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Lecture Notes in Mathematics 2217 Sergey Bezuglyi · Palle E. T. Jorgensen Transfer Operators, Endomorphisms, and Measurable Partitions Lecture Notes in Mathematics 2217 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryBoard: MichelBrion,Grenoble CamilloDeLellis,Zurich AlessioFigalli,Zurich DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GáborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,NewYork AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Sergey Bezuglyi (cid:129) Palle E. T. Jorgensen Transfer Operators, Endomorphisms, and Measurable Partitions 123 SergeyBezuglyi PalleE.T.Jorgensen DepartmentofMathematics DepartmentofMathematics UniversityofIowa UniversityofIowa IowaCity IowaCity Iowa,USA Iowa,USA ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-319-92416-8 ISBN978-3-319-92417-5 (eBook) https://doi.org/10.1007/978-3-319-92417-5 LibraryofCongressControlNumber:2018944127 MathematicsSubjectClassification(2010):Primary:47B65,37B45,28D05;Secondary:37C30,82C41, 28D15 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface While the fundamental notions, transfer operators, and Laplacians over decades haveplayedacrucialroleindiverseapplications,theirmutualinterconnectionsstill deserveasystematicstudy. Webeginbypresentingasuitablemeasuretheoreticframeworkforsuchastudy of transfer operators, arising as part of the wider context of positive operators in measurespaces.Fromalonglistofapplications,weshallstressadetailedanalysis of endomorphisms, of measurable partitions, and of Markov processes. Indeed, a numberofearlierstudiescoverahostofspecialcases.Papersofspecialrelevance to our present book include Vershik [VF85], [Ver00], [Ver01], [Ver05]; Baladi et al. [BER89], [Bal00], [BB05]; the first named author et al., e.g., [BKMS10], [BK16];Alpayandthesecondnamedauthor,e.g.,[AJL13],[AJK15],[AJLM15]; andHawkinsetal.,e.g.,[HS91],[Haw94]. Transfer operators arise in dynamical systems, and in a number of related applications; and a choice of transfer operator may serve different purpose from one application to the next. But typically a transfer operator serves to encode information about an iterated map, or iterated substitution systems. Here our focus will be an endomorphism in a fixed measure space and their iterations. Transferoperatorsareusedinordertostudythebehaviorofassociateddynamical systems,andtheyoccurinforexamplestochasticprocesses,instatisticalmechanics, pioneered by David Ruelle, in quantum chaos, and in the study of such fractals that arise in iterated function systems (IFS). Each of these cases will be studied inside the book. Since transfer operators arise in multiple contexts, they also go by other names, for example Ruelle operators (after David Ruelle) or Ruelle- Perron-Frobenius operators. They are infinite-dimensional analogues of positive matrices and the Frobenius-Perron theorem. As in the matrix case, key questions arethedeterminationoftheeigenvalues,thespectralradius,andPerron-Frobenius eigenvectorspace.Theroleofeigenvaluesandspectrumisalsoimportantininfinite dimensions,butthenquestionsaboutspectrumaremoresubtle. Fromouranalysisoftransferoperators,wepasstotheirconnectionstothetheory ofLaplaceoperators.ThetraditionalsettingofLaplaciansrangesfromthediscrete domaintosuchmeasuretheoreticframeworksthatarisenaturallyinpotentialtheory v vi Preface andinmanydiverseapplications.ThepointofviewofLaplacianshasbeenstudied inearlierpapers,forexample[JP11,JP13,JT16b,JT16a].Inthepresentbook,we havestressedthe“transferoperatorpointofview.” Ourapproachtotransferoperatorshasbeenmotivatedbyconnectionstorecent studies of Laplace operators. Starting with the discrete case, recall that a discrete Laplace operator is an analogue of the better known continuous PDE-Laplace operator,thediscretevariantsarisingfromsuitablediscretizationalgorithms.They areusedextensivelyinnumericalanalysis.AdiscreteLaplaceoperatoristypically definedsothatithasmeaningonagraphG,i.e.,adiscretegridofverticesandedges. OurpresentframeworkisthecasewhenGisinfinite,evenpossiblyameasurespace. Inthevariousapplicationsdiscussedinthebook,westresstheconnectiontotransfer operatortheory. For the case of finite graphs(G having a finite number of edges and vertices), the associated discrete Laplace operators are also called Laplacian matrices. In interestingapplications,theanalysisofinfiniteGmaybeachievedassuitablelimits ofthefinitecases.However,recentlyLaplaceoperatorshaveplayedanincreasingly importantroleoutsidethesettingofPDEtheory. ApplicationsofLaplaciansoccurinphysicsproblems,suchasinIsingmodels, in loop quantum gravity, and in the study of discrete dynamical systems. Other applicationsincludeimageprocessing,intheformofLaplacefilters,andinmachine learning for clustering and semi-supervised learning on neighborhood graphs. In addition to considering the connectivity of vertices (nodes) and edges in a graph, mesh Laplace operators take into account the underlying geometry (e.g., vertex angles).Differentdiscretizationsexist,someofthemareanextensionofthegraph operator,whileotherapproachesarebasedonthefiniteelementmethodandallow forhigherorderapproximations. IowaCity,IA,USA SergeyBezuglyi IowaCity,IA,USA PalleE.T.Jorgensen April2018 Acknowledgments The first named author is thankful to Professors Jane Hawkins, Olena Karpel, Konstantin Medynets, and Cesar Silva for useful discussions on properties of endomorphisms. The second named author gratefully acknowledges discussions, on the subject of this book, with his colleagues, especially helpful insight from ProfessorsDanielAlpay,DorinDutkay,JudyPacker,ErinPearse,Myung-SinSong, andFengTian. The authors thank the members in the Operator Theory and Mathematical PhysicsseminarsattheUniversityofIowaforenlighteningdiscussions. Theauthorsaregratefultoseveralanonymousreviewersforkindlypreparinglists ofcorrectionsandformakingconstructivesuggestions.Wehavefollowedthemall. Thebookisbetterforit.Remainingflawsaretheresponsibilityoftheauthors. vii Contents 1 IntroductionandExamples................................................ 1 1.1 Motivation............................................................ 4 1.2 ExamplesofTransferOperators..................................... 6 1.3 DirectionsandMotivationalQuestions ............................. 11 1.4 MainResults.......................................................... 12 2 EndomorphismsandMeasurablePartitions ............................ 13 2.1 StandardBorelandStandardMeasureSpaces...................... 13 2.2 EndomorphismsofMeasurableSpaces ............................. 14 2.3 MeasurablePartitionandSubalgebras .............................. 17 2.4 SolenoidsandApplications.......................................... 20 3 Positive,and Transfer, OperatorsonMeasurable Spaces: GeneralProperties ......................................................... 23 3.1 TransferOperatorsonBorelFunctions.............................. 24 3.2 Classification ......................................................... 27 3.3 KernelandRangeofTransferOperators............................ 29 3.4 MultiplicativePropertiesofTransferOperators .................... 32 3.5 HarmonicFunctionsandCoboundariesforTransferOperators ... 34 4 TransferOperatorsonMeasureSpaces.................................. 39 4.1 TransferOperatorsandMeasures ................................... 39 4.2 ErgodicDecompositionofTransferOperators ..................... 48 4.3 PositiveOperatorsandPolymorphisms............................. 51 5 TransferOperatorsonL1andL2 ........................................ 59 5.1 PropertiesofTransferOperatorsActingonL1 andL2............. 59 5.2 TheAdjointOperatorforaTransferOperator...................... 67 5.3 MoreRelationsBetweenRandσ ................................... 72 6 ActionsofTransferOperatorsontheSetofBorelProbability Measures..................................................................... 77 ix x Contents 7 Wold’sTheoremandAutomorphicFactorsofEndomorphisms...... 85 7.1 HilbertSpaceDecompositionDefinedbyanIsometry............. 85 7.2 AutomorphicFactorsandExactEndomorphisms.................. 88 8 Operatorson the Universal Hilbert Space Generatedby TransferOperators......................................................... 93 8.1 DefinitionoftheUniversalHilbertSpaceH(X).................... 93 8.2 TransferOperatorsonH(X)......................................... 95 9 TransferOperatorswithaRieszProperty............................... 105 10 TransferOperatorsontheSpaceofDensities........................... 113 11 PiecewiseMonotoneMapsandtheGaussEndomorphism............ 119 11.1 TransferOperatorsforPiecewiseMonotoneMaps................. 119 11.2 TheGaussMap....................................................... 126 12 IteratedFunctionSystemsandTransferOperators.................... 133 12.1 IteratedFunctionSystemsandMeasures ........................... 133 12.2 TransferOperatorforx (cid:2)→2x mod1............................... 138 13 Examples .................................................................... 143 13.1 TransferOperatorandaSystemofConditionalMeasures......... 143 References......................................................................... 151 Index............................................................................... 159