinuous}0andforeveryα>0,letHα={g∈H0:Vα(g)<∞}.ThesetHαbecomesaBanachspacewhenendowedwiththenormgα=g0+Vα(g).ThesubclassofHαconsistingofsummablefunctionswillbedenotedbyHs.Themaintechnicalresultofthissection,calledtheIonescu-TulceaαandMarinescuinequality,isthefollowing.Lemma2.4.1Letf:E∞→IRbeasummableH¨oldercontinuousfunc-tionwithanexponentβ>0andsupposefhasaGibbsstate.ThenthenormalizedoperatorL0:H0→H0preservesthespaceHβ.Moreover,thereexistsaconstantC>0suchthatforeveryn≥1andeveryg∈Hβ,||Ln(g)||≤Qe−βn||g||+C||g||.0ββ0Proof.Letτ,ρ∈E∞,τ|=ρ|andτ=ρforsomek≥1.Thenkkk+1k+1foreveryn≥1Ln(g)(ρ)−Ln(g)(τ)00

=exp(Snf(ωρ)−P(f)n)g(ωρ)−ω∈En:Aωnρ1=1ω∈En:Aωnρ1=1×exp(Snf(ωτ)−P(f)n)g(ωτ)
=exp(Snf(ωρ)−P(f)n)g(ωρ)−g(ωτ)ω∈En:Aωnρ1=1
+g(ωτ)exp(Snf(ωρ)−P(f)n)ω∈En:Aωnρ1=1−exp(Snf(ωτ)−P(f)n).(2.22) 2.4Ionescu-TulceaandMarinescuinequality33But|g(ωρ)−g(ωτ)|≤V(g)e−β(n+k),andtherefore,employingThe-βorem2.3.4,weobtain
exp(Snf(ωρ)−P(f)n)|g(ωρ)−g(ωτ)|ω∈En:Aωnρ1=1n−β(n+k)(2.23)≤L0(11)(ρ)Vβ(g)e−β(n+k)−βn≤QVβ(g)e≤eQ||g||βdβ(ρ,τ)NownoticethatthereexistsaconstantM≥1suchthat|1−ex|≤M|x|forallxwith|x|≤log(T(f)).SincebyLemma2.3.1|Snf(ωρ)−Snf(ωτ)|≤dβ(ρ,τ)log(T(f))≤log(T(f)),wecanmakethefollowingestimate:|exp(Snf(ωρ)−P(f)n)−exp(Snf(ωτ)−P(f)n)|=exp(Snf(ωρ)−P(f)n)|1−exp(Snf(ωτ)−Snf(ωρ))|≤Mexp(Snf(ωρ)−P(f)n)|Snf(ωρ)−Snf(ωτ)|≤Mexp(Snf(ωρ)−P(f)n)log(T(f))dβ(ρ,τ)=Mlog(T(f))exp(Snf(ωρ)−P(f)n)dβ(ρ,τ).So,usingTheorem2.3.4again,weget
|g(ωτ)||exp(Snf(ωρ)−P(f)n)−exp(Snf(ωτ)−P(f)n)|ω∈En:Aωnρ1=1
≤||g||0Mlog(T(f))dβ(ρ,τ)exp(Snf(ωρ)−P(f)n)ω∈En:Aωnρ1=1n=||g||0Mlog(T(f))dβ(ρ,τ)L0(11)(ρ)≤MQlog(T(f))||g||0dβ(ρ,τ).Combiningthisinequality,(2.23)and(2.22),weget|Ln(g)(ρ)−Ln(g)(τ)|≤e−βnQ||g||d(ρ,τ)+MQlog(T(f))||g||d(ρ,τ).00ββ0βCombininginturnthisandTheorem2.3.4weget||Ln(g)||≤Qe−βn||g||+Q(Mlog(T(f))+1)||g||.0ββ0
Remark2.4.2WeremarkthattheproofofLemma2.4.1usedonlya“weaker”propertyofGibbsstates,namelytheright-handsideinequalityof(2.3).IftheunitballinHβwerecompactasasubsetoftheBanachspaceH0withthesupremumnorm||·||0,wecouldusenowthefamousIonescu-TulceaandMarinescutheorem(see[ITM])toestablishsome 34SymbolicDynamicsusefulspectralpropertiesofthePerron–FrobeniusoperatorL0.ButthisballiscompactonlyinthetopologyofuniformconvergenceoncompactsubsetsofE∞andweneedtoprovethesespectralpropertiesdirectlyaswecannotapplytheIonescu-TulceaandMarinescutheorem.Next,weintroduceandimportantfixedpointψforL0.Theorem2.4.3SupposeasummableH¨oldercontinuousfunctionf:E∞→IRwithanexponentβ>0hasaGibbsstateandtheoperatorconjugatetothenormalizedPerron–FrobeniusoperatorL0fixesaBorelprobabilitymeasurem˜.ThentheoperatorL0:Hβ→Hβhasafixedpointψ≤Qsuchthatψdm˜=1.If,inaddition,theincidencematrixAisfinitelyprimitivethenψ≥R,whereRistheconstantproducedinTheorem2.3.5.Proof.InviewofLemma2.4.1,Ln(11)≤Q+Cforeveryn≥0.0βThus,n
−11jL0(11)≤Q+C(2.24)nj=0βforeveryn≥1.Therefore,bytheAscoli–Arzel´atheorem,thereexistsastrictlyincreasingsequenceofpositiveintegers{nk}k≥1suchthatthe1nk−1jsequencenkj=0L0(11)convergesuniformlyoncompactsetsk≥1ofE∞toψ:E∞→IR.Obviously,||ψ||≤Q+Cand,inparticularβψ∈Hβ.Since˜misafixedpointoftheoperatorconjugatetoL0,j1n−1jL0(11)dm˜=1foreveryj≥0.Consequently,nj=0L0(11)dm˜=1foreveryn≥1.So,applyingLebesgue’sdominatedconvergencetheoremalongwithTheorem2.3.4,weconcludethatψdm˜=1andψ≤Q.AssuminginadditionthattheincidencematrixAisfinitelyprimitive,usingTheorem2.3.5,wesimultaneouslygetψ≥R.ItremainstoshowthatL0(ψ)=ψ.Indeed,usingTheorem2.3.4,wegetforeveryk≥1thatn
k−1n
k−1L1Lj(11)−1Lj(11)=1Lnk(11)−L(11)|0n0n0n000kj=0kj=0k01≤Q.nk 2.4Ionescu-TulceaandMarinescuinequality35Thus,1n
k−11n
k−1LLj(11)−Lj(11)→0(2.25)000nknkj=0j=0uniformly.Therefore,inordertoconcludetheproofitsufficestoshowthatifasequence{g}∞⊂Hisuniformlyboundedandconvergeskk=10uniformlyoncompactsetsofE∞toafunctiong,thenL(g),k≥1,0kconvergesuniformlyoncompactsetsofE∞toL(g).Toseethis,first0noticethatg0≤B,whereBisanupperboundofthesequence{g}∞.Now,fix">0.SincefhasaGibbsstate,theseriesM=kk=1i∈Iexpsup(f|[i])−Pconverges,wherePisthepressureoffandthereforethereexistsafinitesetI+⊂Isuchthat
"2Bexpsup(f|[i])−P<.(2.26)2i∈IIFixanarbitrarycompactsetK⊂E∞.Thenforeveryi∈I,thesetiK={iω:ω∈KandAiω1=1}isalsocompactandsoisthesetiK.Since{g}∞convergesuniformlyoncompactsetstog,i∈Ikk=1thereexistsq≥1suchthatforeveryn≥q,(g−g)|≤+.ni∈IiK02MApplyingthis,Theorem2.3.4and(2.26),wegetforeveryn≥qandeveryω∈Kthat|L0(g)(ω)−L0(gn)(ω)|=|L0(g−gn)(ω)|
≤|gn(iω)−g(iω)|exp(f(iω)−P)i∈I:Aiω=11
+|gn(iω)−g(iω)|exp(f(iω)−P)i∈II:Aiω=11
"≤exp(f(iω)−P)2Mi∈I:Aiω=11
+|gn(iω)−g(iω)|exp(f(iω)−P)i∈II:Aiω=11"

≤expsup(f|[i])−P+2Bexpsup(f|[i])−P2Mi∈Ii∈II:Aiω=11""≤+=".22
36SymbolicDynamicsFromnowonweassumeinthissectionthattheincidencematrixAisfinitelyprimitive.Thenψ≥RandthereforetheweightednormalizedPerron–FrobeniusoperatorT:H0→H0givenbytheformula1T(g)=L0(gψ)(2.27)ψiswelldefined.ItisstraightforwardtocheckthatT(Hβ)⊂Hβ(i.e./1/ψandtheproductofanytwofunctionsinHβareagaininHβ).ThebasicpropertiesoftheoperatorTwhichfollowfromLemma2.4.1,Theorem2.3.5andTheorem2.4.3arelistedbelow.Theorem2.4.4SupposeasummableH¨oldercontinuousfunctionf:E∞→IRwithanexponentβ>0hasaGibbsstateandthematrixAisfinitelyprimitive.Definingthebalancingfunctionsψ(ω)un(ω)=exp(Snf(ω)−Pn),n≥1,ψ(σn(ω))wehaveforallg∈Hβandalln≥1(a)Tn(g)(ω)=1Ln(gψ)(ω)=u(τω)g(τω).ψ(ω)0τ∈En:Aτnω1=1n(b)Tn(11)=11and||Tn||=1.0(c)M=sup{||Tn||}<∞.n≥1β(d)T∗(˜µ)=˜µ.InparticulartheclosedsubspacesH0={g∈H:ff00µ˜(g)=0}andH0={g∈H:˜µ(g)=0}areT-invariant.fββf(e)H=IR11⊕H0(g=˜µ(g)11+(g−µ˜(g)11)).ββffDenote0,10Hβ={g∈Hβ:||g||β≤1}.Weshallprovethefollowing.Lemma2.4.5Foreachn≥0letn0,1bn=sup{||T(g)||β:g∈Hβ}.Thenlimn→∞bn=0.Proof.Defineforeveryn≥0n0,1an=sup{||T(g)||0:g∈Hβ}.ItimmediatelyfollowsfromTheorem2.4.4(b)thatthesequence{an}n≥1isnonincreasing.Weshallshowfirstthatliman=0.n→∞ 2.4Ionescu-TulceaandMarinescuinequality37LetΛbeafinitesetofwordsoflengthqwitnessingtheprimitivityoftheincidencematrixA.Foreveryα∈Λfixawordα∈E∞suchthatα|q=α.Supposenowonthecontrarythata=limn→∞an>0.ByTheorem2.4.4(c),supn≥0bn≤M<∞.Therefore,thereexistsk≥0suchthatifω|k=τ|k,thennn|T(g)(τ)−T(g)(ω)|≤a/2(2.28)0,1forallg∈Hβandalln≥0.Leta=min{a,1}.Sincethemeasure˜µfisregular,thereexistsacompactsetY⊂E∞suchthat˜µ(Y)≥1−a.f8Fixnowg∈H0,1andn≥0.SupposethatTn(g)(τ)>a/4forallβτ∈Y.SinceTngdµ˜=0andsince||Tn(g)||≤||g||≤1,wefindf00nnaaaaaa0=Tgdµ˜f+Tgdµ˜f≥1−−≥1−−YYc488448aa2=−>0.816ThiscontradictionshowsthatTn(g)(τ)≤a/4forsomeτ∈Y(depend-ingongandn).Fixthisτ.Noticenowthatforeveryω∈E∞thereexistsα(ω)∈Λsuchthatρ(ω)=τ|α(ω)ωisinE∞.Itfollowsfromk(2.28)thatnna3aT(g)(ρ(ω))≤T(g)(τ)+≤a≤an−.(2.29)244Putp=q+k.Sinceρ(ω)|k=τ|k,applyingLemma2.3.1wegetup(ρ(ω))ψ(ρ(ω))ψ(σp(τ))=expSq+kf(ρ(ω))−Sq+kf(τ)pup(τ)ψ(τ)ψ(σ(ρ(ω)))R2≥expSkf(ρ(ω))−Skf(τ)Q2×expSf(ρ(σk(ω)))−Sf(σk(τ))qq≥R2Q−2T(f)−1expSf(α(ω)ω)exp(−qsup(f))q≥R2Q−2T(f)−2expSf(α(ω)exp(−qsup(f))q≥R2Q−2T(f)−2expmax{Sf(α):α∈Λ}exp(−qsup(f)),qwheresup(f)<+∞dueto(2.16).Denotingthelast(constant)ex-pressionappearinginthisformulabyU,wegetup(ρ(ω))≥Uup(τ)≥Uinf(up|Y)>0sinceupispositivecontinuousandYiscompact.Thus, 38SymbolicDynamicsusing(2.29),weget
Tp(Tng)(ω)=Tng(ρ)u(ρ(ω))+Tng(η)u(η)ppη∈σ−p(ω){ρ(ω)}a
≤an−up(ρ(ω))+anup(η)8η∈σ−p(ω){ρ(ω)}aa=an−up(ρ(ω))≤an−Uinf(up|Y).88SimilarlywegetTp(Tng)(ω)≥−a+aUinf(u|)andinconsequence,n4pY||Tp+ng||≤a−aUinf(u|)or0n2pYna||Tg||0≤an−p−Uinf(up|Y)20,1foreveryn≥p.Takingthesupremumoverallg∈H,wefindβa≤a−aUinf(u|).So,a=lima≤lima−nn−p4pYn→∞nn→∞n−paUinf(u|)=a−aUinf(u|)0andthenanintegerv≥1solargethata≤+andv2CQe−βnM≤"/2foralln≥v.Then,inviewofLemma2.4.1forevery0,1n≥2vandeveryg∈Hwegetβnn−vv−β(n−v)vv||Tg||β≤||T(Tg)||β≤Qe||Tg||β+C||Tg||0"≤+Cav≤".2MSo,bn≤".
Theorem2.4.6Supposethatf:E∞→IRisasummableH¨oldercon-tinuousfunction,saywithanexponentβ>0.SupposealsothatthenormalizedconjugatePerron–FrobeniusoperatorL∗hasaBorelprob-0abilityfixedpointm˜f.AssumethattheincidencematrixAisfinitelyprimitive.ThenthereexistconstantsM>0and0<γ<1suchthatforeveryg∈Hβandeveryn≥0(a)Tn(g)−gdµ˜≤Mγngfββand(b)nnL0(g)−(gdm˜f)ψβ≤Mγgβ 2.4Ionescu-TulceaandMarinescuinequality39whereTistheoperatordefinedby(2.27),ψisthefixedpointforL0andµ˜fistheuniqueinvariantGibbsstateofthepotentialfwhoseexistenceanduniquenessfollowfromTheorem2.2.4andTheorem2.3.3.Proof.Lemma2.4.5saysthatlimT|n=0.Thus,thereex-n→∞H0ββqqnistsq≥1suchthatT|H0β≤(1/2).ByinductionwehaveT|H0β≤ββ(1/2)n.Considernowanarbitraryn≥0andwriten=pq+r,0≤r≤q−1.Then,usinginadditionTheorem2.4.4(c),wegetforeveryζ∈H0βn−rnpqrprpTζβ=T(Tζ)β≤(1/2)Tζβ≤M(1/2)=M(1/2)qn−q+11−qn≤M(1/2)q=M(1/2)q(1/2)q1−qandthereforeforeveryn≥0,T|n≤M(1/2)qγn,whereγ=H0ββ(1/2)1/q<1.Ifnowg∈H,theng−µ˜(g)∈H0andg−µ˜(g)≤βfβfβgβ+µ˜f(g)β≤2gβ.Thus,foreveryn≥0Tn(g−µ˜(g))≤22q−1/qMγngfββandtheproofofTheorem2.4.6(a)iscomplete.Part(b)isanimmediateconsequenceofpart(a).
Thenextproposition,thelastresultofthissection,explainstherealdy-namicalmeaningofthefixedpointsofthenormalizedPerron–FrobeniusoperatorL0.Proposition2.4.7Assumethattheoperatorconjugatetothenormal-izedPerron–FrobeniusoperatorL=e−P(f)LhasaBorelprobability0ffixedpointm˜.LetFix(L0)={g∈L1(˜m):L0(g)=g,gdm˜=1,andg≥0}andlet−1AI(˜m)={g∈L1(˜m):gm˜◦σ=gm,˜gdm˜=1,andg≥0}denotetheinvariantabsolutelycontinuousprobabilitydensityfunctions.ThenFix(L0)=AI(˜m).Proof.Itfollowsfrom(2.21)thatforeveryi∈Iandeveryω∈E∞withAiω1=1,wehavedm˜◦i(ω)=expf(iω)−P(f),dm˜ 40SymbolicDynamicswherewetreati:{ω∈E∞:A=1}→E∞asthemapdefinediω1bytheformulai(ω)=iω.Therefore,thePerron–FrobeniusoperatorL0sendsthedensityofameasure˜µabsolutelycontinuouswithrespecttom˜tothedensityofthemeasure˜µ◦σ−1.Thepropositionfollows.
2.5StochasticlawsInthissection,inafashionsomewhatsimilartosection3of[DU1],wereapsomeofthefruitscomingfromtheworkdoneinthetwopreced-ingsections.LetΓbeafiniteorcountablemeasurablepartitionofaprobabilityspace(Y,F,ν)andletS:Y→Ybeameasurepreservingtransformation.For0≤a≤b≤∞,setΓb=S−lΓ.Theaa≤l≤bmeasureνissaidtobeabsolutelyregularwithrespecttothefiltrationdefinedbyΓ,ifthereexistsasequenceβ(n)0suchthatasupsup|ν(A|Γ0)−ν(A)|dν≤β(n).YaA∈Γ∞a+nThenumbersβ(n),n≥1,arecalledcoefficientsofabsoluteregularity.LetαbethepartitionofE∞intoinitialcylindersoflength1.UsingTheorem2.4.6,andproceedingexactlyasintheproofof[Ry,§3ofTheorem2.5]wederivethefollowing(withthenotationofprevioussections).Theorem2.5.1Themeasureµ˜fisabsolutelyregularwithrespecttothefiltrationdefinedbythepartitionα.Thecoefficientsofabsoluteregularitydecreaseto0atanexponentialrate.Theorem2.5.1saysinparticularthatthedynamicalsystem(σ,µ˜f)isweak-Bernoulli(see[Or]).Asanimmediateconsequenceofthistheoremandtheresultsprovedin[Or]wegetthefollowing.Theorem2.5.2Thenaturalextensionofthedynamicalsystem(σ,µ˜f)isisomorphicwithsomeBernoullishift.Itfollowsfromthistheoremthatthetheoryofabsolutelyregularpro-cessesapplies([IL],[PS]).Wesketchthisapplicationbriefly.Wesaythatameasurablefunctiong:E∞→IRbelongstothespaceL∗(σ)if2+δthereexistconstantsδ,γ,M>0suchthatg0dµ˜f<∞andg−Eg|(α)n2+δdµ˜≤Mn−2−γµ˜f0f 2.5Stochasticlaws41foralln≥1,whereEg|(α)ndenotestheconditionalexpectationµ˜fofgwithrespecttothepartition(α)nandthemeasure˜µ.L∗(σ)isfalinearspace.ItfollowsfromTheorem2.5.1,[IL]and[PS]thatwithµ˜f(g)=gdµ˜f,theseriesσ2=σ2(g)=(g−µ˜(g))2dµ˜ffE∞
∞+2(g−µ˜(g))(g◦σn−µ˜(g))dµ˜fffE∞n=1isabsolutelyconvergentandnon-negative.Thereadershouldnotbeconfusedbytwodifferentmeaningsofthesymbolσ:thenumberdefinedaboveandtheshiftmap.Thentheprocess(g◦σn:n≥1)exhibitsanexponentialdecayofcorrelations,andifσ2>0itsatisfiesthecentrallimittheorem.Theorem2.5.3Ifu,v∈L∗(σ)thenthereareconstantsC,θ>0suchthatforeveryn≥1wehave(u−Eu)(v−Ev)◦σndµ≤Ce−θn,fwhereEu=udµ˜fandEv=vdµ˜f.Theorem2.5.4Ifg∈L∗(σ)andσ2(g)>0,thenforallrn−1g◦σj(ω)−nEg∞j=0√µ˜fω∈E:0wehave˜µf-a.e.
!"1j2−λg◦σ−µ˜f(g)−B(σt)=O(t2).0≤j≤tLeth:[1,∞)−→IRbeapositivenon-decreasingfunction.Thefunction 42SymbolicDynamicshissaidtobelongtothelowerclassif∞h(t)1exp−h(t)2dt<∞t21andtotheupperclassif∞h(t)12exp−h(t)dt=∞.1t2Well-knownresultsforBrownianmotionimply(seeTheoremAin[PS])thefollowinggeneralizationofthelawoftheinteratedlogarithm.Theorem2.5.5Ifg∈L∗(σ)andσ2(g)>0thenn
−1∞jµ˜f{ω∈E:g(σ(ω))−µ˜f(g)j=0√>σ(g)h(n)nforinfinitelymanyn≥1}0ifhbelongstothelowerclass=1ifhbelongstotheupperclass.OurlastgoalinthissectionistoprovideasufficientconditionforafunctionψtobelongtothespaceL∗(σ).Lemma2.5.6EachH¨oldercontinuousfunctionwhichhassomefinitemomentgreaterthan2belongstoL∗(σ).Proof.ItsufficestoshowthatanyH¨oldercontinuousfunctionψ:E∞→IRsatisfiestherequirementψ−Eψ|(α)n3dµ˜≤Mn−2−γ.µ˜f0fSo,givenn≥1supposethatω,τ∈AforsomeA∈αn.Inparticu-larω|=τ|.Hence|ψ(ω)−ψ(τ)|≤V(ψ)e−βnwhichmeansthatnnβψ(τ)−V(ψ)e−βn≤ψ(ω)≤ψ(τ)+V(ψ)e−βn.Integratingthesein-ββequalitiesagainstthemeasure˜µfandkeepingωfixed,weobtain−βn−βnψdµ˜f−Vβ(ψ)eµ˜f(A)≤ψ(ω)˜µf(A)≤ψdµ˜f+Vβ(ψ)eµ˜f(A).AADividingtheseinequalitiesby˜µf(A)wededucethat1ψ(ω)−ψdµ˜≤V(ψ)e−βn.µ˜(A)fβfAThus,ψ(ω)−Eψ|(α)n3dµ˜≤V(ψ)3e−3βnandwearedone.
µ˜f0fβ 2.6AnalyticpropertiesofpressureandthePerron–Frobenius432.6AnalyticpropertiesofpressureandthePerron–FrobeniusoperatorInthissectionKβisthespaceofallcomplex-valuedH¨oldercontinuousfunctionsoforderβonE∞and
Ks={f∈K:expsup(Re(f|))<∞}.ββ[i]i∈IL(Kβ)denotesthespaceofallbounded(continuous)operatorsonKβ.WegiveacompletedetailedproofoftherealanalyticityofthePerron–Frobeniusoperator,cf.[EM],[HMU]and[UZi].Westartwiththefollowing.Lemma2.6.1Ifforeveryω∈E∞,thefunctiont→f(ω)∈CIistholomorphiconadomainG⊂CIandthemapt→Lft∈L(Kβ)iscontinuousonG,thenthemapt→Lft∈L(Kβ)isholomorphiconG.Proof.Letγ⊂Gbeasimpleclosed,contractibleinG,rectifiablecurve.Fixg∈Kandω∈E∞.LetW⊂Gbeaboundedopensetβsuchthatγ⊂W⊂W⊂G.Sinceforeache∈IsuchthatAeω1=1,thefunctiont→g(eω)expft(ω)∈CI,t∈G,isholomorphicandsinceforeveryt∈W

g(eω)expft(ω)∞≤g(eω)expft(ω)βe:Aeω=1e:Aeω=111≤gβsup{Lfzβ:z∈W}<∞bycompactnessofWandcontinuityoft→Lft,weconcludethatthefunction
t→Lft(g)(ω)=g(eω)expft(ω)∈CI,t∈W,e:Aeω=11isholomorphic.HencebyCauchy’stheoremγLftg(ω)dt=0.Sincethefunctiont→Lftg∈Kβiscontinuous,theintegralγLftgdtexists,andforeveryω∈E∞,wehaveLgdt(ω)=Lg(ω)dt=0.HenceγftγftγLftgdt=0.Now,sincet→Lft∈L(Kβ)iscontinuous,theintegralγLftdtexists,andforeveryg∈Kβ,γLftdt(g)=γLftgdt=0.ThusγLftdt=0andinviewofMorera’stheoremthemapt→Lft∈L(Kβ)isholomorphiconG.
Inordertoprovethemainresultofthissectionweneedseveralelement-arylemmas.Inordertoformulatethemweneedtodefineacertainclass 44SymbolicDynamicsofmappings.Namely,giveni∈I,wedefinethemappingi:{ω∈E∞:A=1}→E∞iω1bysettingi(ω)=iω.Ifg:E∞→CI,thenbyg◦i:E∞→CIwemeanthefunctiondefinedbythefollowingformula.g(iω)ifAiω1=1g◦i(ω)=0ifAiω1=0.Lemma2.6.2Ifi∈Iandρ∈KβthentheoperatorAi,ρgivenbytheformulaAi,ρ(g)(ω)=ρ◦i(ω)·g◦i(ω)actsonthespaceKβ,iscontinuousandAi,ρβ≤3ρ◦iβ.Proof.Fixg∈K,ω∈E∞andsupposethatA=1.Thenβiω1|Ai,ρ(g)(ω)|=|ρ(iω)g(iω)|≤ρ◦i0g0≤ρ◦iβgβ.(2.30)Fixnowinadditionτ∈E∞{ω}suchthat|ω∧τ|≥1.Then|Ai,ρ(g)(ω)−Ai,ρ(g)(τ)|=|ρ(iω)g(iω)−ρ(iτ)g(iτ)|=|ρ(iω)(g(iω)−g(iτ))+g(iτ)(ρ(iω)−ρ(iτ))|≤ρ◦i0|g(iω)−g(iτ)|+g0|ρ(iω)−ρ(iτ)|≤ρ◦ige−β|ω∧τ|+gρ◦ie−β|ω∧τ|.ββββHenceVβ(Ai,ρ(g))≤2ρ◦iβgβand,combiningthiswith(2.30),weconcludethatAi,ρ(g)≤3ρ◦iβgβ.consequentlyAi,ρactsonthespaceKβ,iscontinuousandAi,ρβ≤3ρ◦iβ.
Similarly(butmoreeasily)oneprovesthefollowing.Lemma2.6.3Ifρ,g∈Kβ,thenρg∈Kβandρgβ≤3ρβgβ.Lemma2.6.4Iff∈Kthenef∈K,andifρ:Y→KisaβββcontinuousmappingdefinedonacompactsetYtheneρ:Y→Kisβalsocontinuous.Proof.ByLemma2.6.3,fn≤3nfn.Hence,theseriesef=ββ∞fnn=0n!convergesinKβandthefirstpartofourlemmaisproved. 2.6AnalyticpropertiesofpressureandthePerron–Frobenius45Thesecondpartfollowsnowfromtheremarkthatforeveryy∈Y,∞ρ(y)nρ(y)β≤sup{ρ(x)β:x∈Y}<∞andtheseriesn=0n!con-vergesuniformlyonY.
Lemma2.6.5ForeveryR>0thereexistsM=MR≥1suchthatif|z−ξ|≤R,then|eξ−ez|≤MeRez|z−ξ|.Proof.LookingattheTaylor’sseriesexpansionoftheexponentialfunctionabout0,weseethatthereexistsaconstantM≥1such|ew−1|≤M|w|,if|w|≤R.Hence|eξ−ez|=|ezez−ξ−1|≤eRezM|z−ξ|.
Lemma2.6.6Iff∈Kβ,thenforeveryi∈Ief◦i≤2MexpsupRef|f.βfβ[i]βProof.Fixω∈E∞suchthatA=1.Then|ef(iω)|=eRef(iω)≤iω1expsupRef|[i],whenceef◦i≤expsupRef|.(2.31)β[i]Fixnowinadditionτ∈E∞{ω}with|τ∧ω|≥1.UsingtheLemma2.6.5,weget|ef(iω)−ef(iτ)|≤MeRef(τ)|f(iω)−f(iτ)|≤Mfβfβ−β|τ∧ω|×expsupRef|[i]fβe.Thus,V(ef◦i)≤expsupRef|f.Combiningthisand(2.31)β[i]βcompletestheproof.
Lemma2.6.7Ifρ:Y→KβisacontinuousmappingdefinedonametricspaceY,thenforeveryi∈I,thefunctiony→Ai,ρ(y)∈L(Kβ),y∈Y,iscontinuous.Proof.Fixy0∈Yandtakeδ>0sosmallthatforeveryy∈B(y0,δ),ρ(y)−ρ(y0)β≤"/3.Thenfory∈B(y0,δ),wehaveinviewofLemma2.6.2thefollowing.Ai,ρ(y)−Ai,ρ(y0)β=Ai,ρ(y)−ρ(y0)β≤3ρ(y)−ρ(y0)β≤3.
Ourmaintheoreminthissectionisthefollowing.Theorem2.6.8IfGisanopenconnectedsubsetofCI,thefunctiont→f∈Ks,t∈G,iscontinuousandthefunctiont→f(ω)∈CI,t∈G,tβt 46SymbolicDynamicsisholomorphicforeveryω∈E∞,thenthefunctiont→L∈L(K),ftβt∈G,isholomorphic.Proof.InviewofLemma2.6.2itsufficestodemonstratethatthefunctiont→Lft∈L(Kβ),t∈G,iscontinuous.So,fixt0∈Gandδ>0sosmallthatB(t0,2δ)⊂Gandft−ft0∞≤ft−ft0β≤1forallt∈B(t0,2δ).ByLemmas2.6.7and2.6.4,foralli∈I,thefunctiont→Ai,eft∈L(Kβ),t∈B(t0,δ),iscontinuous.Since
Lft=Ai,eft,i∈Iitthereforesufficestodemonstratethattheseriesi∈IAi,eftconvergesuniformlyonB(t0,δ).Andindeed,inviewofLemma2.6.2andLemma2.6.6,foreveryi∈Iandeveryt∈B(t0,δ)wehaveAi,eftβ≤3exp(ft◦i)β≤6MexpsupReft|[i]M1,whereM1=sup{ftβ:t∈B(t0,δ)}isfinitebycontinuityofthefunctiont→f∈KsonthecompactsetB(t,δ),andM=Minthetβ0M1senseofLemma2.6.5.Now,inviewofourchoiceofδ,wecancontinuetheaboveestimatesasfollows.Ai,eftβ≤6MM1expsupReft0|[i]+ft−ft00≤6MM1expsupReft0|[i]+1≤6MM1expsupReft0|[i].Sincebysummabilityofthefunctionft0,theseriesi∈IexpsupReft0|[i]converges.
AsanimmediateconsequenceofTheorem2.6.8,wegetthefollowing.Corollary2.6.9IfGisanopenconnectedsubsetofCI,andthefunctiont→f∈Ks,t∈G,isholomorphic,thenthefunctiont→L∈L(K),tβftβt∈G,isalsoholomorphic.Inthesequelwewillneedthefollowingmuchmorespecialresultfollow-ingimmediatelyfromCorollary2.6.9.Corollary2.6.10IfGisanopenconnectedsubsetofCI,andforeacht∈G,thepotentialf=f+tψ∈Ksforsomef,ψ∈K,thenthetββfunctiont→Lft∈L(Kβ),t∈G,isholomorphic. 2.6AnalyticpropertiesofpressureandthePerron–Frobenius47Remark2.6.11Sinceeachrealanalyticfunctiont→f∈Ksdefinedtβonanopeninterval∆⊂IRextendsuniquelytoaholomorphicfunctiondefinedonanopenconnectedneighborhoodof∆inCIandsinceKsisβanopensubsetofKβ,Corollary2.6.9and2.6.10remaintrueifGisreplacedby∆andholomorphicitybyrealanalyticity.AsanimmediateconsequenceofRemark2.6.11,Corollary2.6.9,Theorem2.3.3andTheorem2.4.6(implyingthateP(f)isanisolatedsimpleeigenvalueofthePerron–FrobeniusoperatorLf)andtheper-turbationtheoryofanalyticdependenceofanisolatedsimpleeigenvalue(see[Ka]),wegetthefollowingremarkableresult.Theorem2.6.12If∆⊂IRisanopenintervalandt→f∈Ks,tβt∈∆,isareal-analyticfamilyofreal-valuedfunctions,thenthefunctiont→P(ft)∈IR,t∈∆,isalsorealanalytic.Proposition2.6.13Ifq∈IRandf,ψ∈Karesuchthatqf+ψ∈Ks0β0βand−(qf+ψ)dµ˜q0<∞forallqinanopenneighborhoodofq0,thendP(q0)=fdµ,˜dqwhereµ˜=˜µq0f+ψ.Proof.SinceKsisanopensubsetofK,byTheorem2.6.12weknowββthatthederivativedP(q)exists.Sincethefunctionq→P(q)isconvex,dq0inordertocompletetheproofitisthereforeenoughtodemonstratethatP(q)≥P(q0)+fdµ˜(q−q0)onanopenneighborhoodofq0.Andindeed,inviewofourassumptions,Theorem2.1.8andTheorem2.2.9,thereexistsanopenneighborhoodUofq0suchthatforeveryq∈UwehaveP(q)≥hµ˜+(qf+ψ)dµ˜=hµ˜+q0fdµ˜+ψdµ˜+fdµ˜(q−q0)=P(q0)+fdµ˜(q−q0).
Fixnowf,ψ∈Kand(q,t)∈IR2suchthatthefunctionqf+tψisβsummable.Usingthenotation,˜µq,t=˜µqf+tψandgdµ˜q,t=˜µq,t(g), 48SymbolicDynamicsset
∞σ2(f,ψ)=µ˜(f·ψ◦σk)−µ˜(f)˜µ(ψ)q,tq,tq,tq,tk=0
∞=µ˜(ψ·f◦σk)−µ˜(f)˜µ(ψ).qq,tqk=0Iff=ψwesimplywriteσ2(f)forσ2(f,f).Thelastresultinthisq,tq,tsectionisthefollowing,whichcanbeprovedbyproceedingasinthecaseofasubshiftoffinitetypeoverafinitealphabet(see[Ru],[PU]).Proposition2.6.14Iff,ψ∈Kβ,q0f+t0ψissummableand−(qf+tψ)dµ˜q0,t0<∞,forallpairs(q,t)inanopenneighborhoodof(q0,t0)inIR2,then∂2P1|(q0,t0)=limSn(f−fdµ˜q0,t0)Sn(ψ−ψdµ˜q0,t0)∂q∂tn→∞n=σ2(f,ψ).q0,t02.7TheexistenceofeigenmeasuresoftheconjugatePerron–FrobeniusoperatorandofGibbsstatesSofarthecharacterofthischapterhasbeenslightlyconditional.ToremedythisweproveherethemainresultconcerningtheexistenceofeigenmeasuresoftheconjugatePerron–Frobeniusoperator.Through-outthesectionweassumethatf:E∞→IRisasummableH¨oldercontinuousboundedfunctionwithsomeexponentβ>0whichallowsustodefinethePerron–FrobeniusoperatorLanditsconjugateL∗.Inffordertosimplifynotationwewilldropthesubscriptf.Webeginwiththefollowingresult,whosefirstproofcanbefoundin[B1].Lemma2.7.1IfthealphabetIisfiniteandtheincidencematrixisir-reducible,thenthereexistsaneigenmeasurem˜oftheconjugateoperatorL∗.fProof.ByourassumptionLfisastrictlypositiveoperator(inthesensethatitmapsstrictlypositivefunctionsintostrictlypositivefunc-tions).InparticulartheformulaL∗(ν)fν→L∗(ν)(11)f 2.7Theexistenceofeigenmeasuresoftheconjugate49definesacontinuousmapofthespaceofBorelprobabilitymeasuresonE∞intoitself.SinceE∞isacompactmetricspace,theSchauder–Tikhonovtheoremapplies,andasitsconsequence,weconcludethatthemapdefinedabovehasafixedpoint,say˜m.ThenL∗(˜m)=λm˜,wherefλ=L∗(˜m)(11).
fInTheorem2.7.3,themainresultofthissection,wewillneedasimplefactaboutirreduciblematrices.Wewillprovideashortproofforthesakeofcompleteness.Itismorenaturalandconvenienttoformulateitinthelanguageofdirectedgraphs.Letusrecallthatadirectedgraphissaidtobestronglyconnectedifandonlyifitsincidencematrixisirreducible.Inotherwords,itmeansthatanytwoverticescanbejoinedbyapathofadmissibleedges.Lemma2.7.2IfΓ=E,V!isastronglyconnecteddirectedgraph,thenthereexistsasequenceofstronglyconnectedsubgraphsEn,Vn!ofΓsuchthatalltheverticesV⊂VandalltheedgesEarefinite,{V}∞isnnnn=1anincreasingsequenceofvertices,{E}∞isanincreasingsequenceofnn=1∞∞edges,n=1Vn=Vandn=1En=E.Proof.Indeed,letV={vn:n≥1}beasequenceofallverticesofΓ.andletE={en:n≥1}beasequenceofedgesofΓ.Wewillproceedinductivelytoconstructthesequences{V}∞and{E}∞.Inordernn=1nn=1toconstructE1,V1!letαbeapathjoiningv1andv2(i(α)=v1,t(α)=v2)andletβbeapathjoiningv2andv1(i(β)=v2,t(β)=v1).ThesepathsexistsinceΓisstronglyconnected.WedefineV1⊂VtobethesetofallverticesofpathsαandβandE1⊂Etobethesetofalledgesfromαandβenlargedbye1ifthisedgeisamongalltheedgesjoiningtheverticesofV1.ObviouslyE1,V1!isstronglyconnectedandthefirststepoftheinductiveprocedureiscomplete.SupposenowthatastronglyconnectedgraphEn,Vn!hasbeenconstructed.Ifvn+1∈Vn,wesetVn+1=VnandEn+1isthendefinedtobetheunionofEnandalltheedgesfrom{e1,e2,...,en,en+1}thatareamongalltheedgesjoiningtheverticesofVn.Ifvn+1∈/Vn,letαnbeapathjoiningvnandvn+1andletβnbeapathjoiningvn+1andvn.WedefineVn+1tobetheunionofVnandthesetofallverticesofαnandβn.En+1isthendefinedtobetheunionofEn,alltheedgesbuildingthepathsαnandβnandalltheedgesfrom{e1,e2,...,en,en+1}thatareamongalltheedgesjoiningtheverticesofVn+1.Sincewasstronglyconnected,soisEn+1,Vn+1!.Theinductiveprocedureiscomplete.It 50SymbolicDynamicsimmediatelyfollowsfromtheconstructionthatVn⊂Vn+1,En⊂En+1,∞∞n=1Vn=Vandn=1En=E.
Ourmainresultisthefollowing.Theorem2.7.3Supposethatf:E∞→IRisaH¨oldercontinuousboundedfunctionsuchthat
exp(sup(f|[e]))<∞,e∈Iandtheincidencematrixisirreducible.ThenthereexistsaBorelprob-abilityeigenmeasurem˜oftheconjugateoperatorL∗.fProof.WithoutlossofgeneralitywemayassumethatI=IN.Sincetheincidencematrixisirreducible,itfollowsfromLemma2.7.2thatwecanreorderthesetINsothatthereexistsanincreasingunboundedse-quence{ln}n≥1suchthatforeveryn≥1thematrixA|{1,...,ln}×{1,...,ln}isirreducible.Then,inviewofLemma2.7.1,thereexistsaneigenmeas-ure˜moftheoperatorL∗,conjugatetoPerron–FrobeniusoperatornnL:C(E∞)→C(E∞)nlnlnassociatedtothefunctionf|E∞,where,foranyq≥1,lnE∞=E∞∩{1,...,q}∞q={(ek)k≥1:1≤ek≤qandAekek+1=1forallk≥1}.OccasionallywewillalsotreatLasactingonC(E∞)andL∗asactingnnonC∗(E∞).Ourfirstaimistoshowthatthesequence{m˜}istight,nn≥1where˜m,n≥1,aretreatedhereasBorelprobabilitymeasuresonE∞.nLetPn=P(σ|E∞,f|E∞).ObviouslyPn≥P1foralln≥1.Foreverylnlnk≥1letπ:E∞→INbetheprojectionontothek-thcoordinate,ki.e.π({(e)})=e.ByTheorem2.3.3,ePnistheeigenvalueofL∗kuu≥1kncorrespondingtotheeigenmeasure˜mn.Therefore,applying(2.18),weobtainforeveryn≥1,everyk≥1,andeverye∈INthat

−1m˜n(πk(e))=m˜n([ω])≤expsup(Skf|[ω])−Pnkω∈Ek:ωk=eω∈Ek:ωk=elnln
≤e−Pnkexpsup(Sf|)+sup(f|))k−1[ω][e]ω∈Ek:ωk=elnk−1
≤e−P1kesup(f|[i])esup(f|[e]).i∈IN 2.7Theexistenceofeigenmeasuresoftheconjugate51Therefore,k−1

m˜(π−1([e+1,∞)))≤e−P1kesup(f|[i])esup(f|[j]).nki∈INj>eFixnow">0andforeveryk≥1chooseanintegernk≥1suchthatk−1

"e−P1kesup(f|[i])esup(f|[j])≤.2ki∈INj>nkThen,foreveryn≥1andeveryk≥1,˜m(π−1([n+1,∞)))≤+.nkk2kHence#
m˜E∞∩[1,n]≥1−m˜(π−1([n+1,∞)))nknkkk≥1k≥1
"≥1−=1−".2kk≥1$SinceE∞∩[1,n]isacompactsubsetofE∞,thetightnessofk≥1kthesequence{m˜n}n≥1isthereforeproved.Thus,inviewofProhorov’stheoremthereexists˜m,aweak-limitpointofthesequence{m˜n}n≥1.LetnowL=e−PnLandL=e−P(f)Lbethecorrespondingnormalized0,nn0operators.Fixg∈C(E∞)and">0.Letusnowconsideranintegerbn≥1solargethatthefollowingrequirementsaresatisfied.
"g0expsup(f|[i])−P(f)≤,(2.32)6i>n
egexpsup(f|)e−P1(P(f)−P)≤,(2.33)0[i]n6i≥1"|m˜n(g)−m˜(g)|≤,(2.34)3and"L0(g)dm˜−L0(g)dm˜n≤.(2.35)3Itispossibletomakecondition(2.33)satisfiedsince,duetoTheorem2.1.5,limn→∞Pn=P(f).Letgn=g|E∞.Thefirsttwoobservationsln 52SymbolicDynamicsarethefollowing.
L∗m˜(g)=g(iω)exp(f(iω)−P)dm˜(ω)0,nnnnE∞i≤n:Aiωn=1
=g(iω)exp(f(iω)−Pn)dm˜n(ω)E∞lni≤n:Aiωn=1(2.36)
=gn(iω)exp(f(iω)−Pn)dm˜n(ω)E∞lni≤n:Aiωn=1∗=L0,nm˜n(gn)=˜mn(gn)andm˜n(gn)−m˜n(g)=(gn−g)dm˜n=0dm˜n=0.(2.37)E∞E∞lnlnUsingthetriangleinequalitywegetthefollowing.|L∗m˜(g)−m˜(g)|≤|L∗m˜(g)−L∗m˜(g)|000n+|L∗m˜(g)−L∗m˜(g)|0n0,nn(2.38)+|L∗m˜(g)−m˜(g)|0,nnnn+|m˜n(gn)−m˜n(g)|+|m˜n(g)−m˜(g)|Letusnowlookatthesecondsummand.Applying(2.33)and(2.32)weget∗∗|L0m˜n(g)−L0,nm˜n(g)|
=g(iω)exp(f(iω)−P(f)E∞i≤n:Aiωn=1−exp(f(iω)−Pn))dm˜n(ω)
+g(iω)exp(f(iω)−P(f))dm˜n(ω)E∞i>n:Aiωn=1
(2.39)≤gef(iω)e−Pn(P(f)−P)0ni≤n
+g0expsup(f|[i]−P(f)i>n
"≤gexpsup(f|)e−P1(P(f)−P)+0[i]n6i≥1"""≤+=.663 2.7Theexistenceofeigenmeasuresoftheconjugate53Combiningnowinturn(2.35),(2.39),(2.36),(2.37)and(2.34)wegetfrom(2.38)that∗"""|L0m˜(g)−m˜(g)|≤++=".333Letting"0wethereforegetL∗m˜(g)=˜m(g)orL∗m˜(g)=eP(f)m˜(g).0fThus,L∗m˜=eP(f)m˜.
fAsanimmediateconsequenceofthistheoremandTheorem2.3.3,wegetthefollowingimportantresult.Corollary2.7.4Supposethatf:E∞→IRisaH¨oldercontinuousboundedfunctionsuchthat
exp(sup(f|[e]))<∞e∈Iandtheincidencematrixisfinitelyirreducible.ThenthereexistsaGibbsstateforf.AsanimmediateconsequenceofTheorem2.7.3,Theorem2.3.6,Theorem2.3.3,andTheorem2.2.4,wegetthefollowing.Corollary2.7.5Supposethatf:E∞→IRisaH¨oldercontinuousboundedfunctionsuchthat
exp(sup(f|[e]))<∞e∈Iandtheincidencematrixisfinitelyirreducible.Then(a)Thereexistsauniqueeigenmeasurem˜foftheconjugatePerron–FrobeniusoperatorL∗andthecorrespondingeigenvalueisequalftoeP(f).(b)Theeigenmeasurem˜fisaGibbsstateforf.(c)Thefunctionf:E∞→IRhasauniqueσ-invariantGibbsstateµ˜f.Incasethematrixisfinitelyprimitive,thisGibbsstateiscompletelyergodicandthestochasticlawspresentedinSection2.5aresatisfied. 3H¨olderFamiliesofFunctionsandF-ConformalMeasuresInthischapterwecomebacktothesettingfromChapter1.OuraimhereistodefinesummableH¨olderfamiliesoffunctionsandwiththehelpofthemachinerydevelopedinChapter2toconstructF-conformalmeas-ures,whichinturnwillbeusedinChapter4toconstructgeometricallysignificantconformalmeasures.3.1SummableH¨olderfamiliesLetF={f(e):X→IR:e∈I}beafamilyofreal-valuedfunctions.t(e)Foreveryn≥1andβ>0let(ω1)(ω1)β(n−1)Vn(F)=supsup{|f(φσ(ω)(x))−f(φσ(ω)(y))|}e.ω∈Enx,y∈Xt(ω)Wehavemadetheconventionsthattheemptyword∅istheonlywordoflength0andφ∅=IdX.Thus,V1(F)<∞simplymeansthediametersofthesetsfi(X)areuniformlybounded.ThecollectionFiscalledaH¨olderfamilyoffunctions(oforderβ)ifVβ(F)=sup{Vn(F)}<∞.(3.1)n≥1TheH¨olderfamilyFiscalledsummable(oforderβ)if(3.1)issatisfiedand
(e)fe0<∞.(3.2)e∈EWedefinethetopologicalpressureP(F)ofFbysettingn1

P(F)=limlogexpfωj◦φσjωn→∞nω∈nj=1054 3.1SummableH¨olderfamilies551

nsupωj,=limlogexpf◦φσjωn→∞nXω∈Enj=1where·denotesthesupremumnorm.Givenn≥1andω∈Enwe0definethefunctionSω(F):Xt(ω)→IRbydeclaring
n(ωj)Sω(F):=f◦φσjω.j=1RepeatingwithobviousmodificationstheproofofTheorem2.1.3wefindthefollowing.Theorem3.1.1Wehave
P(F)=inft∈IR:expsup(S(F))e−t|ω|<∞.ωω∈E∗Letusnowprovethefollowingversionoftheboundeddistortionprop-erty.Lemma3.1.2Ifω∈E∗andx,y∈X∩φ(X)forsomeτ∈E∗,t(ω)τt(τ)thenV(F)eβ−β|τ||Sω(F)(x)−Sω(F)(y)|≤e1−e−βProof.Letn=|ω|.Writex=φτ(u),y=φτ(w),whereu,w∈Xt(τ).By(3.1)weget
n
n(ωj)(ωj)f(φσjω(x))−f(φσjω(y))j=1j=1
n
n(ωτ)j(ωτ)j=f◦φσjωτ(u)−f◦φσjωτ(w)j=1j=1
n(ωτ)j(ωτ)j≤f◦φσjωτ(u)−f◦φσjωτ(w)j=1
n≤V(F)e−β(n+|τ|−j−1)j=1V(F)eβ−β|τ|≤e.1−e−β
56H¨olderFamiliesofFunctionsandF-ConformalMeasuresSetV(F)eβT(F)=exp.1−e−βInordertoconnectwiththepreviouschapterwenowintroduceapo-tentialfunctionoramalgamatedfunction,f:E∞→IR,inducedbythefamilyoffunctionsFasfollows.f(ω)=f(ω1)(π(σ(ω))).OurconventionwillbetouselowercaselettersforthepotentialfunctioncorrespondingtoagivenH¨oldersystemoffunctions.ThefollowinglemmaisastraightforwardconsequenceofLemma3.1.2.Lemma3.1.3IfFisaH¨olderfamily(oforderβ)thentheamalgamatedfunctionfisH¨oldercontinuous(oforderβ).Thefollowingresultdemonstratesthattheconceptoftopologicalpres-sureP(F)can,infact,bereducedtothetopologicalpressureintroducedinthepreviouschapter.Proposition3.1.4P(F)=P(f).Proof.Firstnoticethatforeveryn≥0,everyω∈Enandeveryτ∈E∞withA=1ωnτ1
n
n(ωj)j−1Sω(F)(τ)=f◦φσjω(τ)=f(σωτ)j=1j=1n
−1=f(σjωτ)=Sf(ωτ).nj=0Hence,expS(F)≥expsup{Sf(ωτ):τ∈E∞,A=1}ω0nωnτ1=expsup(Snf|[ω]).AndthereforeP(F)≥P(f).Fixnowanarbitraryτ∈E∞withA=ωnτ11andconsideranarbitraryx∈X.Then,byLemma3.1.2wehaveexpS(F)(x)≤eT(F)expS(F)(π(τ))ωω≤eT(F)Sf(ωτ)≤eT(F)expsup(Sf|).nn[ω] 3.2F-conformalmeasures57ThereforeexpsupS(F)≤eT(F)expsup(Sf|)andconsequently,ωn[ω]P(F)≤P(f).
3.2F-conformalmeasuresABorelprobabilitymeasuremissaidtobeF-conformalprovideditissupportedonthelimitsetJandthefollowingtwoconditionsaresatisfied.Foreveryω∈E∗andforeveryBorelsetA⊂Xt(ω)m(φω(A))=expSω(F)−P(F)|ω|dm(3.3)Aandforallincomparablewordsω,τ∈E∗mφω(Xt(ω))∩φτ(Xt(ω))=0.(3.4)Asimpleinductiveargumentshowsthatinsteadof(3.3)and(3.4)itisenoughtorequirethatforeveryi∈IandforeveryBorelsetA⊂Xt(i)m(φ(A))=expf(i)−P(F)dm(3.5)iAandforalli,j∈I,i=jm(φi(Xt(i))∩φj(Xt(j))=0.(3.6)Givena∈IsetXa=φb(Xt(b)).b∈I:Aab=1Weshallprovethefollowingauxiliaryresult.Lemma3.2.1IfmisaBorelprobabilitymeasureonJsuchthat(3.4)holdsand(3.3)issatisfiedforallBorelsetsA⊂Xi,i∈I,thenmisF-conformal.Proof.Sinceforeveryi∈I,(Xt(i)Xi)∩J=∅,wehavem(Xt(i)Xi)=0.(3.7)Takenowanarbitraryelementi∈Iandconsideranarbitraryelementx∈φXX∩J.Thenx=π(ω)=φ(π(σ(τ)))forsomeω∈E∞it(i)iω1withω1=i.HenceJ∩φiXt(i)Xi⊂j=iφjXt(j)∩φiXt(i)andconsequentlyby(3.6)mφiXt(i)Xi=mJ∩φiXt(i)Xi
≤mφjXt(j)∩φiXt(i)=0.j=i 58H¨olderFamiliesofFunctionsandF-ConformalMeasuresThereforeusing(3.7)weobtainbyinduction,foreveryω∈E∗andeveryA⊂Xt(ω),m(φω(A))=m(φω(A∩Xω|ω|))=expSω(F)−P(F)|ω|dmA∩Xω|ω|=expSω(F)−P(F)|ω|dm.A(3.8)
Weshallnowprovidesomesufficientconditionsfortheexistence(anduniqueness)ofF-conformalmeasures.Ourfirstconditioncomesfromthefollowingdefinition.Namely,wesaythataniteratedfunctionsystem{φi:i∈I}satisfiesthestrongseparationconditionifφi(X)∩φj(X)=∅foralli,j∈I,i=j.Oursecondconditionissomewhattechnicalandmayseemstrangeatfirstsight.Itsrealgeometricalmeaningwillbecomeclearinthenextchapter,whichisdevotedtoconformalsystems.Definition3.2.2Givenq≥1,wesaythattwodifferentwordsρ,τ∈E∗ofthesamelength,sayn>q,formapairofq-codesofapointx∈Xifx∈φρ(Xt(ρ))∩φτ(Xt(τ))andρ|n−q=τ|n−q.TheGDMSSissaidtobeconformal-likeifforeveryq≥1thereisnopointinX(orequivalentlyinJ)witharbitrarilylongpairsofq-codes.Ofcourseeachsystemsatisfyingthestrongopensetconditionisconfor-mal-like.Itwillturnoutinthenextchapterthateachconformalsystemisconformal-like.Theorem3.2.3SupposethattheGDMS{φi:i∈I}isconformal-likeandthattheincidencematrixisfinitelyprimitive.IfFisasummableH¨olderfamilyoffunctions,thenthereexistsauniqueF-conformalmeas-urem.Moreoverm=˜m◦π−1,wherem˜istheeigenmeasureofFFFFtheconjugatePerron–FrobeniusoperatorL∗.fProof.ByCorollary2.7.5(a)thereexists˜m,aneigenmeasureoftheconjugateoperatorL∗associatedwiththeamalgamatedfunctionf:fE∞→IR,andthecorrespondingeigenvalueisequaltoeP(f).Letm=˜m◦π−1.WeshallshowthatmisanF-conformalmeasure.FFFAndindeed,supposeonthecontrarythatmF(φρ(Xt(ρ))∩φτ(Xt(τ)))>0 3.2F-conformalmeasures59forsometwoincomparablewordsρ,τ∈I∗withi(ρ)=i(τ)=vforsomev∈V.Wemayassumewithoutlossofgeneralitythatρandτareofthesamelength,sayq≥1.LetE=φρ(Xt(ρ))∩φτ(Xt(τ))andforeveryn≥1,letEn=ω∈En:t(ω)=vφω(E).SinceeachelementofEnadmitsatleasttwoq-codesoflengthn+q,weconcludethat∞∞En=∅.k=1n=kOntheotherhand,E=φ(E)⊃π(σ−n(π−1(E))),whichnω∈En:t(ω)=vωimpliesthatπ−1(E)⊃σ−n(π−1(E)).InviewofCorollary2.7.5(c)nthereexistsaσ-invariantGibbsstate˜µforfwhich,byProposition2.2.2,isequivalentwith˜m=˜m.Hence˜µ(π−1(E))≥µ˜(σ−n(π−1(E))=Fnµ˜◦π−1(E)>0.Thus,∞∞−1−1µ˜◦πEn≥µ˜◦π(E)>0.k=1n=k∞∞Inparticular,k=1n=kEn=∅.Thiscontradictionshowsthatprop-erty(3.4)holds.Inordertoprove(3.3)fixω∈E∗,sayω∈En,andaBorelsetB⊂Xt(ω).Put−1−1πω(B)={τ∈π(B):Aωnτ1=1}andconsideranelementτ∈π−1(B)π−1(B).ThenA=0and,onωωnτ1theotherhand,π(τ)∈B⊂Xt(ω)whichimpliesthatπ(τ)=φi(Xt(i))forsomei∈IsuchthatAωni=1.Hencei=τ1andπ(τ)∈φi(Xt(i))∩φτ1(Xt(τ1)).Therefore−1−1ππ(B)πω(B)⊂φi(Xt(i))∩φj(Xt(j))i=jand,using(3.4),weconsequentlygetm˜π−1(B)π−1(B)≤m˜◦π−1φ(X)∩φ(X)ωit(i)jt(j)i=j(3.9)=mFφi(Xt(i))∩φj(Xt(j))=0i=jItisstraightforwardtocheckthatπ−1(φ(B))⊃[ωπ−1(B)].ωω 60H¨olderFamiliesofFunctionsandF-ConformalMeasuresConsiderthenanelementρ∈π−1(φ(B))[ωπ−1(B)].Thenπ(ρ)=ωωφ(x),whereA=1,φ(x)∈Bandρ/∈[ωπ−1(B)].Ifρ|=ωi,ωiωniiωn+1thenx=π(σn+1ρ)andwededucethatπ(ρnρ)=φ(π(σn+1ρ))=φ(x),iiandconsequentlyρ=ωiσn+1(ω)∈[ωπ−1(B)].Thereforeρ|=ωiωn+1andwegetthatπ−1(φ(B))[ωπ−1(B)]⊂π−1φ(X)∩φ(X).ωωτt(τ)ηt(η)τ,η∈En+1:τ=ηUsing(3.4)andthedefinitionofthemeasuremFwethereforededucethatm˜π−1(φ(B))[ωπ−1(B)]=0.ωωCombiningthisequality,(3.9)and(2.18),wecanwritem(φ(B))=˜m◦π−1(φ(B))=˜m[ωπ−1(B)]Fωωω=expSnf(ωρ)−P(f)ndm˜(ρ)−1πω(B)=expSωF(π(ρ))−P(F)ndm˜(ρ)−1πω(B)=expSωF(π(ρ))−P(F)ndm˜(ρ)π−1(B)=expSF(x)−P(F)ndm˜◦π−1(x)ωB=expSωF−P(F)ndmF.BSo,byLemma3.2.1,theproofoftheexistencepartiscomplete.InordertoproveuniquenesssupposethatνisanotherF-conformalmeasure.Inviewof(3.4)thereexistsauniqueprobabilitymeasure˜νonE∞suchthatν=˜ν◦π−1.Fixn≥1,ω∈EnandaBorelsetA⊂E∞suchthatAωnτ−1=1foreveryτ∈A.Thenπ(A)⊂Xω,andusing(3.3),wegetν˜([ωA])=ν(π([ωA]))=ν(φω(π(A)))=exp(SωF−P(F)n)dνπ(A)=exp(Snf(ωρ)−P(f)n)dν.˜ATherefore,inviewofRemark2.3.2,˜νisaneigenmeasureoftheoperatorL∗correspondingtotheeigenvalueeP(f).Hence,byTheorem2.3.6,fν˜=˜m.Thusν=˜ν◦π−1=˜m◦π−1=m.
F 3.2F-conformalmeasures61Weshallnowprovearesultwhich,thoughsimple,isrichingeometricconsequences.Lemma3.2.4WiththesameassumptionsasinTheorem3.1.7wehaveM=min{mF(Xv):v∈V}>0.Proof.SincethegraphΓisstronglyconnected,thereexistse∈Isuchthati(e)=v.Using(2.20)wethengetm(X)≥m˜([e])≥CT(f)−1(#Λ)−1expsupf|−P(F)>0.FvF[e]
KeepingtheassumptionthatthesystemSisconformal-likeandFisasummableH¨olderfamilyoffunctionsweput−1µF=˜µf◦π.(3.10)SinceµFisequivalentwithmFwithRadon-Nikodymderivativesboun-dedawayfromzeroandinfinity,wecallµFtheS-invariantversionofmF.Remark3.2.5Wewouldliketodrawthereader’sattentiontothefactthatinprovinguniquenessintheprevioustheoremwehave,infact,demonstratedmore.Namely,ifweassumethatameasuremsupportedonJsatisfies(3.4)and(3.3)withanarbitraryconstantLreplacingtheconstantP(F),thenm◦πisaneigenmeasureoftheconjugatePerron–FrobeniusoperatorL∗correspondingtotheeigenvalueeL.IttheninturnffollowsfromTheorem2.3.3thatL=P(f)=P(F),andconsequently,misF-conformal. 4ConformalGraphDirectedMarkovSystemsThisisthecentralchapterofourbook.WepresentherethebasicandmorerefinedgeometricpropertiesoflimitsetsofconformalGraphdirectedMarkovsystems.4.1SomepropertiesofconformalmapsinIRdwithd≥2Inthissectionwestudyindetailanalytic,geometricandespeciallydis-tortionpropertiesofconformalmapsinanydimensiond≥3.WesaythataC1diffeomorphism(homeomorphism)φ:U→IRd,d≥1,fromanopenconnectedsetU⊂IRntoIRdisconformalifitsderivativeateverypointofUisasimilaritymap.Byφ
(z):IRd→IRdwedenotethederivativeofφevaluatedatthepointzandby|φ
(z)|itsnorm,whichintheconformalcasecoincideswiththecoefficientofsimilarity.Notethatford=1C1-conformalitymeansthatallthemapsφ,e∈I,areeC1-diffeomorphismsandmonotone,ford≥2thewordsC1-conformalmeanholomorphicorantiholomorphic,andford≥3themapsφe,e∈IareM¨obiustransformations.MorepreciselyeachC1-conformalhomeo-morphismφdefinedonanopenconnectedsubsetofIRd,d≥3,extendstotheentirespaceIRdandtakestheformφ=λA◦ia,r+b,(4.1)where0<λ∈Risapositivescalar,AisalinearisometryinIRd,ia,riseitherinversionwithrespecttothespherecenteredatapointaandwithradiusr,ortheidentitymap,andb∈IRd.Inthesequelλwillbecalledthescalarfactoranda=φ−1(∞)willbecalledthecenterofinversion.IfAistheidentitymap,φwillbecalledaconformalaffinehomeomorphism.TheproofofthischaracterizationofC1-conformalhomeomorphismsinthedimension≥3canbefoundin[BP]forexample,62 4.1SomepropertiesofconformalmapsinIRdwithd≥263whereitiscalledLiouville’stheorem.Inparticularitiseasytocomputethatλr2φ
(z)
|φ(z)|=and=Id−2Q(z−a),(4.2)z−a2|φ
(z)|whereQ(z)isthematrixgivenbytheformulazizjQ(z)ij=.z2Indimensiond=2wewillextensivelyusethefollowingcelebratedKoebe’sdistortiontheorem(seee.g.[CG]).Theorem4.1.1Ifw∈CIandg:B(w,R)→CIisaunivalentholo-morphicfunction,thenforallz∈B(w,R),1−|z−w|
|g(z)|1+|z−w|R≤≤.
3|g
(w)|
3|z−w||z−w|1+1−RRFix0<γ≤1/2sosmallthat1+γ1−γ−1√Kγ=max,≤2.(4.3)(1−γ)3(1+γ)3Wewillneedthefollowingfact,whichisinasenseanimprovementofKoebe’sdistortiontheorem(Theorem4.1.1).Theorem4.1.2ThereexistsaconstantK3≥1suchthatifx∈CI,R>0andφ:B(x,R)→CIisaunivalentconformalmap,then

K3
|φ(y)−φ(x)|≤|φ(x)y−x|Rforally∈Bx,(minγ,π)R.24Proof.Sinceeachconformalmapintheplaneiseitherholomorphicorantiholomorphicandsincecomplexconjugationisanisometry,wemayassumewithoutlossofgeneralitythatφisholomorphic.Aholomorphicfunctiong:ID→CI(ID={z∈CI:|z|<1})iscalledaBlochfunctionifg=sup{(1−|z|2)|g
(z)|}<∞.Bz∈IDIff:ID→CIisaunivalentholomorphicunivalentmap,then(see[Po2,p.73])logf
isaBlochfunctionandlogf
≤6foreverybranchofB 64ConformalGraphDirectedMarkovSystemslogarithmoff
.Sincethefunctionz→φ(x+Rz)isunivalentandholomorphiconID,usingthechainruletwice,weconcludethat

8|(logφ)(w)|≤(4.4)R

φ(w)forallw∈B(x,R/2).Denotebylogφ
(x)thebranchofthelogar-
ithmofthefunctionw→φ(w)definedbytheformulaw→logφ
(w)−φ
(x)logφ
(x).Sincelogφ
(y)−logφ
(x)=y(logφ
)
(z)dzforally∈xB(x,R),using(4.4)wegetφ
(y)8log=|logφ
(y)−logφ
(x)|≤|y−x|(4.5)φ
(x)Rforally∈B(x,R/2).Ifnowy∈B(x,γR),theninviewofTheorem4.1.1andthechoiceofγ,wefind1φ
(y)√√≤≤2.(4.6)2φ
(x)Ifnowy∈Bx,πR,thenusing(4.5)weobtain24φ
(y)8πRπArg≤·=(4.7)φ
(x)R243Iflogw,forwintherighthalfplaneisthebranchofthelogarithmsending1to0,thenthefunctionw−1isanalyticontherightplane.logwThereforeputting1√F={z∈CI:√≤|z|≤2and|Arg(z)|≤π/3}2wegetthatw−1C:=sup:w∈F<∞.(4.8)logwSinceallbranchesofthelogarithmdifferby2πin,n∈Z,weconcludethat(4.8)remainstrue(perhapswithalargerconstantC)ifbylogwwemeananyvalueofthelogarithm.Combining(4.5)–(4.8),wegetφ
(y)φ
(y)12C−1≤Clog≤1|y−x|φ
(x)1φ
(x)Rforally∈Bx,(min{γ,π})RTherefore,puttingK=8Ccompletes2431theproof.
Inthedimensiond≥3wehavethefollowing. 4.1SomepropertiesofconformalmapsinIRdwithd≥265Theorem4.1.3SupposethatYisaboundedsubsetofIRd,d≥3,andWisanopensubsetofIRdcontainingY.ThenthereexistsaconstantK4≥1suchthatifφ:IRd→IRdisaconformaldiffeomorphismsuchthatφ(W)⊂W,then|φ
(y)|−|φ
(x)|≤K|φ
(x)·|y−x4forallx,y∈Y.Inparticular|φ
(y)|≤1+K4diam(Y).|φ
(x)|Proof.Inviewof(4.1)therethenexistλ>0,alinearisometryA:IRd→IRd,aninversion(ortheidentitymap)i=iandavectora,rb∈IRdsuchthatφ=λA◦i+b.Wheniistheidentitymapthestatementofourtheoremisobvious.So,wemayassumethatiisaninversion.Sinceφ(W)⊂W⊂IRd,a/∈W.Thereforeforallx,y∈Yx−ax−y+y−ax−y≤=1+y−ay−ay−a(4.9)diam(Y)≤1+.dist(Y,∂W)Thus,using(4.2),weget
2|φ(y)|diam(Y)≤1+.|φ
(x)|dist(Y,∂W)Theproofofthesecondpartofourtheoremiscomplete.Inordertoprovethefirstpartwemayassumewithoutloosinggeneralitythat|φ
(x)|≤|φ
(y)|.Using(4.2)and(4.9)wethenget|φ
(y)|x−a2|φ
(y)|−|φ
(x)|≤|φ
(x)|−1=|φ
(x)|−1|φ
(x)|y−a2
x−ax−a=|φ(x)|−1+1y−ay−a
diam(Y)x−y≤|φ(x)|2+dist(Y,∂W)y−a
diam(Y)x−y≤|φ(x)|2+dist(Y,∂W)dist(Y,∂W) diam(Y)1
≤2+|φ(x)|·y−xdist(Y,∂W)dist(Y,∂W)Theproofisthuscomplete.
66ConformalGraphDirectedMarkovSystemsAsanimmediateconsequenceofTheorem4.1.2andTheorem4.1.3,weobtainthefollowingboundeddistortionresult.Corollary4.1.4SupposethatYisaboundedsubsetofIRd,d≥2,andWisanopensubsetofIRdcontainingY.Thenforevery%>0thereexistsδ>0suchthatifφ:IRd→IRdisaconformaldiffeomorphismsuchthatφ(W)⊂W,then|φ
(y)|(1+%)−1≤≤1+%|φ
(x)|forallx∈Yandally∈B(x,δ).AsanimmediateconsequenceofTheorem4.1.2alonewegetthefollowing.Theorem4.1.5Ifx∈CI,R>0andφ:B(x,R)→CIisaunivalentholomorphicmap,then
K3
2|φ(y)−φ(x)−φ(x)(y−x)|≤|φ(x)·y−xRforally∈Bx,(minγ,π)R,whereγisdefinedby(4.3)andK243comesfromTheorem4.1.2.Proof.InviewofTheorem4.1.2weget
|φ(y)−φ(x)−φ(x)(y−x)|yyy



=|φ(z)dz−φ(x)dz|=(φ(z)−φ(x))dz|xxxK3
K3
2≤|φ(x)|·|y−x|·|y−x|=|φ(x)y−x|.RRTheproofisfinished.
Inthecased≥3wehaveasimilarresult.Theorem4.1.6SupposethatYisaboundedsubsetofIRd,d≥3,andWisanopensubsetofIRdcontainingY.ThenthereexistsaconstantK5≥1suchthatifφ:IRd→IRdisaconformaldiffeomorphismsuchthatφ(W)⊂W,thenforallx,y∈Ywehaveφ(y)−φ(x)−φ
(x)(y−x)≤K|φ
(x)|·y−x2.5 4.1SomepropertiesofconformalmapsinIRdwithd≥267Proof.Writeφ=λA◦ia,r+b.Ifia,ristheidentitymap,thentheleft-handsideoftheclaimedinequalityisequalto0andwearedone.Otherwise,puttingi=ia,randusing(4.2),wegetφ(y)−φ(x)−φ
(x)(y−x)=λA◦i(y)−λA◦i(x)−λA◦i
(x)(y−x)=λA(i(y)−i(x)−i
(y−x))2y−a2x−a=λAr−ry−a2x−a2(4.10)r2−(Id−2Q(x−a))(y−x)x−a22y−ax−a(Id−2Q(x−a))(y−x)=λrA−−.y−a2x−a2x−a2Sinceitisstraightforwardtocheckthatx−a,uQ(x−a)u=(x−a),x−a2where·,·isthestandardinnerproductinIRd,wecanwritey−ax−a(Id−2Q(x−a))(y−x)∆:=−−y−a2x−a2x−a2y−xx−ax−ay−x=+−−y−a2y−a2x−a2x−a2x−a,y−x+2(x−a)x−a411=−(y−x)y−a2x−a211x−a,y−x+−+2(x−a).y−a2x−a2x−a4Writenowx−a=α(y−x)+β(y−x)⊥,where(y−x)⊥isavectorperpendiculartoy−xwith(y−x)⊥=y−x.Thenx−a2=(α2+β2)y−x2,y−a2=((α+1)2+β2)y−x2and(4.11)x−a,y−x=αy−x2. 68ConformalGraphDirectedMarkovSystemsThus1111∆=−+α−y−a2x−a2y−a2x−a22αy−x2+×(y−x)x−a4112αy−x2⊥+β−+(y−x)y−a2x−a2x−a41α2+β2α2+β2=−1+α−1x−a2(α+1)2+β2(α+1)2+β22α2α2+β22α+(y−x)+β−1+(y−x)⊥α2+β2(α+1)2+β2α2+β21(1+α)(α2+β2)2α2=−(1+α)+(y−x)x−a2(α+1)2+β2α2+β2α2+β22α⊥+β−1+(y−x).(α+1)2+β2α2+β2(4.12)Now,adirectcalculationusing(4.11)showsthat(1+α)(α2+β2)2α2|α3+α2−3αβ2−β2|−(1+α)+=(α+1)2+β2α2+β2(α2+β2)((α+1)2+β2)y−x4=|α3+α2−3αβ2−β2|x−a2y−a2y−x4≤(|α|3+α2+3|α|β2+β2)x−a2y−a2y−x4x−a3x−a2≤4+x−a2y−a2y−xy−x2y−x4x−ay−x≤+y−ay−ay−ay−x4(y−x+y−a)y−x≤+y−ay−ay−ay−xy−x≤5+4y−ay−a15diam(Y)≤+4y−x,RR 4.1SomepropertiesofconformalmapsinIRdwithd≥269whereR=dist(Y,∂W)andα2+β22αβ−1+(α+1)2+β2α2+β2|β||3α2−β2+2α||β|(3α2+β2+2|α|)4=≤y−x(α2+β2)((α+1)2+β2)x−a2y−a2321x−ax−a4≤4+2y−xx−a2y−a2y−xy−xy−xy−x≤(4x−a+2y−x)≤(4y−a+6y−x)y−a2y−a21y−x16diam(Y)≤4+6y−x≤4+y−x.y−ay−aRRTherefore,combiningthesetwoestimateswith(4.12),wefindthereisaconstantK5suchthat12∆≤K5y−x.x−a2Itnowfollowsfromthisalongwith(4.10)and(4.2)thatλr2φ(y)−φ(x)−φ
(x)(y−x)≤Ky−x2=K|φ
(x)|·y−x2.x−a255
Givenapointx∈IRd,avectoru∈IRdandanangleα∈(0,π/2)weputCon(x,α,u)={y∈IRd:∠(y−x,u)≤αandy−x≤u}d={y∈IR:cos∠(y−x,u)>cosαandy−x≤u}={y∈IRd:y−x,u>cosαu·y−xandy−x≤u}andwecallCon(x,α,u)theconegeneratedbyx,αandu.Weprovethefollowinggeometricalfactaboutconformalmaps.Theorem4.1.7SupposethatYisaboundedsubsetofIRd,d≥2,andWisanopensubsetofIRdcontainingY.Thenforevery%∈(0,π/2)thereexistsδ>0suchthatify∈Y,u<δandα∈(0,π/2−%),andφ:W→WisaC1conformaldiffeomorphism,thenφ(Con(x,α,u))⊂Con(φ(x),α+%,2φ
(x)u). 70ConformalGraphDirectedMarkovSystemsProof.Sinceinthecasewhend=2,theconformaldiffeomorph-ismiseitherholomorphicorantiholomorphicandsincecomplexcon-jugacyisanisometry,wemayassumethatifd=2,thenφisholo-morphic.InviewofTheorem4.1.6andTheorem4.1.5,puttingK6=maxK,K3weseethatifx∈Yandy−x0.InviewofTheorem4.1.3andTheorem4.1.2thereexists9<δ.Ify∈Con(x,α,u),thenusing(4.14),wegetφ
(x)(y−x),(1+η)φ
(x)u
2=(1+η)|φ(x)|y−x,u(4.16)
2≥(1+η)|φ(x)|cosαuy−x≥cosαφ
(x)uφ(y)−φ(x).Ontheotherhand,itfollowsfrom(4.13)that|φ(y)−φ(x)−φ
(x)(y−x),φ
(x)u|≤φ(y)−φ(x)−φ
(x)(y−x)·φ
(x)u(4.17)≤K|φ
(x)·y−x2·φ
(x)u.6 4.2Conformalmeasures;Hausdorffandboxdimensions71Assumingδ>0tobesmallerthan(2K)−1,weobtainfrom(4.13)that6φ(y)−φ(x)≥|φ
(x)·y−x−K|φ
(x)|·y−x26≥|φ
(x)|·y−x−K|φ
(x)|·y−x(2K)−1661
≥|φ(x)|·y−x.2Therefore,wecancontinue(4.17)asfollows|φ(y)−φ(x)−φ
(x)(y−x),φ
(x)u|≤2Kφ
(x)u·φ(y)−φ(x)·y−x6−1
≤2K6δ(1+η)((1+η)φ(x)u)φ(y)−φ(x).Combiningthisand(4.15)wegetφ(y)−φ(x),(1+η)φ
(x)u−1
≥(1+η)(cosα−2K6δ)(1+η)φ(x)u)φ(y)−φ(x).Takingnow0<η<1/2andδ>0sosmallthatarccos1+η)−1(cosα−2K6δ)<α+%,thisinequalityalongwith(4.14)showsthatφ(Con(x,α,u))⊂Con(φ(x),α+%,(1+η)φ
(x)u)⊂Con(φ(x),α+%,2φ
(x)u).
4.2Conformalmeasures;HausdorffandboxdimensionsWearenowinpositiontointroduceandexplorethecentralobjectofthisbook.Namely,wecallaGDMSconformal(CGDMS)ifthefollowingconditionsaresatisfied.(4a)Foreveryvertexv∈V,XvisacompactconnectedsubsetofaEuclideanspaceIRd(thedimensiondcommonforallv∈V)andXv=Int(Xv).(4b)(Opensetcondition)(OSC)Foralla,b∈E,a=b,φa(Int(Xt(a))∩φb(Int(Xt(b))=∅.(4c)Foreveryvertexv∈VthereexistsanopenconnectedsetWv⊃Xvsuchthatforeverye∈Iwitht(e)=v,themapφeextendstoaC1conformaldiffeomorphismofWintoW.vi(e) 72ConformalGraphDirectedMarkovSystems(4d)(Coneproperty)Thereexistγ,l>0,γ<π/2,suchthatforeveryx∈X⊂IRdthereexistsanopenconeCon(x,γ,l)⊂Int(X)withvertexx,centralangleofmeasureγ,andaltitudel.(4e)TherearetwoconstantsL≥1andα>0suchthatφ
(y)|−|φ
(x)|≤L(φ
)−1−1y−xαeeeforeverye∈Iandeverypairofpointsx,y∈X,where|φ
(x)|t(e)ωmeansthenormofthederivative.Sinceforevery00thereexistsδ>0suchthatifω∈E∗,x∈X,||u||≤δandα∈(0,π/2−%),thent(ω)φ(Con(x,α,u))⊂Con(φ(x),α+%,2φ
(x)u).ωωWeshallprovethefollowing.Thereexistsβ>0suchthatifD≥1issufficientlylarge,thenforallω∈E∗andx∈X,wehavet(ω)φ(X)⊃Con(φ(x),β,D−1φ
(x)u)ωt(ω)ωx(4.24)⊃Con(φ(x),β,D−2diam(φ(W))u).ωωt(ω)xAndindeed,ifd=1,thefirstinclusionisanimmediateconsequenceof(4.21).So,inprovingthisinclusionwemayassumethatd≥2.Noticethatthenthereexistsanintegerq≥1suchthatforeveryx∈Xthereexistunitvectorsu1(x),...,uq(x)suchthatalltuples 4.2Conformalmeasures;Hausdorffandboxdimensions75(ux,u1(x),...,uq(x))aremutuallyisometric,qCon(x,γ,IRu)∪Con(x,γ,IRu(x))⊂IRd(4.25)+x+jj=1andqCon(x,γ/2,IR+ux)∩Con(x,γ,IR+uj(x))=∅.j=1Then,thereexistsan%>0suchthatqCon(x,γ/4,IR+u˜x)∩Con(x,γ+%,IR+u˜j(x))=∅(4.26)j=1forallx∈Xandalltuples(˜ux,u˜1(x),...,u˜q(x))equivalentwith(ux,u1(x),...,uq(x))byasimilaritymap.Let0<δqforeveryk≥1.Weshallconstructbyinductionwithrespecttok≥1asequence{Ck}k≥1suchthatforeveryk,Ckconsistsofatleastk+1incomparablewordsfrom{ρ(j),τ(j):j≤k}.Indeed,setC={ρ(1),τ(1)}.SupposenowthatChasbeendefined.1kIfρ(k+1)doesnotextendanywordinC,thenweformCbyaddingkk+1ρ(k+1)toC.Wecandoasimilarthingincaseτ(k+1)doesnotextendkanywordinC.If,ontheotherhand,ρ(k+1)extendssomewordκinkCandτ(k+1)extendsawordηinC,thenκandηarebothextendedkkbyρk+1sinceforj≤k,|ρ(j)|,|τ(j)|≤n≤n,n>n+q,andjkk+1kρ(k+1)|=τ(k+1)|.SincethewordsinCareincomparable,nk+1−qnk+1−qkκ=ηandthisistheonlywordinCwhichisextendedbybothρ(k+1)kandτ(k+1).InthiscaseweformCbytakingawayκandaddingk+1bothρ(k+1)andτ(k+1).Nowthesets{φ(X):κ∈C}arenon-κt(κ)koverlappingsincethewordsareincomparable.By(4.24)wegetk+1pairwisedisjointopenconeseachwithvertexxandopeningangleβ.Sincethisisimpossibleifkislargeenough,theproofisfinished.
Wewillfrequentlymakeuseofthefollowingsimpleestimate.ThissortofestimatewasusedbyMoran[Mo]inthestudyofgeometricconstruc-tionsusingsimilaritymaps.ThisincludestheiterationoffinitelymanysimilaritymapsandwasusedbyHutchinson[Hu].Theestimatewasextendedtotherandomcasein[GMW],todirectedgraphconstructionsusingsimilaritymapsin[MW2]andtotheiterationofinfinitelymanyconformalmapsin[MU1].Thisestimateremainsvalidwhentheconepropertyisreplacedintheseargumentsbythemoregeneral“neighob-hoodboundednessproperty”asdefinedandusedin[GMW]. 4.2Conformalmeasures;Hausdorffandboxdimensions77Lemma4.2.6IfSisaCGDMS,thenforeveryx∈X,everyκ>0andeveryr>0,thecardinalityofanycollectionofmutuallyincomparablewordsω∈E∗satisfyingB(x,r)∩φ(X)=∅anddiam(φ(X))≥ωt(ω)ωt(ω)κrisboundedfromabovebythenumberVκ−dD2dβ−1(1+κD−2).dProof.LetFbesuchacollection.Itfollowsfrom(4.24)thatφ(X)⊃Con(φ(x),β,D−2κr)⊂B(x,(1+κD−2)r)ωt(ω)ωωforallω∈F,whereφω(xω)isontheboundaryofφω(Xt(ω))∩B(x,r).HencetheconesCon(φ(x),β,κD−2r),ω∈F,aremutuallydisjointωωandthereforeV(1+κD−2)drd=λ(B(x,(1+κD−2)r))dd≥λCon(φ(x),β,κD−2r)dωωω∈F−2d=#Fβ(κDr).Now,therequiredestimatefollows.
Weshallnowintroducethemostimportanttechnicaltoolinourex-plorationsofgeometryoflimitsets.ItwasoriginatedinthecontextofFuchsiangroupsbyS.Pattersonin[Patt],adoptedbyD.Sullivanin[Su1]tothecaseofrationalfunctionsandbytheauthorstothecaseofconformaliteratedfunctionsystems(see[MU1]).So,givent≥0aBorelprobabilitymeasuremissaidtobet-conformalprovideditissupportedonthelimitsetJandthefollowingtwoconditionsaresatisfied.Foreveryω∈E∗andforeveryBorelsetA⊂Xt(ω)
tm(φω(A))=|φω|dm(4.28)Aandforallincomparablewordsω,τ∈E∗mφω(Xt(ω))∩φτ(Xt(τ))=0.(4.29)Asimpleinductiveargumentshowsthatinsteadof(4.28)and(4.29)itisenoughtorequirethatforeverye∈IandforeveryBorelsetA⊂Xt(e)m(φ(A))=|φ
|tdm(4.30)eeAandforalla,b∈I,a=bm(φa(Xt(a))∩φb(Xt(b))=0.(4.31) 78ConformalGraphDirectedMarkovSystemsGivent≥0andn≥1,wedenote
tZn(t)=||φω||andP(t)=P(tLog),|ω|=nwhereLog={log|φ
|}.ee∈IWethensimplycallP(t)thepressurefunction.ThefollowingsimplebutcrucialfactaboutthefamilyLogisanimmediateconsequenceofLemma4.2.2.Proposition4.2.7LogisaH¨olderfamilyoforderαlogs.LetnowF(S)={t≥0:P(t)<∞}andθ(S)=inf(F(S)).Thebasicstraightforwardpropertiesofthepressurefunctionarecontainedinthefollowing(thefirstandthelastoneimpliedbyProposition2.1.9).Proposition4.2.8SupposethattheincidencematrixAisfinitelyprim-itive.Then(a)inf{t:Z1(t)<∞}=θ(S).(b)ThetopologicalpressurefunctionP(t)isnon-increasingon[0,∞),strictlydecreasingon[θ,∞)tonegativeinfinity,convexandcon-tinuousonF(S).(c)P(0)=∞ifandonlyifIisinfinite.(d)P(t)=infu≥0:
||te−u|ω|<∞.ω∈E∗||φωFollowingtheterminologyintroducedin[MU1]inthecontextofiteratedfunctionsystemswecallthesystemSregularifthereexistst≥0suchthatP(t)=0.ItfollowsfromProposition4.2.8thatthereexistsatmostonesucht.Weshallprovethefollowingusefulcharacterizationofregularity.Theorem4.2.9AfinitelyprimitiveCGDMSisregularifandonlyifthereexistsat-conformalmeasure.Ifsuchameasureexists,thenthismeasureisunique,P(t)=0andthet-conformalmeasureistLog-conformal. 4.2Conformalmeasures;Hausdorffandboxdimensions79Proof.Supposethatat-conformalmeasureexists.TheninviewofRemark3.2.5,P(t)=0andmistLog-conformal.InviewofTheorem3.2.3(uniqueness)itisthereforelefttoshowthatifP(t)=0,thenat-conformalmeasureexists.Andindeed,ifP(t)=0,theninparticular,P(t)<∞,andinviewofProposition4.2.7alongwithPro-position2.1.9,tLogisasummableH¨olderfamilyoffunctions.Hence,weconcludefromTheorem3.2.3andProposition4.2.5thatthereex-istsauniquetLog-conformalmeasure.SinceP(t)=0,thismeasureist-conformal.
Theuniquet-conformalmeasurewillbedenotedinthesequelbym,itsinvariantversionbyµandtheirliftstothecodingspacerespect-ivelyby˜mand˜µ.AsanimmediateconsequenceofthistheoremandLemma3.2.4,wegetthefollowing.Lemma4.2.10IfafinitelyprimitiveCGDMSisregular,P(h)=0,andmistheuniqueh-conformalmeasure,thenM=min{m(Xv):v∈V}>0.Thenextresultestablishesastronggeometricmeaningofconformalmeasureinthefinitecase.Itsfirstversioncanbefoundin[B2]andinthecontextofconformaliteratedfunctionsystemsitessentiallyappearedin[Be].Theorem4.2.11IfSisafiniteCGDMSwhoseincidencematrixisprimitive,thenSisregularandthereexistsC≥1suchthat−1m(B(x,r))C≤≤Crhforallx∈Jand00suchthatP(h)=0.IttheninturnfollowsfromTheorem4.2.9thatthereexistsanh-conformalmeasureforS.SinceSisfinite,ξ=inf{||φ
||:i∈I}>0.iConsiderx=π(ω),ω∈E∞,0ξ.Then,using(4.20),foreveryintegern≥1sufficientlylargewehavediam(φ(X))t≤Dt||φ
||t≤Dtexp(nP(t)/2).ωt(ω)ωω∈Enω∈En 82ConformalGraphDirectedMarkovSystemsSincethefamilyφ(X),ω∈En,isacoverofJandsinceitsdia-ωt(ω)metersconvergeto0asn→∞,itfollowsfromtheestimateobtainedthatHt(J)=0.ThusHD(J)≤ξ.LetΛandqbegivenbythefiniteprimitivityofthesystemS.Setη=sup{hF:F∈Fin(I)}.Inordertodemonstratethatξ≤ηweneedanimprovementofLemma4.2.12.Foreveryv∈Vfixev∈Isuchthati(ev)=v.ConsiderthenanarbitraryfinitesubsetFofIsuchthatGq⊃ΛandG⊃{e:v∈V}.SinceGvisfiniteandfinitelyprimitive,thesystemSGisregularandthereexistsanhG-conformalmeasuremGforSG.Using(2.21)andproceedingasintheproofofLemma4.2.12,wefindforeveryn≥1hhGm(φω(Xt(ω)))||φω||≤Kω∈En∩Gnω∈En∩Gnm(Xt(ω))≤KhGmin{K−hG||φ
||hG:α∈Λ}−1αω∈En∩Gn×T(hLog)#Λ||φ||−hGm(φ(X))Get(ω))ωt(ω)≤K2hGKhG)#Λmax{||φ
||−hG:α∈Λ}α×max{||φ||−hG:v∈V}m(φ(X))evωt(ω)ω∈En∩Gn≤K3h#Λmax{||φ
||−h:α∈Λ}max{||φ||−h:v∈V}.αevDenotethislastnumberbyMandnoticethatitisindependentofG.DenotebyH⊂ItheminimalsetsuchthatHq⊃ΛandH⊃{e:vv∈V}.Obviouslyη=sup{hF:F∈Fin(I)andF⊃H}.SincehF≤h=hIforeveryF⊂IwehaveMF≤MIforeveryF∈Fin(I)suchthatFq⊂Λ.Usingtheaboveestimatewecanwriteasfollowsforanyt>ηandanyn≥1.||φ
||t=sup||φ
||tωωω∈EnF∈Fin(I):F⊃Hω∈Fn∩En≤sup{||φ
||hFsn(t−hF)}ωFω∈Fn∩En≤s(t−η)nsup{||φ
||hF}≤Ms(t−η)n.ωFnω∈FHenceP(t)≤(t−η)logs<0,whichgivest≥ξandconsequentlyη≥ξ.Obviously(cf.Lemma4.2.10)η≤HD(J),andsincewehaveprovedthatHD(J)≤ξ,theproofofthe“equality”partofthetheoremiscompleted.Theinequalityθ≤ξfollowsimmediatelyfromthedefinitionsofboth 4.2Conformalmeasures;Hausdorffandboxdimensions83numbers.Finally,thelaststatementofthetheoremistruesinceP(t)iscontinuousandstrictlydecreasingon(θ,∞).
IteasilyfollowsfromTheorem4.2.11thatifthesetofverticesisfinitethenthebox-countingdimensionandtheHausdorffdimensionofthelimitsetcoincide.AsExample5.2.3shows,thisequalityfailsintheinfinitecase.WeshallhowevernowprovideacompletesolutionoftheproblemofdeterminingtheMinkowskiorbox-countingdimensionandthepackingdimension(seeAppendix2,[Ma],or[Fa2])ofthelimitsetsforallconformalprimitiveGDMS.Ourfirststepistodemonstratethatthesenumbersareequalevenifweassumeonlythatthecontrac-tionsformingthesystemSarebi-Lipschitzcontinuous.Theorem4.2.14IfSisaCGDMSandallcontractionsφiarebi-Lipschitzcontinuous,thenPD(J)=BD(J)=PD(J)=BD(J).Proof.TheinequalitiesPD(J)≤PD(J)≤BD(J)andPD(J)≤BD(J)=BD(J)areobvious.ThustocompletetheproofitsufficestoshowthatPD(J)≥BD(J).Indeed,fixtM.ByTheorem4.2.13P(t)<0.ThusthereissomeQsuchthatifq≥Q,thenZ(t)<4−tandif|ω|≥Q,then||φ
||≤1/4.qωFixq≥QandchooseAsuchthatforallD≥r>0,Nr(Lq(Y))≤Ar−t.Now,chooseBsuchthatif1≤r≤D,thenN(J)≤Br−tandrsuchthatB≥4tA/(1−4tZ(t)).Wewillshowbyinductionthatforqeachn∈IN,if1/n≤r≤D,thenN(J)≤Br−t.Thisinequalityrholdsforn=1.Supposeitholdsfornand1/(n+1)≤r<1/n.LetC={ω∈Eq:diam(φ(J))≤1/(2(n+1))}.SinceJ=
n+1
ωt(ω)ω∈Cn+1φω(Jt(ω))∪ω∈EqCn+1φω(J(Jt(ω),wegetNr(J)≤N1/(n+1)(J)≤N1/(n+1)φω(J)ω∈Cn+1+N1/(n+1)(φω(J)).ω∈EqCn+1Forω∈EqC,wehaven+1N1/(n+1)(φω(Jt(ω)))≤N1/((n+1)||φ
||)(J)≤N1/(2(n+1)||φ
||)(J).ωωSince||φ
||≤1/4≤(1/2)(n/n+1),wehave1/n≤1/(2(n+1)||φ
||).ωωSince1/(2(n+1))t−s>BD(L1(xw))≥BD(L1(xv)foreveryv∈V.Thenfix0<%<ρ/2andforeveryv∈VconsiderthesetI(%)={e∈I:t(e)=vandK%ρ−1≤||φ
(x)||≤2K%ρ−1}.vevSinceforgivenv∈V,theballsB(φi(xv),%)withe∈Iv(%)aredisjoint,N,(L1(xv))≥#Iv(%).SincethesetVisfinite,wehaveforallv∈Vandall%>0smallenough,%s≥%tN((L(x)).Therefore,forallk,1vlargeenough,sayk≥k0,wegetobtain||φ
||t≤2tKtρ−t2−kt#I(2−k)ivk≥k0v∈Vi∈Iv(2−k)k≥k0v∈V≤2tKtρ−t2−ktN(x))2−k(L1k≥k0v∈V−1t−ks≤(2Kρ)2v∈Vk≥k0−1t#V≤(2Kρ)<∞.1−2−sSincelim||φ
||=0,thesetII(2−k)isfinite,andthere-i∈Iiv∈Vk≥k0foret≥θ(S).Lettingt→BD(L1(xw)),wegetθ(S)≤BD(L1(xw)).Now,fixt>BD(L1(xw))again.Weshallshowbyinductionthatforalln≥1thereexists0BD(L1(xw))≥BD(L1(xv)).Supposethat0θ(S),ωt(ωωN(L(x))≤2t#VAr−t+Ar−t||φ
||trn+1n1ωω∈EnI1t−t−t≤2#VAnr+A1Zn(t)r=(2t#VA+AZ(t))r−t.n1nTheproofof(4.34)iscompletedbysettingA=2t#VA+AZ(t).n+1n1nHenceBDn(v)=BDn(Ln(xv))≤tandthereforeBDn(v)≤OD(S).
AsaconsequenceofLemma4.2.15andLemma4.2.17,wehaveasimplemeansofobtainingthepackingandupperbox-countingdimensionsofthelimitset.Theorem4.2.18Let{φi:i∈I}beaconformalGDMS.ThenPD(J)=BD(J)=OD(S).4.3Stronglyregular,hereditarilyregularandirregularsystemsInthissectionfollowingtheterminologyintroducedin[MU1]inthecon-textofiteratedfunctionsystemsweclassifyGDMSintoregular,hered-itarilyregularandirregularones.Thelatterdifferentiatethemselvesbyoddgeometricfeatures.Definition4.3.1ACGDMSissaidtobestronglyregularifthereexistst≥0suchthat0θ.Proof.If{t:P(t)<∞}=(θ,∞),thenSishereditarilyregularinviewofLemma4.3.3,Theorem4.2.9,andProposition4.2.8.Ifψ(θ)<
∞,thenthereexistsacofinitesubsystemSofSsuchthatψS
(θ)<1,
whencePS
(θ)<0.ThereforeSisnotregularinviewofTheorem4.2.9,Theorem4.2.13andProposition4.2.8.AllotherequivalencesinvolvedinthistheoremnowfollowfromLemma4.3.3.
Theorem4.3.5Eachstronglyregularsystemisregularandeachhered-itarilyregularsystemisstronglyregular.Inaddition,foreachstronglyregularsystemS,hS>θS.Proof.ItimmediatelyfollowsfromProposition4.2.8thateachstronglyregularsystemSisregularandthathS>θSforthissystemS.IfSishereditarilyregular,thenθS=∞byTheorem4.3.4,andSisthereforestronglyregularbyProposition4.2.8.
WeshallprovethefollowinginterestingcharacterizationoftheθSnumber.Theorem4.3.6limhIT=infhIT=θS.T∈Fin(I)T∈Fin(I)Proof.InviewofLemma4.3.3,θS
=θSforeverycofinitesubsystemS
ofS.Therefore,usingTheorem4.2.13,weconcludethatlimT∈Fin(I)hIT≥θS.Inordertoprovetheoppositeinequalityfixt>θS.ThenψS(t)<∞,andthereforethereexistsF∈Fin(I)suchthatψIT(t)<1foreveryfinitesubsetTofIcontainingF.HencePIT(t)<0foreveryfinitesubsetTofIcontainingFwhichshowsthatlimT∈Fin(I)hIT≤t.
Definition4.3.7IfaCGDMSSisnotregularwecallitirregular.FromTheorem4.2.9andProposition4.2.8wegetthefollowingcharac-terizationofirregularsystems. 4.3Stronglyregular,hereditarilyregularandirregularsystems89Theorem4.3.8ACGDMSSisirregularifandonlyifP(h)<0⇔P(θ)<0.Asthenexttheoremshows,irregularsystemsalsoexhibitsomehered-itaryfeatures.Theorem4.3.9IfSisirregular,theneverycofinitesubsystemS
ofSisirregularandhS
=θS.Proof.InviewofTheorem4.3.8hS=θS.InviewofLemma4.3.3andTheorem4.3.8,PS
(θS
)=PS
(θS)≤PS(θS)<0andthereforeitfol-lowsfromTheorem4.3.8thatS
isirregular.Thus,usingLemma4.3.3,hS
=θS
=θS.
Weendthissectionwiththefollowinginterestingcharacterizationofstronglyregularsystems.Theorem4.3.10SupposethatS={φi:i∈I}isaCGDMS.Thenthefollowingconditionsareequivalent.(a)Sisstronglyregular.(b)hS>θS.
(c)ThereexistsapropercofinitesubsystemSofSsuchthathS
0foreveryj∈II
.
j4.4DimensionsofmeasuresLetusobservefirstthattheargumentgivenatthebeginningoftheproofofTheorem3.2.3alsoyieldsthefollowingremarkablefact,whichcanbecalledameasuretheoreticopensetcondition.Theorem4.4.1IfµisaBorelshift-invariantergodicprobabilitymeas-ureonE∞,thenµ◦π−1φ(X)∩φ(X)=0(4.35)ωt(ω)τt(τ)forallincomparablewordsω,τ∈E∗.RecallthatifνisafiniteBorelmeasureonX,thenHD(ν),theHausdorffdimensionofν,istheminimumofHausdorffdimensionsofsetsoffullνmeasure.Asbefore,byα={[i]:i∈I},wedenotethepartitionofE∞intoinitialcylindersoflength1.IfµisaBorelshift-invariantergodicprobabilitymeasureonE∞,byh(σ)wedenoteitsentropywithµrespecttotheshiftmapσ:E∞→E∞andbyχ(σ)=ζdµ>0itsµcharacteristicLyapunovexponent,whereζ(ω)=−log|φ
(π(σ(ω)))|.ω1Westartwiththefollowingmainresultofthissectionversionsofwhichwereestablishedinvariousconformalcontexts.Theorem4.4.2(Volumelemma)SupposethatµisaBorelshift-invariantergodicprobabilitymeasureonE∞suchthatatleastoneofthenumbersHµ(α)orχµ(σ)isfinite,whereHµ(α)istheentropyofthepartitionαwithrespecttothemeasureµ.Then−1hµ(σ)HD(µ◦π)=.χµ(σ)Proof.SupposefirstthatHµ(α)<∞.Thentheseriesi∈I−µ([i])log(||φ
||)convergesandusing(4e)and(4f)weconcludethatthefunc-i0tionζisintegrable.SinceHµ(α)<∞andsinceαisageneratingpartition,theentropyhµ(σ)=hµ(σ,α)≤Hµ(α)isfinite.Thus,inviewoftheBirkhoffergodictheoremandtheBreimann–Shannon–McMillan 4.4Dimensionsofmeasures91theoremthereexistsasetZ⊂E∞suchthatµ(Z)=1,n−11j−log(µ([ω|n])limζ◦σ(ω)=χµ(σ)andlim=hµ(σ)(4.36)n→∞nn→∞nj=0forallω∈Z.Fixnowω∈Zandη>0.Forr>0letn=n(ω,r)≥0betheleastintegersuchthatφω|n(Xt(ωn))⊂B(π(ω),r).Thenlogµ◦π−1(B(π(ω),r))≥logµ◦π−1(φ(X))≥log(µ([ω|])≥ω|nt(ωn)n−(hµ(σ)+η)nforeveryrsmallenough(whichimpliesthatn=n(ω,r)islargeenough)anddiamφω|n−1(Xt(ωn−1))≥r.Thelastinequalityimpliesthatlogr≤logdiam(φ(X))≤logD|φ
(π(σn−1(ω)))|ω|n−1t(ωn−1)ω|n−1n−1≤logD+log|φ
(π(σj(ω)))|≤logD−(n−1)(χ(σ)−η)ωjµj=1forallrsmallenough.Therefore,fortheserlogµ◦π−1(B(π(ω),r))−(hµ(σ)+η)n≤logrlogD−(n−1)(χµ(σ)−η)hµ(σ)+η=.−logD+n−1(χ(σ)−η)nnµHencelettingr→0,andconsequentlyn→∞,weobtainlogµ◦π−1(B(π(ω),r))hµ(σ)+ηlimsup≤.r→0logrχµ(σ)−ηSinceηwasanarbitrarypositivenumberwefinallyobtainlogµ◦π−1(B(π(ω),r))hµ(σ)limsup≤r→0logrχµ(σ)forallω∈Z.Hence(seeAppendix,SectionA.2),asµ◦π−1(π(Z))=1,HD(µ◦π−1)≤h(σ)/χ(σ).LetnowJ⊂JbeanarbitraryBorelsetµµ1suchthatµ◦π−1(J)>0.Fixη>0.Inviewof(4.36)andEgorov’s1theoremthereexistn≥1andaBorelsetJ˜⊂π−1(J)suchthat021µ(J˜)>µ(π−1(J))/2>0,21µ([ω|n])≤exp(−hµ(σ)+η)n(4.37)and|φ
(π(σn(ω))|≥exp(−χ(σ)−η)nforalln≥nandallω∈J˜.ω|nµ02By(4.23),thelastinequalityimpliesthatthereexistsn1≥n0suchthatdiamφ(X)≥D−1e(−χµ(σ)−η)n≥e−(χµ(σ)+2η)n(4.38)ω|nt(ωn) 92ConformalGraphDirectedMarkovSystemsforalln≥n1andallω∈J˜2.Givennow0n1,hencen(ω,r)≥n1anddiamφω|n(Xt(ωn))≥r.InviewofLemma4.2.6thereexistsauniversalconstantL≥1suchthatforeveryω∈J˜2and00suchthatZ(h−η)<∞,whichmeansthat||φ
||h−η<1i∈Ii∞.Since||φ
||−η≥−hlog||φ
||forallbutperhapsfinitelymanyi∈iiI,theseries−hlog(||φ
||)||φ
||hconverges.Hence,bytheGibbsi∈Iiiproperty,i∈I−log(˜µ([i])˜µ([i])<∞,whichmeansthatHµ˜(α)<∞.So,supposethatSisregularandHµ˜(α)<∞.Sinceµandmareequivalent,HD(µ)=HD(m).Sinceforeveryω∈E∞,n−1j−log(˜µ([ω|n]))hj=0ζ◦σ(ω)nlim→∞n−1j=limn→∞n−1j=h,ζ◦σ(ω)ζ◦σ(ω)j=0j=0usingBirkhoff’sergodictheoremandtheBreimann–Shannon–McMillantheorem,weseethathµ/χµ=h.TheproofisnowconcludedbyinvokingTheorem4.4.2andRemark4.4.3.
Weendthisshortsectionwiththeproofofthefollowingtwofactsshow-ingthatessentially˜µistheonlyinvariantmeasureonE∞whose−hζprojectionontoJShasthemaximaldimensionHD(JS). 94ConformalGraphDirectedMarkovSystemsTheorem4.4.7SupposethatS={φi}i∈Iisaregularconformalsystemsuchthatχµ˜−hζ<∞.Supposealsothatµisashift-invariantergodicBorelprobabilitymeasureonE∞suchthatH(α)<∞.IfHD(µ◦µπ−1)=h:=HD(J),thenµ=˜µ.−hζProof.Ifχµ=∞,thenitfollowsfromRemark4.4.3thath=HD(µ◦π−1)=0,whichisacontradiction.So,χ<∞anditfollowsfromµRemark4.4.3thathµ−hχµ≥0.Since,inviewofTheorem4.2.9andTheorem4.2.13,P(−hζ)=P(h)=0,wethereforededucefromTheorem2.2.9withf=−hζ,thatµ=˜mu−hζ.
Corollary4.4.8SupposethatS={φi}i∈Iisaregularconformalsystemsuchthatχ<∞.SupposealsothatF={f(i):i∈I}isasummableµ˜−hζH¨olderfamilyoffunctionssatisfyingtheassumptionsofCorollary4.4.5(orequivalentlyHµ˜F(α)<∞).IfHD(µF)=h:=HD(J),thenµ˜F=µ˜andthedifferencebetweentheamalgamatedfunctionf:E∞→IR−hζandthefunction−hζ:E∞→IRiscohomologoustoaconstantintheclassofboundedH¨oldercontinuousfunctionsonE∞.Proof.Sinceµ=˜µ◦π−1,alltheassumptionsofTheorem4.4.7areFFsatisfied.Itthereforefollowsfromthistheoremthat˜µF=˜µ−hζ.AsanimmediateapplicationofTheorem2.2.7wenowconcludethatf+hζiscohomologoustoaconstantintheclassofboundedH¨oldercontinuousfunctions.
4.5Hausdorff,packingandLebesguemeasuresWestartthissectionwiththefollowingtwogeneralresultsshowingthatdespiteExamples5.2.6,5.2.7and5.2.8somethingpositivecanprovedaboutHausdorffandpackingmeasureseveninanentirelygeneralcon-formalsetting.Theorem4.5.1Ifmisat-conformalmeasureonJ,thentheHausdorffmeasureHtrestrictedtoJisabsolutelycontinuouswithrespecttomand||dHt/dm||0<∞.Inparticular,Ht(J)isfinite.Proof.Inviewof(4.20)diam(φ(X))≤D||φ
||foreveryω∈E∗.ωt(ω)ωHence,byt-conformalityofthemeasuremandtheboundeddistortionproperty,wegetm(φ(X))≥K−tm(X)||φ
||t≥MK−t||φ
||t,ωt(ω)t(ω)ωω 4.5Hausdorff,packingandLebesguemeasures95whereMistheconstantcomingfromLemma4.2.10.Hencediam(φ(X))t≤M−1(DK)tm(φ(X)).ωt(ω)ωt(ω)LetnowAbeaclosedsubsetofJandforeveryn≥1putAn={ω∈En:φ(J)∩A=∅}.Thenthesequenceofsetsφ(X)isωω∈Anωt(ω)decreasingandn≥1ω∈Anφω(Xt(ω))=A.ThereforetHt(A)≤liminfdiam(φω(Xt(ω)))n→∞ω∈An≤liminfM−1(DK)tm(φ(X))ωt(ω)n→∞ω∈An=M−1(DK)tliminfmφ(J)=M−1(DK)tm(A).ωn→∞ω∈AnSinceJisaseparablemetricspace,themeasuremisregularandthere-foretheinequalityH(A)≤M−1(DK)tm(A)extendstoallBorelsub-tsetsofJ.
Letusnowproveananalogousresultforpackingmeasures.Theorem4.5.2Ifmisat-conformalmeasureforaCGDMSSandeitherthealphabetIisfiniteorJ∩Int(X)=∅,thenmisabsolutelycontinuouswithrespecttoΠt.Moreover,theRadon-Nikodymderivativedm/dΠtisuniformlyboundedawayfrominfinity.InparticularΠh(J)>0.Proof.IfIisfinite,thentheresultfollowsfromTheorem4.2.11.So,supposethatJ∩Int(Xv)=∅forsomev∈V.Thenthereexistsq≥1andτ∈Eqwithi(τ)=vandsuchthatφ(X)⊂Int(X).Setτt(τ)vγ=dist(φτ(X),∂X).Let∞R={ω∈E:ω|[n+1,n+q]=τforinfinitelymanyn’s}andletRbethesetofthoseelementsofE∞whichcontainnosubword0τ.Since[τ]∩R=∅,wegetµ(R)<1,andsinceσ−1(E∞R)⊂000E∞R,itfollowsfromtheergodicityofσprovedinTheorem2.2.40thatµ(R)=0.AsE∞R=σ−n(R),weobtainµ(E∞R)=0.0n≥00Therefore,using(4.29),wegetµ(Jπ(R))=˜µ◦π−1(Jπ(R))≤µ˜(E∞R)=0.Takenowω∈Randanintegern≥1suchthatω|=τ.Putx=π(ω)andconsidertheballB(x,K−1||φ
||γ).[n+1,n+q]ω|nSinceby(4.21)B(x,K−1||φ
||γ)⊂φB(π(σn(ω)),γ)andsinceω|nω|n 96ConformalGraphDirectedMarkovSystemsB(π(σn(ω)),γ)⊂Int(X)⊂X,usingtheconformalityofmwegetvvmB(x,K−1||φ
||γ)≤||φ
||tm(B(π(σn(ω)),γ))≤||φ
||tω|nω|nω|n=(Kγ−1)t(K−1||φ
||γ)t.ω|nSincem(Jπ(R))=0,applyingTheoremA2.0.13(1)wethusgetΠt(E)≥(K−1γ)tp(t)m(E)foreveryBorelsubsetEofJ.
1TheassumptionJ∩Int(X)=∅isknownintheliteratureconcerningiteratedfunctionsystemsasthestrongopensetcondition(abbreviated(SOSC)).WeshallnowprovidecharacterizationsofpositivityoftheHausdorffmeasureandfinitenessofthepackingmeasureofthelimitsetofaregularCGDMS.Thesecharacterizationsreducetheprocedureofcomparingtheratiosofmeasuresofballsandtheirradiitotheballs“containing”thefirstlevelsets.Thisisinthespiritofgoingtoa“largescale”.ThisideaisalsoemployedinTheorem4.6.2andTheorem6.3.1.Theorem4.5.3IfSisaregularCGDMS,thenthefollowingconditionsareequivalent.(a)Hh(J)>0.(b)ThereexistsaconstantL>0suchthatforeveryi∈I,everyr≥diam(φ(X)),andeveryy∈φ(V),m(B(y,r))≤Lrh.it(i)it(i)(c)TherearetwoconstantsL>0,γ≥1suchthatforeveryi∈Iandeveryr≥γdiam(φi(Xt(i)))thereexistsy∈φi(Vt(i))suchthatm(B(y,r))≤Lrh.Proof.(a)⇒(b).Inordertoprovethisimplicationsupposethat(b)hfails.ThenforeveryL>1/dist(X,∂V)thereexistsj∈Isuchthatm(B(x,r))>Lrhforsomex∈φ(X)andsomer≥diam(φ(X)).jt(j)jt(j)LetJbetheimageunderπofallwordsofE∞thatcontaineachele-1mentofIinfinitelyoften.Considerz∈J,z=π(ω)∈E∞suchthat1ω=jforsomen≥1.Setz=π(σn(ω)).Thenz=φ(z)n+1nω|nnandz∈B(x,r).Sincer≤1/L1/h≤dist(X,∂V),allthegeometricnconsequencesoftheboundeddistortionproperty(4e),(4f)and(4.19)–(4.24)areapplicabletotheballB(x,r).Inparticular,weget|φω|n(zn)−φ(x)|≤||φ
||randBφ(x),||φ
||r⊃φ(B(x,r)).There-ω|nω|nω|nω|nω|nfore,B(z,2||φ
||r)⊃φ(B(x,r)).Byconformalityand(4f)thisω|nω|n 4.5Hausdorff,packingandLebesguemeasures97impliesthatmB(z,2||φ
||r)≥K−h||φ
||hm(B(x,r))≥K−hL||φ
||hrhω|nω|nω|n
hL=2||φω|n||r(2K)h.UsingTheoremA2.0.16,wegetHh(J1)≤C/L,forsomeconstantCindependentofL.Now,lettingL→∞weconcludethatHh(J1)=0.ByTheorem2.2.4,m(JJ1)=0.Thisinturn,inviewofTheorem4.5.1,showsthatHh(JJ1)=0.Thus,Hh(J)=0andthereforetheproofoftheimplication(a)⇒(b)isfinished.Theimplication(b)⇒(c)isobvious.(c)⇒(a).Setη=dist(X,∂V).IncreasingDorKifnecessary,wemayassumethat2KDη−1≥1.Takeanarbitraryx∈Jandra-diusr>0.Set˜r=2KDη−1r.Foreveryz∈B(x,r)∩Jconsiderashortestwordω=ω(z)suchthatz∈π([ω])andφω(Xt(ω))⊂B(z,r˜).Thendiam(φω||ω|−1(Xt(ω|ω|−1))≥r˜.LetR={ω(z)||ω(z)|−1:z∈J∩B(x,r)}.NoticethatRisfinitesincelimi∈Idiam(φi(Xt(i)))=0andsincelimsup{diam(φ(X)):ω∈In}=0.Thereforewecann→∞ωt(ω)findafiniteset{z,z,...,z}⊂J∩B(x,r)suchthatthefamilyR∗=12k{ω(zj)||ω(zj)|−1:j=1,...,k}⊂Rconsistsofmutuallyincomparablewordsandthefamily{π(ω(zj)||ω(zj)|−1:j=1,...,k}coversB(x,r)∩J.Now,temporarilyfixanelementz∈{z1,z2,...,zk},setω=ω(z),q=|ω|,andψ=φω|q−1.Sincediam(ψ(Int(Xt(ωq−1)))≥r˜,itfollowsfromLemma4.2.6that#R∗≤M,aconstantdependingonlyonthesystemS.Bythechoiceofω,(4.23)and(4f)wehaveD−1K−1||ψ
||·||φ
||≤2˜r,ωqwhence,by(4.20),2KD2||ψ
||−1r˜≥D||φ
||≥diam(φ(X)).So,ωqωqt(ωq)ify∈φωq(Xt(ωq))isthepointfromtheassumptionscorrespondingtotheradius4γKD2||ψ
||−1r˜≥2γdiam(φ(X)),using(4.21),theωqt(ωq)inequality2rK||ψ
||−1≤2rKDr˜−1=ηandtherelationψ−1(z)∈φωq(Xt(ωq)),wecanwriteB(x,r)∩ψ(Xt(ωq−1))⊂B(z,2r)∩ψ(Xt(ωq−1))⊂ψB(ψ−1(z),2rK||ψ
||−1)⊂ψB(y,2˜rK||ψ
||−1+2˜rK||ψ
||−1D2)⊂ψB(y,4D2K||ψ
||−1r˜). 98ConformalGraphDirectedMarkovSystemsSo,bytheassumptionsofthelemma,2
−1mB(x,r)∩ψ(Xt(ωq−1))≤mψ(Xt(ωq−1))∩B(y,4DK||ψ||r˜)≤||ψ
||hmB(y,4D2K||ψ
||−1r˜)h≤||ψ
||hL4D2K||ψ
||−1r˜=L(8D3K2η−1)hrh.Thereforem(B(x,r))≤#R∗L(8D3K2η−1)hrh≤ML(8D3K2η−1)hrhandapplyingTheoremA2.0.12(2)finishestheproofofthisimplicationandsimultaneouslythewholethorem.
Remark4.5.4Itisobviousthatitsufficestheforconditions(b)and(c)ofTheorem4.5.3tobesatisfiedforacofinitesubsetofI.Theorem4.5.5IfS={φi:i∈I}isaregularCGDMS,thenthefollowingconditionsareequivalent.(a)Πh(J)<+∞.(b)TherearetwoconstantsL>0,ξ>0suchthatforeveryi∈I,everydiam(φi(X))≤r≤ξandeveryy∈φi(Wt(i)),m(B(y,r))≥Lrh.(c)TherearethreeconstantsL>0,ξ>0,andγ≥1suchthatforeveryi∈Iandeveryγdiam(φi(X))≤r≤ξthereexistsy∈φ(W)suchthatm(B(y,r))≥Lrh.it(i)Proof.(a)⇒(b).Inordertoprovethisimplicationsupposethat(b)failsand(a)holds.FixL,ξ>0.Thentherearei∈Ianddiamφi(X)≤r≤ξsuchthatforsomex∈φi(Wt(i)),wehavem(B(x,r))≤Lrh.Sincethesystemisregular,thereisaBorelsubsetBofJwithm(B)=1andsuchthateachpointzofBhasauniquecode,ω,andπ(σn(ω))isintheballB(x,r/2)forinfinitelymanyn’s.Forsuchapointzandintegern≥1,wehavebyconformalityofmand(4f)m(φ(B(π(σn(ω)),r/2)))≤φ
hm(B(π(σn(ω)),r/2))ω|nω|n≤φ
hm(B(x,r))≤φ
hLrh.ω|nω|nBut,by(4.21),φ(B(π(σn(ω)),r/2))⊃B(z,φ
K−1r/2).ω|nω|n 4.5Hausdorff,packingandLebesguemeasures99So,m(B(z,φ
r/2K))≤(φ
r/2K)h(2K)hL.UsingTheoremω|nω|nA2.0.13(1),wegetΠh(J)≥Πh(J∩B)≥(2K)−hL−1p(t).Now,letting1L→0wegetΠh(J)=∞.Thiscontradictionfinishestheproofoftheimplication(a)⇒(b).Theimplication(b)⇒(c)isobvious.(c)⇒(a).FirstnoticethatdecreasingLifnecessary,theassumptionofthelemmacontinuestobefulfilledifthenumberξisreplacedbyanyotherpositivenumber,forexamplebyη/2,whereη=dist(X,∂V).Fix0D−1||ψ
||.Hence,by(4.20),B(x,r)⊃B(x,D||ψ
||)⊃ψ(W)andtherefore,t(ωk+1)usingtheconformalityofm,wegetm(B(x,r))≥K−h||ψ
||hm(X))t(ωk+1)≥MK−h||ψ
||hwhereMcomesfromLemma4.2.10.IfnowγDK||ψ
||≥(2D3)−1ηr,thenm(B(x,r))h42−h≥Mη(2DKγ).rhOtherwise,γDK||ψ
||<(2D3)−1ηr.(4.40)Setg=φωk+1andletybeanarbitrarypointing(Wt(ωk+1)).Sincediam(g(W))≤D||g
||andsinceγ≥1,itfollowsfrom(4.40)thatt(ωk+1)3−1
−1k−3
−1By,(2D)η||φω|k||r⊂Bπ(σ(ω)),Dη||φω|k||r.(4.41)From(4.19)φB(π(σk(ω)),D−3η||φ
||−1r)ω|kω|k(4.42)−3
⊂Bx,Dηr||φω|k(g(z))||⊂B(x,r).Inviewof(4.40)wehave(2D3)−1η||φ
||−1rγD||g
(z)||≥γdiamω|k(g(W)).By(4.39)and(4.20),D−3||φ
||−1r≤1;hence(2D3)−1t(ωk+1)ω|kη||φ
||−1rγD||g
(z)||≤η/2.Asthenumber(2K)−1||φ
(g(z))||−1rω|kω|kdoesnotdependonthechoiceofy∈g(Wt(ωk+1)),wecanassumethatysatisfiestheassumptionofourlemma.So,usingthisassumption,itfollowsfrom(4.41)and(4.42)thatm(B(x,r))≥K−h||φ
||hL(2D3)−h||φ
||−hrh=L(2D3K)−hrh.ω|kω|k
100ConformalGraphDirectedMarkovSystemsRemark4.5.6Itisobviousthatitsufficesfortheconditions(b)and(c)ofTheorem4.5.5tobesatisfiedforacofinitesubsetofI.AsaconsequenceofTheorem4.5.3,wegetthefollowing.Corollary4.5.7IfS={φi:i∈I}isaregularCGDMSandthereexistasequenceofpointszj∈S(∞)andasequenceofpositivereals{rj:j≥1}suchthatm(B(zj,rj))limsup=∞,j→∞rhjthenHh(J)=0.Proof.Indeed,bythedefinitionofS(∞),foreveryj≥1thereexistsi(j)∈Isuchthatφi(j)(Xt(i(j)))⊂B(zj,rj).Thenforeveryx∈φi(j)(Xt(i(j)))wehavem(B(x,2rj)m(B(zj,rj))≥(2rj)h(2rj)handlettingj∞weseethatcondition(b)ofTheorem4.5.3isnotsatisfied.
AsaconsequenceofTheorem4.5.3,wegetthefollowing.Corollary4.5.8IfS={φi:i∈I}isaregularCGDMSandthereexistasequenceofpointszj∈S(∞)andasequenceofpositivereals{rj:j≥1}suchthatm(B(zj,rj))liminf=0,j→∞rhjthenΠh(J)=∞.Proof.Indeed,bythedefinitionofS(∞),foreveryj≥1thereex-istsi(j)∈Isuchthatφi(j)(Xt(i(j)))⊂B(zj,rj/4).Thenforeveryx∈φi(j)(Xt(i(j))),wehaveφi(j)(Xt(i(j)))⊂B(x,rj/2)⊂B(zj,rj).Thereforem(B(x,rj/2)m(B(zj,rj))≤(rj/2)h(rj/2)handlettingj∞weseethatcondition(b)ofTheorem4.5.5isnotsatisfied.
4.5Hausdorff,packingandLebesguemeasures101Nowforeachn≥0putXn=φω(Xt(ω)).ω∈EnWeshallnowprovetworesultsconcerningthed-dimensionalLebesguemeasureλdofthesesets,theLebesguemeasureofthelimitsetJ,andanestimateontheHausdorffdimensionofJ.Proposition4.5.9IfSisaCGDMSandλdInt(X)X1>0,thenthereexists0<γ<1suchthatλ(X)≤γnλ(X)foralln≥1.Indndparticularλd(J)=0.Proof.Foreveryv∈VputG=Int(X)Xandξ=K−d1min{λd(Gv)/λd(Xv)}>0.Inviewoftheboundeddistortionprop-ertywehaveλd(φω(Gt(ω)))≥ξλd(φω(Xt(ω))).Inviewoftheopensetconditionwehaveφω(Gt(ω))∩φτ(Xt(ω))=∅ifω=τand|ω|=|τ|.Thusforalln≥0Xn+1=φω(X1)⊂φω(Xt(ω)Gt(ω))ω∈Enω∈En=φω(Xt(ω))φω(Gt(ω))=Xnφω(Gt(ω)).ω∈Enω∈Enω∈EnThereforeλd(Xn+1)=λd(Xn)−λφω(Gt(ω))ω∈En=λd(Xn)−λd(φω(Gt(ω)))ω∈En≤λd(Xn)−ξλd(φω(Xt(ω)))ω∈En≤λd(Xn)−ξλd(Xn)=(1−ξ)λd(Xn).So,puttingγ=1−ξfinishestheproof.
Theorem4.5.10IfSisaregularCGDMSandλd(Int(X)X1)>0,thenh=HD(J)0,inparticularHD(J)=d,andλd/λd(X)istheonlyconformalmeasure.Proof.Inordertoprovethefirstpartsupposetothecontrarythath=d.Thenforeveryω∈E∗andeveryBorelsetA⊂Xwitht(ω) 102ConformalGraphDirectedMarkovSystemsλd(A)>0wehave
d−d
dm(A)dm(φω(A))≤||φω||m(A)=K||φω||λd(A)Kλd(A)(4.43)dm(A)≤λd(φω(A))K.λd(A)Foreveryn≥1andeveryω∈EndefineY=φ(X)∩ωωt(ω)τ∈En{ω}φ(X).Thenthesetsφ(X)Y,ω∈En,aremutuallydisjoint,τt(ω)ωt(ω)ωm(Y)=0forallω∈Eninviewof(4.29),andφ−1(Y)⊂∂Xbyωωωtheopensetcondition.Therefore,using(4.43),wegetthefollowingestimate:m(X)=mφ(X)Y=mφ(Xφ−1(Y))nωt(ω)ωωt(ω)ωωω∈Enω∈Enm(Xφ−1(Yω))−1dt(ω)ω≤λdφω(Xt(ω)φω(Yω))K−1ω∈Enλd(Xt(ω)φω(Yω))dm(X)≤Kλdφω(Xt(ω))Yωmin{λd(Int(Xv)):v∈V}ω∈EnKd≤λd(Xn).min{λd(Int(Xv)):v∈V}Thus,byProposition4.5.9,m(J)=limn→∞m(Xn)=0.Thiscompletestheproofofthefirstpart.Movingtotheotherpartofthetheoremnoticefirstthatforeveryn≥0wehaveXnXn+1=φω(Xt(ω))φω(X1)ω∈Enω∈En⊂φω(Xt(ω))φωX1∩Xt(ω)ω∈En=φω(Xt(ω)X1)ω∈EnSinceλd(XX1)=0,wethereforeobtainλd(XnXn+1)=0orequival-entlyλd(Xn)=λd(Xn+1).Henceλd(J)=limn→∞λd(Xn)=λd(X)>0.Inparticularh=d.Sinceobviouslyλ(φ(A))=|φ
|ddλdωAωdforalln≥0andallBorelsubsetsAofX,andsinceλdφω(Xt(ω))∩φ(X)=0forallincomparablewordsω,τ∈E∗bytheconditionτt(τ)λd(∂X)=0,weconcludethatλd/λd(X)isaconformalmeasureforthesystemS.
4.6Porosityoflimitsets103Letusfinishthissectionwiththefollowingresultconcerningirregularsystems.Theorem4.5.11IfSisirregular,theneachmeasureHg(J)orΠg(J)iseitherzeroorinfinityforeverygaugefunctiongoftheformthL(t),whereL(t)isaslowlyvaryingfunction.AdditionallyHh(J)=0.Proof.SupposethatameasureHg(J)orΠg(J)(callitGg)isfinite.ThentheJacobian(Radon-Nikodymderivative)ofamapφ,ω∈I∗,ωwithrespecttothemeasureGisequalto|φ
|h.Bythedefinitionofgωpressurethereexistsn≥1suchthat
||h0suchthateachopenballBcenteredatapointofXandofanarbitraryradius00suchthateachopenballBcenteredatapointofXandofanarbitraryradius0<κr≤1containsanopenballofradiuscrdisjointfromX.Forafixedκ,ciscalledaporosityconstantofX.ItiseasytoseethateachporoussethasboxcountingdimensionlessthanthedimensionoftheEuclideanspaceitiscontainedin.Furtherrelationsbetweenporosityanddimensionscanbefoundforexamplein[Ma2]and[Sal].Amuchweakerproperty,alsocalledporosity,wasintroducedin[De].Forasurveyconcerningthisconceptsee[Zar].Here 104ConformalGraphDirectedMarkovSystemswewillonlybeinterestedinthenotionofporositydescribedinthefirstparagraphofthissection.Inthissection,followingtheapproachfrom[U2],wedealwiththeproblemofporosityoflimitsetsofconformalgraphdirectedMarkovsystems.Weprovideanecessaryandsufficientconditionforthelimitsetsofthesesystemstobeporousandweshowthatthelimitsetofeachnon-trivialfinitesystemisporous.In[U2]thereadermayfindsomeexamplestowhichtheresultsofthissectionapply.OneofthemostinterestingisprovidedbythearithmeticcharacterizationofsubsetsIofpositiveintegerssuchthatthesetofallrealsin[0,1],allofwhoseentriesinthecontinuedfractionexpansionbelongtoI,isporous.In[JJM],conformaliteratedfunctionssystemsyieldinglimitsetswithpositiveporosityarealsocharacterizedandapplicationsaregiventorandomfractals.Westartthissectionwiththefollowingstraightforwardresult.Theorem4.6.1IfS={φi}i∈IisaCGDMSandIntXJ=∅,thenJ⊂IRdisanowhere-denseset.Proof.Consideranarbitrarypointx∈Jandaradiusr>0.Bythedefinitionofthelimitsetthereexistsω∈E∗suchthatφ(X)⊂ωt(ω)B(x,r).Bytheopensetconditionφω(Int(Xt(ω))J)⊂B(x,r)isthenanopensetdisjointfromJandwearedone.
Themainresultofthissectionisthefollowingcharacterizationofporos-ityoflimitsetsofconformalGDMSsintermsofthe“largescale”behavior.Theorem4.6.2LetS={φi}i∈IbeaCGDMSsuchthattheconecon-ditionissatisfiedforeveryx∈X.Thenthefollowingthreeconditionsareequivalent.(a)ThelimitsetJisporous.(b)∃c>0∃ξ>0∀i∈I∀00∃ξ>0∃β≥1∀i∈I∀00ifnecessary,wemayassumethatitholdswithξ≥2KD3β.Fixanarbitraryx=π(ω)∈J,ω∈E∞,andapositiveradiusr<2KD3β.Letn≥1betheleastintegersuchthatrφω|n(Xt(ωn))⊂Bx,2.2KDβSupposefirstthatn=1.Thenr≥βdiam(φω1(Xt(ω1)))and,asr<2KD3β,weconcludefrom(c)thatB(x,cr)∩J=∅.Sincealsoω1B(xω1,cr)⊂B(x,cr+κr)⊂B(x,(c+κ)r),wearedoneinthiscasewiththeporosityconstant≤c/2.So,supposeinturnthatn≥2.Thenrrdiam(φω|n(X))≤KD2βanddiam(φω|n−1(X))≥2KD2β.(4.44)Thereforeby(4.20)and(4f)||φ
||
ω|ndiam(φωn(Xt(ωn)))≤D||φωn||≤DK
||φ||ω|n−1≤DK||φ
||−1Ddiam(φ(X))ω|n−1ω|nt(ωn)22−1
−1−1
−1≤DKr(βKD)||φω|n−1||=βr||φω|n−1||.Hence,r||φ
||−1≥βdiam(φ(X)).(4.45)ω|n−1nωnt(ωn)Alsoby(4.44)r||φ
||−1≤Drdiam−1(φ(X))≤2KD3β.(4.46)ω|n−1nω|n−1t(ωn−1)Hence,condition(c)isapplicablewithi=ωnandtheradiusr||φ
||−1.Using(4.45)wegetω|n−1nφBx,cr||φ
||−1ω|n−1ωnω|n−1
−1
−1⊂φω|n−1Bφω|n(X),cr||φω|n−1||+κr||φω|n−1||⊂φBπσn−1(ω),β−1r||φ
||−1+cr||φ
||−1ω|n−1ω|n−1ω|n−1
−1+κr||φω|n−1||⊂B(x,(2+κ)r).SinceBx,cr||φ
||−1maynotbecontainedinX,weneedtheωnω|n−1i(ωn)followingreasoningtoconcludetheproof.Inviewof(4.46)andthecone 106ConformalGraphDirectedMarkovSystemscondition(4d),wegetforsomey∈Conx,α,min{c,l(2KD3β)−1}ωnr||φ
||−1,ω|n−1φBx,cr||φ
||−1ω|n−1ωnω|n−1n⊃φConx,α,min{cr||φ
||−1,l}ω|n−1ωnω|n−1⊃φConx,α,min{c,l(2KD3β)−1}r||φ
||−1ω|n−1ωnω|n−1⊃φBy,c
min{c,l(2KD3β)−1}r||φ
||−1ω|n−1ω|n−1−1
3−1⊃Bφω|n−1(y),Kcmin{c,l(2KDβ)}r,where00∃ξ>0∀i∈I∀00∃ξ>0∃β≥1∀i∈I∀0P(t).Thenv||φ
||te−u|ω|ωω∈E∗2≤K2tmax{||φ
||−t:α∈Λ}||φ
||te−u|ω|αα(ω1)ωβ(ω|ω|)(4.47)ω∈E∗22t2qu
−t
t−u|τ|v≤Kemax{||φα||:α∈Λ}||φτ||e<∞,τ∈I∗vwherewecouldwritethesecondinequalitysignsinceα(ω1)ωβ(ω|ω|)∈I∗,sincethefunctionω→α(ω)ωβ(ω)is1-to-1,andsince|τ|≤v1|ω|v|τ|foreveryτ∈I∗,where|τ|denotesthelengthofτwrittenasavvconcatenationoflettersfromthealphabetI∗.ThelastinequalitysignVfollowsfromTheorem3.1.1sinceu>Pv(t).ConsequentlyPv(t)≥P(t)andtheproofiscomplete.
SupposethatF={f(i):X→IR}isaH¨olderfamilyoffunctionst(i)i∈Iofsomeorderβ>0.Givenv∈VdefineFv,afamilyofmapsfromXt(v)toXt(v),asfollows.Ifω∈Iv,thensetf(ω)=S(F)−P(F)|ω|:X→IR.vωvAsanimmediateconsequenceofLemma3.1.2wegetthefollowing.Proposition4.7.5IfF={f(i):X→IR}isaH¨olderfamilyoft(i)i∈Ifunctionsofsomeorderβ>0,thenforeveryvertexv∈V,FvisalsoaH¨olderfamilyoffunctionsoforderβ.AsaconsequenceofthedefinitionofF-conformalmeasures,andthedefinitionsoffamiliesFv,wegetthefollowing.Theorem4.7.6IfF={f(i):X→IR}isaH¨olderfamilyoft(i)i∈IfunctionsandmistheF-conformalmeasure,thenm(J)−1m|isF-vJvvconformalforSvforeveryvertexv∈V.Proof.Sincetheshift-invariantmeasure˜µisergodicandsince˜µispositiveonallcylinders,wegetforeveryv∈Vthatµ˜(I∞)=˜µ({ω∈E∞:i(ω)=v}).v1 4.8Refinedgeometry,F-conformalmeasures109Hence˜m(I∞)=˜m({ω∈E∞:i(ω)=v})andconsequentlyv1∞∞m(JIv)=˜m(Iv)=˜m({ω∈E:i(ω1)=v})=m(Jv).Thatthemeasurem(J)−1m|isF-conformalfollowsimmediatelyvJvvfromthedefinitionofFconformalityandthedefinitionoffamiliesFv.
Inparticularwegetthefollowing.Corollary4.7.7IfthesystemSisregular,thensoisSvforeveryv∈V.Inaddition,ifmistheuniqueh-conformalmeasureforS,thenm(J)=m(J)andm(J)−1m|isauniqueh-conformalmeasureforIvvvJvSv.4.8Refinedgeometry,F-conformalmeasuresversusHausdorffmeasuresInthissectionweestablishrelationsbetweenGibbsstatesandappro-priategeneralizedHausdorffmeasures,cf.[DU!],[PUZ]and[U1].Letψ=f+κζ−P(F),whereκ=HD(µF)and,asintheprevioussections,
ζ(ω)=−log|φω1◦π◦σ(ω)|.Throughoutthissectionweassumethat|f|2+γdµ˜<∞and|ζ|2+γdµ˜<∞(4.48)FFforsomeγ>0.InviewofLemma2.5.6andsinceL∗(σ)isalinearspace,ψ∈L∗(σ)and,inparticular,σ2=σ2(ψ)exists.Thefollowinglemmahasbeenprovedin[DU1]asLemma4.3.Weprovideaformulationandshortproofofitforthesakeofcompleteness.Lemma4.8.1Letη,χ>0andletρ:[(χ+η)−1,∞)→IRbelong+totheupper(lower)class.Letθ:[(χ+η)−1,∞)→IRbeafunction+suchthatlimt→∞ρ(t)θ(t)=0.Thenthereexistsanupper(lower)classfunctionρ+:[1,∞)→IR+,(ρ−:[1,∞)→IR+)withthefollowingproperties.(a)ρ(t(χ+η))+θ(t(χ+η))≤ρ+(t),(t≥1)(b)ρ(t(χ−η))−θ(t(χ−η))≥ρ−(t),(t≥1). 110ConformalGraphDirectedMarkovSystemsProof.Sincelimt→∞ρ(t)θ(t)=0,thereexistsaconstantMsuchthat(ρ(t)+θ(t))2≤ρ(t)2+M.Letρbelongtotheupperclass.Thent→ρ(t/(χ+η))alsobelongstotheupperclass.Hencewemayassumethatχ+η=1.Defineρ(t)2=inf{u(t)2:uisnon-decreasingandu(t)≥ρ(t)+θ(t)}.+Thenρ+(t)≥ρ(t)+θ(t)fort≥1andρ+isnon-decreasing.Sinceρ(t)2≤ρ(t)2+M,wealsoget+∞ρ+(t)2exp−(1/2)ρ+(t)dt1t∞ρ+(t)2≥exp(−M/2)exp−(1/2)ρ(t)dt=∞.1tTheproofinthecaseofafunctionofthelowerclassissimilar.
Afunctionh:[1,∞)→IRissaidtobeslowlygrowingifh(t)=o(tα)+forallα>0.Letχ=χµ˜F(σ)=ζdµ˜F.Firstweshallprovethemaingeometricallemma.Lemma4.8.2(Refinedvolumelemma)Supposethatσ2=σ2(ψ)>0.Ifaslowlygrowingfunctionhbelongstotheupperclass,thenforµF-a.e.x∈J,µF(B(x,r))limsup√=∞.r→0rκexpσχ−1/2h(−logr)−logrIf,ontheotherhand,hbelongstothelowerclass,thenforevery%>0thereexistsaBorelsetJ,⊂JsuchthatµF(J,)≥1−%,andthereexistsaconstantk(%)≥1suchthatforallx∈J,andall00letn=n(ω,r)betheleastintegersuchthatφω|n(Xt(ωn))⊂B(x,r).Thenr≤diam(φω|n−1(Xt(ωn−1)))andmF(B(x,r))≥mF(φω|n(X))=expSω|n(F)−P(F)n(z)dmF(z).Xt(ωn) 4.8Refinedgeometry,F-conformalmeasures111Hence,usingLemma3.2.4,Lemma3.1.2and(4.20),wegetthefollowing.mF(B(x,r))√rκexpσχ−1/2h(−logr)−logrXexpSω|n(f)−P(F)n(z)dmF(z)t(ωn)≥√rκexpσχ−1/2h(−logr)−logrQ−1expS(F)−P(F)n(x)ω|n≥√rκexpσχ−1/2h(−logr)−logrMT(F)−1expS(F)−P(F)n(x)ω|n≥√diam(φω|n−1(X))κexpσχ−1/2h(−logr)−logr(4.49)MT(F)−1expn−1f◦σj(ω)−P(F)nj=0≥κn−2j−1/2√Dexp−κg◦σ(ω)expσχh(−logr)−logrj=0n−1=exp(f◦σj(ω)+κg◦σj(ω))−P(F)nj=0!−σχ−1/2h(−logr)−logr−κζ(σn−1(ω))T(F)M−1Dκ.InviewoftheBirkhoffergodictheoremthereexistsaBorelsetY⊂E∞1of˜µFmeasure1suchthatforeveryη>0,everyω∈Y1andeverynlargeenough,sayn≥n1(ω,η),−logr≤−logdiam(φω|n(Xt(ωn)))+log2≤(χ+η)n.(4.50)Infact,inwhatfollowswewillneedabetterupperestimateon−logr.Inordertoprovideit,noticethatinviewofLemma2.5.6and(4.48)thefunctionζisamemberofL∗(σ).Letτ2denotetheasymptoticvarianceofζ.Ifτ2=0,thenby[IL]giscohomologoustotheconstantχbyanL1coboundary.Itturnsoutthatthefollowingproof,whereweassumeτ2>0becomesmuchsimplerwhenτ2=0.Sincethefunction√t→2tloglogtbelongstothelowerclassthereexistsY2⊂Y1of˜µFmeasure1suchthatforallω∈Y2,τ1>τ,andallnlargeenough,say 112ConformalGraphDirectedMarkovSystemsn≥n2(ω)≥n1(ω,η),−logr≤−logdiam(φω|n(Xt(ωn))+log2n−1≤logD+log2+g(σj(ω))j=0(4.51)≤χn+2τnloglogn+logD+log2≤χn+2τ1nloglognItisasimpleexerciseinmeasuretheorytocheckthatift>0and|g|tdµ<∞,thenforeverya>0,µ(|g|≥a)≤a−t|g|tdµ.Sinceby(4.48)|ζ|2+4γdµ˜<∞forsome0<γ<1/2,forallη>0wegetF√µ˜{ω∈E∞:κ|ζ(ω)|≥σ(h((χ+η)n)n)1−γ}F√1−γ−2−4γ2+4γ≤σ(h((χ+η)n)n)|ζ|dµ˜F.Sinceforeveryn≥1,h((χ+η)n)≥h(χ+η)>0,1(1−γ)(−2−4γ)<−12andthemeasure˜µFisσ-invariant,weget∞∞n−1µ˜F{ω∈E:κ|ζ(σ(ω))|n=1√1−γ≥σ(h((χ+η)n)n)}∞√1−γ−2−4γ2+4γ≤σ(h((χ+η)n)n)|ζ|dµ˜F<∞n=1Therefore,inviewoftheBorel–Cantellilemma,thereexistsasetY3⊂Y2of˜µFmeasure1suchthatforallω∈Y3thereexistsn3(ω)≥n2(ω)suchthatforalln≥n3(ω)√n−11−γκ|ζ(σ(ω))|≤σ(h((χ+η)n)n).(4.52) 4.8Refinedgeometry,F-conformalmeasures113Combiningthis,(4.51),(4.50)and(4.49)wegetforalla<01/4m(B(x,r))eσanF√rκexpσχ−1/2h(−logr)−logrn−1Mj−1/2≥expψ◦σ(ω)−σχh((χ+η)n)T(F)Dκj=0"√1/4×χn+2τnloglogn−σ(h((χ+η)n)n)1−γeσan1n−1√Mj=expψ◦σ(ω)−σnh((χ+η)n)T(F)Dκj=0#$2τ1loglogn−1/4√1−γ−γ/2×1+−an+h((χ+η)n)n)n.χn(4.53)Now,considerthefunction&'#'(2τ1loglog(t(χ+η)−1)θ(t)=h(t)1+−1χt(χ+η)−1h(t)1−γ−1−1/4−a(t(χ+η))+.(t(χ+η)−1)γ/2Thus,θ(t)>0andsinceh(t)isslowlygrowing,limt→∞h(t)θ(t)=0.ThereforeitfollowsfromLemma4.8.1(a)thatthereexistsh+(t)intheupperclasssuchthath+(t)≥h(t(χ+η))+θ(t(χ+η)).Since,by(4.48),Theorem2.2.9andTheorem4.4.2,hµ˜Fψdµ˜F=fdµ˜F+χ−P(F)=fdµ˜F+hµ˜F−P(F)=0,χitfollowsfromTheorem2.5.5thatforinfinitelymanyn’sn−1√0≤ψ◦σj(ω)−σnh(n)+j=0n−1√≤ψ◦σj(ω)−σnh(n(χ+η))+θ(n(χ+η))j=0#$n−1√j2τ1loglogn≤ψ◦σ(ω)−σnh(n(χ+η))1+χnj=0−an−1/4+h((χ+η)n)1−γn−γ/2. 114ConformalGraphDirectedMarkovSystemsCombiningthisand(4.53)weseethatmF(B(x,r))≥M(T(F)Dκ−11/4√)exp(−σan)rκexpσχ−1/2h(−logr)−logrfor˜µF-a.e.ωandinfinitelymanyn’sprovidedtheyareoftheformn(ω,r).Butsincethereexistsn3(ω)suchthateachn≥n3(ω)isoftheformn(ω,r),fixinga<0,weeventuallygetmF(B(x,r))limsup√=∞.r→0rκexpσχ−1/2h(−logr)−logrforµFa.e.x∈J.SincethemeasuresµFandmFareequivalentwithboundedRadon-Nikodymderivatives,theproofofthefirstpartofLemma4.8.2iscomplete.Letusnowprovethesecondpartofthelemma.Foreveryω∈E∞andeveryr>0letn=n(ω,r)≥0betheleastintegernsuchthatdiam(φω|n+1(Xt(ωn+1)))0,everyω∈Y1andeverynlargeenough,sayn≥n1(ω,η),−logr≥−logdiam(φω|n(Xt(ωn)))≥(χ−η)n.(4.56)Infact,inwhatfollowswewillneedabetterupperestimateon−logr.InordertoprovideitrecallthatthefunctionζisamemberofL∗(σ).Letτ2denotetheasymptoticvarianceofζ.Ifτ2=0,thenby[IL]ζiscohomologoustotheconstantχbyanL1coboundary.Itturnsoutthatthefollowingproof,whereweassumeτ2>0,becomesmuchsimpler√whenτ2=0.Sincethefunctiont→2tloglogtbelongstothelowerclassthereexistsY2⊂Y1of˜µFmeasure1suchthatforallω∈Y2,τ1>τ,andallnlargeenough,sayn≥n2(ω)≥n1(ω,η),n−1j−logr≥−logdiam(φω|n(Xt(ωn))≥−log(DK)+ζ(σ(ω))j=0≥χn−2τnloglogn−log(DK)≥χn−2τ1nloglogn.(4.57)Thesameargumentasthatleadingto(4.52)showsthatthereexistsaBorelsetY3⊂Y2of˜µFmeasure1suchthatforallω∈Y3thereexistsn3(ω)≥n2(ω)suchthatforalln≥n3(ω),√κ|ζ(σn(ω))|≤σ(h((χ−η)n)n)1−γ.Combiningthis,(4.55),(4.56)and(4.57)wegetforalla>01/4m˜([ω|])eσanFn√rκexpσχ−1/2h(−logr)−logrn−1"κj−1/2≤QDexpf◦σ(ω)−σχh((χ−η)n)χn−2τ1nloglognj=0√1/41−γσan+σ(h((χ−η)n)n)e#$n−1√κj2τ1loglogn=QDexpψ◦σ(ω)−σnh((χ−η)n)1−χnj=0√−1/41−γ−γ/2−an−h((χ−η)n)n)n(4.58) 116ConformalGraphDirectedMarkovSystemsNow,considerthefunction&'#'(2τ1loglog(t(χ−η)−1)θ(t)=h(t)1−1−χt(χ−η)−1h(t)1−γ−1−1/4+a(t(χ−η))+.(t(χ−η)−1)γ/2Thus,θ(t)>0andsinceh(t)isslowlygrowing,limt→∞h(t)θ(t)=0.ThereforeitfollowsfromLemma4.8.1(b)thatthereexistsh−(t)inthelowerclasssuchthath−(t)≤h(t(χ−η))−θ(t(χ−η)).Since,by(4.48),Theorem2.2.9,andTheorem4.4.2,hµ˜Fψdµ˜F=fdµ˜F+χ−P(F)=fdµ˜F+hµ˜F−P(F)=0,χitfollowsfromTheorem2.5.5thatthereexistsaBorelsetY4⊂Y3of˜µFmeasure1suchthatforallω∈Y4andallnlargeenough,sayn≥n4(ω)≥n3(ω),n−1√0≥ψ◦σj(ω)−σnh(n)−j=0n−1√≥ψ◦σj(ω)−σnh(n(χ−η))−θ(n(χ−η))j=0#$n−1√j2τ1loglogn≥ψ◦σ(ω)−σnh(n(χ−η))1−χnj=0−an−1/4−h((χ−η)n)1−γn−γ/2.Combiningthisand(4.58)weconcludethatforeveryω∈Y4andeveryn≥n4(ω)m˜F([ω|n])κ−σan1/4√≤QDe.rκexpσχ−1/2h(−logr)−logrInotherwords,foreveryω∈Y4andeveryr>0smallenough,sayr≤r(ω)≤r1(ω),m˜F([ω|n(ω,r)])≤QDκ−σan(ω,r)1/4√e.(4.59)rκexpσχ−1/2h(−logr)−logr1/4Fixnow%>0andtakeqsolargethatQDκe−σaq≤%.Then,sincelimr0n(ω,r)=∞,thereexistsk(ω)≥1suchthatforall00σh˜(t)=tκexp√h(−logt)−logt.χOneoftheresultsweannouncedatthebeginningofthissectionishere. 118ConformalGraphDirectedMarkovSystemsTheorem4.8.3Supposethatσ2(ψ)>0andthath:[1,∞)→(0,∞)isaslowlygrowingfunction.(a)Ifhbelongstotheupperclass,thenthemeasuresµFandHh˜onJaresingular.(b)Ifhbelongstothelowerclass,thenµFisabsolutelycontinuoush˜withrespecttotheHausdorffmeasureH.Proof.Supposefirstthathbelongstotheupperclass.Foreveryn≥1andevery%>0,byLemma4.8.2thereexistsaBorelsetEn⊂Jsuchthatµ(E)>1−%2−nandsuchthatforeveryx∈EandsomeFnnclosedballB(x)centeredatxandwithdiameter<1/n,µF(B(x))>nh˜(B(x)).ByBesicovitch’scoveringtheoremthereexistsauniversalconstantC>0suchthatfromthecover{B(x):x∈En}onecanchooseacountablesubcover{B(xj):j≥1}ofmultiplicity≤C.Sinceforeveryj≥1,diam(B(xj))<1/n,weget∞h˜1CCH(En,1/n)≤µF(B(xj))≤µF(J)=.nnnj=1h˜SettingE,=n≥1EnwethenhaveH(E,)=0andµF(E,)≥1−%.h˜FinallythesetE=q≥1E1/qsatisfiesH(E)=0andµF(E)=1.Theproofofitem(a)isthereforecomplete.SupposeinturnthathbelongstothelowerclassandconsideraBorelsetE⊂JwithµF(E)>0.Take%=µF(E)/2.ThenbyLemma4.8.2µF(J,∩E)≥µF(E)−%=%.Fix0<δ≤1/k(%)andconsiderB={B(xi,ri)},thecoverofJ,∩EbyballscenteredatpointsofJ,∩Eandwithradii≤δ.ThenbyLemma4.8.211h˜(ri)≥µF(J,∩B(xi,ri))≥µF(J,∩E)≥1.%%iih˜HenceHδ(E)≥B>0,whereBisauniversalconstant(seeThe-oremA2.0.13).ThusH˜(E)≥B>0andwearedone.
hRemark4.8.4Takingh:=0itfollowsfromTheorem4.8.3(a)thatthemeasureµFissingularwithrespecttotheκ-dimensionalHausdorffmeasureHκonJ.Werecallthattwofunctionsf,f:E∞→IRarecohomologousina12classHifthereexistsafunctionu∈Hsuchthatf2−f1=u◦σ−u. 4.8Refinedgeometry,F-conformalmeasures119AsacomplementaryresulttoTheorem4.8.3weshallprovethefollowing.Theorem4.8.5Ifσ2(ψ)=0,thenκ=HD(µ)=HD(J):=h,theFfunctions−hgandf−P(F)arecohomologousintheclassofH¨oldercontinuousboundedfunctions,thesystem{φi:i∈I}isregularandmFisequivalentwiththeh-conformalmeasureonJ,thatiswithm−hg,withboundedRadon-Nikodymderivatives.Inaddition,theinvariantmeasuresµ˜Fandµ˜hgareequal.InordertoproveTheorem4.8.5weneedsomepreparations.First,letαbethepartitionofthetwo-sidedshiftspaceEZ={{ω}}into−nn∈Zelementsoftheformω×E{1,2,...},whereω∈E{...,−2,−1,0}.Given−∞≤m≤n≤+∞letω|n=ωω...ωandletmmm+1n[ω]n={τ∈EZ:τ=ωforallm≤k≤n}.mkkFinallyletµFbeRokhlin’snaturalextensionoftheinvariantmeasureµ˜ontothetwo-sidedshiftspaceEZ.LetusrecallthatµisdefinedFFonacylinderC=π−1(C)∩π−1(C)∩···∩π−1(C)n11n22nkkwithn1≤n2≤···≤nk,bytheformulaµ(C)=˜µσ−(n1+1)(C)|∞,FF1whereforeveryk∈Z,π:EZ→Iistheprojectionontothekthkcoordinategivenbytheformulaπ(ω)=ωandforeverysetB⊂EZ,kkB|∞istheprojectionofBontoE{1,2,...}denotedalsobyπ(B).Let1usrecallthatthemeasureµFisshift-invariant.Weshallprovethefollowing.Lemma4.8.6If{µ:ω∈EZ}istheRokhlincanonicalsystemofα−(ω)measuresofthemeasureµFonthepartitionα−,thenforµF-a.e.ω∈EZtheconditionalmeasureµconsideredonEINisequivalentwithα−(ω)µ˜F.Moreover,theRadon-Nikodymderivativedµα−(ω)/dµ˜FisboundedfromaboveandfrombelowrespectivelybyQ2T(F)2andQ−2T(f)−2,whereQcomesfromtheGibbspropertyofthemeasureµ˜F.Proof.Bythemartingaletheorem,forµ-a.e.ω∈EZandeveryFBorelsetB⊂EZ,µ(B∩[ω]0)F−nµα−(ω)(B)=lim0.n→∞µF([ω]−n) 120ConformalGraphDirectedMarkovSystemsItthereforesufficestoshowthatforeveryτ∈E∗µ([ω|0τ])−4α−(ω)−∞4Q≤≤Q.µF([τ])Andindeed,inviewoftheGibbspropertyofthemeasure˜µF,wegetµ([ω]0)=˜µ(σ−(n+1)([ω]0)|∞)F−nF−n1≤QexpsupSσ−(n+1)([ω]0)|n+1(F)−P(F)(n+1)−n1=QexpsupSω−n...ω0(F)−P(F)(n+1).UsinginadditionLemma3.1.2andputtingk=|τ|,wegetµ([τ]∩[ω]0)F−n=˜µ(σ−(n+1)([τ]∩[ω]0)|∞))F−n1−1≥QinfexpSσ−(n+1)([τ]∩[ω]0)|n+1(F)−P(F)(n+1+k)−n1−1=QexpinfSω−n...ω0τ1...τk(F)−P(F)(n+1+k)−1≥QexpinfSω−n...ω0(F)+infSτ1...τk(F)−P(F)(n+1+k)≥Q−1T(f)−2expsupS(F)−P(F)(n+1)ω−n...ω0×expsupSτ1...τk(F)−P(F)k.ApplyingtheGibbspropertyofthemeasure˜µFagain,wethereforeobtainµ([τ]∩[ω]0)F−n≥Q−1T(f)−2expsupS(F)−P(F)k0τµF([ω]−n)≥Q−2T(f)−2µ([τ]).FHenceµ([ω|0τ])≥Q−2T(f)−2µ([τ]).Similarcomputationsα−(ω)−∞Fshowthat022µα−(ω)([ω|−∞τ])≤QT(f)µF([τ]).
Asanimmediateconsequenceofthislemmawegetthefollowing.Corollary4.8.7If{µ:ω∈EZ}istheRokhlincanonicalsystemα−(ω)ofmeasuresofthemeasureµFonthepartitionα−,thenforµF-a.e.ω∈EZ,supp(µ)=α(ω),whereα(ω)istheonlyatomofαα−(ω)−−−containingω.Lemma4.8.8Ifη:E∞→IRisaH¨oldercontinuousfunctionofsomeorderβ>0suchthat|η|2+γdµ˜<∞,ηdµ˜=0andσ2(η)=0,FF 4.8Refinedgeometry,F-conformalmeasures121thenthereexistsaboundedH¨oldercontinuousfunctionuoforderβ>0suchthatη=u−u◦σ.Inparticularηturnsouttobebounded.Proof.ItfollowsfromTheorem2.5.1and[IL]thatthereexistsu∈L2(˜µF)suchthatη=u−u◦σ(4.62)µ˜F-a.e.OuraimistoshowthatuhasaH¨oldercontinuousversionoforderβ.Wefirstextendηanduonthetwo-sidedshiftspaceEZbydeclaringη(ω)=η(ω|∞)andu(ω)=u(ω|∞)11whereveru(ω|∞)isdefined.Thecohomologicalequation(4.62)remains1satisfiedsinceu(ω)−u◦σ(ω)=u(ω|∞)−u(σ(ω)|∞)=u(ω|∞)−u(σ((ω|∞)))=η(ω).1111(4.63)InviewofLuzin’stheoremthereexistsacompactsetD⊂EZsuchthatµF(D)>1/2andthefunctionu|Discontinuous.InviewofBirkhoff’sergodictheoremthereexistsaBorelsetB⊂EZsuchthatµ(B)=1,Fforeveryx∈B,σn(x)visitsDwiththeasymptoticfrequency>1/2,uiswelldefinedonσ−n(B)and(4.62)holdsonσ−n(B).Bythen∈Zn∈ZdefinitionofconformalmeasuresandbyLemma4.8.6thereexistsaBorelsetF⊂EZsuchthatµ=1,forallω∈F,µ(B∩α(ω))=1,andFα−(ω)−supp(µα−(ω))=α−(ω).Inparticular,foreveryω∈F,thesetB∩α−(ω)isdenseinα−(ω).Fixoneω∈Fandconsidertwoarbitraryelementsρ,τ∈α−(ω).Thenthereexistsacontinuousincreasingunboundedsequence{n}suchthatσ−nj(ρ),σ−nj(τ)∈Dforallj≥1.Usingj(4.62)weget|u(ρ)−u(τ)|njnj−nj−k−nj−k=u(σ(ρ))−η(σ(ρ))−u(σ(τ))−η(σ(τ))k=1k=1nj≤|u(σ−nj(ρ))−u(σ−nj(τ))|+|η(σ−k(ρ))−η(σ−nk(τ))|.k=1(4.64)Now,sincelimdist(σ−nj(ρ),σ−nj(τ))=0,sincebothσ−nj(ρ)andj→∞σ−nj(τ)belongtoDandsinceu|isuniformlycontinuous(asDisD 122ConformalGraphDirectedMarkovSystemscompact),weconcludethatlim|u(σ−nj(ρ))−u(σ−nj(τ))|=0.j→∞SinceηisH¨oldercontinuousoforderβ,wegetnjnjVe−β|η(σ−k(ρ)−η(σ−nk(τ)|≤V(η)e−βkd(ρ,τ)≤βd(ρ,τ).ββ1−e−ββk=1k=1Therefore,itfollowsfrom(4.64)thatVe−ββ|u(ρ)−u(τ)|≤dβ(ρ,τ).1−e−βHence,asα−(ω)∩Bisdenseinα−(ω),uhasaboundedH¨oldercontinu-ousextensionfromα(ω)∩Bonα(ω)=ω×EIN,whereω=ω|0.−−−∞Denotethisextensionbyu:α(ω)→IRandforeveryτ∈EINset−u(τ)=u(ωτ).ThisobviouslydefinesaboundedH¨oldercontinuousfunctionu:EIN→IR.DefinenowthesetBωtobedeterminedbytheconditionωBω=α−(ω)∩B.Thefunctionu:EIN→IRisaversionofu.Indeed,sinceµ(ωB)=α−(ω)ω1,itfollowsfromLemma4.8.6that˜µF(Bω)=1andadditionally,foreveryτ∈Bω,u(τ)=u(ωτ)=u(τ).Sincethemeasure˜µFisshift-invariant,˜µ(B∩σ−1(B))=1.TakenowanarbitraryelementFωωρ∈B∩σ−1(B).Thenσ(ω)∈Bandwehaveη(ρ)=u(ρ)−u(σ(ρ))=ωωωu(ρ)−u(σ(ρ)).Butsincesupp(˜µ)=EIN,thesetB∩σ−1(B)isdenseFωωinEINandthereforeη=u−u◦σonEIN.
ProofofTheorem4.8.5FirstnoticethatinviewofTheorem4.4.2,The-orem2.2.9andLemma2.2.8hµ˜Fψdµ˜F=fdµ˜F+χµ˜F−P(F)χµ˜F=fdµ˜F+hµ˜F−P(F)=P(F)−P(F)=0.HencetheassumptionsofLemma4.8.8aresatisfiedwithη=ψandthereforethereexistsaboundedfunctionu∈Hβsuchthatf−P(F)+κζ=u−u◦σ, 4.9Multifractalanalysis123thatisthefunctions−κζandf−P(F)arecohomologousintheclassofboundedfunctionsofHβ.ItfollowsfromthisequationthattheconstantRappearinginTheorem2.2.7(2)(withf:=f−P(F)andg=−κζ)isequaltozero.Therefore,itfollowsfromthistheoremthatP(−κζ)=P(f)−P(F)=0.Hence,thesystemSisregular,κ=handitfollowsfromTheorem2.2.7(1)that˜µF=˜µ−κζ.TheequivalenceofmeasuresmFandm−κζwithboundedRadon-NikodymderivativesfollowsnowfromthefactthatboththesemeasuresareGibbsstatesofthefunctionsf−P(F)and−κζrespectively.
4.9MultifractalanalysisThemultifractalformalismarosefromvariousconsiderationsinphysicsandmathematics(seee.g.[Man],[FP],[Gr],[Ha]).Inthislastpaperaformulationofthescenariosofmultifractaltheorywaselaboratedinwhichtherewerestronghintsofparallelstothetheoryofstatisticalphysics.Someofthefirstrigorousmathematicalresultsconcerningthisformalismarein[CM]and[Ra].Sincethentherehavebeenmanypa-perswrittenverifyingsomeaspectsofthisformalism(seeforexample[O1]–[O4],[PW],[Pat1],[Pat2]).Recently,Pesinpresentedageneralformulationofthesettingformultifractaltheory[Pe].Also,manymorereferencesconcerningthistopicmaybefoundinhisbook.Inthissec-tiondealingwithmultifractalanalysisofconformalGDMSswedevelopSection7of[HMU].Wewouldliketoemphasizethatouranalysisisper-formedonlyforcylinders(andalsoundersomeothertechnicalassump-tions),whereasthequestionconcerningtheanalysisofballs,successfullytakencareof(seethepaperscitedabove)inthecaseoffiniteiteratedfunctionsystems,remainsopenforinfiniteiteratedfunctionsystems.InthissectionS={φe:Xt(e)→Xi(e):e∈I}isaregularconformalGDMSsuchthatφi(Xt(i))∩φj(Xt(j))isatmostcountable(4.65)foralli=j∈IandF={f(i):X→IR:i∈I}isasummableH¨olderfamilyoffunctions.Subtractingfromeachofthefunctionsf(i)thetopologicalpressureofFwemayassumethatP(F)=0.Weconsideratwo-parameterfamilyofH¨oldercontinuousfamiliesoffunctionsG={g(i):=qf(i)+tlog|φ
|}.q,tq,ti 124ConformalGraphDirectedMarkovSystemsLet.Fin(F)={q∈IR:LqF(11)<∞}={q∈IR:P(qF)<∞}andθ(F)=infFin(F),wherethesecondequalityfollowsfromProposition2.1.9.Bythedefin-itionofsummableH¨olderfamiliesoffunctions,1∈Fin(F)and,inparticular,{i:supf(i)>0}isfinite.Beforedealingwithsmoothnesspropertiesweshallprovethefollowingresult,whichwillbeneededinthenextsection.Lemma4.9.1Thefunction(q,t)→P(q,t):=P(Gq,t)isdecreasingwithrespecttobothvariablesq≥0andt≥0.Proof.Considertwopairs(q1,t1)and(q2,t2)suchthatq1≤q2andt1≤t2.IfP(q1,t1)=∞,thereisnothingtobeproved.So,supposethatP(q1,t1)<∞.ThenbyProposition2.1.9Gq1,t1isasummableH¨olderfamilyoffunctions.Sincetheset{i:supf(i)>0}isfiniteandsinceallthefunctionslog|φ
|arenegative,thisimpliesthatGiq2,t2alsoformsasummableH¨olderfamilyoffunctions.ItthenfollowsfromTheorem2.1.8thatforevery%>0thereexistsaBorelprobabilitymeasureµonE∞suchthat−(qf−tζ)dµ<∞(whichimpliesthat22−(q1f−t1ζ)dµ<∞)andP(q2,t2)≤hµ+(q2f−t2ζ)dµ+%=hµ+(q1f−t1ζ)dµ+(q2−q1)fdµ+(t1−t2)ζdµ+%≤hµ+(q1f−t1ζ)dµ+%≤P(q1,t1)+%,wherethelastinequalityfollowsfromTheorem2.1.8.Letting%"0wethusgetP(q2,t2)≤P(q1,t1).Theproofiscomplete.
Givenq≥0letFin(q)=inf{t:LGq,t(11)<∞}=inf{t:P(Gq,t)<∞}≤θ(S)andletθ(q)=infFin(q).Noticethatifq∈Fin(F),then0∈Fin(q).Weassumethatforeveryq∈Fin(F)thereexistsu∈Fin(q)suchthat0θ(F).Sinceforeveryn≥1thefunctiont→nωj|ω|=n||expj=1φq,t◦φσjω||,t∈Fin(q),islogarithmicconvex,thefunctiont→P(Gq,t)isconvexandhencecontinuousin(θ(q),∞).Since0θ(q)andδ>0.Wethenhave1P(G)=limlog|||φ
|t+δexp(S(qφ))||q,t+δωωn→∞n|ω|=n1≤limlog||φ
||δ|||φ
|texp(S(qF))||ωωωn→∞n|ω|=n1≤limlogsnδ|||φ
|texp(S(qF))||ωωn→∞n=δlogs+P(Gq,t)0whereverthissecondderivativeexists.TheLegendretransformofkisthefunctionlofanewvariablepdefinedbyl(p)=max{px−k(x)}Ieverywherewherethemaximumexists.Itcanbeprovedthatthedomainofliseitherapoint,anintervalorahalf-line.ItisalsoeasytoshowthatlisstrictlyconvexandthattheLegendretransformisaninvolution.WethensaythatthefunctionskandlformaLegendretransformpair.The 4.9Multifractalanalysis127followingtheorem(see[Ro]forexample)givesausefulcharacterizationofaLegendretransformpair.Theorem4.9.3TwostrictlyconvexdifferentiablefunctionskandgformaLegendretransformpairifandonlyifl(−k
(q))=k(q)−qk
(q).Ourmainresultinthissectionisthefollowing.Theorem4.9.4Supposethatcondition(4.66)issatisfiedforallq∈Fin(F).Supposealsothatthereexistsaninterval∆1⊂Fin(F)suchthat1∈∆1andforeveryq∈∆1andalltinsomeneighborhoodofT(q)(containedin(θ(q),∞)),2+γ2+γ(|f|+|ζ|)dµ˜q<∞and(|f|+|ζ|)dµ˜q,t<∞forsomeγ>0.Supposefinallythathµq(σ)/χµq(σ)>θ(S)forallq∈∆2⊂∆1forsomeinterval∆2⊂∆1.Then(a)ThenumberDµF(x)existsforµF-a.e.x∈Jand−fdµ˜FDµF(x)=.ζdµ˜F(b)ThefunctionT:∆1→IRisreal-analytic,T(0)=HD(J),andT
(q)<0,T

(q)≥0forallq∈∆.1(c)Foreveryq∈∆,f(−T
(q))=T(q)−qT
(q).2µF(d)Ifµ˜F=˜µ−HD(J)ζ,thenthefunctionα→fµF(α),α∈(α1,α2)isreal-analytic,wheretheinterval(α1,α2),0≤α1<α2≤∞istherangeofthefunction−T
(q)definedontheinterval∆.2OtherwiseT
(q)=HD(J)foreveryq∈(θ(F),∞).(e)Ifµ˜F=˜µ−HD(J)ζ,thenthefunctionsfµF(α)andT(q)formaLegendretransformpair.(f)Foreveryq∈∆1thenumberT(q)isuniquelydeterminedbythepropertythatthereexistsaconstantC≥1suchthatforeveryn≥1C−1≤µq([ω])diamT(q)(φ(X))≤C.Fω|ω|=nProof.Since1∈∆1and|f|dµ˜F<∞,part(a)isacombinedcon-sequenceofBirkhoff’sergodictheorem(alongwith(4f),(4.20)and(4.23)),theBreimann–Shannon–McMillantheoremandtheassumption 128ConformalGraphDirectedMarkovSystemsthatP(F)=0.Weshallnowprovepart(b).Andindeed,sincebyProposition2.6.13,∂P|=−ζdµ˜<0foreveryq∈∆andallt∂tq,tq,t1inaneighborhoodofq,andsinceT(q)isuniquelydeterminedbytheconditionP(q,T(q))=0,itfollowsfromTheorem2.6.12andtheim-plicitfunctiontheoremthatthemapq→T(q)isreal-analyticon∆1.SincethesystemFisregular,P(−HD(J)ζ)=0,whichmeansthatT(0)=HD(J).ItfollowsfromProposition2.6.13thatforeveryq∈∆1dP∂P∂P

0=(q,T(q))=|(q,T(q))+|(q,T(q))T(q)=φdµ˜q−gdµ˜qT(q)dq∂q∂tandtherefore
fdµ˜qT(q)==−α(q).(4.67)ζdµ˜qSinceP(f)=0andfdµ˜q<∞,wededucefromTheorem2.1.8thatfdµ˜+h(σ)≤0andthereforeitfollowsfrom(4.67)thatT
(q)≤qµ˜q−h(σ)/ζdµ˜≤0.ThustoprovethatT
(q)<0itsufficestonoticeµ˜qqthathµ˜q(σ)>0,whichfollowsimmediatelyfromTheorem2.5.2.Hence,tocompletetheproofofTheorem4.9.4(b)itislefttoshowthatT

(q)≥0forallq∈∆1.Thisisdoneinthefollowing.Lemma4.9.5Thefunctionq→T(q),q∈∆1isconvex.Itisnotstrictlyconvexifandonlyifµ˜fisequaltoµ˜−HD(J)ζ.Proof.Differentiatingtheformula∂P(q,t)
∂P(q,t)0=|(q,T(q))·T(q)+|(q,T(q))∂t∂qandusingProposition2.6.13weobtain222T
(q)2∂P(q,t)+2T
(q)∂P(q,t)+∂P(q,t)

∂t2∂q∂t∂q2T(q)=−∂P(q,t)∂t222T
(q)2∂P(q,t)+2T
(q)∂P(q,t)+∂P(q,t)∂t2∂q∂t∂q2=,χµ˜qwhere,letusrecall,χµ˜q=ζdµ˜qistheLyapunovcharacteristicexpo-nentofthemeasure˜µq.InvokingProposition2.6.14weseethat∂2P∂2P∂2P=σ2(−ζ),=σ2(ζ,f),=σ2(f).∂t2µ˜q∂q∂tµ˜q∂q2µq 4.9Multifractalanalysis129Hence,wecanwrite∂2P∂2P∂2P
2
T(q)+2T(q)+∂t2∂q∂t∂q2∞∞=T
(q)2µ˜(ζ·ζ◦σk)−χ2+T
(q)µ˜(−ζ·f◦σk)qµ˜qqk=0k=0∞+χµ˜(f)+T
(q)µ˜(f(−ζ◦σk))µ˜qqqk=0∞+χµ˜(f)+µ˜(f·f◦σk)−µ˜(f)2µ˜qqqqk=0∞=µ˜−T
(q)ζ(−T
(q)ζ◦σk+f◦σk)qk=0∞+µ˜f(−T
(q)ζ(−T
(q)ζ◦σk+f◦σk)qk=0∞
2−−T(q)χµ˜q+˜µq(f)k=0∞=µ˜(−T
(q)ζ+f)(−T
(q)ζ+f)◦σk−(−T
(q)χ+˜µ(f))2qµ˜qqk=02
=σµ˜q(−T(q)ζ+f).Itfollowsthenfrom(4.67)that(−T
(q)ζ+f)dµ˜=0.Weknowthatqσ2(−T
(q)ζ+f)≥0,andbyLemma4.8.8σ2(−T
(q)ζ+f)=0ifµ˜qµ˜qandonlyifthefunction−T
(q)ζ+fiscohomologousto0intheclassofboundedH¨oldercontinuousfunctions.ThereforeT
(q)ζiscohomologoustofand,asP(f)=0,alsoP(T
(q)ζ)=0.Thus,byTheorem4.2.13,T
(q)=−HD(J)andconsequentlyfiscohomologoustothefunction−HD(J)ζ.Thisimpliesthat˜µf=˜µ−HD(J)ζ,thelatterbeingtheequi-librium(invariantGibbs)stateofthepotential−HD(J)ζ.Theproofiscomplete.
So,item(b)ofTheorem4.9.4isnowanimmediateconsequenceofLemma4.9.5.Weshallnowfocusonacontributiontowardtheproofofparts(c)–(e).Givenα≥0wedefinen−1f◦σj(x)K˜(α)=x∈J:limj=0=α.n→∞n−1−ζ◦σj(x)j=0 130ConformalGraphDirectedMarkovSystemsLemma4.9.6Foreveryα≥0,K˜(α)=Kµf(α).Proof.Inordertoprovethislemmaitsufficestoshowthatforallx∈Jlogµf(φx|n(Xt(xn)))logdiam(φx|n(Xt(xn)))nlim→∞n−1j=1andnlim→∞n−1j=1.f◦σ(x)−ζ◦σ(x)j=0j=0AndinordertoprovethefirstequalityitsufficestodemonstratethatlogmF(φx|n(Xt(xn)))lim=1n→∞n−1f◦σj(x)j=0andthisfollowsimmediatelyfromtheF-conformalityofthemeasuremFandthefactthatP(F)=0.Thesecondinequalitytobeprovedisanimmediateconsequenceoftheboundeddistortionproperty.
Lemma4.9.7Ifx∈K˜(α)andlog|φ
(σn(x))|xnliminf=0,n→∞log|φ
(σn−1(x))|x|n−1thenforeveryq∈IRlog|φ
(σn(x))|qf(σn−1x)xnliminf+≤0n→∞log|φ
(σn−1(x))|log|φ
(σn−1(x))|x|n−1x|n−1Proof.Ifq≤0,thenourinequalityfollowsimmediatelyfromthefactthatq∈Fin(F),ourassumptionandtheformulalimlog||φ
||=n→∞x|n−1−∞.So,wemayassumethatq>0.Let{n}∞beanincreasingin-kk=1finitesequencesuchthatlog|φ
(σnk(x))|xnlimk=0.(4.68)k→∞log|φ
(σnk−1(x))|x|n−1kInordertoconcludetheproofitsufficestosowthatf(σnk−1x)lim≤0.k→∞log|φ
(σnk−1(x))|x|n−1kSo,supposeonthecontrarythatf(σnk−1x)limsup≥2b>0k→∞log|φ
(σnk−1(x))|x|n−1kforsomepositiveb.Passingtoasubsequenceofthesequence{n}∞kk=1f(σnk−1x)wemayassumethatthelimitlimk→∞log|φ
(σnk−1(x))|existsandx|n−1k 4.9Multifractalanalysis131isgreaterthanorequalto2b(perhaps+∞).This,(4.68)andthefactthatx∈K˜(α)implytheexistenceofanintegerl0≥1suchthatforeveryl≥l0ljlljj=0−f(σx)bf(σx)j=0ζ(σx)lj≥α−3,lj≥bandl+1j≥1−δ,ζ(σx)ζ(σx)ζ(σx)j=0j=0j=0whereδissosmallthat(α−b)(1−δ)≥α−b.Butthen,takingkso32largethatnk−2≥l0,wegetnk−1jnk−2jnk−2jn−1j=0−f(σx)j=0−f(σx)j=0ζ(σx)−f(σkx)nk−1j=nk−2j·nk−1j+nk−1jζ(σx)ζ(σx)ζ(σx)ζ(σx)j=0j=0j=0j=0bbb≥(α−)(1−δ)+b≥α−+b=α+.322Thishoweverimpliesthatnjj=0−f(σx)bliminfn≥α+>αn→∞ζ(σjx)2j=0whichisacontradictionsincex∈K˜(α).
Lemma4.9.8WiththesameassumptionsasinTheorem4.9.4(a)µq(KµF(α(q)))=1forallq∈∆1.(b)dµq(x)≤T(q)+qα(q)forallq∈∆1andforeveryx∈KµF(α(q))butasetofHausdorffdimension≤θ(S).(c)fµF(α(q))=T(q)+qα(q)foreveryq∈∆2.Proof.Fixq∈∆1.Sincethefunctions|f|and|ζ|areintegrablewithre-specttothemeasureµq,part(a)followsimmediatelyfromLemma4.9.6andBirkhoff’sergodictheorem.Inordertoprovepart(b)fixx∈KµF(α(q))andr>0.Letn=n(x,r)betheleastintegersuchthatφx|n(Xt(xn))⊂B(x,r).Thenµq(B(x,r))≥µq(φx|n(Xt(xn))andφx|n−1(Xt(xn−1))isnotcontainedinB(x,r).Fromthelatter,diamφx|n−1(Xt(xn−1))≥r.Hence,byLemma3.1.2logµq(B(x,r))logµq(φx|n(X))≤logrlogdiam(φx|n−1(X))n
jn−1jT(q)j=1log|φxj(σ(x))|+qj=0f◦σ(x)+M1≤n−1log|φ
(σj(x))|+Mj=1xj2 132ConformalGraphDirectedMarkovSystemsforsomeconstantsM1andM2.Sincetherangeofthefunctionr→n(x,r),r∈(0,1],isoftheformIN∩[A,∞),itfollowsfromthelastinequality,Lemma4.9.7andLemma4.9.6thatiflog|φ
(σn(x))|xnliminf=0,n→∞log|φ
(σn−1(x))|x|n−1thendµq(x)≤T(q)+qα(q).Considerthesetlog|φ
(σn(x))|xnBad=x∈J:liminf>0.n→∞log|φ
(σn−1(x))|x|n−1WeshallshowthatHD(Bad)≤θ(S).So,givenγ>0definelog|φ
(σn(x))|xnBad(γ)=x∈J:∃q≥1∀n≥q≥γlog|φ
(σn−1(x))|x|n−1andgivenn≥1putlog|φ
(σk(x))|Bad(γ)=x∈J:xk≥γ∀k≥n.nlog|φ
(σk−1(x))|x|k−1Fixη>θ(S).Bythedefinitionofθ(S)thereexistsk≥1solargethatforalll≥k1
η||φi||0≤.(4.69)2{i∈I:||φ
||0≤Ksγl}iFixn≥1.Foreveryl≥p=max{n−1,k}letΩ={ω∈El:lφω(Xt(ω))∩Badn(γ)=∅}.Weshallprovebyinductionthatforeveryl≥p1l−p||φ
||η≤||φ
||η,(4.70)i02i0ω∈Ωlω∈Ωpwhereasη>θ(S),||φ
||η≤||φ
||η<∞.(4.71)i0i0ω∈Ωpω∈IpIndeed,forl=pweevenhaveequality.So,supposethat(4.70)holdsforsomel≥p.Fixω∈Ωl+1.Thenω|l∈Ωlandthereexistsx∈φ(X)∩Bad(γ).Sincex=φ(σl+1(x)),itfollowsfrom(4.65)ωt(ω)nx|l+1thatω=x|l+1.Sincel≥n−1andx∈Badn(γ),wethereforeget||φ
||≤K|φ
(σl+1(x))|=K|φ
(σl+1(x))|ωl+10ωl+1xl+1≤K|φ(σl(x))|γ≤Ksγl.x|l 4.9Multifractalanalysis133Thus,using(4.69)and(4.70)forlwecanwrite||φ
||η≤||φ
||η||φ
||ηi0ω0i0ω∈Ωl+1ω∈Ωl{i∈I:||φ
||0≤Ksγl}i=||φ
||η||φ
||ηω0i0ω∈Ωl{i∈I:||φ
||0≤Ksγl}i11l+1−p≤||φ
||η=||φ
||η.2ω02i0ω∈Ωlω∈ΩpTheinductiveproofof(4.70)isfinished.By(4.20)wethereforegetforalll≥k1l−pdiamη(φ(X))≤D||φ
||ηωt(ω)i02ω∈Ωlω∈Ωpandusing(4.71)weconcludethatHη(Bad(γ))=0.ThusHD(Badnn(γ))≤ηwhichimpliesthatHD(Badn(γ))≤θ(S).SinceBad(γ)=n≥1Badn(γ),HD(Bad(γ))≤θ(S)andsinceBad=m≥1Bad(1/m),HD(Bad)≤θ(S).Theproofof(b)iscomplete.Sinceµq(KµF(α(q)))=1,itfollowsfromTheorem4.4.2thatfµF(α(q))=HD(KµF(α(q))≥HD(µq)=hµq(σ)/χµ˜q(σ).SinceP(Gq,T(q))=0,usingTheorem2.2.9,wecontinuewritinghµ˜q(σ)−gq,T(q)dµ˜q(−T(q)ζ+qf)dµ˜qfµF(α(q))≥==χµ˜q(σ)χµ˜q(σ)−χµ˜q(σ)(4.72)−T(q)χµ˜q(σ)+qfdµ˜q==T(q)+qα(q).−χµ˜q(σ)Thisprovesonehalfof(c).Ifnowq∈∆2,thenourassumptionsgivehµq(σ)/χµq(σ)>θ(S).Applyingthisalongwith(a)and(b),itfollowsfromTheoremA2.0.16thatfµF(α(q))=HD(KµF(α(q)))≤T(q)+qα(q).Thisprovestheotherpartof(c).
Part(c)ofTheorem4.9.4isanimmediateconsequenceofLemma4.9.8(c)andformula(4.69).Part(d)isacombinedconsequenceofLemma4.9.5andpart(c)ofTheorem4.9.4.Part(e)ofTheorem4.9.4followsfromLemma4.9.5,part(c)ofTheorem4.9.4andTheorem4.9.3.WeendtheproofofTheorem4.9.4bydemonstratingpart(f).Andindeed,sincethediametersoftheimagesφω(Xt(ω))tendtozerouniformly(exponen-tially)fastwithrespecttothelengthofω,weconcludethatthereexists 134ConformalGraphDirectedMarkovSystemsatmostonevaluet∈IRsuchthatforsomeC≥1andeveryn≥1C−1≤µq([ω])diamt(φ(X))≤C.Fωt(ω)|ω|=nSo,weonlyneedtoshowthatthedisplayappearinginpart(f)ofTheorem4.9.4istrue.Andindeed,ifω∈I∗,say|ω|=nandρ∈[ω],thenitfollowsfromthedefinitionofmeasures˜µqand˜µFthatn−1n−1µ˜([ω])#expqf◦σj(ρ)−T(q)ζ◦σj(ρ)qj=0j=0qT(q)n−1n−1=expf◦σj(ρ)exp−ζ◦σj(ρ)j=0j=0qT(q)#µ˜F([ρ|n])diam(φ|ρ|n(Xt(ρn)))qT(q)=˜µF([ω])diam(φ|ω(Xt(ω))).Sinceµ˜([ω])=1,summingtheabovedisplayoverallω∈In|ω|=nqweobtainthedesiredinequalities.TheproofofTheorem4.9.4iscomplete.
Letusrecallthatin[MU2]wehaveintroducedtheclassofabsolutelyregularconformaliteratedfunctionsystemsSbytherequirementthatθ(S)=0.ThesamedefinitionextendstoconformalGDMSs.ForthesesystemswecanrewriteTheorem4.9.4,relaxingtheassumptionhµ˜q(σ)/χµ˜q>θ(S)sincewealreadyknow(seetheparagraphpreceed-ingLemma4.9.5)thattheentropyhµ˜q(σ)isalwayspositive.Itthenreadsasfollows.Theorem4.9.9Supposethatcondition(4.66)issatisfiedforallq∈Fin(F).Supposealsothatthereexistsaninterval∆1⊂Fin(F)suchthat1∈∆1andforeveryq∈∆1andalltinsomeneighborhoodofT(q)(containedin(θ(q),∞)),(|f|2+γ+|ζ|2+γ)dµ˜<∞and(|f|+|ζ|)dµ˜<∞qq,tforsomeγ>0.SupposefinallythatthesystemSisabsolutelyregular.Then 4.9Multifractalanalysis135(a)ThenumberDµF(x)existsforµF-a.e.x∈Jand−fdµ˜FDµF(x)=.ζdµ˜F(b)ThefunctionT:∆1→IRisreal-analytic,T(0)=HD(J),andT
(q)<0,T

(q)≥0forallq∈∆.1(c)Foreveryq∈∆,f(−T
(q))=T(q)−qT
(q).2µF(d)Ifµ˜F=˜µ−HD(J)ζ,thenthefunctionα→fµF(α),α∈(α1,α2)isreal-analytic,wheretheinterval(α1,α2),0≤α1<α2≤∞istherangeofthefunction−T
(q)definedontheinterval∆.2OtherwiseT
(q)=HD(J)foreveryq∈(θ(F),∞).(e)Ifµ˜F=˜µ−HD(J)ζ,thenthefunctionsfµF(α)andT(q)formaLegendretransformpair.(f)Foreveryq∈∆1thenumberT(q)isuniquelydeterminedbythepropertythatthereexistsaconstantC≥1suchthatforeveryn≥1C−1≤µq([ω])diamT(q)(φ(X))≤C.Fω|ω|=n 5ExamplesofGDMSs5.1ExamplesofGDMSsinotherfieldsofmathematicsInthissectionwemainlyprovidesomeclassesofconformalGDMSnat-urallygeneratedinotherareasofmathematics.Wewouldespeciallyliketocallthereader’sattentiontotheclassofKleiniangroupsofSchottkytype.WestarthoweverwiththefollowingtwooperationsoniteratedfunctionsystemswhichleadtographdirectedMarkovsystems.Example5.1.1(Gluing)Supposethatwearegivenfinitelymanyiteratedfunctionsystems{S}inthesameEuclideanspace,sayIRd,andletSbeaGDMSvv∈VwithverticesVandedgesE.WeformanewGDMSSˆbyaddingtoS(v)allthecontractions{φi:Xv→Xv}i∈Iv,whereIvisthealphabetofthesystemSv,andbydeclaringthatthenewincidencematrixAˆcon-tainstheoldmatrixandhasadditionalentriesAˆa,bequalto1ifeithert(a)=vandb∈Ivora∈Ivandi(b)=v.Example5.1.2(Restrictions)LetS={φi:X→X}i∈Ibeaconformaliteratedfunctionsystem(e.g.generatedbyacontinuedfractionalgorithmwithrestrictedentriesseeExample5.1.4formoredetails)andletA:I×I→{0,1}.ThesystemSˆgeneratedbytakingthesetofverticestobeasingleton{v}withXv=X,thesetofedgesequaltoIandtheincidencematrixequaltoAisaGDS.Example5.1.3(Expandingmaps)Eachdistanceexpandingmapf:X→X(see[Ru],cf.[PU])hasMarkovpartitionsR={Xt}t∈Twitharbitrarilysmalldiameters.It136 5.1ExamplesofGDMSsinotherfieldsofmathematics137givesrisetoaGDSwithT,thesetofvertices,thecontractionsformedbycontinuousinversebranchesoffdefinedonXt,t∈T,andtheincidencematrixdeterminedbytheMarkovpartitionR.Example5.1.4(Continuedfractions)Thissystemisgivenbythemapsφn:[0,1]→[0,1]definedbytheformulae1φn(x)=.x+nItiseasytoseethat1φ(ω)=1ω1+1ω2+ω3Thecontinuedfractionsystemwithrestrictedentries(seeExample5.1.2)hasbeenthoroughlyexploredfromthegeometricviewpointin[MU2].Comparealso[HeU]and[U2].Example5.1.5(KleiniangroupsofSchottkytype)Fixfinitelymany,sayq≥1,closedballsB,B,...,B⊂IRd.Foreach12qj=1,2,...,qletgjbetheinversionwithrespect∂Bj,theboundaryofBj.ThegroupG=g1,g2,...,gqistheKleiniangroupofSchottkytypegeneratedbytheinversionsg1,g2,...,gq.RecallthatL(G),thelimitsetofaKleiniangroupGisthesetoflimitpointslimn→∞gn(z),wheregn∈Garemutuallydistinct.Thevalueofthislimitdoesnotdependonthepointz∈IRd.OurgoalistorepresentL(G),thelimitsetoftheSchottkygroupGasthelimitsetonanappropriateconformalGDS.LetusfirstconstructthisGDS.SetV={1,2,...,q},E=V×V{(i,i):i∈{1,2,...,q}}andXv=Bvforallv∈V.Sincegi(IRd)=Int(Bi),weseethatforevery(i,j)∈Ethemapg(i,j)=gi|Bj:Bj→Biiswelldefined.Theassociatedincidencematrixisdefinedasfollows.
1ifk=jA(i,j),(k,l)=0ifk=j.ThesystemSG=V,E,{ge}e∈EobviouslysatisfiesalltherequirementsofaconformalGDSexceptthatthemaps{ge}e∈Eneednotbeuniformcontractions.Butsincethediametersofthesetsgω(Btω)convergetozerouniformlywithrespecttothelengthofthewordω,thebounded 138ExamplesofGDMSsdistortionproperty(see[PU]fordetails)impliesthatallmapsfromasufficientlyhighiterateofthesystemSGareuniformlycontracting.Andthisispreciselywhatweneed.Weshallprovethefollowing.Theorem5.1.6IfGisaKleiniangroupofSchottkytype,thenL(G)=JSG.Proof.TheinclusionJSG⊂L(G)isobvious.Inordertoprovetheoppositeinclusion,fixasequence{g}∞ofmutuallydistinctelementsnn=1ofGsuchthatlimn→∞gn(z)existsforsome(andequivalentlyall)z∈IRd.Withappropriatenkj∈{1,2,...,q}writegn=gnkn◦gnkn−1◦···◦gn2◦gn1initsuniqueirreducibleform,thatiswithgnj=gnj+1forall1≤j≤kn−1.Passingtoasubsequencewemayassumethatn1=iforalln≥1andsomei∈{1,2,...,q}.Fixz∈Bi.Weshallshowbyinductionthatgnl◦gnl−1◦···◦gn1(z)∈Bnlforalll=1,2,...,kn.Andindeed,forl=1thisfollowsfromthefactthatz/∈Bn1andgn1(IRdBn1)=Int(Bn1).So,supposethatgnl◦gnl−1◦···◦gn1(z)∈Bnlforsome1≤l≤kn−1.Since,byirreducibilityofthewordgnkn◦gnkn−1◦···◦gn2◦gn1,Bn∩Bn=∅andsincegn(IRdBn)=Int(Bn),weconcludel+1ll+1l+1l+1thatgnl+1◦gnl◦···◦gn1(z)∈Bnl+1.Theinductiveproofisfinished.Hence,wecanwritegn(z)=g(nkn,nkn−1)◦g(nkn−1,nkn−2)◦···g(n2,n1)◦g(n1,i)(z).Thuslimn→∞gn(z)∈JSG=JSG,wherewecouldwritetheequalitysignsincethesystemSGisfinite.
WeshallnowgeneralizetheaboveconsiderationsbyestablishingnaturalrelationsbetweenthelimitsetofeachsubgroupofaKleiniangroupofSchottkytypeandthelimitsetofanaturallyassociatedconformalGDS,perhapswithinfinitelymanyedges.Indeed,letΓ⊂G=g1,...,gqbeasubgroupofaSchottkygroup.WefirstlookatthesetΓ0⊂Γformedbythoseelementsg=gin◦···◦gi1(ij∈{1,2,...,q})writteninirreducibleformaselementsofG,withnoproperinitialblockbelongingtoΓ.InordertoassociatewithΓtheconformalGDSSΓwetakeasbeforeV={1,2,...,q}andXv=Bvforallv∈V.TheelementsofthesystemSΓareformedbythemapsφg,j=gin◦···◦gi1:Bj→Bin 5.2Exampleswithspecialgeometricfeatures139foreveryelementg=gin◦...◦gi1∈Γ0andeveryj∈{1,2,...,q}{i1}.Theincidencematrixisdeterminedbytherequirementthatthecompositionsφg,j◦φγ,iareallowedifandonlyifj=im,whereγ=gim◦...◦gi1∈Γ0isrepresentedinitsirreducibleform.Asinthecaseofa(full)Schottkygroup,thesystemSΓisaconformalGDSbut,recallingthediscussioninthecaseofa(full)Schottkygroup,weseethattheonlypointwhichmayrequiresomeexplanationistheopensetcondition.So,supposethatφg,j=φγ,i,bothbelongingtoSΓ.Ifg=γ,thenj=iandφg,j(Bj)∩φγ,i(Bi)=g(Bj)∩g(Bi)=∅sincegis1-to-1andBj∩Bi=∅.Ifg=γ,thenφg,jandφγ,iaretwoincomparableelementsofSG,andthereforeφg,j(Bj)∩φγ,i(Bi)=∅.Thus,wehaveprovedthatSΓisaconformalGDS.Repeatingessentiallytheproofoftheprevioustheorem,weshalldemonstratethefollowing.Theorem5.1.7IfΓisasubgroupofaKleiniangroupofaSchottkytype,thenJSΓ⊂L(Γ),L(Γ)=JSΓandL(Γ)=JSΓifΓ0isfinite.Proof.TheinclusionJSΓ⊂L(Γ)isobvious.InordertoprovethatL(Γ)⊂Jfixasequence{γ}∞ofmutuallydistinctelementsofΓSΓnn=1suchthatlimn→∞γn(z)existsforsome(andequivalentlyall)z∈IRd.Writeγn=gnkn◦gnkn−1◦···◦gn2◦gn1,initsuniqueirreducibleforminGanddecomposethisrepresentationofγnintoblocksγnl◦...◦γn1ofelementsfromΓ0.Passingtoasubsequencewemayassumethatt(n1)=iforalln≥1andsomei∈{1,2,...,q}.Fixz∈Bi.Then,asintheproofofTheorem5.1.6,weseethatγn(z)=φγn,i(γn)◦φγn,i(γn)◦···◦φγn,i(γn)◦φγn,i(z)ll−1l−1l−2211andthereforelimn→∞γn(z)∈JSΓ.ThustheproofoftheinclusionL(Γ)⊂JSΓiscomplete.IfnowΓ0isfinite,thenSΓisalsofiniteandconsequentlyJSG=JSΓ=L(G),whichfinishestheproof.
5.2ExampleswithspecialgeometricfeaturesInthissectionweprovideanumberofexamplesofinfiniteconformaliteratedfunctionsystemsshowinghowflexibletheyare,andhowlargeavarietyoffractalfeaturesonecanalreadymeetfindamongthem,nottomentiongeneralGDMSs.WebeginwithanextremelysimpleexampleofaconformalgraphdirectedMarkovsystemwhoselimitsetcannotberepresentedasthelimitsetofaconformaliteratedfunctionsystem. 140ExamplesofGDMSsExample5.2.1AconformalfiniteirreduciblegraphdirectedMarkovsystemwhoselimitsetcannotberepresentedasthelimitsetofacon-formaliteratedfunctionsystem.LetthesetofverticesconsistoftwoelementsvandwandletE={vw,vv,wv,ww}.WeputX=[2,4]×[1,3]⊂IR2andX=[0,2]×vw[3,5]⊂IR2.Letφ:X→X,φ:X→X,φ:X→X,vwwvwwwwwvvwφvv:Xv→Xvbesimilaritymapswithscalingfactor1/2mappingrespectivelythesetXwontothesquare[2,3]×[2,3],Xwonto[1,2]×[3,4],Xvonto[1,2]×[4,5]andXvonto[3,4]×[2,3].ObviouslythisgraphdirectedsystemisirreducibleanditslimitsetJis[2,4]×{3}∪{2}×[3,5].SinceJhasHausdorffdimensionisequalto1anditisnotananalyticarc,itfollowsfromthesecondparagraphfollowingTheorem6.4.1thatJcannotberepresentedasthelimitsetofaconformaliteratedfunctionsystem.Example5.2.2ThelimitsetJisanFσδbutnotaGδ.DenotebyQthesetofallrationalnumbersin[0,1].LetX=[0,1]×[0,1]andlet∆={(x,x)∈X}bethediagonalofX.Consideraconformaliteratedfunctionsystem{φi:X→X:i∈Q∪{−1}}consistingoflinearmappingsandsuchthat(a)φi(X)∩∆={φi(0,1)}={(i,i)}foralli∈Q(b)φ−1(x,y)=(x/2,(y+1)/2)(c)Thesetsφi(X),i∈Q∪{−1},aremutuallydisjoint.ThenJ∩∆=QisnotGδ,soneitherisJ.Letusalsonotethatthissystemisnotlocallyfinite.
Example5.2.3AniteratedfunctionsystemforwhichPD(J)≥BD(J)>HD(J).Takeanysequenceofpositivenumbers{ri:i≥1}(forexampleoftheformbi,01/2andbyTheorem4.3.4Sishereditarilyregular.Letmbethecorrespondingconformalmeasure.Thenforeveryn≥1h∞2h−11−h−2h−h11mB(0,1/n)=≥3xdx=3.3k2n2h−1nk≥nTakingnowforany01.LetI(r)={k≥1:1≤1and1≥1−r}.n2nknk+1nNoticethat#I(r)≥(1/n−r)−1−n=n2r/(1−nr)≥n2r.Thereforehh11−h−2h2mB(1/n,r)≥≥#I(r)≥(12)nnr≥3k23(2n)2k∈I(r)h−1−h1−hh−1−hh=(12)r≥(12)rr=(12)r.n2Nowsupposethat1/(2n)≤r≤2/n.Then1/n2≤r/4≤1/(2n)andinviewofthepreviouscasem(B(1/n,r))≥m(B(1/n,r/4))≥(12)−h(r/4)h=(48)−hrh.Finallysupposethatr≥2/n.ThenB(1/n,r)⊃B(0,r/2)≥C(r/2)2h−1=2C4−hrhrh−1.ThustheassumptionsofLemma4.5.5aresatisfiedandthereforeΠh(J)<∞.
WeshouldmentionherethatinthenextsectiontheCIFSinducedbycomplexcontinuedfractionswillbeconsidered,whichisalsohereditar-ilyregularandwhoselimitsethash-dimensionalHausdorffmeasure0andh-dimensionalfinitepackingmeasure.TheideaforprovingthesepropertieswillbethesamethereasinExample5.2.6.Example5.2.7HereditarilyregularlinearsystemwithΠh(J)=∞andHh(J)>0.LetX=[0,1]andletS={φn:X→X:n≥1}betheCIFSconsistingofsimilaritiesφ(x)=2−2nx+2−n−2−2nsothatφ(0)=2−n−2−2nnnandφ(1)=2−n.Thus||φ||=2−2nandψ(t)=||φ||t=nnn≥1n2−2nt.Henceh=1/2andbyTheorem4.3.4Sishereditarilyn≥1regular.Letmbethecorrespondingconformalmeasure.Thenforeveryn≥1wehavemB(0,2−n)=2−2kh=2(2−2nh).Takingnowfork≥nany00.
Example5.2.8HereditarilyregularlinearsystemwithΠh(J)=∞andHh(J)=0.ThisexampleismadeupbygluingtogetherExamples5.2.6and5.2.7.NamelyletX=[0,2]andS={φn,0,φn,1:n≥2},whereφn,0(x)=1x+1−1andφ(x)=2−2nx+2−n−2−2n+1.Thenφ([0,2])=3n22n3n2n,12n,0[1−1,1]⊂[0,1/2]andφ([0,2])=[2−n−2−2n+1,2−n+1]⊂[1,2]n3n2nn,1andψ(t)=(||φ||t+||φ||t)=6−tn−2t+2−t2−2nt.So,n≥2n,0n,1n≥2theintervalofconvergenceofψ(t)is(1/2,∞).Thus,inviewofThe-orem3.20,Sishereditarilyregularandh=HD(J)>1/2.WeseethatX(∞)={0,1}and,ifmisthecorrespondingh-conformalmeasure,thenasinExample5.2.6wegetm(B(0,1/n))≥(2h−1)−1(1/n)2h−1.InviewofTheorem4.3.10thisimpliesthatHh(J)=0andasinEx-ample5.2.7wegetm(B(1,2−n))≤4h(4h−1)−1(2−n)2handthisinviewofCorollary4.5.8impliesthatΠh(J)=∞.
Example5.2.9One-dimensionalsystems.HerewewanttodescribehoweverycompactsubsetFoftheintervalX=[0,1]canonicallygivesrisetoalinearCIFSonXsuchthatS(∞)=(∂F)d={x∈X:xisanaccumulationpointof∂F},theCantor–Bendixsonderivedsetof∂F.Indeed,letRbethefamilyofallconnectedcomponentsofXFandforeveryC∈RletφC:X→XbetheuniquelinearmapsuchthatφC(0)istheleftendpointoftheclosureofCandφC(1)istherightendpointoftheclosureofC.ThesystemS={φC:X→X:C∈R}hasthepropertyrequired.
6ConformalIteratedFunctionSystemsInthischapterwedealwithconformaliteratedfunctionsystemsCIFS.RecallthismeansthatweassumeStobeaCGDMSsuchthatthesetofverticesisasingleton,thecorrespondingspacesaredenotedbyXandW,andalltheentriesoftheincidencematrixareequalto1.WewouldliketonoteattheverybeginningthatinthecontextofCIFSs,indeedinthewidercontextofconformal-likeiteratedfunctionsystems,thenameS-invariantbecomesmeaningfulsinceeachmeasureµF,whereFisasummableH¨olderfamilyoffunctions,enjoysthefollowingtwopropertiesexpressedforaBorelprobabilitymeasureνsupportedonJS.
ν(φi(A))=ν(A)(6.1)i∈Iandν(φi(X)∩φj(X))=0(6.2)foralli∈I,j∈I{i},andallBorelsetsA⊂X.Anymeasureνsatisfying(6.1)and(6.2)willbeinthesequelcalledS-invariant.dµ6.1TheRadon-Nikodymderivativeρ=dmInthissectionwestudyanalyticpropertiesoftheRadon-Nikodymde-dµrivativeρ=wheremistheh-conformalmeasureofaregularCIFSSdmandµisitsS-invariantversion.WefirstintroduceanauxiliaryPerron–FrobeniusoperatorF:C(X)→C(X)andthenweshowthatρhasauniquecontinuousextensiononthewholesetXsuchthatF(ρ)=ρonX.ThiswillplayacrucialroleinSection6.2.Ourultimateaiminthepresentsection,playinganimportantroleinSection6.4andSection6.7istoshowthatindimensiond≥2(andalsoindimensiond=1ifallthecontractionsformingthesystemSarereal-analytic)thedensityρhas144 dµ6.1TheRadon-Nikodymderivativeρ=145dmareal-analyticextensiononaneighborhoodofJS.So,noticefirstthatifthesystemSisregularthenbyLemma4.2.12theseries||φ||hi∈Iiconverges,andthereforetheoperatorF:C(X)→C(X)definedbytheformulahF(g)=|φi|g◦φii∈IactscontinuouslyonC(X).Letusrecallfrom[Lj]thataboundedoperatorL:B→BdefinedonaBanachspaceBissaidtobealmostperiodicifforeveryx∈Btheorbit{Lnx}∞isrelativelycompactinn=0B.WestartwiththefollowingresultestablishingalmostperiodicityoftheoperatorF:C(X)→C(X).Lemma6.1.1IfSisaregularCIFS,thentheoperatorF:C(X)→C(X)isalmostperiodicandthesequence{Fn(11)}∞isuniformlyn=0boundedbetweenKhandK−h.Proof.Theuniformboundfromaboveofthesequence{Fn(11)}∞isann=0immediateconsequenceofLemma4.2.12.Thelowerboundfollowsfromthislemmacombinedwith(4f).AlthoughtheproofofalmostperiodicityoftheoperatorF:C(X)→C(X)issimilartotheproofofLemma2.4.1,following[MU1],weprovideithereforthesakeofcompletenessandtheconvenienceofthereader.Andindeed,fixg∈C(X)and>0.Sincegisuniformlycontinuous,thereexistsη1>0suchthat|g(y)−g(x)|<ifx,y∈Xand||y−x||≤η1.SinceitfollowsfromtheproofofProposition4.2.7thatthefamily{log|φω|}ω∈I∗isequicontinuous,thereexistsη2>0sosmallthat1|log|φω(y)|−log|φω(x)||≤min{,}2hforallω∈I∗andallx,y∈Xwith||y−x||≤η.Putη=min{η,η}212andconsidertwopointsx,y∈Xwith||y−x||≤η.ThenusingLemma4.2.12weobtainforeveryn≥1|Fn(g)(y)−Fn(g)(x)|=g(φ(y))|φ(y)|h−g(φ(x))|φ(x)|hωωωωω∈Inω∈In≤|g(φ(y))|φ(y)|h−g(φ(x))|φ(x)|h|ωωωωω∈In≤|g(φ(y))||φ(y)|h−|φ(x)|hωωωω∈In+|φ(x)|h|g(φ(y))−g(φ(x))|ωωωω∈In≤||g|||φ(y)|h−|φ(x)|h+Kh.0ωωω∈In 146ConformalIteratedFunctionSystemsLookingattheTaylor’sseriesexpansionofetabout0,wededucethatthereexistsM>0suchthat|et−1|≤M|t|ifonly|t|≤1/2.Hence|φ(x)|h|φ(y)|h−|φ(x)|h=|φ(y)|h1−ωωωω|φ(y)|hω=|φ(y)|h1−exphlog|φ(x)|−log|φ(y)|ωωω≤|φ(y)|hMh|log|φ(x)|−log|φ(y)||ωωωh≤|φω(y)|Mh.Consequently,usingLemma4.2.12again,weget|φ(y)|h−|φ(x)|h≤M|φ(y)|h≤KhM.ωωωω∈Inω∈InThus,finally|Fn(g)(y)−Fn(g)(x)|≤Kh(M||g||+1).0DuetotheAscoli–ArzelatheoremalongwithLemma4.2.12thisdemon-stratesthatthefamily{Fn(g)}∞isrelativelycompactinthesupnormn=1onC(X).
Wearenowinpositiontoprovethefollowingresult,whichcanbefoundin[MU4].PerhapsitsmostimportantandratherunexpectedpartisthatthedensityfunctionρhasacanonicalextensiontothewholespaceX.Theorem6.1.2SupposethatSisaregularCIFSandmisthecorres-pondingconformalmeasure.Then(a)Thereexistsauniquecontinuousfunctionρ:X→[0,∞)suchthatFρ=ρandρdm=1.(b)K−h≤ρ≤Kh.(c)Thesequence{Fn(11)}∞convergesuniformlytoρonX.n=1dµ(d)ρ|J=dm,whereµistheS-invariantversionoftheconformalmeasurem.Proof.Supposethatρ:X→[0,∞)isacontinuousfunctionsuchthatFρ=ρandρdm=1.SinceFρ(πω)=Lhζ(ρ◦π)(ω),whereLhζistheoperatorconsideredinChapter2,wethusgetLhζ(ρ◦π)=ρ◦π.ItthereforefollowsfromProposition2.4.7andTheorem2.4.6that dµ6.1TheRadon-Nikodymderivativeρ=147dmdµ˜dµρ◦π=dm˜andconsequentlyρ|J=dm.So,item(d)isprovedandifρ1,ρ2:X→[0,∞)aretwofunctionssatifyingtherequirementsofitem(a),thenρ1|J=ρ2|J.Denotethiscommonrestrictionbyˆρ.Fix>0andconsiderη>0sosmallthatforeachi=1,2,|ρi(y)−ρi(x)|<ifx,y∈Xand||y−x||≤η.Takeanarbitraryn≥1solargethatDsn≤η.Finallyfixanarbitraryz∈Xandconsideranω∈In.Thendiam(φ(X))≤Dsn≤η.Choosex∈J∩φ(X).Thenωω|ρ2(φω(z))−ρ1(φω(z))|≤|ρ2(φω(z))−ρˆ(x)|+|ρˆ(x)−ρ1(φω(z))|≤+=2.Hence,usingLemma4.2.12,weget|ρ(z)−ρ(z)|=|Fnρ(z)−Fnρ(z)|=|Fn(ρ−ρ)(z)|212121≤|ρ(φ(z))−ρ(φ(z))|·|φ(z)|h2ω1ωω|ω|=n≤2||φ||h≤2Kh.ω|ω|=nTherefore,letting0weconcludethatρ2(z)=ρ1(z)andtheunique-nesspartofitem(a)isproved.SincebyLemma6.1.1thesequence{Fn(11)}∞isuniformlyXn=1boundedbetweenK−δandKδandisequicontinuous,thesequence1n−1jnj=0F(11X)hasthesameproperties.Letρbeanaccumulationpointofthissequenceofaverages.Thenobviously,ρiscontinuousandFρ=ρ,ρdm=1andK−h≤ρ≤Kh.Thusitems(a)and(b)arealsoproved.Itremainstodemonstrateitem(c).Andindeed,sincebyLemma6.1.1thePerron–FrobeniusoperatorF:C(X)→C(X)isalmostperiodic,itfollowsfromaLyubich’sresult(see[Lj])thatC(X)=E0⊕Eu,whereE={f:||Fn(f)||→0}andEistheclosedspanof{f:F(f)=0uλfforsomeλwithλ|=1}.WeshalldemonstratefirstthatEu={cρ:c∈CI}.(6.3)Indeed,supposeF(ψ)=λψwith|λ|=1.SinceFisapositiveoperatorontheBanachlattice,C(X),itfollowsfromLemma18,Theorem4.9andExercise2in[Sc](p.326–327)thatthespectrumofFmeetstheunitcircleinacycliccompactgroup.Therefore,thegroupisfiniteandthereissomepositiveintegerrsuchthatλr=1.Thus,Fr(ψ)=ψandFr(Reψ)=ReψFr(Imψ)=Imψ.LetussupposeReψ=0.Fix 148ConformalIteratedFunctionSystemsM∈RsolargethatReψ+Mρ>0.But,bylemma2.2.4,σrisergodicwithrespectto˜µ.Thismeansthatρ◦πm˜istheonlyinvariantmeasureforσrequivalentto˜m.Therefore,thereisaconstantc>0suchthatReψ◦π+Mρ◦π=cρ◦π.So,Reψ◦π=(c−M)ρ◦πandReψdm=c−M.RepeatingthisargumentforImψ,weobtainψ◦π=(ψdm)ρ◦π.SinceFρ=ρ,thisimpliesthatλ=1andsinceπ(E∞)isdenseinJ,ψ|=ρ|.ButrepeatingnowtheargumentψdmJJoftheproofofuniquenessinitem(a),weconcludethatψ=ψdmρonX.Theproofof(6.3)iscomplete.Representingthefunction11asauniquesumofanelementfromEuandE0,itfollowsfrom(6.3)thatthereexistsc∈CIsuchthat11−cρ∈E0.ButsincetheoperatorFpreservesintegrationwithrespecttothemeasurem,gdm=0foreveryg∈E0.Consequentlyc=1.Therefore||Fn(11)−ρ||=||Fn(11−ρ)||→0whenn→∞.Wearedone.
Themainresultofthissectioniscontainedinthefollowing.Theorem6.1.3Ifd≥2andthesystemSisregular,thentheRadon-dµNikodymderivativeρ=hasareal-analyticextensiononanopendmconnectedneighborhoodUofXinW.Proof.Weshalldealfirstwiththecased≥3.Inviewof(4.1),therethenexistλω>0,alinearisometryAω,aninversion(ortheidentitymap)i=iandavectorb∈IRdsuchthatφ=λA◦i+b.ωaω,rωωωωωωωThenλr2|φ(z)|=ωωifi=Idω||z−a||2ωωand|φ(z)|=λifi=IdωωωSinceφω(W)⊂W,aω∈/W.Fixξ∈Xandconsiderthefunctionρ:CId→CIgivenbytheformulaω||ξ−a||2ωρω(z)=d2orρω(z)=1ifiω=Id.j=1zj−(aω)jWeshallshowthatthereexistaconstantB>0andaneighborhoodU˜ofXinCIdsuchthat|ρω(z)|≤B(6.4)foreveryω∈I∗andeveryz∈U˜.Indeed,otherwisethereexistse-quencesω(n)∈I∗andz(n)∈CId,n≥1suchthatlimdist(z(n),X)=n→∞ dµ6.1TheRadon-Nikodymderivativeρ=149dm0and|ρ(n))|≥nforeveryn≥1.Passingtoasubsequenceweω(n)(zmayassumethatthelimitw=limz(n)exists.Thenw∈X⊂IRdn→∞andforeveryn≥1wehavedd(n)2(n)2zj−(aω(n))j=(zj−wj)+(wj−(aω(n))j)j=1j=1dd(n)2(n)=(zj−wj)+2(zj−wj)(wj−(aω(n))j)(6.5)j=1j=1d+(w−(a)2.jω(n))jj=1Now,sinceaω(n)∈/Wandw∈X,wegetd(w−(a)2=||w−a2≥dist2(X,∂W).(6.6)jω(n))jω(n)||j=1Fixnowq≥1solargethatforeveryn≥qd(n)212|zj−wj|≤dist(X,∂W)(6.7)4j=1andd(n)121|zj−wj|≤mindist(X,∂W),.(6.8)8d8dj=1So,if|wj−(aω(n))j|≤1,thenby(6.8)(n)122|zj−wj||wj−(aω(n))j|≤dist(X,∂W)(6.9)4dandif|wj−(aω(n))j|≥1,thenbytheotherpartof(6.8)(n)(n)22|zj−wj||wj−(aω(n))j|≤2|zj−wj||wj−(aω(n))j|12≤|wj−(aω(n))j|(6.10)4d12≤||w−aω(n)||.4d 150ConformalIteratedFunctionSystemsApplyingnow(6.5)alongwith(6.6),(6.7),(6.9),and(6.10),wegetforeveryn≥qddd(n)22(n)2zj−(aω(n))j≥(wj−(aω(n))j)−|zj−wj|j=1j=1j=1d(n)−2|zj−wj||wj−(aω(n))j|j=1212≥||w−aω(n)||−||w−aω(n)||41212−d||w−aω(n)||=||w−aω(n)||.4d2Andtherefore,usingalso(4f)and(4.2),weget||ξ−a2||ξ−a2n≤|ρ(n))|=ω(n)||≤2ω(n)||≤2K.ω(n)(zd(n)22j=1zj−(aω(n))j||w−aω(n)||Thiscontradictionfinishestheproofof(6.4).DecreasingU˜ifnecessary,wemayassumethatthissetisconnected.Now,foreveryn≥1definethefunctionbn:U˜→CIbysettingb(z)=ρh(z)|φ(ξ)|h.nωω|ω|=nSinceeachtermofthisseriesisananalyticfunctionandsince,byLemma6.1.1and(6.4),|ρh(z)||φ(ξ)|h≤Bh||φ||h≤BhKh,ωωω|ω|=n|ω|=nweconcludethatallthefunctionsbn:U˜→CIareanalyticand||bn||∞≤BhKhforeveryn≥1.Hence,inviewofMontel’stheorem,wecanchooseasubsequence{b}∞convergingonaconnectedneighborhoodnkk=1U˜1ofX(withclosureU˜1containedinU˜)toananalyticfunctionb:U˜1→CI.Sinceforeveryn≥1andeveryz∈X,b(z)=|φ(z)|h=n|ω|=nωLn(11),itthereforefollowsfromTheorem6.1.2(c)thatb|=ρ=dµ.XdmHence,puttingU=Pr(U˜),wherePr:CId→IRdistheorthogonal1projectionfromCIdtoIRd,completestheproofinthecased≥3. dµ6.1TheRadon-Nikodymderivativeρ=151dmFollowing[MPU]weshalldealnowwiththecased=2.Westartbydefiningthesequenceoffunctionsbn:V→(0,∞)bysettingb(z)=|φ(z)|h=|ψ(z)|h,(6.11)nωω|ω|=n|ω|=nwhereψω=φωifφωisholomorphicandψω=φωifφωisantiholo-morphic.InviewofLemma6.1.1|b(z)|=b(z)≤Khforallz∈Xandnnalln≥1.Hence,applyingtheKoebedistortiontheorem,weconcludethatthereexistsT>0suchthatforeachpointw∈Xthereexistsaradiusr=r(w)>0suchthatB(w,2r)⊂Vandforallz∈B(w,2r)andalln≥1|bn(z)|=bn(z)≤T.(6.12)NowequateCI,whereourcontractionsφ,i∈I,act,toIR2withco-iordinatesx,y,therealandcomplexpartofz.EmbedthisintoCI2withx,ycomplex.DenotetheaboveCI=IR2byCI.Wemayassumethat0w=0inCI.Givenω∈I∗definethefunctionρ:B(0,2r)→CIby0ωCI0settingφ(z)ωρω(z)=.φ(0)ωSinceBCI0(0,2r)⊂CI0issimplyconnectedandρωnowherevanishes,allthebranchesofthelogρωarewell-definedonBCI0(0,2r).Choosethebranchthatmaps0to0anddenoteitalsobylogρω.ByKoebe’sdistor-tiontheorem|ρω|and|argρω|areboundedonB(0,r)byuniversalcon-stantsK1,K2respectively.Hence|logρω|≤K=logK1+K2.Wewrite∞mlogρω=amzm=0andnotethatbyCauchy’sinequalities|a|≤K/rm.(6.13)mWecanwriteforz=x+iyinCI0∞mRelogρω=Ream(x+iy)m=0∞p+q=Reaiqxpyq:=cxpyq.p+qp,qqp,q=0Inviewof(6.13)wecanestimate|c|≤|a|2p+q≤Kr−(p+q)2p+q.p,qp+q 152ConformalIteratedFunctionSystemsHenceRelogρextends,bythesamepowerseriesexpansioncxpyq,ωp,qtoacomplex-valuedfunctiononthepolydiskIDCI2(0,r/2)and|Relogρω|≤4KonIDCI2(0,r/4).(6.14)Noweachfunctionbn,n≥1,extendstothefunctionB(z)=|φ(0)|hehRelogρω(z),(6.15)nω|ω|=nwhosedomain,likethedomainsofthefunctionsRelogρω,containsthepolydiskIDCI2(0,r/2).Finally,using(6.12)and(6.14)wegetforalln≥0andallz∈IDCI2(0,r/4)|B(z)|≤|φ(0)|heRe(hRelogρω(z))≤|φ(0)|heh|Relogρω(z)|nωω|ω|=n|ω|=nKhhKh≤e|φω(0)|≤eT.|ω|=nNowbyCauchy’sintegralformula(inIDCI2(0,r/4))forsecondderivat-ivesweprovethatthefamilyBnisequicontinuouson,say,IDCI2(0,r/5).HencewecanchooseauniformlyconvergentsubsequencewhoselimitfunctionGiscomplexandanalyticandextendsρonX∩B(0,r/5),inthemannerdescribedinTheorem6.1.2.Thuswehaveprovedthatforeveryw∈X,thedensityρextendstoacomplexandanalyticfunctioninanopenballcontainedinCI2andcenteredatthepointw∈X.SinceX=IntX,theseextensionsrestrictedtoCI0,coincideonintersectionsoftheseballs.Thisfinishestheproofinthedimensiond=2.
Asanimmediateconsequenceofthistheoremwegetthefollowing.Corollary6.1.4Supposethatd=1,thesystemSisregularandallcon-tractionsarereal-analytic.Moreover,supposethatthereexistsanopenconnectedsetU⊂IR2containingXandinvariantunderallelementsdµofS.ThentheRadon-Nikodymderivativeρ=hasareal-analyticdmextensiononanopenconnectedneighborhoodofXinIR.Foreveryω∈I∗denotebyD=dµ◦φωtheJacobianofthemapφωdµφω:J→Jwithrespecttothemeasureµ.AsanimmediateconsequenceofTheorem6.1.3,thefollowingcomputationdµ◦φωdµ◦φωdm◦φωdmdµδdm=··=◦φω·|φω|·dµdm◦φωdmdµdmdµandtheobservationthat|φ|δisreal-nalyticonV,wegetthefollowing.ω 6.2RateofapproximationoftheHausdorffdimension153Corollary6.1.5Foreveryi∈ItheJacobianDφihasareal-analyticextensionD˜φiontheneighborhoodUofXproducedeitherinThe-orem6.1.3orCorollary6.1.4.6.2RateofapproximationoftheHausdorffdimensionbyfinitesubsystemsLetusrecallthatinviewofProposition4.2.7,foreveryt≥0thecol-lectionζ={tlog|φ|}isaH¨olderfamilyoffunctions.Ift>θ,thentii∈IζtisasummableH¨olderfamilyoffunctions.DenotethecorrespondingmeasuremζtbymtandthepressureP(ζt)byP(t).AsintheprevioussectionwedefinetheoperatorFt:C(X)→C(X)bytheformulaF(g)=|φ|tg◦φ.tiii∈IFtactscontinuouslyonC(X).InthissectionweprovideaneffectiveestimateontheperturbationoftopologicalpressureifweformiteratedfunctionsystemsconsistingofbiggerandbiggerfinitesubsetsofI.FromTheorem4.2.13weknowthatthelimitofHausdorffdimensionsofthelimitsetsofthesefinitesystemsconvergestotheHausdorffdimensionofthelimitsetoftheoriginalsystem.Inthissection,asourmainresult,weestimate(seeTheorem6.2.3)therateofthisconvergence,thusmakingpossiblenumericalcalculationsofthisHausdorffdimension.ProceedinginamannersimilartotheproofofTheorem6.1.2wecandemonstratethefollowinggeneralizationofthattheorem.Theorem6.2.1IfSisaCIFSandt>θS,then(a)Thereexistsauniquecontinuousfunctionρt:X→[0,∞)suchthatFρ=eP(t)ρandρdm=1.tttt(b)K−t≤ρ≤Kt.t(c)Thesequence{Fn(11)}∞convergesuniformlytoρonX.tn=1tdµt(d)ρt|J=dmt,whereµtistheS-invariantversionofthemeasuremt.(e)C(X)=C(X)0⊕CIρ,tt 154ConformalIteratedFunctionSystemswhereC(X)0={f∈C(X):||Fn(f)||→0}={f∈C(X):ttfdmt=0}.IfinsteadofSweconsiderasystemSG={φi}i∈GforsomesubsetGofI,weaddthesubscriptGtotheobjectsF,P(t),m,ρ,andC0ttttresultingrespectivelyinF,P(t),m,ρ,andC0.WearenowG,tGG,tG,tG,tinapositiontoprovethefollowing.Theorem6.2.2IfS={φi}i∈IisaCIFS,thenforallG⊂Iandt∈(θS,d]wehave|eP(t)−ePG(t)|≤K3d(2+Kd)||F−F||.tG,tProof.Considerafunctionψ=rρ+u,r∈IR,u∈C0(X)andassumettthat||ψ||=1.Then1=||ψ||≥|ψdmt|=|r|ρtdmt=|r|.Hence,usingalsoTheorem6.2.1(b),weconcludethat||u||≤||ψ||+|r|·||ρ||≤1+Kd.Thust1d1||rρt+u||=1=(1+K+1)≥(||u||+|r|).(6.16)2+Kd2+KdIfwenowrelaxtheassumption||ψ||=1butstillkeepψ=0,thenitfollowsfrom(6.16)thatψ1|ur≥+||ψ||2+Kd||ψ||||ψ||andtherefore||ψ||≥1(||u||+|r|),sothat(6.16)remainstrueinthis2+Kdcase.Ifψ=0,thenr=0,u=0,and(6.16)isobviouslysatisfiedinthiscase.InviewofTheorem6.2.1(e)thereisauniquerepresentationρG,t=rρt+uwithappropriater∈IRandu∈C0(X).InviewofTheorem6.2.1(b)tweget−dK≤ρG,tdmt=(rρt+u)dmt=rρtdmt=r.(6.17)Writeλ=eP(t)andλ=ePG(t).ApplyingTheorem6.2.1(b),tG,tTheorem6.2.1(a),L-invariantnessofthespaceC0(X),(6.16)and(6.17),tt 6.2RateofapproximationoftheHausdorffdimension155wegetd||Ft−FG,t||K≥||Ft−FG,tρG,t||=||rFt(ρt)+Ft(u)−λG,tρG,t||=||(rλtρt−rλG,tρt)+Ft(u)−λG,tu||1≥dr|λt−λG,t|·||ρt||+||Ft(u)−λG,tu||2+K1−d−d≥KK|λt−λG,t|.2+KdThus|λ−λ|≤K3d(2+Kd)||F−F||andtheproofistG,ttG,tcomplete.
Let1χ=supinf−log||φω||.n≥1ω∈InnThemainresultofthissection,comingmainlyfrom[HeU]andculmin-atingtheapproachbegunin[GM]andcontinuedin[MU1],establishingtherateofapproximationoftheHausdorffdimensionh=HD(JS)ofthelimitsetJSofthesystemSbytheHausdorffdimensionsofthelimitsetsofitsfinitesubsystems,isincludedinthefollowing.Theorem6.2.3Ifγ>θSandhG≥γforsomefinitesetG⊂I,then0≤h−h≤χ−1K3d(2+Kd)||φ||γ.Gii∈IGProof.ApplyingProposition2.6.13alongwithBirkhoff’sergodicthe-oremforthefunctionf(ω)=log|φ(π(σ(ω)))|,ω∈I∞,wegetω1dP(t)=log|φω1(π(σ(ω)))|dµ˜t≤−χ,dtwhere˜µisthelifttothecodingspaceI∞ofthemeasureρm.HencettthGhGP(hG)P(hG)P(h)dP(t)P(t)e−1=e−e=edt=P(t)edthdth(6.18)hh=−−P(t)eP(t)dt≥χdt=χ(h−h).GhGhG 156ConformalIteratedFunctionSystemsOntheotherhand,usingTheorem6.2.2,weobtaineP(hG)−1=eP(hG)−ePG(hG)≤K3d(2+Kd)||F−F||hGG,hG=K3d(2+Kd)||φ||hGii∈IG≤K3d(2+Kd)||φ||γ.ii∈IGCombiningthisinequalityand(6.18)completestheproof.
Noticethattheassumptionsγ>θSandhG≥γimplythath≥hG>θS,andconsequentlythesystemSisstronglyregular.Ontheotherhand,ifthethesystemSisstronglyregular,thenthereexistafiniteG⊂Iwiththeseproperties.6.3UniformperfectnessAcompactsetL⊂CIiscalleduniformlyperfectifthereexistsaconstant0nw.Letpbealeastsuchindexk.Ifdiam(φ(J))≥r/8D2C,thenusingthefactthatφ(J)⊂B(w,r),wejpjpconcludethatA(z,r/16D2C,r)∩φ(J)=∅.Butsinceφ(J)⊂J,wejpjpgetA(z,r/16D2C,r)∩J=∅andwearedoneinthiscasewithanyconstantc≤1/16D2C.Sosupposethatdiam(φ(J))≤r/8D2C.Thenby(6.19),diam(φ(X))≤r/8C.jpjpSo,bythedefinitionofw,rrdist(φjp(X),φjp−1(X))0andc=min{1/2,1/16D2C}.wa1Considernowanarbitrarypointz∈φ(J)forsomei∈Iand4ic1diam(φi(X))0212appropriatelysmaller,ifnecessary,wehavelocaluniformperfectnessatthepointzforeveryrsatisfying4diam(φ(X))0.2(6.22)InparticularA(x,(2C2)−1/hr,r)∩J=∅andtheproofiscomplete.
S6.4GeometricrigidityInthissection,following[MMU]weexplorethestructureoflimitsetsJofinfiniteconformaliteratedfunctionsystemswhoseclosureisacon-tinuum(compactconnectedset).Underanaturaleasilyverifiabletech-nicalcondition(alwayssatisfiedifthesystemisfinite),wedemonstratethefollowingdichotomy.EithertheHausdorffdimensionofJexceeds1orelseJisapropercompactsegmentofeitherageometriccircleorastraightlineifd≥3orananalyticintervalifd=2(cf.Theorem6.4.1).Fromtheviewpointofconformaldynamics,thisresultcanbethoughtofasafargoinggeneralizationofresultsoriginatedin[Su1]and[B2],whichareformulatedintheplanecase.TheproofscontainedthereusetheRiemannmappingtheoremandcanbecarriedoutonlyintheplane.Theproofpresentedinourpaperisdifferentandholdsinanydimen-sion.Thereaderisalsoencouragedtonoticeananalogybetweenourresultandaseriesofotherpapers(seee.g.([B2],[FU],[MU2],[Ma],[Pr],[Ru],[Su1],[U1],[UV],[Z1],[Z2])whichareaimedtowardestab-lishingasimilardichotomy.However,toourknowledge,alltheseresultslikethosein[B2]and[Su1],wereformulatedintheplaneandusedthe 6.4Geometricrigidity161Riemannmappingtheorem,exceptthosein[MU2].Thecurrentresultishowevermuchstrongerthanthatin[MU2]andinparticularwithourpresentapproachthemainresultof[MU2]canbestrengthenedasde-scribedattheendofthissection.Anothercorollaryofourresultisthefollowing:ifacontinuumCinIRdistheself-conformalsetgeneratedbyfinitelymanyconformalmappingssatisfyingtheopensetcondition,theHausdorff1-measureofCisfiniteandoneofthemappingsisasim-ilarity,thenthecontinuumisaline-segment.Inparticular,thisholdsifallthemapsaresimilarities,aresultobtainedearlyonbyMattila[Ma3].Themainresultofthissectionisthefollowing.Theorem6.4.1Ifd≥3,S={φi}i∈IisaCIFS,Jisacontinuum(compactconnectedset)anddimH(S(∞))1or(b)Jisapropercompactsegmentofeitherageometriccircleorastraightline.Inaddition,ifanyoneofthemapsφiisasimilaritymapping,thenJisalinesegment.WenotethatthetechnicalconditioninTheorem6.4.1isnecessary.AmodificationofExample5.2of[MU1]showsthatthedichotomyofThe-orem6.4.1ingeneralfailsifdimH(S(∞))≥dimH(J).Hereissuchanexample.Takeanysequenceofpositivereals{ri:i≥1}(forexampleoftheformbi,01or(b)Jisapropercompactsegmentofeitherageometriccircleorastraightline.Inaddition,ifanyoneofthemapsφiisasimilaritymapping,thenJisalinesegment.TheproofofTheorem6.4.1willconsistofseveralsteps.FirstofallweassumefromnowonthroughouttheentiresectionthattheassumptionsofTheorem6.4.1aresatisfiedanddimH(J)=1.Ourgoalistoshowthatthenitem(b)issatisfied.SincedimH(S(∞))0.ItthereforefollowsfromTheorem4.5.11thatthesystemSisregular.Letmbethecorresponding1-dimensionalmeasure.ByTheorem4.5.1andsincedimH(S(∞))0,weputCon(x,θ,γ)=Con(x,η,γ)∪Con(x,−η,γ),whereη∈IRdisarepresentativeofθ∈IPIRd.WerecallthatasetYhasatangentinthedirectionθ∈IPIRdatapointx∈Yifforeveryγ>0H1Y∩(B(x,r)Con(x,θ,γ))lim=0.r→0rSincewewillconsideronlytangentsof1-sets(thesetJabove),thisdefinitioncoincideswiththedefinitiongivenonp.31of[Fa1].WesaythatasetYhasastrongtangentinthedirectionθ∈IPIRdatapointxprovidedforeach0<β≤1thereissomer>0suchthatY∩B(x,r)⊂Con(x,θ,β).Weshallprovethefollowing.Theorem6.4.4IfYislocallyarcwiseconnectedatapointxandYhasatangentθatx,thenYhasstrongtangentθatx. 6.4Geometricrigidity163Proof.Supposethereissome0<β<1andpointsxninYsuchthatforeachn,|xn−a−θ|>β|xn−x|.Foreachn,letαn:[0,1]→Ybeanarcfromxtoxnwithdiam(αn)→0.Foreachn,notethatsinceYhasatangentθ∈IPIRdatx,thereissomet,0yn−x/2,then,consideringtheprojectionofthissubarconthespherewithcenteraandradiusyn−a/2,wegetH1(X∩B(x,yn−x))≥(π/2)(β/2)yn−a.Con(x,θ,β/2)Thus,Ydoesnothaveatangentatx.
Wecallapointτ∈I∞transitiveifω(τ)=I∞,whereω(τ)istheω-limitsetofτundertheshifttransformationσ:I∞→I∞.WedenotethesetofthesepointsbyI∞andputt∞Jt=π(It).WecalltheJtthesetoftransitivepointsofJandnoticethatforeveryτ∈I∞,theset{π(σnτ):n≥0}isdenseinJ(orJifthisisthespacetunderconsideration).Lemma6.4.5IfJhasastrongtangentatapointx=π(τ),τ∈I∞,thenJhasastrongtangentateverypointofπ(ω(τ)).Proof.SupposeonthecontrarythatJdoesnothaveastrongtangentatsomepointy∈π(ω(τ)).Letθ∈IPIRdbethetangentdirectionofJatxandlet{n}∞beanincreasingsequenceofpositiveintegerssuchkk=1thatlimπ(σnkτ)=y.Passingtoasubsequence,wemayassumek→∞that−1φ(x)ω|nklimθ=ξk→∞|φ−1(x)|ω|nk 164ConformalIteratedFunctionSystemsforsomeξ∈IPIRd.SinceJdoesnothaveastrongtangentaty,thereexists0<β≤1suchthatforeveryr>0J∩B(y,r)J∩Con(y,ξ,β)=∅.ThenJ∩B(π(σnkτ),r)J∩Con(π(σnkτ),ξ,β/2)=∅(6.23)kforallklargeenoughwhere−1φ(x)ω|nkξk=θ.−1|φ(x)|ω|nkButinviewofProposition4.2.3,forallr>0smallenoughwehaveφB(π(σnkτ),r)Con(π(τ),ξ,β/2)⊂ω|nkkφ(π(σnkτ))ω|nkβ⊂Bx,r||φω|nk||Conx,|φ(π(σnkτ))|ξk,4ω|nk=Bx,r||φω|n||Con(x,θ,β/4).kSinceinviewof(6.23),J∩φB(π(σnkτ),r)=∅,weconcludethatω|nkforeveryklargeenough,J∩Bx,r||φ||Con(x,θ,β/4)=∅.ω|nkSincelim||φ||=0,thisimpliesthatθisnotthestrongdensityk→∞ω|nkdirectionofJatx.Thiscontradictionfinishestheproof.
Corollary6.4.6ThecontinuumJhasastrongtangentateverypoint.Proof.SinceH1(J)<∞,Corollary3.15from[Fa1]showsthatJhasatangentatH1-a.e.pointinJ,andthereforeatasetofpointsofpositivemmeasure.Sincem(Jt)=1,therethusexistsatleastonetransitivepointxinJhavingatangentofJ.ByTheorem6.4.4andLemma6.4.3Jhasastrongtangentatx,anditthenfollowsfromLemma6.4.5thatJhasastrongtangentateverypoint.
Now,thefollowinglemmafinishestheproofofTheorem6.4.1.Lemma6.4.7Supposethatφ:IRd→IRd,d≥3,isaconformaldiffeomorphismthathasanattractingfixedpointa(φ(a)=a,|φ(a)|<1).SupposethatacompactconnectedsetMhasastrongtangentata,thatφ(M)⊂Mandthatlimφn(x)=aforallx∈M.ThenMn→∞ 6.5Refinedgeometricrigidity165isasegmentofaφ-invariantlineorcircle.Ifφisaffine(φ(∞)=∞),thentheformerpossibilityholds.Proof.Sinceaisanattractingfixedpointofφ,thereexistsaradiusdddr>0sosmallthatφ−1RB(a,r)⊂RB(a,r),whereRistheAlexandrovcompactificationofIRddonebyaddingthepointatinfinity.dSinceRB(a,r)isatopologicalclosedball,inviewofBrouwer’sfixed−1dpointtheoremthereexistsafixedpointbofφinRB(a,r).Hencebisalsoafixedpointofφandb=a.Thenthemapψ=ib,1◦φ◦ib,1(ib,1equalsidentityifb=∞)fixes∞,whichmeansthatthismapisaffine,andw=ib,1(a)isanattractingfixedpointofψ.Inadditionψ(M˜)⊂M˜,whereM˜=ib,1(M),w∈M˜,andM˜hasastrongtangentatw.LetlbethelinethroughwdeterminedbythestronglytangentdirectionofM˜atw.Sinceψ(w)=w,sinceψ(l)isastraightlinethroughwandsinceψ(M˜)⊂M˜,weconcludethatψ(l)=l.SupposenowthatM˜isnotcontainedinl.Considerx∈M˜l.Thenforeveryn≥0ψn(x)∈ψ(M˜)ψ(l)⊂M˜landsincethemapψisconformalandaffine∠(ψn(x)−w,l)=∠(ψn(x−w),ψn(l))=∠(x−w,l).Sincelimψn(x)=w,wethereforeconcludethatlisnotastronglyn→∞tangentlineofM˜atw.ThiscontradictionshowsthatM˜⊂l.SinceinadditionM˜isacontinuum,itisasegmentofl.
AndindeedtoconcludetheproofofTheorem6.4.1itsufficestopickanarbitraryindexi∈I(affineifexists)andtoputφ=φi,M=Janda=xi,theonlyattractingfixedpointofφibelongingtoJ.
6.5RefinedgeometricrigidityThroughoutthesectionweassumethatd≥2.ByTDwewilldenotethetopologicaldimension(wewillonlydealwithsubsetsofIRdsoallHausdorffandtopologicaldimensionsarefinite).Thefollowingisthemainresultofthissection.Itrefinesthemainresultoftheprevioussection,cf.[MyU]. 166ConformalIteratedFunctionSystemsTheorem6.5.1Ifd≥3,S={φi}i∈IisaCIFSandHD(S(∞))TD(J)or(b)Jisapropercompactsubsetofeitherageometricsphereofdi-mensionTD(J)oraTD(J)-dimensionalaffinehyperspace,bothcontainedinIRd.Inaddition,ifanyoneofthemapsφiisasimilaritymap,thenthelattercaseholds.SinceinthefinitecasethesetS(∞)isempty,wegetimmediatelythefollowing.Corollary6.5.2Ifd≥3,S={φi}i∈IisafiniteCIFS,theneither(a)HD(J)>TD(J)or(b)Jisapropercompactsubsetofeitherageometricsphereofdi-mensionTD(J)oraTD(J)-dimensionalaffinehyperspace,bothcontainedinIRd.Inaddition,ifanyoneofthemapsφiisasimilaritymap,thenthelattercaseholds.Remark6.5.3Putk=TD(J).Sinceacompactsubsetofak-dimen-sionalsphereorhyperspaceGhastopologicaldimensionkifandonlyifitsinteriorinGisnotempty,weseethatinthesecondalternativeofTheorem6.5.1andCorollary6.5.2,Jcontainsanopenballintheappropriatesphereorhyperspaceand,fordynamicalreasons,itturnsoutthatthereisanopensubsetΩofthatsphereorhyperspacesuchthatJ=Ω.Wefirstdiscusssomeconceptsandresultsfromrectifialibitytheory.Abeautifulexpositionofthiscanbefoundin[Ma1].AsetQ⊂IRdiscalledk-rectifiableifHk(Q)>0andthereexistLipschitzmapsg:IRk→IRd,ii=1,2,...,suchthat
∞HkQg(IRk)=0.ii=1AsetT⊂IRdiscalledpurelyk-unrectifiableifandonlyifHk(Q∩T)=0foreveryk-rectifiablesetQ.kItfollowsfromTheorem15.19in[Ma1]thatforH-a.e.pointzinak-rectifiablesetQ⊂IRdthereisauniqueapproximatetangentk-plane 6.5Refinedgeometricrigidity167forQatz.ThistangentplanewillbedenotedinthesequelbyTzQasasubsetofG(d,k).WerecallthatG(d,k)istheGrassmannianmanifoldofallk-dimensionallinearsubspacesofIRdandthattheexistenceofatangentk-planeTzQforQatzimpliesthat,forevery0k0,HD(J)=TD(J),thenthesystemSisregular,m=H|istheHk(J)Jk-conformalmeasureonJandtheclosureJSisTD(JS)-rectifiable.kProof.Putk=TD(J).SinceH(J)>0andsinceHD(J)=k,weconcludefromTheorem4.5.11thatthesystemSisregularand,usingkkTheorem4.5.1,wededucethatH(J)<∞andm=H|istheHk(J)Jk-conformalmeasureonJ.ItfollowsfromFederer’stheoremonp.545in[Fe]thattheintegralgeometricmeasureIk(J)>0.Since(see[Ma1],1p.86)Ik(J)=H0(J∩P−1(a))dHk(a)dγ(V),1Vd,kG(d,k)VwethereforeconcludethatthereexistsaBorelsetG⊂G(d,k)with0−1γd,k(G)>0suchthatH(J∩PV(a))>0foreveryV∈GandallakinsomesetWV⊂VwithH(WV)>0.InparticularPV(J)⊃WVkandthereforeH(PV(J))>0forallV∈G.Hence,itfollowsfromTheorem18.1(2)onp.250in[Ma1]thatJisnotpurelyk-unrectifiable.Therefore,combiningTheorem17.6(noticethatalthoughthisisnotkindicatedinMattilas’sbook,weneedtoknowthatH(J)>0forthistheoremtomakeactuallysense),Theorem6.2(1)in[Ma1]andthefactthatHk(J)=Hk(J)>0,weconcludethatΘk(J,x)=1forallxinsomesetF⊂JwithHk(F)>0,wherethedensityfunctionsΘkaswellasΘkandΘ∗kweredefinedonp.89in[Ma1].Fixnowx∈J.It∗ 168ConformalIteratedFunctionSystemsfollowsfromtheboundeddistortionproperty(4f)thatforalli∈Iandallr>0smallenoughHk(J∩B(φ(x),|φ(x)|r))≥Hk(φ(J∩B(x,K−1r)))iiir≥K−k|φ(x)|hHk(J∩B(x,K−1r)),rirwhereKrissuchthatlimr→0Kr=1.HenceHk(J∩B(φ(x),|φ(x)|r))K−k|φ(x)|kii≥riHk(J∩B(x,K−1r))(2|φ(x)|r)k(2|φ(x)|r)kriiHk(J∩B(x,K−1r))=K−2kr.r(2K−1krr)andlettingr0weconcludethatΘk(J,φ(x))≥Θk(J,x).(6.24)∗i∗Let˜mbetheliftoftheconformalmeasuremtothecodingspaceI∞andlet˜µbeitsshift-invariantversionproducedinTheorem2.2.4.Sincebythistheoremthedynamicalsystem(σ,µ˜)isergodic,itthereforefollowsfromBirkhoff’sergodictheoremand(6.24)thatthefunctionω→Θk(J,π(ω))isconstant˜µ-a.e.Since˜µ(π−1(F))>0,wetherefore∗concludethatΘk(J,π(ω))=1for˜µ-a.e.ω∈I∞.ThusΘk(J,x)=1∗∗kforH-a.e.x∈J.CombiningthiswithTheorem6.2(1)in[Ma1]weseethatΘk(J,x)existsandisequalto1forHk-a.e.x∈J.InvokingnowTheorem17.6(1)in[Ma1]finishestheproof.
WenowpassdirectlytotheproofofTheorem6.5.1.Putk=TD(JS)andsupposethatHD(J)≤k.SinceHD(S(∞))0.ThustheassumptionsofLemma6.5.4aresat-isfied.InviewofthislemmathesetJisk-rectifiable.ByTheorem15.19kin[Ma1]thisequivalentlymeansthatforH-a.e.z∈Jthereisauniqueapproximatetangentk-planeTzJforJatz.Wefixnowsuchapoint,sayz=π(ω)∈J,ω∈I∞,andmakethe0followingrenormalization.Setλ=|φ(π(σn(ω)))|−1anddefinethennω|nβn(z)=λn(z−z0).Itfollowsfromtheboundeddistortionproperty(4f)thateachmap-pingβ◦φ:X→IRdislocallyLipschitzcontinuouswithLipschitznω|n 6.5Refinedgeometricrigidity169constantKandfrom(4.20)thatβn◦φω|n(X)⊂B(0,DK).There-fore,theAscoli-Arzel`atheoremappliesandthereexistsanunboundedincreasingsequenceofpositiveintegers{n}∞suchthatthesequencejj=1ψ:X→IRdconvergesuniformlytoacontinuousfunctionΨ:X→IRd,jwhereψj=βnj◦φω|n.ThelimitfunctionΨ:X→Xisconformal.jWeshallprovethefollowing.Claim6.5.5Ψ(J)⊂Tz0J.Proof.Supposeonthecontrarythattheclaimdoesnothold.ThenthereexistsanopenboundedsetΩ⊂Ψ(J)suchthatη=dist(Ω,IRk)>0.(6.25)SinceΩisanopensubsetofJ,wegetHk(Ω)>0.PutU=Ψ−1(Ω)andUj=fj(U).Then00andj0≥1suchthat0<τ≤Hk(β(U))=λkHk(U)(6.26)njjnjjforallj≥j0.Dueto(6.25)andtheboundednessofΩwecanchoose00njjjnjjandthiscontradicts(6.27).Wethusprovedtheclaimandthereforethe“smoothorfractal”dichotomyannouncedinTheorem6.5.1.Wearelefttoshowthatifoneofthemapsφiisasimilaritymap(λiAi+ai,0<λi<1),thenJiscontainedinak-dimensionalhy-perspaceofIRd.Andindeed,supposeonthecontrarythatJ⊂Q, 170ConformalIteratedFunctionSystemsageometricsphereinIRd.Sinceφ(Q)=λA(Q)+aisageomet-iiiiricsphereofdimensionkandthesphereQ∩φi(Q)containsthek-dimensionalsetJ,thisintersectionisak-dimensionalsphere,andthere-foreequaltobothQandφi(Q).Thiscontradictsthefactthatφi:Q→QisastrictcontractionwithaLipschitzconstantequaltoλi.
7DynamicalRigidityofCIFSsInthischapterwedealwithdynamicalrigiditystemmingfromtheworkofSullivan(see[Su3])onconformalexpandingrepellersinthecomplexplane.Weaskthefundamentalquestionwhentwotopologicallycon-jugateinfiniteiteratedfunctionsystemsareconjugateinasmootherfashion.Theansweristhatsuchconjugacyextendstoaconformalcon-jugacyonsomeneighborhoodsoflimitsetsifandonlyifitisLipschitzcontinuous.Thisturnsouttoequivalentlymeanthatthisconjugacyexchangesmeasureclassesofappropriateconformalmeasuresorthatthemultipliersofcorrespondingfixedpointsofallcompositionsofourgeneraterscoincide.7.1GeneralresultsInthissectionwepresentthegeneralrigidityresults.WemakenoassumptionaboutthespaceXandthedimensiond.WecalltwoiteratedfunctionsystemsF={fi:X→X,i∈I}andG={gi:Y→Y,i∈I}topologicallyconjugateifandonlyifthereexistsahomeomorphismh:JF→JGsuchthath◦fi=gi◦hforalli∈I.Thenbyinductionweeasilygetthath◦fω=gω◦hforeveryfinitewordω.Thefirstresultisthefollowing.Theorem7.1.1SupposethatF={fi:X→X}i∈IandG={gi:Y→Y}i∈Iaretwotopologicallyconjugateconformaliteratedfunctionsystems.Thenthefollowingfourconditionsareequivalent.171 172DynamicalRigidityofCIFSs(1)∃C≥1∀ω∈I∗−1diam(gω(Y))C≤≤C.diam(fω(X))(2)|g(y)|=|f(x)|forallω∈I∗,wherexandyaretheonlyωωωωωωfixedpointsoffω:X→Xandgω:Y→Yrespectively.(3)∃E≥1∀ω∈I∗||g||E−1≤ω≤E.||f||ω(4)ForeveryfinitesubsetTofI,HD(JG,T)=HD(JF,T)andtheconformalmeasuresmandm◦h−1areequivalent.G,TF,TSupposeadditionallythatbothsystemsFandGareregular.Thenthefollowingconditionisalsoequivalenttothefourconditionsabove.(5)HD(J)=HD(J)andtheconformalmeasuresmandm◦h−1GFGFareequivalent.Proof.Letusfirstdemonstratethatconditions(2)and(3)areequi-valent.Indeed,supposethat(2)issatisfiedandletKFandKGdenotethedistortionconstantsofthesystemsFandGrespectively.Thenforallω∈I∗,||g||≤K|g(y)|=K|f(x)|≤K||f||andsimilarlyωGωωGωωGω||f||≤K||g||.Sosupposethat(3)holdsand(2)fails,thatisthatωFωthereexistsω∈I∗suchthat|g(y)|=|f(x)|.Withoutloossofωωωωgeneralitywemayassumethat|g(y)|<|f(x)|.Foreveryn≥1letωωωωnnωbetheconcatenationofnwordsω.Thengωn(yω)=gω(yω)=yω∞andsimilarlyfωn(xω)=xω.So,xωn=xω=πF(ω)andyωn=yω=π(ω∞).Moreover|g(y)|nand|f(x)|n.Gωn(yω)|=|gωωωn(xω)|=|fωωHence|gωn(yω)|lim=0.n→∞|fωn(xω)|Ontheotherhand,by(3)andtheboundeddistortionproperty|gK−1||gωn(yω)|≥Gωn||≥E−1K−1|f)|||fGωn(xωωn||foralln≥1.Thiscontradictionfinishestheproofofequivalenceofcon-ditions(2)and(3).Sincetheequivalenceof(1)and(3)followsimmedi-atelyfromtheboundeddistortionproperty,theproofoftheequivalenceofconditions(1)–(3)isfinished.Weshallnowprovethat(3)⇒(5).Indeed,itfollowsfrom(3)thatE−1ψ(t)≤ψ(t)≤Eψ(t)forG,nF,nG,nallt≥0andalln≥1.HencePG(t)=PF(t)andthereforebyThe-orem4.2.13,HD(JG)=HD(JF).Denotethiscommonvaluebyh. 7.1Generalresults173AlthoughwekeepthesamesymbolforthehomeomorphismestablishingconjugacybetweenthesystemsFandG,itwillnevercausemisunder-standings.Supposenowthatbothsystemsareregular(infactassuming(3)regu-larityofoneofthesesystemsimpliesregularityoftheother).Thenforeveryω∈I∗K−h||f||hm(f(J))||f||h(KE)−h≤Fω≤FωF≤ω≤(EK)h.F||g||hm(g(J))−hhGωGωGKG||gω||So,themeasuresmandm◦h−1areequivalent,andevenmoreGFdm◦h−1(KE)−h≤F≤(EK)h.FGdmGLetusshownowthat(5)⇒(3).Indeed,if(5)issatisfiedthenthemeasureµ◦h−1isequivalenttoµ.Sinceadditionallyµ◦h−1FGFandµGarebothergodic,theyareequal.Hence,usingtheequalityHD(JF)=HD(JG):=h,weget
hh||gω|||gω|dmG=mG(gω(JG))µG(gω(JG))=µ◦h−1(g(J))=µ(f(J))m(f(J))FωGFωFFωF
hh=|fω|dmF ||fω||andraisingthefirstandthelasttermofthissequenceofcomparabilitiestothepower1/h,wefinishtheproofoftheimplication(5)⇒(3).Theequivalenceof(4)andconditions(1)–(3)isnowarelativelysimplecorollary.Indeed,toprovethat(3)implies(4)fixafinitesubsetTofI.By(3)E−1≤||f||/||g||≤Eforallω∈T∗,andaseveryfinitesystemωωisregular,theequivalenceofmeasuresmandm◦h−1followsG,TF,Tfromtheequivalenceofconditions(3)and(5)appliedtothesystems{f:i∈T}and{g:i∈T}.Ifinturn(4)holdsandω∈I∗,theniiω∈T∗,whereTisthe(finite)setoflettersmakingupthewordω,andthemeasuresmandm◦h−1areequivalent.Hence,bytheG,TF,Tequivalenceof(2)and(5)appliedtothesystems{fi}i∈Tand{gi}i∈Tweconcludethat|g(y)|=|f(x)|.Thus(2)followsandtheproofofωωωωTheorem7.1.1isfinished.
WesaythataCIFS{φi:X→X:i∈I}isofboundedgeometryifandonlyifthereexistsC≥1suchthatforalli,j∈I,i=jmax{diam(φi(X)),diam(φj(X))}≤Cdistφi(X),φj(X). 174DynamicalRigidityofCIFSsThenexttheoremprovidesanecessaryandsufficientconditionfortwosystemsofboundedgeometrytobebi-Lipschitzequivalent.Theorem7.1.2IfbothsystemsF={fi:X→X}i∈IandG={gi:Y→Y}i∈Iareofboundedgeometry,thenthetopologicalconjugacyh:JF→JGisbi-Lipschitzcontinuousifandonlyifthefollowingtwoconditionsaresatisfied.−1diam(fω(X))Q≤≤Q(7.1)diam(gω(Y))forsomeQ≥1andallω∈I∗.−1distgi(Y),gj(Y)D≤≤D(7.2)distfi(X),fj(X)forsomeD≥1andalli,j∈IN,i=j.Proof.Firstnoticethat(7.1)and(7.2)remaintrue,withmodifiedconstantsQandDifnecessary,ifXisreplacedbyJFandYisreplacedbyJG.Supposenowthatx∈fi(JF)andy∈fj(JF)withi=j.Then|h(y)−h(x)|≤diam(gi(JG))+dist(gi(JG),gj(JG))+diam(gj(JG))≤Qdiam(fi(JF))+Ddist(fi(JF),fj(JF))+Qdiam(fj(JF))≤2QCdist(fi(JF),fj(JF))+Ddist(fi(JF),fj(JF))≤(2QC+D)dist(fi(JF),fj(JF))≤(2QC+D)|y−x|.Supposeinturnthatx=ybothbelongtothesameelementfk(JF).Thenthereexistω∈I∗(|ω|≥1)andi=j∈INsuchthatx,y∈f(J),ωFx∈fωi(JF)andy∈fωj(JF).Fromwhathasbeenprovedsofarweknowthat|g−1(h(y))−g−1(h(x))|≤(2QC+D)|f−1(y)−f−1(x)|.Sinceωωωω|y−x| ||f||·|f−1(y)−f−1(x)|and|h(y)−h(x)| ||g||·|g−1(h(y))−ωωωωωg−1(h(x))|,wegetω||g||ω|h(y)−h(x)||y−x| |y−x|,||f||ωwherethecomparabilitysignwecanwritebecauseof(7.1)andequival-enceofconditions(1)and(3)ofTheorem7.1.1.Inthesamewayweshowthath−1isLipschitzcontinuouswhichcompletestheproofofthefirstpartofourtheorem. 7.1Generalresults175Sosupposenowthathisbi-Lipschitzcontinuous.Weshallshowthatconditions(7.1)and(7.2)aresatisfied.Indeed,toprove(7.1)supposethataandbinJaretakensothat|h(a)−h(b)|≥1diam(g(J)).F2ωGThendiam(gω(JG))≤2|h(a)−h(b)|≤2L|a−b|≤2Ldiam(fω(JF)),whereLisaLipschitzconstantofhandh−1.Inthesamewayitcanbeshownthatdiam(fω(JF))≤2Ldiam(gω(JG))whichcompletestheproofofproperty(7.1).Inordertoprovetheright-handsideofproperty(7.2)weproceedasfollows.Fixi,j∈I,i=janda=b∈JF.Thendist(gi(Y),gj(Y))≤dist(gi(JG),gj(JG))≤|gi(h(a))−gj(h(b))|≤L|fi(a)−fj(b)|≤Ldiam(fi(X))+dist(fi(X),fj(X))+diam(fj(X))≤L(2C+1)dist(fi(X),fj(X)),wherethelastinequalityfollowsfromboundednessofgeometryofthesystem{fi}i∈I.
Remark7.1.3SupposenowthatI=INandthatthemapsi→φi(X)aremonotone,thatissupposethatforalliandj,i0andn≥nsolargethatforalln≥nandallω∈In101−δ−δsup{|ψλ(ω)◦φω|}−inf{|ψλ(ω)◦φω|}<;/M.Then,using(7.12),weconcludethatforalln≥n1andallz1,z2∈Uβcδ|ψ(φ(z))|−δ−cδ|ψ(φ(z))|−δ≤;β,ωλ(ω)ω2β,ωλ(ω)ω1|ω|=nandthereforelimcδψ(φ(z))|−δ−cδψ(φ(z))|−δ=0.n→∞β,ωλ(ω)ω2β,ωλ(ω)ω1|ω|=nCombiningthis,(7.12)and(7.13)weconcludethatthereexistsacon-stantcβ≥0suchthatforallz∈Uβlimcδ|ψ(φ(z))|−δ=c.β,ωλ(ω)ωβn→∞|ω|=nCombininginturnthis,(7.12)and(7.13)weconcludethatforallz∈Uβρ(z)=c|ψ(z)|δ.(7.15)ββFixnowi∈I,w∈Uβ∩J,andchooseλ∈Λsuchthatφi(w)∈UλandaconnectedneighborhoodVw⊂Uβofwsuchthatφi(Vw)⊂Uλ.Thenforeveryz∈VwD˜(z)=ρ◦φ(z)|φ(z)|δρ(z)−1=c|ψ(φ(z))|δ·|φ(z)|δ·c−1|ψ(z)|−δφiiiλλiiββ−1−1δ=cλcβ|ψλ(φi(z))|·|φi(z)|·|ψβ(z)| 188DynamicalRigidityofCIFSsandtherefore,sinceoursystemSisaffine,D˜φiisconstantonVw.Since,byTheorem2.2,D˜φiisreal-analyticonU,wethusconcludethatD˜φiisconstantonU.Theproofoftheimplication(d2)⇒(a)isfinished.•(d2)⇒(eh).WecanassumethesetsUtappearingincondition(d2)areopenballs.SinceJiscompact,wemaychoosefromthefamily{Ut}afinitesubcover{Bλ}λ∈ΛofJ.Definethenforeveryλ∈Λthemapγ:B→IRtobeacontinuousbranchofargψandadditionallyforλλλeveryi∈I,argφ:B→IRtobeacontinuousbranchofargumentλiλofφ.ThesebranchesexistsinceBissimplyconnectedandψandiλλφnowherevanish.Ofcourseallthemapsγ,λ∈Λ,areharmonic.iλConsidernowtwoindicesλ,λ∈ΛsuchthatBλ∩Bλ=∅.Since−1ouratlasisaffine,ψλ(z)=ψλ◦ψλ(ψλ(z))=a(ψλ(z))+bforallz∈Bλ∩Bλandsomea,b∈CI.Weconcludethatγλ−γλisonBλ∩Bλequaltoarg(a)uptoanintegermultipleof2π.Thismeansthat(7.7)issatisfied.Sinceallthecontractions{φi}i∈Iareaffineintheatlasψ:B→CI,weconcludethatgivenλ,λ∈Λ,i∈Ithereexistλλ−1constantsd,c∈CIsuchthatforeveryz∈φi(Bλ∩φi(Bλ))−1ψλ◦φi(z)=ψλ◦φi◦ψλ(ψλ(z))=dψλ(z)+c.Weconcludethatargλφi−γλ+γλ◦φiisequaltoarg(d)uptoaninteger−1multipleof2πontheconnectedsetφi(Bλ∩φi(Bλ)).Thismeansthat(7.8)issatisfied.Thustheproofoftheimplication(d2)⇒(eh)iscomplete.•Theimplications(eh)⇒(er)⇒(ec)areobvious.•(ec)⇒(d2).Thegeneralideaisherethesameasintheproofoftheimplication(c)⇒(d1).Surprisingly,wedonotgetdirectly(c)⇒(d1).Forthisweneedtogovia(d2)⇒(a)⇒(d1).Let4δ>0beaLebesguenumberofthecover{Bλ}λ∈ΛofJ.BycompactnessofJthereexistsafinitesetTandpointsvt∈J,t∈T,suchthatthefamily{B(vt,δ)}t∈TisacoverofJ.Since4δisaLebesguenumberofthecover{Bλ}λ∈Λ,foreveryt∈Tthereexistsatleastoneelementλ(t)∈ΛsuchthatB(v,2δ)⊂B.Fixnowt∈T,τ∈I∞,tλ(t)0thatissimilarlyasintheimplication(c)⇒(d1).Thenforeachintegern≥1choosetn∈Tsuchthatφτ|n(vt0)∈B(vtn,δ).Sinceφτ|nonB(vt0,δ)shrinksdistancesbyafactoratleasts<1forn≥1,wegetφτ|n(B(vt0,δ))⊂B(vtn,(1+s)δ).Now,foreveryi∈Iandeveryλ∈Λletargφ:B→IRbeacontinuousbranchofargumentofφ.Itλiλi 7.3Two-dimensionalsystems189followsfrom(4.5)that|arg(t)φ(y)−arg(t)φ(x)|≤L|y−x|λiλiforallt∈T,alli∈Iandallx,y∈B(vt,δ),whereL=6/d.Henceforallz∈B(vt0,δ)|argφ(φ(z))−argφ(φ(v))|λ(tn−1)τnτ|n−1λ(tn−1)τnτ|n−1t0n≥1α(n−1)α≤Ls|z−vt0|(7.16)n≥1α1≤Ldiam(V)<∞.1−sαIteratingformula(7.8)weobtainforeveryn≥1andeveryz∈B(vt0,δ)γλ(t0)(z)−γλ(t0)(vt0)n=arg(φ(φ(z))−argφ(φ(v))λ(tk−1)τkτ|k−1λ(tk−1)τkτ|k−1t0k=1+γλ(tn)(φτ|n(z))−γλ(tn)(φτ|n(vt0)).Sinceforallt∈T,B(vt,(1+s)δ)⊂B(vt,2δ)⊂Bλ(t),allthefunctionsγλ(t)|B(vt,(1+s)δ)areuniformlycontinuous.Therefore,sincethesetTisfinite,sinceφτ|n(z),φτ|n(vt0)∈B(vtn,(1+s)δ)andsince|φτ|n(z)−φ(v)|≤δsn,applying(7.16)weconcludethatforallz∈B(v,δ)τ|nt0t0γλ(t0)(z)∞=γ(v)+arg(φ(φ(z))λ(t0)t0λ(tk)τkτ|k−1k=1−argφ(φ(v)).λ(tk)τkτ|k−1t0Thusthefunctionγλ(t0)|B(vt,δ)asthesumofanabsolutelyuniformly0convergentseriesofharmonicfunctionsisharmonic.So,allthefunctionsγλ(t):B(vt,δ)→IR,t∈T,areharmonic.RemarkthatinthecasewhenSisnot1-dimensionaltheequation(ec)assumedonlyonJ(analogouslyto(c))wouldbesufficientforγλexten-dedbytheformulaabovetosatisfy(ec)onV;inparticular(eh)wouldbeproved.However,ifSisone-dimensionaltheexistenceofγλsatisfying(ec)onJisalwaystrue.JusttakeforγanargumentofthedirectiontangenttoM,theunionofafinitefamilyofreal-analyticcurvescontainingJ. 190DynamicalRigidityofCIFSsNow,foreveryt∈Tbylt:B(vt,δ)→IRdenotetheharmonicconjugatetoγλ(t).ThusthefunctionGt=exp(lt+iγλ(t)):B(vt,δ)→CIisholomorphic.Denotebyψt:B(vt,δ)→CIaprimitiveofGt.Fixw∈Jandchooset∈Tsuchthatw∈B(v,δ).Sinceψ(w)=ttexp(lt(w)+iγλ(t)(w))=0,thereexistsadiskUw⊂B(vt,δ)suchthatψt|Uwisinjective.ApplyingTheorem7.3.3asbefore,wemayassumethedisksUwtobesosmallthatallthesetsφi(Uw)areconvex.Weclaimthatthefamily{ψw:Uw→CI}w∈JformsanaffineatlasfortheiteratedfunctionsystemS.Indeed,fixw,v∈Jandconsidert,t∈TsuchthatUw⊂B(vt,δ)⊂Bλ(t)andUv⊂B(vt,δ)⊂Bλ(t).Thenforeveryz∈Uw∩Uvweget(ψ◦ψ−1)(ψ(z))=ψ(z)(ψ(z))−1=G(z)G−1(z)wvvwvλ(t)λ(t)=explt(z)+iγλ(t)(z)−lt(z)−iγλ(t)(z)=expi(γλ(t)(z)−γλ(t)(z)explt(z)−lt(z).Sinceby(7.7)γλ(t)−γλ(t)isconstantonz∈Uw∩Uv⊂Uλ(t)∩Uλ(t)andsinceltandltdifferonUλ(t)∩Uλ(t)byanadditiveconstantasharmonicconjugatestoharmonicfunctionsγλ(t)andγλ(t)respectively,weconcludethat(ψ◦ψ−1)isconstantonψ(U∩U).wvvwv−1Nowfixw,v∈J,i∈I,andwriteC=φi(φi(Uw)∩Uv)).Sinceφi(Uw)∩Uv))isaconvexsetandthereforeconnected,itscontinu-ousimageCisalsoconnected.Thentherearet,t∈TsuchthatUw⊂B(vt,δ)⊂Bλ(t),Uv⊂B(vt,δ)⊂Bλ(t)andCiscontainedin−1aconnectedcomponentofBλ(t)∩φi(Bλ(t)).Usingthechainrulewethengetforallz∈C(ψ◦φ◦ψ−1)(ψ(z))=ψ(φ(z))φ(z)(ψ(z))−1viwvviiw−1=Gt(φi(z))φi(z)Gt(z)=expi(γλ(t)(φi(z)))+lt(φi(z))+log|φi(z)|+iargλ(t)φi(z)−iγλ(t)(z)−lt(z)=explt(φi(z))+log|φi(z)|−lt(z)×expi(argλ(t)φi(z)−γλ(t)(z)+γλ(t)(φi(z)).Hence,using(7.8)weconcludethatthederivative(ψ◦φ◦ψ−1)hasaviwconstantargumentonψ(C)andconsequently(ψ◦φ◦ψ−1)isconstantvviwonψv(C).Theproofoftheimplication(ec)⇒(d2)iscomplete.•Theimplication(a)⇒(f)isobvious. 7.3Two-dimensionalsystems191•(f)⇒(er).SupposefirstthatthesystemSis1-dimensional.Thenthecondition∇D˜φi≡0onJissimilar(formallyweaker)toD˜φiconstantin(a).Weprove(er)similarly,via(c)⇒(d1)⇒(eh).AssumenowthatSisnotone-dimensional.Supposethat∇D˜φi=0onJforalli∈I.SinceSisnotone-dimensional,itimpliesthat∇Dφi=0onUforalli∈I.ThusD˜φi=0isconstantonUforalli∈I,sinceUisconnected.So,item(a)isprovedinthiscaseandtherefore,inviewofwhatwehavealreadyproved,sois(er).So,wemayassumethatthereexistsj∈Iandw∈Jsuchthat∇Dφj(w)=0.Bycontinuityofthefunction∇D˜φjtherethusexistsaneighborhoodW⊂Vofw∈CIonwhich∇D˜φjnowherevanishes.LetusconsideronWthelinefieldlorthogonalto∇D˜φj.BythedefinitionofthelimitsetJ,foreveryz∈Jthereexistsτ∈I∗suchthatφ(z)∈τJ∩W.Thendefine−1l(z)=(φτ)φτ(z)(l(φτ(z))),(7.17)where,temporarilychangingnotation,(φ−1)denotesthederivativeτφτ(z)ofthemapφ−1evaluatedatthepointφ(z)andthedisplayaboveττexpressesitsactiononalineelement.WewanttoshowfirstthatinthismannerwedefinealinefieldonJ.So,weneedtoshowthatifφτ(z),φη(z)∈J∩W,then(φ−1)(l(φ(z)))=(φ−1)(l(φ(z))).(7.18)τφτ(z)τηφη(z)ηSupposeonthecontrarythat(7.18)failswithsomez,τ,ηasrequiredabove.Thenthereexistsapointx∈W∩Jandγ∈I∗(infactforeveryx∈Wthereexistsγ)suchthatφγ(x)issoclosetozthat(φ−1)(l(φ(φ(x))))=(φ−1)(l(φ(φ(x)))).τφτ(φγ(x))τγηφη(φγ(x))ηγHence(φ−1)l(φ(x))=(φ−1)l(φ(x)).τγφτγ(x)τγηγφηγ(x)ηγSo,either−1(φτγ)φτγ(x)l(φτγ(x))=l(x)or−1(φηγ)φηγ(x)l(φηγ(x))=l(x).Supposeforexamplethefirstincompatibilityofl’sholds.Thendet(∇D˜φj◦φτγ(x),∇D˜φj(x))=0 192DynamicalRigidityofCIFSscontrarytoourassumption.ThusthelinefieldliswelldefinedonJanditimmediatelyfollowsfromthemethodthisfieldisconstructedthatitisinvariantwithrespecttoallthecontractionsφi,i∈I.Noticethatformula(7.17)definesaninvariantlinefieldonV.Wecanuseanyτ∈I∗suchthatφ(V)⊂W.Theresultingldoesnotdependonττbecauseforanyothersuchη(7.18)holdsforz∈J,soitholdsonthewholeofV.Otherwisethesystemwouldbeone-dimensionalbecauselisreal-analyticsotheequationholdsonareal-analyticset.Theargumentarglisofcoursedefineduptointegermultiplicityofπ.UsingagainTheorem7.3.3,onecanfind{Bλ},afinitecoverofJbydiskscontainedinV,smallenoughthatalltheimagesφi(Bλ),i∈I,areconvex.ThenalltheintersectionsBλ∩BλandBλ∩φi(Bλ)areconnected.DefineγλasanarbitrarybranchofarglonBλ.Then(7.7)and(7.8)followfromtheinvarianceoflunderS,withconstantsc(λ,λ)andc(λ,λ,i)thataremultiplesofπ.Thus(er)isproved.•(d2)⇒(d3).Let{ψt:Ut→CI}t∈Tbetheatlasproducedby(d2).Fixx∈J,chooses∈Tsuchthatx∈Uandthenρ∈I∗suchthatSsx∈φρ(J)⊂φρ(W)⊂Us.ConsidernowtheiteratedfunctionsystemS={ψ◦φ◦φ◦φ−1◦ψ−1},ρsρiρsi∈IwheretheroleofXisplayedbyψs(φρ(X))andtheroleofWisplayedbyψ(φ(W)).Itfollowsfrom(d2)thateachmapψ◦φ◦φ◦φ−1◦ψ−1,sρsρiρsi∈I,isaffineoneachsufficientlysmallneighborhoodofeachpointofψs(φρ(J)).Hence,asholomorphic,thismapmustbeaffineonthewholeconnecteddomainψs(φρ(W)).
Letusprovenowatechnicalfactabout1-dimensionalsystems.Proposition7.3.7SupposethatF={fi:X→X}i∈IandG={gi:Y→Y}i∈Iaretwonotessentiallyaffinetopologicallyconjugatesystems.Supposealsothatthemeasuresmandm◦h−1areequivalent.IfoneGFofthesesystemsis1-dimensional,thensoistheotherone.Proof.SupposeonthecontrarythatGisnot1-dimensional.ThenitfollowsfromTheorem7.3.6thatthereexisty∈J,j∈I,ω∈I∗andaGneighborhoodW2⊂CIofysuchthatthemapG=(D˜gj◦γω,D˜gj) 7.3Two-dimensionalsystems193isinvertibleonW.Sincethemeasuresmandm◦h−1areequivalent,2GFafteranappropriatenormalizationµF=µG◦hmeaningthatDh=dµG◦h=1.Sinceh◦f=g◦hforallτ∈I∗andsinceD=1,dµFττhG◦h=FonJ,whereF=(D˜◦γ,D˜).Writex=h−1(y).Thenh=G−1◦FfjωfjonW1∩JFforsomeopenneighborhoodW1ofxinCIsuchthatF(W1)⊂G(W).SinceF,G−1arereal-analytic,theimageG−1◦F(W∩M)fora21FsmallenoughsuchWisareal-analyticcurveandG−1◦F(W∩M)∩J11FGcontainsanopenneighborhoodofyinJG.UsingnowLemma7.3.4weconcludethatGis1-dimensional.
Themainresultofthissectioniscontainedinthefollowing.Theorem7.3.8IftwoconformalregulariteratedfunctionsystemsF={fi:X→X}i∈I}andG={gi:Y→Y}i∈I}satisfyingtheopensetconditionarenotessentiallyaffineandareconjugatebyahomeomorph-ismh:JF→JG,thenthefollowingconditionsareequivalent.(a)TheconjugacybetweenthesystemsFandGextendsinacon-formalfashiontoanopenneighborhoodofJF.(b)TheconjugacybetweenthesystemsFandGextendsinareal-analyticfashiontoanopenneighborhoodofJF.(c)TheconjugacybetweenthesystemsFandGisbi-Lipschitzcon-tinuous.(d)|g(y)|=|f(x)|forallω∈I∗,wherexandyaretheonlyωωωωωωfixedpointsoffω:X→Xandgω:Y→Yrespectively.(e)∃S≥1∀ω∈I∗−1diam(gω(Y))S≤≤S.diam(fω(X))(f)∃E≥1∀ω∈I∗||g||E−1≤ω≤E.||fω||(g)HD(J)=HD(J)andthemeasuresmandm◦h−1areGFGFequivalent.(h)Themeasuresmandm◦h−1areequivalent.GFProof.Theimplications(a)⇒(b)and(b)⇒(c)areobvious.That(c)⇒(d)resultsfromthefactthat(c)impliescondition(1)ofThe-orem7.1.1,whichinviewofthattheoremisequivalentwithcondition 194DynamicalRigidityofCIFSs(2)ofTheorem7.1.1,whichfinallyisthesameascondition(d)ofThe-orem7.3.8.Theimplications(d)⇒(e)⇒(f)⇒(g)havebeenprovedinTheorem7.1.1.Theimplication(g)⇒(h)isagainobvious.Wearelefttoprovethat(h)⇒(a).Weshallfirstprovethat(h)⇒(b).So,supposethat(h)holds.Then,afteranappropriatenormalizationµF=µG◦hdµG◦hmeaningthatDh=dµF=1.IfFisone-dimensional,thenbyPro-position7.3.7,soisGandtheimplication(h)⇒(b)followsfromThe-orem7.2.4andthefactthateachreal-analyticmapbetweenreal-analyticcurvesextendstoa(complex)analyticmapdefinedonsomeoftheirneighborhoodsinCI.Hence,wemayassumethatneithersystemForGis1-dimensional.Therefore,sinceGisnotessentiallyaffine,thereexisty∈J,j∈I,ω∈I∗andaneighborhoodW⊂CIofysuchthattheG2mapG=(D˜gj◦gω,D˜gj)isinvertibleonW.Sinceh◦f=g◦hforallτ∈I∗andsinceD=1,2ττhG◦h=FonW1∩Jf,whereF=(D˜fj◦gω,D˜fj)andW1isaneighborhoodofx=h−1(y)⊂CI.SinceGisinvertibleonW,G(y)=F(x)andFis2continuous,wemayassumethatF(W)⊂G(W).HenceG−1◦Fiswell12definedonWandG−1◦F|=h.Considernowω∈I∗suchthat1W1∩JFfω(JF)⊂W1.SinceG−1◦F(f(J))=h◦f(J)=g◦h(J)=g(J)⊂g(V),ωFωFωFωGωGsinceg(W)isopen,andsincefandG−1◦Farecontinuous,thereω2ωexistsanopenneighborhoodV1⊂VFofJFsuchthatfω(V1)⊂W1andG−1◦F(f(V))⊂g(W).Hence,themapω1ω2g−1◦(G−1◦F)◦f:V→CIωω1iswelldefinedandbyCorollary6.1.5isreal-analytic,andg−1◦(G−1◦ωF)◦fω|JF=h.Thus,theproperty(b)isproved.(b)⇒(a).LetHbethisreal-analyticextensionofhonsomeneighbor-hoodWFofJFinCI.IfFisone-dimensionalandDFisthefamilyofreal-analyticcurvescomingfromtheone-dimensionalityofF,then,inviewofProposition7.3.7,Gisalsoone-dimensionalandthereal-analyticmapH|MF:DF→DGhasa(complex)analyticextensiontosomeneighborhoodofDFinCI.So,wemayassumethatFisnotone-dimensional.WemayalsoassumeWtobesosmallthatHisalinearF 7.4Rigidityindimensiond≥3195isomorphismateverypointofWF.Definethefunctionψ:WF→IRbytheformula||H(z)||ψ(z)=.||(H(z))−1||Supposethatψ(ξ)=1forsomepointξ∈W.Sinceforeveryω∈I∗F||H(f(ξ))||||g(H(ξ))·H(ξ)·(f(ξ))−1||ωωωψ(fω(ξ))=||(H(f(ξ)))−1||=−1−1ω||gω(H(ξ))·H(ξ)·(fω(ξ))||||H(ξ)||==ψ(ξ)||(H(ξ))−1||andsince{fω(ξ):ω∈I∗}⊃JF,weconcludethatψ=1identicallyonJF.Sinceψisreal-analyticandsinceFisnot1-dimensional,usingLemma7.3.4,weconcludethatψ=1onanopenneighborhoodofJF.ButthismeansthatHisconformal.So,wemayassumethatψ(z)=1foreveryz∈WF.Definethenthefield{Ez}z∈WFonWFasfollows.|H(z)w|Ez=w∈CI:=|H(z)|∪{0}.|w|Foreveryz∈WF,thesetEzisalinearsubspaceofCIofdimension≥1.Itscodimensionis≥1sinceψ(z)=1.Inconclusiondim(Ez)=1forallz∈WF.ObviouslyEzdependscontinuouslyonz.Sincethemapsfi:CI→CI,i∈,areconformal,f(z)(E)=Eforalli∈Iandputtingizfi(z)locallyγλ(z)=arg(Ez),itthereforefollowsfromTheorem7.3.6(ec)thatthesystemFisessentiallyaffine.Thiscontradictionfinishestheproof.
7.4Rigidityindimensiond≥3Inthissectionwestrengthentherigidityresultsoftheprevioussection,cf.[U5].Webeginwiththefollowing.Lemma7.4.1Supposethatφ:IRd→IRd,d≥3,isaconformaldiffeomorphismthathasanattractingfixedpointa(φ(a)=a,|φ(a)|<1).IfMisanopenconnectedC1-submanifoldofIRdsuchthatφ(M)⊂Manda∈M,thenMiseitherasubsetofaφ-invariantaffinesubspaceofthesamedimensionasM,orasubsetofaφ-invariantgeometricsphereofthesamedimensionasM.Proof.Sinceaisanattractingfixedpointofφ,thereexistsaradiusdddr>0sosmallthatφ−1RB(a,r)⊂RB(a,r),whereRis 196DynamicalRigidityofCIFSstheAlexandrovcompactificationofIRdobtainedbyaddingthepointdatinfinity.SinceRB(a,r)isaclosedtopologicalball,inviewofBrouwer’sfixedpointtheoremthereexistsafixedpointbofφ−1indRB(a,r).Hencebisalsoafixedpointofφandb=a.Thenthemap−1ψ=ib,1◦φ◦ib,1(ib,1equalsidentityifb=∞)fixes∞,whichmeansthatthismapisaffine,andw=ib,1(a)isanattractingfixedpointofψ.Inadditionψ(M˜)⊂M˜,whereM˜=i(M),w∈M˜,andψ:IRd→IRd,asanb,1affinemap,canbewrittenintheformλA+c,whereλ>0andAisanorthogonalmatrix.Sinceψ(M˜)⊂M˜,andsinceψisadiffeomorphism,ψ(z)(TM˜)=TM˜.Inparticularψ(w)E=E,whereE=TM˜.zψ(z)wWithoutloossofgeneralitywemayassumethatM˜iscontainedinthebasinofimmediateattractiontow.WeshallshowthatTzM˜=Eforeveryz∈M˜.Andindeed,takeanarbitrarypointpointz∈M˜.Sinceψ(x)=λAforallx∈IRdandsinceλAisconformal,wegetforalln≥0that∠(TM,E˜)=∠(An(TM˜),AnE)zznM˜),E)=∠(TM,E˜),=∠((ψ)(z)Tzψn(z)where∠denotestheanglebetweenlinearhyperspaces.Sincelimn→∞M˜=TM˜=E,weconcludethat∠(TM,E˜)=0,orequivalentlyTψn(z)wzTzM˜=E.Sincetheonlyintegralmanifoldsofaconstantfieldoflinearsubspacesareaffinesubspaces,weconcludethatM˜iscontainedinanaffinesubspace.SinceM˜isanopensubsetofit,thisaffinesubspaceisφ-invariant.SinceM=ib,1(M˜),wearedone.
WecallthesystemS={φi}i∈Iatmostq-dimensional,1≤q≤d,ifthereexistsM,eitheraq-dimensionallinearsubspaceofIRdoraSq-dimensionalgeometricspherecontainedinIRd,suchthatJ⊂MSandφi(MS)=MSforalli∈I.WecallthesystemS={φi}i∈Iq-dimensionalifqistheminimalnumberwiththisproperty.Lemma7.4.2Ifanon-emptyopensubsetofJiscontainedinaq-dimensionalreal-analyticsubmanifold,thenthesystemSisatmostq-dimensional. 7.4Rigidityindimensiond≥3197Proof.Theassumptionsofthelemmastatethatthereexistsapointx∈J,anopenballB(x)centeredatxandM,ap-dimensionalopenconnectedreal-analyticsubmanifoldMcontainingJ∩B(x),where1≤p≤qistheminimalintegerwiththisproperty.Fixnowanarbitraryauxiliarypointz∈J.Sincex∈J,thereexistsω∈I∗suchthatφω(z)∈J∩B(x);moreoverφω(V)⊂B(x).Thenthesetφω(V)∩Mcontainsφω(V)∩J,anopenneighborhoodofφω(z)inJ,andconsistsofcountablymanyconnectedp-dimensionalreal-analyticsubmanifolds.Takingthelengthofωlargeenoughwemayassumethatthiscountablefamilyisasinglemanifold.ThenN=φ−1(φ(V)∩M)isaconnectedωωp-dimensionalreal-analyticsubmanifold(therearenobranchingpointssinceφ−1is1-to-1)containingJandcontainedinV.ForthepurposeofωthisproofitisnotimportantwhetherNisinfactindependentofωornot.Fixanarbitraryi∈I.Letxi∈Jbetheonlyattractingfixedpointofφ.SincetheconnectedcomponentCofφn(N)∩Ncontainingxisai,niireal-analyticmanifoldofdimension≤dim(N)≤dim(M)=p,itfollowsfromthedefinitionofp(itsminimality)thatdim(Ci,n)=p.Hence,thetwoconnectedp-dimensionalreal-analyticopenmanifoldsφn(N)andiNareextensionsofthesamep-dimensionalreal-analyticmanifoldCi,n.Therefore,sincelimdiam(φn(N))=0andsincex∈φn(N),wen→∞iiiconcludethatforalln≥1solargethatdiam(φn(N))0denoterespectivelyitsinversioncenterandthescalarcoefficient.Then(7.19)takestheformλ||z−w||−2=ecionG(J)or2−ci||z−w||=λeonG(J).√So,G(J)iscontainedinthesphereS(w,λe−ci)centeredatwandof√radiusλe−ci.Sinceforeveryn≥0,G◦φn◦G−1(G(J))=G◦φn(J)⊂iiG(J),weconcludethatallthedescendingsetsG◦φn◦G−1(G(J))arei 200DynamicalRigidityofCIFSs√√containedinthesphereS(w,λe−ci).LetH⊂S(w,λe−ci)beaminimalsphere(inthesenseofinclusion)containingatleastoneofthesetsG◦φn◦G−1(G(J)),n≥0.Thusthereexistsk≥0suchthatiG◦φk◦G−1(G(J))⊂H.TheniG◦φk+1◦G−1(G(J))⊂H∩(G◦φ◦G−1(H)).(7.20)iiSinceG◦φ◦G−1(H)iseitherasphereoranaffinesubspaceofIRdandisinceG◦φk+1◦G−1(G(J))containsatleastthreepoints(isuncountableiinfact),theintersectionH∩(G◦φ◦G−1(H))isasphere(atleast1-idimensional)again.ThereforebytheminimalityofHandby(7.20)weconcludethatH∩(G◦φ◦G−1(H))⊃H,whichmeansthatG◦φ◦iiG−1(H)⊃H.Therefore,sincedim(G◦φ◦G−1(H))=dim(H),weiconcludethatG◦φ◦G−1(H)=H.(7.21)iLetxbetheuniquefixedpointofthemapφ:V→V.SinceG◦φn◦iiiG−1(z)→G(x)uniformlyonG(V)⊂G(J),itfollowsfrom(7.19)thati√ci>0.So,foreveryz∈S(w,λe−ci)⊃H,|(G◦φ◦G−1)(z)|=λ||z−w||−2=λλ−1eci=eci.iiiiThisimpliesthatG◦φ◦G−1isauniformcontractiononHandthereforeiG◦φn◦G−1(z)→G(x)uniformlyonH.Thishowevercontradicts(7.21)iiandfinishestheproofoftheimplication(c)⇒(d).•(d)⇒(a).LetG:IRd→IRdbeaconformalhomeomorphismprovid-ingconjugacyofSwithasystemconsistingonlyofconformalaffinecontractions.Thenforeveryi∈I−1gi=|(G◦φi◦G)(z)|isanumberindependentofz∈G(V).Bythechainrulewehaveforeveryz∈VnLn(11)(z)=|φ(z)|h=|G(z)|h|G(φ(z))|−δgh.ωωωi|ω|=n|ω|=ni=1 7.4Rigidityindimensiond≥3201Fixnowj∈I.Thenforeveryn≥1andallz∈Vwegetn|G(φ(z))|δ|G(φ(φ(z)))|−δngδL(11)(φj(z))δ|ω|=njωji=1ωiLn(11)(z)|φj(z)|=|G(z)|δ|G(φ(z))|−δngδ|ω|=nωi=1ωi×gδ|G(z)|δ|G(φ(z))|−δjj|G(φ(z))|δ|ω|=nωδ=|G(φ(φ(z))|δgj.|ω|=nωjSince||φ(x)−x||≤consts|ω|,wherexistheonlyfixedpointofωωωφω:V→V,weconcludethatLn(11)(φ(z))j|φ(z)|h→ghLn(11)(z)jjuniformlyonV.Hence,applyingTheorem7.1.1,weconcludethatLn(11)(φ(z))D˜=ρ(φ(z))ρ(z)|φ(z)|h=limj|φ(z)|h=ghφjjjnjjn→∞L(11)(z)onX.SinceD˜isreal-analyticonU,weconcludethatD˜=ghonφjφjjU.Theproofoftheimplication(d)⇒(a)iscomplete.•(d)⇒(er).DefineGγ=.|G|Sinceforeveryi∈I,G◦φ◦G−1isaffine,weconcludethatG◦φ◦iiG−1·φ◦G−1·(G)−1◦G−1=k∈LC(d).Henceii−1GφGk◦φ·i·◦G−1=i|G|i|φ||G||k|iianditsufficestotakeγ=G.Thustheproofoftheimplication|G|(d)⇒(er)iscomplete.•Theimplication(er)⇒(ec)isobvious.•(ec)⇒(d).Ifallthemapsφi,i∈Iareaffine,thereisnothingtoprove.So,assumethatthereisj∈Isuchthatφjisnotaffine.Thennoiterate(n)φjnisaffine.Letadenotetheinversioncenterofφjn.Fixv∈J.By 202DynamicalRigidityofCIFSs(ec)thefollowingholdsforeveryz∈Vandeveryn≥1−1(γ(v))γ(z)−1φn(v)−1φn(z)jn−nj=·γ◦φjn(v)kjkjγ◦φjn(z)|φjn(v)||φjn(z)|−1φn(v)−1φn(z)jj=·γ◦φjn(v)γ◦φjn(z)|φjn(v)||φjn(z)|−1−1=(Tn(v))γ◦φjn(v)γ◦φjn(z)Tn(z),whereT(w)=Id−2Q(w−a(n))andinthecanonicalcoordinatesQisngivenbythematrixxixjQ(x)=.||x||2Weshallnowprovethatthesequence{a(n)}∞doesnotconvergeton=1∞.Indeed,supposeonthecontrarythatlima(n)=∞.Sincen→∞a(n)=φ−1−1)n(∞),wethereforegetjn(∞)=(φj∞=lima(n−1)=limφ((φ−1)n)=φ(lim(φ−1)n)jjjjn→∞n→∞n→∞(n)=φj(lima)=φj(∞)n→∞whichmeansthatφjisaffine.Thiscontradictionshowsthatthereexistsasubsequence{k}∞suchthata(kn)→aforsomea∈IRd.Thenfornn=1everyn≥1,−1−1−1(γ(v))γ(z)=(Tkn(v))γ◦φjnk(v)γ◦φjnk(z)Tkn(z),andtakingthelimitwhenn→∞,weobtain−1−1(γ(v))γ(z)=Id−2Q(v−a)Id−2Q(z−a)or,equivalently,−1γ(z)=γ(v)Id−2Q(v−a)Id−2Q(z−a).Define−1G=γ(v)Id−2Q(v−a)◦ia,1.Then−11G(z)=γ(v)Id−2Q(v−a)Id−2Q(z−a).||z−a||2 7.4Rigidityindimensiond≥3203HenceG(z)−1=γ(v)Id−2Q(v−a)Id−2Q(z−a).=γ(z)|G(z)|Thereforeforeveryi∈I,(ec)takestheformG◦φ(z)φ(z)G(z)ii|G◦φ(z)|·|φ(z)|·|G(z)|=ki.iiSupposethatG◦φ◦G−1isnotaffine.Thenfory,theinversioncenteriofG◦φ◦G−1,wegetincanonicalcoordinatesthati(zm−ym)(zn−yn)δmn−22=(ki)mn||z−y||forallz∈Vandallm,n∈{1,2,...,d},whereδdenotestheKroneckersymbolhere.ButthisisimpossibleandweconcludethatG◦φ◦G−1iisaffine.Theproofoftheimplication(ec)⇒(d)iscomplete.•Theimplication(a)⇒(g)followsfromtheimplication(a)⇒(d).•(g)⇒(f).ConjugatingthesystemSbyaconformaldiffeomorphismwemayassumethatM=IRq.Giveni∈Iand(ω(j))q∈(I∗)qletSj=1A=(i,ω(1),...,ω(q))andletH:IRq→IRqbethemapdefinedbytheformulaAHA(z)=D˜φi◦φω(1)(z),...,D˜φi◦φω(q)(z).Supposefirstthatforeveryi∈IthereexistsAsuchthatH=0onAJ.SinceSisnot(q−1)-dimensional,thisimpliesthatH=0onaAneighborhoodofJinIRq.ButthenD˜isconstantonanopensubsetφiofIRqhavinganon-emptyintersectionwithJ.SincebyCorollary6.1.5D˜φiisreal-analytic,itisthereforeconstantontheappropriatesetUqproducedinthiscorollary.Hence,inviewofalreadyprovedimplication(a)⇒(d),thesystem{φ:IRq→IRq}isconjugatebyaconformalidiffeomorphismρ:Rq→IRqwithanaffinesystem.SinceρextendstoaconformaldiffeomorphismfromIRdtoIRdandsinceanextensionofanaffinemapinIRqtoanaffinemapinIRdisalsoaffine(ifq≤1weneedtobecertainthattheseextensionsareoftheformλA+b),wearedoneinthiscase.So,supposethatthereexistsi∈IsuchthatforeveryAwiththefirstelementequaltoithereexistsx∈JsuchthatH(x)=0.ChooseAw∈JandA=(i,ω(1),...,ω(q))suchthatdimKerH(w)isminimal,A 204DynamicalRigidityofCIFSssayequaltop≤q−1.Bytheassumptionsof(g),dimKerH(w)≥1.ASo1≤dimKerHA=p≤q−1onW,aneighborhoodofwinIRq.BythedefinitionofthelimitsetJ,foreveryz∈Vthereexistsτ∈I∗suchthatφ(z)∈W.Thendefineτl(z)=(φ−1)(KerH(φ(z))),τφτ(z)Aτwhere,temporarilychangingnotation,(φ−1)denotesthederivativeτφτ(z)ofthemapφ−1evaluatedatthepointφ(z).WewanttoshowfirstthatττwedefineinthismanneralinefieldonV.So,weneedtoshowthatifφτ(z),φη(z)∈W,then−1−1(φτ)φτ(z)(l(φτ(z)))=(φη)φη(z)(l(φη(z))).(7.22)Supposeonthecontrarythat(7.22)failswithsomez,τ,ηasrequiredabove.Thenthereexistsapointx∈Wandγ∈I∗(infactforeveryx∈Wthereexistsγ)suchthatφγ(x)issoclosetozthat(φ−1)(l(φ(φ(x))))=(φ−1)(l(φ(φ(x)))).τφτ(φγ(x))τγηφη(φγ(x))ηγHence(φ−1)l(φ(x))=(φ−1)l(φ(x)).τγφτγ(x)τγηγφηγ(x)ηγSo,either(φ−1)l(φ(x))=KerH(x)τγφτγ(x)τγAor(φ−1)l(φ(x))=KerH(x).ηγφηγ(x)ηγAWithoutlossofgeneralitywemayassumethatthefirstinequalityholds.Since(H◦φ)(x)=H(φ(x))φ(x),wegetKer(H◦φ)(x)=AτγAτγτγAτγφ(x)−1(KerH(φ(x)))andthereforeτγAτγKer(HA◦φτγ)(x)=KerHA(x).(7.23)Ifnowφγ(x)issufficientlyclosetoz,thenφτγ(x)issoclosetoφτ(z)thatφτγ(x)∈W.Then+dimKerH(φ(x))=p=dim(KerH(x)).(7.24)AτγAConsidernowlinearlyindependentvectors∇Dφ˜i◦φω(k1)(x),...,∇Dφ˜i◦φ,t=q−dim(KerH(x)).Ifv∈KerH(x),then<∇Dφ˜◦ω(kt)(x)AAiφ(kj)(x),v>=0forallj=1,2,...,t.Supposethateachvectorω∇Dφ˜i◦φω(j)τγ(x),j=1,...,q,isalinearcombinationofthevectors 7.4Rigidityindimensiond≥3205∇Dφ˜◦φDφ˜◦φ,t=q−dim(KerH(x)).Theniω(k1)(x),...,∇iω(kt)(x)A<∇Dφ˜◦φ(x),v>=0forallj=1,...,qandallv∈KerH(x).iω(j)τγAHenceKer(H◦φ)(x)⊃KerH(x).Thususing(7.24)wecon-AτγAcludethatKer(H◦φ)(x)=KerH(x).Thiscontradicts(7.23)AτγAandshowsthatthereexists1≤u≤qsuchthatthevectors∇Dφ˜i◦tφω(kj)(x)j=1togetherwiththevector∇Dφ˜i◦φω(u)τγ(x)formalin-earlyindependentset.Hence,ifB=(i,ω(u)τγ,ω(k1),...,ω(kt),i,...,i)((q−(t+1))i’sattheend),thentherankofH(x)isgreaterthanBorequaltot+1.ThusKerH(x)=q−rank(H(x))≤q−(t+1)=BBq−q+dim(KerH(x))−1=p−1,whichisacontradictionwiththeAdefinitionofpandfinishestheproofoftheimplication(g)⇒(f).•(f)⇒(d).InordertoprovethisimplicationsupposethatthereexistsafieldoflinearsubspacesExinTMSofdimensionandco–dimensiongreaterthanorequalto1definedonaneighborhoodofJinMSandinvariantundertheactionofderivativesofallmapsφi,i∈I.Con-jugatingoursystembyaconformaldiffeomorphism,wemayassumethatM=IRq.Fixanelementj∈I.InthecourseoftheproofSofLemma7.4.1wehaveshownthatbesidesoneattractingfixedpointx∈X,themapφhasadifferentfixedpointy∈IRd.ConjugatethejjjsystemSbytheinversioniyj,1(equaltotheidentityifyj=∞)anddenotetheresultingsystembyS1.Putψi=iyj,1◦φi◦iyj,1foralli∈I.ThefieldF=i(E)isdefinedonaneighborhoodWofJanditxyj,1xS1isS-invariant.Sinceψ:IRq→IRqislinear,inspectingtheappropri-1jatepartoftheproofofLemma7.4.1weseethatthefield{Fx}x∈Wisconstant,sayequaltoF.So,thefieldofaffinesubspaces{x+F}x∈W,astheuniquefieldofintegralmanifoldsoftheS1-invariantfield{F}oflinearsubspaces,isS1-invariant,whichmeansthatψ(x+F)=ψi(x)+Fforeveryi∈Iandeveryx∈W.So,ifψiisnotaffineforsomei∈I,−1thenx+Fmustcontainψi(∞),thecenterofinversionofψi,foreveryx∈W.SinceWisopeninIRqandsincedim(F)≤q−1,thisisim-possibleandprovesthatψiisaffine.Theimplication(f)⇒(d)isthusproved.
Ournextresultisthefollowingratherunexpectedfact.Proposition7.4.5SupposethatF={fi:X→X}i∈IandG={gi:Y→Y}i∈Iaretwonotessentiallyaffinetopologicallyconjugatesystems.Ifthemeasuresmandm◦h−1areequivalent,thenthesystemsFGFandGareofthesamedimension. 206DynamicalRigidityofCIFSsProof.SupposeonthecontrarythatthedimensionsofFandGarenotequal.Withoutlossofgeneralitywemayassumethatp=dimF0suchthatforeveryx∈∂X⊂IRdthereexistsanopenconeCon(x,α,l)⊂Int(X)withvertexx,centralangleofLebesguemeasureα,andaltitudel.(8)ThereexistsaconstantL≥1suchthat|φ
(y)|−|φ
(x)|≤L||φ
||·|y−x|iiiforeveryi∈Iandeverypairofpointsx,y∈V.(9)Foreveryi∈Ωput˜Xi=φj(X).j∈I{i}Wewillalsoneedthefollowing.||φ
α<∞.in||Xin≥0WecallsuchasystemofmapsS={φi:i∈I}asubparaboliciteratedfunctionsystem.Letusnotethatconditions(1),(3),(5)–(7)aremodeledonsimilarconditionswhichwereusedtoexaminehyperbolicconformalsystems.IfΩ=∅,wecallthesystem{φi:i∈I}parabolic.Asdeclaredin(2)theelementsofthesetIΩarecalledhyperbolic.Weextendthisnametoallthewordsappearingin(5)and(6).ByI∗wedenotethesetofallfinitewordswithalphabetIandbyI∞allinfinitesequenceswithtermsinI.Itfollowsfrom(3)thatforeveryhyperbolicwordω,φ(W)⊂W.Notethatourconditionsinsurethatφ
(x)=0,forωialliandx∈V.Weprovidebelowwithoutproofsallthegeometricalconsequencesoftheboundeddistortionproperties(5)and(8),derivedinSection4.1whichremaintrueintheparaboliccase.Wehaveforallhyperbolicwordsω∈I∗andallconvexsubsetsCofWdiam(φ(C))≤||φ
||diam(C)(8.1)ωωand
diam(φω(V))≤D||φω||,(8.2)wherethenorm||·||isthesupremumnormtakenoverVandD≥1isauniversalconstant.Moreover,diam(φ(X))≥D−1||φ
||(8.3)ωωandφ(B(x,r))⊃B(φ(x),K−1||φ
||r),(8.4)ωωω 8.1Preliminaries211foreveryx∈X,every00suchthatφω(Int(X))⊃Conφω(x),β,l(ω,x).(8.6)Theimportantpointin(8.6)isthatbyconformalitywecangetaconewithvertexxandopeningangleβlyinginφω(X),butwecannotsayanythingabouttheheightofthisconeunlessωisahyperbolicword,inwhichcasewehave(8.5).Foreachω∈I∗∪I∞,wedefinethelengthofωbytheuniquelydeterminedrelationω∈I|ω|.Ifω∈I∗∪I∞andn≤|ω|,thenbyω|nwedenotethewordω1ω2...ωn.Ourfirstaiminthissectionistoprovetheexistenceofthelimitset.Moreprecisely,webeginwiththefollowinglemma.
Lemma8.1.1Forallω∈I∞theintersectionφ(X)isan≥0ω|nsingleton.Proof.Sincethesetsφω|n(X)formanestedsequenceofcompactsets,
theintersectionn≥0φω|n(X)isnotempty.Moreover,itfollowsfrom(4)thatifωisoftheformτi∞,τ∈I∗,i∈Ω,thenthediametersofthe
kintersectionn=0φω|n(X)tendto0and,intheothercase,thesame
conclusionfollowsimmediatelyfrom(6).Inanycase,n≥0φω|n(X)isasingleton.
Improvingslightlytheargumentjustgiven,wegetthefollowing.Lemma8.1.2limn→∞sup|ω|=n{diam(φω(X))}=0.Proof.Letg(n)=maxi∈Ω{diam(φin(X))}.SinceΩisfiniteitfollowsfrom(4)thatlimg(n)=0.Letω∈I∞.Givenn≥0considern→∞thewordω|n.Lookatthelongestblockofthesameparabolicelement√appearinginω|n.Ifthelengthofthisblockexceedsnthen,sincedueto(2)allthemapsφj,j∈I,areLipschitzcontinuouswithaLipschitz√constant≤1,wehavediam(φω|n(X))≤g(n).Otherwise,wecanfind 212ParabolicIteratedFunctionSystems√√n−ninω|natleast√=n−1distincthyperbolicindices.Itthenfollowsnfrom(6)(andLipschitzcontinuitywithaLipschitzconstant≤1ofall√themapsφ,i∈I)thatdiam(φ(X))≤sn−1.
iω|nWeintroduceonI∞thestandardmetricd(ω,τ)=e−n,wherenisthelargestnumbersuchthatω|n=τ|n.ThecorollarybelowisnowanimmediateconsequenceofLemma2.2.
Corollary8.1.3Themapπ:I∞→X,π(ω)=φ(X),isn≥0ω|nuniformlycontinuous.ThelimitsetJ=JSofthesystemS={φi}i∈IsatisfiesJ=π(I∞)=φ(J).ii∈IWerecallthatthesetJisnotcompactiftheindexsetIisinfinite.Thisofcourseisoneofthemaintechnicalissuestohandle.Lemma8.1.4IfXisatopologicaldiskcontainedinCI,theneveryparabolicpointliesontheboundaryofX.Proof.Supposeonthecontrarythataparabolicpointxi∈Int(X).LetD1={z∈CI:|z|<1}andletR:D1→Int(X)betheRiemannmap(conformalhomeomorphism)suchthatR(0)=xi.ConsiderthecompositionR−1◦φ◦R:D1→D1.Then|(R−1◦φ◦R)
(0)|=ii|R
(0)|−1|R
(0)|=1.ThusbySchwarz’slemmaR−1◦φ◦Risarotation.iSinceφ=R◦(R−1◦φ◦R)◦R−1,itfollowsthatφ(X)=R◦(R−1◦iiiφ◦R)◦R−1(X)=X.Thiscontradictionfinishestheproof.
i8.2TopologicalpressureandassociatedparametersOurfirstgoalistorelatethepressureofthe“volumepotential”func-tionζtothewaypressurewasdefinedinSection3.1.Weconsiderthefunction:I∞→IRgivenbytheformula
ζ(ω)=−log|φω1(π(σ(ω)))|.Usingheavilycondition(8),weshallprovethefollowing.Proposition8.2.1Thefunctionζdefinedaboveisacceptable. 8.2Topologicalpressureandassociatedparameters213Proof.Fixn≥1andω,τ∈I∞suchthatω|=τ|.Itthenfollowsnnfrom(8)that|g(ω)−g(τ)|=log|φ
(π(σ(ω)))|−log|φ
(π(σ(τ)))|ω1ω1|φ
(π(σ(ω)))|−|φ
(π(σ(τ)))|ω1ω1≤min{|φ
ω(π(σ(ω)))|,|φ
(π(σ(ω)))|}1ω1||φ
||ω1≤Lmin{|φ
ω(π(σ(ω)))|,|φ
(π(σ(ω)))|}1ω1×|π(σ(ω))−π(σ(τ))|.Ifω1isahyperbolicindex,thenusingtheboundeddistortionproperty,weget|g(ω)−g(τ)|≤LK|π(σ(ω))−π(σ(τ))|α.Ontheotherhand,sincethereareonlyfinitelymanyparabolicindices,thereisapositiveconstantMsuchthatifω1isparabolic,then|g(ω)−g(τ)|≤LM|π(σ(ω))−π(σ(τ))|α.LetL
=Lmax{K,M}.SinceXbeingcompactisbounded,takingn=1,itfollowsfromthelastinequalitiesthatmaxi∈I{supg|[i]−infg|}≤L
diamα(X)<∞.Theuniformcontinuityofgfollowsfrom[i]inequality|g(ω)−g(τ)|≤L
|π(σ(ω))−π(σ(τ))|αandCorollary8.1.3.
WedefineforeachW⊂X1P(t)=lim||φ
||t,WωWn→∞n|ω|=nwhere||φ
||=sup{|φ
(x)|:x∈W}.LetusnotethatωWωP(t)=inf{s:||φ
||te−sn<∞}.WωWn≥1|ω|=nWenowintroducesomenotation.Foreachi∈Ω,letIp={ω∈Ip:giωp=i}.Lemma8.2.2P(σ,−tζ)=P(t). 214ParabolicIteratedFunctionSystemsProof.First,weshowP(t)=PJ(t).Clearly,PJ(t)≤P(t).Toprovetheconverseinequality,supposePJ(t)qforeveryk≥1.Weshallconstructbyinductionwithrespecttok≥1asequence{Ck}k≥1suchthatforeveryk,Ckconsistsofatleastk+1incomparablewordsfrom{ρ(j),τ(j):j≤k}.Indeed,setC={ρ(1),τ(1)}.SupposenowthatChasbeendefined.1kIfρ(k+1)doesnotextendanywordinC,thenweformCbyaddingkk+1ρ(k+1)toC.Wecandoasimilarthingincaseτ(k+1)doesnotextendkanywordinC.If,ontheotherhand,ρ(k+1)extendssomewordκinkCandτ(k+1)extendsawordηinC,thenκandηarebothexten-kkdedbyρ(k+1)sinceforj≤k,|ρ(j)|,|τ(j)|≤n≤n,n>n+q,jkk+1kandρ(k+1)|=τ(k+1)|.SincethewordsinCareincom-nk+1−qnk+1−qkparable,κ=ηandthisistheonlywordinCkwhichisextendedbybothρ(k+1)andτ(k+1).InthiscaseweformCbytakingawayκandk+1addingbothρ(k+1)andτ(k+1).Nowthesets{φ(X):κ∈C}arenon-κkoverlappingsincethewordsareincomparable.By(8.6)wegetk+1pair-wisedisjointopenconeseachwithvertexxandopeningangleβ.Thisisclearlyimpossibleifkislargeenough.Soitisimpossibletohavesuchapointx.Letµbeashift-invariantprobabilitymeasureandsupposethatµ◦π−1(φ(X)∩φ(X))>0forsomeincomparablewordsτ,ρ∈I∗.Withoutτρlossofgeneralitywemayassumethat|τ|=|ρ|.PutE=φτ(X)∩φρ(X)andset∞∞E∞=φω(E).k=0n=k|ω|=nInviewofwhatwehavejustprovedE∞=∅.Ontheotherhand,by(8.6),foreveryn≥0,eachelementofXbelongstoatmost1/βelementsofφ(X),ω∈In(weassumethatλ(Sd−1)=1).Usingωd−1 216ParabolicIteratedFunctionSystemsthisandtheσ-invarianceofmeasureµ,wegetforeveryn≥0µ◦π−1φ(E)≥β−1µ◦π−1φ(E)ωω|ω|=n|ω|=n≥β−1µωπ−1(E)|ω|=n−1−1=βµ◦π(E),whereωA={ωκ:κ∈A}foreverysetA⊂I∞.Hence,foreveryk≥0,∞µ◦π−1φ(E)≥β−1µ◦π−1(E),ωn=k|ω|=nandthereforeµ◦π−1(E)≥β−1µ◦π−1(E)>0.Thiscontradiction∞finishestheproof.
Letθ=θ(S)=inf{t≥0:P(t)<∞}.FollowingthepreviouschapterwecallθthefinitenessparameterofthesystemS.Recallthatα={[i]:i∈I}isthepartitionofI∞intoinitialcylindersoflength1.ThefollowingresulthasbeenprovedinthehyperboliccontextasTheorem4.4.2.Theorem8.2.4Ifµisashift-invariantergodicBorelprobabilitymeas-ureonI∞suchthatH(α)<∞,χ(σ)=ζdµ<∞andeitherµµχµ(σ)>0orhµ(σ)>0(hµ(σ)>0impliesχµ(σ)>0),then−1hµ(σ)HD(µ◦π)=.χµ(σ)Thesameproofgoesthrough.Let−1β=β(S)=sup{HD(µ◦π)},wherethesupremumistakenoverallergodicfinitelysupported(soshift-invariant)measuresofpositiveentropy.Ofcoursetherearemanysuchmeasures.Weshallprovethefollowing.Proposition8.2.5ThepressurefunctionP(t)hasthefollowingprop-erties:(1)P(t)≥0forallt≥0 8.2Topologicalpressureandassociatedparameters217(2)P(t)>0forall0≤t<β.(3)P(t)=0forallt≥β.(4)P(t)isnon-increasing.(5)P(t)isstrictlydecreasingon[θ,β].(6)P(t)iscontinuousandconvexon(θ,∞).Proof.(1)Letibeaparabolicindexandletxibethecorrespond-ingparabolicpoint.Thenπ(i∞)=x.LetµbetheDiracmeas-iuresupportedoni∞.Ofcourse,µisergodic,finitelysupported,andtgdµ=tlog|φ
(x)|=0.Hence,byTheorem2.1.6andProposi-iition8.2.1,P(σ,tg)≥hµ(σ)+tgdµ=0and(1)isproved.(2)Supposethat0≤t<β.Thenthereexistsanergodic,σ-invariant,andfinitelysupportedmeasureµofpositiveentropysuchthatHD(µ◦π−1)>t.So,Theorem8.2.4appliestogivet0.(3)SupposethatP(t)>0forsomet≥0.TheninviewofTheorem2.1.6andProposition8.2.1thereexistsanergodicσ-invariantfinitelysuppor-tedmeasureµsuchthathµ(σ)−tχµ(σ)>0.Thereforehµ(σ)>0andhence,byTheorem8.2.3,t0.ByTheorem2.1.6andProposition8.2.1thereexistsanergodicfinitelysupportedmeasureµ2suchthathµ2(σ)+t2gdµ2≥P(σ,t2g)−4.ThenbyTheorem2.1.6andProposition8.2.1,P(σ,t1g)≥hµ2(σ)+t1gdµ2=hµ2(σ)+t2gdµ2+(t1−t2)gdµ2≥hµ2(σ)+t2gdµ2≥P(σ,t2g)−4.Letting40,wearedone.(5)Supposeθ≤t10,inviewofThe-orem2.1.6andProposition8.2.1thereexistsanergodicσ-invariantandfinitelysupportedmeasureµ2suchthat1t2−t1hµ2(σ)+t2gdµ2≥max,1−P(σ,t2g).(8.7)24βhµ(σ)Thenh(σ)≥P(σ,tg)/2>0andthereforebyTheorem8.2.4,2=µ22χµ(σ)2HD(µ◦π−1)≤β.Hence−gdµ≥h(σ)/β≥P(σ,tg)/2β.Thus,22µ22 218ParabolicIteratedFunctionSystemsusing(8.7),Theorem2.1.6andProposition8.2.1,wegetP(σ,t1g)≥hµ2(σ)+t1gdµ2=hµ2(σ)+t2gdµ2+(t1−t2)gdµ2t2−t1t2−t1≥P(σ,t2g)−P(σ,t2g)+P(σ,t2g)4β2βt2−t1=P(σ,t2g)+P(σ,t2g)>P(σ,t2g).4βAnapplicationofH¨older’sinequalityshowsthateachfunctionn−1t→expsupg(σj(τ))τ∈[ω]|ω|=nj=0islogconvex.Thereforethemapt→P(t),t∈(θ,∞),isconvexandconsequentlycontinuous.
Letusremarkthatitispossiblethatβ=θ.Wewillcallsuchsystems“strange”anddealwiththeminmoredetailinsections8.4and8.5.Also,althoughitcanhappenthatθS=0,wealwayshaveP(0)≥log2andthereforeh>0.8.3Perron–Frobeniusoperator,semiconformalmeasuresandHausdorffdimensionItfollowsfromProposition8.2.5thatβisthefirstzeroofthepressurefunction.Weshallprovidebelowmorecharacterizationsofthisnumber.Givent>θ(S)wedefineinafamiliarfashiontheassociatedPerron–FrobeniusoperatoractingonC(X)asfollowsL(f)(x)=|φ
(x)|tf(φ(x)).tiii∈INoticethatthenthcompositionofLsatisfiesLn(f)(x)=|φ
(x)|tf(φ(x)).tωω|ω|=nConsiderthedualoperatorL∗actingonthespaceoffiniteBorelmeas-turesonXasfollows∗Lt(ν)(f)=ν(Lt(f)).Noticethatthemapν→L∗(ν)/L∗(ν)(1)sendingthespaceofBorelttprobabilitymeasuresintoitselfiscontinuousandbytheSchauder- 8.3Perron–Frobeniusoperator,semiconformalmeasures219Tikhonovtheoremithasafixedpoint.InotherwordsL∗(ν)=λν,tforsomeprobabilitymeasureν,whereλ=L∗(ν)(1)>0.Aprobabilitytmeasuremissaidtobe(λ,t)-semiconformalprovidedthatL∗(m)=λm.tIfλ=1wesimplyspeakaboutt-semiconformalmeasures.RepeatingashortargumentfromtheproofofTheorem3.6of[MU1]weshallfirstprovethefollowing.Lemma8.3.1Ifmisa(λ,t)-semiconformalmeasureforthesystemSwithλ>0,thenm(J)=1.Proof.Foreachn≥1let
Xn=|ω|=nφω(X).ThesetsXnformadescendingfamilyandn≥1Xn=J.Noticethat11X|ω|◦φω=11Xforallω∈I∗andtherefore,usingthe(λ,t)-semiconformalityofm,weobtainforeveryn≥1λnm(X)=11dL∗n(m)=Ln(11)dmnXnttXn=|φ
|t(11◦φ)dmωXnω|ω|=n=|φ
|tdm=11dL∗n(m)ωXt|ω|=n=λn11dm=λn.X
Thus,m(Xn)=1andthereforem(J)=mn≥1Xn)=1.
Wesetψ(t)=||φ
||t.nω|ω|=nWenotethatθ(S)=inf{t:ψ(t)=ψ1(t)<∞}.Inordertodemon-stratetheexistenceof(eP(t),t)-semiconformalmeasuresweshallprovethefollowing.Lemma8.3.2Ift>θ(S)andL∗(m)=λmforsomemeasuremontX,thenλ=eP(t). 220ParabolicIteratedFunctionSystemsProof.Wefirstshowtheeasierpartthatλ≤eP(t).Indeed,foralln≥1λn=Ln(11)dm=|φ
(x)|tdm(x)tXω|ω|=n≤||φ
||tdm=||φ
||tωω|ω|=n|ω|=nandtherefore1
tlogλ≤limlog||φω||=P(t).(8.8)n→∞n|ω|=nInordertoprovetheoppositeinequality,foreachp≥1,letTp=
tppω∈Ip||φω||,whereIgisthesetofthosewordsω∈Isuchthatgωp−1,ωparenotthesameparabolicelement.Foreachn,ψ(t)=||φ
||t≤||φ
||t+||φ
||t||φ
||tnωωωi|ω|=nω∈Ini∈Ωω∈In−1ggn+||φ
||t||φ
||t+···+||φ
t≤#ΩT,ωiiin||ki∈Ωω∈In−2i∈Ωk=0gwhereT0=1.Take0≤q(n)≤nthatmaximizesTk.Thenψn≤(n+1)#ΩTq(n)andtherefore1P(t)=limlogψnn→∞nlog(n+1)q(n)11≤liminf+·logTq(n)+log#Ω(8.9)n→∞nnq(n)n1≤max0,limsuplogTn.n→∞nLetL˜n(1)=|φ
|t.tωω∈IngItfollowsfromcondition(5)ofasubparaboliciteratedfunctionsystemthatforalln≥1,ω∈Inandallx∈Xg||φ
||t≤Kt|φ
(x)|t.ωωSummingwehaveT≤KtL˜n(1)(x)andintegratingthisinequalitywithntrespecttothemeasurem,wegetT≤KtL˜n(1)(x)dm(x)≤Ktλn.nt 8.3Perron–Frobeniusoperator,semiconformalmeasures221Thus,by(8.9)1P(t)≤max{0,limsuplogTn}≤max{0,logλ}.n→∞nIfnowt<β(S),thenbyProposition8.2.5(2),P(t)>0,andwethereforegetP(t)≤logλ.Thus,wearedoneinthiscase.So,supposethatt≥β(S).ThenbyProposition8.2.5(3),P(t)=0andinviewof(8.8)wearelefttoshowthatλ≥1.Inordertodoitfixanarbitrary0<η<1.Itfollowsfromconditions(4)and(2)thatforallnlargeenough,sayn≥n,|φ
nforalli∈Ωandallx∈X.Fix0in(x)|≥ηj∈Ω.Wethenhaveforalln≥n0λn=11dL∗n(m)=|φ
|tdm≥|φ
tdm≥ηtndm=ηtn.ωjn||ω|=nThusλ≥ηtandlettingη1wegetλ≥1.
Lemma8.3.3Foreveryt>θ(S)a(P(t),t)-semiconformalmeasureexists.Proof.InviewofLemma8.3.2itsufficestoprovetheexistenceofaneigenmeasureoftheconjugateoperatorL∗.ButthishasbeendoneinttheparagraphprecedingLemma8.3.1.
Lete=e(S)betheinfimumoftheexponentsforwhichat-semiconformalmeasureexists.Weshallshortlyseethisinfimumisaminimum.Also,leth=hSbetheHausdorffdimensionofthelimitsetJ.Asanimme-diateconsequenceofProposition8.2.5(3)andLemma8.3.3wegetthefollowing.Lemma8.3.4e(S)≤β(S).Now,supposethatmist-semiconformal,orequivalently|φ
|t(f◦φ)dm=fdm,(8.10)ωωω∈Inforeverycontinuousfunctionf:X→IR.Sincethisequalityextendstoallboundedmeasurablefunctionsf,wegetm(φ(A))=|φ
|t(1◦φ)dm≥|φ
|tdm(8.11)ωτφω(A)τωτ∈InAforalln≥1,ω∈InandallBorelsubsetsAofX. 222ParabolicIteratedFunctionSystemsOurnexttaskinthissectionistonotethath≤e.Butthisfollowsimmediatelyfromthefollowinglemmawhoseproof,using(8.10),isthesameastheproofofTheoremt4.4.1.Lemma8.3.5Ifmisat-semiconformalmeasure,thenHt|mandJttheRadon-NikodymderivativedHisuniformlyboundedfromabove.dmSinceobviouslyβ≤h,wehavethusprovedthefollowingcharacteriz-ationoftheHausdorffdimensionofthelimitset;cf.Theorem4.2.13,whichistrueinthehyperboliccontext,andthediscussionprecedingit.Theorem8.3.6e=β=h=theminimalzeroofthepressurefunction.AsanimmediateconsequenceofLemma8.3.5,Lemma8.3.3,Proposi-tion8.2.5(3)andTheorem8.3.6wegetthefollowing.Corollary8.3.7Theh-dimensionalHausdorffmeasureofthelimitsetJisfinite.8.4Theassociatedhyperbolicsystem.ConformalandinvariantmeasuresInthissectionwedescribehowtoassociatetoourparabolicsystemanewsystemwhichishyperbolicandwhosepropertiesweapplytostudytheoriginalsystem,inparticulartoprovetheexistenceofh-conformalmeasures.However,webeginthissectionwitharesultdescribingthestructureoft-semiconformalmeasureswithexponentst>h.LetΩ={φ(x):i∈Ω,ω∈I∗}.∗ωiSo,Ω∗isthesetoforbitsofparabolicpoints.Thefollowingthe-oremallowsustoconcludethatat-semiconformalmeasureisconformalprovidedtheparabolicorbitsdonotmix.Theorem8.4.1Ift>handmtisat-semiconformalmeasure,thenmtissupportedonΩ,thatism(Ω)=1.Ifforeveryω∈I∗andevery∗t∗i∈Ω,π−1(φ(x))=ωi∞,theneacht-semiconformalmeasure(t>h)ωiist-conformal. 8.4Theassociatedhyperbolicsystem223Proof.Foreveryr>hletmrbeanr-semiconformalmeasure.Notethattheexistenceofatleastonesuchmeasure(foreveryr>h)hasbeenprovedinLemma8.3.3,cf.alsoProposition8.2.5(3)andTheorem8.3.6.Itisthennotdifficulttoseethatforeveryr>hthereexistsaBorelprobabilitymeasure˜monI∞suchthat˜m◦π−1=mand˜m([ω])=rrrr|φ
|rdm,forallω∈I∗.Now,fixt>handh0.Wethengeti|φ
(y)|−|φ
(x)|injinj|φ
inj(y)||φ
inj(y)|=|φ
(x)|1−≤||φ
||loginj
inj
|φinj(x)||φinj(x)|n−1≤log|φ
(y)|−log|φ
(x)|+log|φ
(φ(y))|−log|φ
(φ(x))|jjiikjiikjk=0n−1K1≤|φ
(y)|−|φ
(x)|+|φ
(φ(y))|−|φ
(φ(x))|φ
jjtiikjiikjjk=0n−11≤KL|y−x|+L|φikj(y)−φikj(x)|tk=0n−1L
≤KL|y−x|+||φik||Xi|φj(y)−φj(x)|tk=0∞L
≤KL+||φik||Xi|y−x|tk=0T≤LK+|y−x|.t
InparticularallthefamiliestLog={tlog|φ
|}areH¨older.bb∈I∗ThelimitsetgeneratedbythesystemS∗isdenotedby.ThenextlemmashowsthataslongasweareinterestedinthefractalgeometryofthelimitsetJ,wecanreplacethissetbyJ∗.SSLemma8.4.3ThelimitsetsJandJ∗ofthesystemsSandS∗respect-ivelydifferonlybyacountableset:J∗⊂JandJJ∗iscountable.Proof.Indeed,itisobviousthatJ∗⊂J.Ontheotherhand,theonlyinfinitewordsgeneratedbySbutnotgeneratedbyS∗areoftheformωi∞,whereωisafinitewordandiisaparabolicelementofS.
Definition8.4.4IfSisaniteratedfunctionsystemwithlimitsetJ,thenameasureνsupportedonJissaidtobeinvariantforthesystem 8.4Theassociatedhyperbolicsystem225Sprovidedν(E)=νφi(E)i∈IandνissaidtobeergodicforthesystemSprovidedν(E)=0orν(JE)=0wheneverν(Ei∈Iφi(E))=0.Letussetupsomenotation.LetJ0⊂JconsistofallpointswithauniquecodeunderS.Foreachx=π(ω)∈Jexpressω=inτ,where0iisaparabolicelement,n≥0,τ1=ianddefinen(x)=n.Foreachk≥0,putBk={x∈J0:n(x)=k}andDk={x∈J0:n(x)≥k}.Thenexttheoremshowshowtoobtaininvariant(σ-finite)measuresfortheparabolicsystemSprovidedthataprobabilityS∗-invariantmeasureisgiven.Theorem8.4.5Supposethatµ∗onJ∗isaprobabilitymeasureinvari-antunderS∗andµ∗(J)=1.Definethemeasureµbysettingforeach0BorelsetE⊂J0,µ∗(φ(E)∩D)=µ∗(φ∗µ(E)=ωkik(E))+µ(E).(8.13)k=0|ω|=kk≥1i∈ΩThenµisaσ-finiteinvariantmeasureforthesystemSandµ∗isab-solutelycontinuouswithrespecttoµ.If,foreachi∈I,themeasureµ∗◦φisabsolutelycontinuouswithrespecttothemeasureµ∗,thenµiandµ∗areequivalent;andifµ∗isergodicforthesystemS∗,thenµisergodicforthesystemS.Moreover,inthislastcaseµisuniqueuptoamultiplicativeconstant.Proof.LetuscheckfirstthatµisS-invariant.Indeed,∞µφ(E)=µ∗φφ(E))+µ∗(φ(E))jik(jjj∈Ik≥1i∈Ωj∈Ij∈I∞=µ∗(φ(E))+µ∗(φ(E))ikjjk=1i∈Ωj∈Ij∈I 226ParabolicIteratedFunctionSystems∞∞=µ∗(φ∗ikj(E)∩Dk+1)+µ(φikj(E)∩Bk)k=1i∈Ωj∈Ik=1i∈Ωj∈I+µ∗(φ(E))jj∈I∞∞=µ∗(φ∗∗ik(E))+µ(φikj(E))+µ(φj(E))k=2i∈Ωk=1i∈Ωj=ij∈I∞=µ∗(φ∗∗ik(E))+µ(E)+µ(φj(E))=µ(E),k=2i∈Ωj∈Ωwherethelastequalityholdsduetotheinvarianceofµ∗underS∗.Theinvarianceofµhasbeenproved.SinceJ0=n≥0Bn,inordertoshowthatµisσ-finiteitsufficestodemonstratethatµ(Bn)<∞foreveryn≥0.Andindeed,givenn≥0wehave∗∗µ(Bn)=µ(φik(Bn))+µ(Bn).(8.14)k≥1i∈ΩNow,foreveryi∈Ω,φik(Bn)⊂Bk∪Bn+k.∞∗∗∞∗Hence,µ(Bn)≤2#Ωk=0µ(Bk)=2#Ωµ(k=0Bk)=2#Ωµ(X)=2#Ω.Thus,µisσ-finite.Itfollowsinturnfrom(8.13)thatµ(E)=0impliesµ∗(E)=µ∗(E∩D)=0.So,µ∗isabsolutelycontinuouswith0respecttoµ.Nowsupposethatforeachi∈I,themeasureµ∗◦φisabsolutelycon-itinuouswithrespecttothemeasureµ∗.Ifµ∗(E)=0,thenµ∗(φ(E))=ω0forallω∈I∗.Thus,itfollowsfrom(8.13)thatµ(E)=0andtheequivalenceofµandµ∗isshown.SupposenowthatEisS-invariant,implyingthatφ(E)⊂E.Thenφ(E)⊂Eandsinceµ∗isi∈Iiω∈I∗ωergodic,eitherµ∗(E)=0orµ∗(Ec)=0.Sinceµisabsolutelycontinu-ouswithrespecttoµ∗,thisimpliesthateitherµ(E)=0orµ(Ec)=0.Henceµisergodicandtheproofiscomplete.
WecanprovidethefollowingnecessaryandsufficientconditionfortheS-invariantmeasureproducedinthistheoremtobefinite. 8.4Theassociatedhyperbolicsystem227Theorem8.4.6IftheassumptionsofTheorem8.4.5aresatisfied,thentheσ-finitemeasureµproducedthereisfiniteifandonlyif∗nµ(Bn)<∞.n≥1Proof.LetussetBi={x∈J:x=π(jnτ),j∈Ω{i},τ∈I∞,τ=j}n01andDi=Bi.By(8.14),wecanwriten≥0nµ(J)=µ(B)=µ∗(φ))nik(Bnn≥0n≥0k≥0i∈Ω∗∗i=µ(Bk+n)+µ(φik(Bn))k≥0n≥0k≥0n≥0i∈Iµ∗(B)+µ∗(φi=k+nik(D))k≥0n≥0k≥0i∈Ω=(n+1)µ∗(B)+µ∗(B)=(n+2)µ∗(B).nnnn≥0n≥0n≥0
Themainresultofthissection,relatingconformalmeasuresofthesys-temsSandS∗,isprovidedbythefollowing.Theorem8.4.7SupposethatSisaparabolicconformaliteratedfunc-tionsystemandtheassociatedhyperbolicsystemS∗isregular.Thenm,theh-conformalmeasureforS∗,isalsoh-conformalforSandmistheonlyh-semiconformalmeasureforS.Proof.Wewillfirstshowthatmish-conformalforthesystemSoverthelimitsetJ.WewillthenassociatewithSonemorehyperbolicsystemS∗∗andusesomepropertiesofthissystemtoverifythatmish-conformalforS.Sincem(J∗)=1,theprobabilitymeasuremclearlysatisfiesthefirstconditionforconformality:m(J)=1.Next,wewillshowthatmsatisfiesequation(4.30)forallBorelsubsetsAofJ.SinceJJ∗iscountableandmisatomless,itsufficestoshowthat(4.30f)holdsforBorelsubsetsofJ∗.Also,since(4.30)holdswheneveriisahyperbolicindexevenforBorelsubsetsofX,weonlyneedtoverify(4.30)forparabolicindices.LetG={A:AisaBorelsubsetofJ∗and(4.30)holds∀i∈Ω}. 228ParabolicIteratedFunctionSystemsSinceGisclosedundermonotonelimits,itsufficestoshowthat(4.30)holdsforeverysubsetUofJ∗whichisrelativelyopen.Let∞Γ={ω∈I∗:ω=(a1b1),(a2b2),(a3b3),...;∀nan,bn∈Ω,bn=an,an+1}.LetW=π(Γ).UsingTheorem2.2.4,Theorem3.2.3,Proposition4.2.5andBirkhoff’sergodictheorem,weseethatm(W)=0andm(φi(W))=0,∀i∈Ω.Letusdemonstratethatifi∈Ωandω=(ω1,ω2,ω3,...)∈I∞Γ,thenthereissomelsuchthatforeveryk≥l,(ω,...,ω)∈I∗∗1k∗andtheconcatenationi∗ω1∗···∗ωkcanbeparsed(orregrouped)sothatitrepresentsanelementofI∗.Toseethis,firstsupposethatω∈IΩ.∗1Thenl=1sincei∗ω1∗···∗ωkcanbeparsedasiω1,ω2,...,ωkwhichisanelementofI∗.Now,supposeω=pnqwherep∈Ωandp=q.∗1Ifp=i,thenagainl=1,sincei∗ω1∗···∗ωkcanbeparsedasin+1q,ω,...,ωwhichisanelementofI∗.Ifi=pandn>1,then2k∗i∗ω∗···∗ωcanbeparsedas(ip,pn−1q,ω,ω,...,ω)∈I∗and1k23k∗alsointhiscasel=1.If,ontheotherhand,n=1andp=i,thenω1=a1b1,wherea1∈Ωandb1=a1.Ifb1∈IΩ,theni∗ω1∗···∗ωkcanbeparsedas(ia,b,ω,ω,...,ω)∈I∗andl=1.So,suppose1123k∗thatb1∈Ω.Now,considerω2.Ifω2∈IΩ,thentheconcatenationi∗ω∗···∗ωcanbeparsedas(ia,bω,ω,...,ω)∈I∗andl=2.1k1123k∗Otherwiseω=pnq,wherep∈Ω,q=pandn≥1.Ifp=b,21theni∗ω∗···∗ωcanbeparsedas(ia,bn+1q,ω,...,ω)∈I∗and1k113k∗l=2.Ifp=b1andn>1,theni∗ω1∗···∗ωkcanbeparsedas(ia,bp,pn−1q,ω,...,ω)∈I∗andl=2.If,ontheotherhand,n=1,113k∗thenω2=a2b2,wherea2=b1,b2.Ifb2∈IΩ,theni∗ω1∗···∗ωkcanbeparsedas(ia,ba,b,ω,...,ω)∈I∗andl=2.So,wemay11223k∗assumethatb2∈Ω.Now,excludinginductivelyinthismannerthecaseswheni∗ω1∗···∗ωkcanbeparsedinsuchafashionthatitwouldbelongtoI∗,wewouldendupwiththeconclusionthatω∈Γ,∗contrarytoourassumption.Now,letU⊂J∗berelativelyopen.ThenthereisasetM⊂I∗,consistingofincomparablewordsandsuchthat∗UW⊂φ(J∗)⊂U,andifτ∈Mtheni∗τ∈I∗.Thus,τ∈Mτ∗∗∗∗m(φi(U))=m(φi(φτ(J)))∪(Uφτ(J))=m(φi(φτ(J)))τ
h
h
h=|(φi◦φτ)|dm=|φi|dm=|φi|dm,τJ∗τφτ(J∗)Uwherethethirdequalityfollowssincemish-conformalforthesystemS∗andinthefourthequalityweadditionallyemployedthechangeof 8.4Theassociatedhyperbolicsystem229variablesformula.Now,wewanttoshowm(φi(J)∩φj(J))=0wheneveri=j.Again,itsufficestoverifythiswhenJisreplacedbyJ∗andatleastoneoftheindicesiandjisparabolic.AsbeforethereisasetM⊂I∗ofincomparablewordsandsuchthatJ∗W⊂i∗φ(J∗)⊂J∗,andifτ∈Mtheni∗τ∈I∗.Also,letM⊂I∗τ∈Miτi∗j∗havesimilarpropertieswithrespecttotheindexj.Thenm(φ(J)∩φ(J))=m(φ(J∗)∩φ(J∗))ijiτjρτ,ρ∈Mi×Mj∗∗≤m(φiτ(J)∩φjρ(J))=0.Mi×MjFinally,toshowthatmisconformal,wemustdemonstratethat(4.30)and(4.31)holdwheneverAisaBorelsubsetofX.Notethatitsufficestoshowthatm(A)=0impliesm(φi(A))=0,forallBorelsubsetsAofXandallparabolicindicesi.Inordertoprovethis,weintroduceanewhyperbolicsystem.TheindexsetforthissystemisI=I3{(i,i,i):∗∗i∈Ω}∪{pnq:p∈Ω,q=p,n≥2}.LetusprovethatthesystemS∗∗satisfiestheboundeddistortionproperty.Toseethis,readawordω∈I∗asawordinI∗:ω=(ω,ω,...,ω).Ifω∈IΩ,thenwe∗∗12nnhaveboundeddistortionbyproperty(5)ofthesystemS.Ifωn∈Ωandωn−1=ωn,thenagainbyproperty(5)wehaveboundeddistortionwithconstantK.Ifω=ω,thenω=ω,bythedefinitionofI∗.n−1nn−2n−1∗∗Thenthewordω|n−1satisfiesthehypothesisofcondition(5)andso
|φ
(φ(y))||φ
(y)||φω(y)|ω|n−1ωnωn=|φ
ω(x)||φ
(φωn(y))||φ
ω(y)|ω|n−1n||φ
||i≤Kmax:i∈Ω,min{φ
i(x):x∈X}wherethelastnumberisfinitesinceΩis.ToseethatS∗∗satisfiestheopensetcondition,noticethatφijk(Int(X))∩φpqr(Int(X))=∅forallijk=pqr.Nextconsiderφinj(Int(X))∩φpmq(Int(X)),wheren,m≥2.Ifi=p,thisintersectionisempty.Alsoifi=pandn=m,theintersectionisempty.Otherwise,q=jandtheintersectionisempty.Finally,considerφinj(Int(X))∩φpqr(Int(X)),wheren≥2.Ifi=porifi=pandq=i,theintersectionisempty.Otherwise,i=p=qandinthiscaser=isincetheword(i,i,i)isnotallowedinI∗∗.Finally,thehyperbolicityofthesystemS∗∗isanimmediateconsequenceofproperty 230ParabolicIteratedFunctionSystems(6).So,S∗∗isahyperbolicconformaliteratedfunctionsystem.Also,sinceeachelementofI∞canbeparsedintoanelementofI∞,wehave∗∗∗J∗∗=J∗=J{eventuallyparabolicpoints}.AlsonoticethatifthesystemS∗isregular,thenthesystemS∗∗isregular.ToseethisnotethatwehavealreadyshownthatifmisconformalforS∗,thenmisconformalforSoverJ.ThusmisconformalforS∗∗overJ.So,foreachn,1=dm=|φ
(x)|dm.But,foreachx∈J,weJJω∈Inω∗∗have||φ
||h≥|φ
(x)|h≥(K∗∗)−h||φ
||h,ωωωω∈Inω∈Inω∈In∗∗∗∗∗∗whereK∗∗isthedistortionconstantforthesystemS∗∗overX.Integ-ratingthisformulaagainstthemeasuremweget||φ
||h≥1≥(K∗∗)−h||φ
||h.ωωω∈Inω∈In∗∗∗∗∗∗FromthisitimmediatelyfollowsthatP(h)=0.But,thisisequivalenttosayingthatthereisanh-conformalmeasurem∗∗forthesystemS∗∗overX.Weonlyneedtoprovethatm∗∗=m.LetGbeopenrelativetoJ∗.LetWbeacollectionofincomparablewordsinIsuchthat∗∗G=φ(J∗).SincemisconformalforS∗∗overJ,ω∈Wωm(G)=|φ
|dm≤Kh||φ
||≤KhK∗∗h|φ
|dm∗∗ωωωω∈WJω∈Wω∈WJ=KhK∗∗hm∗∗(G).Interchangingmandm∗∗intheaboveestimate,weget(KhK∗∗h)−1m∗∗(G)≤m(G)≤KhK∗∗hm∗∗(G).Fromthisitfollowsthatmandm∗∗areequivalent.Toshowthatm=m∗∗,letAbeaBorelsubsetofX.Thenm(φ(A))=m(φ(A∩ωωJ))+m(φ(AJ)).But,sincem∗∗isconformaloverX,m∗∗(AJ)=ω|φ
|hdm∗∗=0.So,sincemisconformalforSoverJ,wehaveAJωm(φ(A))=|φ
|hdm=|φ
|hdm.Also,onecanshowthatωA∩JωAω(4.30)holdsusingthesameprocedure.Thus,misconformalforS∗∗overX.Finally,toseethatmisconformalfortheentiresystemSoverX,leti∈Ωandchooseanarbitraryq=i,q∈I.Theniq∈I∗andiqi∈I∗∗.Thus,|φ
|hdm=m(φ(φ(A))=m(φ(A)))=|φ
|hdm.iqiqiiqiiqiφi(A)A 8.4Theassociatedhyperbolicsystem231So,ifm(A)=0,thensince|φ
|hispositiveonφ(A),wehaveiqim(φi(A))=0.InordertoprovethesecondpartofourtheoremsupposethatνisanarbitrarymeasuresupportedonJandsatisfyingν(φ(A))≥|φ
|hdν(8.15)ωωAforallBorelsetsA⊂Xandallω∈I∗.Weshowthatmisabsolutelycontinuouswithrespecttoν.Indeed,foreveryω∈I∗wehave∗ν(φ(X))≥|φ
|hdν≥K−h||φ
||hωωωX≥K−h|φ
|hdm=K−hm(φ(X)).ωωXNext,consideranarbitraryBorelsetA⊂Xsuchthatν(A)=0.Fix4>0.SinceνisregularthereexistsanopensubsetGofXsuchthatA∩J∗⊂Gandν(G)≤4.TherenowexistsafamilyF⊂I∗of∗mutuallyincomparablewordssuchthatA∩J∗⊂φ(X)⊂G.ω∈FωIteasilyfollowsfromLemma4.2.4thatthereexistsauniversalupperboundMonthemultiplicityofthefamily{φω(X):ω∈F}.Hence,usingthefactthatmissupportedonJ∗,weobtain∗m(A)=m(A∩J)≤mφω(X)≤m(φω(X))ω∈Fω∈F≤Khν(φ(X))≤KhMνφ(X)≤KhMν(G)ωωω∈Fω∈Fh≤KM4.Thus,letting40,wegetm(A)=0,whichfinishestheproofoftheabsolutecontinuityofmwithrespecttoν.Ournextaimistoshowthatν(JJ∗)=0.Supposeonthecontrarythatν(JJ∗)>0.SetP={φ(x):i∈Ω,ω∈I∗}.SinceJJ∗⊂P,ν(P)>0.Writeωiν=ν0+ν1,whereν0|XP=0andν1|P=0.Thusν0(P)=ν(P)>0.Sinceφ(P)⊂Pforallω∈I∗,wegetforeveryBorelsetA⊂Xandωeveryω∈I∗ν0(φω(A))≥ν0(φω(A∩P))=ν(φω(A∩P))
h
h≥|φω|dν=|φω|dν0.A∩PA 232ParabolicIteratedFunctionSystemsHence,multiplyingν0by1/ν0(X),weconcludefromwhathasbeenprovedthatmisabsolutelycontinuouswithrespecttoν0.Sinceν0(XP)=0,thisimpliesthatm(XP)=0,andconsequentlym(P)=1.Thisisacontradictionsinceanyconformalmeasureofahyperbolicsystemiscontinuous.Thusν(J∗)=1.Supposenowinadditionthatνish-conformalforS.Then,aswehavejustproved,νish-conformalforS∗(thisstatementincludesthatfactthatν(J∗)=1).Theequalityν=mfollowsnowfromTheorem4.2.9.Finally,noticethatusingtheargumentfromtheproofofLemma3.10from[MU1]andproceedingasintheproofofTheorem5.7from[MU7]onecouldprovethatmistheuniqueh-semiconformalmeasureforS.
Followingthecaseofhyperbolicsystemswecallaparabolicsystemregularifthereexistsanh-conformalmeasureforSsupportedonJ∗.Sincesuchameasureish-conformalforS∗,asanimmediateconsequenceofTheorem8.4.7wegetthefollowing.Corollary8.4.8Theparabolicsystemisregularifandonlyiftheas-sociatedsystemS∗isregular.Intryingtosaysomethingaboutparabolicsystemswhicharenotreg-ular,weareledtointroducetheclassof“strange”systems,whichbydefinitionarethosesystemsforwhichthereisnotwith02max:i∈Ω.βi+1Proof.TheproofisanimmediateconsequenceofCorollary8.4.12.
8.5ExamplesThissectioncontainsexamplesillustratingsomeoftheideasdevelopedinthischapter.Example8.5.1Apollonianpacking.Consideronthecomplexplanethethreepointsz=e2πij/3,j=0,1,2j√andthefollowingadditionalthreepoints:a0=3−2,a1=(2−√√3)eπij/6anda=(2−3)e−πij/6.Letf,f,andfbetheM¨obius2012transformationsdeterminedbythefollowingrequirements:f0(z0)=z0,f0(z1)=a2,f0(z2)=a1,f1(z0)=a2,f1(z1)=z1,f1(z2)=a0,f2(z0)=a1,f2(z1)=a0,andf2(z2)=z2.SetX=B(0,1),theclosedballcenteredattheoriginofradius1.Itisstraightforwardthattheimages 8.5Examples235f0(X),f1(X)andf2(X)aremutuallytangent(atthepointsa0,a1anda2,respectively)diskswhoseboundariespassthroughthetriples(z0,a1,a2),(z1,a0,a2)and(z2,a0,a1)respectively.Ofcourseallthethreedisksf0(X),f1(X)andf2(X)arecontainedinXandaretangenttoXatthepointsz0,z1andz2respectively.LetS={f0,f1,f2}betheiteratedfunctionsystemonXgeneratedbyf0,f1andf2.Noticethatallthemapsf0,f1andf2areparabolicwithparabolicfixedpointsz0,z1andz2respectively.Itisnotdifficulttocheckthatalltherequirementsofaparabolicsystemaresatisfied.ObservethatthelimitsetJoftheparabolicsystemScoincideswiththeresidualsetoftheApollonianpackinggeneratedbythecurvilineartrianglewithverticesz0,z1,z2.In[MU8],usingaslightlydifferentiteratedfunctionsystem,wehavedealtwithgeometricalpropertiesofJ,provingthat11=21,itthereforefollowsfromi1+1Corollary8.4.13thatµisfinite.
Example8.5.3AlargeclassofexamplesappearsalreadyinthecasewhenXisacompactsubintervalofthereallineIR.Wecallsuchsystemsone-dimensional.Iftheparabolicelementsφiofaone-dimensionalsystemShave,aroundparabolicfixedpointsxi,arepresentationoftheform1+βi1+βiφi(x)=x+a(x−xi)+o((x−xi))(8.16)then(seee.g.,[U4])βi+1|φ
−βi(8.17)in(x)|#noutsideeveryopenneighborhoodofxi.HencethefollowingtheoremisaconsequenceofTheorem8.4.7andCorollary8.4.13.Theorem8.5.4IfSisaone-dimensionalparabolicsystemwithfinitealphabetandsatisfying(8.16),thenSisregularandanyS-invariantmeasureµequivalentwiththehS-conformalmeasureisfiniteifandonlyβiifh>2max{:i∈Ω}.βi+1Proof.TheregularityofS∗ischeckedinexactlythesamewayasinExample8.5.1.So,thesystemSisregularbyCorollary8.4.8.SincetheotherassumptionsofCorollary8.4.13aresatisfiedby(8.17),theproofofthistheoremisanimmediateconsequenceofCorollary8.4.13.
Corollary8.5.5IfSisaone-dimensionalparabolicsystemwithfinitealphabet,andifforalli∈Ω,βi≥1(orequivalentlyifallφi’saretwicedifferentiableatxi),thenSisregularandthecorrespondinginvariantmeasureµequivalentwiththehS-conformalmeasureisinfinite.Proof.TheproofisanimmediateconsequenceofTheorem8.5.4andthefactthath≤1.
Wewouldliketoclosethissectionwithsomeexamplesofstrangesys-tems.Example8.5.6Ouraimhereistodescribeaclassofone-dimensionalsystemswhicharestrange.Towardsthisendconsideranarbitraryhyperbolicsystem 8.5Examples237S={φi:i∈I}ontheintervalX=[0,1]suchthatψ(θ(S))<∞orequivalentlyP(θ(S))<∞(examplesofsuchsystemsmaybefoundinthesectionExamplesof[MU1]);wemayassumethatthereisanintervalG=[0,γ)withG⊂Xi∈Iφi(X).Consideralsoaparabolicmapφ:X→Gsuchthat0isitsparabolicpointandφhasthefollowingrepresentationaround0:φ(x)=x−axβ+1+o(xβ+1),β+1whereθ(S)>1anda>0.Weshallprovethefollowing.βTheorem8.5.7IfF⊂Iisasufficientlylargefiniteset,thenthesystemSF={φ}∪{φi:i∈IF}isstrange.Proof.Inviewof(6.2)andtherelationbetweenθ(S)andβthereexistsaconstantC≥1suchthatforeachi∈I,||(φn◦φ)
||θ(S)≤n≥1iC||φ
||θ(S).Sinceψ(θ(S))<∞,foreverysufficientlylargefinitesetiSF⊂Iwehave(C+1)||φ
||θ(S)<1.Hencei∈IFiψ∗(θ(S))=||φ
||θ(S)+||(φn◦φ)
||θ(S)SFiii∈IFi∈IFn≥1≤||φ
||θ(S)+C||φ
||θ(S)iii∈IFi∈IF
θ(S)=(C+1)||φi||<1.i∈IF∗HencePSF(θ(S))<0andtherefore,ashS∗=hSF,PSF(t)=0forallFt≥θ(S).Ontheotherhand,sinceforeveryt<θ(S),ψS(t)=∞andsinceFisfinite,ψ(t)=||φ
||t+||φ
||t=∞.HenceSFi∈IFiPSF(t)=∞.
9ParabolicSystems:HausdorffandPackingMeasuresInthischapterwecontinueourstudiesofaparaboliciteratedfunctionsystemS={φi}i∈I,followingclosely[MU9].Wekeepallthenotationintroducedinthepreviouschapter.Ourmaingoalistocharacterizecon-formalmeasuresoffiniteparabolicsystemsintermsofHausdorffandpackingmeasures.Thissimultaneouslyprovidestheanswertotheques-tionaboutnecessaryandsufficientconditionsforthesetwogeometricmeasurestobefiniteandpositive.9.1PreliminariesForeveryintegerq≥1wedenoteSq={φ:ω∈Iq}.ωOfcourseJSq=JSandsometimesinthesequelitwillbemorecon-venienttoconsideranappropriateiterateSqofSratherthanSitself.Thefollowingpropositionisanimmediateconsequenceofcondition(4)fromthepreviouschapter.Proposition9.1.1IfthealphabetIisfinite,thenS∗(∞)={x:i∈Ω},thesetofparabolicfixedpoints.iInSection4weshallprovethefollowing.Theorem9.1.2IfSisafiniteparabolicIFS,thenthesystemS∗ishereditarilyregularandconsequentlyanh-conformalmeasureforS∗exists.238 9.1Preliminaries239Fromnow,unlessotherwisestated,wewillassumethatthealphabetIisfiniteandmwilldenotetheh-conformalmeasureproducedinThe-orem9.1.2.CombiningthistheoremalongwithCorollary8.3.7andCorollary8.4.10(duetotheexistenceofparabolicelementsthestrongopensetconditionissatisfied)wegetthefollowing.Theorem9.1.3IfSisafiniteparabolicIFSsatisfyingthestrongopenthsetcondition,thenH(J)<∞andP(J)>0.Thefollowingmaintheoremofthischaptercontainsacompletedescrip-tionoftheh-dimensionalHausdorffandpackingmeasuresofthelimitsetofafiniteparabolicIFS.Theorem9.1.4SupposethatSisafiniteparabolicIFSsatisfyingthestrongopensetcondition.Thenhh(a)Ifh<1,then01,then00,Disanorthogonalmatrix,andc∈IRd.Fromnowon,withoutlossofgenerality,wewillassumethatω=0,i.e.,ωistheorigin,andwewillwriteifori0,1.Lemma9.2.2IfA:IRd→IRdisaparabolicconformalmapandifλisthescalarinvolvedintheformulaforA˜,thenλ=1.Proof.Ifλ<1,thenA˜:IRd→IRdisastrictcontractionandduetoBanach’scontractionprinciple,ithasafixedpointb∈IRdsuchthatlimn→∞A˜n(z)=bforeveryz∈IRd.However,thisisacontradiction, 9.2TheCased≥3241sincelimn→∞A˜n(i(ξ))=∞.Thus,λ≥1.Assumeλ>1.Thenforeveryz∈IRd{0},||A(z)||=||i(A˜(i(z))A˜(i(z))i(z)||=λ||A˜(i(z))||−2||z||−2=λ||z||−2||λD(||z||−2(z))+c||−2−1−2=λ||(λ||z||D(z)+c||z||)||=λ−1[||D(z/||z||)+(||z||/λ)c||]−2.Sincelim||z||=0andsince||D(z)||=||z||,wededucethat||A(0)||=z→0lim||A(z)||=λ−1<1.Thiscontradictionshowsthatλ≤1,andz→0consequentlyλ=1.
Next,wewanttoestimatetherateatwhichA˜n(z)goesto+∞.Lemma9.2.3IfA:IRd→IRdisaparabolicconformalmap,thenthereexistsanon-zerovectorb∈IRdandapositiveconstantκsuchthatforeveryz∈IRdandeverypositiveintegern||A˜nz−nb||≤||z||+κ.Proof.Byastraightforwardinduction,wegetn
−1A˜nz=Dnz+Dj(c).j=0Writec=b+a,wherebisafixedpoint(aprioriperhaps0)ofDandabelongstoW,theorthogonalcomplementofthevectorspaceofthefixedpointsofD.Sincelimn→∞A˜n(i(ξ))=∞,WisnotthetrivialsubspaceofIRd.Inaddition,D(W)=WandD−Id:W→Wisinvertible.Sincen
−1(D−Id)Dj(a)=Dna−aj=0andsince||Dna−a||≤2||a||,wethereforeconcludethatforeveryn≥1n
−1Dj(a)≤2||a||·(D−Id)|−1.Wj=0Hence,n
−1||A˜nz−nb||=Dnz+Dj(a)≤||z||+2(D−Id)|−1·||a||.Wj=0 242ParabolicSystems:HausdorffandPackingMeasuresAgain,sincelimn→∞A˜n(i(ξ))=∞,wefinallyconcludethatb=0.
Asanimmediateconsequenceofthislemmawegetthefollowing.Corollary9.2.4LetA:IRd→IRdbeaparabolicconformalmap.ForeverycompactumF⊂IRd,thereexistsaconstantB≥1andintegerFMF∈INsuchthatforeveryn≥MFandeveryz∈FB−1n≤||A˜nz||≤Bn.FFLemma9.2.5LetA:IRd→IRdbeaparabolicconformalmap.ForeverycompactumL⊂IRd{0},thereexistaconstantC≥1andL,1integerNL∈INsuchthatforeveryn≥NLandeveryz∈LC−1n−2≤||(An)(z)||≤Cn−2anddiam(An(L))≤Cn−2.L,1L,1L,1Proof.Bythechainrule,wefindforeveryz∈IRd{0}nnn||(A)(z)||=||i(A˜(i(z)))||·||(A˜)(i(z))||·||i(z)||n−2−2=||A˜(i(z))||||z||.Foreveryz∈L,Dist−2(0,L)≤||z||−2≤dist−2(0,L),andinviewofCorollary9.2.4,ifn≥M,thenB−1n≤||A˜nz||≤Bn.Con-i(L)i(L)i(L)sequently,ifz∈Landn≥Mi(L),wehave−2−2n2−2−2Bi(L)Dist(0,L)n≤||(A)(z)||≤Bi(L)dist(0,L)n.
Lemma9.2.6LetA:IRd→IRdbeaparabolicconformalmap.ForeverycompactumL⊂IRd{0},thereexistsaconstantC≥1suchL,2thatforallintegersk,nwithn≥k≥1,Dist(Ak(L),An(L))≤Ck−1−(n+1)−1L,2andDist(An(L),0)≤Cn−1.L,2Proof.Letusstartwiththesecondinequality.Ifn≥Mi(L)andz∈L,then,byCorollary9.2.4,weget||Anz||=||A˜n(i(z))||−1≤Bn−1andi(L)thesecondinequalityfollowsprovidedCL,2issufficientlylarge.Towardsobtainingthefirstinequality,foreverysetY⊂IRd,letconv(Q)denotetheconvexhullofY.Obviously,conv(Y)⊂ 9.2TheCased≥3243B(Y,diam(Y))anddiam(conv(Y))=diam(Y).ByusingLemma9.2.3,wehaveforeveryu∈Landn∈IN,n+1n||A˜(i(u))−A˜(i(u))||n+1n≤||A˜(i(u))−(n+1)b−(A˜(i(u))−nb)+b||≤2(||i(u)||+κ)+||b||≤2(Dist(0,i(L))+κ)+||b||.Next,chooseapositiveintegerN0suchthatDist(0,conv(t≥N0A˜t(i(L))))=H>0andN0||b||>Dist(0,i(L))+κ+||b||:=M.WeclaimthereisapositiveconstantCsuchthatifu,v∈L,k≥N0andj≥0,thenk+j+1k+j1||A(v)−A(u)||≤C.(k+j+1)2Inordertoseethis,notethatk+j+1k+j||A(v)−A(u)||k+j+1k+j+1≤||i(A˜(i(v)))−i(A˜(i(u)))||k+j+1k+j+||i(A˜(i(u)))−i(A˜(i(u)))||k+j+1k+j+1≤sup{||i(w)||:w∈[A˜(i(v)),A˜(i(u))]}||×A˜k+j+1i(v)−A˜k+j+1i(u)||+sup{||i(w)||:w∈[A˜k+j(i(u)),A˜k+j+1(i(u))]}||×A˜k+j+1i(u)−A˜k+ji(u)||≤diam(i(L))sup{||w||−2:w∈[A˜k+j+1(i(v)),A˜k+j+1(i(u))]}+2Msup{||w||−2:w∈[A˜k+j(i(u)),A˜k+j+1(i(u))]}.Now,ifw∈[A˜k+j+1(i(v)),A˜k+j+1(i(u))],thenbyLemma9.2.3,||w−(k+j+1)b||≤Dist(0,i(L))+κand||w||≥(k+j+1)[||b||−(Dist(0,i(L))+κ)/N].Also,since||A˜k+j(i(u))−(k+j+1)b||≤||i(u)||+κ+||b||,if0w∈[A˜k+j(i(u)),A˜k+j+1(i(u))],then||w−(k+j+1)b||≤Dist(0,i(L))+κ+||b||and||w||≥(k+j+1)[||b||−(Dist(0,i(L))+κ+||b||)/N0]≥(k+j+1)[||b||−M/N0].Combiningtheseinequalitiesestablishesourclaim.Therefore,ifN0≤k≤nwehaven−
k−1knk+j+1k+jDist(A(L),A(L))≤DistA(L),A(L)j=0n
−k−2−1−1≤C(k+j)≤CL,2(k−(n+1))j=0 244ParabolicSystems:HausdorffandPackingMeasuresforsomeconstantCL,2≥1.Clearly,increasingCL,2appropriately,weseethatthelastinequalityisalsotrueforall1≤k≤n.Theproofofthefirstpartofourlemmaisthuscomplete.
Lemma9.2.7ForeverycompactumL⊂IRd{0}thereexistaconstantCL,3≥1andanintegerq≥0suchthatforallk≥1andalln≥k+qkn−1−1dist(A(L),A(L))≥CL,3k−nanddist(An(L),0)≥Cn−1.L,3Proof.First,noticethatitfollowsfromLemma9.2.3thatifw,z∈i(L)andk,n∈N,thennk(n−k)||b||−2(Dist(0,i(L))+κ)≤||A˜(w)−A˜(z)||.Therefore,thereisapositiveintegerq0suchthatifn−k≥q0,then||A˜n(w)−A˜k(z)||≥(1/2)||b||(n−k).LetNbeasintheproofof0Lemma9.2.6andMi(L)beasinCorollary9.2.4.Letk,n≥N1=max{N0,Mi(L)}.Considertwoarbitrarypointsz,w∈i(L)andpara-metrizethelinesegmentγjoiningA˜k(z)andA˜n(w)asγ(t)=A˜k(z)+t(A˜n(w)−A˜k(z)),t∈[0,1].Thecurvei(γ)isasubarcofeitheracircleoraline;letl(i(γ))beitslength.Wehavel(i(γ))11=||(i◦γ)(t)||dt=||i(γ(t))||·||γ(t)||dt001=||A˜n(w)−A˜k(z)||||γ(t)||−2dt01=||A˜n(w)−A˜k(z)||||A˜k(z)+t(A˜n(w)−A˜k(z))||−2dt01nkknk−2≥||A˜(w)−A˜(z)||||A˜(z)||+t||A˜(w)−A˜(z)||dt0 9.2TheCased≥3245=||A˜n(w)−A˜k(z)||·||A˜n(w)−A˜k(z)||−1||A˜k(z)||+||A˜n(w)−A˜k(z)||×u−2du||A˜k(z)||(9.1)=||A˜k(z)||−1−||A˜k(z)||+||A˜n(w)−A˜k(z)||||A˜n(w)−A˜k(z)||=.||A˜k(z)||·||A˜k(z)||+||A˜n(w)−A˜k(z)||Wehaveknk11||A˜(z)||+||A˜(w)−A˜(z)||≤Bi(L)k+Ci(L),1−.kn+1So,thereisaconstantUsuchthat||A˜k(z)||+||A˜n(w)−A˜k(z)||≤Un.InviewofCorollary9.2.4,thereisaconstantQ0suchthat||A˜n(w)−A˜k(z)||l(i(γ))≥Q0.knThus,thereisaconstantQsuchthatifk≥N1andn≥k+q0,thenl(i(γ))≥Q(k−1−n−1).(9.2)Ifi(γ)isalinesegment,thennk−1−1||A(i(w))−A(i(z))||=l(i(γ))≥Q(k−n).(9.3)If,however,i(γ)isanarcofacircle,thenconsidertherayg(t)=A˜k(z)+t(A˜n(w)−A˜k(z)),t∈(−∞,0].Proceedingexactlyasintheformula(9.2)andusingtheestimate||g(t)||≤||A˜k(z)||−t||A˜n(w)−A˜k(z)||,weget∞l(i(g))≥u−2du=||A˜k(z)||−1.||A˜k(z)||AndapplyingCorollary9.2.4wegetl(i(g))≥B−1k−1≥B(k−1−i(L)i(L)n−1).Therefore,invoking(9.2),wededucethatbotharcsjoiningthepointsAk(i(z))andAn(i(w))onthecirclei({A˜k(z)+t(A˜n(w)−A˜k(z)):t∈IR∪{∞}})havethe≥min{B,Q}(k−1−n−1).Thus,alsotakingi(L)intoaccount(9.3),weseethereisaconstantP0suchthatifk,n≥N1andn−k≥q0,thendist(Ak(L),An(L))≥P(k−1−n−1).0N1jSince0isnotanelementofA(L),andsinceitfollowsfromj=1Lemma9.2.3thatAk(L)→0ask→∞,thereisaconstantCL,3 246ParabolicSystems:HausdorffandPackingMeasuressuchthatthefirstpartoftheconclusionofthelemmaholds.Applyingtheprovedpartofthelemma,weconcludethatdist(An(L),0)=limdist(An(L),Ak(L))k→∞−1−1−1≥limCL,3(n−k)=CL,3n.k→∞
WeendthissectionbyprovingthefollowingresultconcerninggeneralparabolicIFSsindimensiond≥3.Thefirstisastraightforwardcon-sequenceofLemma9.2.3.First,letusnotethatLemma9.2.6showsthataconformalparabolicmapinIRd,d≥3hasauniquefixedpoint.Proposition9.2.8If{φi:X→X}i∈Iisanatleastthree-dimensionalparabolicconformalIFS(Iisallowedtobeinfinite),thenxi,theonlyfixedpointofaparabolicmapφi,belongsto∂X.Proof.InviewofLemma9.2.3,foreveryR>0largeenoughandeveryn≥1,thesetφ˜i({z:||z||>R})isnotcontainedin{z:||z||>R}.Consequently,foreveryneighborhoodUofx,thesetφn(U)doesnotiiconvergetox.Sincehoweverlimφn(X)=x,thepointxcannotin→∞iiibelongtoInt(X).
9.3Theplanecase,d=2Wecallaholomorphicmapφ,definedaroundapointω∈CI,simpleparabolicifφ(ω)=ω,φ(ω)=1andφisnottheidentitymap.Thenonasufficientlysmallneighborhoodofω,themapφhasthefollowingTaylorseriesexpansion:φ(z)=z+a(z−ω)p+1+b(z−ω)p+2+···withsomeintegerp≥1anda∈CI{0}.Beinginthecircleofideasre-latedtoFatou’sflowertheorem(see[Al]forextendedhistoricalinforma-tion),wenowwanttoanalyzequalitativelyandespeciallyquantitativelythebehaviorofφinasufficientlysmallneighborhoodoftheparabolicpointω.Letusrecallthattherayscomingoutfromωandformingtheset{z:a(z−ω)p<0}arecalledattractingdirectionsandtheraysformingtheset{z:a(z−ω)p>0} 9.3Theplanecase,d=2247arecalledrepellingdirections.Fixanattractivedirection,sayA=√ω+p−a−1(0,∞),wherep·isaholomorphicbranchofthepthrad-icaldefinedonCIa−1(0,∞).Inordertosimplifyouranalysisletuschangethesystemofcoordinateswiththehelpoftheaffinemap√ρ(z)=p−a−1+ω.Wethengetφ(z)=ρ−1◦φ◦ρ(z)=z−zp+1+bp−a−1zp+2+···0andρ−1(A)=(0,∞)isanattractivedirectionforφ.Wewantto0analyzethebehaviorofφ0onanappropriateneighborhoodof(0,5),for5>0sufficientlysmall.Inordertodoit,similarlyasintheprevioussection,weconjugateφ0onCI(−∞,0]toamapdefined“near”infinity.√Precisely,weconsiderp·,theholomorphicbranchofthepthradicaldefinedonCI(−∞,0]andleavingthepoint1fixed.Thenwedefinethemap1H(z)=√pzandconsidertheconjugatemapφ˜=H−1◦φ◦H.0Straightforwardcalculationsshowthat−1φ˜(z)=z+1+O(|z|p)(9.4)andp+1φ˜(z)=1+O(|z|−p).(9.5)Givennowapointx∈(0,∞)andα∈(0,π),letS(x,α)={z:−αmax:iisparabolic,pi+1wherepiistheintegerindicatedin(9.20).piProof.Using(9.17),ifwetaketslightlylargerthan,thenψ(t)canpi+1∗∗bemadeaslargeaswelike.SinceP(t)≥−tlogK+logψ(t),P(t)>0.piTherefore,h=HD(JS∗)>p.Itthereforeimmediatelyfollowsfromi+1Lemma8.4.3thatpiHD(JS)=HD(JS∗)>max:iisparabolic.pi+1
9.4Proofsofthemaintheorems257IfinadditionSisfinite,thenweconcludefrom(9.17)thatpiθS∗=max:iisparabolicpi+1∗andψ(θS∗)=∞.ThismeansthatthesystemSishereditarilyregularandwehaveprovedTheorem9.1.2.Lemma9.4.3Foreveryparabolicindexi∈I,thereexistsanopenconeCi⊂Xwithvertexxiandsuchthatxi∈J∩Ci.Proof.Incased≥3thisisanimmediateconsequenceofLemma9.2.3.Incased≥3thisisanimmediateconsequenceof(9.9)andLemma9.3.8.
InviewofTheorem9.1.3,inordertoproveTheorem9.1.4itsufficestodemonstratethefollowingfourlemmasassumingthefiniteparabolicsystemSsatisfiesthestrongopensetcondition.Lemma9.4.4Ifh<1,thenHh(J)=0.Lemma9.4.5Ifh≤1,thenPh(J)<∞.Lemma9.4.6Ifh>1,thenPh(J)=∞.Lemma9.4.7Ifh≥1,thenHh(J)>0.ProofofLemma9.4.4.Leti∈Ibeaparabolicindex.Fixj∈I{i}.Sinceφinj(X)⊂B(xi,r)ifandonlyifDist(xi,φinj(X))(Qr−1)pipi+1≥Q−h(const)r−h(Qpir−pi)1−pih≥(const)r−hr−pi+(pi+1)h=(const)rpi(h−1). 258ParabolicSystems:HausdorffandPackingMeasuresSinceh<1,thisimpliesthatlimr−hm(B(x,r))=∞.ByPro-r→0iposition9.1.1,x∈S∗(∞).ItthereforefollowsimmediatelyfromThe-ihhorem4.5.3thatH(JS)=H(JS∗)=0.
ProofofLemma9.4.5.Fixaparabolicindexi∈I,j∈I{i},n≥1andfixr,2diam(φinj(X))2diam(φinj(X))thatif−1−1k≤nandQkpi−npi0suchthatif0(Qr)−pi

pi+1≤Qhr−hn−pihj=in>(Qr)−pip+1pi+1≤(const)#IQhih−1r−h(Qr)(−pi)1−pihpi−h+(pi+1)h−pipi(h−1)=(const)r=(const)r.Sinceh>1,thisimpliesthatlimr−hm(B(x,r))=0.Applyingr→0iTheorem4.5.5alongwithLemma9.4.3andProposition9.1.1,wecon-cludethatPh(J)=∞.
ProofofLemma9.4.7.Fixaparabolicindexi∈I,j∈I{i},n≥max{2q,q+1}andx∈φinj(X).Given1≥r>diam(φinj(X))andusing(9.17)twiceweobtain
n
+q
n
+qΣ:=m(φh1ika(X))≤||φika||a=ik=n−qa=ik=n−q(9.23)
n
+qpi+1pi+1≤Qhk−pih≤#IQh2q(n−q)−piha=ik=n−q 260ParabolicSystems:HausdorffandPackingMeasurespi+1hnpipi+1=2#IqQhn−pihn−qpi+1pi+1hpihhh+1pihh≤2qQ#I2Qdiam(φinj(X))≤2qQ2#Ir.1−pi1−−Putl=Enpi−Qr+1ifQr0define
∞δHg(A)=infg(diam(Ui))i=1wheretheinfimumistakenoverallcountablecovers{Ui:i=1,2,...}ofAofwiththediameterofeachUinotexceedingδ.ThefollowinglimitH(A)=limΛδ(A)=supΛδ(A)gφφδ→0δ>0δexists,butmaybeinfinite,sinceHg(A)increasesasδdecreases.SinceallδthefunctionsHgareoutermeasures,Hgisanoutermeasure.Moreover,HgturnsisametricoutermeasureandthereforeallBorelsubsetsofXareHg-measurable.Aparticularroleisplayedbyfunctionsgoftheformt→tα,t,α>0andinthiscasethecorrespondingoutermeasureHgisdenotedbyHα.Itiseasytoseethatthereexistsauniquevalue,HD(A),calledtheHausdorffdimensionofA,suchthat∞if0≤t0let
∞Π∗r(A)=supg(r)gii=1wherethesupremumistakenoverallpackings{(xi,ri):i=1,2,...}ofAofradiusnotexceedingr,wheretheradiusofapackingissup{ri:i∈I}.LetΠ∗(A)=limΠ∗r(A)=infΠ∗r(A)gggr→0r>0ThelimitexistssinceΠ∗r(A)decreasesasrdecreases.Π∗neednotbeφganoutermeasure.Inordertoconstructanoutermeasure,calledthepackingmeasureassociatedwiththefunctiong,weputΠ(A)=inf{Π∗(A)}.φφiItisthistwostageprocessinvolvedinthedefinitionofpackingmeasurethatmakesitmorecomplicatedtodealwith(cf.[MM]).Now,inexactlythesamewayasHausdorffdimensionHD,onecandefinethepacking∗dimensionPD∗andpackingdimensionPDusingrespectivelyΠ∗(A)andtΠt(A)insteadofHt(A).Onehasmonotonicityandσ-stabilityforthepackingdimensionalso.SomebasicsufficientconditionsforfinitenessandpositivityofHaus-dorffandpackingmeasuresaredescribedasfollows.LetνbeaBorelprobabilitymeasureonXandlett≥0bearealnumber.Definethefunctionρ=ρt(ν):X×(0,∞)→(0,∞)byν(B(x,r))ρ(x,r)=.rt 266GeometricmeasuretheoryThefollowingtwotheoremsareforouraimssomekeyfactsfromgeo-metricmeasuretheory.TheoremA2.0.12AssumethatXisacompactsubspaceofad-dimen-sionalEuclideanspace.Thenforeveryt≥0thereexistconstantsh1(t)andh2(t)withthefollowingproperties.(1)IfAisaBorelsubsetofXandC>0isaconstantsuchthatforall(butcountablymany)x∈Alimsupρ(x,r)≥C−1,r→0thenforeveryBorelsubsetE⊂AwehaveHt(E)≤h1(t)Cν(E)and,inparticular,Ht(A)<∞.(2)IfAisaBorelsubsetofXandC>0isaconstantsuchthatforallx∈Alimsupρ(x,r)≤C−1,r→0thenforeveryBorelsubsetE⊂AwehaveHt(E)≥Ch2(t)ν(E).TheoremA2.0.13AssumethatXisacompactsubspaceofad-dimen-breaksionalEuclideanspace.Thenforeveryt≥0thereexistconstantsp1(t)andp2(t)withthefollowingproperties.(1)IfAisaBorelsubsetofXandC>0isaconstantsuchthatforallx∈Aliminfρ(x,r)≤C−1,r→0thenforeveryBorelsubsetE⊂AwehaveΠt(E)≥Cp1(t)ν(E).(2)IfAisaBorelsubsetofXandC>0isaconstantsuchthatforallx∈Aliminfρ(x,r)≥C−1,r→0thenforeveryBorelsubsetE⊂AwehaveΠt(E)≤p2(t)Cν(E)and,consequently,Πt(A)<∞.(1’)Ifνisnon–atomicthen(1)holdsundertheweakerassumptionthatthehypothesisofpart(1)issatisfiedonthecomplementofacountableset. Geometricmeasuretheory267Passingtoball-countingdimensions,foreveryr>0considerthefamilyofallcollections{B(xi,r)}whichcoverAandarecenteredatA,meaningthatallxiareinA.PutN(A,r)=∞ifthisfamilyisempty.OtherwisedefineN(A,r)tobetheminimumofallcardinalitiesofelementsofthisfamily.Thelowerball-countingdimensionandupperball-countingdimensionofAaredefinedrespectivelybylogN(A,r)logN(A,r)BD(A)=liminfandBD(A)=limsup.r→0−logrr→0−logrIfBD(A)=BD(A),thecommonvalueiscalledsimplytheball-countingdimensionofAandisdenotedbyBD(A).Intheliteraturethenamesbox-countingdimension,Minkowskidimensionandcapacityarealsofre-quentlyusedfortheball-countingdimension.Thebasicrelationbetweenthedimensionswehaveintroducedisprovidedbythefollowing.TheoremA2.0.14ForeverysetA⊂XHD(A)≤min{PD(A),BD(A)}∗≤max{PD(A),BD(A)}≤BD(A)=PD(A).Wefinishthissectionwiththefollowingdefinition.DefinitionA2.0.15LetµbeaBorelmeasureon(X,ρ).ThentheHausdorffdimensionHD(µ)ofthemeasureµisdefinedasHD(µ)=inf{HD(Y):µ(XY)=0}andananalogousdefinitioncanbeformulatedforpackingdimension.Anextendedexpositionofthematerialcontainedinthissectioncanbefoundforexamplein[Fa2]or[Ma1].Anappropriatetoolusefulforcal-culatingHausdorffdimensionsofmeasuresisprovidedbythefollowing.TheoremA2.0.16SupposethatµisaBorelprobabilitymeasureonIRn,n≥1.(a)Ifthereexistsθsuchthatforµ-a.e.x∈IRn1logµ(B(x,r))liminf≥θ1r→0logrthenHD(µ)≥θ1. 268Geometricmeasuretheory(b)Ifthereexistsθsuchthatforµ-a.e.x∈IRn2logµ(B(x,r))liminf≤θ2r→0logrthenHD(µ)≤θ2. GlossaryofNotationAI(˜m)setofdensitiesofabsolutelycontinuousinvariantprobabilitymeasures,39nαstandardpartitionintocylindersoflengthn,10TzQapproximatetangentk-plane,167CGDMSabbreviationforconformalgraphdirectedMarkovsystem,71χµ(σ)characteristicLyapunovexponent,90Con(x,α,u)centralconewithvertexx,measureαandtheaxisparalleltothevectorU,69dλA◦ia,r+bformofanyconformalmapinIRwithd≥3,62dαmetriconcodingspace,4Dist(A,B)sup{||a−b||:(a,b)∈A×B},240dist(A,B)inf{||a−b||:(a,b)∈A×B},240∞EFadmissiblewordswithentriesformthesetF,5∗EAfiniteadmissiblewordsonanalphabetA,1∞EAinfiniteadmissiblewordsonanalphabetA,1nEAadmissiblewordsoflengthnonanalphabetA,1Hµ˜(β)entropyofµoverthepartitionβ,10ωEnsetofwordsoflengthnwhoselastentryconnectstothefirstentryofω,13ZEtwo-sidedshiftspace,119Fin(I)familyofallfinitesubsetsofI,81Fin(F)finitesetoftheone-parameterfamilygeneratedbyF,124Fin(q)inf{t:LGq,t(11)<∞},124Fix(L0)setofintegrablefunctionsfixedbyL0,39fµ(α)HausdorffdimensionofthelevelsetKµ(α),126F(S)finitenessinterval,78G(i):=qf(i)+tlog|φ|}appearinginmul-q,tfamilyoffunctions{gq,titifractalanalysis,123G(d,k)Grassmannianmanifold,167HαfunctionsinH0withfiniteαvariation,32sHαsummablefunctionsinHα,32269 270GlossaryofNotationHD(ν)Hausdorffdimensionofthemeasureν,90∞H0boundedcontinuouscomplexvaluedfunctionsonE,320HβfunctionsinHβwithintegral0,360,10HβunitballinHβ,360H0functionsinH0withintegral0,36inversion,62ia,rinversionwithrespecttothespherecenteredatthepointaandwithradiusr,62I∗thealphabetofthehyperbolicsystemassociatedwithapara-bolicsystem,223∗Jlimitsetgeneratedbythehyperbolicsystemassociatedwithaparabolicsystem,224Jvthepartofthelimitsetcodedbywordsstartingwithvertexv,2∞KβcomplexH´’oldercontinuousfunctionsoforderβonE,43sKβsummablefunctionsinKβ,43Kµ(α)α-levelsetofthemeasureµ,126J=JSthelimitset,2L0normalizedPerron–Frobeniusoperator,29Nr(E)minimumnumberofballsofradius≤rcoveringasetE.,83µFS-invariantversionofmF,61γd,knaturalmeasureontheGrassmannianmanifoldG(d,k),167||g||αnormonHα,32OD(S)maximumofupperbox-countingdimensionsoffirstlevelorbitsofpointsinX,85|ω|lengthofthewordω,1n[ω]mcylindergeneratedbyωwithcoordinatesbetweenmandn,119ω|nthewordωrestrictedtoitsfirstnentries,1Ω∗thesetoforbitsofparabolicpoints,222ω∧τthelongestinitialblockcommontoωandτ,4osc(f)thesupofthevariationoffoverbasiccylinders,8Dµ(x)upperlocalcylindricaldimensionofthemeasureµatthepointx,126dµ(x)upperlocaldimensionofthemeasureµatthepointx,126µFRokhlin’snaturalextensionoftheinvariantmeasure˜µF,119Lf(g)Perron–Frobeniusoperatoronspaceofboundedcontinuousfunc-tions,27nLf(g)nthcompositionofthePerron–Frobeniusoperatoronspaceofcontinuousfunctions,27∗LfconjugatePerron–Frobeniusoperatoractingondualspaceofcontinuousfunctions,27PF(f)thetopologicalpressureoffoverthesubsytemdeterminedbyF,7φωthemapcodedbythewordω,2 GlossaryofNotation271πthecodingmapfromthecodingspacetothelimitset,2P(F)pressureofthefamilyF,54dIPIRd−1dimensionalprojectivespace,162∗ψ(t)ψfunctionofthehyperbolicsystemassociatedwithaparabolicsystem,232Rlowerboundonsumsofiteratesof11,29S(∞)theasymptoticboundaryofthesystem,2Sv(∞)theasymptoticboundaryfromthevertexv,2Snfthenthpartialorbitsumoff,7Sω(F)iteratedsumofthefamilyF,55S(ω,K)preparatoryquantitityneededtodefinescalingfunctions,176∗Shyperbolicsystemassociatedwithaparabolicsystem,223w∞S({ωn}0,i)weakerscalingfunction,177T(g)boundingconstant,26θ(F)infimumofFin(F),124θ(q)infimumofFin(q),124θ(S)finitenessparameter,78T(F)boundeddistortionconstant,56T(q)temperaturefunctionassociatedwiththefamilyGq,t,125TDtopologicaldimension,165T:H0→H0weightednormalizedoperator,36un(ω)balancingfunctionforthenthiterateofT,36Dµ(x)lowerlocalcylindricaldimensionofthemeasureµatthepointx,126dµ(x)lowerlocaldimensionofthemeasureµatthepointx,126Vα(f)totalvariationoforderα,19Vα,n(f)variationoforderαoncylindersetsoflengthn,19ζdistinguishedpotentialfuncionassociatedwithaconformalgraphdirectedMarkovsystem,90Zn(F,f)nthpartitionfunction,6Zn(t)partitionfunction,78 Bibliography[Al]D.S.Alexander,AHistoryofComplexDynamics,Vieweg1994.[Be]T.Bedford,Hausdorffdimensionandboxdimensioninself-similarsets,Proc.TopologyandMeasureV,Ernst-Moritz-ArndtUniversit¨atGreisfwald(1988).[BP]Benedetti,Petronio,LecturesonHyperbolicGeometry,Springer-Verlag,NewYork,1991.[B1]R.Bowen,EqulibriumStatesandtheErgodicTheoryofAnosovDiffeomorphisms,Lect.NotesinMath.470(1975),SpringerVerlag.[B2]R.Bowen,Hausdorffdimensionofquasi-circles.Publ.IHES50(1980),11–25.[CG]L.Carleson,T.W.Gamelin,ComplexDynamics,Springer(1993).[CM]R.Cawley,R.D.Mauldin,MultifractaldecompositionsofMoranfractals,Adv.Math.92(1992),196–236.[De]A.Denjoy,Surunepriopri´et´edess´eriestrigonom´etriques,Verlagv.d.G.V.derWis-enNatuur.Afd.,30Oct.(1920).[DKU]M.Denker,G.Keller,M.Urba´nski,Ontheuniquenessofequi-libriumstatesforpiecewisemonotonemaps,StudiaMath.97(1990),27–36.[DU1]M.Denker,M.Urba´nski,RelatingHausdorffmeasuresandhar-monicmeasuresonparabolicJordancurves,Journalf¨urdieReineundAngewandteMathematik,450(1994),181–201.[DU2]M.Denker,M.Urba´nski,Hausdorffandconformalmeasureson272 Bibliography273Juliasetswitharationallyindifferentperiodicpoint,J.LondonMath.Soc.43(1991),107–118.[DU3]M.Denker,M.Urba´nski,Geometricmeasuresforparabolicra-tionalmaps,Ergod.Th.Dynam.Sys.12(1992),53–66.[E]G.A.Edgar,Measure,TopologyandFractalGeometry,Springer,NewYork,1990.[EM]G.A.Edgar,R.D.Mauldin,Multifractaldecompositionsofdi-graphrecursivefractals,Proc.LondonMath.Soc.65(1992),604–628.[EH]S.Eilenburg,O.G.Harrold,Jr.,Continuaoffinitelinearmeasure,Amer.J.Math.,65(1943),137–146.[Fa1]K.J.Falconer,TheGeometryofFractalSets,CambridgeUniver-sityPress,1986.[Fa2]K.J.Falconer,FractalGeometry,J.Wiley,NewYork,1990.[Fa3]K.J.Falconer,TechniquesinFractalGeometry,J.Wiley,NewYork,1997.[Fe]H.Federer,Dimensionandmeasure,Trans.Amer.Math.Soc.,62(1947),536–547.[FU]A.Fisher,M.Urba´nski,Oninvariantlinefields,Bull.LondonMath.Soc.32,no.5(2000),555–570.[FP]U.Frisch,G.Parisi,Onthesingularitystructureoffullydevelopedturbulence.appendixtoU.Frisch,Fullydevelopedturbulenceandin-termittency,in“Turbulenceandpredictabilityingeophysicalfluiddy-namcisandclimatedynamics,”(Proc.Int.Sch.Phys.“EnricoFermi,”CourseLXXXVIII),84–88,North-Holland,Amsterdam,1985.[GM]R.J.Gardner,R.D.Mauldin,OntheHausdorffdimensionofasetofcontinuedfractions,IllinoisJ.Math,27(1983),334–345.[GL]S.GrafandH.Luschgy,FoundationsofQuantizationforProbabil-ityDistributions,LectureNotesinMathematics1730,Springer,Berlin,2000.[GMW]S.Graf,R.D.MauldinandS.C.Williams,TheexactHausdorffdimeninsioninrandomrecursiveconstructions,MemiorsAmer.Math.Soc.381,1988.[Gr]P.Grassberger,Generalizeddimensionofstrangeattractors.Phys.Lett.A97(1983),346–349. 274Bibliography[GP]P.Grassberger,I.Procaccia,Characterizationofstrangeattractors,Phys.Lett.50(1983),346–349.[Gu]B.M.Gurevich,ShiftentropyandMarkovmeasuresinthepathspaceofadenumerablegraph,Dokl.Akad.Nauk.SSSR,192(1970);Englishtrans.inSovietMath.Dokl.,11(1970),744–747.[GS]B.M.Gurewich,S.V.Savchenko,ThermodynamicformalismforcountablesymbolicMarkovchains,Russ.Math.Surv.53:2(1998),245–344.[HMU]P.Hanus,R.D.Mauldin,M.Urba´nski,Thermodynamicform-alismandmultifractalanalysisofconformalinfiniteiteratedfunctionsystems,ActaMath.Hungarica96(2002),27–98.[HU]P.Hanus,M.Urba´nski,Anewclassofpositiverecurrentfunctions,Contemp.Math.SeriesAmer.Math.Soc.246(1999),123–136.[HU1]P.Hanus,M.Urba´nski,Rigidityofinfiniteiteratedfunctionsys-tems,RealAnal.Ex.,241998/9,275–288.[Ha]T.C.Halsey,M.H.Jensen,L.P.Kadanoff,I.ProcacciaB.J.Shraiman,Fractalmeasuresandtheirsingularities:Thecharacterizationofstrangesets,Phys.Rev.A33(1986),1141–1151.[HeU],S.Heinemann,M.Urba´nski,Hausdorffdimensionestimatesforinfiniteconformaliteratedfunctionsystems,Nonlinearity,15(2002),727–734.[Hu]J.E.Hutchinson,Fractalsandself-similarity,IndianaMath.J.30(1981),713–747.[IL]I.A.Ibragimov,Y.V.Linnik,IndependentandStationarySequencesofRandomVariables,Wolters-NoordhoffPubl˙,Groningen1971.[JJM]E.J¨arvenp¨a¨a,M.J¨arvenp¨a¨a,R.D.Mauldin,Deterministicandrandomaspectsofporosities,DiscreteContinuousDynam.Sys.8(2002),121–136.[Ka]T.Kato,PerturbationTheoryforLinearOperators,Springer(1995).[KH]A.Katok,B.Hasselblatt,IntroductiontotheModernTheoryofDynamicalSystems,CambridgeUniversityPress,1995.[KS],M.Keesb¨ohmer,B.Stratmann,AmultifractalformalismforgrowthratesandapplicationstogeometricallyfiniteKleiniangroups,Preprint2001. Bibliography275[LM]L.LindsayandR.D.Mauldin,Quantizationdimensionforcon-formaliteratedfunctionsystems,Nonlinearity15(2002),189–199.[Lj]V.Ljubich,EntropypropertiesofrationalendomorphismsoftheRiemannsphere,Ergod.Th.Dynam.Sys.3(1983),351–385.[Man]B.B.Mandelbrot,Intermittentturbulenceinself-similarcascades:divergenceofhighmomentsanddimensionofthecarrier,J.FluidMech.62(1974),331–358.[MM]A.Manning,H.McCluskey,Hausdorffdimensionforhorseshoes,Ergod.Th.Dynam.Sys.3(1983),251–260.[Ma1]P.Mattila,GeometryofSetsandMeasuresinEuclideanSpaces,FractalsandRectifiability,CambridgeUniversityPress,1995.[Ma2]P.Mattila,Distributionofsetsandmeasuresalongplanes,J.LondonMath.Soc.(2)38(1988),125–132.[Ma3]P.Mattila,Onthestructureofself-similarfractals,Ann.Acad.Sci.Fenn.Ser.AIMath.7(1982),189–192.[MM]P.Mattila,R.D.Mauldin,Measureanddimensionfunctions:measurabilityanddensities,Math.Proc.Camb.Phil.Soc.121(1997),81–100.[MMU]R.D.Mauldin,V.Mayer,M.Urba´nski,RigidityofconnectedlimitsetsofconformalIFS,MichiganMath.J.49(2001),451–458.[MPU]R.D.Mauldin,F.Przytycki,M.Urba´nski,Rigidityofconformaliteratedfunctionsystems,CompositioMath.129(2001),273–299.[MU1]R.D.Mauldin,M.Urba´nski,Dimensionsandmeasuresininfiniteiteratedfunctionsystems,Proc.LondonMath.Soc.(3)73(1996),105–154.[MU2]R.D.Mauldin,M.Urba´nski,Conformaliteratedfunctionsystemswithapplicationstothegeometryofcontinuedfractions,Trans.Amer.Math.Soc.351(1999),4995–5025.[MU3]R.D.Mauldin,M.Urba´nski,Gibbsstatesonthesymbolicspaceoveraninfinitealphabet,IsraelJ.Math.125(2001),93–130.[MU4]R.D.Mauldin,M.Urba´nski,Ontheuniquenessofthedensityfortheinvariantmeasureinaninfinitehyperboliciteratedfunctionsystem,PeriodicaMath.Hung.37(1998),47–53. 276Bibliography[MU5]R.D.Mauldin,M.Urba´nski,Jordancurvesasrepellors,PacificJ.Math.166(1994),85–97.[MU6]R.D.Mauldin,M.Urba´nski,Thedoublingpropertyofconformalmeasuresofinfiniteiteratedfunctionsystems,J.NumberTh.,toappear.[MU7]R.D.Mauldin,M.Urba´nski,Paraboliciteratedfunctionsystems,Ergod.Th.Dynam.Sys.20(2000),1423–1447.[MU8]R.D.Mauldin,M.Urba´nski,Dimensionandmeasuresforacurv-linearSierpinskigasketorApollonianpackings,Adv.Math.136(1998),26–38.[MU9]R.D.Mauldin,M.Urba´nski,FractalmeasuresforparabolicIFS,Adv.Math.168(2002),225–253.[MW1]R.D.Mauldin,S.C.Williams,Randomrecursiveconstructions:asymptoticgeometricandtopologicalproperties,Trans.Amer.Math.Soc.259(1986),325–346.[MW2]R.D.Mauldin,S.C.Williams,Hausdorffdimensioningraphdirectedconstructions,Trans.Amer.Soc.309(1988),811–829.[MyW]V.Mayer,M.Urba´nski,FinergeometricrigidityoflimitsetsofconformalIFS,Proc.Amer.Math.Soc,toappear[Mo]P.A.P.Moran,AdditivefunctionsofintervalsandHausdorffmeas-ure,Proc.Cambridge,Philos.Soc.42(1946),15–23.[Ne]Z.Nehari,ConformalMapping,DoverPublications,1975.[O1]L.Olsen,Amultifractalformalism,Adv.Math.116(1995),82–196.[O2]L.Olsen,RandomGeometricallyGraphDirectedSelf-similarMul-tifractals,PitmanResearchnotesinMath.Series,Vol.307,LongmanScientific,1994.[O3]L.Olsen,Multifractalgeometry,ProceedingsFractalGeometryandStochasticsII,Greifswald,GermanyAug.1998.ProgressinProbability,Vol.46(EditorsC.Bandt,S.Graf,andM.Z¨ahle),1–37.Birkh¨auser,2000.[O4]L.Olsen,MultifractalanalysisofdivergencepointsofdeformedmeasuretheoreticalBirkhoffaverages,preprint.[Or]D.Orstein,ErgodicTheory,RandomnessandDynamicalSystems,YaleUniversityPress,NewHavenandLondon1974. Bibliography277[Par]W.Parry,EntropyandGeneratorsinErgodicTheory,W.A.Benja-min,Inc.,NewYork,1969.[Pat]S.J.Patterson,ThelimitsetofaFuchsiangroup,ActaMath.136(1976),241–273.[Pat1]N.Patzschke,Self-conformalmultifractalmeasures,Adv.Appl.Math.19(1997),486–513.[Pat2]N.Patzschke,Self-conformalmultifractalmeasuresandself-similarrandomthermodynamics,Preprint2001.[Pe]Y.B.Pesin,DimensionTheoryinDynamicalSystems,Chicago1997.[PP]Y.Pesin,B.Pitskel,Topologicalpressureandvariationalprinciplefornon-compactsets,Funct.Anal.Appl.18(1985),307–318.[PW]Y.B.Pesin,H.Weiss,ThemultifractalanalysisofequilibriummeasuresforconformalexpandingmapsandMarkovMorangeometricconstructions,J.Sta.Phys.86(1997),233–275.[Po1]C.Pommerenke,UniformlyperfectsetsandthePoincar´emetric,Arch.Math.32(1979),192–199.[Po2]C.Pommerenke,BoundaryBehaviourofConformalMaps,Springer1992.[PS]W.Philipp,W.Stout,Almostsureinvarianceprinciplesforpartialsumsofweaklydependentrandomvariables,MemoirsAmer.Math.Soc.161(2),(1975)[Pr]F.Przytycki,Onholomorphicperturbationsofz→zn,Bull.PolishAcad.Sci.Math.24(1986),127–132.[PT]F.Przytycki,F.Tangerman,Cantorsetsintheline:Scalingfunc-tionsoftheshiftmap,Preprint1992.[PU1]F.Przytycki,M.Urba´nski,FractalsinthePlane-ErgodicTheoryMethods,tobepublishedbytheCambridgeUniversityPress,availableonUrba´nski’swebpagehttp://www.math.unt.edu/∼urbanski.[PUZ]F.Przytycki,M.Urba´nski,A.Zdunik,Harmonic,GibbsandHausdorffmeasuresonrepellersforholomorphicmapsI,Ann.Math.130(1989),1–40.[Ra]D.Rand,thesingularityspectrumf(α)forcookie-cutters,Ergod.Th.Dynam.Sys.9(1989),527–541. 278Bibliography[Ru]D.Ruelle,ThermodynamicFormalism,EncyclopediaofMathem-aticsanditsApplications.vol.5,Addison-Wesley1978.[Sal]A.Salli,OntheMinkowskidimensionofstronglyporousfractalsinIRn,Proc.LondonMath.Soc.(3)62(1991),353–372.[Sar]O.Sarig,TheormodynamicformalismforcountableMarkovshifts,Ergod.Th.Dynam.Sys.19(1999),1565–1593.[Sc]H.Schaefer,BanachLatticesandPositiveOperators,Springer,NewYork,1974.[SU]B.Stratmann,M.Urba´nski,RealAnalyticityofTopologicalPres-sure,forparabolicallysemihyperbolicgeneralizedpolynomial-likemaps,Preprint2001,IndagationesMath.,toappear.[Su1]D.Sullivan,SeminaronconformalandhyperbolicgeometrybyD.P.Sullivan(NotesbyM.BakerandJ.Seade),PreprintIHES(1982).[Su2]D.Sullivan,Differentiablestructuresonfractal-likesets,determ-inedbyintrinsicscalingfunctionsondualCantorsets,TheMathemat-icalHeritageofHermanWeyl,Proc.Symp.PureMath.48(1988).[Su3]D.Sullivan,Quasiconformalhomeomorphismsindynamics,topo-logy,andgeometry,Proc.InternationalCongressofMathematicians,A.M.S.(1986),1216–1228.[Su4]D.Sullivan,Entropy,Hausdorffmeasuresoldandnew,andlimitsetsofgeometricallyfiniteKleiniangroups,ActaMath.153(1984),259–277.[U1]M.Urba´nski,Hausdorffmeasuresversusequilibriumstatesofcon-formalinfiniteiteratedfunctionsystems,PeriodicaMath.Hung.37(1998),153–205.[U2]M.Urba´nski,Porosityinconformalinfiniteiteratedfunctionsys-tems,J.Numb.Th.88No.2(2001),283–312.[U3]M.Urba´nski,OntheHausdorffdimensionofaJuliasetwitharationallyindifferentperiodicpoint,StudiaMath.97(1991),167–188.[U4]M.Urba´nski,ParabolicCantorsets,Fund.Math.151(1996),241–277.[U5]M.Urba´nski,Rigidityofmulti-dimensionalconformaliteratedfunc-tionsystems,Nonlinearity14(2001),1593–1610. Bibliography279[U6]M.Urba´nski,Rationalfunctionswithnorecurrentcriticalpoints,Ergod.Th.Dynam.Sys.14(1994),391–414.[U7]M.Urba´nski,Thermodynamicformalism,topologicalpressureandescaperatesfornon-recurrentconformaldynamics,Preprint2000,Fund.Math.,toappear.[UV]M.Urba´nski,A.Volberg,Arigiditytheoremincomplexdynam-ics,inFractalGeometryandStochastics,ProgressinProbability37,Birkh¨auser,1995.[UZd]M.Urba´nski,A.Zdunik,Hausdorffdimensionofharmonicmeas-ureforself-conformalsets,Adv.Math.171(2002),1–58.[UZi]M.Urba´nski,M.Zinsmeister,ThegeometryofhyperbolicJulia-Lavaurssets,IndagationesMath.12(2001)273–292.[Wa]P.Walters,AnIntroductiontoErgodicTheory,Springer1982.[Wh]G.T.Whyburn,Concerningcontinuaorfinitedegreeandlocalseparatingpoints,Amer.J.Math.57(1935),11–18.[Zaj]L.Zajicek,Porosityandσ-porosity,RealAnal.Exchange13(1987/8),314–350.[Zar]A.Zargaryan,VariationalprinciplesfortopologicalpressureinthecaseofMarkovchainwithacountablenumberofstates,Math.NotesAcad.Sci.USSR40(1986),921–928.[Z1]A.Zdunik,Parabolicorbifoldsandthedimensionofthemaximalmeasureforrationalmaps,Invent.Math.99(1990),627–649.[Z2]A.Zdunik,HarmonicmeasureversusHausdorffmeasuresonre-pellorsforholomorphicmaps,Trans.Amer.Math.Soc.326(1991),633–652. IndexF-conformal,57entropy,10σ-finite,225entropywithrespecttoapartition,10k-rectifiable,166equilibriumstate,22t-conformal,77essentiallyaffine,182expandingmaps,136finitelyprimitive,5exponentialdecayofcorrelations,41stronglyconnected,5finitelyirreducible,5finitelysupported,9absolutelyregularalongafiltration,40acceptablefunction,8generalizedHausdorffmeasures,109almostsureinvarianceprinciple,41geometricrigidity,160almost-periodic,145Gibbsstate,12amalgamatedfunction,56graphdirectedsystem(GDS),3anglebetweenlinearhyperspaces,196GraphDirectedMarkovApollonianpacking,234System(GDMS),1approximatetangentk-plane,166Grassmannianmanifold,167associatediteratedfunctionsystem,107guling,136bi-Lipschitz,174H¨oldercontinuouswithexponentα>0,Blochfunction,6319boundeddistortionlemma,26H¨olderfamilyoffunctions,54hereditarilyregular,87centrallimittheorem,41characteristicLyapunovexponent,90invariantGibbsstate,12codingmapπ,2Ionescu-TulceaandMarinescucoefficientsofabsoluteregularity,40inequality,32cofinitesubsystem,87irreducible,5cohomologous,19irregular,88Coneproperty,72iteratedfunctionsystem(IFS),3conformalmappingofIRd,62conformalaffinecontractions,182Jacobian,152conformalaffinehoeomorphism,62conformalparaboliciteratedfunctionKleiniangroupsofSchottkytype,136systems,209Koebe’sDistortionTheorem,63conformal-like,58Continuedfractions,137lawoftheiteratedlogarithm,41cookie-cutterCantorset,176Legendretransform,126Legendretransformpair,127dynamicalrigidity,170limitset,2280 Index281linefield,191restrictions,136Liouville’stheorem,63RigidityinDimensiond≥3,195lowerclass,42Rokhlincanonicalsystemofmeasures,119metricdα,4Rokhlin’snaturalextension,119Minkowski,83Montel’stheorem,150scalingfunction,176MultifractalAnalysis,123standardpartition,10multifractalformalism,123strongopensetcondition(SOSC),96strongseparationcondition,58normalizedPerron–Frobeniusoperator,strongtangent,16229strongerscalingfunction,177stronglyconnected,49Opensetcondition,71stronglyregular,87orbitsum,7summable,27partitionfunction,6tangent,162Perron–Frobeniusoperator,27topologicaldimension,165porosity,103topologicalpressure,7porous,103topologicallymixing,5potential,22two-sidedshiftspace,119primitive,5purelyk-unrectifiable,166uniformlyperfect,156upperclass,42Radon-Nikodymderivative,144variationalprinciple,9RateofapproximationoftheHausdorffVolumelemma,90dimension,153real-analyticextension,148weak-Bernoulli,40RefinedGeometry,109weakerscalingfunction,177refinedvolumelemma,110weightednormalizedPerron–Frobeniusregular,78operator,36