Geometric and Ergodic Aspects of Group Actions (Infosys Science Foundation Series) PDF
Preview Geometric and Ergodic Aspects of Group Actions (Infosys Science Foundation Series)
Infosys Science Foundation Series Infosys Science Foundation Series in Mathematical Sciences S. G. Dani Anish Ghosh Editors Geometric and Ergodic Aspects of Group Actions Infosys Science Foundation Series Infosys Science Foundation Series in Mathematical Sciences Series Editors Irene Fonseca, Carnegie Mellon University, Pittsburgh, PA, USA Gopal Prasad, University of Michigan, Ann Arbor, USA Editorial Board Manindra Agrawal, Indian Institute of Technology Kanpur, Kanpur, India Weinan E, Princeton University, Princeton, USA Chandrashekhar Khare, University of California, Los Angeles, USA Mahan Mj, Tata Institute of Fundamental Research, Mumbai, India Ritabrata Munshi, Tata Institute of Fundamental Research, Mumbai, India S. R. S. Varadhan, New York University, New York, USA The Infosys Science Foundation Series in Mathematical Sciences is a sub-series of The Infosys Science Foundation Series. This sub-series focuses on high-quality contentinthedomainofmathematicalsciencesandvariousdisciplinesofmathematics, statistics, bio-mathematics, financial mathematics, applied mathematics, operations research,appliedstatisticsandcomputerscience.Allcontentpublishedinthesub-series arewritten,edited,orvettedbythelaureatesorjurymembersoftheInfosysPrize.With this series, Springer and the Infosys Science Foundation hope to provide readers with monographs, handbooks, professional books and textbooks of the highest academicqualityoncurrenttopicsinrelevantdisciplines.Literatureinthissub-series willappealtoa wide audienceofresearchers, students, educators, and professionals acrossmathematics,appliedmathematics,statisticsandcomputersciencedisciplines. More information about this subseries at http://www.springer.com/series/13817 S. G. Dani Anish Ghosh (cid:129) Editors Geometric and Ergodic Aspects of Group Actions 123 Editors S. G.Dani AnishGhosh UM-DAE, Centerfor Excellence Schoolof Mathematics in BasicSciences TataInstitute of FundamentalResearch Mumbai,Maharashtra, India Mumbai,Maharashtra, India ISSN 2363-6149 ISSN 2363-6157 (electronic) Infosys Science FoundationSeries ISSN 2364-4036 ISSN 2364-4044 (electronic) Infosys Science FoundationSeries in MathematicalSciences ISBN978-981-15-0682-6 ISBN978-981-15-0683-3 (eBook) https://doi.org/10.1007/978-981-15-0683-3 ©SpringerNatureSingaporePteLtd.2019 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface The recent decades, especially since the 1990s, have witnessed a surge in research activity in the group of interrelated areas of geometry, actions and representations ofLiegroups,ergodictheoryandDiophantineapproximation.Increasinglyoverthe years, a variety of new techniques have emerged from the activity and led to new insights into resolution of classical problems in the areas, on the one hand, and widening of horizons of exploration in the respective areas. AttheTataInstituteofFundamentalResearch,Mumbai,inanendeavourtokeep abreast of the developments and to disseminate on a broader scale some of the significant developments in this respect, two major events were held, among other related pursuits, in recent years—made possible by generous financial assistance from the Tata Institute and the University Grants Commission, India: 1. GeometricandErgodicAspectsofGroupActions.Thismeetingwasheldduring 20–24 April 2015 and organised by S. Bhattacharya and A. Ghosh. It featured three series of lectures, in parallel, delivered by F. Maucourant, M. Mj and O. Sarig, which were complemented with talks by several researchers in the general area. 2. DistinguishedLecturesinDynamics.Thiswasafollow-upmeetingheldduring 10–14 April 2017, organised by S. G. Dani and A. Ghosh, featuring three lecture series, by A. Gorodnik, Y. Guivarc'h and A. Pollicott, and individual lectures on other topics. Based onresponsestothe expositionsatthese lectures, andawareof theefforts the speakers had put in, we were convinced that systematic notes of the lectures wouldbebeneficialtothebroadercommunityofresearchersandstudentsengaged in pursuing the group of areas, and persuaded the speakers to make available suitablenotesforpublication.Therewasaverygoodresponse,thoughlimiteddue to certain circumstances that need not be gone into here, and the present book represents the fruit of the endeavour. This book comprises articles of M. Mj, O. Sarig, A. Gorodnik and M. Pollicott and covers a range of interrelated topics. Here is a brief overview of the contents: v vi Preface The article by Mj provides an introduction to Kleinian groups from dynamical andgeometricperspectives. ThesearediscretesubgroupsofPSL ðCÞ,andinMj’s 2 article,heprovidesafast-pacedintroductiontothebasictheoryofKleiniangroups including their limit sets and then proceeds to review the highlights of this rich theory.Geodesiclaminations,thetamenessconjecture,theendinglaminationtheorem and Cannon-Thurston maps are discussed. The article shows how deeply interwoven dynamics and geometry are in the setting of Kleinian groups. Mj goes a long way in providinglegitimacytotheslogan(putoutinthearticle)that“DynamicsontheLimit Set determines Geometry in the Interior”. The article by Sarig is a natural successor to Mj’s introduction to Kleinian groups. In his article, Sarig provides an introduction to the ergodic theory of horocycle flows on hyperbolic surfaces with infinite genus. We recall that the horocycle flow on finite volume hyperbolic surfaces has been extensively studied and possesses remarkable rigidity properties. In particular, thanks to the work of Furstenberg,DaniandSmillie,onehasaclassificationofallorbitclosuresandfinite invariant measures for such flows. The more general finite-genus (infinite volume) casehasbeenstudiedbyBurgerandRoblin.Thesituationchangesquitedrastically for surfaces of infinite genus, where all the dynamics are captured by infinite invariant Radon measures. In his article, Sarig provides a careful and detailed exposition of the subject covering the construction of invariant Radon measures (duetoBabillot),theirergodic-theoreticpropertiesandpathologies,equidistribution of horocycle flows and finally a sketch of the fundamental theorem, due substan- tially to Sarig, that all ergodic invariant Radon measures for the horocycle flow arise from extremal positive eigenfunctions via Babillot's construction. Two appendices provide the relevant background on Busemann functions and the cocycle reduction theorem. Gorodnik’s article surveys higher-order correlations for dynamical systems arising from Lie group actions, especially on homogeneous spaces. This is a very classicaltopicstartingfromtheworkofHarish-Chandraandiscloselyconnectedto theanalyticbehaviourofmatrixcoefficientsforrepresentationsofLiegroups.Inhis notes, Gorodnik presents a complete and detailed proof of quantitative bounds for higher-order correlations of actions of simple Lie groups. Several applications are also presented: asymptotic formulas for counting lattice points using decay, at infinity, of matrix coefficients and the existence of approximate configurations in lattice subgroups as well as a central limit theorem for multi-parameter group actions using higher-order correlations. The decay of matrix coefficients discussed in Gorodnik’s article is intimately related to mixing for one-parameter flows associated with surfaces of constant negative curvature with finite Riemannian area. Pollicott’s article is a natural continuation and describes the rates of mixing for various types of hyperbolic systems, for example, geodesic flows on surfaces of variable negative curvature. The transfer operators, which constitute the main technical tool, are discussed in detail, and a simplified account of the important results of Dolgopyat and Liverani isprovided.Thearticleendswithsomeapplicationsofmixingestimates,including Preface vii counting closed geodesics, skew products and a very interesting application to Euclidean algorithms. We hope that these lectures will serve as a useful guide to advanced graduate students as well as a convenient reference to professional mathematicians. Mumbai, India S. G. Dani Anish Ghosh Contents Lectures on Kleinian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Mahan Mj Horocycle Flows on Surfaces with Infinite Genus. . . . . . . . . . . . . . . . . . 21 Omri Sarig Higher Order Correlations for Group Actions. . . . . . . . . . . . . . . . . . . . 83 Alexander Gorodnik Exponential Mixing: Lectures from Mumbai . . . . . . . . . . . . . . . . . . . . . 135 Mark Pollicott ix About the Editors S.G.Dani hasbeenadistinguishedprofessorattheCentreforExcellenceinBasic Sciences at the University of Mumbai and the Department of Atomic Energy, Mumbai, India, since July 2017, following a fruitful career at the Tata Institute of FundamentalResearch(TIFR),Mumbai,from1969to2012.Hehasmadesignificant contributions in many areas of mathematics, including ergodic theory, dynamics, number theory, Lie groups, and measures on groups, and has published in many leading international journals. He also has written on the history of mathematics, in ancient as well as recent times in India. A recipient of the Shanti Swarup Bhatnagar Prize, the Srinivasa Ramanujan Medal,andtheMathematicalSciencesPrizeofTWASbytheAcademyofSciences oftheDevelopingWorld,ProfessorDaniisaFellowofTWASandthethreemajor academiesofscienceinIndia(INSA,IASc,andNASI) andserved onthecouncils of INSA and IASc. He was an invited speaker at ICM 1994. He has been on the editorial boards of several international journals and was the editor of the ProceedingsMathematicalSciencesfrom1987to2000.Heiscurrentlytheeditorof Ganita Bharati, the Bulletin of the Indian Society for History of Mathematics. He was a member of the National Board of Higher Mathematics from 1996 to 2015 anditschairmanfrom2006to2011.HewasalsothepresidentoftheCommission forDevelopmentandExchangeoftheInternationalMathematicalUnionfrom2007 to2010andamemberoftheexecutivecommitteeoftheInternationalCommission for History of Mathematics (ICHM) during 2015–2018. He is currently the presi- dent of the recently founded Mathematics Teachers’ Association, India. AnishGhosh isAssociateProfessorattheSchoolofMathematics,TataInstituteof Fundamental Research, Mumbai, India. Earlier, he was a lecturer at the University of East Anglia, UK; a research fellow at the University of Bristol, UK; and an instructor at the University of Texas at Austin, USA. He earned his Ph.D. from Brandeis University, USA, in 2006. He has co-edited a book on Recent Trends in Ergodic Theory and DynamicalSystems and published over36 research articles in xi
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