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Springer Monographs in Mathematics José F. Alves Nonuniformly Hyperbolic Attractors Geometric and Probabilistic Aspects Springer Monographs in Mathematics Editors-in-Chief Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul,SouthKorea;MathematicalInstitute,UniversityofWarwick,Coventry,UK Katrin Wendland, Research group for Mathematical Physics, Albert Ludwigs University of Freiburg, Freiburg, Germany Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NY, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NY, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan IsabelleGallagher,DépartementdeMathématiquesetApplications,EcoleNormale Supérieure, Paris, France Sinan Güntürk,Courant Institute ofMathematical Sciences,New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France AlbertoA.Pinto,DepartmentofMathematics,UniversityofPorto,Porto,Portugal GabriellaPinzari,DepartmentofMathematics,UniversityofPadova,Padova,Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK ShmuelWeinberger,DepartmentofMathematics,UniversityofChicago,Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK Thisseriespublishesadvancedmonographsgivingwell-writtenpresentationsofthe “state-of-the-art”infieldsofmathematicalresearchthathaveacquiredthematurity neededforsuchatreatment.Theyaresufficientlyself-containedtobeaccessibleto morethanjusttheintimatespecialistsofthesubject,andsufficientlycomprehensive to remain valuable references for many years. Besides the current state of knowledgeinitsfield,anSMMvolumeshouldideallydescribeitsrelevancetoand interaction with neighbouring fields of mathematics, and give pointers to future directions of research. More information about this series at http://www.springer.com/series/3733 é Jos F. Alves Nonuniformly Hyperbolic Attractors Geometric and Probabilistic Aspects 123 JoséF.Alves Department ofMathematics University of Porto Porto, Portugal ISSN 1439-7382 ISSN 2196-9922 (electronic) SpringerMonographs inMathematics ISBN978-3-030-62813-0 ISBN978-3-030-62814-7 (eBook) https://doi.org/10.1007/978-3-030-62814-7 MathematicsSubjectClassification: 37A05,37A25,37C40,37D30 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2020 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. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To Romery, Lavínia & Leonardo “And after all is said and done And the numbers all come home The four rolls into three The three turns into two And the two becomes a one” Paul Simon Preface I started writing this text without even suspecting that I was starting to write a textbook.Isuppose,likemanyotherscientificbooks,itstartedoutaslecturenotes, designed to support a short course I lectured at SUSTech, Shenzhen—China, in July2016.Sincethen,severalothershortcoursesaroundtheworldhavemotivated many improvements. The initial lecture notes were essentially a compilation of results published over the past two decades in various research articles by several authors. This text goes far beyond a simple compilation of articles. Idaresaythatwritinganarticleistheartofconvincing,writingabookistheart of explaining. Articles are rarely self-contained and often not presented in great detail. This sometimes contributes to the appearance of a certain folklore which is hard to understand (and impossible to reach in the literature) for beginners in the field. To write a textbook like this it is mandatory to go beyond the content of articles, for otherwise, it makes little sense to write a new text. The price to pay is thenumberofpages.ThistextwasinitiallyscheduledfortheSpringerBriefsseries. Due to restrictions on the length of the texts in that series (up to 120 pages, supposedly) and to make the presentation of this theory self-contained, examples and applications were discarded in the end, making the first version of this text merelythepresentation ofan abstract theory.However,ontherecommendation of anonymous referees (with which I unconditionally agreed from the first hour), the initial version was promoted to the normal Springer series and recommended examples and applications were included. This meant an increase of about one hundred pages. Of course, only a few of the possible examples and applications fromthevastlistofpublicationsonthesetopicsinrecentyearshavebeenincluded. Theserepresentonlytheauthor’stasteandviewofthetheory.Manyotherscanbe found in the bibliographic references left here, and in other references not left, by mere forgetfulness or ignorance. This textbook is the production of a single author, but it has a godfather: indelibly associated to its genesis, Wael Bahsoun convinced Springer’s publishing editor and myself (and himself, I presume) that turning the lecture notes available on my webpage into a book would be a good idea (for both the publishing house andtheauthor),withtheattractive sideofbeingaquickandeasytask:inacouple vii viii Preface of months a new book would see daylight. In the end, this possible couple of months became my whole sabbatical leave at Loughborough University, from October 2018 to August 2019, and effectively, a couple more months after my return to the University of Porto. Inadditiontomyco-authorsinseveralscientificarticlesthatmuchcontributedto this text, I would like to thank M. Benedicks, M. Carvalho, J. M. Freitas, S.Luzzatto,D.Mesquita,I.Melbourne,F.J.Moreira,V.Pinheiro,P.Varandasand H. Vilarinho for many fruitful conversations and valuable exchanges of ideas. I particularly wish to thank Wael Bahsoun, not only for pushing me towards this projectthatbroughtmealotofpleasure(andalittlebitsufferingaswell),butalso for the many helpful conversations and advice on these topics, especially after reading a preliminary version of the text. Special thanks also to M. Viana for his availability in frequent exchanges of ideas, carried out exclusively through online chats—this efficient way of communicating in modern times. A final acknowledgement to Loughborough University for the excellent condi- tions provided and The Leverhulme Trust for the financial support through the Visiting Professorship VP2-2017-004. Also, partial financial support from CMUP (UID/MAT/00144/2013&UIDB/00144/2020)andtheprojectsPTDC/MAT-CAL/ 3884/2014 and PTDC/MAT-PUR/28177/2017 funded by FCT with national and European structural funds through the program FEDER, under the partnership agreement PT2020, must be recognised. Loughborough, England José F. Alves Porto, Portugal August 2019/July 2020 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Physical Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 SRB Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Decay of Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Generating Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Measurable Partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Expanding Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Gibbs-Markov Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 Bounded Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.2 A Space for the Densities. . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.3 Equilibrium Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Induced Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Tower Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.1 Tower Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.2 Convergence to the Equilibrium . . . . . . . . . . . . . . . . . . . . 48 3.3.3 Decay of Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4 Lifting Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 ix x Contents 3.5 Application: Intermittent Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.5.1 Neutral Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.5.2 Interval Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.5.3 Circle Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4 Hyperbolic Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.1 Young Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.1.1 Quotient Return Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.1.2 Bounded Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 SRB Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.1 Return Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.2 Original Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.3 Tower Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.3.1 Quotient Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.4 Decay of Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.4.1 Reducing to the Quotient Tower. . . . . . . . . . . . . . . . . . . . 126 4.4.2 Regularity of the Discretisation. . . . . . . . . . . . . . . . . . . . . 129 4.4.3 Specific Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.4.4 The Non-exact Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.5 Regularity of the Stable Holonomy . . . . . . . . . . . . . . . . . . . . . . . 137 4.5.1 Absolute Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.5.2 The Density Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.6 Application: A Solenoid with Intermittency . . . . . . . . . . . . . . . . . 146 4.6.1 Partially Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.6.2 Positive Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . 151 4.6.3 Young Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5 Inducing Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.1 A General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.1.1 Bounded Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.2 The Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.2.1 Inductive Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.2.2 Key Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.2.3 Metric Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.3 Inducing Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.3.1 Integrability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.3.2 Tail Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6 Nonuniformly Expanding Attractors. . . . . . . . . . . . . . . . . . . . . . . . . 189 6.1 Nonuniform Expansion and Slow Recurrence. . . . . . . . . . . . . . . . 189 6.1.1 Hyperbolic Times and Preballs. . . . . . . . . . . . . . . . . . . . . 192