Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov (Contemporary Mathematics) 1470425602, 9781470425609

Citation preview

692

Modern Theory of Dynamical Systems A Tribute to Dmitry Victorovich Anosov Anatole Katok Yakov Pesin Federico Rodriguez Hertz Editors

American Mathematical Society

Modern Theory of Dynamical Systems A Tribute to Dmitry Victorovich Anosov Anatole Katok Yakov Pesin Federico Rodriguez Hertz Editors

692

Modern Theory of Dynamical Systems A Tribute to Dmitry Victorovich Anosov Anatole Katok Yakov Pesin Federico Rodriguez Hertz Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss

Kailash Misra

Catherine Yan

2010 Mathematics Subject Classification. Primary 37Bxx, 37Cxx, 37Dxx, 37Exx, 37Gxx, 37Jxx.

Library of Congress Cataloging-in-Publication Data Names: Katok, A. B., editor. | Pesin, Ya. B., editor. | Rodriguez Hertz, Federico, 1973- editor. Title: Modern theory of dynamical systems : a tribute to Dmitry Victorovich Anosov / Anatole Katok, Yakov Pesin, Federico Rodriguez Hertz, editors. Description: Providence, Rhode Island : American Mathematical Society, [2017] — Series: Contemporary mathematics ; volume 692 | Includes bibliographical references. Identifiers: LCCN 2016052689 | ISBN 9781470425609 (alk. paper) Subjects: LCSH: Anosov, D. V. | Differentiable dynamical systems. | Hyperbolic spaces. | Boundary value problems. | AMS: Dynamical systems and ergodic theory – Topological dynamics – Topological dynamics. msc | Dynamical systems and ergodic theory – Smooth dynamical systems: general theory – Smooth dynamical systems: general theory. msc | Dynamical systems and ergodic theory – Dynamical systems with hyperbolic behavior – Dynamical systems with hyperbolic behavior. msc | Dynamical systems and ergodic theory – Low-dimensional dynamical systems – Low-dimensional dynamical systems. msc | Dynamical systems and ergodic theory – Local and nonlocal bifurcation theory – Local and nonlocal bifurcation theory. msc | Dynamical systems and ergodic theory – Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems – Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems. msc Classification: LCC QA614.8 .M645 2017 | DDC 515/.39–dc23 LC record available at https://lccn.loc.gov/2016052689 DOI: http://dx.doi.org/10.1090/conm/692

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Dmitry Victorovich Anasov Photograph courtesy of the Steklov Mathematical Institute of Russian Academy of Sciences

Contents

Preface

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Dmitry Viktorovich Anosov: His life and mathematics Anatole Katok

1

D.V. Anosov and our road to partial hyperbolicity Michael Brin and Yakov Pesin

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Escape from large holes in Anosov systems Valentin Afraimovich and Leonid Bunimovich

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A dynamical decomposition of the torus into pseudo-circles ¨ ger Franc ¸ ois B´ eguin, Sylvain Crovisier, and Tobias Ja

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On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups Artem Dudko and Rostislav Grigorchuk 51 Isolated elliptic fixed points for smooth Hamiltonians Bassam Fayad and Maria Saprikina

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Nonlocally maximal and premaximal hyperbolic sets T. Fisher, T. Petty, and S. Tikhomirov

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Rotation numbers for S 2 diffeomorphisms John Franks

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Path connectedness and entropy density of the space of hyperbolic ergodic measures Anton Gorodetski and Yakov Pesin

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Around Anosov-Weil theory V. Grines and E. Zhuzhoma

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Attractors and skew products Yu. Ilyashenko and I. Shilin

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Thermodynamic formalism for some systems with countable Markov structures Michael Jakobson 177 Non-uniform measure rigidity for Zk actions of symplectic type Anatole Katok and Federico Rodriguez Hertz

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On a differentiable linearization theorem of Philip Hartman Sheldon E. Newhouse

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CONTENTS

Time change invariants for measure preserving flows Marina Ratner

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Spectral boundary value problems for Laplace-Beltrami operator: Moduli of continuity of eigenvalues under domain deformation A. Stepin and I. Zilin

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Measure-theoretical properties of center foliations Marcelo Viana and Jiagang Yang

291

Preface This volume of the “Contemporary mathematics ” series is dedicated to the achievements and memory of Dmitry Viktorovich Anosov (1936–2014), one of the founders of the modern dynamical systems theory . While Anosov lived and worked all his life in the Soviet Union and Russia, his work beginning from 1960s, had great international resonance. Anosov’s name is forever connected with hyperbolic dynamics, the area where he made his most important contributions. S. Smale named one of the central objects of this area, originally introduced by Anosov as U-systems, Anosov systems, and this name quickly came into the universal use. The features captured by that notion are so striking that various derivative and related objects were given names that still refer to Anosov. Another important contribution of Anosov is the discovery of a very flexible and rather paradoxical AbC (Approximation by Conjugation) method of constructing smooth dynamical systems with interesting, often unexpected, properties. In the literature this method, that is still widely used, is often called AK (Anosov-Katok) method. The composition of this volume reflects both the influence of Anosov’s contributions and his personal legacy. Two leading articles contain personal recollections; the first of them also includes an informal partial survey of Anosov’s work. The remaining fifteen papers are primarily original research papers; several among them are fully or partially surveys dedicated primarily to various aspects of Anosov’s work. Thematically hyperbolic dynamics in a broad sense appears as the subject in nine of those papers. Four of those are fairly directly connected with the themes and contents of Anosov’s work. Two more papers include new applications of the AbC method. The authors of this volume can be approximately divided into three groups: (i) long-term friends, colleagues, students and collaborators of Anosov from the “Russian school”, some of them still in Russia, others now permanently living in the United States; (ii) senior Western mathematicians directly influenced by Anosov’s work, and (iii) mathematicians of younger generation who did not know Anosov personally but have been influenced by his work or by the developments directly based on that work. The editors hope that this volume will serve as a fitting memorial to one of the outstanding mathematicians of the second half of the twentieth century. Anatole Katok Yakov Pesin Federico Rodriguez Hertz

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Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13925

Dmitry Viktorovich Anosov: His life and mathematics Anatole Katok

Dmitry Viktorovich Anosov (Dima for his friends) died on August 7, 2014 at the age of 77. This article is a tribute to his memory. It consists of two parts, different in style and purpose.

D.V. Anosov in 1977 Photo by Konrad Jacobs. Archives of the Mathematisches Forschungsinstitut Oberwolfach, licensed under CC BY-SA 2.0 DE

The first part contains personal recollections touching on both professional and social matters. All information there is first-hand; all opinions are strictly my own. I did not try to ask other people or do any research to provide any kind of coherent narrative. My goal is to preserve memories of events and attitudes that not many people have ever known, and even fewer remain in possession with passage of time. I tried to present and preserve an image of a pretty remarkable man who lived through complex times and whose views of the world and people around him were lucid and free of illusions without becoming cynical. He followed a certain implicit code of honor more strictly than many of his contemporaries, even some of those who had reputations of being more progressive and liberal. 2010 Mathematics Subject Classification. Primary 01A60, 01A65. c 2017 A. Katok

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I’d like to emphasize that my recollections concern the man I knew in the late Soviet period, namely from mid-1960s till 1982 (the latter year is the date of our meeting in Germany, the first after my 1978 emigration from the Soviet Union). I also met Anosov several times in the post-Soviet period in the US and in Russia. I do not include any recollections of those meetings; in fact I do not remember much of great interest. I’d like to point out though that, judging by Anosov’s later writings, especially the historical surveys [8] and [9], his outlook changed in later years, probably in the direction somewhat away from the picture I try to present. The second part is a brief sketch of Anosov’s work, primarily from the same period that is covered in the first part, and its influence on the broader mathematical community. 1. Personal recollections Algebraic topology course. I first met Anosov at some time in the midsixties when he was already a very accomplished and well-established mathematician and I was still an undergraduate student although our age difference is less that eight years. I do not remember the first meeting or first introduction. What I do remember is that our first serious interaction was in 1966 (I believe in the fall of that year) when my thesis adviser Ja. G. Sinai asked me to take what now would be called a “reading course” in algebraic topology with Anosov. I was not a novice in algebraic topology at the time. Although there was no regular undergraduate course on the subject at the Moscow State University when I was a student, algebraic topology was considered then and there the “queen of mathematics” (or at least one of very few principal ladies) and every self-respecting student was supposed to learn quite a bit of the subject somehow. I sat through the remarkable special (topics) course given by D.B. Fuks attended for most of the semester by 200-250 people, and fairly carefully read few books, both classical and modern. We used the classical book by Hilton and Wiley as a text. In fact, my task was to solve independently all problems/exercises from that book (that I did successfully) and also ponder about a specific then unsolved problem relating topology and dynamics: rationality of ζ-function for Anosov diffeomorphisms. Most of the time during our meetings was taken by discussions of various topics and issues emerging from those problems. So I had an ample opportunity to develop my views of Anosov as a topologist. He was a master of the subject in full possession of all essential results, topics and techniques. The reader should keep in mind that topology was never Anosov’s principal mathematical area; he published only one expository paper on the subject [10], albeit in the prestigious Uspehi. This first impression is consistent with the opinion that I formed and held later when we interacted closely and extensively. If Anosov claimed to know a major or minor mathematical subject, he knew all its ins and outs, otherwise he would either profess ignorance or dismiss the topic. Anosov-Katok method. If our interaction during the topology course gave me an impression of Anosov as a scholar, some time after that I had a superb opportunity to observe and appreciate his creativity. In retrospect this was the high point of original creative thinking that Anosov displayed during the period of our close contacts, from 1966 till early 1978, and, I believe, also afterwards. In front of my eyes Anosov invented the core of what has become known as “Anosov-Katok

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method”1 for construction of dynamical systems with interesting, often exotic properties. I will tell the story with minimal but necessary mathematical technicalities in the second part of this paper. This joint work published as [12] is considered by many as the (probably distant) second most important mathematical contribution by Anosov after his major role in the creation of the modern theory of dynamical systems with hyperbolic behavior immortalized by ascribing his name to several important classes of such systems. It’s a pity I do not remember exact date of Anosov’s inspired invention; I am pretty sure this was during the second half of 1968. I very quickly added my essential and extensive contributions that greatly extended the power of the method and several weeks of discussions followed. Then I remember vividly having written a complete draft of the paper just from my head in three successive evenings on Friday, Saturday and Sunday (I never reached a comparable level of productivity in my life, before or after) and having extensive discussions with Anosov that lasted for many weeks and resulted in the final version. Typing (by a professional typist) from the manuscript and inserting formulas and drawing pictures by hand (the last task was performed by my wife) was not a very fast process either and the only hard date is that of the journal submission: May 20, 1969. We published a short announcement in Uspehi [11], but I think it was written after the main text and in fact it appeared in print the same year (1970) as the main text. For more than ten years that preceded my emigration from the Soviet Union in February 1978 my contacts with Anosov, both professional and social, were frequent and extensive. In fact, my wife and I became close friends with Anosov. I will not follow exact chronology but rather try to address various facets of Anosov’s personality, his attitudes and characteristic actions or sometimes inaction. Mathematically there were strong mutual influences. Our joint work at the beginning of the period owes its framework to the theory of periodic approximations that we developed few years earlier with A.M. (Tolya) Stepin. On the other hand, my own interests during the period were moving more and more toward hyperbolic dynamics and its variations and was greatly influenced, directly and indirectly, by Anosov and his work. Conversely, an observation in one of my papers [20] that developed new applications of our method led Anosov to his next major interest, variational methods in Finsler geometry. Our seminar. The principal vehicle around which our professional interaction was organized was a weekly afternoon seminar that most likely met at the university during the 1969-70 academic year, and then definitely at the Steklov institute from the fall of 1970 till 1975, and through the spring of 1977 at CEMI (The Central Economics-Mathematics Institute of the USSR Academy of Sciences) where I worked. I had an opportunity earlier to write in detail about this seminar including Anosov’s role in it [22]; see also [14]. Anosov was already perceived as a “senior statesman” although he only turned forty toward the end of this period. Here is a relevant quote from [22] about Anosov’s role: “He was brilliant and quick, and possessed a very perceptive and critical mind. Everybody, around him, including myself, greatly benefited from from his comments, criticism, and help with pointing out and correcting errors.” 1 Since I still use this method in my work I prefer to call it descriptively the “approximation by conjugation method”

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Anosov’s moral code. Let me try to describe Anosov’s moral position in professional life at the time. Mathematics was not exempt from general trends that dominated the life of the country during the late Soviet (post Khrushchev– pre Gorbachev) period. Two of those tendencies, most relevant for the life of the mathematical community, were, first, discrimination against Jews, and, to a lesser extent, other groups, ethnic, social or professional, and, second, pervasive corruption that led to erosion of professional standards. Any mathematician who had a formal or informal standing and influence faced issues related to those tendencies constantly, and had to determine his or her position and line of behavior. Needless to say, many people behaved inconsistently and opportunistically so often there was a gap between convictions (openly stated or not) and actual behavior. To Anosov’s great credit, his position at the time was both consistent and explicit and his behavior on all specific occasions, known to me, fully agreed with this position. It can be summarized like that: (1) do no evil; (2) do right things if there is no danger of direct clash with authorities immediately responsible for the matter; (3) act within established institutional structures; (4) do not try any endeavor that is either doomed or would require a moral compromise beyond certain narrowly defined bounds. This code of behavior may not look heroic; many intellectuals, including mathematicians, at the time declared more radically progressive views. From their standpoint Anosov was a conformist and to a certain extent a “collaborator”. The problem with this position is that most of those holding and declaring progressive views were not able to act according to those views and ended up doing nothing at best or making grave moral compromises at worst. Before illustrating this general description with examples let me formulate my own attitude toward Anosov’s moral code. I was greatly impressed by (1); he followed this fundamental principle rigidly. I do not know whether its origins had a Christian admixture or were fully secular. He was aware of the necessity of moral compromises to advance a good cause (I will mention an example later) but stayed within strict limits, and, while a few of these compromises had harmful consequences, those were results of mistakes in his original judgement or unforeseen outside circumstances. (2) resulted in many good things in practice, some of which I will mention in due time. It is important to emphasize that Anosov always considered as “authorities” only people occupying particular positions on whom relevant decisions depended: an editor-in-chief of a journal, a department head in an institute, a dean in a university or the chair of a scientific council responsible for accepting dissertations. Thus his apprehensions were strictly limited and based on the knowledge of concrete persons and their positions. To the best of my knowledge, he was never prevented from acting for a good cause by any general trends or policies by academic, let alone party, authorities. (3) I took matter-of-factly; boldness in organizational matters, that was not completely impossible under the circumstances of the time, was not in Anosov’s character. (4) annoyed me a bit, since sometimes Anosov’s sober estimate of difficulties of a certain undertaking led to inertia. To his credit, he never resisted initiatives by others that concerned him; moving of our seminar to CEMI after Pontryagin’s attempt to sabotage it that is described in [22] is an example. Helping young mathematicians. Anosov’s personality was less engaging or flamboyant than that of several of his contemporaries. Besides, his main position

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was in a non-teaching institution so he did not have constant contact with students. As a result of this there are not many mathematicians for whom Anosov was a primary Ph.D. adviser in the way that is usually understood. During the period covered by this recollections Anosov had two such students: A.B. Krygin and A.A. Blohin. The former published several very good papers related to our approximation by conjugation method and to Anosov’s program on cylindrical cascades during 1970s but unfortunately stopped publishing soon afterwards. The latter showed early promise but published only one paper and did not even defend his Ph. D. due to illness. There are however several successful and even highly accomplished mathematicians whom Anosov helped both with their mathematics and with their careers and for whom this help was crucial. Two best known and most impressive among those are of course M. (Misha) Brin and Ya. (Yasha) Pesin. Their names will appear later in this article. More information can be found in their article in this volume [14] and in my article [22]. For both of them, aside from very significant help with their work and acting as the official thesis adviser, Anosov provided invaluable help that was needed to overcome the difficulties of being “out of the system” due to their Jewish origin and strong anti-semitic tendencies of the time. E.A. Sataev. Now I’d like to tell a story of another mathematician, E.A. (Zhenya) Sataev, who died in 2015 whose accomplishments are somewhat less known than they deserve. This story shows that Jewish decent was not the only source of difficulties young people faced in the Soviet Union. Sataev was a student at the Moscow State University; he came from a village or a small town in the Volga area and was not an ethnic Russian but belonged to one of the Finnish peoples native to the area. Unlike the majority of successful students in the university he really came from the midst of ordinary people. Sataev’s first undergraduate adviser was Stepin who left for Egypt around 1970 for what was supposed to be a multi-year appointment that was cut short by the famous expulsion by Anvar Sadat of all Soviet personnel from Egypt. Stepin left his two students, R.I. Grigorchuk (later of groups of intermediate growth or “Grigorchuk groups” fame) and Sataev, to me. When Stepin returned, Grigorchuk continued to work with him but Sataev stayed with me. Sataev did not have a residence permit for Moscow or Moscow district so he was not able to get a job in Moscow or nearby. My standing at the time of his graduation (1972) was not sufficient to recommend Sataev for the graduate program at the university. In any event Sataev decided to go to work in one of the notorious Soviet “secret towns” (Arzamas-16) that paid a good wage and was located not too far from his family home. Very quickly Sataev realized that this had been an unwise decision since it greatly restricted whatever limited freedoms ordinary Soviet citizens enjoyed. At the end of his initial contract Sataev was able to leave for a graduate program in mathematics. But I still was not able to help him with admission to the university program2 . Anyway, I mentioned Sataev’s situation to Anosov and he generously offered to have Sataev admitted to the graduate program at the Steklov Institute on the understanding that Sataev will continue 2 I am not sure why I did not try to have him admitted to the program at my place of work, CEMI (Central Economics and Mathematics Institute of the USSR Academy of Sciences). This was feasible. Maybe I already started to think about emigration and did not want to leave Sataev unprotected. Or, more likely, since the pressure to do work related to the institute mission increased, I did not feel I could have a student working in pure mathematics.

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to work with me. Unlike the cases of Brin and Pesin, where Anosov was a genuine co-adviser, this was a “cover”: Sataev worked then on topics unrelated to Anosov’s interests. This “cover” was at least as valuable for Sataev as theirs was for Brin and Pesin. In the event, Sataev did an excellent thesis work on Kakutani equivalence theory that came earlier than a similar project by the giants of ergodic theory D. Ornstein, D. Rudolph and B. Weiss and was only marginally weaker than theirs. This followed his similarly impressive Masters thesis (on a quite different topic); both were published in Izvestija. I like to believe that Sataev’s Ph.D. defense took place on February 15, 1978, the day I left The Soviet Union for good. I still have a pretty huge twelve-layers matryoshka as his parting present. While this date may be wrong by a couple of weeks, having Anosov as an official adviser guaranteed that Sataev was not tainted by too close an association with an emigre.3 Sataev still had no residence permit for Moscow or the district so the best he could do was to get a job in Obninsk, a restricted but not fully secret town less than a hundred miles from Moscow. He had a successful career there becoming a Doctor of Science and Department chair and consistently doing good work in a particular area of hyperbolic dynamics, but I have reasons to believe judging by his brilliant early work that the relative isolation of the place prevented him from realizing his full potential. Anosov as journal editor. In practical terms, the great majority of situations where Anosov had to exercise his professional judgement and face moral dilemmas appeared in connection with consideration of papers for publication and defense and approval of dissertations, both candidate (equivalent to Ph. D.) and the higher level Doctor of Science. At the time Anosov was on editorial boards of two journals: Izvestija of the USSR Academy of Sciences, and Zametki (Mathematical Notes of the Academy). Those days virtually all papers by mathematicians in the Soviet Union were published in domestic journals; the leading journals were quickly translated into English cover-to-cover and published in the West. Hierarchy of journals was quite important. The top tier consisted of Izvestija, Sbornik, Uspehi, and Doklady to which is often added irregularly published Trudy MMO (Transactions of the Moscow Mathematical Society).4 Among those only the first two (and Trudy) were regular vehicles for publishing complete original papers; Izvestija in general was considered the most prestigious among the three. Doklady published research announcements with the strict limit of four printed pages and Uspehi at least officially was dedicated to publishing surveys, although in reality often those “surveys” contained a high proportion of original results. Zametki was among the best journals in the next tier. Both discrimination and corruption issues appeared in the context of publication of mathematical papers. Discrimination pressure was felt more acutely in the journals published by the Academy due to pronounced anti-semitic attitudes of the academy bureaucracy and some leading academicians, including I.M. Vinogradov, 3 We had published a joint paper in Zametki in 1976 but this degree of association was not harmful at the time. 4 “Functional Analysis and its application”, established in the sixties, de facto maintained as high and occasionally even higher level than any of those journals; it, however, had a reputation of being the mouthpiece of Gelfand’s school.

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L.S. Pontryagin and A.N. Tykhonov.5 Influence of corruption was felt through preferential treatment of papers by certain authors, and, conversely, keeping away the work of their competitors, loose refereeing standards, and suchlike. Thus on boards of two leading Academy journals Anosov faced not the most friendly environment. Still during the ten-year period I observed him in this capacity he never swayed from the highest standard on research and scholarship and never hesitated to promote high quality papers, independently of personalities of their authors. Careers of Brin and Pesin were launched by the publication of their major paper on partial hyperbolicity in Izvestija [13], a landmark in the field. This occurrence is hard to imagine under standard circumstances existing at the time with two unknown young Jewish authors who were not even graduate students but worked in institutions unrelated to mathematics (for more on their circumstances at the time see [22]). Another paper of Pesin, [29], that contained the core technical results of celebrated “Pesin theory” appeared in Izvestija a couple of years later. A convincing, albeit indirect, illustration of my thesis comes from difficulties both Brin and Pesin faced in submitting and defending their Ph.D.’s based on their world class work (two Izvestija papers, one Uspehi paper plus publications in other first-rate journals, including Zametki) that under normal circumstances would qualify each of them for the Doctor of Science degree. In both cases anti-semitic attitudes and policies prevented them from having their dissertations accepted for defense in the leading places such as Moscow State University, Steklov Institute or even other places in the capital, and forced them and their backers to look for places outside of Moscow. Brin was able to defend his thesis in Kharkov in 1975, of course with strong backing by Anosov, but using some key connections of his own to overcome even more pronounced anti-semitism that existed in the Ukraine at the time. Pesin had to wait till 1979, when his work already became world famous, and it was entirely due to Anosov, that he was able to defend his dissertation in Gorky (now Nizhnij Novgorod). Very interesting story of Pesin’s defense is told in another article in this volume [14]; notice in particular the moral compromise that Anosov consciously made to achieve success. As far as I know, a previous attempt by Anosov to have Pesin’s dissertation accepted for defense in the university of Rostov was unsuccessful, but this setback did not discourage Anosov. I published two major papers in Izvestija during the period [20, 21]. The story of publication of the second paper is worth telling since it vividly illustrated some of the features of Anosov’s approach to the difficulties exiting at the time that I described above. The paper presents a core of what I called “monotone equivalence theory” in ergodic theory and is now commonly called Kakutani equivalence theory. It is based on results that I obtained in 1975 and early 1976. It was an extensive body of work and by no means exhausted the subject by that time. A Doklady announcement of key results was published in 1975 and I asked Anosov about feasibility of submitting the paper with complete proofs (some of which were not yet written then) to Izvestija. Anosov had a very high opinion of the results and asked me how long the paper would be. I answered that after everything is completed it

5 Uspehi and Trudy were published by the Moscow Mathematical Society; Sbornik was a joint publication of the Society and the Academy; and Doklady, while an Academy publication, published papers communicated by individual academicians without additional reviews.

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would be about 100 pages. Anosov explained that he would not be able to have a paper of such length published in Izvestija. The editor-in-chief of the journal was elderly I.M.Vinogradov, then 84 years of age but still director of the Steklov institute and the N1 in the hierarchy of mathematicians in the Soviet Union. As I already mentioned, he was a convinced and inveterate anti-semite. His anti-semitic attitude was apparently not of opportunistic nature as was the case with many Soviet officials and even scientists but was a deeply held conviction stemming from attitudes of the hard right in the late czarist times. His deputy was I.R. Shafarevich, a great mathematician who held and in fact expressed strong anti-communist views, and, surprisingly, was only mildly reprimanded for that. Of course, later he became infamous for his chauvinistic and anti-semitic writings and became an icon of the Russian nationalist hard right. Still, in fairness, his anti-semitism at the time (and, I believe, later too) was of a theoretical nature and did not descend into hatred of individual Jews simply because they were Jews. In any event, as a de facto editor-in-chief of Izvestija, Shafarevish followed a high-minded and fair policy. He could approve all articles with strong positive recommendation by other editors up to a certain length independently of the authors’ nationality and other extraneous features without showing them to Vinogradov. But for exceptionally long articles (and 100 pages was over the limit) Vinogradov’s direct approval was required. Anosov was ready to argue merits of my work with Shafarevich and other members of the editorial board (not all of them friendly to Jews) but not ready to confront Vinogradov who would almost surely veto publication. Anosov’s suggested solution was simple: to split the work into two papers about 50 pages each and publish them with some time interval. For that, approval of Shafarevich, which would be forthcoming, was sufficient. In order for this scheme to succeed an interval of about a year between publication of two articles was required; if the interval was too short Vinogradov may be informed that splitting of the paper was a ploy. This conversation took place at the end of 1975 or at the beginning of 1976. By then my tolerance of life in the Soviet Union was wearing thin and I already decided to leave the country after some necessary preparations. When in 1971-72 for the first time emigration became a realistic possibility with a tolerable level of risk and our friends and colleagues started leaving, my wife and I seriously discussed the possibility and decided to stay put. This changed by the late 1975 due to a variety of factors.6 Thus I knew that I may not have time to publish two papers with an interval of a year. On the occasion, the 54-page long paper [21] was submitted to Izvestija on March 3, 1976 and appeared in print in the first issue of 1977, i.e. around February of that year. On February 15 of 1978 I left the Soviet Union with my family for good as stateless persons stripped of the Soviet citizenship. I applied for emigration in July of 1977 and this quickly became public knowledge. Thus if I submitted the second part early in 1977 or even late in 1976 it would not have appeared in print by that time.

6 I hope to discuss the matter of emigration in detail on another occasion since it has only tangential relation to my present subject.

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What happened with submitted or even accepted papers of would-be emigrants was well-known. A paper by B.G. Moishezon, a brilliant algebraic geometer, one of the favorite students of Shafarevich and my CEMI colleague, who emigrated in 1973, that was not only accepted but already typeset for an issue of Izvestija, was removed and the issue came out thiner than usual. My wife Svetlana Katok, whose short paper was scheduled to appear in the more friendly Uspehi, was asked by a very honorable and decent editorial board member to withdraw the paper when we applied for emigration. The motivation was not to expose the journal and the society to attacks and criticism by the party watchdogs. Thus I decided to call off the still unfinished second part of my paper, pack into the last section of the fist (and, as it turned out, the only) paper the announcements of remaining results, and submit it as fast as I could. As is seen from my narrative, I had just a few months to spare. I got a consolation from simultaneous publication by Izvestija in the first issue of 1977 of Sataev’s thesis paper that was in a way a continuation of mine albeit in a quite different direction than my projected second part. Refereeing and approving dissertations. Anosov was a member of the Higher Attestation Board that had to approve all mathematics dissertations defended in the USSR. For readers not familiar with the Soviet/Russian academic system here is a brief summary that ignores issues related to discrimination and corruption. The candidate degree is usually considered an equivalent to Ph.D. but signifying a slightly higher level of achievement and accordingly carrying more prestige. The main requirement was a dissertation accompanied by publications. The dissertation was to be submitted to a scientific council with fixed membership (typically 15 to 25 members) attached to a particular university or a research institute. It has to be accepted for defense that was public with two referees (somewhat misleadingly called “official opponents”) and outside review from a “leading organization in the field”. Defense included a presentation by the candidate, speeches by the opponents, reading of outside review, usually also a speech by the thesis adviser, and a freeformat discussion in which both the council members and guests could participate. The vote was secret and a two-third majority of positive votes was required for approval. Active mathematicians usually defended their candidate theses within 3-8 years after graduation from the university. Three-year (post)-graduate studies program was meant to prepare the candidate thesis; it could be taken even right after graduation of after an interval of time. It was not a precondition for submitting the thesis. In fact, both Brin and Pesin did not attend the graduate program [22]. Doctor of Science degree was very prestigious, at the time there were no preparatory courses, defense procedure was similar but with three opponents, all of whom naturally should hold the degree. There were fewer councils that were authorized to accept Dr. S. dissertations. All dissertations have to be approved by a central authority called the Higher Attestation Board (VAK). Under normal circumstances its function was to exercise quality control that was necessary due to great variation of standards across the degree-granting institutions. I believe in the seventies it performed this function to a certain extent but in addition to that the Board (or its particularly eager individual members) also watched after dissertations whose authors were Jewish and on a

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number of occasions rejected them. Needless to say, that Anosov’s position as a member of the VAK’s mathematics panel was based on strict professional standards. While to guarantee a successful outcome of Pesin’s defense in Gorky Anosov had to make a promise to see a sub-standard dissertation through VAK [14], he was fully aware of the moral cost and realized that this was the only way to help Pesin whose work by then was several levels above the accepted standards for a PH.D. Overall Anosov looked after a number of excellent dissertations by Jewish mathematicians (they were mostly people who had some local connections to pass the hurdle of defense) that would not have passed VAK without his intervention.

Warsaw 1977 conference. Here is another episode from the same period that demonstrates Anosov’s attitude. In the summer of 1977 an international dynamical systems conference was organized in Warsaw by a group of Polish mathematicians among whom Wieslaw Szlenk played the principal role. An explicit purpose of that conference was to arrange a major encounter between “the East” and “the West”. This was spectacularly successful. There were two groups of participants from Moscow. An officially approved “delegation” was headed by Anosov and included also Stepin and E.B. Vul. Three other participants, Brin, M.V.(Misha) Jakobson, and myself, came ostensibly on private invitations of our Polish colleagues. This was the only realistic way for us to travel outside of the Soviet Union; any attempt to obtain permission to go on official business would be blocked by one of the numerous bureaucratic offices or party committees whose approval was required. The principal but unstated reason would be that the applicants were Jewish. One that might be stated and did have relevance, was that the subject of the conference did not fit with the principal specialties of our places of work. Of course, form the point of view of the conference organizers we were fully-fledged participants of the conference; maybe even somewhat more interesting than the official delegates since we had not traveled to the West before and were new for the Western participants. So I come to the punchline. Anosov, as the head of the official delegation, was supposed to write a report to appropriate authorities in Moscow, I presume the administration of the mathematics division of the academy, or the Steklov Institute. He was in a bind: to acknowledge the presence of unauthorized participants from Moscow who spoke at the conference with a considerable success, or to lie. He found an imaginative solution, very much in his style. While he socialized and closely interacted with us during the conference, he was conspicuously absent from our talks. In a sense this was mocking his official status and obligations. But looking from another viewpoint, he took a certain risk. Obviously, he planned to ignore our presence in his official report. But, if there was a KGB informer in the audience, Anosov could be denounced for socializing with unauthorized conference participants from Moscow and not admitting their presence. Such a possibility could not be excluded given a large number of local people (and probably some Soviet visitors unrelated to the conference) in the audience. This is a good example of “passive resistance” that Anosov practiced in a variety of situations.

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Relations with Pontryagin. Anosov was a student of L.S. Pontryagin, one of the greatest among the first great generation of mathematicians of the postrevolutionary (Soviet) period. Pontryagin’s evolution, both as a mathematician and and as human being, presents a somewhat sad sight. In the 1930s and 1940s he was a brilliant creative and very broad mathematician who made a great impact in algebraic topology during its formative period, created duality theory for locally compact abelian groups and, together with the physicist A.A. Andronov, became one of the creators of modern theory of dynamical systems. He was also known for his independent and on the whole honest and courageous behavior in professional life that is not a small compliment for someone who lived in the Soviet Union through that terrible period. Sometime during the 1950s, when he was still in his mid-forties, his mathematical interests moved toward more applied direction and he made a major impact in that area as one of the creators of the modern theory of optimal control. After that, his mathematical standards dropped and he was in general surrounded by people of less than stellar quality, if not outright hacks. Pontryagin wrote a textbook in ordinary differential equations and he taught the regular ODE course to my sophomore class. His presentation was heavy and not very illuminating and it was hard to believe that was the same man who more than twenty years earlier had written the masterpiece “Continuous groups” that still remains as good and illuminating presentation of the Pontryagin duality and related subject as any. It was also strange that Pontyagin tried to supersede then standard in the Soviet Union ODE text by the great I.G. Petrovsky that, even though a bit outdated by now, still makes an excellent and lucid reading.7 One had a nagging feeling that Pontryagin pushed his approach and his book out of spite of Petrovsky and that feeling unfortunately has some support in the story of evolution of Pontryagin’s personal views. More or less simultaneously with his turn toward applied mathematics Pontryagin’s views and behavior became quite retrograde and reactionary. He became a pronounced anti-semite and this attitude found an expression in his professional behavior.8 Anosov was Pontryagin’s graduate student at the Steklov Institute on the ODE side. He was clearly Pontryagin’s favorite. He was greatly influenced by AndronovPontryagin seminal work on structural stability and his presentation, both in the early papers on averaging and in the classical work on hyperbolic dynamics, was clearly influenced by Pontryagin’s style of the period. Pontryagin was the head of the ODE department of the Steklov Institute and Anosov stayed in the department after the graduate school. Pontryagin not only highly valued Anosov’s work but also from the beginning considered Anosov his trusted lieutenant. Steklov Institute as a whole and its individual members, especially those in senior positions, wielded considerable influence, if not outright control, over research enterprise in mathematics. Anosov, despite his young age and a somewhat reticent personality, quickly became a member of this senior elite. Even though I do not know details of the inner workings of the Steklov elite at the time, it is clear (and is confirmed 7 Petrovsky’s approach indeed needed some updating and that was brilliantly accomplished a bit later by V.I. Arnold 8 My opinion of Pontryagin’s course and his text is at variance with what Anosov wrote several decades later in [9]. Anosov, while acknowledging tensions between Petrovsky and Pontryagin, and certain decline in Pontryagin’s stsndards, takes Pontryagin’s side and in fact makes some disparaging remarks about Petrovsky.

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by Anosov’s occasional remarks) that this quick accent was due to patronage and protection of Pontryagin. That at the time Anosov disapproved of Pontryagin’s attitudes and behavior on the Jewish issue was quite obvious and he expressed this disapproval in private conversations, as well as in some actions. Here is what I wrote on another occasion [22]:“Anosov, a former student of Pontryagin, considered for a while as his picked successor, refused to follow the hard line of his bosses, and, short of open rebellion, was sabotaging their agenda with considerable success.9 ” I continue with details about the move of our seminar from Steklov to CEMI that Anosov approved after Pontryagin refused to authorize the list of persons for admission to the building with many Jewish names. At the time I was under the impression that cooling down of relationships between Anosov and Pontryagin went further and came close to a formal break. Unfortunately, I never asked Anosov about that when I met with him in post-Soviet times. According to S.P. Novikov, who possesses lots of inside information that is not always 100% reliable, no visible cooling down or a break has ever taken place. So I rest here. Private life. Anosov came from an academic family. Both of his parents were chemistry professors/researchers of considerable repute. In fact, there is an article about his father Viktor Yakovlevich Anosov in a respectable Russian series “Scientific heritage of Russia” where V.Y.Anosov is called “one of the most important specialists in the area of physical chemistry analysis”. The mother Nina Konstantinovna Voskresenskaya also held a Dr. S. degree. I have known a number of Soviet academic families of that generation when material rewards for upper-crust academics were very high compared to the overall living standards, e.g. the base pay of two professors/Dr. S.’s was about 12-15 times the average earning of a person with a college degree. Such families in the 1940-50s and, to a lesser extent, in 1960-70s typically enjoyed lifestyle with pronounced bourgeois overtones: a spacious well-furnished city apartment, with valuable items, often even good works of art, good food, a car, often a country house (dacha), until about 1960 a live-in maid; later a part-time maid/cook. The Anosov family that at the time I got to know it, had three, not just two, high-earners, presented a great contrast to that stereotype. They did live in a large by the Moscow standards, apartment in a good (but not great) location and they did have a rather sorry looking woman helper (the parents were well in their seventies), but there it stopped. The furniture was spartan, to put it charitably, or plain shabby, the food very simple. The lifestyle of the family can be described as ascetic. When I first got to know them the father was still alive but looked very old and fragile but the mother was still quite active. The father died in 1972, the mother around 1975. The only luxury was an excellent for the time stereo-system and a collection of high quality records of classical music, mostly foreign made. Dima was a great lover and connoisseur of classical music. Those records were practically the only things Dima brought from his relatively frequent foreign trips. At the time when Levi’s blue jeans were both the badge of distinction and almost an alternative currency it is extremely remarkable that he did not buy abroad any clothing items and dressed in an old-fashioned and somewhat awkward way in domestically made clothing. 9 Anosov did become the department head but already under somewhat different circumstances

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I see two reasons for this contrast. The principal one is the difference between the old Russian and Soviet “intelligencia”. The former defined itself mostly by moral and intellectual attitudes and very often, although not always, was indifferent to the material comforts, let alone luxury. It was characterized by great sensitivity to the plight of the poor and the disadvantaged and often made material sacrifices to alleviate it. The latter, to allow for a certain oversimplification, thought of itself as an elite of the middle class whose material and spiritual interests were in a sort of balance or equilibrium. Thus, those of its members (by no means a majority) who could afford good things in life, usually went for those, conforming to the generalized picture presented above. Anosov’s family belonged to intelligencia at least in the third generation and it quickly became clear to me, that the parents spiritually belonged to the old Russian intelligencia although their careers spanned the Soviet period. I heard that Anosov’s parents used to support poor students and other destitute people in keeping with the Russian intelligencia traditions. I will comment on Dima’s generosity later. Still I believe the expenditures of the family were much smaller than their earnings. While the parents may have been genuinely disinterested in material comforts Dima, who belonged to a different generation, was not averse to enjoying some of those. And here comes another subsidiary reason. At the time (the late Soviet period) there was great scarcity of quality items of almost any kind (food, clothing, furniture, books etc) through regular distribution channels. Money as such could buy little. One needed in addition “connections” in the form of access to official (special stores) semi-official (wholesale distribution chain) or unofficial (black market) alternative distributions channels. Dima lacked skills necessary to obtain such access to an astonishing degree. Only when he married shortly after his mother’s death was this problem alleviated. Let me finish this sketch by describing an instance of Dima’s generosity. By 1971 I and my wife Svetlana had two children and we lived in a single room (about 220-240 sq. ft.) in a communal apartment with two more families and the fourth room occupied until 1970 by Svetlana’s grandparents and idle after her grandmother’s death and grandfathers’ move with her parents. That was obviously inadequate, even more so since I grew up in a single-family apartment and was accustomed to better living conditions. So we started to look for a co-op apartment. Although those were built by various enterprises, leftovers, that could not be filled by the employees, were available to the general public. There were also restrictions that prevented most people, even those who had money, from buying larger apartments. Fortunately a Ph.D. holder had a considerable extra allowance so our family was eligible for approximately an 750-800 sq. ft apartment. And larger apartments were often left out due to administrative restrictions and cost so we were able to quickly find a decent apartment of about that size. But then money became an issue. Downpayment was strictly fixed at 40% and that amounted to 4500 rubles or one-and-a-half year gross earnings of myself and my wife at the time. After a relatively routine promotion that was expected in a year or two that would go down to just my own gross earnings for the same period. We had no savings to speak of and neither Svetlana’s parents nor my mother could help us. While mortgages existed and the rate for remaining 60% of the cost was quite low (we were able to afford monthly payments) there was no way to borrow money for the

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downpayment, and besides, there were no realistic chances of repaying, even the principal, within several years. So we pondered this problem. I had an aunt who had considerable savings and who obviously loved me but I could not approach her with a request of that magnitude. At the end I asked and received from her 1000 rubles, nominally as a loan. We were quite close with Anosov at the time and once when he visited us in our room we started to discuss the issue in his presence not having in mind to ask him for anything. We were astounded when Dima offered to borrow the whole amount from him, naturally without interest and with an indefinite term of repayment. Needless to say, we accepted and later decreased the amount by borrowing 1000 rubles from my aunt. The fact is that, had we stayed in the USSR, we would probably have not been able to repay the money before serious inflation started. As it was, we repaid in full from the money we received after selling our apartment before leaving the USSR in 1978. So what was the reason for such an extraordinary generosity? Yes, we were friends but our friendship at the time was only three years old and our closeness was considerable but still not very great. I believe the answer is this: in Anosov’s value system the welfare and comfort of a family he cared about stood much higher than this amount of money for which after all he did not have an immediate use. And the love of money as such was completely alien to him. 2. Mathematical legacy Chronology of Anosov’s principal publications and his expository and historical writing. Control theory and averaging in ODE: three papers in 1959-60 and a paper in 1996. Hyperbolic dynamics: Six works (including a monograph) published in 1962-70 although they mostly cover work done before 1964-5. Approximation in smooth dynamics: four papers in 1970-74. Various aspects of geodesics in Riemannian and Finsler geometry: six papers in 1975-85. Nielsen numbers: a 1985 paper. Behavior of lifts of orbits of flows on compact surfaces to the universal cover: twelve papers in 1987-2005. Return to hyperbolic dynamics: a 1996 paper and five papers in 2010-14. Anosov’s output, especially in his later years, contains a substantial amount of expository, historical and biographical writing. Those range from numerous jubilee articles and obituaries (usually signed by many people), to presentations of some classical topics, to analysis of historical developments, to attempts at broad surveys of recent history. Anosov possessed an excellent and very lucid understanding of many subjects as well as a somewhat peculiar wit and this makes some of his writings very interesting. A good representative example is [7]. As a matter of general fairness to the dynamical community, I feel compelled to comment on Anosov’s most ambitious attempts in this genre, two historical surveys [9] and [8] written in his later years. The former gives his view of the “hyperbolic

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revolution” of the 1960s and his personal participation in it. While the personal recollections are of obvious interest, the general picture is somewhat distorted and, as I will explain later, not necessarily in Anosov’s favor. Worse, his evaluation of contributions of various people then and later is distorted by omission of some essential names that were well known to him. The long survey [8] that covers the last quarter of the twentieth century suffers from similar deficiencies even more. Selection and even more omissions of names and topics for the survey make a strange impression; see the detailed critical MR review [17]. Evidently, Anosov’s outlook and some of his principles evolved in the post-Soviet period when in Russia he received a lot of recognition and achieved fairly high visibility even outside of the mathematics community (in contrast with restrained reception in the West), and did not have to face moral dilemmas of the previous period. Anosov’s contribution to hyperbolic dynamics. In this paper I will only discuss Anosov’s works on hyperbolic dynamics and approximation in smooth dynamics. One reason is that those works are the most influential among Anosov’s contributions. Another is that they and their immediate aftermath correspond to the period of my close interaction with Anosov described in the first section. I apologize for patchy character of my bibliography. I tried not to overload this completely non-technical paper with references. In the era of internet in general and Google Scholar and MathSciNet in particular an interested reader can fill the gaps with only moderate efforts. Anosov’s name is forever connected with hyperbolic dynamics, one of the principal parts of the theory of dynamical systems. Hundreds of papers study and thousands mention Anosov systems, various versions of this notion such as Anosov diffeomorphisms and Anosov flows, as well as later variations: pseudo-Anosov maps, Anosov group actions, and so on. Anosov originally named this object U -systems, but soon afterwards Smale re-christened them Anosov systems, and this term immediately took off in the English language literature, and in a few years substituted the original terminology in the Russian language publications as well. Is this an accidental luck or are there deeper reasons? I think that popularity of these mathematical objects, and hence their names after their discoverer, is not accidental. The origins of the modern view of dynamical systems with finite-dimensional phase space can be traced to the works of H. Poincar´e at the end of the 19th century. During the first half of the 20th century development of the theory followed several, mostly parallel, courses. Here are key names from that period: J. Hadamard, G.D. Birkhoff, O. Perron, M. Morse, G. Hedlund, A. Denjoy, A.A. Andronov, L.S. Pontryagin, J. von Neumann, E. Hopf. Fundamental progress that led to a new synthesis that created foundations of the modern theory of dynamical systems took place in the 1950s and early 1960s. Its principal elements are two discoveries of Kolmogorov ( KAM theory and entropy in dynamics) and “hyperbolic revolution”, as Anosov called it in his historical survey [9]. Retrospectively, that was not that much of a revolution if one properly synthesizes the points of view of Hadamard, Morse, Hedlund and Hopf. Principal actors of this revolution came from different mathematical backgrounds: S. Smale from topology, Anosov from the theory of differential equations, Sinai from probability theory, V. M. Alexeyev from classical mechanics. As is well known, the first impulse came from Smale, as Anosov colorfully describes in [9]. One should not, however, overestimate the role of this

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impulse that provided a description of a somewhat artificial example of a structurally stable diffeomorphism that contains the “Smale horseshoe” as an invariant set. Almost immediately the leadership in the “hyperbolic revolution” passed to a group of young mathematicians from Moscow (Alexeyev, Anosov, Sinai; V.I. Arnold also showed lively and fruitful interest). The reason for this (other than obvious talents of the main actors) was that Moscow mathematicians brought to this new area deep understanding and intuition from different areas of analysis, differential equations and probability theory while Smale’s motivation came almost exclusively form topology. Anosov’s principal contributions to hyperbolic dynamics are contained in his monograph [3] based on his 1965 Doctor of Science thesis and published in 1967. Most of its results, including the main ones are announced in two Doklady notes [1] and [2] published in 1962 and 1963 correspondingly. Notice that the first of these papers that contains, among other things, the structural stability of Anosov systems, was submitted for publication in March of 1962, i.e. less than 6 months after Smale’s famous appearance at the Kiev non-linear oscillations conference that Anosov describes in [9]. Smale at the time had no ideas how to prove structural stability of hyperbolic toral automorphisms, let alone geodesic flows on negatively curved manifolds, while Anosov’s results go far beyond Smale’s structural stability program. So, from my point of view, in his own account, Anosov gives too much credit to Smale and puts his own achievements into the shadow. The reasons for that are not clear to me; while modesty may have played a role, there may have been some additional motives. Anosov developed principal technical tools of hyperbolic dynamics, first of all, the theory of stable and unstable foliations, and approach based on considerations of ε-orbits (pseudo-orbits) and their families. Let me try to describe briefly the essence of these two main discoveries of Anosov in this area. An Anosov system is characterized by the presence of two invariant sub-bundles of the tangent bundle to the phase space that (in the continuous time case, together with the orbit direction) generate the tangent bundle. Vectors in one sub-bundle (called contracting or stable) are contracted with exponential speed under the time evolution in the positive direction, while in the other one (called expanding or unstable) are contracted with exponential speed under the time evolution in the negative direction. A priori those sub-bundles are not even assumed to be continuous, but continuity, and even H¨ older continuity, follow; see [4] for the proof of the H¨ older property. On the other hand, even for infinitely differentiable or analytic systems they are not even C 1 .10 Still, those sub-bundles are uniquely integrable to two foliations with smooth leaves that, however, often do not change in a differentiable way in the transverse direction.Their integrability follows from classical results of Hadamard and Perron (Anosov emphasizes that); and, at least in the analytic case, can be traced even farther back to even earlier work of Darboux, Poincar´e and Lyapunov. Existence of these foliations is very useful for the investigation of topological properties of Anosov systems, including structural stability. However, for the study of more subtle analytic as well as ergodic properties, the absence of differentiability, from the 10 While the C 1 property holds for large open sets of Anosov systems, C 2 is already highly exceptional and related to phenomena of rigidity; Anosov pioneering insight in this direction is contained in Section 24 of [3].

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first glance, presents and an unsurmountable obstacle. E. Hopf in [18] proved the ergodicity of the geodesic flows on the surfaces of variable negative curvature.11 In this case stable and unstable foliations are constructed geometrically as horocyclic foliations. Hopf used the fact that the horocyclic foliations are C 1 .12 This is also true in higher dimension if the curvature is “pinched”: the ratio of the minimal (largest in absolute value) curvature to the maximal one is strictly less than 4. Otherwise the horocyclic foliation on negatively curved manifolds of dimension greater than 2 are usually not C 1 . The proof of ergodicity of geodesic flows on such manifolds was one of the principal goals of Anosov’s work. He discovered a property that holds in this case and is sufficient for applicability of the Hopf argument, and hence allows to prove ergodicity. In a somewhat simplified way, the property of absolute continuity of foliations, discovered and proved (for general Anosov systems) by Anosov, states that the holonomy map between two nearby stable leaves along the unstable (weak-unstable in the continuous time case) foliation is absolutely continuous with respect to Lebesgue measure. Here the weak-unstable foliation is obtained by integrating jointly the unstable sub-bundle and the orbit direction. This property of absolute continuity and its various versions and generalizations played a central role in the development of the hyperbolic dynamics in the last half-century. Anosov’s second discovery provides a convincing explanation of “chaotic” behavior of trajectories in an even more general class of systems with hyperbolic behavior than Anosov systems. I am talking about hyperbolic sets introduced by Smale, where existence of contracting and expanding sub-bundles is postulated not for the whole phase space but only on a closed invariant set. To avoid non-essential technicalities I will discuss the discrete time case only. If in such a system one can find a sequence of points (not even in the hyperbolic set itself but in its small neighborhood) such that every subsequent point is close to the image of the previous one (such a sequence is called an ε-orbit or a pseudo-orbit) then close to this sequence there exists a unique genuine orbit of the system. Furthermore, if a family of ε-orbits, naturally parametrized and continuous by elements of a topological space then the corresponding family of orbits is also continuous in that topology. This principle of shadowing is one of the most important, if not the most important organizing principle of hyperbolic dynamics. It directly implies the widely known Anosov closing lemma, structural stability, existence of Markov partitions and many other things. This principle was present implicitly in [3], was explicitly formulated in [5], became the central element of the fundamental series of papers by R. Bowen (1947-1978), the most brilliant representative of the Smale school, was made the centerpiece of the presentation of the hyperbolic dynamics by the author [19] from where it found its way to [23] that became a standard text. Early influence of Anosov’s ideas can be seen both from the development of the Smale school and from work of such outstanding mathematicians as Ju.Moser [27] and J.Mather [26] who interpreted and developed some of those ideas. Two mathematicians of the next generation, who played a central role in the development of hyperbolic dynamics, M. Brin and Ja.Pesin, were joint students of 11 For the constant negative curvature case ergodicity had been proved earlier by Hedlund using geometric methods. 12 His method that is justifiably known as the “Hopf argument” still plays the central role in the study of ergodic properties of various classes of systems with hyperbolic behavior.

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Anosov and the author. Their work is foundational for two major directions of hyperbolic dynamics that continue as active research areas to this day: partially hyperbolic dynamics [13] and non-uniformly hyperbolic dynamics [30] which is often called the “Pesin theory”. Anosov’s role in their mathematical development included strong conceptual influence, constructive criticism and editing of their work, as well as great help at early stages of their mathematical careers mentioned in the first section that was highly non-trivial in the complicated and unfriendly environment of 1970s. Interestingly, the most important direct follower of Anosov was Smale’s student J. Franks. In his thesis [16] he proved basic results about global structure of Anosov diffeomorphisms and formulated a program for their further study that greatly influenced subsequent work on the subject. Important early contributions to the realization of this program are due to S. Newhouse [28] and A. Manning [25]. The problem of global topological classification of Anosov diffeomorphisms remains one of the most interesting open problems in the theory of dynamical systems. In my 2004 Berkeley-MSRI lecture I listed it among “Five most resistant problems in dynamics”. Brin and Pesin found important applications of their general work on partially hyperbolic systems but otherwise the area lay dormant for a while, Twenty years after the pioneering work of Brin and Pesin [13] the next big development in the theory of partially hyperbolic systems appeared in the work of C. Pugh, M. Shub and their students among whom A. Wilkinson stands out. Notice that Anosov’s concept of absolute continuity plays the central role in these developments. This has become a major research area with many outstanding practitioners. Some of the top names are M. Viana, F. Rodriguez Hertz and Wilkinson. Non-uniformly hyperbolic dynamics developed by Pesin immediately attracted great attention. Among the early work there are important papers by D. Ruelle, R. Ma˜ n´e, Pugh and Shub, M. Herman, A. Fathi and J.-C.Yoccoz This area remains one of the central in the theory of dynamical systems and listing even most important papers will take too much space. Smooth ergodic theory deals with ergodic properties of smooth conservative dynamical systems on smooth manifolds (usually compact) that preserve volume or, more generally, an absolutely continuous measure. Anosov made two major contribution into this area. The first of them is a description of ergodic properties of Anosov systems preserving a smooth measure. The foundation of this work is a deep analysis of properties of stable and unstable foliations that were discussed above. Those properties imply that in the discrete time case an Anosov system is a K-system and, by using later results of Ornstein and his school, also a Bernoulli system. In the continuous time case, Anosov’s result is even more remarkable since it connects ergodic properties of Anosov flows with topology. Namely, a volumereserving Anosov flow either has a continuous spectrum or is a suspension over an Anosov diffeomorphism f defined over a certain global smooth transverse section S. Here suspension is understood as in algebraic topology, i.e. the return time to the section S is constant. Using the terminology of the theory of dynamical systems, this is the special flow of the diffeomorphism f with a constant roof function. This is the famous “Anosov alternative” that, in particular, implies that any eigenfunction of an Anosov flow is smooth. As in the case of discrete time, an Anosov flow with continuous spectrum is a K-flow and a Bernoulli flow. Anosov’s approach

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to the study of ergodic properties of conservative dynamical systems based on the investigation of deep and subtle properties of stable and unstable foliations is the foundation of several central directions of this area that greatly developed in the last 50 years. Approximations in smooth dynamics and ergodic systems on arbitrary manifolds. Anosov’s second contribution to smooth ergodic theory is of a different nature. Beginning from the appearance of ergodic theory as a method of analysis of classical dynamical systems in the early 1930s and till the end of 1960s ergodic properties had been successfully studied only for a limited class of systems of algebraic origin (translations and linear flows on the torus, nil-flows, horocycle flows on surfaces of constant negative curvature, and so on), for some systems that are closely related to those algebraic systems (e.g. flows obtained by a time change in a linear flow on a 2-torus), and, of course, for Anosov systems. In all these cases, topology of the phase space is rather special. For example, there were no methods of constructing area-preserving ergodic diffeomorphisms on the 2-dimensional sphere or the 2-dimensional disk. This problem was solved in our joint work with Anosov [11] that was mentioned in the first part of this article. We developed a new method of constructing systems with interesting, often exotic, ergodic properties on every manifold that allows a non-trivial action of the circle, i.e. the compact group R/Z. As I described above, the original highly unusual idea of this method that does not have analogues in dynamics or in analysis, was suggested by Anosov. Later both authors developed applications of this method that became the central tool in the study of so-called Liouvillean phenomena in dynamics. Anosov [6] published a proof that every manifold of dimension greater than 2 admits a volume-preserving ergodic flow. Results like that require another ingredient that allows to pass from some simple manifolds that allow a circle action to arbitrary manifolds. For a brief description of the method we follow [15]. Let M be a differentiable manifold with a nontrivial smooth circle action S = {St }t∈R , St+1 = St . Every by taking any smooth S 1 action preserves a smooth volume ν which can be obtained 1 volume μ and averaging it with respect to the action: ν = 0 (St )∗ μdt. Similarly S preserves a smooth Riemannian metric on M obtained by averaging of a smooth Riemannian metric. Fairly representative models are rotations of the disc around its center and of the two-dimensional sphere around any fixed axis that preserve Lebesgue measure. Denote by Cq the subgroup of S 1 with q elements, i.e the qth roots of unity. Volume preserving maps with various interesting topological and ergodic properties are obtained as limits of volume preserving periodic transformations f = lim fn , where fn = Hn Sαn+1 Hn−1

(2.1) with αn = (2.2)

n→∞

pn qn

∈ Q and Hn = h1 ◦ ... ◦ hn ,

where every hn is a volume preserving diffeomorphism of M that satisfies (2.3)

hn ◦ Sαn = Sαn ◦ hn .

Equivalently, hn has to commute with the action of the finite group Cqn . To achieve that one maps a fundamental domain for this group to another fundamental domain

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(e.g. to itself in the simplest case that already leads to highly non-trivial results) and then extends the diffeomorphism periodically on the rest of the space. Usually at step n, the diffeomorphism hn is constructed first, and αn+1 is chosen afterwards close enough to αn to guarantee convergence of the construction. For example, it is easy to see that for the limit in (2.1) to exist in the C ∞ topology it is largely sufficient to ask that 1 |αn+1 − αn | ≤ n (2.4) . 2 qn ||Hn ||C n The power and fruitfulness of the method depend on the fact that the sequence of diffeomorphisms fn is made to converge while the conjugacies Hn diverge often “wildly” albeit in a controlled (or prescribed) way. Dynamics of the circle actions and of their individual elements is simple and well–understood. In particular, no element of such an action is ergodic or topologically transitive, unless the circle action itself is transitive, i.e M = S 1 . To provide interesting asymptotic properties of the limit typically the successive conjugacies spread the orbits of the circle action S, and hence also those of its restriction to the subgroup Cq for any sufficiently large q (that is of course will be much larger than qn ) across the phase space M making them almost dense (Anosov’s original idea, when he invented this scheme; he took just one S orbit at each step), or almost uniformly distributed (my first improvement; here one needs to control a majority of orbits simultaneously), or approximate another type of interesting asymptotic behavior. Due to the high speed of convergence this remains true for sufficiently long orbit segments of the limit diffeomorphism. To guarantee an appropriate speed of approximation extra conditions on convergence of approximations in addition to (2.4) may be required. References [1] D. V. Anosov, Roughness of geodesic flows on compact Riemannian manifolds of negative curvature (Russian), Dokl. Akad. Nauk SSSR 145 (1962), 707–709. MR0143156 [2] D. V. Anosov, Ergodic properties of geodesic flows on closed Riemannian manifolds of negative curvature (Russian), Dokl. Akad. Nauk SSSR 151 (1963), 1250–1252. MR0163258 [3] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature (Russian), Trudy Mat. Inst. Steklov. 90 (1967), 209. MR0224110 [4] D. V. Anosov, Tangential fields of transversal foliations in U -systems (Russian), Mat. Zametki 2 (1967), 539–548. MR0242190 [5] D.V. Anosov, On a class of invariant sets in smooth dynamical systems (Russian), Proceedings of the Fifth International conference on non-linear oscillations, vol. 2, 39–45, Inst. Mat. Akad. Nauk Ukr. SSR, Kiev, 1970. [6] D. V. Anosov, Existence of smooth ergodic flows on smooth manifolds (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 518–545. MR0358863 [7] D. V. Anosov, A note on the Kepler problem, J. Dynam. Control Systems 8 (2002), no. 3, 413–442, DOI 10.1023/A:1016386605889. MR1914450 [8] D. V. Anosov, On the development of the theory of dynamical systems during the past quarter century, Surveys in modern mathematics, London Math. Soc. Lecture Note Ser., vol. 321, Cambridge Univ. Press, Cambridge, 2005, pp. 70–185, DOI 10.1017/CBO9780511614156.006. MR2166925 [9] D. V. Anosov, Dynamical systems in the 1960s: the hyperbolic revolution, Mathematical events of the twentieth century, Springer, Berlin, 2006, pp. 1–17, DOI 10.1007/3-540-294627 1. MR2182776 [10] D. V. Anosov and V. L. Golo, Vector fiberings and the K-functor (Russian), Uspehi Mat. Nauk 21 (1966), no. 5 (131), 181–212. MR0202152 [11] D. V. Anosov and A. B. Katok, New examples of ergodic diffeomorphisms of smooth manifolds (Russian), Uspehi Mat. Nauk 25 (1970), no. 4 (154), 173–174. MR0281228

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[12] D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms (Russian), Trudy Moskov. Mat. Obˇsˇ c. 23 (1970), 3–36. MR0370662 [13] M. I. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212. MR0343316 [14] M.I. Brin and Ja. B. Pesin, D.V. Anosov and our road to partial hyperbolicity, in this volume. [15] Bassam Fayad and Anatole Katok, Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1477–1520, DOI 10.1017/S0143385703000798. MR2104594 [16] John Franks, Anosov diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 61–93. MR0271990 [17] B. Hasselblatt, Review of “On the development of the theory of dynamical systems during the past quarter century.” by D.V. Anosov, MR2166925 (2007k:37002) [18] Eberhard Hopf, Statistik der geod¨ atischen Linien in Mannigfaltigkeiten negativer Kr¨ ummung (German), Ber. Verh. S¨ achs. Akad. Wiss. Leipzig 91 (1939), 261–304. MR0001464 [19] A. B. Katok, Dynamical systems with hyperbolic structure (Russian), Ninth Mathematical Summer School (Kaciveli, 1971), Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1972, pp. 125–211. Three papers on smooth dynamical systems. MR0377991 [20] A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 539–576. MR0331425 [21] A. B. Katok, Monotone equivalence in ergodic theory (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 1, 104–157, 231. MR0442195 [22] Anatole Katok, Moscow dynamic seminars of the nineteen seventies and the early career of Yasha Pesin, Discrete Contin. Dyn. Syst. 22 (2008), no. 1-2, 1–22, DOI 10.3934/dcds.2008.22.1. MR2410944 [23] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR1326374 [24] Anthony Manning, Axiom A diffeomorphisms have rational zeta functions, Bull. London Math. Soc. 3 (1971), 215–220. MR0288786 [25] Anthony Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974), 422–429. MR0358865 [26] John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math. 30 (1968), 479–483. MR0248879 [27] J. Moser, On a theorem of Anosov, J. Differential Equations 5 (1969), 411–440. MR0238357 [28] S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math. 92 (1970), 761–770. MR0277004 [29] Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1332–1379, 1440. MR0458490 [30] Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory (Russian), Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287. MR0466791 Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802 E-mail address: [email protected] URL: http://www.personal.psu.edu/axk29

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13921

D. V. Anosov and our road to partial hyperbolicity Michael Brin and Yakov Pesin Abstract. This paper provides a brief historical account of our interaction with our advisor and mentor Dmitry Victorovich Anosov and in particular, our scientific activity in the well-known Anosov- Katok seminar during the period from 1968 till 1979. We also comment on our joint work on partial hyperbolicity.

We met Dmitry Viktorovich Anosov in 1968 as third year Mekh-Mat (mathematics department of the Moscow State University) students and we started attending the Anosov–Katok seminar since its first meeting in the fall of 1969. For a long time since then our mathematical lives have been tied to and heavily influenced by these two great mathematicians whom we consider our mentors and advisers and not only in mathematics. In fact it would have been next to impossible for us to become professional mathematicians without their constant support and guidance. In this note we will briefly describe our interaction with Anosov and comment on our joint work in mathematics during the decade from 1969 till 1979. M.B. emigrated from the former USSR in 1979 and while Ya.P. stayed in Russia for another decade his interaction with Anosov has gradually decreased. Our lives in mathematics started rather well. In 1965 we both graduated with honors from elite mathematical high schools and successfully passed the entrance exams to Mekh-Mat which at the time was arguably the best mathematical center in the world. Indeed, the decade from the end of the 50s until the end of the 60s is generally considered the golden age of Moscow mathematics [9]. We were lucky to have our math courses taught by such distinguished mathematicians and scholars as Efimov (linear algebra), Manin (algebra), Arnold (ordinary differential equations), Vishik (partial differential equations), Shabat (complex analysis), Shilov (real analysis). In addition, during our 3rd through 5th years at MekhMat we greatly benefited from many topics courses and special seminars offered by stellar faculty. Our undergraduate adviser was Yakov Grigorievich Sinai who was very popular among students. Naturally, we started to attend the Sinai–Alexeyev seminar which was then the central arena for those interested in dynamics. At this time our close interaction with Katok began. He was a graduate student of Sinai and got his PhD in 1968. On his own initiative he gave topics courses in dynamical systems and ergodic theory, and this was the only course work in dynamical systems 2010 Mathematics Subject Classification. Primary 37D25, 37D35. The second author is partially supported by NSF grant DMS-1400027. c 2017 American Mathematical Society

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we ever had. In the Fall of 1969 we started attending the Anosov–Katok seminar which became essentially our only connection to mathematics for a decade; for a more detailed description and history of this seminar see, [12, 13]. By the end of the 60s Mekh-Mat’s golden years were over. Foremost it manifested itself in increased anti-Semitism and general oppression against liberal thought [7, 8]. Almost no Jews were accepted as either undergraduate or graduate students at Mekh-Mat, and no Jewish faculty were hired. This was the reason Katok did not have a chance of getting a position at Mekh-Mat and was “lucky” to get a job at the Central Economics-Mathematics Institute where he was rather free to do research of his choice. However, he could not continue teaching or run seminars at Mekh-Mat, but Anosov could, and this is how the Anosov–Katok seminar started. Anti-Semitism at Mekh-Mat affected both of us enormously. Although we graduated from Mekh-Mat with honors, and were recommended for the graduate school by our adviser Sinai and by the Mekh-Mat administration, the department communist party bureau rejected our applications. In the end M.B. got a job at the Research Economics Institute of the State Planning Committee and Ya.P. at the Research Institute of Optical-Physical Measurements. Here we faced a very hard choice – either to quit mathematics (as many of our classmates did) or to combine it with our meaningless full time jobs. Since we were not affiliated with any mathematical institution, our resources to carry out research in mathematics were very limited and our mathematical future was quite uncertain. The Anosov–Katok seminar was the main reason and, in fact, the only possibility for us to stay in mathematics since it allowed us to be abreast of current developments in dynamical systems, helped us navigate our own research and discuss our results. The personal qualities of the seminar leaders created an open and democratic intellectual atmosphere which for us was a kind of escape from the unpleasant reality of our day-to-day duties at work. Anosov was the official PhD adviser for both of us and played a vital role in our mathematical lives. Since he was a student of Pontryagin (who was at the very top of the Soviet mathematical hierarchy) and obtained spectacular results early in his career, he quickly advanced to the higher tiers of the Soviet mathematical establishment. Anosov was a professor of the Moscow State University, a member of the Steklov Mathematical Institute, a recipient of the prestigious State Prize, a member of the editorial boards of two top mathematical journals, and a member of the Higher Attestation Board (the state body charged with certifying higher scientific degrees). For many mathematicians to reach such a level and stay at it meant getting involved to a higher or lesser degree in some unethical activities. Anosov was one of the very few who never compromised on moral issues and, in fact, often used his influence to correct the wrong. He was one of the best representatives of the Russian intelligentsia with high self-imposed moral principles. It was his conscientious decision to always keep his “hands clean”; by the standards of the time this was a hard choice to make. There was not a drop of anti-Semitism in him and in fact, he helped quite a few Jewish mathematicians, us in particular. Although our research at the beginning of the 70s was rather successful, since we were Jewish and were not affiliated with any mathematical institution, if not for Anosov it would have been virtually impossible for us to publish our results in any major mathematical journal and get a PhD.

D. V. ANOSOV AND OUR ROAD TO PARTIAL HYPERBOLICITY

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We would like to emphasize that Anosov’s advising style was not of the type that is common and expected in Western schools: he never proposed any problem to us to work on and we did not expect him to help us work out technical difficulties should we face some. Perhaps partly this may be due to the fact that we were not students at any graduate school. However, when we obtained some interesting results that we were eager to present and discuss he would be always willing to listen and express his opinion. From time to time he would provide us with some relevant recent papers or preprints which otherwise we would not be aware of or able to find. Most important he was instrumental in helping us publish our major papers and he did it purely because he considered our results to be a major achievement in dynamical systems. Unfortunately many other people “in power” acted differently and did not feel embarrassed to ask for something in return. For example, publishing one of our papers with the help of a person “in power” was conditioned on explaining some results of the paper to a student of this person, so that the student could claim and publish these results “independently”. Actually this was not considered outrageous at that time; or, as Anosov put it ”An evil world begets evil morality.” Soon after we graduated from Mekh-Mat, the Anosov–Katok seminar was thrown out of the university and moved to the Steklov Mathematical Institute. We met once a week starting at 5 p.m. to accommodate many of the participants with full time jobs. It lasted for about 2 hours. At the beginning the entrance to the building and seminar participation were not controlled. Before long, however, as part of its anti-Semitic policy, the institute administration demanded that the list of participants of each seminar be submitted for approval. Anosov’s problem was that many participants of his seminar were Jewish. He found a way around it by adding (to the seminar list) fictitious Russian sounding names to please the eye of the administration. In 1971 after looking through papers [11, 18] Katok noted to us that it would be interesting to consider dynamical systems with stable and unstable directions of not complementary dimensions and pointed out that the frame flow on a manifold of negative curvature was a natural example of this situation. In about 2 years we obtained the results which are now considered the foundation of partial hyperbolicity. This was a rather bumpy road with progress often followed by setbacks. We could only work after hours or on weekends thus taking time away from our families which required understanding, sacrifice and strong support of our wives. On the positive side, we had frequent long discussions with Anosov and Katok who were genuinely interested in our work, and this kept us going. As an example, Brin visited Anosov at his dacha to present a “proof” of ergodicity for a system with accessibility. The argument was long and convoluted and took about half an hour. Sharp-minded Anosov thought for about 5 minutes and pointed out a subtle mistake – one of the sets considered did not need to be measurable which ruined the argument. We would like to make a few comments on our work on partial hyperbolicity [5]. At the beginning we followed the path which Anosov and Sinai developed for hyperbolic systems [2]: establish the H¨ older continuity of the stable and unstable distributions, their integrability, and the absolute continuity of the stable and unstable foliations. However, in the partially hyperbolic case this already required a substantial modification of the known techniques and introduction of some new

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methods. For example, in proving integrability we used a proper version of Perron’s method which works better in the setting of partial hyperbolicity than more standard Hadamard’s method, and in proving the absolute continuity property we had to deal with a serious obstacle of possible expansions (albeit at a slower rate) in the direction transverse to the unstable. Further, since the stable and unstable invariant foliations are not smooth in general, we realized that to prove ergodicity one needed a generalization of the notion of commutator of vector fields for the non-differentiable setting. Eventually this led us to the notion of accessibility which we called transitivity of foliations. Anosov systems obviously have this property. In modern language, the non-differentiable commutator leads to accessibility through the Brin argument [3]. We understood from the beginning that the classical Hopf argument needed a substantial modification to work in the settings of partially hyperbolic systems. Our main result states that accessibility implies ergodicity under the following additional assumptions: Lipschitz continuity of the central distribution and its integrability, dynamical coherence (i.e., integrability of center-stable and centerunstable distributions), and the Lipschitz continuity of the stable and unstable holonomy maps along the central leaves. While these requirements are strong, it was not difficult to verify them in the situations we were interested in – group extensions and, in particular, frame flows. Later major progress in partial hyperbolicity to a large extent involved removing and/or weakening these conditions [6, 10, 17]. There are three lines at the end of the introduction in our joint paper [5] which an unprepared reader may consider strange: “The results of Sec. 3 as well as Theorems 4.1 and 4.2 belong to M. Brin. The results of Sec. 2 belong to Ya. Pesin. The rest of the results are joint.” The explanation is that when we discussed our PhD theses with Anosov, he recommended that we explicitly split the results between the two of us and then added that for two non-Jewish authors this joint paper would be more than enough for two dissertations but “two Jewish authors of one paper” must each write an additional separate paper on a different subject. Following his advice Brin started working on the generisity of ergodicity for frame flows resulted in [3, 4] and Pesin began working on what later has evolved as the non-uniform hyperbolicity theory [14–16]. To defend a PhD thesis, someone not in a graduate school needed to have an official scientific adviser as well as to find a mathematical institution which would accept his/her thesis for defense. In our case an obvious adviser would be either Katok or Anosov. Although Katok was more involved with our research, the chances of finding an institution which would agree to consider a thesis with a Jewish adviser and a Jewish student were zero. So Anosov was the only choice, and this was a huge commitment on his behalf. It still took several years after our theses were completed to get our PhDs. Brin eventually got his PhD in 1975 from the Kharkov State University thanks to great efforts by Naum Ilyich Akhiezer. Pesin got his PhD in 1979 from the Gorky State University with support from Leonid Pavlovich Shilnikov. Here is the story of how it happened. After several unsuccessful attempts to find a place for the defense, Anosov made a deal with O, the head of the Scientific Council at Gorky. O had two graduate students with theses ready to be defended and he agreed to arrange the three defenses on the same day in exchange for Anosov ensuring a safe passage of the theses through the Higher Attestation Board. For

D. V. ANOSOV AND OUR ROAD TO PARTIAL HYPERBOLICITY

27

Anosov it was a rare and very serious compromise with his principles, but he decided it was worth it. As he put it “Yasha, I traded you for two”. As it turned out the thesis of one of O’s students claimed three theorems of which two were completely wrong and the third one needed very serious corrections. As a result that student dropped out and, as Anosov remarked: “it became a fair trade”. The deal notwithstanding, the positive outcome was not guaranteed. Unexpected help came from Evgeniya Aleksandrovna Leontovich-Andronova, the widow of a famous mathematician Andronov and a prominent member of the Scientific Council. At the end of the defense proceedings she said: “When my husband was alive, this work would result in a Doctor of Science degree1 , years later it would be considered an outstanding PhD thesis, and now we are thinking whether to vote yes or no”. Actually, there were still two negative votes. We remember with pleasure the many hours we spent talking to Anosov at his home about mathematics as well as many other subjects. The atmosphere was very welcoming, and often his mother would bring out tea and cookies. Anosov was raised in a family of prominent scientists and had a large library at home. He was interested in and knew history very well, appreciated art and music. On occasion one could observe some elements of a “nobleman-among-peasants” in his demeanor which became more pronounced in later years. Anosov possessed a great sense of humor, and his remarks were often sharp and ironic. Some examples of this can be found in the introduction to his famous book on geodesic flows [1]. For instance, commenting on the Hadamard–Perron theorem he writes: “Every five years or so, if not more often, someone “discovers” the theorem of Hadamard and Perron, proving it either by Hadamard’s method or by Perron’s. I myself have been guilty of this”. In 1991 the University of Maryland held a dynamical systems conference. Fresh from Russia Anosov entered the room in the middle of a talk, and after the talk was over many participants (quite a few of them Russian) rushed to greet the master. His immediate remark was: “let us rename the conference Anosov’s seminar and make Russian the conference language”. Anosov’s next stop was Penn State where he gave a colloquium talk. When he was running overtime, the colloquium chair interrupted him saying that his time was up. Anosov asked if there was time for questions. “Certainly”, replied the chair, “OK”, said Anosov, “then I will ask myself a question”. He then continued for another 15 minutes uninterrupted. There was a common perception among quite a few Russian mathematicians in the 60s and 70s that a real mathematician should know if not all of mathematics then at least most of its major branches. Although this was hardly possible, some came rather close to this ideal. Anosov was one of them, his knowledge of mathematics was amazingly broad and deep. This manifested itself in his role as an editor. The books and papers he edited range from topology to geometry to dynamical systems and contain numerous footnotes, long and substantive introductions and remarks. For Anosov this was a way to express his mathematical views. Anosov’s name is forever a part of the theory of dynamical systems. We are proud and fortunate that Dmitry Victorovich Anosov was our teacher and mentor.

1 The

second degree after PhD, see [13].

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References [1] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature., Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969. MR0242194 [2] D. Anosov and Y. Sinai, Some smooth ergodic systems, Russian Math. Surveys, 22:5 (1967) 103–167. [3] M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature (Russian), Funkcional. Anal. i Priloˇzen. 9 (1975), no. 1, 9–19. MR0370660 [4] M. I. Brin, The topology of group extensions of C-systems (Russian), Mat. Zametki 18 (1975), no. 3, 453–465. MR0394764 [5] M. I. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212. MR0343316 [6] Keith Burns and Amie Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171 (2010), no. 1, 451–489, DOI 10.4007/annals.2010.171.451. MR2630044 [7] You failed your math test, comrade Einstein, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Adventures and misadventures of young mathematicians or test your skills in almost recreational mathematics; Edited by M. Shifman. MR2145211 [8] Edward Frenkel, Love and math, Basic Books, New York, 2013. The heart of hidden reality. MR3155773 [9] Golden years of Moscow mathematics: History of Mathematics, 6, AMS, Edited by S. Zdravkovska and P. Duren. [10] Matthew Grayson, Charles Pugh, and Michael Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2) 140 (1994), no. 2, 295–329, DOI 10.2307/2118602. MR1298715 [11] Leon W. Green, Group-like decompositions of Riemannian bundles, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Springer, Berlin, 1973, pp. 120–139. Lecture Notes in Math., Vol. 318. MR0400310 [12] Anatole Katok, Moscow dynamic seminars of the nineteen seventies and the early career of Yasha Pesin, Discrete Contin. Dyn. Syst. 22 (2008), no. 1-2, 1–22, DOI 10.3934/dcds.2008.22.1. MR2410944 [13] A. Katok, Dmitry Victorovich Anosov: His life and mathematics, in this volume. [14] Ya. Pesin, Lyapunov characteristic exponents and ergodic properties of smooth dynamical systems with an invariant measure, Sov. Math. Dokl., 17:1 (1976) 196–199. [15] Ya. Pesin, Families of invariant manifolds corresponding to non-zero characteristic exponents, Math. USSR Izvestija, 40:6 (1976) 1261–1305. [16] Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory (Russian), Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287. MR0466791 [17] F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math. 172 (2008), no. 2, 353–381, DOI 10.1007/s00222-007-0100-z. MR2390288 [18] Richard Sacksteder, Strongly mixing transformations, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R. I., 1970, pp. 245– 252. MR0415684 Department of Mathematics, University of Maryland, College Park, Maryland 20742 E-mail address: [email protected] Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 E-mail address: [email protected]

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13916

Escape from large holes in Anosov systems Valentin Afraimovich and Leonid Bunimovich Abstract. In the article we obtain estimates for the amount of points that go trough a hole in the phase space of an Anosov system, provided that this hole is an element of a Markov partition. Moreover, we describe invariant sets that remain to stay out of the hole for all instants of time and estimate the survival probability. For algebraic automorphisms of torus these estimates become exact formulas.

1. Introduction A traditional problem in dynamical systems (DS) theory is to study the process of escape through a hole when the size of the hole is negligibly small (see [?BY1], [8], [11], and references therein). The problem was studied mainly by using ideas and methods from probability theory. If the size of the hole is large, a topological approach was proposed in [2] and successfully applied to dynamical systems generated by Markov maps of the interval with a hole being an element of a Markov partition. The problem is closely related to the study of perturbation propogation through dynamical networks ([1], [6],[3]) so the study of the escape rate problem can be applied to analysis of dynamical networks, which makes the problem even more attractive. Let us also remark that in [5] fractal (and multifractal) properties of escape time were studied for DS generated by Markov maps of the interval. In this article we generalize results from [2] to the case of Anosov systems. We consider here DS with discrete time. Let us recall that a DS generated by a differentiable map f : M Ñ M of a smooth manifold M is Anosov if: (i) for all x P M the tangent space Tx M splits into stable and unstable subspaces Tx M “ Exs ‘ Exu and (ii) there are constants C ą 0 and 0 ă λ ă 1 independent of x such that for all n ě 0 ||Dx f n ξ||Tf n pxq M ď Cλn ||ξ||Tx M if ξ P Exs , ||Dx f ´n η||Tf ´n pxq M ď Cλn ||η||Tx M if η P Exu . A Bowen theorem ([7]) tells us that for any sufficiently small ą 0 there exists a finite generating Markov partition whose elements have diameter less than (see [9] for a short proof). Thus, as in [2] we may (and will) use topological Markov chains to study the escape time problem. 2010 Mathematics Subject Classification. Primary 37C99. c 2017 American Mathematical Society

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VALENTIN AFRAIMOVICH AND LEONID BUNIMOVICH

Let us recall some notions related to Markov partitions for Anosov maps. Denote by Ws pxqpWu pxqq the -ball on the stable (unstable) manifold of the point x centered at x. Proposition 6.4.13 of the book [12] tells us that: (i) there exists δ ą 0 such that if distpx, yq ă δ then Ws pxq X Wu pyq ‰ H, and (ii) there exists ą 0 such that this intersection, denoted by rx, ys, contains no more than one point. Furthermore, (see, for instance, Definition 18.7.1 in [12] and/or Definition  and rx, ys P R for each 4.73 in [9]) a set R is called a rectangle if diam R ă 10 pair x, y P R. A rectangle is proper if R “ clospint Rq. A Markov partition of M is a finite cover R “ tR0 , . . . , Rm´1 u of M by proper rectangles Ri with disjoint interiors such that if x P intpRi q and f pxq P intpRj q then WRuj pf pxqq Ă f pWRui pxqq and f pWRs i pxqq Ă WRs j pf pxqq where WRi pxq “ Wi pxq X R, i “ u, s. If M “ T2 and f is an algebraic automorphism then rectangles Ri are just parallelograms (or standard rectangles), (see, for instance, [12]). 2. Set up We will follow the scheme of the article [2]. Let f : M Ñ M be an Anosov diffeomorphism and R “ tR0 , . . . , Rm´1 u be a generating Markov partition. Introduce an oriented graph G containing m vertices v0 , v1 , . . . , vm´1 such that there exists an edge starting at vi and ending at vj iff f pint Ri q X int Rj ‰ H. Consider now m ˆ m transition matrix A with entries aij P t0, 1u such that aij “ 1 iff there exists an edge in the graph G starting at vi and ending at vj . Therefore, a two-sided topological Markov chain (tMc) pσ, ΩA q is well-defined where ΩA is the set of all admissible sequences i “ p. . . i´1 i0 i1 . . . ik . . . q, ik P t0, 1, . . . , m ´ 1u, i.e. aik ik`1 “ 1, k P Z. ř8 k| Moreover, ΩA is endowed with a metric dpi, jq “ ´8 |ikb´j , b ą 1, preserving the |k| direct product topology, so that ΩA is a complete metric space and the shift map σ : ΩA Ñ ΩA , pσiqk “ ik`i , is continuous. The map f is semi-conjugated to σ. In other words, the following proposition holds (see, for instance, [12], [9]). Proposition 1. There exists a continuous finite-to-one coding map χ : ΩA Ñ M such that χ ˝ σ “ f ˝ χ. The map χ is one-to-one on the set χ´1 pΛq where Λ “ M z YiPZ f i pB s R Y B u Rq and B s R “ Yi B s Ri , B u R “ Yi B u Ri It is well known that many features of the dynamical system generated by f have their counterparts in the symbolic system pσ, ΩA q, in particular if the symbolic system has positive topological entropy then the same is true for the system generated by f , and if M “ T2 and f is an algebraic automorphism then both systems have the same topological entropy. 3. Induced open dynamical system We introduce a new map fi : M Ñ M as follows " f pxq, x R Ri fi pxq “ (1) x, x P Ri i.e. each point inside Ri is a fixed point of fi . Thus we obtain m open dynamical systems generated by fi “ M Ñ M, i “ 0, 1, . . . , m´1, induced by f . The partition R will remain Markov for fi but the corresponding graph, say Gi , and the transition m´1 matrix Ai “ tasr piqus,r“0 become different.

ESCAPE FROM LARGE HOLES IN ANOSOV SYSTEMS

31

In fact, its entry is aij “ δij , the Kronecker symbol, j “ 0, . . . , m ´ 1. In other words, the state “i” is the sink. We will use below the topological Markov chain tMC pσ, ΩAi q. Given n ě 0 let Xpi, nq “ “

tx P M |f n x P Ri , f k x R Ri , 0 ď k ď nu tx P M |fin P Ri , fik x R Ri , 0 ď k ď nu

Y pi, nq “

tx P M |f k x P Ri , for some k, 0 ď k ď nu

“

tx P M |fik x P Ri , for some k, 0 ď k ď nu

So Xpi, nqpY pi, nqq consists of the points in M such that the trajectory of the DS generated by f going through each of them intersects Ri for the first time at the instant n (at the instant k, 0 ď k ď n). It is clear that, Xpi, n1 qXXpi, n2 q “ H, n1 ‰ n2 , and Y pi, nq “ Ynl“0 Xpi, lq, Y pi, nq “ tx P M, fin pxq P Ri u. Therefore, for an arbitrary measure on M n ÿ μpY pi, nqq “ μpXpi, lqq. l“0

The measure μpXpi, nqq can be treated as the probability that an orbit of f hits the hole Ri for the first time at the instant n, and μpY pi, nqq is the probability that an orbit hit the hole Ri at some instant k, 0 ď k ď n. Moreover, the quantity Pn pfi q “ 1 ´ μpY pi, nqq can be treated as a survival probability at the instant n. Let Γpi, nq “ Yri0 i1 . . . in´1 is be the set of all points of all non-empty cylinders in ΩAi of length n ` 1 ended by the symbol i. It follows from the definition that (2)

χpΓpi, nqq “ Y pi, nq

and since (3) (4)

Xpi, nq “ Y pi, nqzY pi, n ´ 1q, χpΓpi, nqzΓpi, n ´ 1qq “ Xpi, nq 4. Algebraic automorphism of the torus

To clarify main ideas of our approach we consider algebraic automorphisms of T2 . Such an automorphism is determined by a map f px, yq “ pb11 x ` b12 y, b21 x ` b22 yq mod 1 of the torus where bij P Z and | det B| “ 1, B “ pbij q2i,j“1 . Let λ1 , λ2 be the eigenvalues of B so |λ1 λ2 | “ 1. We assume that |λ1 | ą 1, |λ2 | ă 1, i.e. f is an Anosov diffeomorphism. m´1 such that qsr “ asr piqpr , pi “ 1, pj “ Introduce the matrix Qi “ pqsr qs,r“0 |λ2 |, j ‰ i, i.e. Qi “ Ai diagpp0 , p1 , . . . , pm´1 q. Each element Rj of the partition R is a parallelogram with sides parallel to the eigenvectors of the matrix B corresponding to the eigenvalues λ1 and λ2 . Denote by lk pRj q the length of the side of the parallelogram Rj that is parallel to the eigenvector related to λk , k “ 1, 2, so that μpRj q “ l1 pRj ql2 pRi q sin α, where μ is the Lebesgue measure (area) on T2 . Theorem 1. (5)

μpY pi, nqq “

m´1 ÿ

pnq

qji μpRj q

j“0

where

pnq qsk

are the entries of the matrix Qn .

l1 pRi q l1 pRj q

32

VALENTIN AFRAIMOVICH AND LEONID BUNIMOVICH

Proof. We consider, first, the case n “ 1 and show that μpY pi, 1qq “

m´1 ÿ

qij μpRj q

j“0

l1 pRi q . l1 pRj q

Indeed, (a) qii “ aii piq “ 1; (b) if aji piq “ 0 then f ´1 pint Ri q X int Rj “ H, so μpfi´1 pRi q X Rj q “ 0; l1 pRi q (c) if aji “ 1 then μpfi´1 pRi q X Rj q “ |λ´1 1 |l2 pRj ql1 pRi q sin α “ |λ2 |μpRj q l1 pRj q . Thus μpY pi, 1qq “ μpRi Yj pfi´1 pRi q X Rj qq m´1 ÿ l1 pRi q “ μpRi q ` aji piq|λ2 |μpRj q l 1 pRj q j“0 j‰i m´1 ÿ

“

qji μpRj q

j“0

l1 pRi q l1 pRj q

We can rewrite it as (6)

μpY pi, 1qq “ μpY pi, 0qq ` |λ2 |

ÿ

aji piqμpRj q

rj,is

l1 pRi q l1 pRj q

For n “ 2, we have (taking into account that ais piq “ qis “ 0) ˛ ¨ m´1 m´1 m´1 ÿ p2q ÿ ˚ÿ l1 pRi q l1 pRi q ‹ “ qji μpRj q |λ2 |2 ajs piqasi piq ` |λ2 |aji piq‚μpRj q ˝ l1 pRj q l1 pRj q s“0 j“0 j“0 s‰i

j‰i

`

m´1 ÿ

qis qsi μpRi q

s“0

“

m´1 ÿ

l1 pRi q l1 pRj q

|λ2 |2 ajs piqasi piqμpRj q

j,s“0 j,s‰i

l1 pRi q ` μpY pi, 1qq l1 pRj q

“ μpY pi, 2qq The last equality holds because of the fact that μpRj X fi´1 pRs X fi´1 pRi qqq “ pRi q |λ2 | l2 pRj ql1 pRi q sin α “ |λ2 |2 μpRj q ll11pR if ajs piqasi piq “ 1. Thus (5) is true for jq n “ 2. We can rewrite it as 2

(7)

μpY pi, 2qq “ μpY pi, 1qq ` |λ2 |2

ÿ rj s is

ajs piqasi piqμpRj q

l1 pRi q l1 pRj q

where the sum is taken over all non-empty cylinders of the tMc pσ, ΩAi q such that j ‰ i, s ‰ i.

ESCAPE FROM LARGE HOLES IN ANOSOV SYSTEMS

33

In the same way, by induction, we obtain the formula (5) by an arbitrary n ą 0. The formula can be rewritten as μpY pi, nqq “ μpY pi, n ´ 1qq ` ˜ ¸ n´2 ÿ ź l1 pRi q n (8) ajs1 piq ask sk`1 piq asn´1 i piqμpRj q |λ2 | l 1 pRj q k“1 rjs1 s2 ...sn´1 is

where the sum is taken over all nonempty cylinders of the tMc pσ, ΩAi q such that  j ‰ i and sk ‰ i, k “ 1, . . . , n ´ 1 Corollary 1. Since Y pi, n ´ 1q Ă Y pi, nq and because of the formula (3) we obtain from (8) that μpXpi, nqq “ μpY pi, nqq ´ μpY pi, n ´ 1qq ˜ ¸ n´2 ÿ ź l1 pRi q (9) ajs1 piq ask sk`1 piq asn´1i piqμpRj q “ |λ2 |n l 1 pRj q k“1 rjs1 s2 ...sn´1 is

Corollary 2. (10)

μpXpi, nqq “

m´1 ÿ

pnq

pn´1q

pqji ´ qji

qμpRj q

j“0

l1 pRi q . l1 pRj q

So, we have obtained exact formulas for the escape probabilities. The formula (9) can be treated in another way. Introduce pm ´ 1q ˆ pm ´ 1qmatrix that is obtained from the matrix Ai (or A) by removing the i-th column ´ and the i-th row. Denote this matrix by A´ i . The entries of Ai are 0 or 1, so it ´ corresponds to a graph, say Gi , with m ´ 1 vertices vk , k “ 0, . . . , m ´ 1, k ‰ i, and the edges joining vertices vk and vs iff aks “ 1. We call the vertex vk to be an α-vertex if there is no vertex vs such that ask “ 1, and an ω-vertex if theres is no vertex vs such that aks “ 1. Remove ´ α and ω vertices from G´ i and obtain a new graph, say Gi p1q, again perform ´ this procedure and obtain Gi p2q, etc. After finitely many steps we obtain either empty set or a graph, say Γi in which there are no α or ω-vertices. We ignore the first possibility and denote the corresponding matrix by Hi , and the tMc by pσ, ΩHi q. Let us recall that the symbolic complexity Cn pσ, ΩHi q is the number of all admissible non-empty cylinders of the length n. The following formula holds (see, for instance [4]) Cn pσ, ΩHi q “ E T Hin´1 E T where E “ p1 1 . . . 1q and E is the corresponding column. Corollary 3. The following estimate holds (11)

K2 |λ2 |n Cn pσ, ΩHi q ď μpXpi, nqq ď K1 |λ2 |n Cn pσ, ΩHi q

where K1 , K2 are a constants independent of n. Proof. Since Un :“

ÿ rjs0 ...sn´1 is

ajsi piq

n´1 ź

ask sk`1 piq

i“1

is the number of all admissible words determined by the matrix A´ i (the corremight contain α and ω-vertices) and Cn pHi q is sponding paths in the graph G´ i

34

VALENTIN AFRAIMOVICH AND LEONID BUNIMOVICH

the number of all admisible words determined by the matrix Hi (the corresponding paths in Γi contain neither α nor ω-vertices) then evidently Un ě Cn pHi q. Therefore, the formula (9) implies that the estimate from below in (11) holds with K2 “ min μpRj q

(12)

j

l1 pRi q . l1 pRj q

To estimate Un from above, one first should take into account the fact that if j or sk in Un , k ď n ´ 1, corresponds to an ω-vertex then the corresponding term in the sum Un equals 0. So without loss of generality one may consider in Un only cylinders rjs1 , s2 . . . sn´1 is with symbols different from those denoting ω-vertices. Second, let us remark that the number of steps, say m0 , needed to obtain the final graph Γ from inductively determined graphs G´ i pkq, k “ 1, . . . , can not be greater than pm ´ 1q!. Indeed, the number of α-vertices in G´ i can not be greater than m ´ 1. If there exist s such vertices, s ě 1, then the number of α-vertices in the graph G´ i (1) can not be greater than pm ´ s ´ 1q, etc. So, m0 ď pm ´ 1q! Third, every cylinder rj0 , . . . jn´1 s is a concatenation of the cylinders rj0 , . . . jm0 ´1 s and rjm0 , jm0 `1 , . . . , jn´1 s provided that n ´ 1 ą m0 . Thus, (13)

Un ď Km0 ¨ Cn´1´m0 pHi q

where Km0 is the number of admissible words of the lenght m0 with respect to the matrix Ai . Evidently Km0 ď mm0 . Let Sm0 “

Cn´1´m0 pHi q Cn pHi q nąm0 `1 sup

then (14)

Un ď Km0 ¨ Sm0 ¨ Cn pHi q

and (11) holds with (15)

K1 “ Km0 ¨ Sm0 ¨ max μpRj q j

l1 pRi q . l1 pRj q 

Remark 1. If there are no α-vertices in the graph G´ i then (11) holds with K1 “ max μpRj q j

l1 pRi q l1 pRi q , K2 “ min μpRj q . j l1 pRj q l1 pRj q

One can check that exactly such a case occurs for the example 1 below. Corollary 3 implies that if htop pσ, ΩHi q :“ hi ą 0 then Cn pσ, ΩHi q behaves asymptotically as n Ñ 8 like ehi n , and (11) implies (16)

˜ 2 pλ2 ehi qn ď μpXpi, nqq ď K ˜ 1 pλ2 ehi qn K

By using (11) (or (16)) one may compare different elements of the Markov partition according to their ability to produce different rates of escape.

ESCAPE FROM LARGE HOLES IN ANOSOV SYSTEMS

35

5. The survival probabilities and escape rates ` ˘ ´k In our case the survival probability is μn :“ μ Xn´1 pM zRi q and the k“0 f ` ˘ ´k pM zRi q , see [10] and escape rate is ´γi where γi “ limnÑ8 n1 log μ Xn´1 k“0 f references therein. For Anosov systems generated by algebraic automorphisms one may obtain exact formulas. Theorem 2. The survival probability (17)

μn “ |λ2 |n

ÿ

n´2 ź

ask sk`1 μpRs0 q

rs0 ...sn´1 s k“0 sk ‰i

l1 pRsn´1 q l1 pRs0 q

where the sum is taken over all words that do not contain the symbol i. The proof of Theorem is the same as the one for Theorem 1. Corollary 4. The following estimates hold ˜ 1 |λ2 |n Cn pσ, ΩH q ˜ 2 |λ2 |n Cn pσ, ΩH q ď μn ď K (18) K i i ˜ ˜ where K1 , K2 are constants and Cn pσ, ΩHi q is the complexity function for tMc pσ, ΩpHi qq. ˜ 2, K ˜ 1 can be The prove is the same as that for Corollary 3 and constants K determined similarly to K2 , K1 . In particular, if there are no α-vertices in the ˜ ˜ graph G´ i then K1 “ K1 , K2 “ K2 Corollary 5. If the topological entropy hi of tMc pσ, ΩHi q is positive then the escape rate is ´γi where (19)

γi “ ln |λ2 | ` hi .

The result is consistent with [10] where the escape rate is presented through a pressure and here (19) is, in fact, the topological pressure over tMc pσ, ΩHi q with respect to a constant potential. Thus, the more complex the invariant set consisting of trajectories that do not go to the hole is the greater the survival probability is (and the smaller the escape rate). Example 1. For the Arnold cat map generated by the matrix ˆ ˙ 2 1 B“ 1 1 ?

?

(and having Lyapunov exponents λ1 “ lnp 3`2 5 q, λ2 “ lnp 3´2 5 q) there exists a Markov partition {Δ0 , Δ1 , . . . , Δ4 } corresponding to the transition matrix ¨ ˛ 1 1 0 1 0 ˚1 1 0 1 0‹ ˚ ‹ ‹ B“˚ ˚1 1 1 1 0‹ ˝0 0 0 1 1‚ 0 0 0 1 1 ?

see, for instance [13]. One can check that hi “ ln 2, i ‰ 1, and h1 “ lnp 32 ` 25 q, so, the escape through the hole Δ1 occurs with maximal survival probability. Moreover γ1 “ 0, therefore μn decreases sub-exponentially as n Ñ 8.

36

VALENTIN AFRAIMOVICH AND LEONID BUNIMOVICH

6. Multidimensional automorphisms The theorem and its corollaries may be generalized for multidimensional automorphisms of Tk , k ą 2.¯ The Anosov system here is generated by f px1 . . . xn q “ ´ř řk k mod 1, bij P Z, | det B| “ 1, B “ pbij qki,j“1 . s“1 b1s xs , s“1 bks xs We assume that this automorphism is hyperbolic, so that the eigenvalues of B, say ρ1 , ρ2 , . . . , ρk are partitioned by two groups, say, ρ1 , . . . , ρt , with |ρs | ă 1, s “ śt śk 1, . . . , t and |ρs | ą 1, s “ t ` 1, . . . , k. Let λ1 “ s“1 ρs , λ2 “ s“t`1 ρs . Again |λ1 λ2 | “ 1. Then we proceed exactly as for k “ 2: (i) introduce the matrices Ai , Qi “ Ai diagpp0 , p1 , . . . , pm´1 q, pi “ 1, pj “ |λ2 |, j ‰ i. Elements of the Markov partition, so called rectangles, are no not necessarily ”good” geometric bodies, so an analogue to the Theorem 1 will be formulated in a slightly different way: Theorem 3. (20)

μpY pi, nqq “

m´1 ÿ

pnq

qji νji

j“0

where νpi, jq are positive constants depending on the Markov partition Scheme of the proof. It is well known that the Lebesque measure is the only measure of maximal entropy of the DS generated by a hyperbolic automorphism of the torus and the entropy h “ ln |λ2 |. Moreover, given a Markov partition and the corresponding tM cpσ, ΩA q, the only measure of maximal entropy for the tM cpσ, ΩA q is the Parry measure and the entropy also equals ln |λ2 | (see [12] and references therein). In particular, |λ2 | is the maximal eigenvalue of the transition matrix A. For the sake of convenience let |λ2 | “ λ. Recall that if q “ pq1 , . . . , qm qT is a positive right eigenvector of A, Aq “ λq and v “ pv, . .ř . , vm q is a positive left eigenvector of A, vA “ λv, normalized in such a ś way that ni“1 qi vi “ 1 then one aij vi defines entries πij “ λv of the stochastic matrix “ pπij qni,j“1 and its eigenvector j ś p “ pp1 , . . . , pm q, pi “ qi vi , p “ p. The Parry measure of the cylinder rjs is pj , and the measure, say, of the cylinder i´n , . . . , i0 is (21)

μP pri´n , . . . , i0 sq “

´1 ź

πik ik`1 pi0 “ λ´n p

k“´n

´1 ź

aik ik`1 qvi´n ¨ qi0 .

k“´n

Taking into account the fact that μP pri´n , . . . , i0 sq is exactly the Lebesque measure of the intersection Xnk“0 f ´k Rk´n we apply the formula (21) and repeat the proof pRi q of Theorem 1 replacing the constants μpRj q ll11pR by vj qi . So, in the formation of jq the theorem vji “ vj ¨ qi . Corollary 2 becomes Corollary 6. (22)

μpXpi, nqq “

m´1 ÿ

pnq

pn´1q

pqji ´ qji

qνji

j“0

Formulas (11), (16) are exactly the same as for k “ 2, but now ¨ ˛´1 ź ź |λ2 | “ |ρs | “ ˝ |ρt |‚ |ρs |ă1

|ρt |ą1

ESCAPE FROM LARGE HOLES IN ANOSOV SYSTEMS

37

Proofs of all these statements are similar to those for k “ 2, so we do not present them here. For dynamical systems generated by multidimensional automorphisms of tori an analogue of Theorem 2 holds when the eigenvalue λ2 is replaced by the product of eigenvalues that are less than on by absolute values and the constants μpRs0 q ¨ l1 pRsn´1 q l1 pRs0 q

are replaced by vs0 ¨ qsn´1late . The formulations of corollaries are exactly the same as in Section 4. 7. Nonlinear systems When we deal with a general Anosov system one can not expect to get exact formulas for the probabilities μpY pi, nqq and μpXpi, nqq. Nevertheless there is a way to obtain reasonable estimates of these quantities by using the topological pressure technique, as we did in [2]. Indeed, given the non-empty intersection f ´1 Ri X Rj let ||Bf pxqξ|| λ1 pj, iq “ inf ||ξ|| ||Bf pxqξ|| λ1 pj, iq “ sup ||ξ|| where infimum (supremum) is taken over all ξ P Exu and all x P f ´1 Rj X Ri . Then, from the matrix Ai we obtain the transition matrix Hi as has been described above and introduce the one-sided topological Markov chain pσ, Ω` Hi q where Ω` “ tpi i . . . i qu is the collection of infinite (in one direction) H -admissible 0 1 n´1 i Hi sequences endowed with the standard distance and σpi0 i1 i2 . . . q “ pi1 i2 . . . q is the řn´1 k shift map. Given a function ϕ : Ω` Hi Ñ R let Sn piq “ k“0 ϕpσ iq. For any non` empty cylinder wn “ ri0 . . . in´1 s Ă ΩHi one can define Sn pwn q “ supiPwn Sn piq. Denote by Wn the collection of all non-empty cylinders of length n and consider a partition function ÿ Γpϕ, nq “ exp Sn pwn q. wn PWn

Then the topological pressure ppϕq is lim

nÑ8

ln Γpϕ, nq “ ppϕq n

see, for instance, [12]. For the functions φpi0 i1 . . . q :“ ´ ln λ1 pi0 , i1 q, φpi0 i1 . . . q :“ ´ ln λ1 pi0 , i1 q let p “ ppϕq, p “ ppϕq. Denote by h the topological entropy of the tMc pσ, ΩHi q. Conjecture. If h ą 0, p ‰ 0, p ‰ 0 then the following inequalities hold Keph`pqn ď μpXpi, nqq ď Keph`pqn where K, K are constants independent of n. To prove it, maybe, one should impose additional conditions, like smoothness of the SRB-measure, etc.

38

VALENTIN AFRAIMOVICH AND LEONID BUNIMOVICH

Acknowledgement The authors would like to thank Ya. B. Pesin for useful remarks. References [1] Valentin S. Afraimovich and Leonid A. Bunimovich, Dynamical networks: interplay of topology, interactions and local dynamics, Nonlinearity 20 (2007), no. 7, 1761–1771, DOI 10.1088/0951-7715/20/7/011. MR2335082 [2] V. S. Afraimovich and L. A. Bunimovich, Which hole is leaking the most: a topological approach to study open systems, Nonlinearity 23 (2010), no. 3, 643–656, DOI 10.1088/09517715/23/3/012. MR2593912 [3] V. S. Afraimovich, L. A. Bunimovich, and S. V. Moreno, Dynamical networks: continuous time and general discrete time models, Regul. Chaotic Dyn. 15 (2010), no. 2-3, 127–145, DOI 10.1134/S1560354710020036. MR2644325 [4] Valentin Afraimovich and Sze-Bi Hsu, Lectures on chaotic dynamical systems, AMS/IP Studies in Advanced Mathematics, vol. 28, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2003. MR1956214 [5] V. Afraimovich and R. Vasquez, Spectrum of dimensions for escape time, Discontinuity, Nonlinearity and Complexity, 2 (2013), 247-262. [6] Michael Blank and Leonid Bunimovich, Long range action in networks of chaotic elements, Nonlinearity 19 (2006), no. 2, 329–344, DOI 10.1088/0951-7715/19/2/005. MR2199391 [7] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR0442989 [8] L.A. Bunimovich and A. Yuvchenko, Where to place a hole to achieve fastest escape rate, Israel J. Math, (2011). [9] Pierre Collet and Jean-Pierre Eckmann, Concepts and results in chaotic dynamics: a short course, Theoretical and Mathematical Physics, Springer-Verlag, Berlin, 2006. MR2266984 [10] Mark F. Demers and Paul Wright, Behaviour of the escape rate function in hyperbolic dynamical systems, Nonlinearity 25 (2012), no. 7, 2133–2150, DOI 10.1088/0951-7715/25/7/2133. MR2947939 [11] Mark F. Demers and Lai-Sang Young, Escape rates and conditionally invariant measures, Nonlinearity 19 (2006), no. 2, 377–397, DOI 10.1088/0951-7715/19/2/008. MR2199394 [12] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR1326374 ´ ´ n en Comunicacio ´ n Optica, ´ noma de San Instituto de Investigacio Universidad Auto Luis Potos´ı, Karakorum 1470, Lomas 4a. 78220, San Luis Potos´ı, M´ exico ABC Math Program and School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13915

A dynamical decomposition of the torus into pseudo-circles Fran¸cois B´eguin, Sylvain Crovisier, and Tobias J¨ager To the memory of Dmitri V. Anosov Abstract. We build an irrational pseudo-rotation of the 2-torus which is semiconjugate to an irrational rotation of the circle in such a way that all the fibres of the semi-conjugacy are pseudo-circles. The proof uses the well-known ‘fast-approximation method’ introduced by Anosov and Katok.

1. Introduction It is well known that continua (connected compact metric spaces) with complicated structure naturally appear in smooth surface dynamics. A striking example is provided by the pseudo-circle, introduced by Bing [Bi1] and characterized by Fearnley [Fe]. It is a continuum which: – can be embedded in S2 and separates, – is circularly chainable: it admits coverings into compact subsets (Ai )i∈Z/nZ whose diameter are arbitrarily small, such that Ai ∩ Aj = ∅ if and only if if i = j ± 1 or i = j, – is indecomposable: it cannot be written as the union of two proper continua, – and whose non-trivial proper subcontinua are indecomposable, homogeneous (any point can be sent on any other point by some homeomorphism) and all homeomorphic to the same topological space (called the pseudoarc). Handel [Ha] has built a smooth diffeomorphism of S2 preserving a minimal invariant set homeomorphic to the pseudo-circle. Later, Prajs [Pr] has constructed a partition of the annulus into pseudo-arcs, and likewise his method could be used to produce partitions of the torus into pseudo-circles. It was not known, however, if such a pathological foliation could be ‘dynamical’, that is, invariant under the dynamics of a torus homeomorphism or diffeomorphism that permutes the leaves of the foliation. Conversely, if a homeomorphism of the two-torus is semiconjugate to an irrational rotation of the circle, one may wonder whether most, or at least some, of the fibres of the semi-conjugacy must have a simple structure or even be topological circles. We give a positive answer to the first and a negative to the second of these questions. Denote by Td = Rd /Zd the d-dimensional torus and by 2 2 Diff ω vol,0 (T ) the space of real-analytic diffeomorphisms of T that are isotopic to the identity and preserve the canonical volume. 2010 Mathematics Subject Classification. Primary 37E30, 37E45. c 2017 American Mathematical Society

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2 Theorem. There exists a minimal diffeomorphism f ∈ Diff ω vol,0 (T ) whose rotation set is reduced to a unique totally irrational1 vector and which preserves a partition C of T2 into pseudo-circles. Moreover there exists a continuous map π : T2 → T1 which semi-conjugates f to an irrational rotation. The elements of C are the pre-images π −1 (x). This result has implications for a number of questions that naturally come up in the rotation theory on the torus and, more specifically, the dynamics of totally irrational pseudo-rotations. We discuss these issues in more detail in Section 4, alongside with the uniqueness of the semi-conjugacy. Idea of the construction. The diffeomorphism f is obtained as limit of a sequence of diffeomorphisms fn that are conjugated to rational rotations Rαn by diffeomorphisms Hn isotopic to the identity, following the celebrated Anosov-Katok method, see [FK1]. As a side effect, this means that its dynamics can be made uniquely ergodic, although we will not expand on this. Note that most of the constructions using this method deal with the C ∞ category; some cases, as [FK2] allow to work in the real-analytic category. The sequence fn = Hn−1 ◦ Rαn ◦ Hn is obtained inductively. At stage n, the diffeomorphism fn preserves the foliation Vn which is the pre-image under Hn of the foliation by vertical circles {x} × T1 . The main requirement is to have the circles of the foliation Vn+1 arbitrarily close to the circles of Vn in the Hausdorff topology, but more crooked. Then the partitions Vn will converge to the partition into pseudo-circles. The foliation Vn+1 will be the pre-image of the foliation Vn under a new homeomorphism hn+1 . In order to obtain Vn+1 , one first builds a leaf of Hn (Vn+1 ), which is crooked with respect to the vertical circles (recall that these vertical circles are the leaves of the foliation Hn (Vn )), and transverse to a periodic linear flow ϕn+1 . The complete foliation Hn (Vn+1 ) is obtained by pushing this initial leaf by the flow ϕn+1 (see Figure 1). The homeomorphism hn+1 is chosen so that it maps the foliation Hn (Vn+1 ) to the foliation into vertical circles. Since we then let Hn+1 = hn+1 ◦ Hn , this ensures that the foliation Hn+1 (Vn+1 ) is again the foliation given by vertical circles.

h−1 n+1

ϕn+1

Figure 1. The inverse h−1 n+1 maps straight lines (solid lines on the left) to crooked leaves of Vn+1 (on the right). The latter are translates of each other along the flow lines of ϕn+1 . Since the foliation into vertical circles equals Hn (Vn ), this implies that the leaves of Vn+1 are crooked with respect to Vn . 1A

vector (a, b) ∈ R2 is called totally irrationnal if 1, a and b are independent over Q.

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We note that A. Avila has announced recently the construction of an element 2 of Diff ω vol,0 (T ) whose rotation set is a non-trivial compact interval contained in a line with irrational slope and which does not contain any rational point (a counterexample to one case of a conjecture by Franks and Misiurewicz [FM]). His construction may somewhat be compared to ours: the diffeomorphism is obtained as the limit of diffeomorphisms acting periodically on the leaves of a foliation by circles. In his case however the homotopy class of the leaves is modified at each stage of the construction.

2. A criterion for the existence of a partition into pseudo-circles 2.a–Crooking. The construction of the pseudo-circle uses the following notions. Definitions. A circular chain is a finite family of sets D = {D ,  ∈ Z/N Z} such that Dk intersects D if and only if k −  ∈ {−1, 0, +1}. A circular chain D = {Di , i ∈ Z/N  Z} said to be crooked inside another circular chain D = {D ,  ∈ Z/N Z} if there exists a map  : Z/N  Z → Z/N Z with the following properties. – Di ⊂ D(i) for each i ∈ Z; – if i < j are such that for all i < k < j the element (k) belongs to the same closed interval bounded by (i) and (j) (either positively or negatively oriented in Z/N Z) and the length of this interval is greater than 4, then there exists u, v with i < u < v < j such that d((u), (j)) ≤ 1 and d((v), (i)) ≤ 1. (Here d denotes the canonical distance on Z/N Z.) The pseudo-circle can then be obtained as follows. For the sake of consistency with the later sections, we work in the torus instead of R2 and require that the circular chains – and thus the resulting pseudo-circle – are homologically non-trivial. Theorem ([Bi1, Fe]). Consider a sequence (Dn )n≥0 of circular chains of open topological disks in T2 . Assume that Dn+1 is crooked   inside Dn for each n, the closure of D∈Dn+1 D is contained in D∈Dn D for every n, the maximal  diameter of the elements of Dn goes to zero as n → +∞. the union D∈Dn D contains homotopically non-trivial loops of a unique homotopy type v ∈ Z2 \ {0}.   Then the compact set X := n≥0 D∈Dn D is homeomorphic to the pseudocircle. Moreover, X is an annular continuum of homotopy type v (see [JP, JT]). – – – –

At some point later on, we will have to speak about lifts of circular chains in the torus to the universal covering R2 , and similarly about lifts of circular chains of intervals in the circle to R. Suppose that D is a circular chain of topological disks in T2 as above and denote by p : R2 → T2 the canonical projection. Note that for each D ∈ D, the preimage p−1 (D) consists of a countable number of connected components, each of which is a topological disk homeomorphic to D and disjoint from all its integer translates. Suppose in addition that Diam(D ) < 1/4 for all  ∈ Z/N Z, so that none of the unions D ∪ D+1 is essential in the torus (contains a homotopically non-trivial loop).

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 of D is a sequence of topological disks (D   )∈Z of R2 Definitions. A lift D such that   is a connected component of p−1 (D ); – for all  ∈ Z the disk D   – Dk intersects D if and only if k −  ∈ {−1, 0, 1}.  k+N is the image of D  k by translation by a vector v ∈ Z2 which does The disc D not depend on the lift, nor on k, and is called the homotopy type of D. Note that if D and D are circular chains of topological disks with homotopy  D   are lifts in the type v ∈ Z2 \ {0} as above, D is crooked inside D and D, above sense, then there exists a function ˆ : Z → Z (to which we refer as a lift of  : Z/N  Z → Z/N Z) such that   = D   + v and D  i+N = D  i + v; – D i i+N  ˆ ˆ – (i + N ) = (i) + N ;  (i) for all i ∈ Z;  ⊆ D – D i ˆ belongs to – if i < j < i + N are such that for all i < k < j the integer (k) ˆ ˆ ˆ ˆ the interval bounded by (i) and (j) and |(j)− (i)| > 4, then there exists ˆ − (j)| ˆ ˆ ˆ u, v with i < u < v < j such that |(u) ≤ 1 and |(v), (i)| ≤ 1. All these remarks apply in an analogous way to circular chains of intervals in the circle and their lifts to R. During the construction, we will also use another notion of the crooking. Definition. For ε > 0, a continuous map g : I → R on the interval I is εcrooked if for any a < b in I, there are a < c < d < b such that |g(d) − g(a)| < ε and |g(c) − g(b)| < ε. Note that ε-crooked maps exist for any ε (see [Bi2]). 2.b–Elements of the construction. Let p1 : T2 → T1 be the projection on the first coordinate. Let B(N ) be the covering of the circle by N open intervals defined as follows   i − 5/4 i + 1/4 , . (2.1) B(N ) = {Bi , i ∈ Z/N Z} where Bi = N N We will build inductively: – a sequence of integers (Nn )n≥0 , – a sequence of positive real numbers (εn )n≥0 , 2 – a sequence of conjugating diffeomorphisms (Hn )n≥0 in Diff ω vol,0 (T ), 2 – a sequence of rational rotations (Rαn )n≥0 of T . To Nn , εn , Hn , Rαn , we will associate: – for each x ∈ T1 , the annulus An,x which is the image under Hn−1 of the vertical annulus (x − εn , x + εn ) × T1 , – for each x ∈ T1 , the covering Dn,x of the annulus An,x defined by  Dn,x = Hn−1 ((x − εn , x + εn ) × Bi ) | Bi ∈ B(Nn ) (note that Dn,x is a circular chain with Nn elements), – the projection πn = p1 ◦ Hn : T2 → T1 (where p1 : T2 → T1 is the projection to the first coordinate),

2 – the diffeomorphism fn = Hn−1 ◦ Rαn ◦ Hn ∈ Diff ω vol,0 T .

We will denote by rqnn , sqnn the coordinates of αn , with rn , sn ∈ Z and qn ∈ N−{0}.

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2.c–Inductive properties. The torus T2 is embedded as the subset {(z1 , z2 ) ∈ C2 , |z1 | = |z2 | = 1} of C2 . One will consider homeomorphisms f of T2 such −1 that both functions defined on a neighborhood of    f and f 2 extend as holomorphic Δ = (z1 , z2 ) ∈ C , |z1 |, |z2 | ∈ 12 , 2 . One then introduces the supremum norm .Δ on Δ and the metric d0 (f, f  ) = max(f, f  Δ , f −1 , f 

−1

Δ ) .

The sequences (Nn ), (εn ), (Hn ), (Rαn ) will be constructed inductively so that the following properties hold. (1) For each x ∈ T1 , the circular chain Dn+1,x is crooked inside the circular chain Dn,x (in particular, the annulus An+1,x is contained in the interior of the annulus An,x ), and the supremum of the diameter of the elements 1 , of the coverings Dn+1,x is less than n+1 (2) The angle αn+1 is close, but not equal, to αn . More precisely: both n+1 n+1 − rqnn | and | sqn+1 − sqnn | are smaller than 1/(2n+2 qn ) and the orbits of | rqn+1 1 -dense in T2 . Rαn+1 are 2n+1 1 (3) Every orbit of the diffeomorphism fn+1 is n+1 -dense in T2 . (4) The diffeomorphism fn+1 is (very) close to fn . More precisely, both fn+1 , Hn+1 and their inverses extend holomorphically to (C \ {0})2 and satisfy: i i , fni ) < min( 21 d0 (fni , fn−1 ), n1 ) for i = 1, . . . , qn ; (a) d0 (fn+1 ηn (b) d0 (fn+1 , fn ) < 2 where ηn is choosen such that, for every homeomorphism g in the ball (for d0 ) centered at fn of radius ηn , the rotation set of g is contained in the ball centered at αn of radius n1 . (5) The projection πn+1 is close to πn for the C 0 -topology. More precisely: d0 (πn+1 , πn ) < 21n . Remarks. The existence of the real number ηn used in property 4.b is a consequence of the upper semi-continuity of the rotation set ρ(F ) with respect to F [MZ, Corollary 3.7]. Property 1 (more precisely, the fact that Dn+1,x is crooked inside Dn,x ) implies that the sequence of conjugating diffeomorphisms (Hn ) will necessarily diverge. Nevertheless, the sequence of diffeomorphisms (fn ) = (Hn−1 ◦Rαn ◦Hn ) will converge (Property 4). This convergence is obtained by using the well-known ingredients of the Anosov-Katok method: – one first chooses a conjugating diffeomorphism Hn+1 of the form Hn+1 = hn+1 ◦Hn , where hn+1 might be very wild, but commutes with the rotation −1 Rαn ; this implies that fn = Hn+1 ◦ Rαn ◦ Hn+1 ; – then, choosing αn+1 close enough to αn is enough to ensure that fn+1 = −1 −1 Hn+1 ◦ Rαn+1 ◦ Hn+1 is close to fn = Hn+1 ◦ Rαn ◦ Hn+1 . A specific point in our construction is that, although the sequence of diffeomorphisms (Hn ) will diverge, we require that the sequence of maps (p1 ◦ Hn ) converges (Property 5). Indeed, we want that the fibers of p1 ◦ Hn converge to pseudo-circles “foliating” T2 . 2.d–Proof of the theorem. One can easily check that the theorem follows from the inductive properties stated above. Properties 1 imply that, for every x ∈ T1 , the sequence of annuli (An,x ) decreases and converges in Hausdorff topology to a pseudo-circle Cx . Moreover, the collection of pseudo-circles C = {Cx }x∈T1 is a

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partition of T2 ; this follows from the following fact: – for every n, the collection of annuli {An,x , x ∈ T1 } covers T2 , – for any x = x , the annuli An,x and An,x are disjoint if n is large enough. Property 4.a implies that the sequence (fn ) converges to an holomorphic function f on the interior of Δ. The same holds for (fn−1 ). Consequently, the restriction of f to T2 is a real-analytic diffeomorphism. Since each fn is volume-preserving, f 2 belongs to Diff ω vol,0 (T ). 1 Given x ∈ T , let x := x + p1 (αn ). Then fn (An,x ) = An,x . From property 1, one deduces that An+1,x is mapped by fn inside An,x and An+1,x is mapped by fn−1 inside An,x . Hence, the annulus f (An+1,x ) is contained in the d0 (f, fn )neighbourhood of the annulus An,x and the annulus f −1 (An+1,x ) is contained in the d0 (f, fn )-neighbourhood of the annulus An,x . Since d0 (f, fn ) tends to 0 as n goes to infinity, this implies that f preserves the partition into pseudo-circles C = {Cx }x∈T1 . Property 2 implies that the sequence (αn ) converges towards some α ∈ R2 . It also implies that the qn first iterates of Rα are 1/2n+1 -close to those of Rαn+1 , hence are 21n -dense in T2 . Consequently α is totally irrational. Properties 4 imply that the rotation set of f is reduced to {α} (indeed, they imply that d0 (f, fn ) < ηn and therefore the rotation set of f is contained in the ball of radius n1 centered at αn for every n). Hence f is a totally irrational pseudorotation, i.e. the rotation set of f is reduced to a single totally irrational vector. Consider a point z ∈ T2 . For every n, according to property 3, the orbit of z under fn is n1 -dense in T2 . Using property 4.a, we obtain that the orbit of z under f remains at distance less than n2 of the orbit of z under fn for a time qn . Combined with property 3, this means that the orbit of z under f is n3 -dense in T2 . (Recall here that fn is qn -periodic.) Since n is arbitrary, f is minimal. Property 5 implies that the sequence of maps (πn ) converges in topology C 0 towards a continuous map π. For each n, the map πn semi-conjugates fn to the rotation of T1 with angle p1 (αn ). Passing to the limit, it follows that the map π semi-conjugates f to the rotation of angle p1 (α). So we get all the conclusions of the theorem.  3. Inductive construction Now we explain how to construct a sequence of integers (Nn )n≥0 , a sequence of real numbers (ε)n≥0 , a sequence of conjugating diffeomorphisms (Hn )n≥0 and a sequence of vectors (αn )n≥0 , so that properties 1. . . 5 are satisfied. We assume that the sequences have already been constructed up to rank n. We will construct Nn+1 , εn+1 , Hn+1 , αn+1 .

3.1–Preliminary constructions. Recall that we denote by rqnn , sqnn the coordinates of αn . We introduce a periodic linear flow ϕn+1 : (t, (x, y)) → (x, y) + t · αn + t · (0, rn bn+1 ) on T where bn+1 is an integer which will be specified below. Observe that the time 1 map of this flow is the rotation Rαn . The first return map of ϕn+1 on the vertical circle {x} × T1 is the time qrnn map of ϕn+1 . Denote by mn the period of this first return map, and observe that mn depends on αn , but does not depend on 2

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the choice of the integer bn+1 . We also introduce a C ω map θn : R → R satisfying the following properties. – x → θn (mn x) − mn x is a trigonometric polynomial (hence θn − Id is 1/mn -periodic),

  1 1 = 2, θ 0, = [0, 2] and θn is 4N1 n -crooked – θn (0) = 0, θn 2m n 2mn n   1 on 0, 2m , n     1 1 1 1 = , , 2 and θn is 4N1 n -crooked on – θn mn = m1n , θn 2m m m n n n   1 1 2mn , mn . Indeed by [Bi2], there exists ε-crooked maps of the interval for any ε > 0. This allows to build a map satisfying the two last items above and which is the sum of the identity with a 1/mn -periodic function. Being croocked is an open property. The density of the trigonometric polynomials inside the space of periodic functions allows to get the first item as well. Note that θn − Id induces a function on T1 which extends holomorphically to C \ {0}. Recall that B(N ) denotes the covering of the circle by N compact intervals defined by (2.1). Moreover, θ itself induces a degree one map on the circle, which we denote by θ again for simplicity. Claim. If Nn+1 is large enough, then, for any ω ∈ T1 , the circular chain of intervals {θn (B − ω) + ω, B ∈ B(Nn+1 )} is crooked inside the circular chain B(Nn ). Proof. We work with a lift of the family B(N ), in the sense discussed in  ) of the real line by intervals of the form: Section 2, obtained as a covering B(N i = B



 i − 5/4 i + 1/4 , , N N

i ∈ Z.

i − ω) + ω with B i ∈ B(N  n+1 ) has If Nn+1 is large enough, each interval θn (B 1   of the lenght strictly less than 2Nn and therefore is contained in an interval B ˆ (i) ˆ  family B(Nn ). Since θn has degree 1, one can choose the function  such that ˆ + Nn . ˆ + Nn+1 ) = (i) (i ˆ Let i < j be two integers such that (k) belongs to the interval bounded by ˆ ˆ ˆ − (i)| ˆ (i) and (j) for each i < k < j and such that 4 < |(j) < Nn . Let us choose     ˆ ˆ ˆ ˆ i and i ≤ i < j ≤ j such that (i ) = (i), (j ) = (j) and some points a ∈ B     j  . One considers a ≤ a < b ≤ b such that a, a (resp. b, b ) have the same b∈B image by x →θn (x − ω). One can assume that a − ω, b − ω belong to the same k interval Ik := 2m , k+1 : indeed, for each k, the image under θn of Ik has length n 2mn

k is an end point of θn (Ik−1 ∪ Ik ). > 1 and the point θn 2m n   k k+1 Now the points a − ω and b − ω belong to the same interval Ik = 2m , . 2m n n Since θn is (4Nn )−1 -crooked on this interval, this implies that there exists a < c < d < b such that – θn (a − ω) = θn (a − ω) and θn (d − ω) are (4Nn )−1 -close, – θn (b − ω) = θn (b − ω) and θn (c − ω) are (4Nn )−1 -close.

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ˆ − (i)| ˆ v such that |(v) Hence d is contained in an interval B ≤ 1. Similarly, c is ˆ ˆ  contained in an interval Bu such that |(u) − (i)| ≤ 1. Since a < c < d < b and ˆ − (i)| ˆ 2 < |(j) one gets i < u < v < j. In the projection to T1 , the above shows that the circular chain {θn (B − ω) +  ω, B ∈ B(Nn+1 )} is crooked inside the circular chain B(Nn ) as announced. Since mn only depends on αn , one can fix the map θn but choose bn+1 and Nn+1 arbitrarily large later in the construction. 3.b–Construction of Hn+1 . Consider the map Θn+1 : {0} × T1 → T1 defined by Θn+1 : (0, y) → θn (y) − y. Since θn − Id is 1/mn -periodic, and since the period of the return map associated to the linear flow ϕn+1 on {0} × T1 is equal to mn , this maps extends to a map Θn+1 : T2 → T1 which is constant along the orbits of ϕn+1 . We define Hn+1 by setting Hn+1 := hn+1 ◦ Hn , where   Θn+1 (x, y) hn+1 (x, y) = ϕn+1 − , (x, y) rn bn+1     Θn+1 (x, y) sn = x, y − Θn+1 (x, y) − 1, . qn bn+1 rn Since Θn+1 is constant along the orbits of ϕn+1 , one has     Θn+1 (x, y) sn (3.1) h−1 (x, y) = x, y + Θ (x, y) + 1, . n+1 n+1 qn bn+1 rn 2 Clearly, hn+1 (and hence Hn+1 ) belongs to Diff ω vol,0 (T ). Note also that both hn+1 −1 and hn+1 extend holomorphically to the domain (C \ {0})2 .

3.c–Choice of bn+1 , εn+1 and Nn+1 . Now, we explain how to fix the values of bn+1 , εn+1 , and Nn+1 so that properties 1 and 5 hold. From (3.1), if bn+1 is large 2 enough, the image under h−1 n+1 of every point z = (x, y) ∈ T is arbitrarily close to 1 (x, y + Θn+1 (x, y)). Now observe that, for each x ∈ T , there exists ωx ∈ T1 such that Θn+1 (x, y) = θn (y − ωx ) − y + ωx . As a consequence, if bn+1 is large enough, the image under h−1 n+1 of every point z = (x, y) ∈ T2 is arbitrarily close to (x, θn (y − ωx ) + ωx ). Hence, if bn+1 is large enough and εn+1 is small enough, the image under h−1 n+1 of a rectangle (x − εn+1 , x + εn+1 ) × B is contained in an arbitrary small neighbourhood of the square {x} × (θn (B − ωx ) + ωx ). Using the claim above, this implies that the family Dn+1,x −1 is crooked inside Dn,x , provided Nn+1 is large enough. By continuity of Hn+1 , the 1 diameter of the elements of the covering Dn+1,x is less than n+1 if bn+1 , Nn+1 are large enough and εn+1 is small enough. We have thus checked that property 1 holds, provided that bn+1 , Nn+1 are chosen large enough and εn+1 is chosen small enough. By definition of πn , πn+1 , and Hn+1 , in order to check that property 5 is satisfied, it is enough to check that p1 ◦ hn+1 is close to p1 for the C 0 -topology. This is a direct consequence of the definition of hn+1 , provided that bn+1 is chosen large enough.

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θn(x)  h n+1 ({x} × R)

3

3 2

2

1 1 1 2

0

x 0

1 2

1

0 

 εn+1



Figure 2. Choice of the function θ (on the right) and the image of a vertical line {x} × R under the lift  hn+1 of hn+1 (schematic picuture with mn = 2). Note that the preimages of vertical lines under hn+1 have a similar ‘crookedness’, which is the fact that is needed for our construction (see next section). The horizontal size of images (and preimages) of vertical lines under hn+1 is small compared to εn+1 . This size is controlled by the flow lines of ϕn+1 (dashed lines), whose

direction is given by the almost vertical vector 1 bn+1 qn +(sn /rn ) , 1 .

3.d – Choice of αn+1 . Clearly, one can choose αn+1 arbitrarily close to αn so −1 that property 2 holds. By uniform continuity of Hn+1 and Hn+1 , there exists η −1 1 so that the orbits of fn+1 = Hn+1 ◦ Rαn+1 ◦ Hn+1 are n+1 -dense in T2 provided that the orbits of the rotation Rαn+1 are η-dense in T2 . One can thus choose αn+1 arbitrarily close to αn so that property 3 is satisfied. By construction both −1 fn+1 and fn+1 extend holomorphically to (C \ {0})2 . Moreover the diffeomorphism hn+1 commutes with the flow ϕn+1 , hence with the rotation Rαn . Consequently −1 ◦ Rαn ◦ Hn+1 . This shows that fn+1 is arbitrarily close to fn when αn+1 fn = Hn+1

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is chosen arbitrarily close to αn . In particular, property 4 holds provided that αn+1 is chosen close enough to αn . 4. Uniqueness of the semi-conjugacy, non-existence of wandering curves and further remarks The aim of this last section is to discuss, somewhat informally, the implications of our construction for some questions arising in the context of dynamics and rotation theory on the two-torus. By a totally irrational pseudo-rotation of T2 we mean a homeomorphism of T2 whose rotation set is reduced to a single totally irrational vector. Throughout this section, we assume f is a totally irrational pseudo-rotation of T2 with an invariant foliation of pseudo-circles, consisting of the fibres of a semi-conjugacy π : T2 → T1 to a rigid rotation Rα . Moreover, we will freely add further assumptions on f if these can easily be ensured in the preceeding Anosov-Katok-construction. We first note that the semi-conjugacy in our theorem is unique, modulo postcomposition by rotations. Proposition. The semi-conjugacy π in our main theorem is uniquely determined: any continuous map π  which is homotopic to p and semi-conjugates f to the same circle rotation as p can be written as π  = R ◦ π where R is a rotation of the circle. This follows directly by combining [JP, Corollary 4.3] (uniqueness of the semiconjugacy provided the non-wandering set is externally transitive) with the minimality of the map f . In fact, by [Po, Theorem A], any irrational pseudo-rotation is externally transitive on its non-wandering set, so that minimiality is not strictly required for the above statement. The uniqueness of the semi-conjugacy further allows to see that f does not admit any loop which is wandering (i.e. disjoint from all its iterates) and has the same homotopy type as the pseudo-circles (that is, homotopy vector v2 = (0, 1) in our construction). The reason for this is the fact that the existence of such a loop Γ would allow to construct a semi-conjugacy p˜ to the rotation Rα such that Γ is contained in a single fibre of the semi-conjugacy. Details of this construction can be found in [JP, Lemma 3.2] (the fact that Γ is contained in a single fibre is not mentioned explicitly, but is obvious from the proof). Due to the uniqueness of the semi-conjugacy (modulo rotations) and the fact that none of the pseudo-circles of the foliation contains any non-degenerate curves, this yields a contradiction. More generally, it is even possible to show that f does not admit any loops disjoint of all its iterates, regardless of the homotopy type. This is slightly more subtle, and we only sketch the argument. The crucial observation is the fact that we may construct f such that sup |F n (z) − z − nρ, v| < ∞ iff

(4.1)

v = (1, 0) ,

where F : R → R is a lift of f and ρ ∈ R is the corresponding rotation vector. Now, if there exists a wandering loop of homotopy type w ∈ Z2 \ {0}, then it is not hard to see that   sup  F n (z) − z − nρ, w⊥  < ∞ . 2

2

2

However, according to (4.1) this is only possible if w = (0, 1), and this is exactly the homotopy type of the pseudo-circles which was excluded before. Homotopically

A DYNAMICAL DECOMPOSITION OF THE TORUS INTO PSEUDO-CIRCLES

49

trivial wandering loops cannot exist by minimality, and therefore no loop of any homotopy type can be wandering. Roughly speaking, in order to prove (4.1) one has to use the fact that since the leaves of the foliations Vk are increasingly crooked, connected fundamental domains of these circles in the lift become arbitrarily large in diameter. Since an iterate of fk acts as a rotation on these  leaves, this allows to  see that for suitable integers nk the vertical deviations  Fknk (z) − z − nk ρ, w⊥  become arbitrarily large. If the fk converge to f sufficiently fast, then this carries over to the limit and yields unbounded vertical deviations for f . At the same time horizontal deviations (that is, v = v1 = (1, 0) in (4.1)) are bounded due to the existence of the semi-conjugacy (e.g. [JT, Lemma 3.1]). Together, these two facts yield unbounded deviations for all v = v1 . The fact that f does not admit any wandering loop is of some interest in the context of the Arc Translation Theorem due to Kwapisz [Kw, BCL], which asserts that given an irrational pseudo-rotation and any integer n, there exist essential loops which are disjoint from their first n iterates. It is natural to ask under what additional assumptions this statement can be strengthened by passing from a finite number to all iterates. A natural obstruction is certainly to have unbounded deviations in all directions, as discussed above. However, our example shows that even if the deviations are bounded in some direction, the existence of a wandering curve is not guaranteed. Thus, in this sense the statement of the Arc Translation Theorem is optimal, and essential loops have to be replaced by more general classes of essential continua in order to obtain results in infinite time. Finally, we want to mention a loose connection of our construction to the Franks-Misiurewicz Conjecture [FM]. The latter asserts that if the rotation set of a torus homeomorphism is a line segment of positive length, then either it contains infinitely many rational points, or it has a rational endpoint. As mentioned in the introduction, Avila has recently announced a counterexample to this conjecture for the case where the rotation segment has irrational slope, but the line it defines does not pass through a rational point. Conversely, Le Calvez and Tal have announced the first positive partial result on the conjecture: if the rotation set is a segment with irrational slope, it cannot contain a rational point in its relative interior. One case which is still completely open, however, is whether the rotation set can be a line segment with rational slope, but without rational points – for example, of the form {α} × [a, b] with α ∈ R irrational and a < b. Now, in the situation where there exists a semi-conjugacy π, homotopic to p1 , to the rotation Rα on the circle, the rotation set has to be contained in the line {α} × R [JT, Lemma 3.1]. Hence, this is a natural class of maps to look for counterexamples to this subcase of the Franks-Misiurewicz Conjecture. However, it is known that in order to have a non-degenerate rotation interval, the fibres of the semi-conjugacy need to have a complicated structure – more precisely, they need to be indecomposable [JP]. Our example shows that such a rich fibre structure is possible in principle. Yet, whether a non-degenerate rotation interval can be achieved remains open. Here, the fact that the Anosov-Katok method typically leads to uniquely ergodic examples, resulting in unique rotation vectors, suggests that a different approach would be needed to produce such examples, if these exists at all.

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Added in Proof During the production of this article, A. Koropecki, A. Passeggi and M. Sambarino have announced a result that rules out non-degenerate rotation intervals for torus homeomorphisms semi-conjugate to an irrational circle rotation. References D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms (Russian), Trudy Moskov. Mat. Obˇsˇ c. 23 (1970), 3–36. MR0370662 [BCL] F. B´ eguin, S. Crovisier, F. Le Roux, and A. Patou, Pseudo-rotations of the closed annulus: variation on a theorem of J. Kwapisz, Nonlinearity 17 (2004), no. 4, 1427–1453, DOI 10.1088/0951-7715/17/4/016. MR2069713 [Bi1] R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51. MR0043451 [Bi2] R. H. Bing, Higher-dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 267–273. MR0043452 [FK1] Bassam Fayad and Anatole Katok, Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1477–1520, DOI 10.1017/S0143385703000798. MR2104594 [FK2] Bassam Fayad and Anatole Katok, Analytic uniquely ergodic volume preserving maps on odd spheres, Comment. Math. Helv. 89 (2014), no. 4, 963–977, DOI 10.4171/CMH/341. MR3284302 [FM] John Franks and Michal Misiurewicz, Rotation sets of toral flows, Proc. Amer. Math. Soc. 109 (1990), no. 1, 243–249, DOI 10.2307/2048385. MR1021217 [Fe] Lawrence Fearnley, Classification of all hereditarily indecomposable circularly chainable continua, Trans. Amer. Math. Soc. 168 (1972), 387–401. MR0296903 [Ha] Michael Handel, A pathological area preserving C ∞ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982), no. 1, 163–168, DOI 10.2307/2044419. MR663889 [JP] T. J¨ ager and A. Passeggi, On torus homeomorphisms semiconjugate to irrational rotations, Ergodic Theory Dynam. Systems 35 (2015), no. 7, 2114–2137, DOI 10.1017/etds.2014.23. MR3394110 [JT] T. J¨ ager, F. Tal. Irrational rotation factors for conservative torus homeomorphisms. To appear in Ergodic Theory Dynam. Systems. ArXiv:1410.3662. [Kw] Jaroslaw Kwapisz, A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2865–2895, DOI 10.1090/S0002-9947-02-02952-5. MR1895207 [MZ] Michal Misiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490–506, DOI 10.1112/jlms/s2-40.3.490. MR1053617 [Po] Rafael Potrie, Recurrence of non-resonant homeomorphisms on the torus, Proc. Amer. Math. Soc. 140 (2012), no. 11, 3973–3981, DOI 10.1090/S0002-9939-2012-11249-3. MR2944736 [Pr] Janusz R. Prajs, A continuous circle of pseudo-arcs filling up the annulus, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1743–1757, DOI 10.1090/S0002-9947-99-02330-2. MR1608498 [AK]

LAGA, CNRS - UMR 7539, Universit´ e Paris 13, 93430 Villetaneuse, France LMO, CNRS - UMR 8628, Universit´ e Paris-Sud 11, 91405 Orsay, France Institute of Mathematics, Friedrich-Schiller-University Jena, 07743 Jena, Germany

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13917

On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups Artem Dudko and Rostislav Grigorchuk Abstract. We study Koopman and quasi-regular representations corresponding to the action of a weakly branch group G on the boundary of a rooted tree T . We show that in the case of a quasi-invariant Bernoulli measure on the boundary of T the corresponding Koopman representation of G is irreducible (under some general conditions). We also show that quasi-regular representations of G corresponding to different orbits and Koopman representations corresponding to different Bernoulli measures are pairwise disjoint. A corollary of our results is triviality of the centralizer of G in various groups of transformations on the boundary of T .

1. Introduction The main resources of examples of unitary representations of a countable group are Koopman, quasi-regular and groupoid representations. In the present paper we will study the Koopman and quasi-regular representations corresponding to actions of weakly branch groups on rooted trees. Their relation to the groupoid representation is studied in the second paper of the authors [12] on the subject of spectral properties. Branch groups were introduced by the second author in [17] and play important role in many investigations in group theory and around (see [5]). They possess interesting and often unusual properties. Branch just infinite groups constitute one of three classes on which the class of just infinite groups (i.e. infinite groups whose proper quotients are finite) naturally splits. The class of branch groups contains groups of intermediate growth, amenable but not elementary amenable groups, groups with finite commutator width etc.. Weakly branch groups is a natural generalization of the class of branch groups and keep many nice properties of branch groups (for instance absence of nontrivial laws, see [1]). Weakly branch groups also play important role in studies in holomorphic dynamics (see [28]) and in the theory of fractals (see [20]). Throughout this paper we will assume that G is a countable group. To every subgroup H of G one can associate a quasi-regular representation ρG/H acting in l2 (G/H). In particular, given an action of G on a set X for every x ∈ X one 2010 Mathematics Subject Classification. Primary 20C15, 37A15. The second author was supported by NSF grant DMS-1207699, NSA grant H98230-15-1-0328 and ERC AG COMPASP. . c 2017 American Mathematical Society

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can consider the corresponding quasi-regular representation ρx = ρG/StG (x) , where StG (x) is the stabilizer of x in G. Spectral properties of quasi-regular representations play important role in Random Walks on groups and Schreier graphs (see e.g. [23], [22] and [18]). In the case when H = {e} (i.e. is the trivial subgroup) the quasi-regular representation coincides with the regular representation ρG which is of a special importance. Quasi-regular representations naturally give rise to Hecke algebras and their representations (see e.g. [2] and [29]). A group acting on a rooted tree T is called weakly branch if it acts transitively on each level of the tree and for every vertex v of T it has a nontrivial element g supported on the subtree Tv emerging from v (see e.g. [5] and [18]). Using Mackey criterion of irreducibility of quasi-regular representations Bartholdi and the second author in [4] showed that quasi-regular representations corresponding to the action of a weakly branch group on the boundary of a rooted tree are irreducible. One of the results of the present paper is the following: Theorem 1. Let G be a countable weakly branch group acting on a spherically homogeneous rooted tree T . Then for any x, y ∈ ∂T , where ∂T is the boundary of the tree, such that x and y are from disjoint orbits the quasi-regular representations ρx and ρy are not unitary equivalent. In particular, we obtain a continuum of pairwise disjoint (we will use the word ”disjoint” as a synonym to ”not unitary equivalent”) irreducible representations of a weakly branch group G. Using Theorem 1 we show the following: Theorem 2. Let G be a countable weakly branch group acting on a spherically homogeneous rooted tree T . Then the centralizer of G in the group Bij(∂T ) of all bijections of the boundary of T onto itself is trivial. This gives triviality of centralizers of G in such groups as Homeo(∂T ), Aut(∂T ) etc. (see Corollary 2). As the result, we obtain a new example of dynamical systems for which the generalized Ismagilov’s conjecture fails (see [24]). Another important family of representations gives the Koopman representation κ associated to a dynamical system (G, X, μ), where (X, μ) is a measure space on which G acts by measure class preserving transformations. Such representations are rarely irreducible. In the case when μ is a G-invariant probability measure the subspace of constant functions in L2 (X, μ) is κ(G)-invariant. And moreover, the restriction κ0 of κ on the subspace L20 (X, μ) ⊂ L2 (X, μ) of functions with zero integral (orthogonal complement to constant functions) is also usually reducible. A few known exceptions are listed in [16]. Recently, attention to irreducibility problem of representation κ0 was raised by Vershik in [32]. One of the results of the present paper is constructing new examples of irreducible Koopman representations corresponding to actions with quasi-invariant measures. For a d-regular rooted tree T its boundary ∂T can be identified with a space of sequences {xj }j∈N where xj ∈ {1, . . . , d}. Let (1)

P = {p = (p1 , p2 , . . . , pd ) : pi > 0 for i = 1, 2, . . . , d and

d 

pi = 1}

i=1

be the set of all probability distributions on the alphabet {1, 2, . . . , d} assigning positive probability to every letter and (2)

P ∗ = {p ∈ P : pi = pj for all 1  i < j  d}.

REPRESENTATIONS OF WEAKLY BRANCH GROUPS

For p ∈ P denote by μp = main result is:



53

p the corresponding Bernoulli measure on ∂T . Our

N

Theorem 3. Let G be a subexponentially bounded countable weakly branch group acting on a regular rooted tree and p ∈ P ∗ . Then the following holds: 1) the Koopman representation κp associated to the action of G on (∂T, μp ) is irreducible; 2) this representation is not unitary equivalent to any of the quasi-regular representations ρx , x ∈ ∂T ; 3) Koopman representations associated to different p ∈ P ∗ are pairwise disjoint. Here subexponentially bounded group means a group consisting of subexponentially bounded automorphisms of T . The precise definition will be given in Subsection 2.1. Notice that Theorem 3 gives an additional continuum of pairwise disjoint irreducible representations of a weakly branch group. These representations are faithful (κp (g) = Id if g is not the group unit). Observe also that for the uniform distribution u = ( d1 , d1 , . . . , d1 ) ∈ P the corresponding Bernoulli measure μu is Ginvariant and the Koopman representation κu is a direct sum of countably many finite-dimensional representations (see [4]). In fact, the authors see a possibility of generalizing Theorem 3 to the case of distributions p ∈ P \ {u} (i.e. such that pi = pj for some 1  i < j  d). However, since this leads to further complications of already difficult proof in this paper we focus on distributions from P ∗ . 2. Preliminaries In this section we give necessary preliminaries on groups acting on rooted trees, representation theory and related topics. 2.1. Weakly branch groups. Here we give a brief introduction to actions of groups on boundaries of rooted trees. We refer the reader to [18] for detailed definitions and properties of these actions. A rooted tree is a tree T , with vertex set divided into levels Vn , n ∈ Z+ , such that V0 consists of one vertex v0 (called the root of T ), the edges are only between consecutive levels, and each vertex from Vn , n  1 (we consider infinite trees), is connected by an edge to exactly one vertex from Vn−1 (and several vertices from Vn+1 ). A rooted tree is called spherically homogeneous if each vertex from Vn is connected to the same number dn of vertices from Vn+1 . T is called d−regular, if dn = d is the same for all levels. An automorphism of a rooted tree T is any automorphism of the graph T preserving the root. The boundary ∂T of T is the set of all infinite paths starting at v0 and passing through each level exactly one time. For a vertex v of T denote by ∂Tv ⊂ ∂T the set of paths passing through v. Supply ∂T by the topology generated by the sets ∂Tv . Automorphisms of T act naturally on ∂T by homeomorphisms. If T is spherically homogeneous then ∂T admits a unique Aut(T )-invariant measure μ. This measure is uniform in the sense that 1 μ(∂Tv ) = for any n and any v ∈ Vn . d0 d1 . . . dn−1 Grigorchuk, Nekrashevich and Suschanski showed that this measure is ergodic if and only if the action of G is transitive on each level Vn of T (level transitive). Moreover,

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in this case it is uniquely ergodic. We refer the reader to [21], Proposition 6.5 for details. Definition 1. Let T be a spherically homogeneous tree and G < Aut(T ). Rigid stabilizer of a vertex v is the subgroup ristv (G) = {g ∈ G : supp(g) ⊂ Tv }. Rigid stabilizer of level n is  ristv (G). ristn (G) = v∈Vn

G is called branch if it is transitive on each level and ristn (G) is a subgroup of finite index in G for all n. G is called weakly branch if it is transitive on each level Vn of T and ristv (G) is nontrivial for each v. For each level Vn of a d-regular rooted tree an automorphism g of T can be presented in the form g = σ · (g1 , . . . , gdn ), where σ ∈ Sym(Vn ) is a permutation of the vertices from Vn and gi are the restrictions of g on the subtrees emerging from the vertices of Vn . Definition 2. An element g ∈ Aut(T ) is polynomially bounded (in the sense of Sidki [30]) if the number kn (g) of restrictions gi to the vertices of level n not equal to identity automorphism is bounded by a polynomial of n. We will call g subexponentially bounded if for every 0 < γ < 1 one has lim kn (g)γ n = 0.

n→∞

A group G < Aut(T ) is polynomially (subexponentially) bounded if each g ∈ G is polynomially (subexponentially) bounded. R. Kravchenko showed (see [25]) that for any polynomially bounded automorphism g defined by a finite automaton and any p ∈ P (see (1)) the measure μp is quasiinvariant with respect to the action of g. We will show in Section 4 that in fact the condition that g is defined by a finite automaton is redundant and polynomial boundedness can be weakened to subexponential. 2.2. Quasi-regular representations. Given a countable group acting on a set X and a point x ∈ X one can define the quasi-regular representation ρx in l2 (Gx), where Gx is the orbit of x, by: (ρx (g)f )(y) = f (g −1 y). Notice that the isomorphism class of ρx depends only on the stabilizer StG (x) of x. Recall that two subgroups H1 , H2 of a group G are called commensurable if H1 ∩ H2 is of finite index in both H1 and H2 . The groups H1 and H2 are called quasi-conjugate in G if gH1 g −1 is commensurable to H2 for some g ∈ G. By definition, commensurator of H < G is the subgroup of G defined by commG (H) = {g ∈ G : H ∩ gHg −1 has finite index in H and gHg −1 }. Mackey proved the following (see [27], Corollary 7): Theorem 4. 1) Let H be a subgroup of an infinite discrete countable group G. Then the quasi-regular representation ρG/H is irreducible if and only if commG (H) = H.

REPRESENTATIONS OF WEAKLY BRANCH GROUPS

55

2) Let H1 , H2 be two subgroups of G such that commG (Hi ) = Hi , i = 1, 2. Then ρG/H1 and ρG/H2 are unitary equivalent if and only if H1 and H2 are quasi-conjugate. In [4] the authors showed the following: Proposition 1. Let G be a weakly branch group, T be the corresponding spherically homogeneous rooted tree, x ∈ ∂T and StG (x) be its stabilizer. Then commG (StG (x)) = StG (x). Theorem 4 together with Proposition 1 immediately imply: Corollary 1. For a weakly branch group G and any x ∈ ∂T the quasi-regular representation ρx is irreducible. 2.3. Koopman representation. The most natural representation that one can associate to a measure-preserving action of a group G on a measure space (X, μ), where μ is a probabilty measure, is the Koopman representation κ of G in L2 (X, μ) acting by: (κ(g)f )(x) = f (g −1 x). This representation is important due to the fact that the spectral properties of κ reflect the dynamical properties of the action such as ergodicity and weak-mixing. It is known that for an ergodic action operators κ(g) together with operators of multiplication by functions from L∞ (X, μ) generate in the weak operator topology the algebra of all bounded operators on L2 (X, μ). The representation κ has invariant subspace of constant functions on X. However, it is natural to ask whether κ is irreducible in the orthogonal complement of the constant functions in L2 (X, μ) (κ is called almost irreducible in this case). This question is discussed in E. Glasner’s book [16] and is again raised in Vershik’s paper [32] (Problem 4). The cases when the answer is positive are quite rare. Among few examples is the example of arbitrary dense subgroup of the group Aut([0, 1], μ) of all measure preserving automorphisms of the unit segment with Lebesgue measure μ, supplied by the weak topology (see [16]). More generally, if the measure μ is only quasi-invariant one can still define the Koopman type representation using Radon-Nikodym derivative:  dμ(g −1 (x)) f (g −1 x). (κ(g)f )(x) = dμ(x) If the measure μ is not invariant then constant functions do not form an invariant subspace and Koopman representation can be irreducible in the whole L2 (X, μ). There are several examples of group actions with quasi-invariant measures known for which the Koopman representation is irreducible: 1) actions of free non-commutative groups on their boundaries ([13], [14] and [26]); 2) actions of lattices of Lie-groups (or algebraic-groups) on their PoissonFurstenberg boundaries ([9] and [6]); 3) action of the fundamental group of a compact negatively curved manifold on its boundary endowed with the Paterson-Sullivan measure class ([3]); 4) canonical actions of Higman-Thompson groups on segments endowed with Lebesgue measure ([15] and [10]).

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However, the general case is not well understood and search for new examples is a challenging problem. In the present paper we construct a new class of examples of irreducible Koopman representations (see Theorem 3). Observe that for any spherically homogeneous rooted tree and any G < Aut(T ) the Koopman representation of G corresponding to the invariant probability measure μ on ∂T is a direct sum of countably many finite dimensional irreducible subrepresentations, since for each n the finite-dimensional subspace of functions constant on the subtrees emerging from vertices from Vn is invariant under G (see [4]). Recall that for two groups G < H the centralizer of G in H is defined by: CH (G) = {h ∈ H : hg = gh for all g ∈ G}. Notice that for any group G acting on a rooted tree T we have the following group inclusions: (3)

 G < Aut(T ) < Aut(∂T, μ) < Aut(∂T, μ) and

(4)

Aut(T ) < Homeo(∂T ) < Aut(∂T ) < Bij(T ),

 where Homeo(∂T ) is the group of all homeomorphism of ∂T onto itself, Aut(∂T, μ) is the group of all measure class preserving automorphisms of (∂T, μ), Aut(∂T ) is the group of all Borel automorphisms of ∂T and Bij(T ) is the group of all bijections of ∂T onto itself. One of the results of the present paper (Theorem 2) is related to the question formulated by Kosyak (see [24], Conjecture 0.0.1). For a measure space (X, ν) and a measure class preserving map f : X → X denote by f∗ (ν) the push-forward measure on X. We will use the symbol Id for the identity operator. Question 1 (Kosyak). In which cases for a measure preserving dynamical system (G, X, ν) irreducibility of the associated Koopman representation κ is equivalent to the following two conditions: (G), f = Id; 1) f∗ (ν) ⊥ ν for any f ∈ CAut(X,ν)  2) ν is G-ergodic. As a Corollary of Theorem 2 we obtain: Corollary 2. Let G be a weakly branch group acting on a spherically homogeneous rooted tree T . Then the centralizers of G in the groups Aut(T ), Homeo(∂T ),  Aut(∂T, μ), Aut(∂T, μ) and Aut(∂T ) are trivial. Here μ is the unique G-invariant probability measure on ∂T . Thus, for every weakly branch group the action of G on (∂T, μ) satisfies conditions 1) and 2) of Question 1. Since the corresponding Koopman representation is reducible, this gives a class of dynamical systems for which conditions 1) and 2) are not sufficient to assure irreducibility of κ. 3. Stabilizers and centralizers of weakly branch group actions In this section we will prove Theorems 1 and 2. For simplicity, we denote ∂T by X and ∂Tv by Xv for a vertex v of T . Lemma 1. Let G be a weakly branch group acting on a spherically homogeneous rooted tree T . Then for every pair x, y ∈ X of points from disjoint orbits the groups StG (x) and StG (y) are not quasi-conjugate in G.

REPRESENTATIONS OF WEAKLY BRANCH GROUPS

57

Proof. Assume that for some x, y from disjoint orbits the groups StG (x) and StG (y) are quasi-conjugate in G. By conjugating one of the groups by an appropriate element of g we can assume that StG (x) and StG (y) are commensurable (see Subsection 2.2). Equivalently, the orbits StG (x)y and StG (y)x are finite. Let us show that the latter is false. Choose sufficiently large n such that for the vertices v, w ∈ Vn with x ∈ Xv , y ∈ Xw one has v = w. For any k > n let vk ∈ Vk be the vertex such that Xvk  x. Since G is weakly branch, there exists gk ∈ G, gk = Id (where Id is the trivial automorphism) such that supp(gk ) ⊂ Xvk . Then there exists m > k and a vertex u ∈ Vm such that gk u = u. Since G is level transitive, there exists h ∈ G such that hu = vm . Set hk = hgk h−1 . Then supp(hk ) ⊂ Xv , so in particular hk ∈ StG (y), and hk vm = vm , and thus hk x = x. / Xvkl+1 Finally, construct inductively an increasing sequence kl such that hkl x ∈ for every l. Then hkl x are pairwise distinct, which shows that StG (y)x is infinite. This contradiction finishes the proof of Lemma 1  As a corollary using Mackey Theorem 4 we obtain Theorem 1. Proof of Theorem 2. Assume that the centralizer C of a weakly branch group G in Bij(X) is not trivial. Let c ∈ C, c = Id. Let x ∈ X such that cx = x. Consider two cases: a) cx ∈ Gx. Then c preserves the orbit Gx and the unitary operator C : l2 (Gx) → l2 (Gx) given by: (Cf )(y) = f (c−1 y) commutes with the representation ρx . Since ρx is irreducible, by Schur’s Lemma (see e.g. [7], Proposition 2.2) C is a scalar operator. But this is impossible, since Cδx = δcx and δcx is orthogonal to δx . b) cx ∈ / Gx. Then c maps the orbit Gx onto the orbit Gcx and the corresponding unitary operator C : l2 (Gx) → l2 (Gcx) intertwines representations ρx and ρcx , which is impossible since by Theorem 1 representations ρx and ρcx are disjoint.  4. Irreducibility of Koopman representations of weakly branch groups In this section we will prove Theorem 3. Let G be a weakly branch group acting on a rooted tree T , X = ∂T and p ∈ P (see (1)). First we need to show that the Koopman representation corresponding to the action of G on (X, μp ) is well-defined (i.e. that the measure μp is quasi-invariant). Let Tn be the finite subtree of T composed from levels up to the nth. Observe that for each n the finite group of automorphism Aut(Tn ) can be identified with a subgroup of Aut(T ) consisting of elements g such that all the restrictions of g on subtrees of T emerging from vertices of nth level are trivial. For g ∈ Aut(T ) we denote by g (n) ∈ Aut(Tn ) the automorphism of Tn induced by g. We consider g (n) as an element of Aut(T ). Proposition 2. For any subexponentially bounded automorphism g of a dregular rooted tree T and any p ∈ P the measure μp on X is quasi-invariant with respect to g. Proof. Let g ∈ Aut(T ) be subexponentially bounded and p ∈ P. Denote by An the set of vertices v ∈ Vn such that the restriction gv is not equal to identity. Let

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P = max{p1 , . . . , pd }. Set Mn =



Xv . Then by the definition of subexponential

v∈An

boundedness one has: μp (Mn )  P n |An | → 0 when n → ∞. Clearly, for every n the automorphism g (n) preserves the class of measure μp . Observe that on X \ Mn the automorphism g coincides with g (n) , and thus has welldefined Radon-Nikodym derivative on this set. It follows that g has well-defined Radon-Nikodym derivative for almost all x ∈ X and hence is measure class preserving.  Proposition 3. For any p ∈ P and any subexponentially bounded group G acting level-transitively on a d-regular rooted tree T the measure μp on X is ergodic with respect to the action of G. Proof. Assume that there exists a G-invariant subset A ⊂ T such that 0 < μp (A) < 1. Let v, w ∈ Vn for some n ∈ N. Then there exists g ∈ G such that gv = w. By subexponential boundedness of g for every > 0 there exists m > n, k ∈ N and a collection v1 , . . . , vk ∈ Vm such that the restriction of g on the subtree Tvi is trivial for every i = 1, . . . , k and μp (Bm ) > 1 − , where Bm =

k 

Xvi .

i=1

In particular, g coincides with g (m) on Bm and hence the Radon-Nikodym derivative of g is constant on Xvi for every i. It follows that μp (g(A ∩ Xvi )) μp (g(Xvi \ A)) = . μp (A ∩ Xvi ) μp (Xvi \ A) Since > 0 is arbitrary using G-invariance of A we obtain that (5)

μp (Xw \ A) μp (g(Xv \ A)) μp (Xv \ A) = = . μp (A ∩ Xw ) μp (g(A ∩ Xv )) μp (A ∩ Xv ) μ (X\A)

Since v, w ∈ Vn are arbitrary, the latter is equal to pμp (A) = that for every clopen set B ⊂ X one has:

 μp (B \ A) = μp1(A) − 1 μp (A ∩ B).

1 μp (A)

− 1. It follows

Recall that clopen subsets of X (which is homeomorphic to a Cantor set) approximate all measurable subsets by measure with arbitrary precision. Choosing a clopen subset B such that μp (A ∩ B) > 12 μp (A) and μp (B \ A) < 12 (1 − μp (A)) we obtain a contradiction. This finishes the proof.



The proof of Theorem 3 is based on several technical statements. The following statement is a generalization of Proposition 23 from [11]. The proof is similar. For the reader’s convenience we present it here.

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59

Proposition 4. Let G be a subexponentially bounded countable weakly branch group acting on a d−regular rooted tree T , p ∈ P ∗ (see (2)) and μp be the corresponding Bernoulli measure on X. For any clopen subset A ⊂ X and any > 0 there exists g ∈ G such that supp(g) ⊂ A and μp (A \ supp(g)) < . Let us prove an auxiliary combinatorial lemma. Lemma 2. Let n ∈ N and let H < Sym(n) be a subgroup acting transitively on {1, 2, . . . , n}. Let A be a subset of {1, . . . , n} such that for all g = h ∈ H one has |g(A)Δh(A)|  |A|, where |A| is the cardinality of A. Then |A| > n/2. Proof. Set k = |A|. Then for any two distinct elements h, g ∈ H we have: 1 (|h(A)| + |g(A)| − |h(A)Δg(A)|)  k/2. 2 For every h ∈ H introduce a vector ξh ∈ Cn by:  0, if i ∈ / h(A), ξh = (x1 , . . . , xn ), where xi = 1, if i ∈ h(A). |h(A) ∩ g(A)| =

Denote by (·, ·) the standard scalar product in Cn and by  ·  the corresponding norm. Then for every h = g ∈ H one has: ξh 2 = k and (ξh , ξg ) = |h(A) ∩ g(A)|  k/2. We obtain:   ξh 2  k2 m(m + 1), where m = |H|. h∈H

On the other hand, the group H acts on Cn by permuting coordinates such that g(ξh ) = ξgh for all g, h ∈ H. Using transitivity of H and the fact that the vector ξh is fixed by H we get: h∈H



km km ξh = ( km n , n , . . . , n ), 

h∈H

It follows that km 



ξh 2 =

k 2 m2 n .

h∈H

n m+1 2

and k >

n 2.



Let d  2 be the valency of the regular rooted tree T . The proof of Proposition 4 is based on the following: Lemma 3. If G is weakly branch then for every vertex v there exists g ∈ G with supp(g) ⊂ Xv such that μp (supp(g))  d1 μp (Xv ). Proof. First, let us introduce some notations. For an element h ∈ G set l(h) = max{l : h ∈ StG (l)}, where StG (l) = {g ∈ G : gw = w for each w ∈ Vl } is the stabilizer of the lth level of T . As before, for h ∈ Aut(T ) and n ∈ N symbol h(n) denotes the element of Aut(Tn ) induced by h. We will use the same symbol for the corresponding permutation of Vn . For a vertex v ∈ Vn , n ∈ N and l  n set Vl (v) = Tv ∩ Vl . Fix a vertex v of the tree. Since G is weakly branch there exist g = Id such that supp(g) ⊂ Xv . Set L = min{l(g) : g ∈ G, g = Id, supp(g) ⊂ Xv }.

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ARTEM DUDKO AND ROSTISLAV GRIGORCHUK

For g ∈ G denote by W (g) the set of vertices w from VL (v) such that g induces a nontrivial permutation on VL+1 (w). Set k(g) = |W (g)|, K = max{k(g) : g ∈ G, supp(g) ⊂ Xv }. By the choice of L, K > 0. Fix an element g ∈ G with supp(g) ⊂ Xv such that k(g) = K. Further, since G acts transitively on VL , we can find a number m and a collec(L) (L) tion of elements H = {h1 , h2 , . . . , hm } ⊂ G such that the family S = {h1 , . . . , hm } of transformations of VL forms a group preserving VL (v) and transitive on VL (v). Denote gi = hi gh−1 i . One has: (L)

W (gi ) = hi (W (g)), W (gi gj ) ⊃ W (gi )ΔW (gj ) for all i, j. It follows that the set W (g) together with the action of the group S restricted to VL (v) satisfy the conditions of Lemma 2. Therefore, K = |W (g)| > 1 2 |VL (v)|. For each w ∈ W (g) ⊂ VL (v) the element g induces a nontrivial permutation of VL+1 (w), and thus there exists at least two vertexes w1 , w2 ∈ VL+1 (w) such that supp(g) ⊃ Xw1 ∪ Xw2 . Set W = {wi : w ∈ W (g), i = 1, 2}. We have: |W| > d1 |VL+1 (v)| and supp(g) ⊃



Xu .

u∈W

Now, since G is level transitive, we can find a number r and a collection of ˜ 1, h ˜ 2, . . . , h ˜ r } ⊂ G such that the family S˜ = {h ˜ (L+1) , . . . , h ˜ (L+1) ! = {h elements H } r 1 of transformations of VL+1 forms a group preserving VL+1 (v) and transitive on ˜ i gh ˜ −1 . One has: VL+1 (v). Denote g˜i = h i  Xh˜ (L+1) (u) . supp(˜ gi ) ⊃ u∈W

i

Since S! is a group acting transitively on VL+1 (v), for every u ∈ W the multiset r ˜ (L+1) (u) : i = 1, . . . , r} contains every vertex u1 ∈ VL+1 (v) equal number {h i |VL+1 (v)| times. It follows that r  i=1

μp (supp(˜ gi ))  |W|

r |VL+1 (v)|

μp (Xv ) > dr μp (Xv ).

Therefore, there exists i such that μp (supp(˜ gi )) > proof.

1 d μp (Xv ).

This finishes the 

Proof of Proposition 4. Let A be any clopen set and g0 = Id. Construct by induction elements n gn ∈ G, n = 0, 1, 2 . . . such that supp(gn ) ⊂ A and μp (A \ d supp(gn ))  d+1 . If gn is constructed choose vertices v1 , . . . vk such that Xvj are d disjoint subsets of A \ supp(gn ) and μp (Xvj )  d+1 μp (A \ supp(gn )). Using the lemma construct elements h1 , . . . hk such that supp(hj ) ⊂ Xvj and μp (supp(hj ))  1 d d μp (Xvj ). Set gn+1 = gn h1 h2 . . . hk . Then μp (A \ supp(gn+1 ))  d+1 μp (A \  supp(gn )), which finishes the proof.

REPRESENTATIONS OF WEAKLY BRANCH GROUPS

61

As before, let StG (1) = {g ∈ G : gv = v for each v ∈ V1 } be the stabilizer of the first level of T . For each point x ∈ X denote by Ng (x) ∈ Z+ ∪ {∞} the number of the vertices v on the path defined by x such that gv ∈ / StG (1). Corollary 3. For every clopen set A with μp (A) > 0, any > 0 and k ∈ N there exists g ∈ G such that supp(g) ⊂ A and μp ({x : Ng (x)  k}) > (1 − )μp (A). Proof. We prove the statement by induction on k. The base follows immediately from Proposition 4. Assume that for given clopen set A and > 0 we have an element g such that μp ({x : Ng (x)  k}) > (1 − )μp (A). For n large enough one has μp ({x : Ng(n) (x)  k}) > (1 − )μp (A). Let D = {x : Ng (x)  k}, C = {x : Ng (x) = k}, B = {x ∈ C : Ng(n) (x) = k}. Fix δ > 0. Increasing n if necessary we can assume that μp (C \ B) < δ. Find a ˜ < δ. Using Proposition 4 find an element h ˜ ⊂ A such that μp (BΔB) clopen set B ˜ ˜ with supp(h) ⊂ B such that μp (B \ supp(h)) < δ and h ∈ StG (n + 1). Then one has: Ngh (x)  k + 1 for all x ∈ S = (B ∩ supp(h)) ∪ (D \ (C ∪ supp(h))). By construction, μp (D \ S) < 4δ. It follows that for δ sufficiently small one has μp ({x : Ngh (x)  k + 1})  μp (S) > (1 − )μp (A).  For g ∈ Aut(T ) introduce an element α(g) ∈ Aut(T ) as follows. For arbitrary x ∈ X let v be the vertex of a minimal level from the path defined by x such that g(v) = v. Write x = vy. Set α(g)x = g(v)y, where g(v)y should be understood as a concatenation of a finite word g(v) and an infinite word y. Observe that for every x ∈ X α(g)x differs with x at most by one letter. Define β(g) = gα(g)−1 . Observe that for all x such that gx = x one has (6)

Nα(g) (x) = 1, Nβ(g) (x) = Ng (x) − 1. Recall that pi are pairwise distinct. Set a = a(p) =

min{ ppji

√ 2 a . : pi > pj }, γ = γ(p) = a+1

Lemma 4. Let A be a clopen set and ξA be the characteristic function of the set A. Let g ∈ Aut(T ), g(A) = A and Ng (x)  k for almost all x ∈ A, where k ∈ N. Then (κp (g)ξA , ξA )  γ k μp (A). Proof. We will prove the lemma using induction by k. The base of induction k = 0 is trivial. Assume that the statement is true for given k. Let g be such that Ng (x)  k + 1 for almost all x ∈ A. We will use the presentation g = α(g)β(g) (see (6)). Let v be any vertex such that α(g)v = v and α(g)w = w for the vertex w adjacent to v above v. Denote by v0 = v, v1 , . . . , vl−1 the orbit of v under α(g),

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ARTEM DUDKO AND ROSTISLAV GRIGORCHUK

where α(g)vj = vj+1 for j < l − 1 and α(g)vl−1 = v0 . Observe that β(g)vj = vj for every j. From the induction assumption it follows that (κp (β(g))ξvj , ξvj )  γ k μp (Xvj ). For every j let qj be the weight from {p1 , . . . , pd } corresponding to the last letter of vj . Then dμp (α(g)−1 (x)) qj−1 = dμp (x) qj for all x ∈ Xvj . We obtain:    l−1 " l−1 l−1 l−1 "    qj−1 qj−1 k ξvj , ξvj = (κ (β(g))ξ , ξ )  γ κp (g) p vj vj qj qj μp (Xvj ) = j=0

j=0

γk

l−1 

j=0

√

j=0

qj qj−1 μp (Xw )  γ k+1

j=0

l−1 

qj +qj−1 μp (Xv ) 2

= γ k+1

j=0

l−1 

μp (Xvj ).

j=0

Summing the last inequality over all the orbits {vj } as above (finite or countable number) we obtain the desired inequality.  For a unitary representation of a discrete group Γ in a Hilbert space H denote by Mπ the von Neumann algebra generated by operators π(g), g ∈ Γ (i.e. the closure of linear combinations of operators π(g), g ∈ G in the weak operator topology). Let B(H) be the algebra of all bounded operators on H and Mπ = {R ∈ B(H) : QR = RQ for all Q ∈ Mπ } be the commutant of Mπ . The following fact is folklore: Lemma 5. Let π be a unitary representation of a discrete group Γ in a Hilbert space H. Set H1 = {η ∈ H : π(g)η = η for all g ∈ Γ}. Then the orthogonal projection P onto H1 belongs to Mπ . Proof. Let B ∈ Mπ . Then π(g)Bη = Bπ(g)η = Bη for every η ∈ H1 , g ∈ Γ. This implies that BH1 ⊂ H1 and so BP = P BP . Same argument shows that B ∗ P = P B ∗ P , where ∗ stands for the operation of conjugation in B(H). Conjugating the latter identity we obtain that P B = P BP = BP . By von Neumann Bicommutant Theorem (see e.g. Theorem 2.4.11 in [8]) we get that  P ∈ (Mπ ) = Mπ . For an open set A ⊂ X define (7)

GA = {g ∈ G : supp(g) ⊂ A}, HA = {η ∈ H : π(g)η = η for all g ∈ GA }.

Let PA be the orthogonal projection onto HA . Applying Lemma 5 to the restriction of the representation κp onto the subgroup GA we obtain Corollary 4. For any open subset A ⊂ X one has PA ∈ Mκp . Proposition 5. One has HA = {η ∈ H : supp(η) ⊂ X \ A}.

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Proof. Clearly, every η with supp(η) ⊂ X \ A belongs to HA . Assume that HA is strictly larger than the subspace of functions η with supp(η) ⊂ X \ A. Then there exists a unit vector η ∈ HA such that supp(η) ⊂ A. Fix such a vector. For n ∈ N denote by Vn (A) the set of vertices v from Vn such that Xv ⊂ A. Let > 0. Since locally constant functions are dense in H, one can find a level n and constants αv , v ∈ Vn (A) such that  η − αv ξXv   . v∈Vn (A)

By Corollary 3 and Lemma 4 for each v ∈ Vn (A) there exists a sequence of elements gv,k such that supp(gv,k ) ⊂ Xv and Set hk =



lim (π(gv,k )ξXv , ξXv ) = 0.

k→∞

gv,k . Then supp(hk ) ⊂ A and

v∈Vn (A)

lim (π(hk )

k→∞

 v∈Vn (A)

αv ξXv ,



αv ξXv ) = 0.

v∈Vn (A)

It follows that lim sup |(π(hk )η, η)|  2 + 2 . k→∞

Taking , for instance, to be 13 we obtain a contradiction to the fact that η ∈ HA is a unit vector. This finishes the proof.  Proof of Theorem 3. 1) Proposition 5 means that for each open set A ⊂ X the orthogonal projection PA onto HA is the operator of multiplication by the characteristic function of X \ A. Observe that every function from L∞ (X, μp ) can be approximated arbitrarily well in L2 -norm by finite linear combinations of characteristic functions of open sets. This implies that for every m ∈ L∞ (X, μp ) the operator of multiplication by m H → H, f → mf can be approximated arbitrary well in the strong operator topology by finite linear combinations of projections PA ∈ Mκp (see Corollary 4), and thus belongs to the von Neumann algebra Mκp generated by operators κp (g), g ∈ G. This implies that Mκp contains operators of multiplication by all functions from L∞ (X, μp ). Since for an ergodic measure class preserving action of a group G on a measure space (X, μp ) the algebra generated by group shifts and multiplication by functions coincide with the algebra B(H) of all bounded operators on H (see e.g. [31], Corollary 1.6) we obtain that Mκp coincides with B(H). By Schur’s Lemma (see e.g. [7], Theorem A.2.2) this implies irreducibility of κp . x the subspace of l2 (Gx) 2) Let x ∈ X. For an open subset A of X denote by HA analogous to (7), but corresponding to the representation ρx : x HA = {η ∈ l2 (Gx) : ρx (g)η = η for all g ∈ GA }. x . Assume that κp and ρx are unitary Let PAx be the orthogonal projection onto HA equivalent via intertwining isometry

U : L2 (X, μp ) → l2 (Gx),

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ARTEM DUDKO AND ROSTISLAV GRIGORCHUK

that is U κπ (g) = ρx (g)U for every g ∈ G. Choose a sequence of open covers An of the orbit Gx such that μp (An ) → 0 when n → ∞. From the definition of orthogonal projections PA and subspaces HA we obtain: U PAn U ∗ = PAxn for every n. Since G is weakly branch for every n and every y ∈ Gx the set {gy : g ∈ GAn } is infinite. This implies that PAxn = 0. However, Proposition 5 implies that PAn → Id weakly when n → ∞. This contradiction shows that κp and ρx are disjoint (not unitary equivalent). ˜ A the subspace of 3) Let p˜ ∈ P ∗ , p˜ = p. For an open subset A of X denote by H 2 L (X, μp˜) analogous to (7), but corresponding to the representation κp˜. Let P˜A be ˜ A . Since μp˜ is singular to μp there exists A ⊂ X the orthogonal projection onto H such that μp (A) = 0 and μp˜(A) = 1. Assume that κp and κp˜ are unitary equivalent via intertwining isometry U : L2 (X, μp ) → L2 (X, μp˜). Let An be a sequence of open covers of A such that μp (An ) → 0 when n → ∞. Since A ⊂ An we have that μp˜(An ) = 1 for every n. From the definition of orthogonal projections PA and subspaces HA we obtain: U PA U ∗ = P˜A = Id n

n

for every n. But from Proposition 5 we obtain that PAn → 0 weakly when n → ∞. This contradiction finishes the proof.  Acknowledgement. The authors acknowledge the support of the Swiss NSF. The authors acknowledge Maria Gabriella Kuhn for useful discussions. References [1] Mikl´ os Ab´ ert, Group laws and free subgroups in topological groups, Bull. London Math. Soc. 37 (2005), no. 4, 525–534, DOI 10.1112/S002460930500425X. MR2143732 [2] Anatolij N. Andrianov, Quadratic forms and Hecke operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 286, SpringerVerlag, Berlin, 1987. MR884891 [3] Uri Bader and Roman Muchnik, Boundary unitary representations—irreducibility and rigidity, J. Mod. Dyn. 5 (2011), no. 1, 49–69, DOI 10.3934/jmd.2011.5.49. MR2787597 [4] L. Bartholdi and R. I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova 231 (2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 5–45; English transl., Proc. Steklov Inst. Math. 4 (231) (2000), 1–41. MR1841750 ´ Branch groups, Handbook ˇ [5] Laurent Bartholdi, Rostislav I. Grigorchuk, and Zoran Suni k, of algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 989–1112, DOI 10.1016/S15707954(03)80078-5. MR2035113 [6] M. E. B. Bekka and M. Cowling, Some irreducible unitary representations of G(K) for a simple algebraic group G over an algebraic number field K, Math. Z. 241 (2002), no. 4, 731–741, DOI 10.1007/s00209-002-0442-6. MR1942238 [7] B. Bekka, P. de la Harpe and A. Valette, Kazhdan Property (T), Cambridge University Press 2008. [8] Ola Bratteli and Derek W. Robinson, Operator algebras and quantum statistical mechanics. 1, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987. C ∗ - and W ∗ -algebras, symmetry groups, decomposition of states. MR887100 [9] M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math. 420 (1991), 85–98. MR1124567 [10] A. Dudko, On irreducibility of Koopman representations of Higman-Thompson groups, arxiv:math.RT/1512.02687

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[32] A. M. Vershik, Nonfree actions of countable groups and their characters (Russian, with English and Russian summaries), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 378 (2010), no. Teoriya Predstavlenii, Dinamicheskie Sistemy, Kombinatornye Metody. XVIII, 5–16, 228, DOI 10.1007/s10958-011-0273-2; English transl., J. Math. Sci. (N. Y.) 174 (2011), no. 1, 1–6. MR2749291 Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, ON M5S 2E4, Canada E-mail address: [email protected] Department of Mathematics, Texas A&M University, College Station, Texas 778433368 E-mail address: [email protected]

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13924

Isolated elliptic fixed points for smooth Hamiltonians Bassam Fayad and Maria Saprykina Abstract. We construct on R2d , for any d ≥ 3, smooth Hamiltonians having an elliptic equilibrium with an arbitrary frequency, that is not accumulated by a positive measure set of invariant tori. For d ≥ 4, the Hamiltonians we construct have not any invariant torus of dimension d. Our examples are obtained by a version of the successive conjugation scheme ` a la Anosov-Katok.

Introduction KAM theory (after Kolmogorov Arnol’d and Moser) asserts that generically an elliptic fixed point of a Hamiltonian system is stable in a probabilistic sense, or KAM-stable: the fixed point is accumulated by a positive measure set of invariant Lagrangian tori. In classical KAM theory, an elliptic fixed point is shown to be KAM-stable under the hypothesis that the frequency vector at the fixed point is non resonant (or just sufficiently non-resonant) and that the Hamiltonian is sufficiently smooth and satisfies the Kolmogorov non degeneracy condition that involves its Hessian matrix at the fixed point. Further development of the theory allowed to relax the non degeneracy condition. In [EFK1] KAM-stability was established for non resonant elliptic fixed points under the R¨ ussmann non-planarity condition on the Birkhoff normal form of the Hamiltonian. The problem is more tricky if no non-degeneracy conditions are imposed on the Hamiltonian. In the analytic setting, no examples are known of an elliptic fixed point with a non-resonant frequency ω0 that is not KAM-stable or Lyapunov unstable (none of these two properties implies the other). It was conjectured by M. Herman in his ICM98 lecture [H1] that for analytic Hamiltonians, KAM-stability holds in the neighborhood of a an elliptic fixed point if its frequency vector is assumed to be Diophantine. The conjecture is known to be true in two degrees of freedom [R], but remains open in general. Partial results were obtained in [EFK1] and [EFK2]. Below analytic regularity, Herman proved that KAM-stability of a Diophantine equilibrium holds without any twist condition for smooth Hamiltonians in 2 degrees of freedom (see [H2], [FK] and [EFK2]). In his ICM98 lecture [H1, §3.5], he announced that KAM-stability of Diophantine equilibria does not hold for smooth 2010 Mathematics Subject Classification. Primary 37J40; Secondary 37C75, 70H08. The first author is supported by ANR BEKAM and ANR GeoDyM and the project BRNUH. This work was accomplished while the first author was affiliated to the Laboratorio Fibonacci of the Scuola Normale Superiore di Pisa. c 2017 American Mathematical Society

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Hamiltonians1 in four or more degrees of freedom, without giving any clew about the counter-examples he had in mind. He also announced that nothing was known about KAM-stability of Diophantine degenerate equilibria for smooth Hamiltonians in three degrees of freedom. In this note, we settle this problem by constructing examples of smooth Hamiltonians for any d ≥ 3 having non KAM-stable elliptic equilibria with arbitrary frequency. We now state our results more precisely. Let ω0 ∈ Rd and let  H(x, y) = ω0 , r + O 3 (x, y) (∗) r = (r1 , . . . , rd ), rj = 12 (x2j + yj2 ) be a smooth function defined in a neighborhood of (0, 0). The Hamiltonian system associated to H is given by the vector field XH = (∂y H, −∂x H), namely  x˙ = ∂y H(x, y) y˙ = −∂x H(x, y). The flow of XH denoted by ΦtH has an elliptic fixed point at the origin with frequency vector ω0 . In [EFK2], it was shown that for any ω0 ∈ Rd , d ≥ 4, it is possible to construct C ∞ (Gevrey) Hamiltonians H with a smooth invariant torus, on which the dynamics is the translation of frequency ω0 , that is not accumulated by a positive measure of invariant tori. In this note we adapt the latter construction to the context of elliptic equilibria and we extend it to the three degrees of freedom case. In all our constructions the elliptic equilibria will be degenerate, that is H(x, y) =ω0 , r+O ∞ (x, y). It is a common knowledge that creating instability in the neighborhood of a fixed point is more delicate than in the context of invariant tori, mainly because the action angle coordinates are singular in the neighborhood of the axes {ri = 0}. For instance, when all the coordinates of ω0 are of the same sign, the fixed elliptic point is Lyapunov stable, while it is easy to produce examples of diffusive and isolated invariant tori for any resonant frequency vector, even in the analytic category (see [S]). Definition 1. We say that ΦtH is diffusive if given any A > 0 there exists p and t1 , t2 ∈ R such that |ΦtH1 (p)| ≤ A−1 and |ΦtH2 (p)| ≥ A. Obviously, if the flow is diffusive, the origin is not Lyapunov stable. Theorem A. For any ω0 ∈ Rd , d ≥ 4, there exists H ∈ C ∞ (R2d ) as in (∗), such that ΦtH has no invariant torus of dimension d. More precisely, the manifolds {ri = 0} for i ≤ d, are foliated by invariant tori of dimension ≤ d − 1 and all other obits accumulate on these manifolds or at infinity. Moreover, if the coordinates of ω0 are not all of the same sign, then ΦtH is diffusive. In the case d = 3, our examples will have invariant Lagrangian tori of maximal dimension (equal to 3) that accumulate the origin, but only for r3 in a countable set. 1 Herman

actually raised the problem in the very related context of symplectic maps.

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Theorem B. For any ω0 ∈ R3 , there exists H ∈ C ∞ (R6 ) as in (∗), and a sequence {an }n∈Z of real numbers such that lim an = 0 and lim an = +∞, n→−∞

n→+∞

such that the manifolds {r3 = an }, as well as {ri = 0} for i ≤ 3, are foliated by invariant tori and such that all other obits accumulate on these manifolds or at infinity. Moreover, if the coordinates of ω0 are not all of the same sign, then ΦtH is diffusive. Remark 1. The same construction can be carried out around invariant quasiperiodic tori and gives examples of KAM-unstable tori with arbitrary frequency in 3 degrees of freedom. Remark 2. Our examples are obtained by a successive conjugation scheme ` a la Anosov-Katok [AK], and the flows that we obtain are rigid in the sense that their iterates along a subsequence of time converge to identity in the C ∞ topology. Remark 3. In case all the coordinates of ω0 are of the same sign, there are naturally no diffusive orbits since the equilibrium is Lyapunov stable. In the case where not all the components of ω0 are of the same sign and d ≥ 3, Douady gave in [D] examples of elliptic fixed points with diffusive trajectories. However, his construction, that produces actually examples with an arbitrarily chosen Birkhoff normal form at the fixed point, does not overrule KAM-stability. 1. Notations –A vector ω0 ∈ Rd is said to be non-resonant if for any k ∈ Zd  {0} we have that | k, ω | = 0, where ·, · denotes the usual scalar product. –A vector ω0 ∈ Rd is said to be Diophantine if there exist N > 0 and γ > 0 such that for any k ∈ Zd  {0} we have that | k, ω0  | ≥ γk−N . –A non-resonant vector ω0 is said to be Liouville if it is not Diophantine. T (p) the orbit of length T of the point p by the Hamiltonian flow –We denote by OH ∞ (p). of H. The full orbit of p is denoted by OH –The notation of type {ri < A} should be understood as {(r, θ) | ri < A}. ∞ (p) accumulates on {ri = ∞} –We shall say, with a slight abuse of notation, that OH tj for some i = 1, . . . , d if projri OH (p) → ∞ over a sequence of times (tj ). 2. Orbits accumulating the axis and diffusive orbits 2.1. Two degrees of freedom. As discussed earlier, in 2 degrees of freedom it follows from the Last Geometric Theorem of Herman (see [H2], [FK] and [EFK2]) that if ω0 ∈ R2 is Diophantine then if H ∈ C ∞ (R4 ) is as in (∗), then the origin is KAM-stable for ΦtH . We will be interested in constructing close to integrable non KAM-stable (and diffusive if the frequency vector satisfies ω0,1 ω0,2 < 0) examples in 2 degrees of freedom when the frequency is Liouville. Consider (2.1)

H02 (r) = ω0 , r.

Let U2 be the set of symplectomorphisms U such that U (r, θ) = (r, θ) near the axes {ri = 0}, i = 1, 2, as well as for |r| sufficiently large. Consider the class of conjugates of H02 by the elements of U2 , and denote its C ∞ -closure by H¯2 .

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Theorem 1. Let H02 (r) = ω0 , r for a Liouville vector ω0 ∈ R2 . Let D be ∞ (p) the set of Hamiltonians H ∈ H¯2 such that for each p ∈ R4 we have that OH accumulates on at least one of the following sets: {|r| = ∞}, {r1 = 0} or {r2 = 0}. If ω0,1 ω0,2 < 0, then we assume moreover that for H ∈ D, for every A > 0, ∞  (p ) intersects both {|r| ≤ A−1 } and {|r| ≥ A}. there exists p ∈ R4 such that OH Then D contains a dense (in the C ∞ topology) G δ subset of H¯2 . Remark 4. Note that for H ∈ D and for i = 1, 2, the sets {ri = 0} are foliated by invariant tori of dimension 1 on which the dynamics is the rotation by angle ω0,j j ∈ {1, 2}  {i}. These are the only invariant tori for H. Remark 5. The construction can be extended to any degrees of freedom d ≥ 2 and any Liouville vector ω ∈ Td . 2.2. Four degrees of freedom and higher. We consider d = 4, the case d ≥ 5 being similar. Fix ω0 ∈ R4 . To prove Theorem A we will use the same technique of construction that serves in the Liouville 2 degrees of freedom construction. We will first introduce a completely integrable flow with a fixed point at the origin of frequency ω0 following [EFK2]. It will have the form H04 (r) = ω(r4 ), r = ω0 + f (r4 ), r,

(2.2)

where we use action coordinates rj (x, y) = (x2j + yj2 )/2 and where f (r4 ) = (f1 (r4 ), f2 (r4 ), f3 (r4 ), 0) with f defined as follows. We call a sequence of intervals (open or closed or half-open) In = (an , bn ) ⊂ (0, ∞) an increasing cover of the half line if: (1) (2)

lim an = 0

n→−∞

lim an = +∞

n→+∞

(3) an ≤ bn−1 < an+1 ≤ bn . Proposition 1. [EFK2] Let (ω0,1 , ω0,2 , ω0,3 ) ∈ R3 be fixed. For every > 0 and every s ∈ N, there exist an increasing cover (In ) of (0, ∞) and functions fi ∈ C ∞ (R, [0, 1]), i = 1, 2, 3, such that fi s < and fi (0) = 0, and • For each n ∈ Z, the functions f1 and f2 are constant on I3n : f1 |I3n ≡ f¯1,n ,

f2 |I3n ≡ f¯2,n

• For each n ∈ Z, the functions f1 and f3 are constant on I3n+1 : f1 |I3n+1 ≡ f¯1,n ,

f3 |I3n+1 ≡ f¯3,n

• For each n ∈ Z, the functions f2 and f3 are constant on I3n−1 : f2 |I3n−1 ≡ f¯2,n ,

f3 |I3n−1 ≡ f¯3,n−1

• The vectors (f¯1,n + ω0,1 , f¯2,n + ω0,2 ), (f¯1,n + ω0,1 , f¯3,n + ω0,3 ) and (f¯2,n + ω0,2 , f¯3,n + ω0,3 ) are Liouville. If ω0,1 ω0,2 < 0, we ask that > 0 be sufficiently small so that ω1 (r4 )ω2 (r4 ) < 0 for every r4 . Remark 6. It follows that f1 , f2 , f3 are C ∞ -flat at zero.

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Notice that, as a consequence of Proposition 1, for r4 ∈ In two of the coordinates of (f1 (r1 ) + ω1 , f2 (r1 ) + ω2 , f3 (r1 ) + ω3 ) are constant and form a Liouville vector. This is why we will be able to use a similar construction as in the two dimensional Liouville case. Let U4 be the set of exact symplectic diffeomorphisms of R8 with the following properties: U (r, θ) = (r, θ) in the neighborhood of the axes {ri = 0}, i = 1, . . . , 4, as well as for |r| sufficiently large, and U (r, θ) = (R, Θ) satisfies R4 = r4 . Let H4 be the set of Hamiltonians of the form H04 ◦ U , U ∈ U4 . Finally we denote H¯4 the closure in the C ∞ topology of H4 . We denote I!n = R3 × In × T4 . For H ∈ H4 the flow ΦtH leaves r4 invariant. In particular, for U ∈ U4 we have U (I!n ) = I!n for any n ∈ Z. We shall show how to make arbitrarily small perturbations of H04 inside H4 that create oscillations of the corresponding flow in two of the three directions r1 , r2 , r3 . These perturbations will actually be compositions inside H4 by exact symplectic maps obtained from suitably chosen generating functions. Iterating the argument gives a construction by successive conjugations scheme similar to [AK]. The difference here is that the conjugations will be applied in a ”diagonal” procedure to include more and more intervals In into the scheme. Rather than following this diagonal scheme which would allow to define the conjugations explicitly at each step, we will actually adopt a G δ -type construction (see [FH]) that makes the proof much shorter and gives slightly more general results. Theorem 2. Let H04 (r) = ω(r4 ), r be as in ( 2.2). Let D be the set of Hamiltonians H ∈ H¯4 such that for each p ∈ R8 such that ri (p) = 0 for every i = 1, . . . , 4, ∞ (p) accumulates on at least one of the sets there exists i ∈ {1, 2, 3} such that OH {ri = ∞} and {ri = 0}. If ω0,1 ω0,2 < 0, we assume moreover that for H ∈ D, there exists for every ∞  (p ) intersects both {|r| ≤ A−1 } and {|r| ≥ A}. A > 0, p ∈ R8 such that OH Then D contains a dense (in the C ∞ topology) G δ subset of H¯4 . Proof that Theorem 2 implies Theorem A. Note that for H ∈ D and i = 1, . . . , 4, the set {ri = 0} is foliated by invariant tori of dimension 3 on which the dynamics is that of the integrable Hamiltonian H04 . We want to show that these are the only invariant tori of H. Indeed, let p ∈ R8 such that ri (p) = 0 for every i = 1, . . . , 4. Since the orbit of p accumulates on the axis or at infinity, it cannot lie on an invariant compact set. Note now that if ω0 does not have all its coordinates of the same sign, we can assume that ω0,1 ω0,2 < 0 by possibly renaming the variables. Hence the second part of Theorem A follows form the second part of Theorem 2.  2.3. Three degrees of freedom. The construction of Theorem B for d = 3 will be similar to the case d = 4 but with this difference that we cannot count anymore on an invariant action variable r4 that played the role of a parameter. Instead, one of the action coordinates will both involved in the diffusion, and play the role of the parameter. We choose this variable to be r3 and assume without loss of generality that if the coordinates of ω0 are not all of the same sign then ω0,1 ω0,2 < 0.

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We fix a sequence of intervals In = [an−1 , an ] ⊂ (0, ∞), n ∈ Z, such that lim an = 0, and lim an = +∞, and let I!n = R2+ × In × T3 . We introduce a

n→−∞

n→+∞

completely integrable flow with a fixed point at the origin by (2.3)

H03 (r) = ω(r3 ), r = ω0 + f (r3 ), r,

where f = (f1 , f2 , f3 ) is as in Proposition 1 with r4 replaced by r3 , and we use action-angle coordinates as above. Let U3 be the set of symplectomorphisms U such that U (r, θ) = (r, θ) near the axes {ri = 0}, i = 1, 2, 3, as well as near the sets {r3 = an }, n ∈ Z, and for |r| sufficiently large. Consider the class of conjugates of H03 by the elements of U3 , and denote its C ∞ -closure by H¯3 . Theorem 3. Let H03 (r) = ω(r3 ), r be as in (2.3). Let D be the set of Hamiltonians H ∈ H¯3 such that for each p ∈ R6 satisfying ri (p) = 0 for every i = 1, 2, 3 and ∞ / {an }n∈Z we have that OH (p) accumulates on at least one of the following r3 (p) ∈ sets: {r1 = ∞}, {r2 = ∞}, {r1 = 0}, {r2 = 0}, ∪n∈Z {r3 = an }. If ω0,1 ω0,2 < 0, then we assume moreover that for H ∈ D, there exists for every ∞  A > 0, p ∈ R6 such that OH (p ) intersects both {|r| ≤ A−1 } and {r1 ≥ A} ∩ {r2 ≥ A}. Then D contains a dense (in the C ∞ topology) G δ subset of H¯3 Proof that Theorem 3 implies Theorem B. Note that the axes and the sets {r3 = an } are foliated by invariant tori. The orbit of p ∈ R6 satisfying ri (p) = 0 / {an }n∈Z cannot accumulate on any of these sets if for every i = 1, 2, 3 and r3 (p) ∈ p lies on an invariant torus. Hence the only invariant tori for ΦtH are those foliating the axis and ∪n∈Z {r3 = an }. The second part of Theorem B follows clearly from the second part of Theorem 3.  3. Proof for the case d = 2 All our constructions will be derived from the main building block with two dimensional Liouville frequencies. The construction is summarised in the following Proposition 2 from which Theorem 1 will easily follow. For A > 0, denote R(A) := [A−1 , A] × [A−1 , A],

! R(A) = R(A) × T2 .

We define the ”margins” by: M (A) = {r1 > A} ∪ {r2 > A} ∪ {r1 < A−1 } ∪ {r2 < A−1 }. We shall refer to the individual sets of the above union as margin sets. Proposition 2. For any Liouville vector ω = (ω1 , ω2 ), any > 0, s ∈ N, ! 0 ) we have that A0 > 0, and any symplectic map V that is identity outside R(A for any A > A0 there exist U ∈ U and T > 0 with the following properties for H = H02 ◦ U −1 ◦ V −1 : ! (1) U =Id in the complement of R(2A), 2 −1 (2) H − H0 ◦ V s < , T ! (3) For any P ∈ R(A) we have: OH (P ) intersects M (A). T (4) Moreover, if ω1 ω2 < 0, then there exists p ∈ R4 such that OH (p ) inter2 −1 2 sects both ∩i=1 {ri < 2A } and ∪i=1 {ri > A}.

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Proof of Theorem 1. For n, A, T ∈ N∗ , let $ # T ! OH (P ) intersects M (A) . D(A, T ) := H ∈ H¯2 | ∀P ∈ R(A), It is clear that D(A, T ) are open subsets of H¯2 in any C s topology. Proposition 2 (1)–(3) implies that ∪T ∈N∗ D(A, T ) is dense in H¯2 in any C s topology. Hence the ¯2 ¯ ⊂ D is a dense Gδ set (in any C s topology) in H following set D %  ¯= D D(A, T ). A∈N∗ T ∈N∗

In case ω1 ω2 < 0 we just have to add to the definition of D(A, T ) the existence T (p ) intersects both ∩2i=1 {ri < 2A−1 } and ∪2i=1 {ri > of a point p ∈ R4 such that OH A}. The density of ∪T ∈N∗ D(A, T ) then follows from (1)–(4) of Proposition 2.  The rest of this section is devoted to the proof of Proposition 2. The idea is to construct a conjugacy U that ”wiggles” the invariant tori of H02 and makes them accumulate on the margin sets. Since V in the statement of the proposition is ! assumed to be identity outside R(A) it will be possible to conclude from there that 2 −1 the Hamiltonian H = H0 ◦ U ◦ V −1 satisfies the requirements of the proposition. One has to observe however that since we want H02 ◦U −1 to be very close to H02 , the ”wiggling” will take place almost inside the energy levels of H02 . In particular, one does not get diffusion in the case ω1 ω2 > 0 because in that case the energy lines are compact segments (see Figure 1). To be more precise, fix ω = (ω1 , ω2 ) and define the energy line !p = Ep × T2 . Ep := {(r1 , r2 ) ∈ R2+ : ω1 r1 + ω2 r2 = ω1 r1 (p) + ω2 r2 (p)}, E !p has the form {H 2 = const.}, and is invariant under the flow of H 2 Clearly, E 0 0 (with the same fixed ω). For p = (r, θ), let T (p) = {r} × T2 denote the flat torus passing through p. This is the invariant torus of H02 passing through p. Let H = H02 ◦ U −1 for a symplectic transformation U . Then U (T (p)) is the invariant torus of H passing through the point U (p). The main ingredient in the proof of Proposition 2 is the following lemma in which we construct a symplectic !p essentially for a large set of map U such that U (T (p)) will ”wiggle” inside E starting points p. r1=0

Ep>0

r1=0 Ep=0

Ep 0.

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Lemma 7. For any Liouville vector ω = (ω1 , ω2 ), any > 0, s ∈ N, A0 > 0, and ! 0 ) we have that for any A > A0 any symplectic map V that is identity outside R(A there exist U ∈ U and T > 0 with the following properties: ! (1) U =Id in the complement of R(2A), 2 −1 −1 2 −1 − H0 ◦ V s < (2) H0 ◦ U ◦ V (3) For any p ∈ R2+ × T2 , the torus V ◦ U (T (p)) intersects at least one of the !p ∩ M (A). margin sets, and this intersection is –close to E ! (4) Moreover, for any p ∈ R(A), the torus V ◦ U (T (p)) intersects two of the !p ∩ M (A). margin sets, and this intersection is –close to E Proof of Proposition 2. Let U be as in Lemma 7. The conditions (1) and (2) of the proposition follow directly from (1) and (2) of Lemma 7. The invariant torus of H = H02 ◦ U −1 ◦ V −1 passing through a point P = V ◦ U (p) has the form V ◦ U (T (p)). It follows from (3) of Lemma 7 that the latter torus intersects M (A). Since ω is irrational, the orbit of p is dense on T (p). As a consequence, the orbit of P under the flow of H is dense on V ◦U (T (p)), and conclusion (3) of the proposition follows. Let ω1 · ω2 < 0. In this case, the line Ep has a positive slope of Ω = −ω1 /ω2 . Take a point p such that maxi=1,2 {ri (p)} = 2A−1 , and Ep passes through r = ! the torus U (T (p)) intersects two margin sets close (A−1 , A−1 ). Since p ∈ R(A), to the two components of Ep ∩ M (A). In particular, it contains points in {r1 < !p 2A−1 } ∩ {r2 < 2A−1 }, as well as in {r1 > A} ∪ {r2 > A} (this corresponds to E close to 0 in Figure 1). The same is true for the torus V ◦ U (T (p)) since V preserves the margin sets. Of course, this condition holds for an open set of starting points p. Now, conclusion (4) of the proposition holds for any p ∈ V ◦ U (T (p)) with p as above.  We now turn to the proof of Lemma 7. We will build the conjugacy U by superposing a large number of very slightly wiggling symplectomorphisms which we now present. Sublemma 8. For any Liouville vector ω, for any A > 1, s > 0, Q > 0 and ε > 0, there exists an integer vector q = (q1 , q2 ), |q| > Q, and a symplectic map u ∈ C ∞ with the following properties: ! • u =Id in the complement of R(2A). 2 −1 2 • H0 ◦ u − H0 s < ε, ! • For any p = (r, θ) ∈ R(A), the image (R, Θ) = u(r, θ) satisfies (3.4)

R1 = r1 (1 + br2 cos(2π q, Θ)), R2 = r2 (1 + b(q2 /q1 )r1 cos(2π q, Θ)) where b = (10A)−3 , and

(3.5)

|Θ − θ|0 < 10−3 |q|−1 .

These expressions mean that any small ball gets stretched by u both in r1 and r2 -direction with the amplitude of order br1 r2 ≥ bA−2 , and large frequency |q| (notice that the amplitude is small for small ri , but bounded from below). U will be a composition of a large number of such functions uj , j = 1, . . . , N , constructed with the same A and b, but decaying εj and growing |qj |.

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Here is a heuristic idea. Vector q plays two important roles. On the one hand, having large |q|, we get high frequency of oscillation for R1 and good closeness between θ and Θ (see the formulas above). On the other hand, the key estimate H02 ◦ u−1 − H02 s < ε needs q, ω < ε/pol(q), where pol(q) is a polynomial of q. It is here that we use the assumption of ω being Liouville. It guarantees that there exists a q with sufficiently large components providing the desired smallness of q, ω. Proof of Sublemma 8. Let G(t) ∈ C ∞ be a monotone cut-off function such that |G (t)| ≤ 2 for all t and & 0, for t ≤ 0, (3.6) G(t) = 1, for t > 1. Denote by 1/A the ”margin size”, and by b the scaling constant: b = (10A)−3 . Assume without loss of generality that Ω = −ω1 /ω2 ∈ (0, 1]. Define g(t) = G(2At− 1) − G(t − A). In this case |g  (t)| ≤ 4A for all t, and & 0, for t ≤ (2A)−1 or t ≥ A + 1, g(t) = 1, for t ∈ [A−1 , A]. Given two integers q1 and q2 , whose choice will be specified later, we define the symplectic map u : (r, θ) → (R, Θ) by a generating function b r1 r2 g(r1 )g(r2 ) sin(2π q, Θ). S(r, Θ) = r, Θ + 2πq1 It satisfies ∂S(r, Θ) R1 = = r1 (1 + br2 g(r1 )g(r2 ) cos(2π q, Θ)), ∂Θ1 ∂S(r, Θ) q2 R2 = = r2 (1 + b r1 g(r1 )g(r2 ) cos(2π q, Θ)), ∂Θ2 q1 ∂S(r, Θ) b θ1 = = Θ1 + (r1 g(r1 ))r1 r2 g(r2 ) sin(2π q, Θ) ∂r1 2πq1 ∂S(r, Θ) b  θ2 = = Θ2 + r1 g(r1 ) (r2 g(r2 ))r2 sin(2π q, Θ). ∂r2 2πq1 To see that S defines a diffeomorphism it is enough to verify that the following determinant does not vanish:      2  ∂R ∂θ ∂ S(r, Θ) det = det = det = ∂r ∂Θ ∂r∂Θ   q2 1 + b (r1 g(r1 ))r1 r2 g(r2 ) + r1 g(r1 ) (r2 g(r2 ))r2 cos(2π q, Θ) q1 ≥ 1 − 40bA3 > 0 by the choice of b = (10A)−3 . By a local inverse function theorem, Θ can be expressed as a function of (r, θ). We get Θi = θi + O(|q|−1 ). Plugging in this expression into the first two lines, we get a formula for u in terms of (r, θ). The inverse u−1 exists by the same argument. Since u is a diffeomorphism and equals identity in a neighborhood of {r1 = 0} and {r2 = 0}, we get that the image of u satisfies R1 > 0 and R2 > 0.

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Here we estimate H02 ◦ u−1 (r, θ) − H02 s . For the above coordinate change we have H02 ◦ u(r, θ) − H02 (r, θ) = R, ω − r, ω = 1 b r1 g(r1 )r2 g(r2 ) ω, q cos(2π q, Θ), q1 where Θ = Θ(r, θ). We can estimate H02 ◦ u−1 − H02 s = (H02 − H02 ◦ u) ◦ u−1 s ≤ F (q, A, s) · ω, q , where F (q, A, s) is a polynomial of q1 , q2 whose degree depends only on s, and the coefficients are bounded by functions of A and s. Since vector ω is Liouvillean, there exists (an infinite number of) q such that ω, q < ε/F (q, A, s), and the desired estimate follows.  Proof of Lemma 7. Fix V , s, A0 and ε > 0. Since (V −Id) is compactly ! 0 ), the same holds for any A > A0 . Let b be as in Sublemma supported inside R(A 8, Ω = |ω1 /ω2 | assumed WLOG to lie in (0, 1], and let N be such that (1 + b/(4A))N > A2 ,

(3.7)

(1 − b/(4A))N < 1/A2 .

We shall define uj , j = 1, . . . , N , by Sublemma 8 inductively in j. We choose qj in the construction of u1 so that |qj | > |qj−1 |3 . Moreover, for j = 1, . . . , N , qj are such that uj satisfies a (much stronger) condition 

−1 −1 2 −1 (3.8)  H02 ◦ u−1 s < 2−j ε. j − H0 ◦ uj−1 ◦ · · · ◦ u1 ◦ V Recall from the proof of Sublemma 8 that this is done by choosing vector qj at each step so that ω, qj  is sufficiently small depending on q1 , . . . , qj , s, A and . Define U = u 1 ◦ · · · ◦ uN . ! Then the first statement of the lemma holds since each uj is identity outside R(2A) by construction. The second one follows from (3.8) and the triangle inequality: 

 H02 ◦ U −1 − H02 ◦ V −1 s ≤ N N   

−1 −1 2 −1 ◦ u  H02 ◦ u−1 − H ◦ · · · ◦ u ◦ V  < ε 2−j < ε, s 0 0 j j−1 j=1

j=1

where u0 = id for the uniformity if notations. ! To prove (3), fix p0 = (r 0 , θ 0 ) ∈ R(A), and consider the flat torus Tp0 passing through p0 . Then the invariant torus of H0 ◦ U −1 passing through P 0 = U (p0 ) has the form U (Tp0 ) = u1 ◦ · · · ◦ uN (Tp0 ). !p0 , due It lies on the invariant surface {H02 ◦ U −1 (p) = const}, which is ε-close to E to (2). Since V is identity on the margin sets, we just have to show that U (Tp0 ) intersects the two margin sets. We assume without loss of generality that θ1 (p0 ) = 0. Given θ ∈ {0} × T we define (r 1 , θ 1 ) = uN (r 0 , θ 0 ), (r 2 , θ 2 ) = uN −1 (r 1 , θ 1 ), and in general (3.9)

(r j+1 , θ j+1 ) = uN −j (r j , θ j ) = uN −j ◦ . . . uN (r 0 , θ 0 ), j

j

The lower index indicates the component: (r , θ ) =

j = 0, . . . N − 1.

(r1j , r2j , θ1j , θ2j ).

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So, (3) follows if we prove the following two claims. Claim 1. There exists θˆ ∈ {0} × T and θˇ ∈ {0} × T such that ( ' (3.10) cos(2π θˆj+1 , qN −j ) > 1/2 for all j = 0, . . . N − 1 ' ( cos(2π θˆj+1 , qN −j ) < −1/2 for all j = 0, . . . N − 1 (3.11) ˆ and U (r, θ) ˇ lie close to the two different ends of E !p0 ∩ R(A) Claim 2. U (r, θ) Proof of Claim 1. We will actually restrict ourselves to an interval K(p0 ) in Tp0 and show that even U (K(p0 )) intersects two margin sets. We do so in order to use one dimensional calculus. So, assume without loss of generality that θ1 (p0 ) = 0 and let K(p0 ) := {r 0 } × {0} × T be an interval in θ2 -direction passing through p0 . ˆ 0 and p0 × L ˇ 0, We shall present two subsets of K(p0 ), that we denote by p0 × L 0 0 ˆ ˇ such that U (p × L0 ) intersects {r1 ≥ A} ∪ {r2 ≥ A}, and U (p × L0 ) intersects {r1 ≤ A−1 } ∪ {r2 ≤ A−1 }. j+1 j Recall the notations  j+1 from (3.9) By Sublemma 8, if r ∈ R(A), we have: r1 = j j r1 (1 + br2 cos(2π θ , qN −j )) for all j = 0, . . . N − 1. The simplified idea is the following. Imagine that θ j+1 = θ 0 for all j. Since; by construction, the frequencies |qj | grow very fast with j, there are many points   where cos(2π θ 0 , qj )) > 1/2 for all j ≤ N . This implies that r1j+1 > r1j + br1j r2j /2 for all j = 0, . . . N − 1, and we can prove that r1N > A. In reality though, θ j+1 can be different from θ 0 (but close), and one should be more careful. We want to describe the set of points θˆ0 ∈ {0} × T such that for the images of (r 0 , θˆ0 ) we have: ( ' cos(2π θˆj+1 , qN −j ) > 1/2 for all j = 0, . . . N − 1. We claim that the set

' ( Jˆj+1 = {θ 0 ∈ {0} × T | cos(2π θˆj+1 , qN −j ) > 1/2}

consists of |qN −j | disjoint intervals of size at least C/(10|qN −j |) whose midpoints are at most C/|qN −j | distant from the nearest neighbour, where C is a constant only depending on Ω = −ω1 /ω2 . Since |qj | are assumed to grow very fast with j, ˆ 0 := ∩N Jˆj is nonempty, which proves Claim 1. the above estimates imply that L j=1 To prove the above estimate, notice that it becomes evident if we replace θ j+1 in the definition of Jˆj+1 by θ 0 . In this simplified case, the desired intervals are of size C/(3|qN −j |), and the midpoints are C/|qN −j | distant. By Sublemma 8, |θ 1 − θ 0 |0 < 10−3 |qN |−1 , |θ 2 − θ 1 |0 < 10−3 |qN −1 |−1 , and in general |θ l+1 − θ l |0 < 10−3 |qN −l |−1 . Then   1 1 + ··· + , (3.12) |θ j+1 − θ 0 |0 ≤ 10−3 |qN | |qN −j |     and | θ j+1 , qN −j − θ 0 , qN −j | ≤ 10−3 ( |q1N | + · · · + |qN1−j | )|qN −j | ≤ 10−2 . Hence, ˆ 0 := ∩N Jˆj is nonempty. L j=1

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1 ˇ By the same argument, there exists a nonempty  j+1 subset L0 of {0} × T such 0 ˇ0 0 ˇ 0 we have: cos(2π θˇ , qN −j ) < −1/2 for all j = that for (r , θ ) with θˇ ∈ L 0, . . . N − 1.

Proof of Claim 2. We will treat separately the cases ω1 ω2 < 0 and ω1 ω2 > 0. ˆ lies in {r1 ≥ A} ∪ {r2 ≥ A}, while Case 1 : ω1 ω2 < 0. We will show that U (r, θ) ˇ U (r, θ) lies in {r1 ≤ 1/A} ∪ {r2 ≤ 1/A}. For every j = 1, . . . N , we have the following recursive estimate: ' (

r1j−1 )g(ˆ r2j−1 ) cos 2π qj , θˆj rˆ1j = rˆ1j−1 1 + bˆ r2j−1 g(ˆ ≥ rˆ1j−1 . In the same way, rˆ2j ≥ rˆ2j−1 (here we use that q2 /q1 > 0). Hence, if for some i ∈ {1, 2} and some j < N , rˆij ≥ A, then rˆiN ≥ A. Assume on the contrary that for all j < N , rˆij < A for i = 1, 2. Then, since g ≡ 1 inside R(A), we have, using (3.7) that rˆ1N ≥ r10

N −1 

( ' (1 + bˆ r2j cos(2π qj , θˆj )) ≥ A−1 (1 + b(4A)−1 )N ≥ A.

j=0

We used the fact that R(2A) is invariant, so rˆij ≥ 1/(2A) for i = 1, 2 and j = 1, . . . N . ˆ lies in {r1 ≥ A} ∪ {r2 ≥ A}. In conclusion, U (r, θ) ˇ Now, for θ the argument is similar : For every j = 1, . . . N 

 rˇ1j = rˇ1j−1 1 + bˇ r2j−1 g(ˇ ≤ rˇ1j−1 . r1j−1 )g(ˇ r2j−1 ) cos 2π qj , θˇj In the same way, rˇ2j ≤ rˇ2j−1 (here we use that q2 /q1 > 0). Hence, if for some i ∈ {1, 2} and some j < N , rˇij ≤ 1/A, then rˇiN ≤ 1/A. Assume to the contrary that for all j < N , rˇij > 1/A for i = 1, 2. Then, since g ≡ 1 inside R(A) we have, using (3.7) that rˇ1N ≤ r10

N −1 



 1 + bˇ r2j cos 2π qj , θˇj < A(1 − b(4A)−1 )N < A−1 .

j=0

ˇ lies in {r1 ≤ 1/A} ∪ {r2 ≤ 1/A}. In conclusion, U (r, θ) ˆ lies in {r1 ≥ A} ∪ {r2 ≤ 1/A} Case 2 : ω1 ω2 > 0. We will show that U (r, θ) ˇ while U (r, θ) lies in {r1 ≤ 1/A} ∪ {r2 ≥ A}. Contrary to case 1, here we have that rˆ1j is increasing while rˆ2j is decreasing (this because q1 /q2 < 0). If for some j < N , rˆ2j ≤ 1/A we finish. If not, and if for some j < N , rˆ1j ≥ A we also finish. Hence we assume that for all j < N , rˆ1j < A and rˆ2j > 1/A. In such a situation and due to the fact that g ≡ 1 in R(A) we have that N −1 ( '  rˆ1N ≥ r10 (1 + bˆ r2j cos(2π qj , θˆj )) ≥ A−1 (1 + b(4A)−1 )N ≥ A. j=0

ˇ we just exchange the roles of the coordinates r1 and r2 . To treat U (r, θ)

A

−1

! We proceed now to the proof of (4). Fix p ∈ / R(A). Consider the case r1 (p) < ! , the other cases being similar. Let p¯ be such that Ep = Ep¯ and p¯ ∈ R(A).

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!p , and the torus Tp lies on one side of it, namely The torus Tp¯ separates the space E the one intersecting the set {r1 < A−1 }. Therefore, the torus U (Tp¯) separates the !p ), having U (Tp ) on one side of it. Now, U (Tp¯) intersects two margin space U (E !p is close to U (E !p ), it has to sets, so U (Tp ) has to intersect at least one. Since E −1 intersect {r1 < A }. !p ∩ We have shown that U (T (p)) is ε–close to (one or two components of) E M (A). Finally, notice that V ◦ U (T (p)) has the same property since V =Id outside ! R(A). This finishes the proof of Lemma 7.  4. Proofs for the case d = 4 The proof of Theorem 2 follows from the proposition below that encloses one step of the successive conjugation scheme. Fix In = [an , bn ] for some n. Given a (large) A, we define In (A) := [an + A−1 , bn − A−1 ], I!n4 (A) := [A−1 , A]3 × In (A) × T4 . Define the margins set M4 (A) =

3

  {ri > A} {ri < A−1 } . i=1

Proposition 3. Let n ∈ N, > 0, s ∈ N and V ∈ U4 that is identity outside 4 ! In (A0 ) for some A0 > 0, be given. Then for any A > A0 there exist U ∈ U4 , T > 0 and i ∈ {1, 2, 3} with H = H0 ◦ U −1 ◦ V −1 satisfying the following properties: (1) U = Id in the complement of I!n4 (2A), (2) H − H0 ◦ V −1 s < , T (3) For any P ∈ I!n4 (A) we have that OH (P ) intersects the set M4 (A); (4) Moreover, if ω0,1 ω0,2 < 0 and if n = 3m for some m ∈ Z, then there is a T (p ) intersects both ∩3i=1 {ri < 2A−1 } and point p ∈ I!n4 (A) such that OH {r1 > A} ∪ {r2 > A}. Proof of Theorem 2. For n, A, T ∈ N∗ , let $ # T (P ) intersects M4 (A) D(n, A, T ) := H ∈ H¯4 | ∀P ∈ I!n4 (A), OH It is clear that  D(n, A, T ) are open subsets of H¯4 in any C s topology. Proposition 3 implies that T ∈N∗ D(n, A, T ) is dense in H¯4 in any C s topology. Hence the ¯ ⊂ D is a dense Gδ set (in any C s topology) following set D % %  ¯= D D(n, A, T ). A∈N∗ n∈N∗ T ∈N∗

In case ω0,1 ω0,2 < 0 we add to the definition of D(n, A, T ), when n = 3m for T (p ) intersects some m ∈ Z, the existence of a point p ∈ I!n4 (A) that satisfies OH 3 −1 both ∩i=1 {ri < 2A } and {r1 > A} ∪ {r2 > A}. The second part of Theorem 2 then follows from the fact that if for some fixed A > 0, we consider H ∈ D(n, A, T ) for n sufficiently large, then the orbit of the corresponding point p ∈ I!n4 (A) has its r4 coordinate always smaller than A−1 . The orbit of p hence intersects both ∩4i=1 {ri < 2A−1 } and {r1 > A} ∪ {r2 > A}. 

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Proof of Proposition 3. Since V equals identity near the axes, we can, by increasing A, assume without loss of generality that V = Id. Assume that I = [a, b] = I3n , the other cases being exactly similar. In this case f1 (r4 ) ≡ f¯1 and f2 (r4 ) ≡ f¯2 for r4 ∈ I. Moreover, for ω1 = f¯1 + ω0,1 and ω2 = f¯2 + ω0,2 , the vector (ω1 , ω2 ) is Liouville. We will hence be able to use the two-dimensional Liouville construction of Proposition 2. / I(2A) and a(ξ) = 1 if ξ ∈ I(A) = Let a ∈ C ∞ (R) be such that a(ξ) = 0 if ξ ∈ [a + A−1 , b − A−1 ], and as ≤ C(s, A) where C(s, A) is a constant that depends on s and A (recall that A−1 is assumed to be small compared to the size of I = I3n ). We construct the map U as follows. First, U is independent of (r3 , θ3 ) (i.e., U (r, θ) = U (r1 , r2 , r4 , θ1 , θ2 , θ4 )). Second, r4 acts as a parameter, and for each r4 ∈ I(A) the map U equals the map provided by Lemma 2 (call the latter U2 , where 2 indicates the number of degrees of freedom). More precisely, in the proof of Proposition 2, U2 is constructed as a composition of a certain number of symplectic maps uj2 : R2 × T2 , (r, θ) → (R, Θ), each one given by a generating function of the form S2j (r1 , r2 , Θ1 , Θ2 ) = (r1 , r2 ), (Θ1 , Θ2 ) + g2j (r1 , r2 , Θ1 , Θ2 ), where g2j is some smooth function equal to zero in the neighborhood of the axes. We extend S2j and uj2 to S j and uj defined for (r, θ) ∈ R4 × T4 by letting S j (r, Θ) = r, Θ + a(r4 )g2j (r1 , r2 , Θ1 , Θ2 ). Since a ≡ 0 on I(2A)c , we get (1) of Proposition 3 from (1) of Proposition 2. To check (2), observe that a(r4 ) appears just like a parameter in the construction of S j from that of S2j . Thus, since as ≤ C(s, A) we get (2) by just taking sufficiently small in Proposition 2. Now for P ∈ I!n4 (A) we have that a(r4 (P )) = 1 and since r4 is invariant under the flow we get that the dynamics in (r1 , r2 , θ1 , θ2 ) coordinates is exactly that of Proposition 2, hence (3) holds. To prove (4) of the Proposition, choose p ∈ I!n4 (A) with r4 , θ4 , θ3 arbitrary, with r3 < 2A−1 and with the projection of p on the (r1 , r2 , θ1 , θ2 ) coordinates being the point p2 that satisfies (4) of Proposition 2. Clearly, (4) of Proposition 3  holds for p . 5. Proofs for the case d = 3. Fix In = [an , an+1 ] for some n. Given a (large) A, we define In (A) := [an + A−1 , an+1 − A−1 ],

I!n3 (A) := R(A) × In (A) × T3 .

Define the margins set M3 (A) =

3  i=1

{ri > A} ∪ {r1 < A−1 } ∪ {r2 < A−1 }∪ 

{r3 ∈ In  In (A)}.

n∈Z

The following proposition is an analog of Proposition 3.

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Proposition 4. Let n ∈ N, > 0, s ∈ N and V ∈ U3 that is identity outside I!n3 (A0 ) for some A0 > 0, be given. Then for any A > A0 there exist U ∈ U3 , T > 0 and (i1 , i2 ) ∈ {1, 2, 3} distinct, with the following properties: (1) U = Id in the complement of I!n3 (2A), (2) H03 ◦ U −1 ◦ V −1 − H03 ◦ V −1 s < , T (3) For any P ∈ I!n3 (A) we have that OH (P ) intersects M3 (A). (4) Moreover, if ω0,1 ω0,2 < 0 and if n = 3m for some m ∈ Z, then there is a T (p ) intersects both ∩2i=1 {ri < 2A−1 } and point p ∈ I!n3 (A) such that OH {r1 > A} ∪ {r2 > A}. Proof of Theorem 3. The proof follows exactly the same lines as the proof of Theorem 2. For n, A, T ∈ N∗ , we let # $ T D(n, A, T ) := H ∈ H¯0 | ∀P ∈ I!n3 (A), OH (P ) intersects M3 (A) . ¯ ⊂ D given by and we see that D % ¯= D

%



D(n, A, T ).

A∈N∗ n∈N∗ T ∈N∗

is a dense Gδ set (in any C s topology) in H¯0 . In case ω0,1 ω0,2 < 0 we add to the definition of D(n, A, T ) the existence of a point p satisfying conclusion (4) of Proposition 4. The second part of Theorem 2 then follows from the fact that if for some fixed A > 0, we consider H ∈ D(n, A, T ) for n sufficiently large, then the orbit of the corresponding point p ∈ I!n3 (A) has its r3 coordinate always smaller than A−1 since it lies in In . The orbit of p hence  intersects both ∩3i=1 {ri < 2A−1 } and {r1 > A} ∪ {r2 > A}. Proof of Proposition 4. The proof is similar to that of Proposition 3 (which relies on Lemma 2). We shall only describe the modifications that have to be done in order to get the conjugacy U for d = 3. When r3 ∈ I3n , then (ω1 , ω2 ) is a constant Liouville vector and the construction of Proposition 2 is carried out in the (r1 , r2 , θ1 , θ2 )-space, with r3 acting as a parameter exactly as r4 acted as a parameter in the proof of Proposition 3. The situation for r3 ∈ In for n = 3m + 1or n = 3m + 2 is slightly different since r3 is not invariant anymore. Suppose that n = 3m + 1, the other case being similar. For r3 ∈ In we have that the vector (ω1 , ω3 ) is constant and Liouville. The idea is that since r3 will diffuse but remain inside In , one can still perform the construction of Proposition 2 with r3 playing in the same time the role of a parameter and that of a diffusing action coordinate. The diffusion in the r3 variable is thus limited to accumulating the set {r3 ∈ In  In (A)}. All the rest of the proof is similar to the proof of Proposition 3.  References [AK] [D]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms (Russian), Trudy Moskov. Mat. Obˇsˇ c. 23 (1970), 3–36. MR0370662 ´ R. Douady, Stabilit´ e ou instabilit´ e des points fixes elliptiques (French), Ann. Sci. Ecole Norm. Sup. (4) 21 (1988), no. 1, 1–46. MR944100

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[EFK1] L. H. Eliasson, B. Fayad, and R. Krikorian, KAM-tori near an analytic elliptic fixed point, Regul. Chaotic Dyn. 18 (2013), no. 6, 801–831, DOI 10.1134/S1560354713060154. MR3146593 [EFK2] L. H. Eliasson, B. Fayad, and R. Krikorian, Around the stability of KAM tori, Duke Math. J. 164 (2015), no. 9, 1733–1775, DOI 10.1215/00127094-3120060. MR3357183 [FH] A. Fathi and M. R. Herman, Existence de diff´ eomorphismes minimaux (French), Dynamical systems, Vol. I—Warsaw, Soc. Math. France, Paris, 1977, pp. 37–59. Ast´ erisque, No. 49. MR0482843 [FK] B. Fayad and R. Krikorian, Herman’s last geometric theorem (English, with English and ´ Norm. Sup´ French summaries), Ann. Sci. Ec. er. (4) 42 (2009), no. 2, 193–219. MR2518076 [H1] M. Herman, Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. Extra Vol. II (1998), 797–808 (electronic). MR1648127 [H2] M. Herman, Courbes invariantes sans propri´ et´ e de torsion 1 Archives Herman, to appear. ¨ [R] H. R¨ ussmann, Uber die Normalform analytischer Hamiltonscher Differentialgleichungen in der N¨ ahe einer Gleichgewichtsl¨ osung (German), Math. Ann. 169 (1967), 55–72. MR0213679 [S] M. B. Sevryuk, KAM-stable Hamiltonians, J. Dynam. Control Systems 1 (1995), no. 3, 351–366, DOI 10.1007/BF02269374. MR1354540 IMJ-PRG CNRS, France E-mail address: [email protected] ¨ r Matematik, KTH, 10044 Stockholm, Sweden Institute fo E-mail address: [email protected]

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13914

Nonlocally maximal and premaximal hyperbolic sets T. Fisher, T. Petty, and S. Tikhomirov Abstract. We prove that for any closed manifold of dimension 3 or greater there is an open set of smooth flows that have a hyperbolic set that is not contained in a locally maximal one. Additionally, we show that the stabilization of the shadowing closure of a hyperbolic set is an intrinsic property for premaximality. Lastly, we review some results due to Anosov that concern premaximality.

1. Introduction Since the 1960s the study of hyperbolic sets has been a cornerstone in the field of dynamical systems. These sets are remarkable not only in their complexity, but also in the fact that they persist under perturbations. Additionally, for a point in a hyperbolic set the derivative of the map at this point gives information on the local dynamics for the original nonlinear map. As a reminder, for a diffeomorphism f : M → M , a compact invariant set Λ is hyperbolic for f if TΛ M = Es ⊕ Eu is a Df -invariant splitting such that Es is uniformly contracted and Eu is uniformly expanded by Df . Anosov was one of the pioneers in studying hyperbolic sets. Indeed, if the entire manifold is a hyperbolic set for a diffeomorphism, then the diffeomorphism is called Anosov. This is one of the best understood classes of hyperbolic sets. Another important class of hyperbolic sets are those that are locally maximal. For a compact metric space X and a continuous homeomorphism T : X → X a set K ⊂ X is locally maximal if there exists a neighborhood U of K such that % T n (U ). K= n∈Z

Hence, such sets are the maximal invariant set within U . Locally maximal sets were defined more or less simultaneously by Anosov and Conley (who called these sets isolated) in the 1960s. Alexseev proved in [1] that a 2010 Mathematics Subject Classification. Primary 37D20, 37D05, 37C05. Key words and phrases. Hyperbolic sets, hyperbolic flow, premaximal, locally maximal, isolated. The authors would like to thank the Sixth International Conference on Differential and Functional Differential Equations during which much of the paper was prepared. T.F. is supported by Simons Foundation grant # 239708. S. T. is partially supported by Chebyshev Laboratory under Russian Federation Government grant 11.G34.31.0026, JSC “Gazprom neft”, Saint Petersburg State University research grant 6.38.223.2014, Russian Foundation of Basic Research 15-01-03797a and German-Russian Interdisciplinary Science Center (G-RISC). c 2017 American Mathematical Society

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shift space is locally maximal if and only if it is a shift of finite type. Furthermore, he proved that given any shift space Σ and a neighborhood U of Σ there exists a locally maximal shift Σ containing Σ and contained in U . A question that was posed in the 1960s by Anosov, Alexseev, and others (that is stated for instance in [14, p. 272]), is the following: Question 1.1. If Λ is a hyperbolic set and U is a neighborhood of Λ, then is ˜ such that Λ ⊂ Λ ˜ ⊂ U? there a locally maximal hyperbolic set Λ As stated by Anosov in [3] it was hoped at the time that the answer would be in the affirmative. One reason this was hoped for is that locally maximal hyperbolic sets are more easily classified. Indeed, a hyperbolic set is known to be locally maximal if and only if it has a local product structure (defined in Section 2). A standard assumption used in characterizing topological and/or measure theoretic properties of hyperbolic sets is that the set is locally maximal. For instance, Smale’s Spectral Decomposition Theorem, see for instance [14, p. 575], is valid for locally maximal hyperbolic sets. In fact, quite frequently even when this is not specifically stated in a theorem one finds in the proof that this assumption is required for the result to hold. 1.1. Nonlocally maximal hyperbolic sets. It was shown by Crovisier in [9] that there is a hyperbolic set on the 4-torus that is never included in a locally maximal set. Later, in [13] it was shown that any compact boundaryless manifold with dimension greater than or equal to 2 has a C r open set of diffeomorphisms where 1 ≤ r ≤ ∞ such that each diffeomorphism in the open set contains a hyperbolic set that is not included in a locally maximal one. Our first result is an extension of the results in [13] to hyperbolic flows. As a reminder, for a smooth flow φ : R × M → M , a compact φ-invariant set Λ is hyperbolic for φ if TΛ M = Es ⊕ Ec ⊕ Eu is a flow invariant splitting such that Es is uniformly contracted, Ec is the flow direction, and Eu is uniformly expanded. A set Λ is locally maximal for the flow φ if there is an open set U containing Λ such that Λ = ∩t∈R φt (U ). Theorem 1.2. Let M be a compact, boundaryless C r manifold for 1 ≤ r ≤ ∞ with dim M ≥ 3 and X k (M ) be the set of C k flows on M where 1 ≤ k ≤ r. Then there exists a C k open set of flows on M such that each flow contains a hyperbolic set not contained in a locally maximal one. We notice that if dim M ≤ 2 then every hyperbolic set for a smooth flow is a finite union of hyperbolic closed trajectories and hence it is locally maximal. Also, by suspending the construction in [13] one obtains a smooth flow. To see that this provides an example of a hyperbolic set that cannot be included in a locally maximal one and that this construction can be embedded on on any manifold of dimension 3 or larger is not too complicated, but we include this for completeness. As in [13, Theorem 1.5] we can show the following result. Theorem 1.3. Let Λ be a hyperbolic set for a flow and U be a neighborhood of Λ, then there exists a hyperbolic set Λ with a Markov partition for the flow such that Λ ⊂ Λ ⊂ U .

HYPERBOLIC SETS

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The proof is very similar to that in [13]. Indeed, the necessary theorems used in [13] to prove the similar results for maps hold for hyperbolic sets for flows, and hence the proof is left to the reader. The hyperbolic set constructed in Theorem 1.2 above need not transitive under the flow. However, as in [13, Theorem 1.6] one can construct a flow in higher dimensions with a transitive hyperbolic set that is not contained in a locally maximal one. 1.2. Premaximality. In this paper we also examine conditions under which a hyperbolic set, Λ, is included in a locally maximal hyperbolic set within an arbitrarily small neighborhood of Λ. Following the terminology introduced by Anosov in [3] we define a hyperbolic set Λ for a diffeomorphism to be premaximal if for any open set U containing Λ there ˜ such that Λ ⊂ Λ ˜ ⊂ U . In [3] Anosov proves is a locally maximal hyperbolic set Λ that any zero-dimensional hyperbolic set for a diffeomorphism is premaximal, and in [2] Anosov proves there is an intrinsic property for premaximal hyperbolic sets for diffeomorphisms. Moreover the following holds. Theorem 1.4. [6] Let f : M → M and f  : M  → M  be diffeomorphisms, Λ a hyperbolic set for f , Λ a hyperbolic set for f  , and h : Λ → Λ a homeomorphism such that h ◦ f = f  ◦ h. If U is a neighborhood of Λ and U  is a neighborhood of Λ , then there exists neighborhoods V ⊂ U of Λ and V  ⊂ U  of Λ and continuous injective equivariant maps h1 : If (U ) → M  and h2 : If  (U  ) → M such that h1 |Λ = h, and h1 (If (V )) ⊂ If  (U  ),

h2 (If  (V  )) ⊂ If (U ), f ◦ h2 |If  (V  ) = h2 ◦ f  |If  (V  ) ,

h1 ◦ f |If (V ) = g ◦ h1 |If (V ) , h2 ◦ h1 |If (V ) = id, and

h1 ◦ h2 |If  (V  ) = id.

The above theorem shows that f |Λ defines the set of trajectories that lie in a sufficiently small neighborhood of Λ. However, in [6] the specific intrinsic property for premaximality is not stated. We extend results of [6] to the case of flows and prove the premaximality is an intrinsic property for hyperbolic sets for flows. Let X and X  be vector fields on smooth compact Riemannian manifolds M and M  respectively. Denote by φ and φ flows generated by them. An increasing homeomorphism of the real line α : R → R is called a reparametrization. Let Λ and Λ be hyperbolic sets for X and X  respectively. We say that Λ and Λ are topologically equivalent if there exists a homeomorphism h : Λ → Λ and a continuous map α : M × R → R such that h ◦ φt (x) = φ (α(x, t), h(x)),

x ∈ Λ, t ∈ R

where α(x, ·) is a reparametrization for each x ∈ Λ. In this case there exists a continuous map β : M  × R → R, such that β(h(x), α(x, t)) = t, α(h

−1









x ∈ Λ, t ∈ R

(x ), β(x , t )) = t , and

x ∈ Λ , t ∈ R.

Theorem 1.5. Let Λ and Λ be hyperbolic sets for vector fields X and X  respectively. Assume that Λ and Λ are topologically equivalent. Then Λ is premaximal if and only if Λ is premaximal.

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Below we provide an equivalent condition to premaximality for diffeomorphisms and flows. Before stating the result we define some important terms involving shadowing. Let Φ = φ(t, x) be a dynamical system where t can be taken to be discrete or continuous. If t is discrete, we assume the dynamical system is generated by a diffeomorphism of a compact manifold to itself. If t is continuous, we assume that the dynamical system is generated by a smooth vector field on a compact manifold. Let a > 0 be an expansivity constant for some neighborhood of a hyperbolic set Λ and let δ0 > 0 be such that any δ0 -pseudo orbit in Λ can be a/2-shadowed by an exact trajectory (definitions are given in the next section). Note that due to expansivity the shadowing trajectory is unique (for the case of a flow this is true up to a reparametrization). For Λ a hyperbolic set for a diffeomorphism and δ ∈ (0, δ0 ) the shadowing closure (or δ-shadowing closure) of Λ is sh(Λ, δ) = {y ∈ M : y shadows a δ-pseudo orbit in Λ}. For a fixed δ > 0 we can construct a sequence of shadowing closures Λ0 , Λ1 , . . . , where Λ0 = Λ and Λj = sh(Λj−1 , δ) for j ∈ N. We say a shadowing sequence stabilizes if Λj = Λj+1 for all j ≥ N where N ∈ N. Theorem 1.6. For a hyperbolic set Λ of a dynamical system Φ the following statements are equivalent (1) Λ is premaximal; (2) for any neighborhood U of Λ the shadowing closure stabilizes inside U for some δ > 0. Note that due to Theorem 1.4 and Theorem 1.5 the second property in Theorem 1.6 for diffeomorphisms is intrinsic. The paper proceeds as follows. In Section 2 we review relevant background on hyperbolicity and flows. In Section 3 we prove Theorem 1.2. In Section 4 we review the results of Anosov in [2–6] and prove Theorems 1.5 and 1.6. 2. Background 2.1. Hyperbolic sets for flows. We first review properties of hyperbolic sets for flows. Definition 2.1. Let X be a metric space and φ a continuous flow on X. Then for x ∈ X we define the stable set W s (x) := {y ∈ X : lim d(φt (x), φt (y)) = 0}. t→∞

Further, for ε > 0 the ε-stable set is Wεs (x) := {y ∈ W s (x) : d(φt (x), φt (y)) ≤ ε for all t ≥ 0}. Note that the unstable sets W u (x) and Wεu (x) are defined identically under the flow φ−t . Furthermore, we define the center-stable set  W cs (x) := {φt (W s (x))|t∈R } = W s (y). y∈φt (x) t∈R

The center-unstable set of x is defined to be the center-stable set of x under φ−t . s to mean Wεs for sufficiently small ε (depending We will also use the notation Wloc u on the context) and Wloc similarly to mean Wεu for small ε.

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Let X be a manifold and φ be a C r flow on X. If Λ is a hyperbolic set for φ and p ∈ Λ, then W s (p) is a C r immersed submanifold and is an immersed copy of Rk where k = dim Es (x). Similar statements hold for the unstable sets, center-stable sets, and center-unstable sets. Also note that the stable and unstable manifolds vary continuously on the point p. Definition 2.2. For a metric space X and a flow φ, a set Γ ⊂ X is said to have a local product structure if for all ε > 0 sufficiently small there exists a δ > 0 such that given x, y ∈ Γ with d(x, y) < δ we have, for some real |t| < ε, a unique point S(x, y) := b ∈ Wεu (φt (x)) ∩ Wεs (y) contained in Γ. Remark 2.3. Note that for any hyperbolic set Λ there always exist constants δ and ε sufficiently small such that x, y ∈ Λ and d(x, y) < δ implies S(x, y) = Wεu (φt (x)) ∩ Wεs (y) is a unique point in the manifold, but may not be in Λ. The following lemma is also critical to the paper. Note that this lemma is almost always stated and proved for maps, but is in fact true for flows as well (see [8] and [15]). Lemma 2.4. A hyperbolic set Γ has a local product structure if and only if it is locally maximal. We also need the notion of the shadowing property. For δ > 0 a map g : R → M is an δ-pseudo orbit if the following holds d(g(t + τ ), φτ (g(t))) < δ,

t ∈ R, |τ | < 1.

A δ-pseudo orbit g is ε-shadowed by a point x0 if there exists a reparametrization α : R → R satisfying d(g(t), φα(t) (x0 )) < ε, and     α(t1 ) − α(t2 )  − 1 < ε for t1 = t2 .  t1 − t2 A vector field X is expansive on a compact metric space W if there exist constants a, τ0 > 0 such that if x1 , x2 ∈ W and there exists a reparametrization α : R → R such that the following inequalities hold d(φ(α(t), x1 ), φ(t, x2 )) < a,

t ∈ R,

then x2 = φ(τ, x1 ), where τ ∈ (−τ0 , τ0 ). Theorem 2.5. Let Λ be a hyperbolic set for a flow φ on a compact manifold M . Then the following hold: • there exists a neighborhood W of Λ such that φ is expansive on % Iφ (W ) := φt (W ); t∈R

and • there exists a neighborhood U (Λ) ⊂ W such that for any ε > 0 there exists δ > 0 such that any δ-pseudo orbit g ⊂ U can be ε-shadowed by some point x0 ∈ M . Definition 2.6. For φ a flow on a compact metric space X a nonempty subset A of X is called an attractor if it satisfies the following three conditions.

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(i) A is forward-invariant under φ; i.e., x ∈ A implies φt (x) ∈ A for all t > 0. (ii) There exists a neighborhood of A, called the basin of attraction of A and denoted B(A), which consists of all points that tend towards A under φt as t → ∞. In other words, B(A) consists of all points x such that for any open neighborhood N of A, there exists T > 0 such that φt (x) ∈ N for all t > T . (iii) No proper subset of A satisfies conditions (i) and (ii). When an attractor Λ is (uniformly) hyperbolic we know that • periodic points are dense in Λ, • x ∈ Λ implies W cu (x) ⊂ Λ, and  • for any periodic point x ∈ Λ we know y∈O(x) W cs (y) is dense in B(Λ). We will need the following technical result, known in the literature as the Inclination Lemma, or λ-lemma. The statement can be found in [7]. Note that the statement for hyperbolic periodic points would be similar. Lemma 2.7 (Inclination Lemma). Let p ∈ M be a hyperbolic fixed point for a s u C r flow φ, for r ≥ 1, with local stable and unstable manifolds Wloc (p) and Wloc (p), u respectively. Fix an embedded disk B in Wloc (p) which is a neighborhood of p in u (p), and fix a neighborhood V of this disk in M. Let D be a transverse disk to Wloc s (p) at a point z such that D and B have the same dimension. Write Dt for Wloc the connected component of φt (D) ∩ V which contains φt (z), for t ≥ 0. Then, given ε > 0 there exists T > 0 such that for all t > T the disk Dt is ε-close to B in the C r -topology. 2.2. Hyperbolic sets of diffeomorphisms. Since the statement of Theorem 1.6 refers to diffeomorphisms as well as smooth flows we now review some definitions for discrete dynamical systems. For f : X → X a homeomorphism of a metric space and δ > 0 an δ-pseudo orbit is a sequence {xj }m l where • l ∈ {−∞} ∪ Z, m ∈ Z ∪ {∞}, l < m, and • d(f (xj ), xj+1 ) < δ for all j ∈ [l, m]. For a δ-pseudo orbit {xj }m l we say this sequence is ε-shadowed by a point x ∈ X if d(f j (x), xj ) < ε for j ∈ [l, m]. We say that a homeomorphism f of a compact metric space W is expansive if there exists a constant a > 0 such that if % f n (W ) x1 , x2 ∈ If (W ) := n∈Z

and d(f n (x1 ), f n (x2 )) < a,

n ∈ R,

then x2 = x1 . Theorem 2.8. (Shadowing Lemma) Let Λ be a hyperbolic set for f : M → M a diffeomorphism. Then there exists neighborhood U (Λ) such that for all ε > 0 there exists an δ > 0 such that if {xj }∞ −∞ ⊂ U is an δ-pseudo orbit, then there exists x ∈ M that ε-shadows {xj }∞ −∞ . Let f : M → M be a diffeomorphism and Λ be a hyperbolic set for f . For ε > 0 sufficiently small and x ∈ Λ the local stable and unstable manifolds and stable and unstable manifolds are defined similar to the case for flows and for a

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C r diffeomorphism f the stable and unstable manifolds of a hyperbolic set are C r injectively immersed submanifolds. For Λ a hyperbolic set we know that if ε is sufficiently small and x, y ∈ Λ, then Wεs (x) ∩ Wεu (y) consists of at most one point. For such an ε > 0 define Dε = {(x, y) ∈ Λ × Λ | Wεs (x) ∩ Wεu (y) ∈ Λ} and [·, ·] : Dε → Λ so that [x, y] = Wεs (x) ∩ Wεu (y). We will also need openness of hyperbolicity. Lemma 2.9. Let Λ ⊂ M be a hyperbolic set of the diffeomorphism f : U → M . Then for any open neighborhood V ⊂ U of Λ and every δ > 0 there exists ε > 0 such that if f  : U → M and dC 1 (f |V , f  ) < ε there is a hyperbolic set Λ = f  (Λ ) ⊂ V for f  and a homeomorphism h : Λ → Λ with dC 0 (Id, h) + dC 0 (Id, h−1 ) < δ such that h ◦ f  |Λ = f |Λ ◦ h. Moreover, h is unique when δ is sufficiently small. 2.3. Normal hyperbolicity. For embedding constructions into higher dimensions, we will need the notion of normal hyperbolicity. A normally hyperbolic invariant manifold (NHIM) is a generalization of a hyperbolic fixed point and a hyperbolic set. Fenichel proved that NHIMs and their stable and unstable manifolds are persistent under perturbation [11], [12]. We define NHIMs for maps, but the definition for flows is similar (and more technical). Definition 2.10. Let M be a compact smooth manifold and f : M → M a diffeomorphism. Then an f -invariant submanifold Λ of M is said to be a normally hyperbolic invariant manifold if there exist m ∈ N such that the mapping f m satisfies the following property. There exists a continuous invariant bundle TΛ M = E s ⊕ T Λ ⊕ E u (x), and continuous positive functions ν, νˆ, γ, γˆ : M → R such that ν, νˆ < 1,

ν < γ < γˆ < νˆ−1

and for all x ∈ Λ, v ∈ Tx M , |v| = 1 |Df m (x)v| ≤ ν(x), v ∈ E s (x); γ(x) ≤ |Df m (x)v| ≤ γˆ (x), v ∈ T Λ; |Df m (x)v| ≥ νˆ−1 (x), v ∈ E u (x). Adapting the above for flows gives us an important result ([10, p. 215]) which says that if a C r vector field Y in some C 1 neighborhood of our original vector field X (equated with a flow φ, under which Λ is invariant) there is a C r manifold ΛY invariant under Y and C r diffeomorphic to Λ. An immediate consequence of this is that the dynamics on ΛY under the vector field Y are a perturbation of the dynamics of Λ under X. 3. Nonlocally maximal sets for flows The foundation of the proof of Theorem 1.2 is the Plykin attractor, see for instance [14, p. 537-41] for a construction of the Plykin attractor. The first author used this map with some modifications to prove Theorem 1.3 in [13] on the existence of hyperbolic sets not included in locally maximal ones. To extend the results of [13] we show that the resulting suspension provides an example of a hyperbolic set for a flow not contained in a locally maximal hyperbolic set and that the construction can be embedded into any manifold with dimension greater than or equal to three.

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Proof of Theorem 1.2. We now show how to adapt the construction in [13] to the suspension flow. We sketch the main steps of the construction. For a more detailed proof see [16]. Let T1 be the solid 2-torus used in the construction described above. We first embed the Plykin attractor for the flow in a slightly larger solid 2-torus, T , so that the flow is the identity in a neighborhood of the boundary and then extend the flow to a 3-disk, D, so that it is the identity on D − T . This will allow us to embed the example into any arbitrary 3-manifold. We now modify the flow on T1 to incorporate the construction in [13]. Let Λa be the Plykin attractor and φ be the flow on D. Fix some p ∈ ∂T1 . Take a sufficiently small open neighborhood U of O(p), small enough to be disjoint from ∂T and Λa , and alter φ in U so that p is a hyperbolic saddle periodic point with W s (p) ⊂ U ∩ ∂T1 . Also, W cu (p) \ O(p) contains two components. One of which is contained in T − T1 and the other, denoted W ∗ (p) is contained strictly in the interior of T1 . Let q be a periodic point in Λa . Since W cs (q) = W s (Λa ) = int(T1 ) we know that given any point of W ∗ (p) that there must exist some point in W cs (q) arbitrarily close to it. Fix z ∈ W ∗ (p). Perturb the flow in a neighborhood of z so that z ∈ W cs (q)  W ∗ (p). This can be done since z is a wandering point for the flow. Here we will need two definitions. A hyperbolic set Λ for a C 1 flow has a heteroclinic tangency if there exist x, y ∈ Λ such that W s (x) ∩ W u (y) contains a point of tangency. A point of quadratic tangency for a C 2 flow is defined as a point of heteroclinic tangency where the curvature of the stable and unstable manifolds differs at the point of tangency. Now after a further perturbation to the flow as in [13] there exists a point w ∈ W u (z) and a point q  ∈ W u (q) such that W u (z) and W s (q  ) have a quadratic tangency at w. Let I be the segment of W u (p) from z to w, and let J be the segment of W u (q) from q to q  . The resulting flow will contain a hyperbolic set that cannot be contained in a locally maximal set as we show below. Figure 1 demonstrates a cross section of the constructed flow. Let Λ = Λa ∪ O(p) ∪ O(z). Standard arguments as in [13], adapted to flows, show that this is a hyperbolic set under φ. Now suppose Λ ⊂ Λ , where Λ is a locally maximal hyperbolic set. Modifying the proof in [13] we see from the product structure on Λ that this implies w ∈ Λ , a contradiction. Hence, Λ is not contained in a locally maximal hyperbolic set. We now show the construction is robust under perturbation. Since transversality is trivially open, and hyperbolicity is open by Lemma 2.9, it is sufficient to cu p) ∩ W s (˜ u) for some u ˜ ∈ Wloc (˜ q ). Let show that there remains a point w ˜ ∈ W u (˜ p˜ and q˜ be the continuations of p and q for the perturbed flow. By construction, cu (˜ q ) locally foliate the region, so there must the stable manifolds for all the x ∈ Wloc cu q ) and a point w ˜ ∈ W cs (˜ u) ∩ W u (˜ p) such that the oneexist a point u ˜ ∈ Wloc (˜ u dimensional path W (˜ p) remains tangent to the two-dimensional plane W cs (˜ u) at p)  Tw˜ W cs (˜ u). w. ˜ Specifically, we have Tw˜ W u (˜ Using normal hyperbolicity we can embed our example into any smooth manifold M of dimension greater than 3. Furthermore, normal hyperbolicity implies the construction is still robust in this setting. 2

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Figure 1. Intervals I and J.

4. Premaximality Before we proceed to the proof of Theorems 1.5 and 1.6 let us review results by Anosov on premaximality [2–6]. As was mentioned in Section 1, Theorem 1.4 implies that premaximality is an intrinsic property. Let us recall the main idea of the proof of Theorem 1.4. For δ > 0 denote P (δ) as the set of δ-pseudo orbits {yn ∈ Λ}n∈Z endowed with the Tikhonov product topology. Consider the shift map σ : P (δ) → P (δ) defined as σ({yn }) = {yn+1 }. The Shadowing Lemma implies that for any ε > 0 there exists δ > 0 such that for any {yn } ∈ P (δ) there exists point x such that (1)

d(yn , f n (x)) < ε.

Note that due to the expansivity property we know for small enough ε > 0 that such a point x is unique. Fix such an ε > 0 and a corresponding δ from the shadowing lemma. Consider the map T : P (δ) → M defined by the condition that for {yn } ∈ P (δ) the point x = T ({yn }) is the unique point satisfying (1). It is easy to show that (2)

T ◦ σ = f ◦ T.

Now let us consider c > 0 and a small neighborhood U ⊂ B(c, Λ) of Λ and a point z such that O(z) ⊂ U . There exists a sequence of points {yn } satisfying (3)

d(yn , f n (z)) < c.

For any δ > 0 there exists a c > 0 such that inequality (3) implies {yn } ∈ P (δ) and (4)

T ({yn }) = z.

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We can similarly define for Λ and f  sets P  (δ) and maps T  , σ  . Note that similarly to (2) the equality (5)

T  ◦ σ = f  ◦ T 

holds. For any δ1 > 0 there exists δ1 > 0 such that (6)

h(P (δ1 )) ⊂ P  (δ1 ).

(Recall that h : Λ → Λ is a conjugacy between Λ and Λ .) Similarly for any δ2 there exists δ2 such that (7)

h−1 (P  (δ2 )) ⊂ P (δ2 ).

Equations (2), (4), (5), (6), and (7) allow us to conclude Theorem 1.4. For a detailed exposition of the proof we refer the reader to the original paper [6]. To study premaximality of the zero-dimensional hyperbolic sets we need the following notion. Let A be an “alphabet”: A = {1, . . . , n} and Ω = AZ equipped with the Tikhonov topology and metric  1 d(a, b) = I(ai , bi ), 2|i| i∈Z where a = (ai ), b = (bi ) ∈ Ω and I(ai , bi ) equal to 0 if ai = bi and 1 otherwise. Consider the shift map σ = Ω → Ω defined as (σ(a))i = ai+1 . Consider some set W of admissible words of lengths k ≥ 1 in the alphabet A and consider MW ⊂ Ω such that all subwords of all a ∈ MW of length k are admissible. Theorem 4.1. [1] Consider a set Λ ⊂ Ω. The following holds (1) the set Λ is locally maximal for σ if and only if there exists k ≥ 1 and set of admissible words W such that Λ = MW ; (2) the set Λ is premaximal. In [3] it was proved that zero-dimensional hyperbolic sets are topologically conjugated to Bernoulli shifts, which implies the next result. Theorem 4.2. Let Λ be a zero-dimensional hyperbolic set of a diffeomorphism f . Then Λ is premaximal. Burns and Gelfert were able to extend the above result to prove that a 1dimensional hyperbolic set for a flow is premaximal [8, Proposition 8]. The proof follows an argument provided by Anosov after personal communications. In [4] Anosov obtain the following sufficient condition for a hyperbolic set to not be premaximal. Theorem 4.3. Let Λ be a hyperbolic set of f ∈ C 1 . Assume that there exists a family of exact trajectories ξ : Z × [0, a] → M such that (1) ξn+1,t = f (ξ(n, t)), (2) ξ(0, 0) ∈ Λ, (3) d(ξ(n, t), Λ) → 0 uniformly in t as |n| → ∞, and / Λ. (4) there exists t1 ∈ [0, a] such that ξ(0, t1 ) ∈ Then Λ is not premaximal. Note that the examples of Crovisier and Fisher satisfy these conditions. We now prove an analog of Theorem 1.4 for flows from which Theorem 1.5 will follow.

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Theorem 4.4. Let Λ and Λ be hyperbolic sets for smooth flows φ and φ respectively. Assume that Λ and Λ are topologically equivalent, with the corresponding map h : Λ → Λ . If U is a neighborhood of Λ and U  is a neighborhood of Λ , then there exists neighborhoods V ⊂ U of Λ and V  ⊂ U  of Λ , numbers τ0 , τ0 > 0, and (not necessarily continuous) maps h1 : Iφ (V ) → M  and h1 : Iφ (V  ) → M such that (8)

h1 |Λ = h and h1 |Λ = h−1 .

Furthermore, for any ε > 0 there exists δ > 0 such that if x1 , x2 ∈ Iφ (V ) and d(x1 , x2 ) < δ, then there exists |τ  | < τ0 such that (9)

d(h1 (x1 ), φτ  (h1 (x2 ))) < ε

and for any x1 , x2 ∈ Iφ (V  ) where d(x1 , x2 ) < δ there exists |τ | < τ0 such that (10)

d(h1 (x1 ), φτ (h1 (x2 )))) < ε.

Lastly, we know (11)

h1 (Iφ (V )) ⊂ Iφ (U  ), h1 (Iφ (V  )) ⊂ Iφ (U ),

(12)

Iφ (V  ) ⊂ h1 (Iφ (V )), Iφ (V ) ⊂ h1 (Iφ (V  )),

and for any x ∈ Iφ (U ) and x ∈ Iφ (U  ) we have (13)

h1 (O(x)) ⊂ O  (h1 (x)),

(14)

h1 (O  (x )) ⊂ O(h1 (x )),

and there exists |τ | < τ0 and |τ  | < τ0 such that (15)

h1 ◦ h1 (x) = φ(τ, x),

h1 ◦ h1 (x ) = φ (τ  , x ).

Remark 4.5. Note that (9), (10) are analogs of continuity for the maps h1 , h1 . Relation (15) state that h1 is almost an inverse of h1 . The reason we do not obtain continuous invertible maps is due to the nonuniqueness of shadowing for flows. Proof. Let X be the vector field generating the smooth flow φ. For a point x ∈ M and ε > 0 such that X(x) = 0 denote  L(x, ε) := expx (v). v∈Tx M, |v| 0 and a neighborhood O of Y such that O ⊂ ∪x∈Y L(x, ε), and L(x, ε) ∩ L(φ(t, x), ε) = ∅ for x ∈ M, t ∈ (−τ1 , τ1 ). Let a, a , τ0 , τ0 > 0 be constants from the expansivity property for Λ and Λ . For δ > 0 denote by P (δ) the set of δ-pseudo orbits g(t) contained in Λ. For any τ ∈ R consider the mapping στ : P (δ) → P (δ) defined by the relation (στ g)(t) = g(t + τ ). Shadowing and expansivity imply that for any ε > 0 there exists δ > 0 such that for any g ∈ P (δ) there exists a unique point x ∈ L(g(0), ε) and a (not necessarily unique) reparametrization γ ∈ Rep such that (16)

d(g(t), φ(γ(t), x)) < ε.

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Consider ε = a/2 and the corresponding δ0 . Now define a map T : P (δ0 ) → M such that x = T (g) is the unique point satisfying (16). For η > 0 we say that pseudo orbits g1 , g2 ∈ P (δ) are η-rep-close if there exists a reparametrization γ ∈ Rep such that d(g1 (t), g2 (γ(t))) < η. We will use the following properties of the map T . By shadowing and expansivity we know that for sufficiently small δ that for any g ∈ P (δ) and t ∈ R there exists t ∈ R such that T ◦ σt (g) = φt ◦ T (g). Fix small enough η, δ > 0 such that if g1 , g2 ∈ P (δ) are η-rep-close then there exists |τ | < τ0 such that (17)

T g2 = φ(τ, T g1 ).

Fix δ1 > 0 sufficiently small such that V0 = B(δ1 , Λ) ⊂ U . For any point z ∈ Iφ (V0 ) and t ∈ R let us choose a point g(t) ∈ Λ such that the inclusion φt (z) ∈ L(δ1 , g(t)) holds. We also assume that δ1 is sufficiently small so that the map g(t) is a δ-pseudo orbit. Define a map S : V0 → P (δ) as S(z) := g. We would like to emphasize that the choice of g(t) is not unique, however for any such choice T (g) = z. Again from the expansivity property we have T ◦ S = id. Also, for ε > 0 sufficiently small there exists Θ > 0 such that if g1 , g2 ∈ P (δ) and d(g1 (t), g2 (t)) < δ,

|t| < Θ

then there exists |τ | < τ0 d(S(g1 ), φ(τ, S(g2))) < ε. For δ possibly smaller we can fix η > 0 sufficiently small so that for any g ∈ P (δ) pseudo orbits g and S ◦T g are η-rep-close. Additionally, there exists a neighborhood V1 ⊂ V0 of Λ such that for any x ∈ Iφ (V1 ) the inclusion S(x) ∈ P (δ) holds. Similarly, fix δ0 , δ1 , δ  , η  > 0, maps T  : P (δ0 ) → M  and S  : B(δ1 , Λ ) →  P (δ ), and a neighborhood V1 of Λ . Now we are ready to construct the maps h1 , h1 . Let us choose δ2 ∈ (0, δ) such that for any g ∈ P (δ2 ) the inclusion h(g) ∈ P (δ  ) holds. Now let us choose V ⊂ V1 an open neighborhood of Λ such that for any x ∈ Iφ (V ) the inclusion S(x) ∈ P (δ2 ) holds. Notice that if g is a pseudo-orbit contained strictly in Λ, then we can define a pseudo-orbit in Λ by h(g). We now define the map h1 : Iφ (V ) → M  as h1 := T  ◦h◦S. Similarly, define V  and map h1 : IX  (V  ) → M as h1 = T ◦h−1 ◦S  . Among the properties of maps h1 , h1 the most difficult is relation (15). We will give its proof in full details. Properties (13) and (14) will follow directly from the definitions of the functions h1 and h1 . Relations (8)–(12) can be easily deduced (decreasing V and V  if necessarily) from • properties described above, • expansivity of the vector fields in V, V  , and • continuity of h, h−1 .

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We prove only the second equality in (15). Note that h1 ◦ h1 = T ◦ h−1 ◦ S  ◦ T  ◦ h ◦ S. For η  > 0 perhaps smaller we know that if g1 and g2 are η  -rep-close then h−1 g1 , h−1 g2 are η-rep-close. Also, for δ  > 0 perhaps smaller we know that if g1 ∈ P (δ  ), then pseudo orbits g1 and S  ◦ T  g1 are η  -rep-close. By the continuity of h for V , perhaps smaller, we know that for any x ∈ Iφ (V ) the inclusion h◦Sx ∈ P (δ  ) holds. Let x ∈ V . Set g1 := Sx, g1 := hg1 , x := T  g1 , g2 := S  x , g2 := h−1 g2 , y := T g2 . By construction of V we know g1 ∈ P (δ  ). Note that g2 = S  ◦ T  g1 . Hence, g1 and g2 are η  -rep-close. Also, we know that g1 = h−1 g1 , and g2 = h−1 g2 . Hence, g1 and g2 are η-rep-close. Finally, we have x = T g1 ,

y = T g2 . 

Now we know that (17) implies the second equality (15). 

Proof of Theorem 1.5. We prove that if Λ is premaximal then Λ is premaximal. The converse statement can be proven similarly. Consider neighborhoods V ⊂ U of Λ and V  ⊂ U  of Λ and maps h1 , h1 from Theorem 4.4. Assume that Σ ⊂ V is a locally maximal set with isolating neighborhood W = B(η, Σ), such that Σ = O(h1 (Σ)) ⊂ V  . Below we will prove that Σ is locally maximal. Consider η1 > 0 such that for x ∈ B(η1 , Σ) and |τ | < τ0 the inclusion φτ (x) holds. Using equality (14) and inequality (10) for ε = η1 we find η1 > 0 such that if x ∈ W  := B(η1 , Σ ) then h1 (x ) ∈ B(η1 , Σ). Let us prove that Σ = Iφ W  . Assume contrarily that there exists x ∈ W  \ Σ such that O(x ) ⊂ W  . Relations (13), (14), (15) imply that  φτ (h1 (x1 )). (18) O(h(x )) = |τ | 0 any δ-pseudo orbit consisting of points from Λ can be shadowed by a trajectory from Λ for small enough δ). It easily leads us to the following.

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Proposition 4.6. If sh(Λ, δ) = Λ for some δ > 0. Then Λ is locally maximal. As a reminder if Λ is a locally maximal set, then for ε > 0 sufficiently small we know that Bε (Λ) is an isolating set for Λ. In [9] Crovisier provides an example of a hyperbolic set that is never included in a locally maximal one, and this example shows that to establish premaximality it is not enough to say that a hyperbolic set is included in a locally maximal one or that the shadowing closure stabilizes for some δ > 0. To be more precise, let Let A and B be 2 × 2 hyperbolic toral automorphisms such that A has two fixed points p and q and the hyperbolicity in A dominates the hyperbolicity in B. Let F be a diffeomorphism on the 4-torus defined by F0 (x, y) = (Ax, By) and fix r a fixed point of B. Crovisier proves that if we let V be a sufficiently small neighborhood of (q, r), then the set % Λ= f n (T4 − V ) n∈Z

is a hyperbolic set and the only locally maximal hyperbolic set that Λ can be included in is the entire manifold T4 ; so this example is not premaximal. However, Λ is included in a locally maximal hyperbolic set and for δ > 0 sufficiently large may be in a shadowing closure that stabilizes. Proof of Theorem 1.6. We first consider the case of diffeomorphisms. Suppose that Λ is a hyperbolic set for a diffeomorphism f : M → M and suppose that for any neighborhood U of Λ there exists a δ > 0 such that the shadowing closure of Λ stabilizes inside U . Now the previous proposition shows that the stabilizer is a locally maximal set and Λ is premaximal. To prove the other implication let f : M → M be a diffeomorphism and let Λ be a premaximal set for f and V be a neighborhood of Λ. Then there exists some η > 0 such that  Bη (x) ⊂ V Bη (Λ) = x∈Λ

and Λη = If (Bη (Λ)) is hyperbolic. ˜ be a locally maximal hyperbolic set such that Let Λ ˜ ⊂ Λη ⊂ Bη (Λ) ⊂ V. Λ⊂Λ ˜ is an isolating neighborhood of Λ ˜ and fix δ ∈ (0, ε/2) Fix ε ∈ (0, η) such that Vε (Λ) ˜ is ε shadowed in Λ ˜ by a unique point in Λ. ˜ such that every δ-pseudo orbit in Λ From the choice of constants above we know that Λ ⊂ sh(Λ, δ) ⊂ Λε . Fur˜ so there thermore, we know that each δ-pseudo orbit of Λ is a δ-pseudo orbit of Λ ˜ that is a ε-shadowing point of the pseudo orbit. Hence, exists a unique point y ∈ Λ ˜ Let Λ1 = sh(Λ, δ). sh(Λ, δ) ⊂ Λ. If Λ1 = Λ we know from the previous proposition that Λ is locally maximal. So suppose that Λ1 = Λ and let ν1 = dH (Λ1 , Λ) where dH (·, ·) is the Hausdorff distance between the sets. More generally, let Λj+1 = sh(Λj , δ), νj+1 = dH (Λj+1 , Λj ) for j ∈ N. By the shadowing estimates we know that νj ∈ (0, ε). Claim 4.7. There exists γ > 0 such that for all j ∈ N if Λj = Λj+1 and Λj+1 = Λj+2 , then either νj ≥ γ or νj+1 ≥ γ.

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Proof of Claim. Consider γ ∈ (0, δ/4) such that for all x, y ∈ M if d(x, y) < γ, then d(f (x), f (y)) < δ/4 and d(f −1 (x), f −1 (y)) < δ/4. Suppose that Λj = Λj+1 and Λj+1 = Λj+2 and νj , νj+1 ∈ (0, γ). Fix y ∈ Λj+2 − Λj+1 . We will construct a δ-pseudo orbit in Λj that y ε-shadows. This will show that y ∈ Λj+1 , a contradiction. Then there exists some y0 ∈ Λj+1 and x0 ∈ Λj such that d(y, y0 ) < νj+1 and d(y0 , x0 ) < νj < γ. Then d(f (y), f (y0 )) < δ/4 and d(f (y0 ), f (x0 )) < δ/4. Also, we know there exists points y1 ∈ Λj+1 and x1 ∈ Λj such that d(f (y), y1 ) < νj+2 and d(y1 , x1 ) < νj+1 . Hence, d(f (x0 ), x1 )

≤ d(f (x0 ), f (y0 )) + d(f (y0 ), f (y)) + d(f (y), y1 ) + d(y1 , x1 ) ≤ δ/4 + δ/4 + νj+2 + νj+1 < δ

and d(f (y), x1 ) ≤ d(f (y), y1 ) + d(y1 , x1 ) < νj+2 + νj+1 < δ/2 < ε. Continue inductively, we can construct a forward δ-pseudo orbit (xk )∞ k=0 such that y ε-shadows the pseudo orbit. Also, since the estimates on γ apply for f −1 we can construct a bi-infinite δ-pseudo orbit (xk ) in Λj such that y ε shadows (xk ). Then y ∈ Λj+1 , a contradiction.  We now return to the proof of the theorem. ˜ for all j ∈ N. We Repeating the above arguments for Λj we see that Λj ⊂ Λ know from Proposition 4.6 that if Λj+1 = Λj for some j ∈ N that Λj is locally maximal. To conclude the proof of the theorem we simply need to show that the sequence Λj stabilizes. Suppose that the sequence Λj does not stabilizer. We know from the above claim that for each j the sets Λj+1 or Λj+2 will be a distance of γ from the set Λj using the Hausdorff metric. Inside a compact metric space we know an increasing sequence of compact sets converges in the Hausdorff topology. Hence, if the sequence does not stabilize there exists some N ∈ N where the Hausdorff distance ˜ and from ΛN to Λ is greater than η. This is a contradiction since each Λj ⊂ Λ ˜ Λ ⊂ Vη (Λ). Theorem 1.6 is proved for the case of diffeomorphisms. The proof for the case of flows follows the same ideas. Below we provide a detailed proof of the Claim 4.7, which is the central step of the proof of Theorem 1.6. We will use the following construction. For a sequence of points {xk }k∈Z consider a map g{xk } : R → M defined as g{xk } (t) := φ(t − [t], x[t] ), where [t] is the integer part of t. Consider map r : M → M defined as r(x) = φ(1, x). Note that r does not necessarily satisfy the shadowing property. We will use the following. F1: There exists ε1 > 0 such that for any sequence {xk } and point y satisfying (19)

d(xk , f k (y)) < ε1 ,

k∈Z

the following inequality holds d(g{xk } (t), φ(t, y)) < ε.

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Without loss of generality we can assume that δ < ε1 . F2: There exists δ1 > 0 such that if {xk } with xk ∈ Λj is a δ1 -pseudo orbit of the map r then g{xk } ⊂ Λj is a δ-pseudo orbit of the flow φ. Consider γ ∈ (0, δ1 /4) such that for all x, y ∈ M if d(x, y) < γ, then d(r(x), r(y)) < δ1 /4 and d(r −1 (x), r −1 (y)) < δ1 /4. As in the case of diffeomorphisms suppose that Λj = Λj+1 and Λj+1 = Λj+2 and νj , νj+1 ∈ (0, γ). Fix y ∈ Λj+2 − Λj+1 . Arguing similarly to the case of diffeomorphisms we can construct a δ1 -pseudo orbit {xk } ⊂ Λj of the map r which satisfies (19). Properties F1 and F2 implies that δ-pseudo orbit g{xk } is ε-shadowed by y. This shows that y ∈ Λj+1 , a contradiction.  References [1] V. M. Alekseev, Symbolic dynamics (Russian), Eleventh Mathematical School (Summer School, Kolomyya, 1973), Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1976, pp. 5–210. MR0464317 [2] D. V. Anosov, Intrinsic character of one property of hyperbolic sets, J. Dyn. Control Syst. 16 (2010), no. 4, 485–493, DOI 10.1007/s10883-010-9103-y. MR2739013 [3] D. V. Anosov, Extension of zero-dimensional hyperbolic sets to locally maximal ones, Sb. Math. 201 (2010), no. 7-8, 935–946, DOI 10.1070/SM2010v201n07ABEH004097. MR2907812 [4] D. V. Anosov, On some hyperbolic sets (Russian, with Russian summary), Mat. Zametki 87 (2010), no. 5, 650–668, DOI 10.1134/S0001434610050020; English transl., Math. Notes 87 (2010), no. 5-6, 608–622. MR2766421 [5] D. V. Anosov, Local maximality of hyperbolic sets (Russian, with Russian summary), Tr. Mat. Inst. Steklova 273 (2011), no. Sovremennye Problemy Matematiki, 28–29; English transl., Proc. Steklov Inst. Math. 273 (2011), no. 1, 23–24. MR2893540 [6] D. V. Anosov, On trajectories located entirely near a hyperbolic set (Russian, with Russian summary), Sovrem. Mat. Fundam. Napravl. 45 (2012), 5–17; English transl., J. Math. Sci. (N. Y.) 201 (2014), no. 5, 553–565. MR3087049 [7] V´ıtor Ara´ ujo and Maria Jos´e Pac´ıfico, Three dimensional flows, Publica¸c˜ oes Matem´ aticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matem´ atica Pura e Aplioquio Brasileiro de Matem´ atica. [26th Brazilian cada (IMPA), Rio de Janeiro, 2007. 26o Col´ Mathematics Colloquium]. MR2372421 [8] Keith Burns and Katrin Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces, Discrete Contin. Dyn. Syst. 34 (2014), no. 5, 1841–1872, DOI 10.3934/dcds.2014.34.1841. MR3124716 [9] Sylvain Crovisier, Une remarque sur les ensembles hyperboliques localement maximaux (French, with English and French summaries), C. R. Math. Acad. Sci. Paris 334 (2002), no. 5, 401–404, DOI 10.1016/S1631-073X(02)02274-4. MR1892942 [10] Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/1972), 193–226. MR0287106 [11] Neil Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23 (1973/74), 1109–1137. MR0339276 [12] Neil Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J. 26 (1977), no. 1, 81–93. MR0426056 [13] Todd Fisher, Hyperbolic sets that are not locally maximal, Ergodic Theory Dynam. Systems 26 (2006), no. 5, 1491–1509, DOI 10.1017/S0143385706000411. MR2266370 [14] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR1326374 [15] Yuri Kifer, Random perturbations of dynamical systems, Progress in Probability and Statistics, vol. 16, Birkh¨ auser Boston, Inc., Boston, MA, 1988. MR1015933

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[16] T. Petty, Nonlocally maximal hyperbolic sets for flows, Maters Thesis, Brigham Young University, 2015. [17] Sergei Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, vol. 1706, Springer-Verlag, Berlin, 1999. MR1727170 Department of Mathematics, Brigham Young University, Provo, Utah 84602 E-mail address: [email protected] Department of Mathematics, Brigham Young University, Provo, Utah 84602 E-mail address: [email protected] Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany – and – Chebyshev Laboratory, Saint-Petersburg State University, 14th Line 29B, Vasilyevsky Island, St.Petersburg 199178, Russia E-mail address: [email protected]

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13913

Rotation numbers for S 2 diffeomorphisms John Franks Dedicated to Dmitri Anosov who inspired my first efforts in dynamics Abstract. In these notes we describe the properties of, and generalize, the function R which assigns a number to a 4-tuple of distinct fixed points of an orientation preserving homeomorphism or diffeomorphism of S 2 .

1. The function Rf for fixed points. We let f be an orientation preserving homeomorphism and denote by X the set of fixed points of f . We will assume for the moment that X contains at least four distinct points. Given four distinct points x1 , x2 , x3 , x4 ∈ Fix(f ) we consider the sets A = {x1 , x2 } and B = {x3 , x4 }. One can puncture at A to obtain an annulus and then consider a lift of f to the universal cover of this annulus. The difference of the rotation numbers of the points of B (with respect to this lift) is an element of R which is independent of the choice of lift. This gives a real valued function Rf of the four fixed points {x1 , x2 , x3 , x4 }. This function has some remarkable symmetries under permutations of its arguments which form the content of this note. I am indebted to Patrice Le Calvez for informing me of many of the basic properties described here. We will be interested in the intersection of embedded oriented paths α : [0, 1] → S 2 \ B and β : [0, 1] → S 2 \ A with α running from x1 to x2 , and β running from x3 to x4 . These paths have an algebraic intersection number which we now define. It is determined by the orientation of the paths and the orientation of S 2 . The ends of β, x3 and x4 , are disjoint from α and we can close β to a loop by concatenating with a path δ from x4 to x3 which lies in the complement of the image of α. If δ ∗ β is the loop obtained by concatenating δ and β, then its homology class [δ ∗ β] ∈ H1 (S 2 \ A) ∼ = Z depends on α and β, but is independent of the choice of δ. Definition 1.1. We define the algebraic intersection number α · β ∈ Z by [δ ∗ β] = (α · β)[u], where u is an embedded closed loop in S 2 \ A, positively oriented with respect to x1 , and hence [u] is a generator of H1 (S 2 \ A). The number α · β depends only on the homology classes [α] ∈ H1 (S 2 \ B, A) and [β] ∈ H1 (S 2 \ A, B). There is a standard skew-symmetric intersection pairing i : H1 (S 2 \ B, A) × H1 (S 2 \ A, B) → H0 (S 2 \ A ∪ B) ∼ = Z. 2010 Mathematics Subject Classification. Primary 37C25, 37E30. c 2017 American Mathematical Society

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and the algebraic intersection number of α · β is equal to i([α], [β]). There are three elementary facts we will have occasion to use • The intersection is skew-symmetric: α · β = −β · α. • If α ¯ (t) = α(1 − t), is the path with reverse parametrization, then α ¯·β = −α · β. • If g : S 2 → S 2 is an orientation preserving homeomorphism then (g ◦ α) · (g ◦ β) = α · β. 1.1. Characterization of Rf .. In this section we give a definition of the function Rf and provide several equivalent characterizations (Propositions (1.3), (1.5) and (1.6)). Throughout this section we will assume that f is an orientation preserving homeomorphism and X = {x1 , x2 , x3 , x4 } is a subset of Fix(f ). The final result of this section shows that for a fixed X the function Rf is a homomorphism from the group of orientation preserving homeomorphisms which pointwise fix X to Z. Definition 1.2. Let f : S 2 → S 2 be an orientation preserving homeomorphism and suppose {x1 , x2 , x3 , x4 } is a subset of distinct points of Fix(f ). Let A = {x1 , x2 } and B = {x3 , x4 } and let α be an embedded oriented path in S 2 \ B running from x1 to x2 and let β be an embedded oriented path in S 2 \ A running from x3 to x4 . Then we define Rf (x1 , x2 , x3 , x4 ) = α · (f ◦ β) − α · β. The notation implies that Rf (x1 , x2 , x3 , x4 ) depends only on x1 , x2 , x3 , x4 ∈ Fix(f ) and not on the choice of α and β. This is indeed true and follows immediately from the following proposition which gives an alternate description of Rf . As in the definition above we let {x1 , x2 , x3 , x4 } be a set of distinct points in Fix(f ) and define A = {x1 , x2 } and B = {x3 , x4 }. Proposition 1.3. Let V = S 2 \ A and let f˜ : V˜ → V˜ be the lift of f to the universal covering space of V which fixes x ˜3 , a lift of x3 . Let T : V˜ → V˜ be the generator of the group of covering translations corresponding to a loop in V with intersection number +1 with some (and hence any) oriented path from x1 to x2 . If x4 ) = T n (˜ x4 ). x ˜4 ∈ V˜ is any lift of x4 , then Rf (x1 , x2 , x3 , x4 ) = n ∈ Z where f˜(˜ Proof. Let x ˜3 be a lift of x3 and let f˜ be the lift of f which fixes x ˜3 . If α and β are as in Definition (1.2) choose α ˜ and β˜ lifts of α and β with β˜ running from x ˜3 to a point which is a lift of x4 . We denote this point by z˜4 and define n by z4 ). The value of n would be the same if we used any other lift x ˜4 of f˜(˜ z4 ) = T n (˜ x4 in place of z˜4 . z4 ) = T n (˜ z4 ). It is homotopic to the concatenaThe path f˜β˜ runs from x ˜3 to f˜(˜ z4 ). If γ is the closed tion of the path β˜ from x ˜3 to z˜4 with a path γ˜ from z˜4 to T n (˜ loop in V obtained by projecting γ˜ then [γ] is n times the generator of H1 (V ) and this generator has intersection number +1 with α. Hence α · (f ◦ β) = α · γ + α · β and (1)

Rf (x1 , x2 , x3 , x4 ) = α · (f ◦ β) − α · β = α · γ = n.



Remark 1.4. We note that Rf (x1 , x2 , x3 , x4 ) is sometimes well defined even if the points x1 , x2 , x3 , and x4 are not all distinct. In particular if x1 = x2 but

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{x2 , x3 , x4 } are distinct then Rf (x2 , x2 , x3 , x4 ) = 0 since we may choose the constant path for α. Similarly if x3 = x4 but {x1 , x2 , x3 } are distinct then Rf (x1 , x2 , x3 , x3 ) = 0 since we may choose the constant path for β. If x2 = x3 or x1 = x4 the value of Rf is undefined for the general homeomorphism f . However, if f is a C 1 diffeomorphism we can define Rf when x2 = x3 or x1 = x4 , but the value may no longer be an integer, or even rational. We will do this below in Section (1.3). Since β and f ◦ β have the same endpoints, x3 and x4 , we can concatenate the paths f ◦ β and β¯ = β(1 − t) to form a loop γ in S 2 \ A which, as an immediate consequence of the definition and the properties of intersection number, gives the following characterization of Rf (x1 , x2 , x3 , x4 ). ¯ = β(1 − t) and γ = (f ◦ β) ∗ β¯ then Proposition 1.5. If β(t) Rf (x1 , x2 , x3 , x4 ) = α · γ. An easy consequence of Proposition (1.3) is another useful description of Rf . We use that fact that the space of homeomorphisms fixing three distinct points {x1 , x2 , x3 } is contractible and so there is an isotopy ft (x) f0 = id and f1 = f such that for all t ∈ [0, 1], ft (xi ) = xi for 1 ≤ i ≤ 3. This isotopy is unique up to homotopy as a path in the space of homeomorphisms fixing the points {x1 , x2 , x3 }. Proposition 1.6. Suppose α is a path from x1 to x2 which is disjoint from the set {x3 , x4 }. Let γ0 be the closed loop in S 2 \ {x1 , x2 , x3 } defined by ft (x4 ) for t ∈ [0, 1] then Rf (x1 , x2 , x3 , x4 ) = α · γ0 . Proof. This follows immediately from the proof of Proposition (1.3) because γ0 lifts to a path γ˜0 in the universal covering space of S 2 \ {x1 , x2 } with the same endpoints as the path γ˜ in the proof of Proposition (1.3). Hence the loops γ and γ0 are homotopic in S 2 \ {x1 , x2 } so from equation (1) we have Rf (x1 , x2 , x3 , x4 ) = α · γ = α · γ0 .



An important property of the function Rf is that if G is a subgroup of Homeo(S 2 ) which fixes a set X pointwise then the function which assigns to each f ∈ G the function Rf is a homomorphism. To make this precise, for f, g ∈ Homeo(S 2 ) we will denote their composition by f g. Proposition 1.7. Suppose {x1 , x2 , x3 , x4 } is a set of distinct points in Fix(f )∩ Fix(g). Then Rf g (x1 , x2 , x3 , x4 ) = Rf (x1 , x2 , x3 , x4 ) + Rg (x1 , x2 , x3 , x4 ). Proof. We use Proposition (1.6). As an isotopy from id to f g we choose ft gt , t ∈ [0, 1] where ft and gt are isotopies from id to f and g respectively, each of which fixes the points x1 , x2 , and x3 . The loop γ0 given by ft gt (x4 ) is homotopic to the loop which is the concatenation of γ1 given by gt (x4 ) = f0 gt (x4 ) with γ2 given by ft (x4 ) = ft (g1 (x4 )). Hence Rf g (x1 , x2 , x3 , x4 ) = α · γ0 = α · γ1 + α · γ2 = Rg (x1 , x2 , x3 , x4 ) + Rf (x1 , x2 , x3 , x4 ).



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1.2. Basic Symmetries of Rf . We want to understand the effect permuting the variables of Rf . We first establish basic relations – equations (2), (3), (5), and (6), below. In Section (2) we establish relations applicable to any permutation of the variables of Rf . We will write elements of S4 , the symmetric group of order four, in the standard way as a product of cycles. For example (12)(34) is the product the transposition of 1 and 2 and the transposition of 3 and 4. Recall that the group S4 is generated by the three transpositions σi = (i, i + 1), 1 ≤ i ≤ 3. If x = (x1 , x2 , x3 , x4 ) is a four-tuple of elements of Fix(f ) we will denote the four-tuple (xσ(1) , xσ(2) , xσ(3) , xσ(4) ) by xσ . An element of S4 which we will have occasion to use a number of times is the cyclic permutation (123). We will denote this element by τ. Proposition 1.8. Suppose {x1 , x2 , x3 , x4 } is a set of distinct points in Fix(f ). Then (2)

Rf (x) + Rf (xτ ) + Rf (xτ 2 ) = 0,

(3)

Rf (xσ1 ) = Rf (xσ3 ) = −Rf (x)

Proof. Equation (3) follows immediately from the definition of Rf since reversing the parametrization of α to get α ¯ , a path from x2 to x1 changes the sign of the intersection number. Hence Rf (xσ1 ) = α ¯ · (f ◦ β) − α ¯·β = −α · (f ◦ β) + α · β = −Rf (x). The argument for σ3 is similar. To show equation (2) we choose paths α1 from x1 to x2 , α2 from x2 to x3 , and α3 from x3 to x1 . The concatenation of these three paths is a closed loop η in S 2 . Let γ be as defined in Proposition (1.6). Then Rf (x) + Rf (xτ ) + Rf (xτ 2 ) = α1 · γ + α2 · γ + α3 · γ = η · γ = 0, 

since the intersection number of any two closed loops in S 2 is 0.

Proposition 1.9. Suppose {x1 , x2 , x3 , x4 , w} is a set of distinct points in Fix(f ). Then (4)

Rf (x1 , x2 , x3 , x4 ) = Rf (x3 , x4 , x1 , x2 ),

(5)

Rf (x1 , x2 , x3 , x4 ) = Rf (x1 , w, x3 , x4 ) + Rf (w, x2 , x3 , x4 ),

(6)

Rf (x1 , x2 , x3 , x4 ) = Rf (x1 , x2 , x3 , w) + Rf (x1 , x2 , w, x4 ).

and

Proof. To show equation (4) we note that Proposition (1.7) implies Rf = −Rf −1 . Let α (respectively β) be a path from x1 to x2 (respectively x3 to x4 ) which we choose so that α · β = 0. Then Rf (x1 , x2 , x3 , x4 ) = α · (f ◦ β) = −(f ◦ β) · α = −β · (f

−1

since · is skew-symmetric,

◦ α)

= −Rf −1 (x3 , x4 , x1 , x2 ) = Rf (x3 , x4 , x1 , x2 ).

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105

To show equation (5) we choose paths α1 from x1 to w and α2 from w to x2 and let α be the path from x1 to x2 obtained by concatenating them. Then Rf (x1 , x2 , x3 , x4 ) = α · (f ◦ β) − α · β = α1 · (f ◦ β) − α1 · β + α2 · (f ◦ β) − α2 · β = Rf (x1 , w, x3 , x4 ) + Rf (w, x2 , x3 , x4 ). Equation (6) follows from equations (4) and (5).



1.3. Rf for diffeomorphisms. For a homeomorphism f the value of Rf (x1 , x2 , x3 , x4 ) is generally undefined if either of x1 , x2 is equal to either of x3 , x4 . In the case of C 1 diffeomorphisms we can remove this restriction. This is done essentially by incorporating the infinitesimal rotation number at the fixed point, say x1 = x3 , in question. Suppose p ∈ Fix(f ). Let α : [0, 1] → S 2 and β : [0, 1] → S 2 be C 1 embeddings with α(0) = β(0) = p and with distinct tangent vectors at p. We can blow up the point p to obtain a map fˆ on the compactification of S 2 \ {p}. The action of fˆ on the circle which compactifies S 2 \ {p} is the projectivization of Dfp . We denote the compactified space (which is topologically a closed disk) by M. The points on the boundary of M which correspond to the tangents to α and β at p are distinct and the ends of paths α ˆ and βˆ which agree with α and β on (0, 1]. We define α · β to ˆ be α ˆ · β. We can now define Rf (p, x2 , p, x4 ). Choose embeddings α and β satisfying (1) α(0) = β(0) = p ∈ Fix(f ). (2) α(1) = x2 , β(1) = x4 with x2 , x4 ∈ Fix(f ) (3) If v and w are the tangents at p to α and β respectively then dfpn (v) w = , dfpn (v) w for all n ∈ Z. Lemma 1.10. Suppose the paths α and β satisfy (1) - (3) above. Then the limit α · (f n ◦ β) n→∞ n exists and is independent of the choice of α and β. = lim

Proof. Blow up the points p and x2 to form a closed annulus A. Choose the ˜ →A ˜ which fixes a lift of x4 . Let C be the circle added when p was blown lift f˜ : A up; so C is one component of the boundary of A. Parametrize C so that α(0) ˆ is 0 and let C˜ be the universal cover of C which we may think of as a component of the ˜ boundary of A. ˜ Then f |C˜ is a lift of f |C and it has a well defined translation number (f˜) defined to be p2 ) − p˜2 f˜n (˜ (f˜) = lim n→∞ n where p˜2 is the lift of the endpoint β(0). This limit always exists by the standard theory of rotation numbers for circle homeomorphisms. The annulus A depends only on p and x2 . The lift f˜ is determined by x4 . Finally we note that

 

α · (f n ◦ β) − α · β = − f˜n (˜ p2 ) − p˜2 + K,

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where K satisfies |K| ≤ 2. It follows that α · (f n ◦ β) − α · β = −(f˜). n→∞ n lim



Definition 1.11. We define α · (f n ◦ β) − α · β . n→∞ n

Rf (p, x2 , p, x4 ) = lim

The number Rf (x1 , p, x3 , p) is defined analogously as are the values of Rf when one of first two and one of its second two variables are equal to p. We also observe that if p1 , p2 ∈ Fix(f ) then Rf (p1 , p2 , p1 , p2 ) is also defined and equal to the difference of the rotation numbers of fˆ on the two boundary components of the annulus A obtained by blowing up both p1 and p2 . It is easy to check that Rf as defined still satisfies the symmetries of Propositions (1.9) and (1.9). 1.4. Rf for finite invariant sets. From Proposition (1.7) it is clear that for q ∈ Z we have Rf q = qRf . It follows that if {x1 , x2 , x3 , x4 } is a subset of distinct points in Fix(f ), then Rf (x1 , x2 , x3 , x4 ) =

1 Rf q (x1 , x2 , x3 , x4 ). q

Definition 1.12. Suppose {x1 , x2 , x3 , x4 } is a subset of distinct points in Fix(f q ). We define Rf (x1 , x2 , x3 , x4 ) =

1 Rf q (x1 , x2 , x3 , x4 ). q

Because of the linearity of this definition it is clear that Propositions (1.7), (1.8), and (1.9) all remain valid when the argument of Rf is x = (x1 , x2 , x3 , x4 ), a 4-tuple of distinct points in Per(f ), the set of periodic points of f. If G is a group of homeomorphisms of S 2 and X is a finite set G-invariant set then every G-orbit is finite so from Proposition (1.7) we have the following. Proposition 1.13. Suppose G is a group of homeomorphisms of S 2 and X is a finite G-invariant set. Then for each 4-tuple (x1 , x2 , x3 , x4 ) of distinct points in X there is a homomorphism φ : G → R such that φ(g) = Rg (x1 , x2 , x3 , x4 ). It would be interesting to know if Rf and this proposition could be generalized to a setting where the variables of Rf or some of them are f -invariant measures rather periodic points. The definition for invariant measures with finite support should just be the expected value. The hope might be that by approximating f and the measures one might reduce to this case. One goal is that the Calabi invariant should be a special case. 2. The complete symmetries of Rf . If X is a set we want to investigate the class of functions of four variables F : X 4 → R which possess certain symmetries under permutation of the variables. The simplest example of a function with the symmetries which interest us occurs when X = R and we define the quadratic function q : R4 → R by q(x1 , x2 , x3 , x4 ) = (x1 − x2 )(x3 − x4 ).

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As before we will write elements of S4 in the standard way as a product of cycles and we recall that σi denotes the transposition (i, i + 1), 1 ≤ i ≤ 3. The group S4 is generated by these three transpositions If x = (x1 , x2 , x3 , x4 ) is a four-tuple of elements of some set we will denote the four-tuple (xσ(1) , xσ(2) , xσ(3) , xσ(4) ) by xσ . As before an element of S4 which we will have occasion to use is the cyclic permutation (123). We will denote this element by τ. We are interested in the vector space of all functions F : X 4 → R which satisfy the following three equations. (7)

F(x) + F(xτ ) + F(xτ 2 ) = 0,

(8)

F(xσ1 ) = F(xσ3 ) = −F(x)

(9)

F(x1 , x2 , x3 , x4 ) = F(x1 , w, x3 , x4 ) + F(w, x2 , x3 , x4 ),

for all w ∈ X.

The motivation is, of course, that Propositions (1.8) and (1.9) assert that Rf satisfies these properties. Equation (7) says that if we cyclically permute the first three arguments of F and sum the values, the result is 0. This is a kind of Jacobi identity for F. Equation (8) says that F is skew-symmetric in its first two variables and in its last two. We observe that the symmetries of Equations (7) and (8) are the symmetries of the curvature tensor in Riemannian geometry (see, e.g., §9 of Milnor’s Morse Theory [1]). Equation (9) says that as a function of its first two variables F is a coboundary. We remark that it will be shown below (see Remark (2.2)) that a consequence of these relations is that F(x1 , x2 , x3 , x4 ) = F(x3 , x4 , x1 , x2 ) so it follows that F is also a coboundary in its last two variables, i.e., for all w ∈ X, F(x1 , x2 , x3 , x4 ) = F(x1 , x2 , x3 , w) + F(x1 , x2 , w, x4 ). To understand the symmetries of F it is useful to introduce a function to R3 containing the three values obtained by cyclically permuting the first three arguments of F. Hence we define the function F : X 4 → R3 by

 (10) F(x) = F(x), F(xτ ), F(xτ 2 ) An immediate consequence of equation (7) is the fact that the image of F lies in the subspace of R3 given by y1 + y2 + y3 = 0. The symmetries of F under the action of S4 permuting its arguments are most easily described by means of a representation of S4 in SL(3, Z). Since the group S4 is generated by the three transpositions σi = (i, i + 1), 1 ≤ i ≤ 3, we can define a representation Θ by specifying its value on these three elements of S4 . We define ⎡ ⎤ −1 0 0 0 −1⎦ , Θ(σ1 ) = Θ(σ3 ) = ⎣ 0 0 −1 0 ⎡ ⎤ 0 0 −1 Θ(σ2 ) = ⎣ 0 −1 0 ⎦ . −1 0 0 Proposition 2.1. The function Θ defined for σi , 1 ≤ i ≤ 3, determines a homomorphism Θ : S4 → SL(3, Z).

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Proof. The set σ1 , σ2 , σ3 ∈ S4 is a set of generators. It is easily checked that Θ respects the relations σi2 = id, σ1 σ3 = σ3 σ1 and σi σi+1 σi = σi+1 σi σi+1 , which are a complete set of relations defining S4 so Θ is a homomorphism.  Remark 2.2. Since the transposition (13) = σ1 σ2 σ1 and (24) = σ2 σ3 σ2 it is easy to calculate ⎡ ⎤ 0 −1 0 0 ⎦, Θ((13)) = Θ((24)) = ⎣−1 0 0 0 −1 and hence that Θ((13)(24)) = I. It is easy to check that the kernel of Θ is a group isomorphic to Z/2Z ⊕ Z/2Z which is generated by the elements (12)(34) and (13)(24) of S4 . One can also check that the image of Θ is isomorphic to S3 . Theorem 2.3. Suppose X is a set and Z ⊂ X 4 is an S4 -invariant subset. If a function F : Z → R satisfies equations ( 7) and ( 8) and we define F : Z → R3 by

 F(x) = F(x), F(xτ ), F(xτ 2 ) , then for every σ ∈ S4 and every x ∈ Z (11)

F(xσ ) = Θ(σ)F(x).

We postpone the proof of this result until the next section. Note that this result about the symmetries of F and F requires only that F satisfy equations (7) and (8). For the next result we will additionally use the coboundary condition, equation (9). We note that given any symmetric function of two variables g : X 2 → R, if we define F(x) = g(x1 , x3 ) − g(x1 , x4 ) − g(x2 , x3 ) + g(x2 , x4 ), then it is straightforward to check that F satisfies equations (7) and (8). The next theorem asserts that with the addition of equation (9) the converse of this is also true. Theorem 2.4. Suppose for all x ∈ X 4 the function F : X 4 → R satisfies equations ( 7), ( 8) and ( 9). Then there exists a symmetric function g : X 2 → R such that (12)

F(x1 , x2 , x3 , x4 ) = g(x1 , x3 ) − g(x1 , x4 ) − g(x2 , x3 ) + g(x2 , x4 ).

If we normalize by distinguishing an element a ∈ X and specifying that g(a, w) = g(w, a) = 0 for all w ∈ X then g is unique. Proof. Given F satisfying equations equations (8) and (9), and a distinguished elements a ∈ X we define g(u, v) = F(u, a, v, a). To see that g is symmetric we observe that Remark (2.2), which asserts that Θ((13)(24)) = I, together with Theorem (2.3) implies g(u, v) = F(u, a, v, a) = F(v, a, u, a) = g(v, u). Skew-symmetry in the first two variables and last two variables of F implies F(a, a, v, a) = 0 and F(u, a, a, a) = 0, so g(a, w) = g(w, a) = 0 for all w ∈ X. To show that equation (12) is satisfied we make repeated use of the skewsymmetry in the first two or last two variables (equation (8)) and of the coboundary relation in the first two and last two variables (equation (9)).

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F(x1 , x2 , x3 , x4 ) = F(x1 , a, x3 , x4 ) + F(a, x2 , x3 , x4 ) = F(x1 , a, x3 , x4 ) − F(x2 , a, x3 , x4 ) 

= F(x1 , a, x3 , a) + F(x1 , a, a, x4 )

 − F(x2 , a, x3 , a) + F(x2 , a, a, x4 ) = F(x1 , a, x3 , a) − F(x1 , a, x4 , a) − F(x2 , a, x3 , a) + F(x2 , a, x4 , a) = g(x1 , x3 ) − g(x1 , x4 ) − g(x2 , x3 ) + g(x2 , x4 ). To show uniqueness, suppose g1 : X 2 → R is another function satisfying the conclusion of the theorem. Then for any u, v ∈ X g(u, v) = F(u, a, v, a) = g1 (u, v) − g1 (u, a) − g1 (a, v) + g1 (a, a) = g1 (u, v).



3. Proof of Theorem (2.3) Theorem (2.3) Suppose X is a set and Z ⊂ X 4 is an S4 -invariant subset. If a function F : Z → R satisfies equations ( 7) and ( 8) and we define F : Z → R3 by 

F(x) = F(x), F(xτ ), F(xτ 2 ) , then for every σ ∈ S4 and every x ∈ Z (13)

F(xσ ) = Θ(σ)F(x).

Proof. Since Θ is a homomorphism it suffices to prove that F(xσ ) = Θ(σ)F(x) for each σ in the set of generators of {σ1 , σ2 , σ3 }. Let x = (a, b, c, d) ∈ Z. We define the numbers r and s by F(a, b, c, d) = r and F(b, c, a, d) = s. Then ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ F(a, b, c, d) r F(x) ⎣ F(xτ ) ⎦ = ⎣F(b, c, a, d)⎦ = ⎣ s ⎦ (14) F(xτ 2 ) F(c, a, b, d) −r − s where the equality of the third component follows from equation (7). We next conclude ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ F(xσ1 ) F(b, a, c, d) −r ⎣ F(xτ σ1 ) ⎦ = ⎣F(a, c, b, d)⎦ = ⎣r + s⎦ . (15) F(xτ 2 σ1 ) F(c, b, a, d) −s where the first component comes from equation (8), the third component comes from equation (8) applied to the second component of equation (14) and the second component follows from equation (7). Since ⎡ ⎤⎡ ⎤ ⎡ ⎤ −1 0 0 r −r ⎣0 0 −1⎦ ⎣ s ⎦ = ⎣r + s⎦ 0 −1 0 −r − s −s we have shown F(xσ1 ) = Θ(σ1 )F(x).

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Likewise from equation (15) we conclude ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ F(xσ2 ) F(a, c, b, d) r+s ⎣ F(xτ σ2 ) ⎦ = ⎣F(c, b, a, d)⎦ = ⎣ −s ⎦ . (16) F(xτ 2 σ2 ) F(b, a, c, d) −r Hence F(xσ2 ) = Θ(σ2 )F(x). To handle the case of σ3 we define t = −F(xτ 2 σ3 ) = −F(d, a, b, c) so ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ F(xσ3 ) F(a, b, d, c) −r ⎣ F(xτ σ3 ) ⎦ = ⎣F(b, d, a, c)⎦ = ⎣r + t⎦ (17) F(xτ 2 σ3 ) F(d, a, b, c) −t where equality of the first components follows by equation (8) and the value of the second component again comes from the fact that the sum of the components is 0 (equation (7)). From the last component of equation (15) we get F(c, b, d, a) = s and the second component of (17) shows F(b, d, c, a) = −r − t. By equation (7), F(c, b, d, a) + F(b, d, c, a) + F(d, c, b, a) = 0. and hence s + (−r − t) + F(d, c, b, a) = 0. So F(d, c, b, a) = r + t − s and F(d, c, a, b) = −r − t + s. Also F(c, a, d, b) = r + s by the third component of equation (14). Using equation (7) once again, we see F(c, a, d, b) + F(a, d, c, b) + F(d, c, a, b) = 0, and hence (r + s) + F(a, d, c, b) + (−r − t + s) = 0. So F(a, d, c, b) = −2s + t. But the third component of equation (17) implies F(a, d, c, b) is also equal to −t, so we conclude that −t = −2s + t and hence  s = t. Substituting s for t in equation (17) gives F(xσ3 ) = Θ(σ3 )F(x). We may apply this result to the function Rf to obtain the following result which codifies the value of Rf (xσ ) for all σ ∈ S4 . Theorem 3.1. Let f : S 2 → S 2 be an orientation preserving homeomorphism and suppose x = (x1 , x2 , x3 , x4 ) is a 4-tuple of elements of Fix(f ) for which Rf (xσ ) is defined for all σ ∈ S4 . If we define F : X 4 → R3 by 

F(x) = Rf (x), Rf (xτ ), Rf (xτ 2 ) . Then for every σ ∈ S4 (18)

F(xσ ) = Θ(σ)F(x).

Proof. Proposition (1.8) says that the function Rf satisfies the hypothesis of  Theorem (2.3) with Z being the S4 orbit of x, so the result follows. References [1] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR0163331 Department of Mathematics, Northwestern University, Evanston, IL 60208

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13905

Path connectedness and entropy density of the space of hyperbolic ergodic measures Anton Gorodetski and Yakov Pesin Abstract. We show that the space of hyperbolic ergodic measures of a given index supported on an isolated homoclinic class is path connected and entropy dense provided that any two hyperbolic periodic points in this class are homoclinically related. As a corollary we obtain that the closure of this space is also path connected.

1. Introduction In this paper we consider homoclinic classes of periodic points for C 1+α diffeomorphisms of compact manifolds and we discuss two properties of the space of invariant measures supported on them and equipped with the weak∗ -topology – connectedness and entropy density of the subspace of hyperbolic ergodic measures. The study of connectedness of the latter space was initiated by Sigmund in a short article [32]. He established path connectedness of this space in the case of transitive topological Markov shifts and as a corollary, of Axiom A diffeomorphisms. Sigmund’s idea was to show first that any two periodic measures (i.e., invariant atomic measures on periodic points) can be connected by a continuous path of ergodic measures and second that if one of the two periodic measures lies in a small neighborhood of another one, then the whole path can be chosen to lie in this neighborhood. In order to carry out the first step Sigmund shows that any periodic measure can be approximated by a Markov measure and that any two Markov measures can be connected by a path of Markov measures. We use Sigmund’s idea in our proof of Theorem 1.1. A different approach to Sigmund’s theorem is to show that ergodic measures on a transitive topological Markov shift are dense in the space of all invariant measures. Since the latter space is a simplex and ergodic measures are its extremal points, it means that this space is the Poulsen simplex (which is unique up to an affine homeomorphism). The desired result now follows from a complete description of the Poulsen simplex given in [28] (see also [17]).

2010 Mathematics Subject Classification. Primary 37D25, 37D35, 37A25, 37E30. A. G. was supported in part by NSF grant DMS–1301515. Ya. P. was supported in part by NSF grant DMS–1400027. c 2017 American Mathematical Society

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Since Sigmund’s work the interest to the study of connectedness of the space of hyperbolic ergodic measures has somehow been lost,1 and only recently it has regained attention.2 In [18] Gogolev and Tahzibi, motivated by their study of existence of non-hyperbolic invariant measures, raised a question of whether the space of ergodic measures invariant under some partially hyperbolic systems is path connected. This includes, in particular, the famous example by Shub and Wilkinson [31]. Some results on connectedness and other topological properties of the space of invariant measures were obtained in [8, 17]. All known proofs of connectedness of the space of invariant measures are based on approximating invariant measures by either measures supported on periodic orbits or Markov ergodic measures supported on invariant horseshoes. It is therefore natural to ask whether such approximations can be arranged to also ensure convergence of entropies. If this is possible, the space of approximants is called entropy dense. Some results in this direction were obtained in [23]. We stress that approximating hyperbolic ergodic measures with positive entropy by “nice” measures supported on invariant horseshoes so that the convergence of entropies is also guaranteed, was first done by Katok in [25] (see also [2, 26]). We use this result in the proof of our Theorem 1.5 where we approximate also some hyperbolic ergodic measures with zero entropy as well as non-ergodic measures. We shall now state our results. Consider a C 1+α -diffeomorphism f : M → M of a compact smooth manifold M . Let p ∈ M be a hyperbolic periodic point. By the index s(p) of p we mean the dimension of the invariant stable manifold of p. We say that a hyperbolic periodic point q ∈ M is homoclinically related to p and write q ∼ p if the stable manifold of the orbit of q intersects transversely the unstable manifold of the orbit of p and vice versa.3 Notice that this is an equivalence relation. We denote by H(p) the homoclinic class associated with the point p, that is the closure of the set of hyperbolic periodic points homoclinically related to p. Note that H(p) is f -invariant. Homoclinic classes were introduced by Newhouse in [29]. A basic hyperbolic set gives the simplest example of a homoclinic class, but in general the set H(p) can have a much more complicated structure and dynamical properties. In particular, it can contain non-hyperbolic periodic points, and it can support non-hyperbolic (periodic or not) measures in a robust way, see [3, 5, 11, 27] for a more detailed discussion. 4 It can also contain in a robust way hyperbolic periodic orbits whose index is different than the index of p (i.e. their stable manifolds have different dimensions than the dimension of the stable manifold at p), see [4, 20, 21]. Moreover – and this is of importance for us in this paper – there may exist hyperbolic periodic points in H(p) of the same index as p that are not homoclinically related to p, see [11, 15]. Besides, it can happen that periodic orbits outside the homoclinic class H(p) accumulate to H(p); for example, this is 1 At the time of writing this paper there is no single reference to the paper by Sigmund [32] in MathSciNet. 2 Soon after this paper was completed, several new works related to the subject appeared, see [9, 13] 3 The stable (respectively, unstable) manifold of the orbit of a periodic point is the union of stable (respectively, unstable) manifolds through every point on the orbit. If q ∼ p, then s(q) = s(p). 4 It is conjectured that existence of non-hyperbolic ergodic measures is a characteristic property of non-hyperbolic homoclinic classes, see [3, 11].

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part of the Newhouse phenomena, and also occurs in the family of standard maps, see [16,19]. We wish to avoid both of these complications, and we therefore, impose the following crucial requirements on the homoclinic class H(p): (H1) For any hyperbolic periodic point q ∈ H(p) with s(q) = s(p) we have q ∼ p. (H2) The homoclinic class H(p) is isolated,  i.e., there is an open neighborhood U (H(p)) of H(p) such that H(p) = n∈Z f n (U (H(p))). We stress that these requirements do hold in many interesting cases, see examples in Section 2. In particular, Condition (H2) holds if the map f has only one homoclinic class. This is the case in Examples 1 and 2 in Section 2. We also note a result in [8] that is somewhat related to Condition (H2): if the map f admits a dominated splitting of index s, then a linear combination of hyperbolic ergodic measures of index s can be approximated by a sequence of hyperbolic ergodic measures of index s if and only if their homoclinic classes coincide. The space of all invariant ergodic measures supported on H(p) can be extremely rich and contain hyperbolic measures with different number of positive Lyapunov exponents as well as non-hyperbolic measures. We denote by Mp the space of all hyperbolic invariant measures supported on H(p) for which the number of negative Lyapunov exponents at almost every point is exactly s(p). We say that μ has index s(p). Further, we denote by Mep the space of all hyperbolic ergodic measures in Mp . We assume that the space Mp is equipped with the weak∗ -topology. Theorem 1.1. Under Conditions (H1) and (H2) the space Mep is path connected. Notice that without Conditions (H1) and (H2) the conclusion of Theorem 1.1 may fail, see Subsection 2.2. It follows immediately from Theorem 1.1 that the closure of Mep is connected. In fact, a stronger statement holds. Theorem 1.2. Under Conditions (H1) and (H2) the closure of the space Mep is path connected. Remark 1.3. It is interesting to notice that the closure of Mep is not a Choquet simplex (and hence, not a Poulsen simplex), see Proposition 2.7 in [9]. We shall now discuss the entropy density of the space Mep . Definition 1.4. A subset S ⊆ Mp is entropy dense in Mp if for any μ ∈ Mp there exists a sequence of measures {ξn }n∈N ⊂ S such that ξn → μ and hξn → hμ as n → ∞. Theorem 1.5. Under Conditions (H1) and (H2) the space Mep is entropy dense in Mp . 2. Examples In this section we present some examples that illustrate importance of Conditions (H1) and (H2). 2.1. Non-hyperbolic homoclinic classes satisfying Conditions (H1) and (H2). We describe a class of diffeomorphisms with a partially hyperbolic attractor which is the homoclinic class of any of its periodic points and which satisfies Conditions (H1) and (H2). We follow [10]. Let f be a C 1+α diffeomorphism

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of a compact smooth manifold M and Λ a topological attractor for  f . This means that there is an open set U ⊂ M such that f (U ) ⊂ U and Λ = n≥0 f n (U ). We assume that Λ is a partially hyperbolic set for f , that is for every x ∈ Λ there is an invariant splitting of the tangent space Tx M = E s (x) ⊕ E c (x) ⊕ E u (x) into stable E s (x), central E c (x) and unstable E u (x) subspaces such that with respect to some Riemannian metric on M we have that for some constants 0 < λ1 < λ 2 < λ 3 < λ 4 ,

λ1 < 1,

λ4 > 1

the following holds: (1) df v < λ1 v for every v ∈ E s (x), (2) λ2 v < df v < λ3 v for every v ∈ E c (x), (3) df v > λ4 v for every v ∈ E u (x). If Λ is a partially hyperbolic attractor for f , then for every x ∈ Λ we denote by V u (x) and W u (x) the local and respectively global unstable leaves through x. It is known that for every x ∈ Λ and y ∈ W u (x) one has Ty W u (x) = E u (y), f (W u (x)) = W u (f (x)) and W u (x) ⊂ Λ. Moreover, the collection of all global unstable leaves W u (x) forms a continuous lamination of Λ with smooth leaves, and if Λ = M , then it is a continuous foliation of M with smooth leaves. An invariant measure μ on Λ is called a u-measure if the conditional measures it generates on local unstable leaves V u (x) are equivalent to the leaf volume on V u (x) induced by the Riemannian metric. It is shown in [30] that any partially hyperbolic attractor admits a u-measure: any limit measure for the sequence of measures n−1 1 k μn = f∗ m n k=0

is a u-measure on Λ. Here m is the Riemannian volume in a sufficiently small neighborhood of the attractor (see [30] for more details and other ways for constructing u-measures). In general a u-measure may have some or all Lyapunov exponents along the central direction to be zero.5 Therefore, following [10] we say that a u-measure μ has negative central exponents on an invariant subset A ⊂ Λ of positive measure if for every x ∈ A and v ∈ Tx E c (x) the Lyapunov exponent χ(x, v) < 0. We consider the following requirement on the map f |Λ: (D) for every x ∈ Λ the positive semi-trajectory of the global unstable leaf W u (x) is dense in Λ, that is   f n (W u (x)) = W u (f n (x)) = Λ. n≥0

n≥0

Condition (D) clearly holds if the unstable lamination is minimal, i.e., if every leaf of the lamination is dense in Λ. It is shown in [10] that if μ is a u-measure on Λ with negative central exponents on an invariant subset of positive measure and if f satisfies Condition (D), then 1) μ has negative central exponents at almost every point x ∈ Λ; 2) μ is the unique u-measure for f supported on the whole Λ; and 3) the basin of attraction for μ coincides with the open set U . 5 Clearly, the Lyapunov exponents in the stable direction are negative while the Lyapunov exponents in the unstable direction are positive.

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It is easy to see that in this case: (1) hyperbolic periodic points whose index is equal to the dimension of the stable leaves6 are dense in the attractor Λ; the homoclinic class of each of these periodic points coincides with Λ; (2) the homoclinic class satisfies Conditions (H1) and (H2), and hence, Theorems 1.1, 1.2, and 1.5 are applicable. Let f0 be a partially hyperbolic diffeomorphism which is either 1) a skew product with the map in the base being a topologically transitive Anosov diffeomorphism or 2) the time-1 map of an Anosov flow. If f is a small perturbation of f0 then f is partially hyperbolic and by [24], the central distribution of f is integrable. Furthermore, the central leaves are compact in the first case and there are compact leaves in the second case. It is shown in [10] that f has minimal unstable foliation provided there exists a compact periodic central leaf C (i.e., f  (C) = C for some  ≥ 1) for which the restriction f  |C is a minimal transformation. Furthermore, it follows from the results in [1] that starting from a volume preserving partially hyperbolic diffeomorphism f0 with one-dimensional central subspace, it is possible to construct a C 2 volume preserving diffeomorphism f which is arbitrarily C 1 -close to f0 and has negative central exponents on a set of positive volume. Moreover, if C is a compact periodic central leaf, then f can be arranged to coincide with f0 in a small neighborhood of the trajectory of C. We now consider the two particular examples. Example 1. Consider the time-1 map f0 of the geodesic flow on a compact surface of negative curvature. Clearly, f0 is partially hyperbolic and has a dense set of compact periodic central leaves. It follows from what was said above that there is a volume preserving perturbation f of f0 such that (1) f is of class C 2 and is arbitrary close to f0 in the C 1 -topology; (2) f is a partially hyperbolic diffeomorphism with one-dimensional central subspace; (3) there exists a central leaf C such that the restriction f  |C is a minimal transformation (here  is the period of the leaf); (4) f has negative central exponents on a set of positive volume; (5) the unstable foliation for f is minimal and hence, satisfies Condition (D). We conclude that in this example the whole manifold is the homoclinic class of every hyperbolic periodic point of index two and that this class satisfies Conditions (H1) and (H2). Example 2. Consider the map f0 = A × R of the 3-torus T 3 = T 2 × T 1 where A is a linear Anosov automorphism of the 2-torus T 2 and R is an irrational rotation of the circle T 1 . It follows from what was said above that there is a volume preserving perturbation f of f0 such that the properties (1) – (5) in the previous example hold, and hence the unique homoclinic class satisfies Conditions (H1) and (H2). Remark 2.1. It was shown in [6] that the set of partially hyperbolic diffeomorphisms with one dimensional central direction contains a C 1 open and dense subset of diffeomorphisms with minimal unstable foliation. However, in our examples we use preservation of volume to ensure negative central Lyapunov exponents on a 6 This

dimension is dim E s + dim E c .

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set of positive volume, so we cannot immediately apply the result in [6] to obtain an open set of systems for which Conditions (H1) and (H2) hold, compare with Problem 7.25 from [7]. Remark 2.2. In both Examples 1 and 2 the map possesses a non-hyperbolic ergodic invariant measure (e.g. supported on the compact periodic leaf). We believe that in these examples presence of non-hyperbolic ergodic invariant measures is persistent under small perturbations. Indeed, since the central subspace is one dimensional, the central Lyapunov exponent with respect to a given ergodic measure is an integral of a continuous function (i.e., log of the expansion rate along the central subspace) over this measure, existence of periodic points of different indices combined with (presumable) connectedness of the space of ergodic measures should imply existence of a non-hyperbolic invariant ergodic measure. See [3, 5, 14, 22] for the related results and discussion. 2.2. Homoclinic classes that do not satisfy Conditions (H1) and (H2). There is an example of an invariant set for a partially hyperbolic map with one dimensional central subspace which is a homoclinic class containing two nonhomoclinically related hyperbolic periodic orbits of the same index, hence, not satisfying Condition (H1), see [11, 12, 15]. Moreover, the space of hyperbolic ergodic measures supported on this homoclinic class is not connected due to the fact that the set of all central Lyapunov exponents is split into two disjoint closed intervals, see Remark 5.2 in [12]. As we already mentioned in Introduction, Condition (H2) does not always hold even for surface diffeomorphisms, see for example, [16, 19]. This condition ensures that the hyperbolic horseshoes and periodic orbits that we use to approximate a given hyperbolic ergodic measure do belong to the initial homoclinic class. We do not know whether given a not necessarily isolated homoclinic class, every hyperbolic ergodic invariant measure supported on this homoclinic class can always be approximated in such a way.

3. Proofs The space M of all probability Borel measures on M equipped with the weak∗ topology is metrizable with the distance dM given by  ∞   1  ψk dμ − ψk dν  , (1) dM (μ, ν) =  k 2 k=1

where {ψk }k∈N is a dense subset in the unit ball in C 0 (M ). While the distance defined in this way depends on the choice of the subset {ψk }k∈N , the topology it generates does not. We will choose the functions ψk to be smooth. Proof of Theorem 1.1. By a hyperbolic periodic measure μq we mean an atomic ergodic measure equidistributed on a hyperbolic periodic orbit of q. Lemma 3.1. Let q1 , q2 ∈ H(p) be hyperbolic periodic points with index s(q1 ) = s(q2 ) = s(p). Then the hyperbolic periodic measures μq1 and μq2 can be connected in Mep by a continuous path.

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Proof of Lemma 3.1. By Condition (H1), the points q1 and q2 are homoclinically related. By the Smale-Birkhoff theorem, there is a hyperbolic horseshoe Λ 7 that contains both q1 and q2 . Lemma 3.1 now follows from the results by Sigmund, see the proof of Theorem B in [32].  We wish to approximate a given hyperbolic measure by periodic measures. There are several results in this direction, see, for example, [2, Theorem 15.4.7]. However, we need some specific properties of such approximations that are stated in the following lemma. Lemma 3.2. For any μ ∈ Mep and any ε > 0 the following statements hold: (1) There exists a hyperbolic periodic point q ∈ H(p) with s(q) = s(p) such that we have dM (μq , μ) < ε for the corresponding hyperbolic periodic measure μq ; (2) There exists δ > 0 such that for any hyperbolic periodic points q1 , q2 ∈ H(p) with s(q1 ) = s(q2 ) = s(p) if the corresponding hyperbolic periodic measures μq1 and μq2 lie in the δ-neighborhood of the measure μ, then there exists a continuous path {νt }t∈[0,1] ⊂ Mep with ν0 = μq1 , ν1 = μq2 and such that dM (νt , μ) < ε for all t ∈ [0, 1]. Proof of Lemma 3.2. Let R be the set of all Lyapunov-Perron regular points, and for each  ≥ 1 let R be the regular set (see [2] for definitions). There exists  ∈ N such that μ(R ) > 0. For a μ-generic point x ∈ R , by Birkhoff’s Ergodic Theorem, there exists N ∈ N such that for any n > N / . n−1 ε 1 δf k (x) , μ < . (2) dM n 2 k=0

∞

1 Choose L ∈ N such that k=L+1 2k−1 < 4ε . Let {ψk } be the dense collection of smooth functions {ψk } from the definition (1) of the distance dM and let C = C(ε) be the common Lipschitz constant of the functions {ψ1 , . . . , ψL }. Let U (H(p)) be  a neighborhood of H(p) such that H(p) = n∈Z f n (U (H(p))); its existence is guaranteed by (H2). Let us now choose δ > 0 such that Cδ < 4ε and δ-neighborhood of H(p) is in U . Since μ is a hyperbolic measure, by [2, Theorem 15.1.2], there exists n > N and a hyperbolic periodic point y ∈ M of period n such that distM (f k (x), f k (y)) < δ for all k = 0, . . . , n − 1 and s(y) = s(x). Then the orbit of y is in U , and hence belongs to H(p). Also, we have . n−1 / n−1 1 1 dM δf k (x) , δf k (y) n n k=0 k=0 . n−1 . n−1 / / L ∞    1 1 1  1 (3)  ≤ δ i δ i − ψk d +  ψk d 2k  n i=0 f (x) n i=0 f (y)  2k−1 k=1

k=L+1

ε ε ≤Cδ + < . 4 2 The first statement of the lemma now follows from (2) and (3). To prove the second statement let q1 , q2 ∈ H(p) be any hyperbolic periodic points such that s(q1 ) = s(q2 ) = s(p) and the corresponding hyperbolic periodic 7 By a hyperbolic horseshoe we mean a locally maximal hyperbolic set Λ which is totally disconnected and such that f |Λ is topologically transitive.

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measures μq1 and μq2 lie in the δ-neighborhood of the measure μ. By Conditions (H1) and (H2), the points q1 and q2 are homoclinically related and hence, there is a hyperbolic horseshoe which contains both points. The desired result now follows from [32] (see the proof of Theorem B).  We now compete the proof of Theorem 1.1. Let η and η! ∈ Mep be two hyperbolic ergodic measures. By Statement 1 of Lemma 3.2, there are sequences of hyperbolic qk ) = s(p) such that for the periodic points qk and q!k in H(p) with s(qk ) = s(! corresponding sequences of hyperbolic periodic measures {μqk }k∈N and {μqk }k∈N we have μqk → η and μqk → η!. By Lemma 3.1, there is a path {νt }t∈[ 1 , 2 ] in Mep 3 3 that connects μq1 and μq1 , that is, ν 13 = μq1 and ν 23 = μq1 . By Statement 2 of Lemma 3.2, for any k ∈ N there are paths {νt }t∈[

1 3k+1

,

1 3k

1 ] and {νt }t∈[1− 31k ,1− 3k+1 ]

in Mep that connect measures μqk , μqk+1 and measures μqk , μqk+1 , respectively. Applying again Lemma 3.2, we conclude that the path {νt }t∈[0,1] given by the above choices and such that ν0 = η and ν1 = η! is continuous. The desired result now follows.  Proof of Theorem 1.2. Arguments similar to those used in the proof of Lemma 3.2 (see also the proof of Theorem B in [32]) show that the following statement holds: Lemma 3.3. For any ε > 0 there exists δ > 0 such that for any two measures μ1 , μ2 ∈ Mep with dM (μ1 , μ2 ) < δ there exists a continuous path in Mep connecting μ1 and μ2 of diameter smaller than ε. Now Theorem 1.2 can be obtained using Lemma 3.3 in the same way Theorem 1.1 was obtained using Lemma 3.2.  Proof of Theorem 1.5. Given a (not necessarily ergodic) measure μ ∈ Mp , by the ergodic decomposition, there exists a measure ν on the space Mep such that μ = τ dν(τ ) and hμ = hτ dν(τ ). It follows that for any ε > 0 there are measures τ1 , . . . , τN ∈ Mep and positive coefficients α1 , . . . , αN such that   . N / N       αk τk < ε and hμ − αk hτk  < ε. (4) dMp μ,   k=1

k=1

Given a hyperbolic ergodic measure τ with hτ > 0, there exist a sequence of hyperbolic horseshoes Λn and a sequence of ergodic measures {νn }n∈N supported on Λn such that νn → τ and hνn → hτ as n → ∞, see, for example, Corollary 15.6.2 in [2]. By (H2), one can ensure in the construction of these horseshoes that Λn ⊆ H(p) and that νn are the measures of maximal entropy and hence, Markov measures. Further, for every x ∈ Λn the dimension of the stable manifold through x is equal to the index of p. In the case when hτ = 0 the measure τ can be approximated by a hyperbolic periodic measure supported on an orbit of a hyperbolic periodic point q ∈ H(p) (see Lemma 3.2) with s(q) = s(p). There exists a horseshoe Λq ⊂ H(p) that contains a periodic point q, and one can choose a Markov measure (which is not a

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measure of maximal entropy in this case) supported on Λq that is arbitrary close to the atomic invariant measure supported on the orbit of q, and has arbirary small entropy (notice that the support of this Markov measure does not have to be close to the orbit of q). It follows from what was said above that to each ergodic measure τk we can associate a hyperbolic horseshoe Λk ⊆ H(p) and a Markov measure νk supported on Λk such that for every k = 1, . . . , N we have ε ε and |hτk − hνk | < . (5) dMp (τk , νk ) < N N Notice that all horseshoes Λk have the same index s(p) and that they are homoclinically related (this means that every periodic orbit in one of the horseshoes is homoclinically related to any periodic orbit in the other horseshoe). This implies that there exists a hyperbolic horseshoe Λ ⊂ H(p) that contains all Λk . The Markov measure νk is constructed with respect to a Markov partition of Λk that we denote by ξk . There exists a Markov partition ξ of Λ such that its N restriction on each Λk is a refinement of ξk . The measure k=1 αk νk is a Markov measure on Λ with respect to the partition ξ. Notice that Markov measures as well as their entropies depend continuously on their stochastic matrices. Therefore, given an arbitrarily (not necessarily ergodic) Markov measure, one can produce its small perturbation which is an ergodic Markov measure whose entropy is close to the entropy of the unperturbed one. This gives the required approximation of the N measure k=1 αk νk , which by (4) and (5) is close to the initial measure μ and  whose entropy is close to hμ . References [1] Alexandre T. Baraviera and Christian Bonatti, Removing zero Lyapunov exponents, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1655–1670, DOI 10.1017/S0143385702001773. MR2032482 [2] Luis Barreira and Yakov Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. Dynamics of systems with nonzero Lyapunov exponents. MR2348606 [3] Jairo Bochi, Christian Bonatti, and Lorenzo J. D´ıaz, Robust criterion for the existence of nonhyperbolic ergodic measures, Comm. Math. Phys. 344 (2016), no. 3, 751–795, DOI 10.1007/s00220-016-2644-5. MR3508160 [4] Christian Bonatti and Lorenzo J. D´ıaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. (2) 143 (1996), no. 2, 357–396, DOI 10.2307/2118647. MR1381990 [5] Christian Bonatti, Lorenzo J. D´ıaz, and Anton Gorodetski, Non-hyperbolic ergodic measures with large support, Nonlinearity 23 (2010), no. 3, 687–705, DOI 10.1088/0951-7715/23/3/015. MR2593915 [6] Christian Bonatti, Lorenzo J. D´ıaz, and Ra´ ul Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu 1 (2002), no. 4, 513–541, DOI 10.1017/S1474748002000142. MR1954435 [7] Christian Bonatti, Lorenzo J. D´ıaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR2105774 [8] Ch. Bonatti, K. Gelfert, Dominated Pesin theory: convex sum of hyperbolic measures, preprint (arXiv:1503.05901). [9] Ch. Bonnatti, J. Zhang, Periodic measures and partially hyperbolic homoclinic classes, preprint, arXiv:1609.08489 [10] Keith Burns, Dmitry Dolgopyat, Yakov Pesin, and Mark Pollicott, Stable ergodicity for partially hyperbolic attractors with negative central exponents, J. Mod. Dyn. 2 (2008), no. 1, 63–81, DOI 10.3934/jmd.2008.2.63. MR2366230

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[11] Lorenzo J. D´ıaz and Katrin Gelfert, Porcupine-like horseshoes: transitivity, Lyapunov spectrum, and phase transitions, Fund. Math. 216 (2012), no. 1, 55–100, DOI 10.4064/fm216-1-2. MR2864450 [12] Lorenzo J. D´ıaz, Katrin Gelfert, and Michal Rams, Rich phase transitions in step skew products, Nonlinearity 24 (2011), no. 12, 3391–3412, DOI 10.1088/0951-7715/24/12/005. MR2854309 [13] L. Diaz, K. Gelfert, M. Rams, Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products, preprint. [14] Lorenzo J. D´ıaz and Anton Gorodetski, Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes, Ergodic Theory Dynam. Systems 29 (2009), no. 5, 1479–1513, DOI 10.1017/S0143385708000849. MR2545014 [15] L. J. D´ıaz, V. Horita, I. Rios, and M. Sambarino, Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes, Ergodic Theory Dynam. Systems 29 (2009), no. 2, 433–474, DOI 10.1017/S0143385708080346. MR2486778 [16] Pedro Duarte, Plenty of elliptic islands for the standard family of area preserving maps (English, with English and French summaries), Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 11 (1994), no. 4, 359–409. MR1287238 [17] K. Gelfert, D. Kwietniak, The (Poulsen) simplex of invariant measures, arxiv.org/pdf/1404.0456. [18] Andrey Gogolev and Ali Tahzibi, Center Lyapunov exponents in partially hyperbolic dynamics, J. Mod. Dyn. 8 (2014), no. 3-4, 549–576, DOI 10.3934/jmd.2014.8.549. MR3345840 [19] Anton Gorodetski, On stochastic sea of the standard map, Comm. Math. Phys. 309 (2012), no. 1, 155–192, DOI 10.1007/s00220-011-1365-z. MR2864790 [20] A. S. Gorodetski˘ı, Regularity of central leaves of partially hyperbolic sets and applications (Russian, with Russian summary), Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 6, 19– 44, DOI 10.1070/IM2006v070n06ABEH002340; English transl., Izv. Math. 70 (2006), no. 6, 1093–1116. MR2285025 [21] A. S. Gorodetski˘ı and Yu. S. Ilyashenko, Some new robust properties of invariant sets and attractors of dynamical systems (Russian, with Russian summary), Funktsional. Anal. i Prilozhen. 33 (1999), no. 2, 16–30, 95, DOI 10.1007/BF02465190; English transl., Funct. Anal. Appl. 33 (1999), no. 2, 95–105. MR1719330 [22] A. S. Gorodetski˘ı, Yu. S. Ilyashenko, V. A. Kleptsyn, and M. B. Nalski˘ı, Nonremovability of zero Lyapunov exponents (Russian, with Russian summary), Funktsional. Anal. i Prilozhen. 39 (2005), no. 1, 27–38, 95, DOI 10.1007/s10688-005-0014-8; English transl., Funct. Anal. Appl. 39 (2005), no. 1, 21–30. MR2132437 [23] Hans F¨ ollmer and Steven Orey, Large deviations for the empirical field of a Gibbs measure, Ann. Probab. 16 (1988), no. 3, 961–977. MR942749 [24] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR0501173 [25] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes ´ Etudes Sci. Publ. Math. 51 (1980), 137–173. MR573822 [26] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR1326374 [27] V. A. Kleptsyn and M. B. Nalski˘ı, Stability of the existence of nonhyperbolic measures for C 1 -diffeomorphisms (Russian, with Russian summary), Funktsional. Anal. i Prilozhen. 41 (2007), no. 4, 30–45, 96, DOI 10.1007/s10688-007-0025-8; English transl., Funct. Anal. Appl. 41 (2007), no. 4, 271–283. MR2411604 [28] J. Lindenstrauss, G. Olsen, and Y. Sternfeld, The Poulsen simplex (English, with French summary), Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, vi, 91–114. MR500918 [29] Sheldon E. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc. 167 (1972), 125–150, DOI 10.2307/1996131. MR0295388 [30] Ya. B. Pesin and Ya. G. Sina˘ı, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 417–438 (1983), DOI 10.1017/S014338570000170X. MR721733 [31] Michael Shub and Amie Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508, DOI 10.1007/s002229900035. MR1738057

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[32] Karl Sigmund, On the connectedness of ergodic systems, Manuscripta Math. 22 (1977), no. 1, 27–32. MR0447528 Department of Mathematics, University of California, Irvine, California 92697 E-mail address: [email protected] Department of Mathematics, Penn State University, University Park, Pennsylvania 16802 E-mail address: [email protected]

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13918

Around Anosov-Weil Theory V. Grines and E. Zhuzhoma Abstract. The survey is devoted to the exposition of main results of AnosovWeil Theory that studies nonlocal asymptotic properties of simple curves on a surface with a non-positive constant curvature. This study consists of the lifting these curves to an universal covering and making a “comparison” in a sense with lines of constant geodesic curvature. We review some applications conserning constructions of topological invariants for surface dynamical systems and foliations.

Contents Introduction 1. Mathematical background 2. Historical background 3. Nonlocal behavior of curves on universal coverings 4. Asymptotic properties of special curves 5. Applications to foliations and dynamical systems References

Introduction In 1966 our friend and teacher S.Kh. Aranson met Dmitrii Viktorovich Anosov at Tiraspol (Moldova former part of Soviet Union) at the Symposium on General Topology. It became clear that dynamical systems (even, structurally stable) can have complex dynamics with nontrivially recurrent orbits. This is related to the problem of the topological classification of dynamical systems with complex dynamics beginning with the simplest, in a sense, systems such as surface flows. A classical example of an effective topological invariant is given by the Poincar´e rotation number for fixed-point-free flows on the two-dimensional torus T2 [62]. This number determines the “asymptotical rotation” of trajectories along the meridians and parallels of the torus. It is well known that when all trajectories are nontrivially recurrent,1 the rotation number is a complete topological invariant up to the recalculation by an integer unimodular matrix. 2010 Mathematics Subject Classification. Primary 37E30, 37E35, 37D05. Key words and phrases. Surface, geodesic, dynamical system. 1 Earlier, following Poincar´ e, such trajectories were called nonclosed Poisson stable trajectories [62]. c 2017 American Mathematical Society

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In Tiraspol D.V. Anosov formulated the idea that a clue to the construction of effective topological invariants for dynamical systems with nontrivially recurrent motions (including foliations with nontrivially recurrent leaves) on surfaces consists of studying nonclosed curves without self-intersections that possess certain recurrent properties and in investigating the nonlocal asymptotic behavior of the lifts of these curves to the universal covering by means of the absolute (circle at infinity). Later, the development of this idea led to the topological classification of the basic classes of flows, foliations, 2-webs, nontrivial one-dimensional basic sets, and homeomorphisms with invariant foliations on closed surfaces of constant nonpositive curvature. In 1973, while developing the above-mentioned geometric interpretation of the Poincar´e rotation number, S. Aranson and V. Grines [19] constructed a complete topological invariant for irrational flows on orientable closed surfaces of constant negative curvature. An explicit use of the universal covering in the study of the nonlocal asymptotic behavior of the trajectories of fixed-point-free flows on T2 was first proposed by A. Weil [65] in 1932. Before him, following Poincar´e, mathematicians used a global section and the first-return map on this section. A. Weil proposed an alternative definition for the rotation number. This definition employs the trajectories of a covering flow on the Euclidean plane. Namely, Weil proved that the rotation number is equal to the angular coefficient of a straight line that has the same asymptotic direction as the trajectories of the covering flow [65]. His arguments were based on the fact that the lifts of the trajectories are pairwise disjoint and that each lift divides the Euclidean plane. This fact prompted Weil to suppose that curves without self-intersections, not necessarily defined by differential equations, should possess similar properties. In his lecture delivered at the Moscow International Topological Conference in 1935, Weil formulated two conjectures on the behavior of covering curves for curves without self-intersections [66]. The first conjecture (formulated as a theorem and referred to as the Weil theorem below) stated that the lift of a curve without self-intersections on T2 to the universal covering has an asymptotic direction if this lift is an unbounded curve and goes to infinity. The second conjecture was similar to the first one but referred to closed surfaces of negative Euler characteristic (exact statements of the conjecture and theorem are given below). Unfortunately, Weil’s approach was not developed and was soon forgotten. However, in the early 1960s, interest in this subject was renewed by Anosov within the context of a general upsurge in the theory of dynamical systems. The problem from which Anosov started his studies consisted of determining the common features in the asymptotic behavior of trajectories and geodesics. This problem naturally led Anosov to the investigation of trajectories on the universal covering and to the study of their nonlocal asymptotic behavior. In Tiraspol in 1966, Anosov communicated the theorem stating that the coverings for the trajectories of a smooth flow with a finite number of fixed points on a compact surface of nonpositive Euler characteristic have asymptotic directions. He also formulated several conjectures (one of which generalized the Weil conjecture) on the behavior of coverings for curves without self-intersections. These conjectures, the Anosov theorem, and a number of his subsequent works [2]–[10] catalyzed the development of the whole theory. In view of these circumstances, the field of inquiry in question was called the “Anosov– Weil Problem” or “Anosov–Weil Theory” in the studies of mathematicians from

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Nizhni Novgorod (and later in the studies of other mathematicians). Now, roughly speaking, Anosov-Weil Theory includes the following two parts: • a study of nonlocal asymptotic properties of simple curves on a surface by lifting these curves to an universal covering, and making a “comparison” in a sense with lines of constant geodesic curvature; • an application of nonlocal asymptotic properties for constructions of topological invariants for surface dynamical systems and foliations. Simultaneously with Anosov’s works, Weil’s approach was also considered by N. Markley [50]–[52], who paid more attention to the flows. However, these studies were not so widely recognized in the USA as Anosov’s works in the USSR. Acknowledgments. We thank A. Katok and Ya. Pesin for the invitation to submit this survey. We also thank M. Brin for very careful reading of the manuscript and helpful comments. This work was supported by the Russian Foundation for Basic Research (project nos. 15-01-03687-a, 13-01-12452-ofi-m,16-51-10005 KO-a), Russian Science Foundation (project no 14-41-00044) and was done in the frames of Basic Research Programs at the HSE (project 98 in 2016) and (project T-90 in 2017). 1. Mathematical background We give here the main definitions of Anosov-Weil Theory. We consider surfaces being complete Riemannian manifolds M 2 of constant nonpositive curvature. For simplicity, we restrict ourself to closed surfaces. 1.1. Universal covering and the circle at infinity. The universal covering 2 space M for M 2 is isometric either to the Euclidean plane R2 (in the case of zero curvature and Euler characteristic χ(M 2 ) = 0) or to the hyperbolic plane Δ (in the case of negative curvature and Euler characteristic χ(M 2 ) < 0). Accordingly, M 2 is isometric either to R2 /Γ or to Δ/Γ (hyperbolic surface), where Γ is a properly 2 discontinuous group of isometries. Denote by π : M → M 2 the universal covering map, which is a local isometry. Given a curve C ⊂ M 2 , a lift of C is an arcwise connected component of the pull back π −1 (C). Often the choice of this component is clear from the context. The Euclidean plane R2 endowed with the standard quadratic form ds2 = 2 dx +dy 2 is the simplest flat surface. Sometimes it is convenient to use the unit disk D2 with coordinates ξ, η as a universal covering space: D2 = {(ξ, η) : ξ 2 + η 2 < 1}. One can check that the map (1) is a homeomorphism denoted by τ : D2 → R2 . Then π◦τ is also a universal covering map, see Fig. 1, (a). The boundary S∞ = ∂D2 is called the circle at infinity or absolute. (1)

ξ x= 0 , 1 − ξ 2 − η2

η y= 0 . 1 − ξ 2 − η2

For the hyperbolic plane Δ, we use the Poincar´e model Δ = {z ∈ C : |z| < 1}, Fig. 1, (b). Sometimes we consider Δ as the unit disk on R2 with the topology and metric induced by R2 . Denote by dE (·, ·) (respectively, dNE (·, ·)) the Euclidean (respectively, non-Euclidean) metric on Δ.

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Bk

y=kx+c x

Ak

S

Figure 1. (a) Coverings of the torus; (b) Covering of a hyperbolic surface One introduces the absolute, or the circle at infinity (sometimes, one say circle composed of infinitely remote points) S∞ = {z ∈ C : |z| = 1}. These points do not belong to the hyperbolic plane; however, they play a very important role in hyperbolic geometry. The geodesics are arcs of Euclidean circles and straight lines orthogonal to S∞ . We will suppose that endpoints of geodesics, ideal endpoints, belong to S∞ . Let l+ ⊂ M 2 be a semi-infinite continuous curve endowed with an injective + parametrization t → l+ (t), t ∈ R+ , and l a lift of l+ under a universal covering 2 2 map π : M → M 2 . The universal covering surface M can be thought of as the open unit disk D2 ⊂ R2 provided M 2 is a flat or hyperbolic surface. Let dE (·, ·) be the metric on S∞ ∪ D2 induced by the standard metric of R2 . The parametrization + + of l+ induces the parametrization of t → l (t) such that π(l (t)) = l+ (t). A + point σ ∈ S∞ is called the remote limit point of l if there is a sequence tk , + limk→∞ tk = ∞, such that dE (σ, l (tk )) → 0 as k → ∞. The limit set at infinity + + + + lim∞ (l ) of l is the union of all remote limit points of l . Denote by lim(l ) the set of (ordinary) limit points that belong to M . The union of the limit set that belongs to M and the limit set at infinity is called the complete limit set, + + + Lim(l ) = lim(l ) ∪ lim∞ (l ). Now, we present some ways for specifying points of S∞ . For the flat surfaces (torus and Klein bottle), the universal covering M is R2 . Every point σ ∈ S∞ corresponds to oriented parallel rays y = kx + c, c ∈ R, with the same angular coefficient (including ∞) k. Rationality (irrationality) of k corresponds to rationality (irrationality) of σ. For the sake of generality, assume that ∞ is a rational “number.” We see that pairs of diametrically opposite points are parameterized by angular coefficients k ∈ R ∪ {∞}. This specification of points of S∞ is often quite sufficient for the case of flat surfaces. Any ray with k ∈ Q (respectively, k ∈ / Q) projects to a closed (respectively, unclosed) geodesic, and vise versa, rational points of S∞ are exactly ideal points of lifts of closed geodesics. Now, we consider the description of points of S∞ for the hyperbolic plane Δ, which is the universal covering for hyperbolic surfaces. Let Γ be the group of deck transformations that acts on Δ. Each deck transformation is extended to S∞ . By definition, a fixed point of a deck transformation that belongs to S∞ is called a rational point. Denote by R the union of all rational points. Rational points are exactly ideal endpoints of lifts of all closed geodesics. The remaining points I = S∞ − R are called irrational points. Every point of S∞ corresponds

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to a family of oriented collinear parallel geodesics. However, there does not exist a convenient generally accepted method for assigning a certain number to such a family of geodesics. There are different types of coding that depend on the choice of the generators of the fundamental group of a surface (see [47], [55], [56]). Below, to specify points of S∞ , we’ll consider ideal endpoints of the geodesics that belong to lifts of special geodesic laminations. We’ll see that the description of irrational points is much more rich than in the case of flat surfaces. One may introduce the circle at infinity through families of parallel directed geodesics or geodesic rays, see for example [34, 35]. Any geodesic from this family is called a representative of the point at infinity. Our definition of S∞ gives the same object. 1.2. Asymptotic directions and co-asymptotic geodesics. We describe some possible types of asymptotic behavior of semi-infinite curves. Let l+ = {m(t) : t ≥ 0} ⊂ M 2 be a semi-infinite simple (i.e. without self-intersections) curve and + 2 l = {m(t) : t ≥ 0} its lift to the universal covering M endowed with the metric 2 2 d (d = dE if M = R2 , and d = dNE if M = Δ). + 2 One says that l leaves any compact subset of M , or is unbounded, if lim sup d(a0 , m(t)) = +∞,

(2)

t→+∞ 2

where a0 ∈ M is an arbitrary point, Fig. 2 (a). It is clear that this definition does not depend on the choice of a0 . A curve that belongs to some compact subset of 2 M is called bounded. + We say that l goes to infinity if (3)

lim d(a0 , m(t)) = +∞.

t→+∞

In general, (2) does not imply (3). Obviously, the converse is true: a curve that 2 goes to infinity sooner or later leaves any compact subset of M and never returns to it, Fig. 2 (b). A basic definition of the Anosov-Weil Theory is the following one. + Let l+ = {m(t) : t ≥ 0} ⊂ M 2 be a semi-infinite simple curve and l = {m(t) : + 2 2 t ≥ 0} ⊂ M its lift, where M is either D2 or Δ. If l tends exactly to one + + + point σ ∈ S∞ , Lim(l ) = lim∞ (l ) = σ, then we say that l has an asymptotic direction σ. + 2 Roughly speaking, for an observer situated on M , the curve l goes exactly to one point of the horizon, Fig. 2 (c). The point σ is called a point accessible + + + (or reached) by the curve l . One also says that σ = ω(l ) is attained by l . An asymptotic direction is called rational (irrational) if the point σ ∈ S∞ is rational (respectively, irrational). Clearly, if some lift has an asymptotic direction, then any lift also has an − asymptotic direction. For a curve l that is semi-infinite in the negative direction, − the asymptotic direction and its accessible point α(l ) are defined similarly. Let l = {m(t) ∈ M 2 : −∞ < t < +∞} ⊂ M 2 be a simple infinite continuous 2 curve and l = {m(t) : −∞ < t < +∞} ⊂ M a lift of l. The point m(0) divides + l into two semi-infinite curves: the positive l = {m(t) : t ≥ 0} and negative − + − l = {m(t) : t ≤ 0}. Suppose that l and l have asymptotic directions ω(l) ∈ S∞ and α(l) ∈ S∞ , respectively, and α(l) = ω(l). Then there exists a geodesic g(l) with

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S

(l)= l

(a)

(c)

(b)

Figure 2. Unbounded curve (a); a curve goes to infinity (b); the + + point ω(l ) is reached by l . the same ideal endpoints α(l) and ω(l) oriented from α(l) to ω(l). The geodesic g(l) is called co-asymptotic or corresponding to l, Fig. 3. It is easy to see that the def geodesic π(g(l)) = g(l) on M 2 does not depend on the choice of l ∈ π −1 (l) and is called co-asymptotic or corresponding to l. _

l

_

(l) _ _

g(l) _

(l) Figure 3. Co-asymptotic geodesic. Clearly, a co-asymptotic or corresponding geodesic to a simple non-null-homotopic closed curve is the closed geodesic that is freely homotopic to the closed curve. For the hyperbolic surface, the co-asymptotic geodesic is unique, due to properties of hyperbolic geometry. For the torus, the co-asymptotic geodesic is not unique and must be specified. 2. Historical background Andr´e Weil [65] applied a covering flow to get a geometrical interpretation of the Poincar´e rotation number for a fixed-points-free flow on T2 . On account of the above terminology A. Weil proved that the covering trajectories of such flows must have an asymptotic direction. Weil’s method of the study of asymptotic directions is more geometric than the Poincar´e’s method which consists of the study of the first return maps on global cross-sections to the flows. What is more important Weil apparently inferred that his method works not exclusively for the torus flows but also for the higher genus surface flows and is applicable as well to arbitrary families of curves not necessarily given by the differential equations. This led him to the

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two conjectures (quoted below) on a nonlocal asymptotic behavior of a lift for any curve without topologically transversal self-intersections. We quote the original of his talk at the First international topological conference in Moscow [66] held in 1935: ”Dans la pr´esente communication, l’auteur discute deux m´ethodes pouvant servir a ` l’etude de la question et d’autres analogues. La premi`ere, qui a d´ej` a ´et´e d´evelopp´ee dans un article du [65], consiste ` a consid´erer dans le plan (x, y) en mˆeme temps que la courbe C de la famille, toutes les courbes Cp,q qui s’en d´eduisent par une translation (p, q), p et q ´etant des entiers: la position relative de ces courbes par rapport ` a C permet, non seulement de d´eterminer le nombre de rotation, mais encore la transformation qui ram`ene la famille ´etudi´ee ` a une forme canonique. La m´ethode s’applique dans le cas de Poincar´e, et plus g´en´eralement chaque fois que la famille ne pr´esante pas de ‘col ` a l’infini’ (au sens de Niemytzky). D’ailleurs cette derni`ere circonstance ne peut vraisemblablement pas se pr´esenter si la famille ne ` cette m´ethode se relie encore le th´eor`eme suivant, contient pas de courbe ferm´ee. A d’ailleurs obtenu par une voie quelque peu diff´erente: Soit, sur le tore, une courbe de Jordan, image continue de la demi-droite 0 ≤ t < +∞; on suppose que cette courbe soit sans point double; alors, si l’image de la courbe dans le plan (x, y), surface de recouvrement universelle du tore, tend vers l’infini avec t, elle y tend avec une direction asymptotique bien d´etermin´ee, c’esta-dire que la rapport x(t) ` y(t) tend vers une limite quand t tend vers +∞. Une g´en´eralisation tr`es int´eressante du probll`eme ´etudi´e, qui paraˆıt susceptible d’ˆetre abord´ee par la mˆeme m´ethode, est l’´etude, sur une surface close de genre p, des solutions d’une ´equation diff´erentielle du premier ordre n’ayant d’autres points singuliers que de cols, ou en termes topologiques, d’une famille de courbes dont tous les points singuliers sont d’indice n´egatif. Un premier r´esultat est suivant: Sur le cercle hyperbolique, surface de recouvrement universelle de la surface ´etudi´ee, toute courbe de la famille tend, dans chaque direction, vers un point ` a l’infinie bien d´etetmin´e. ... Actually, A.Weil singled out two conjectures on the behavior of the covering of curves without self-intersections. The first conjecture says that the covering of a curve without self-intersections on the torus has an asymptotic direction, provided this covering goes to infinity. Since A.Weil informed that this statement was proved by Magnier, one formulates this conjecture as a theorem. Theorem 2.1 (Theorem of Weil). Let l = {m(t) : t ≥ 0} be a semi-infinite (continuous) curve without self-intersections on the torus T 2 , and let l = {m(t) : t ≥ 0} be its lift to D2 . If the curve l goes to infinity, it has an asymptotic direction. The second conjecture is similar to the first conjecture and is applied to the higher genus surfaces. Conjecture 2.1 (Conjecture of Weil). Let l = {m(t) : t ≥ 0} be a semi-infinite (continuous) curve without self-intersections on a closed hyperbolic surface M 2 , and let l = {m(t) : t ≥ 0} be its lift to Δ. If the curve l goes to infinity, it has an asymptotic direction. Proof of Conjecture 2.1. Suppose that l does not have an asymptotic direction. Since the curve l goes to infinity, its limit set at infinity contains at least two points and coincides with the complete limit set, Lim(l) = lim∞ (l). The complete limit

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set of the lift of a semi-infinite curve is closed and connected. Therefore, Lim(l) contains a nontrivial interval, which we denote by I ⊂ S∞ . Since the group Γ of deck transformations is a Fuchsian group of the first kind, there exists a hyperbolic isometry γ ∈ Γ such that the ideal endpoints of its axis O(γ) belong to the interval I. Note that O(γ) is projected to a closed geodesic on the surface. Take a sufficiently long interval A ⊂ O(γ) such that one of its endpoints is mapped by γ into A. The interval A divides the axis O(γ) into two subintervals A1 and A2 . Each of these subintervals has one ideal endpoint in I. Since the curve l goes to infinity, it does not intersect A starting from a certain moment. The fact that I belongs to the limit set of the curve l implies that there exists an arc S of the curve l that intersects O(γ) only at the endpoints, such that one of the endpoints is in A1 and the other in A2 , Fig. 4. But then the curve l has self-intersections

_

A1

_

_ S

I

_

A

(S)

_

A2

Figure 4. The arcs S and γ(S) intersect. because S and γ(S) intersect. This contradicts the assumption. 2 Due to unclear reasons, neither Weil nor Magnier have ever published the proof of their statements. Unfortunately Weil’s idea was ignored until the 1960s. In the 60s within the framework of the general progress in dynamical systems Anosov revived the interest to this problem. Anosov’s study was motivated by the common asymptotic behavior which the trajectories of a surface flow and the geodesics curves can exhibit. In the 1960s, on the the American continent apparently under M. Morse’s influence, G. Hedlund brought to the attention of N.G. Markley (who was then his student) all the bunch of problems connected with the area [50]. Unfortunately only a minor part of Markley’s results has been published [51, 52]. N.G. Markley proved independently Weil’s conjecture as well as several related results for the flows on surfaces of constant negative curvature. In 1966 at Tiraspol’s Symposium on General Topology Anosov communicated the theorem stating that unbounded coverings for semitrajectories of smooth flows with a finite number of fixed points on a closed surface of nonpositive Euler characteristic have an asymptotic direction. Besides, Anosov formulated a number of conjectures on the behavior of coverings to the curves without self-intersections. Anosov’s theorem and Anosov’s conjectures sparked the interest to the above area. One of the conjectures of Anosov concerned a deviation of a curve from the coasymptotic geodesic. In 1967, V. Pupko [63] stated the restricted deviation property for the curve without self-intersections but her proof was unclear. About 1972,

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Aranson and Grines came to Moscow to present their results on the classification of transitive flows on hyperbolic surfaces. The essential part of this presentation was the proof of the existence of an asymptotical direction for a nontrivially recurrent trajectory. Anosov asked Grines about the deviation property, and it looked like he doubted Pupko’s statement. Soon, Grines realized that in the example by C. Robinson and F. Williams [64] there are curves with an unbounded deviation. Aranson and Grines constructed a counter-example (described by Anosov in [2]) to Pupko’s statement even if a curve is a semi-trajectory of flow on closed orientable surface of genus g = 2. Later Anosov constructed counter-examples to Pupko’s statement on other surfaces including T2 and the Klein bottle [2, 3, 6]. 3. Nonlocal behavior of curves on universal coverings Weil’s theorem and conjecture say that a lift of a simple (i.e., without selfintersections) curve has an asymptotic direction provided the lift goes to the circle at infinity S∞ . However, this does not imply the existence of an asymptotic direction for an unbounded lift of a semitrajectory because apriori the covering semitrajectory can oscillate. Here we represent Anosov’s results on the existence of asymptotic directions for semitrajectories of surface flows. 3.1. Anosov’s theorems on asymptotic directions. The first theorem is on continuous (or topological ) flows. To formulate this theorem, we need the following definition. A subset F ⊂ M 2 is called contractible to a point if there exists a continuous mapping ϕ : F × [0; 1] → M such that ϕ(m, 0) = m and ϕ(m, 1) = m0 for any m ∈ F , where m0 ∈ F is a certain point. Theorem 3.1. If the set of fixed points of a topological flow f t on a closed surface M 2 of nonpositive Euler characteristic is contractible to a point, then any t 2 semitrajectory of the covering flow f on M is either bounded or has an asymptotic direction. Corollary 3.1. If the set of fixed points of a topological flow f t on a closed surface M 2 of nonpositive Euler characteristic is finite, then any semitrajectory of t 2 the covering flow f on M is either bounded or has an asymptotic direction. Presently, Theorem 3.1 gives the most general sufficient conditions for an unbounded semitrajectory of a topological flow to have an asymptotic direction. Theorem 3.2. If a flow f t on a closed surface M 2 of nonpositive Euler charact 2 teristic is analytic, then any semitrajectory of the covering flow f on M is either bounded or has an asymptotic direction. Theorem 3.2 does not follow from Theorem 3.1 because the set of fixed points of an analytic flow may contain, for instance, homotopically nontrivial closed curves and, hence, may not be contractible to a point. Note that the problem of whether a closed curve has an asymptotic direction is solved without difficulty. Theorem 3.3. Let C be a closed curve on a surface M 2 . Then, 2

(1) if C is null homotopic, then any of its lifts C to M is a closed (and, hence, bounded) curve;

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(2) if C is non-null-homotopic, then any of its lifts C to M is a nonclosed infinite curve both of whose semi-infinite curves have a rational asymptotic direction. Now, we quote some sufficient conditions for the existence of asymptotic directions for so-called widely disposed simple semi-infinite continuous curve. These conditions are then applied to special curves. We’ll consider the cases of flat and hyperbolic surfaces separately. Let T be an arc or a simple closed curve that intersects a semi-infinite curve l+ transversally. The curve l+ is said to be widely disposed with respect to T if there do not exist any T -loops that bound a disk. Recall that T -loop is defined  of l+ as follows. Suppose that l+ intersect T at two points a and b. The arc ab  with endpoints a, b is called a T -arc if T ∩ ab = a ∪ b. The T -arc together with  ∪ ab a subinterval ab ⊂ T between the points a, b forms a simple closed curve ab called a T -loop. On T2 , the concept of wide disposition with respect to a non-null-homotopic simple closed curve coincides with the concept of orientability of the intersection with this curve (orientability means that the index of the intersection is the same at every points of intersection). It can easily be shown that the orientability of the intersection implies the wide disposition on any surface. Theorem 3.4. Let C be a simple closed curve on T2 , and suppose that a simple infinite curve l orientably intersects C infinitely many times in such a way that the positive and negative semi-infinite curves l+ and l− of l also intersect C infinitely many times. Then, any lift l of l to the universal covering R2 is an infinite + − curve whose positive and negative semi-infinite curves l and l have diametrically opposite asymptotic directions. On a hyperbolic surface, one can easily construct an example of a semi-infinite curve that is widely disposed with respect to C and intersects the curve C nonorientably. Let us formulate a sufficient condition for the existence of an asymptotic direction of a widely disposed semi-infinite curve on a hyperbolic surface. Theorem 3.5. Let C be a simple closed curve on a hyperbolic surface M 2 , and suppose that a simple semi-infinite curve l+ is widely disposed with respect to C + and transversally intersects C infinitely many times. Then any lift l ⊂ Δ of l+ + has an asymptotic direction. Moreover, the point of S∞ that is accessible by l is + the topological limit of the lifts C i of C that are successively intersected by l as the parameter increases. This theorem also holds true for noncompact and non-orientable surfaces [12]. The theorem can be conveniently applied to study the existence of asymptotic directions for semi-infinite curves belonging to a simple curve that is infinite in both directions. Theorem 3.6. Let C be a simple closed curve on a hyperbolic surface M 2 , and suppose that a simple infinite curve l is widely disposed with respect to C; moreover, the positive and negative semi-infinite curves l+ and l− of l transversally intersect C infinitely many times. Then any lift l of l on the universal covering is an infinite + − curve whose positive and negative semi-infinite curves l and l have asymptotic + − directions. Moreover, ω(l ) = α(l ).

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3.2. Anosov’s theorems on the approximation of curves. In this section, we represent one of the fundamental results in this field, Anosov’s theorem [3] on the approximation, from the viewpoint of the Frechet distance ρF , of a semiinfinite continuous curve by a semitrajectory of a smooth flow. Theorem 3.7. Let l = {m(t) : t ≥ 0} be a semi-infinite continuous curve without self-intersections on a surface M 2 . Then for any r > 0 there exists a C ∞ flow f t on M 2 such that one of its semitrajectories T = {f t (m0 ) : m0 ∈ M 2 , t ≥ 0} lies at the Frechet distance ≤ r from l; i.e., ρF ([l], [T ]) ≤ r. Recall that the inequality ρF ([l], [T ]) ≤ r means the following: there exists a homeomorphism s : [0; +∞) → [0; +∞) such that supt≥0 d(m ◦ s(t), f t (m0 )) ≤ r, where d(·, ·) is the metric on M 2 . The main idea of the proof of Theorem 3.7 is to approximate the curve l by a C ∞ -embedded curve l∞ that is r-close to l in the sense of the Frechet metric and is obtained by a successive construction of arcs of increasing length. Since l∞ is smoothly embedded, it is embedded into M 2 together with a certain strip. Next, we declare all boundary points of this strip fixed points and construct a C ∞ flow with a semitrajectory l∞ . Note that the initial curve l may contain points of its own limit set or may even completely belong to its own limit set. Therefore, the construction of l∞ must be accompanied by “extruding the tails” of intermediate semi-infinite curves from a certain neighborhood of their initial arcs. In 1995, Anosov [8] generalized Theorem 3.7 and obtained its metric (in the sense of measure theory) version. Theorem 3.8. Let l = {m(t) : t ≥ 0} be a semi-infinite continuous curve without self-intersections on a surface M and μ be a smooth measure on M with everywhere positive C ∞ density. Then, for any r > 0, there exists a C ∞ flow f t that preserves the measure μ and is such that one of its semitrajectories T = {f t (m0 ) : m0 ∈ M 2 , t ≥ 0} lies at the Frechet distance ≤ r from l; i.e., ρF ([l], [T ]) ≤ r. 3.3. Limit sets at infinity. Here, we consider the question concerning possible limit sets at infinity for arbitrary unbounded curves that cover simple semiinfinite curves on a surface. For an observer standing on the universal covering, this question can be reformulated as follows: What regions of the horizon can be covered by an unbounded curve that is a lift of a simple curve? In particular, do + + there exist “wild” covering curves l whose complete limit set Lim(l ) contains the whole absolute? A positive answer to the latter question was obtained by Anosov in [3]. Anosov’s wild curve. We provide a schematic example (in the form of an existence theorem) of a “wild” covering curve whose limit set contains the whole absolute and that is projected to a simple curve on a surface. Theorem 3.9. Let M 2 be a closed surface of nonpositive Euler characteristic. There exists a continuous semi-infinite curve l ⊂ M 2 without self-intersections such 2 that its lift l to the universal covering M contains the whole absolute in its limit set.

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Sketch ofthe proof. Take a countable family of neighborhoods Vn ⊂ M ∪ S∞ such that 1) n Vn ⊃ S∞ ; 2) for any point σ ∈ S∞ and any of its neighborhoods U (σ), there exists Vi such that Vi ⊂ U (σ). Take a smooth semi-infinite curve l0 = {m0 (t) : t ≥ 0} on the universal covering that intersects all Vn . Then, ω(l0 ) ⊃ S∞ . The curve l0 = π(l0 ) ⊂ M , generally speaking, has self-intersections; we will deform it to obtain the required curve without self-intersections. After reparameterizing and slightly jiggling the curve l0 , we can divide it into arcs ln,n+1 = {m0 (t) : n ≤ t ≤ n + 1}, n ≥ 0, that satisfy the following conditions: / {m0 (t) : 0 ≤ t < 1) each arc ln,n+1 has no self-intersections; 2) m0 (n + 1) ∈ n + 1}; 3) there exists a sequence tn ≥ 1 such that m0 (tn ) ∈ Vn for any n ≥ 1.

l‘1,2

l1,2 l 0,1

l0,1

Figure 5. Deleting of self-intersections.  Let us fix the endpoints of the arc l1,2 and deform it into an arc l1,2 so that  the arc l0,1 ∪ l1,2 has no self-intersections, see Fig. 5. Let us subject the curve l2,3  (where the role of l0,1 is now played by the arc l0,1 ∪ l1,2 ) to a similar deformation. Continuing this process, we obtain a curve l without self-intersections on the surface M 2 . By property (3), its lift l contains the whole absolute in its limit set. 2 The following theorem on the existence of a wild semitrajectory follows immediately from Theorems 3.7 and 3.9.

Theorem 3.10. On any closed surface M 2 of nonpositive Euler characteristic, + there exists a C ∞ flow f t that has a positive semitrajectory l+ such that its lift l to the universal covering contains the whole absolute in its limit set at infinnity, + S∞ = lim∞ (l ). Limit sets at infinity of the lifts of curves on the torus T2 . All possible limit sets at infinity for unbounded curves that are the lifts of curves without self-intersections have been described only for T2 . It is obvious that any point of S∞ may serve as the limit set at infinity for a lift of a ray, which is projected to a simple curve on T2 . Moreover, any pair of diametrically opposite points on S∞ can serve as the limit set at infinity. It follows from Theorem 3.9 that the whole absolute may serve as the limit set at infinity. The following theorem, which was proved by Glutsyuk [36] after Anosov’s questions, shows that any (either open or closed) arc of the absolute that covers more than half of S∞ cannot be a limit set. Recall that the absolute S∞ is a unit circle and, hence, has the length 2π. Theorem 3.11. Let Ω ⊂ S∞ be a closed set such that there exists an arc of length strictly less than π among the connected components of the set S∞ \ Ω. Then Ω cannot serve as the limit set at infinity for any curve without self-intersections on T2 . However, any closed arc of S∞ of length at most π is realized as the limit set at infinity for a certain curve without self-intersections on T2 .

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D. Panov [57] constructed a pseudo-Anosov homeomorphism f : T2 → T2 such that a lift for any unstable leaf is dense in the part of the universal covering R2 . For closed hyperbolic surfaces, the question concerning possible limit sets at infinity has not yet been solved in the general case.

4. Asymptotic properties of special curves Now, we are mainly considering curves that have the dynamical origin e.g. trajectories of flows, and leaves of foliations, and one-dimensional invariant manifolds of diffeomorphisms with hyperbolic structure on non-wandering sets. Such curves often form so-called local laminations. The motivation for the definition of local lamination is the statement from theory of differential equations that says that trajectories locally looks like parallel straight lines away from the singularities. Local laminations. Let M ⊂ M 2 be a subset of M 2 (which may coincide  with 2 M ) that contains some closed subset S ⊂ M. Let M be the union S α Lα , where Lα are pairwise disjoint C r -smooth simple curves. We say that the family {Lα } forms a C r,l local lamination if, for any point P ∈ M − S, there exist a neighborhood U (P ) of P , and a C l diffeomorphism ψ : U (P ) → R2 , ψ(P ) = (0, 0), such that any connected component of the intersection U (P ) ∩ Lα (provided that this intersection is nonempty) is mapped by ψ onto the line y = const and the restriction ψ|U(P )∩Lα is a C r diffeomorphism onto its image. The curves Lα are called leaves. Each point of the set S is called a singularity. A point that is not a singularity is called regular. The concept of a local lamination generalizes the classical concepts of lamination and foliation. If ∪α Lα is closed and S = ∅, then M is called a C r,l lamination. An important example of a lamination is a geodesic lamination. Note that a local C r,l lamination without singularities is not always a lamination. If M = M 2 , then M is called a C r,l foliation. One may say that a local lamination with singularities is a “foliation” (with singularities) on a subset. If this subset is closed and there are no singularities, then we obtain a lamination. If this subset coincides with the manifold (and there may be some singularities), then the local lamination is a foliation. It follows from the above that the concept of a local lamination is a quite general concept, which includes, as particular cases, the concepts of lamination and foliation. A foliation on a surface is called transitive if it has at least one everywhere dense leaf. A foliation is called highly transitive if every (one-dimensional) leaf is dense on a surface. Obviously, any highly transitive foliation is a transitive one. One can prove that if a transitive foliation has only isolated singularities, then each singularity is of saddle type (see Fig. 6). In general, a transitive foliation can have separatrix connections, while a highly transitive foliation has no separatrix connections (obviously, a separatrix connection can’t be dense). A highly transitive foliation can have fake saddles whose number could be arbitrary with no connection with the topology of supporting surface. In this sense, fake saddles are artificial. Therefore, it is natural to distinguish transitive foliations without separatrix connections and fake saddles. A highly transitive foliation with no fake saddles is called irrational if it has only isolated singularities. An irrational foliation is called strongly irrational if it is without thorns.

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=1

=2

=3

=4

Figure 6. Singularities of saddle type: the thorn (ν = 1), the fake saddle (ν = 2), the tripod (ν = 3), a saddle with four separatrices (ν = 4), where ν is the number of separatrices.

The question of whether closed leaves and non-closed leaves that tend to a closed leaf have an asymptotic direction is actually solved as follows: • If a closed leaf is null homotopic (as a curve), then it has no asymptotic direction. If a closed leaf is non-null-homotopic, then its lift is an infinite curve both of whose semi-infinite curves have rational asymptotic directions. • If a non-closed leaf tends to a null-homotopic closed leaf, then it has no asymptotic direction. If a non-closed leaf tends to a non-null-homotopic closed leaf, then it has a rational asymptotic direction. It is convenient to consider flows as orientable foliations using the corresponding terminology. A similar statement holds true for a semitrajectory that tends to a loop composed of separatrix connections and saddles. Obviously, a semitrajectory that tends to a single fixed point has no asymptotic direction. It remains to consider the question of whether semitrajectories that tend to trajectories whose limit set contains regular points have an asymptotic direction. According to Maier theorem [48, 49], such semitrajectories tend to nontrivially recurrent trajectories. Therefore, it is natural to consider first the question of whether nontrivially recurrent semitrajectories have an asymptotic direction. Nontrivially recurrent semitrajectories and semileaves. Recall that a nontrivially recurrent semitrajectory is a nonclosed semitrajectory that belongs to its own limit set. Such semitrajectories may exist only on orientable surfaces of genus g ≥ 1 and on non-orientable surfaces of genus g ≥ 3 [13, 49, 62]. The Euler characteristic of these surfaces is nonpositive, and their universal covering is homeomorphic to a disk. The following theorem proved in [19] shows that a nontrivially recurrent semitrajectory of a flow with any set of fixed points has an asymptotic direction, and this asymptotic direction is irrational. Theorem 4.1. Let l be a nontrivially recurrent semitrajectory of a flow f t on a closed surface M 2 of nonpositive Euler characteristic, and let l be its lift to the 2 universal covering M . Then, l has an irrational asymptotic direction. Corollary 4.1. Let l be a nontrivially recurrent trajectory of a flow f t on a closed surface M 2 of nonpositive Euler characteristic, and let l be its lift to the 2 universal covering M . Then, l has irrational asymptotic directions ω(l), α(l) ∈ S∞ ; moreover, ω(l) = α(l).

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An analysis of the proof of Theorem 4.1 shows that similar assertions are valid for local laminations on a hyperbolic surface. Theorem 4.2. Let C be a simple closed curve on a hyperbolic surface M 2 , and suppose that all leaves of a local lamination D are widely disposed with respect to C. Suppose that a nontrivially recurrent leaf l of D transversally intersects C infinitely many times. Then, the positive and negative semileaves of the covering leaf l have different irrational asymptotic directions on the universal covering. 4.1. Dynamical and asymptotical properties. Here we show how properties of remote limit points influence dynamical properties of flows and foliations. For example, the first theorem says that if a foliation (or a flow) with a finite set of singularities has a semi-leaf with an irrational asymptotic direction, then the foliation has a quasiminimal set. Recall that a quasiminimal set is the closure of a nontrivially recurrent semi-leaf. A quasiminimal set is called irreducible if any nontrivially homotopic closed curve on M intersects this quasiminimal set. Theorem 4.3. If a foliation F with finitely many singularities on M 2 has a semi-leaf with an irrational direction, then F has a quasiminimal set (in particular, F has a nontrivially recurrent leaves). Let us introduce some notation. We consider only hyperbolic surfaces here. In this case, using geodesic laminations, we can get a good description for points of the circle at infinity. A lamination whose leaves are geodesics is called a geodesic lamination. One can reformulate this definition in the traditional way: a geodesic lamination is a family of pairwise disjoint simple geodesics such that their union is a closed set. Here, a simple geodesic is either an infinite curve without self-intersections or a simple closed curve. Denote by L(M 2 ) = L the set of geodesic laminations on M 2 . A geodesic lamination is trivial if it consists of closed geodesics and isolated non-closed geodesics. Denote the set of trivial geodesic laminations by Λtriv (M 2 ) = Λtriv . So, it is natural to call a geodesic lamination nontrivial if it contains a non-closed geodesic that is non-isolated in the geodesic lamination. A nontrivial lamination is said to be strongly nontrivial if it consists of non-closed and non-isolated geodesics. Denote by Λ the set of strongly nontrivial geodesic laminations. A lamination is minimal if it contains no proper sub-laminations. A minimal strongly nontrivial geodesic lamination is called weakly irrational. It follows from [32, 35] that if L is a strongly nontrivial geodesic lamination then 1) every geodesic of L is nontrivially recurrent; 2) L is a union of connected pairwise disjoint weakly irrational geodesic laminations; 3) every geodesic of a weakly irrational geodesic lamination is dense in this lamination. So, Λ consists of weakly irrational geodesic laminations. Denote by Λor (respectively, Λnon ) the set of orientable (respectively, non-orientable) weakly irrational geodesic laminations on M , Λ = Λor ∪ Λnon . An important class of geodesic laminations is given by irreducible laminations. A geodesic lamination G ∈ Λ is called irreducible if any closed geodesic on M 2 intersects G. On a closed orientable hyperbolic surface, this condition is equivalent to the fact that any component of the set M − G is simply connected [32]. Denote by Λirr ⊂ Λ the set of irreducible weakly irrational geodesic laminations. We’ll call a geodesic lamination from Λirr strongly irrational (or simply, irrational). Set def

Λor ∩ Λirr = Λirr or ,

def

Λnon ∩ Λirr = Λirr non .

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Let G ∈ L be a geodesic lamination on a hyperbolic surface M . It is clear def that the preimage π −1 (G) = G is a geodesic lamination on the universal covering Δ. If G has a geodesic with an ideal endpoint σ ∈ S∞ , we say that σ is accessible (or, reached, or attained) by the lamination G. Taking a certain liberty, we will also say that σ is accessible by the lamination G, although this lamination lies on the surface. Denote by G∞ ⊂ S∞ the set of points on S∞ that are accessible by the lamination G. Again, taking a certain liberty, we will use the notation G∞ . Sometimes, when the subscript is in use, we will denote the set of accessible points by G(∞) or G(∞). Thus, Λ(∞) ⊂ S∞ is the set of points reached by all laminations from Λ, and Λirr (∞) ⊂ S∞ is the set of points reached by the strongly irrational geodesic laminations. Theorem 4.4. Let F be a foliation with finitely many singularities on M 2 and + l a positive semi-leaf of F such that its lifting l to Δ has the asymptotical direcirr tion σ ∈ S∞ . If σ ∈ Λ(∞) − Λ (∞), then F is not highly transitive and there is a nontrivially homotopic closed curve that is not intersected by any nontrivially recurrent leaf. If σ ∈ Λirr (∞), then F has an irreducible quasiminimal set. Moreover, F is either highly transitive or can be obtained from a highly transitive foliation by a blow-up operation of at least countable set of leaves and by the Whitehead operation. When F is not highly transitive, F has a unique nowhere dense quasiminimal set. +

Take G ∈ Λirr . A point σ ∈ G(∞) is a point of the first kind if there is only one geodesic of G with the endpoint σ. Otherwise, σ is called a point of the second kind. One can prove that this definition does not depend on the choosing of G ∈ Λirr . The following theorem shows that the type of asymptotic direction reflects certain “dynamical” properties of the foliation [28]. Theorem 4.5. Let F be an irrational foliation on M and l+ a positive semi+ leaf of F such that its lifting l to Δ has the asymptotical direction σ ∈ S∞ . Then σ ∈ Λirr (∞). Moreover, (1) If σ is a point of the first kind then l+ belongs to a nontrivially recurrent leaf. (2) If σ is a point of the second kind then l+ belongs to a separatrix of a saddle singularity. We have the following sufficient condition for the existence of a continuum set of fixed points. Theorem 4.6. Suppose that a flow f t on M 2 reaches a point from Λirr non (∞). Then f t has a continuum of fixed points. Furthermore, f t has neither nontrivially recurrent semitrajectories nor closed transversals nonhomotopic to zero. It turns that some points of S∞ attained by C ∞ flows prevent these flows to be analytic. Recall that σ ∈ S∞ is called a point achieved by f t if there is a positive (or negative) semitrajectory l± of f t such that some lift ¯l± of l± has the asymptotic direction defined by σ. Denote by Af l , A∞ , Aan ⊂ S∞ the sets of points achieved by all topological, C ∞ , and analytic flows respectively. Due to the remarkable result by Anosov [4], Af l = A∞ (see Theorem 3.8). Obviously, Aan ⊂ A∞ . It follows from the following theorem that A∞ − Aan = ∅ [29].

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Theorem 4.7. There exists a continual set U (M 2 ) ⊂ A∞ such that any C ∞ flow f t that reaches a point from U (M 2 ), is not analytic. The set U (M 2 ) is dense and has zero Lebesgue measure on S∞ . One can prove that Λtriv (∞) ⊂ Aan ⊂ Λtriv (∞) ∪ Λor (∞), and Λnon (∞) ⊂ A∞ − Aan . 4.2. Geodesic frameworks of local laminations. Applying a medical terminology, one can say that the geodesic framework of local lamination is its geodesic skeleton around which the leaves that have asymptotic directions are grouped. The geodesic framework contains the full information on the asymptotic directions of leaves of a given local lamination. The geodesic framework of a local lamination is defined only if this lamination has a leaf or a generalized leaf (the union of separatrices and their singularities) that has a co-asymptotic geodesic. To be precise, let D be a local lamination on M 2 . Denote by A± (D) the union of all leaves and generalized leaves of D that have co-asymptotic geodesics. The topological closure  def G(D) = clos g(l) l∈A± (D)

is called the geodesic framework of the local lamination D. Since a lamination and a foliation are local laminations, we have defined the concepts of geodesic framework for foliations and laminations. It follows immediately from the definition that a geodesic framework is a geodesic lamination. The geodesic framework of an arbitrary invariant set of a local lamination is defined similarly. On T2 , a geodesic lamination either forms an irrational linear foliation (hence, this lamination fills the whole torus) or is a family of pairwise homotopic closed geodesics. Therefore, below in this section, we’ll consider geodesic frameworks on closed orientable hyperbolic surfaces. Geodesic frameworks of quasiminimal sets. Recall that by Theorem 3.6, a nontrivially recurrent leaf l has a co-asymptotic geodesic g(l) provided l is widely disposed with respect to some simple closed curve C. Lemma 4.1. Let l be a nontrivially recurrent leaf of a local lamination D, and suppose that l is widely disposed with respect to a certain simple closedown curve C and transversally intersects C. Then the co-asymptotic geodesic g(l) is nontrivially recurrent. It follows from a theorem of Cherry [33] (see generalizations in [26,27]) that any quasiminimal set with closed support contains a continuum of nontrivially recurrent leaves each of which is everywhere dense in the quasiminimal set. A quasiminimal set Q is called a Maier quasiminimal set if each semi-leaf from Q that does not tend exactly to one singularity is everywhere dense in Q. In particular, a leaf from Q that is different from a separatrix connection is everywhere dense in Q and is nontrivially recurrent in, at least, one direction. The following theorem describes a geodesic framework of the Maier quasiminimal set. Theorem 4.8. Let Q be a Maier quasiminimal set containing a finitely many singularities and separatrices of a local lamination D with closed support supp D. Suppose that every nontrivially recurrent leaf from Q is widely disposed with respect

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to some simple closed curve C. Then • the geodesic framework G(Q) is equal to clos g(l) for any nontrivially recurrent leaf l ∈ Q; • the geodesic framework G(Q) is a weakly irrational geodesic lamination; • any geodesic from G(Q) is the co-asymptotic geodesic of a certain leaf or a generalized leaf that belongs to Q. Geodesic frameworks of special foliations. Consider a foliation with isolated singularities of negative index (in particular, with saddles of negative index). Then any semileaf of such a foliation that does not tend to a singularity has an asymptotic direction. In addition, any leaf or generalized leaf that is not a separatrix connection has a co-asymptotic geodesic. Theorem 4.9. Let F be an irrational foliation on a closed orientable hyperbolic surface M 2 with singularities that are saddles of negative index. Then, (1) the geodesic framework G(F) of F is an irrational geodesic lamination; (2) any geodesic from G(F) is a co-asymptotic geodesic for a certain leaf or generalized leaf of F. Moreover, (a) any point σ ∈ Λ1,∞ (M 2 ) ∩ G(F)∞ is reached by a leaf projected to an internal nontrivially recurrent leaf on M 2 whose co-asymptotic geodesic is also internal; (b) any point σ ∈ Λ2,∞ (M 2 ) ∩ G(F)∞ is reached by a leaf l that is an α-separatrix of a singularity, and the left and right Bendixson extensions of the leaf l in the negative direction 2 have different asymptotic directions α1 and α2 . Two geodesics that connect σ with the points α1 and α2 are sides of a geodesic polygon with a finite number of sides that belong to G(F), and these geodesics are projected to boundary geodesics on M 2 ; (3) each component of the set M 2 −G(F) is a simply connected domain any of whose lifts to the universal covering is the interior of a geodesic polygon P with a finite number of sides and with vertices lying on S∞ . In this case, the sides of P belong to G(F), and each vertex is reached by exactly one separatrix of a certain saddle of the covering foliation F . Conversely, each saddle of F corresponds to a unique geodesic polygon formed by geodesics from G(F), such that the separatrices of the saddle reach all vertices of the polygon and the number of separatrices is equal to the number of vertices. 4.3. Deviations of curves from co-asymptotic geodesics. Here we focus our attention on the deviation of curves that have asymptotic directions from coasymptotic geodesics on the universal covering. First, we consider examples of curves with unbounded deviation. Historically, the first example with an unbounded deviation was constructed by Aranson and Grines. They constructed a foliation that has a nontrivially recurrent leaf with unbounded deviation from the co-asymptotic geodesic. We now describe this example. On a closed orientable surface Mg21 of genus g1 ≥ 1, consider an irrational foliation F1 that has a topological saddle s1 with k ≥ 3 separatrices (hence, the index of the saddle is equal to ind s1 = 1 − k2 ). Since all saddles of the foliation F1 have a negative index, F1 is a widely disposed foliation with respect to any 2 Without

loss of generality, we may assume that the leaf l is oriented toward the point σ.

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closed transversal and a transversal segment. On another surface Mg22 of genus g2 ≥ 1, take a Denjoy-type foliation F2 with a minimal set Ω(F2 ) such that the set Mg22 − Ω(F2 ) has a component S2 of index ind s1 , Fig. 7.

Figure 7. The saddle s1 and the component S2 of index ind s1 . Let us place the saddle s1 inside a disk D1 whose boundary ∂D1 is transversal to the foliation F1 everywhere except for points a1 , . . . , ak ∈ ∂D1 that are arranged in the order corresponding to the positive (counterclockwise) orientation of ∂D1 . Without loss of generality, we may assume that the leaves passing through the points a1 , . . . , ak are pairwise different and are not separatrices of any saddles. Between the points ai and ai+1 , i = 1, . . . , k (where ak+1 = a1 ), on ∂D1 , there is a unique point of intersection of a separatrix of the saddle s1 with ∂D1 , which we denote by ci , such that the arc (s1 ; ci ) of the separatrix does not intersect ∂D1 (we assume that ck+1 = c1 ). Then the foliation F1 induces in D1 the first-return map φ1 : ∂D1 −

k  i=1

ci → ∂D1 −

k 

ci ,

i=1

φ1 |(ci ;ai ] : (ci ; ai ] → [ai ; ci+1 ), φ1 |[ai ;ci+1 ) : [ai ; ci+1 ) → (ci ; ai ] where i = 1, . . . , k. By the construction, φ21 = id. In the component S2 , take an open disk D2 ⊂ S2 whose boundary ∂D2 intersects Ω(F2 ) only at points b1 , . . . , bk ∈ ∂D2 that are arranged in the order corresponding to the negative (clockwise) orientation of ∂D2 . In addition, let us require that the disk D2 divides S2 into k domains Wi , i = 1, . . . , k, that are homeomorphic to an open strip, Fig. 7. Since the index of the component S2 is ind s1 = 1 − k2 , this can be done. Let us declare that the points ai and ci , i = 1, . . . , k, are the singularities of the foliation F1 , and denote the obtained foliation by F1 . Note that since the leaves passing through the points a1 , . . . , ak are pairwise different and are not separatrices, any one-dimensional leaf of the foliation F1 different from the leaves of the form (s1 ; ci ) is everywhere dense on Mg21 . Let us modify the foliation F2 by placing a Reeb foliation in each strip Wi , i = 1, . . . , k, and declaring each point of the set Ω(F2 ) a singularity. The points di ∈ (bi ; bi+1 ) ⊂ ∂D2 , i = 1, . . . , k, are chosen arbitrarily, where bk+1 = b1 . The foliation is extended arbitrarily into the interior of D2 . Let us glue together the two surfaces Mg21 − Int D1 and Mg22 − Int D2 by Θ : ∂D1 → ∂D2 . As a result, we obtain a closed surface Mg21 +g2 of genus g1 + g2 ≥ 2, which is the connected sum Mg21 Mg22 of the surfaces Mg21 and Mg22 . The foliations F1 and F2 form a foliation on Mg21 +g2 , which we denote by F. It follows from the construction that there is a leaf l of F which is everywhere dense on the surface. Hence, l has an irrational

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asymptotic direction. One can prove that l possesses the property of unbounded deviation. Examples of curves with rational and irrational asymptotic directions and with the property of unbounded deviation on T2 and the Klein bottle were first constructed by Anosov [4, 6]. Note that the construction of a curve with a rational direction is a more complicated. An elegant example of the flow with a semitrajectory that has a rational asymptotic direction and possesses the property of unbounded deviation was constructed in [53] (see also [54]). It is natural to consider conditions under which the deviation from special curves is bounded. The following theorem was proved in [8]. Theorem 4.10. Let f t be a topological flow on T2 and l be a lift to R2 of a semitrajectory l = π(l) that has an asymptotic direction. Suppose that one of the following conditions is fulfilled: 1) the set of fixed points of the flow f t is contractible to a point; 2) l is a nontrivially recurrent semitrajectory. Then l possesses the property of bounded deviation. For flows on hyperbolic surfaces, the following theorem was proved in [24]. Theorem 4.11. Let f t be a topological flow with a finite set of fixed points on a + closed hyperbolic surface M . Let l be a positive semitrajectory of a covering flow t + f on M = Δ that has an asymptotic direction. Then l possesses the property of bounded deviation. As to analytic flows, Anosov [8] proved the following result. Theorem 4.12. Let f t be an analytic flow on a closed orientable surface of constant nonpositive curvature, and let l be a semitrajectory of the covering flow that has an asymptotic direction. Then l possesses the property of bounded deviation. Similar statements hold for surface foliations. Now, we pass on to the local laminations that play an important role in studying surface diffeomorphisms, namely, to one-dimensional stable or unstable manifolds of points that belong to hyperbolic nonwandering sets. The following theorem was proved in [41]. Theorem 4.13. Let f : M → M be an A-diffeomorphism of a closed surface M of nonpositive Euler characteristic. Let Ω be a one-dimensional widely disposed u(s) attractor (repeller) of f, and let lx be the unstable (respectively, stable) manifold u(s) of a point x ∈ Ω. Then both curves lx − x has asymptotic direction and possess the property of bounded deviation. The analysis of the aforementioned example of Robinson and Williams [64] shows that for stable (respectively, unstable) manifolds of points of a one-dimensional attractor (respectively, repeller), Theorem 4.13 is generally incorrect. The above arguments do not work because the theorem on the product structure cannot be applied to all points of stable (respectively, unstable) manifolds of points of a onedimensional attractor (respectively, repeller). However, if we require that f : M 2 → M 2 is a structurally stable diffeomorphism, then we obtain the following result [40, 41]: Theorem 4.14. Let f : M 2 → M 2 be a structurally stable A-diffeomorphism of a closed orientable hyperbolic surface M and let Ω be a one-dimensional widely s(u) disposed attractor (respectively, repeller) of f . Let lx be the stable (respectively,

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unstable) manifold of a point x ∈ Ω and Lσ be one of the connected components of s(u) the set lx − x that does not contain a periodic boundary point. Then Lσ has an asymptotic direction and possesses the property of bounded deviation. On the torus, Theorem 4.14 is valid without the requirement that the diffeomorphism should be structurally stable [38, 41]. Theorem 4.15. Let f : M 2 → M 2 be an A-diffeomorphism of T2 , and let Ω be a one-dimensional widely disposed attractor (repeller) of the diffeomorphism f . s(u) Let lx be the stable (respectively, unstable) manifold of a point x ∈ Ω and Lσ be s(u) one of the connected components of the set lx − x that does not contain a periodic σ boundary point. Then L has an asymptotic direction and possesses the property of bounded deviation. 5. Applications to foliations and dynamical systems Recall that two foliations F1 , F2 on a surface M are topologically equivalent if there exists a homeomorphism h : M 2 → M 2 such that h(Sing (F1 )) = Sing (F2 ) and h sends every leaf of F1 onto a leaf of F2 . One says that h maps the foliation F1 onto the foliation F2 . Orientable foliations (flows) F1 , F2 are orbitally topologically equivalent if the homeomorphism h : M 2 → M 2 above keeps the orientation of leaves (resp., trajectories). In general, the classification assumes the following steps: (1) Find a constructive topological invariant which takes the same values for topologically equivalent foliations. (2) Describe all topological invariants which are admissible, i.e. may be realized in the chosen class of foliations. (3) Find a standard representative in each equivalence class, i.e. given any admissible invariant, one constructs a foliation whose invariant is the admissible one. An invariant is called complete if it takes the same value if and only if two foliations are topologically equivalent. The ‘if’ part only gives a relative invariant. 5.1. Classification of irrational flows and foliations. For completeness, we begin with the classical results on the classification of irrational flows on the torus T2 . After that we present the classification of strongly irrational foliations on a closed hyperbolic surface. Let us recall that an irrational foliation is a foliation with no fake saddles such that every one-dimensional leaf is dense. A strongly irrational foliation is an irrational one with no thorns (saddle type singularities of the index 12 ). We see that an irrational flow which can be considered as an orientable irrational foliation is a strongly irrational foliation automatically. Note that an irrational flow on T2 is a transitive (even minimal) fixed-point-free flow. Irrational flows on 2-torus. A classical example of constructing an effective topological invariant is given by the Poincar´e rotation number for fixed-point-free flows on T2 . Let f t be a flow on T2 . Suppose that f t has a nontrivially recurrent trajectory l. Let π : R2 → T2 be the covering projection, and l a lift of l. By Weil’s theorem, l has an asymptotic direction with the co-asymptotic geodesic a straight line y = kx. Since any straight line divides the plane R2 into two halfplanes, all nontrivially recurrent trajectories of f t have the same co-asymptotic geodesic y = kx. The number k is called the rotation number of f t , denoted by rot (f t ). The existence of nontrivially recurrent trajectory implies the nonexistence

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of periodic trajectories that non-homotopy to zero. Therefore, the rotation number k = rot (f t ) is irrational. Theorem 5.1. Let f1t and f2t be flows on T2 such that the both f1t and f2t have t nontrivially recurrent trajectories. If  f1t and f 2 are topologically equivalent, then a b there is an integer unimodular matrix such that c d (4)

rot (f2t ) =

−c + a · rot (f1t ) , d − b · rot (f1t )

d − b · rot (f1t ) = 0.

Theorem 5.2. Let f t be an irrational flow on T2 . Then f t is orbitally topologically equivalent to a linear flow of the form x˙ = 1, y˙ = μ where μ = rot (f t ). As a consequence, we get the following classical classification result. Theorem 5.3. Let f1t and f2t be irrational flows on T2 . Then f1t and f2t are equivalent if and only if there is the integer unimodular matrix  topologically  a b such that ( 4) holds. Moreover, given any irrational μ ∈ R, the flow c d of the form x˙ = 1, y˙ = μ is irrational and μ = rot (f t ) (clearly, every number calculated by ( 4) is irrational). Irrational foliations on hyperbolic surfaces. Let F be a strongly irrational foliation on a closed orientable surface M 2 . By Theorem 4.9, the geodesic framework G(F) of F is a minimal strongly nontrivial geodesic lamination such that each component of M 2 − G(F) is an open geodesic polygon with a finite number of sides and ideal vertices. Thus, G(F) is a strongly irrational geodesic lamination. The following four theorems obtained by Aranson and Grines [19] give a complete classification of strongly irrational foliations on M 2 (see the survey [21]). This theorems correspond to the three steps of the topological classification. The first and second theorems produce a constructive topological invariant which takes the same values for topologically equivalent foliations. The third theorem describes all topological invariants which are admissible, i.e. may be realized in the chosen class of foliations. The fourth theorem shows that for any admissible invariant there is a strongly irrational foliation whose invariant is the admissible one. Theorem 5.4. Let F1 , F2 be strongly irrational foliations on a closed orientable hyperbolic surface M . Then F1 , F2 are topologically equivalent via a homeomorphism M 2 → M 2 homotopic to identity if and only if their geodesic frameworks coincide, G(F1 ) = G(F2 ). The generalized mapping class group GM is the quotient Homeo (M 2 )/Homeo0 (M 2 ), where Homeo (M 2 ) is the group of self-homeomorphisms of M 2 and Homeo0 (M 2 ) is the subgroup of homeomorphisms homotopic to the identity. It is known that any homeomorphism f : M 2 → M 2 induces a one-to-one map f∗ : L → L, f∗ ∈ GM [32]. Given λ ∈ L, the family GM (λ) = {f∗ (λ) | f∗ ∈ GM } is called an orbit of the geodesic lamination λ.

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Theorem 5.5. Let F1 , F2 be strongly irrational foliations on a closed orientable hyperbolic surface M 2 . Then F1 , F2 are topologically equivalent if and only if the orbits of their geodesic frameworks coincide. We see that the orbit of a geodesic framework is a complete topological invariant for strongly irrational foliations. Thus the geodesic framework is an analog of Poincar´e rotation number for the class of strongly irrational foliations (flows) on T2 . The next theorem shows that the geodesic framework of a strongly irrational foliation is a strongly irrational geodesic lamination. This is completely similar to an irrational Poincare rotation number. Theorem 5.6. Let F be a strongly irrational foliation on a closed orientable hyperbolic surface M 2 . Then its geodesic framework G(F) is a strongly irrational geodesic lamination, G(F) ∈ Λirr . Theorem 5.7. Given any strongly irrational geodesic lamination G ∈ Λirr on a closed orientable hyperbolic surface M , there is a strongly irrational foliation F on M 2 such that G(F) = G. As a consequence, one gets the classification of irrational flows. Note that the classification of irrational flows on closed non-orientable surfaces was obtained in [18]. 5.2. Classification of nontrivial minimal sets. Recall that a minimal set of a flow is a nonempty closed set that is invariant (i.e., consists of trajectories of the flow) and does not contain proper subsets with the above-described properties. A similar definition applies to foliations, provided that “invariant” means a union of leaves and singularities. The trivial minimal sets of flows include fixed points, periodic trajectories, and the minimal set that coincides with a closed surface, which is the torus in this case. The situation for foliations is analogous. Nontrivial minimal sets are nowhere dense and locally homeomorphic to the product of a segment and a Cantor set. A nontrivial minimal set consists of nonclosed trajectories that are recurrent in the Birkhoff sense, in short B-recurrent. Moreover, every B-recurrent trajectory is everywhere dense in the minimal set [17]. Nontrivial minimal sets on T2 . We present here results from [25]. It is obvious that the geodesic framework of a nontrivial minimal set on the torus T2 is a linear irrational flow. Lemma 5.1. Let N be a nontrivial minimal set of a flow f t on T2 and G(N ) the geodesic framework of N . Then there exists a continuous mapping h : T2 → T2 that is homotopic to the identity with the following properties: 1) h(N ) = T2 ; 2) each trajectory from N is homeomorphically mapped by h onto a geodesic of G(N ); 3) if w is the component of the set T2 \ N then h(w) is a geodesic of G(N ). Denote by δ(N ) the boundary of the set N that is accessible from T2 \ N . It can be shown that δ(N ) is invariant and consists of a finite or a countable family of trajectories of N . Therefore, by Lemma 5.1, h(δ(N )) is a finite or a countable family of geodesics from G(N ). This family of geodesics is called a distinguished family of the minimal set N and is denoted by R(N ). Of course, this family depends on the transformation h from Lemma 5.1 and is determined by the set N up to a translation, i.e., up to a homeomorphism of T2 whose covering is given by x → x + x0 , y → y + y0 , where x0 and y0 are certain constants. The following theorem gives a topological classification of nontrivial minimal sets of flows on T2 .

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Theorem 5.8. Let N1 and N2 be nontrivial minimal sets of flows f1t and f2t , respectively, on T2 . Then, N1 and N2 are orbitally topologically equivalent via a homeomorphism T2 → T2 homotopic to the identity if and only if their geodesic frameworks coincide (with regard to the orientation of geodesics) and there exists a translation of T2 that sends the distinguished family of one minimal set to the distinguished family of the other minimal set. The geodesic framework of a nontrivial minimal set on T2 is a linear irrational flow. For any finite or countable family N0 of trajectories of a linear irrational flow, there exists a flow with a nontrivial minimal set N such that R(N ) = N0 . Note that the geodesic framework and the cardinality of the set of distinguished geodesics alone do not provide a complete topological invariant. Moreover, it can be shown that there exists a continuum of pairwise topologically non-equivalent nontrivial minimal sets with the same geodesic framework and any prescribed fixed cardinality ≥ 2 of the set of distinguished geodesics. Nontrivial minimal sets on a hyperbolic surface. The classification below was obtained in [20]. Let N be a nontrivial minimal set of a flow f t on an orientable closed hyperbolic surface M 2 . A component of the set M 2 \N is called a Denjoy cell if it is simply connected and its boundary accessible from M 2 \N consists of exactly two trajectories of N . These two trajectories have the same co-asymptotic geodesic called a distinguished geodesic. Similar to the case of the torus, we’ll call a family of distinguished geodesics a distinguished family of the geodesic framework of the minimal set N . Since the generation or elimination of Denjoy cells do not change the geodesic framework of a nontrivial minimal set, the presence of these cells can be considered, in a sense, artificial. Therefore, we first consider a classification of nontrivial minimal sets without Denjoy cells. Let us recall that a minimal strongly nontrivial geodesic lamination is called weakly irrational. Theorem 5.9. Let N be a nontrivial minimal set of a flow f t on a closed orientable hyperbolic surface M 2 . Suppose that N does not contain Denjoy cells. Then, N is orbitally topologically equivalent, via a homeomorphism M 2 → M 2 homotopic to the identity, to its own geodesic framework G(N ) that is an orientable weakly irrational geodesic lamination, G(N ) ∈ Λor . For any orientable weakly irrational geodesic lamination Λ ∈ Λor , there exists a nontrivial minimal set N without Denjoy cells of a certain flow f t such that G(N ) = Λ.3 We now, consider nontrivial minimal sets with Denjoy cells and describe the type of geodesics that form distinguished families of these minimal sets. Recall that a nontrivially recurrent geodesic may be either left or right improper; i.e., it may approach itself to an indefinitely close distance from either the left or the right side. If a nontrivially recurrent geodesic is improper only from one side, then it is called a boundary one. Otherwise (i.e., if a geodesic is improper from both sides), it is called internal. A weakly irrational geodesic lamination on a closed hyperbolic surface has a finite nonzero number of boundary nontrivially recurrent geodesics and a continuum set of internal ones. The definition of a Denjoy cell and the density of each geodesic in a minimal geodesic lamination imply that each geodesic from a distinguished family is internal. The following two theorems give topological classification of nontrivial minimal sets of flows on a closed orientable hyperbolic surface. 3 In

fact, one can make it so that N = Λ.

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Theorem 5.10. Let N1 and N2 be nontrivial minimal sets of flows f1t and f2t , respectively, on a closed orientable hyperbolic surface M 2 . Then N1 and N2 are orbitally topologically equivalent via a homeomorphism M 2 → M 2 homotopic to the identity if and only if they have identical geodesic frameworks (with regard to the orientation of geodesics) and the same family of distinguished geodesics. Disregarding the orientation of geodesics, we obtain a criterion for topological equivalence. Theorem 5.11. Let N be a nontrivial minimal set of a flow f t on a closed orientable hyperbolic surface M 2 . Then the geodesic framework G(N ) is an orientable weakly irrational geodesic lamination that contains at most a countable distinguished family that consists of internal geodesics. Conversely, let Λ be an orientable weakly irrational geodesic lamination on M 2 and let N be at most a countable family of internal geodesics of Λ. Then there exists a nontrivial minimal set N of a certain flow f t such that G(N ) = Λ and the distinguished family of the geodesic framework G(N ) coincides with N . 5.3. Classification of irrational 2-webs. A 2-web on a surface is a pair of foliations such that they have a common set of singularities and are topologically transversal at all non-singular points. Suppose that two foliations F1 and F2 on a surface M 2 form a 2-web denoted by (F1 , F2 ). The set of singularities of the foliation Fi (for any i) is called the set of singularities of (F1 , F2 ) denoted by Sing (F1 , F2 ). A 2-web is irrational or strongly irrational if it consists of a pair of irrational or strongly irrational foliations respectively. 2-webs (F1 , F2 ) and (F1 , F2 ) are topologically equivalent if there is a homeomorphism ϕ : M 2 → M 2 that maps the foliations Fi (i = 1, 2) to the corresponding foliations Fi and ϕ (Sing (F1 , F2 )) = Sing (F1 , F2 ). All classification results of this subsection was obtained in [23]. On the torus T2 , a strongly irrational 2-web consists of a pair of transversal irrational foliations without singularities. Theorem 5.12. Let (F1 , F2 ) be a strongly irrational 2-web on T2 . Then (F1 , F2 ) is topologically equivalent via a homeomorphism homotopic to the identity to its own geodesic framework, which is a pair of linear transversal irrational foliations. Two strongly irrational 2-webs on T2 are topologically equivalent via a homeomorphism T2 → T2 homotopic to the identity if and only if their geodesic frameworks coincide. Let us pass on to strongly irrational 2-webs on a closed orientable hyperbolic surface. The geodesic framework of a strongly irrational foliation on a closed orientable hyperbolic surface Mh2 , h ≥ 2, is a strongly irrational geodesic lamination. If foliations F1 and F2 form a strongly irrational 2-web, then their geodesic frameworks must satisfy the following consistency conditions: • The sets Mh2 \ supp G(F1 ) and Mh2 \ supp G(F2 ) have the same number of simply connected components, which is equal to the number of singularities of the foliations F1 and F2 (which is the same for these foliations). • For each simply connected component P1 of the set Mh2 \supp G(F1 ), there exists a simply connected component P2 of the set Mh2 \ supp G(F2 ) such that there exist lifts P 1 and P 2 of these components that are polygons with alternating ideal vertices on S∞ , Fig. 8.

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Figure 8. The polygons P 1 and P 2 Theorem 5.13. Let (F1 , F2 ) and (F1 , F2 ) be strongly irrational 2-webs on a closed orientable hyperbolic surface M 2 . Then (F1 , F2 ) and (F1 , F2 ) are topologically equivalent via a homeomorphism M 2 → M 2 that is homotopic to the identity if and only if their geodesic frameworks coincide. For any pair of consistent strongly irrational geodesic laminations, there exists a strongly irrational 2-web whose geodesic framework is equal to this pair of laminations. Sketch of proof. . We restrict ourselves to the first part of the statement. A homeomorphism of M 2 that is homotopic to the identity has a lift that is extended to the identity homeomorphism of S∞ . Therefore, if the webs are topologically equivalent via a homeomorphism homotopic to the identity, then their geodesic frameworks coincide. Suppose that the geodesic frameworks (G(F1 ), G(F2 )) and (G(F1 ), G(F2 )) co  incide, G(F1 ) = G(F1 ), G(F2 ) = G(F2 ). Consider the lifts (F 1 , F 2 ) and (F 1 , F 2 ) of the 2-webs (F1 , F2 ) and (F1 , F2 ), respectively. Let m ∈ Δ be a point that is not a singularity of the 2-web (F 1 , F 2 ). According to Theorem 4.9, semileaves, say l1 and l2 , of F 1 and F 2 passing through m have the asymptotic directions defined by some points σ1 and σ2 of S∞ respectively. Since the foliations F 1 and F 2 are transversal outside the set of singularities, σ1 = σ2 . The points σ1 and σ2 are reached   by the geodesic frameworks of the foliations F 1 and F 2 , respectively. Therefore,   by Theorem 4.9, there exist semileaves l1 and l2 of these foliations that reach the points σ1 and σ2 , respectively. Note that according to Theorem 4.9, if li does not  belong to a separatrix of a singularity, then li does not belong to a separatrix of any  singularity; and conversely, if li belongs to a separatrix of a singularity, then li also belongs to a separatrix of a singularity, i = 1, 2. Since the co-asymptotic geodesics  of the corresponding leaves or semileaves that contain li and li coincide and the    geodesic frameworks of the foliations F 1 and F 2 are transversal, the semileaves l1    and l2 intersect at some point denoted by m . Since F 1 and F 2 form a 2-web, the point m is unique. Denote the mapping m → m by φ. By virtue of Theorem 4.9, φ is extended to all the singularities of the 2-web (F 1 , F 2 ) and maps a singularity   to a singularity of the 2-web (F 1 , F 2 ) with the same number of separatrices that reach the same points on S∞ . One can verify that φ covers a certain homeomorphism φ : M 2 → M 2 that   realizes a topological equivalence of the 2-webs (F 1 , F 2 ) and (F 1 , F 2 ). Since by the

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construction, φ is extended to the absolute as the identity mapping, φ is homotopic to the identity. 2 5.4. Homeomorphisms with invariant local laminations. Let f : M 2 → M be a homeomorphism of a surface M and F a foliation on M 2 that is invariant under f (i.e. f (Sing (F)) = Sing (F) and f maps every leaf onto a leaf). F is said to be contracting if, given any points a and b that belong to same leaf, the distance between f n (a) and f n (b) tends to zero as n → +∞ in the interior metric on the leaves. A foliation F is called expanding if it is contracting under f −1 . Following Anosov and Zhuzhoma [12] a homeomorphisms f : M 2 → M 2 is called almost pseudo-Anosov (AP-homeomorphism) if it satisfies the conditions: • f has invariant foliations F s , F u that form a strongly irrational 2-web. • F s is contractive and F u is expanding under f . AP-homeomorphisms are in sense non-uniform pseudo-Anosov homeomorphisms. The class of AP-homeomorphisms includes pseudo-Anosov ones for which the contraction and expansion satisfy some uniform estimates. Homeomorphisms of T2 and hyperbolic surfaces. Let us recall that on T2 a strongly irrational 2-web actually is a 2-web consisting of a pair of transversal irrational foliations without singularities. The following theorem says that an APhomeomorphism T2 → T2 is Anosov hyperbolic automorphism up to conjugacy (see [44]). 2

Theorem 5.14. Let f : T2 → T2 be AP-homeomorphism. Then f is conjugate to an Anosov hyperbolic automorphism. Let f : Δ → Δ be a lift for f : M 2 → M 2 where M 2 is a closed orientable hyperbolic surface. Due to [56] (see also [46]), f extends continuously to a homeomorphism Δ∪S∞ → Δ∪S∞ denoted again by f . The crucial step in a classification of AP-homeomorphisms is the following theorem (see [39]). Theorem 5.15. Let f1 , f2 : M 2 → M 2 be AP-homeomorphisms of a closed orientable hyperbolic surface M = Δ/Γ. Then f1 and f2 are conjugate via a homotopy trivial homeomorphism if and only if there exist the lifts f 1 , f 2 : Δ → Δ of f1 , f2 respectively whose extensions on S∞ coincide, f 1 |S∞ = f 2 |S∞ . Let G be a group and φ1 , φ2 automorphisms of G. Recall that φ1 , φ2 are conjugate if there is an automorphism ξ : G → G such that φ2 ◦ ξ = ξ ◦ φ1 . It is well known that a homeomorphism f : M 2 → M 2 induces an automorphism f∗ : π1 (M 2 ) → π1 (M 2 ) of the fundamental group π1 (M ). Two homeomorphisms f1 , f2 : M 2 → M 2 are called π1 -conjugate if f1∗ , f2∗ are conjugate automorphisms of the group π1 (M ). If h ◦ f1 = f2 ◦ h then h∗ ◦ f1∗ = f2∗ ◦ h∗ . Therefore, two conjugate homeomorphisms are necessarily π1 -conjugate. Moreover theorem 5.15 and Nielsen [56] imply that the π1 -conjugacy is also a sufficient condition of conjugacy for APhomeomorphisms (see [15, 16, 39] and also [35]). Theorem 5.16. Let f1 , f2 : M 2 → M 2 be AP-homeomorphisms of a closed orientable hyperbolic surface M = Δ/Γ. Then f1 and f2 are conjugate if and only if they are π1 -conjugate. Note that in [14–16], necessary and sufficient conditions for the conjugacy of homeomorphisms f : M → M of a closed hyperbolic surface were obtained in the

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case when one of the invariant foliations is irrational and the other is of Denjoy type, as well as in the case when both invariant foliations are of Denjoy type. Classification of one-dimensional basic sets. Let Ω be a one-dimensional basic set of an A-diffeomorphism f : M 2 → M 2 of a closed orientable hyperbolic surface M 2 . Then Ω is either an attractor or a repeller [58, 67]. Assume, for definiteness, that Ω is an attractor. In this case, Ω is an expanding attractor : its topological dimension coincides with the dimension of unstable manifolds. Profound results on the structure and dynamics of expanding attractors belong to Williams [67]. However, solving the problem of the classification of one-dimensional expanding attractors, one should take into account the character of the embedding of expanding attractors into the surface. Recall that a closed subset Ωc of a basic set Ω is called C-dense if both intersections W s (m) ∩ Ωc and W u (m) ∩ Ωc are everywhere dense in Ωc for any point m ∈ Ωc . It is well known [1, 31] that a basic set consists of a finite number of Cdense components that are cyclically mapped to each other by the diffeomorphism. Passing to an iterate of the diffeomorphism, we can make it so that the diffeomorphism has only C-dense basic sets. Below, unless otherwise stated, we will assume that expanding attractors are C-dense. If Ω is a one-dimensional expanding attractor then the unstable manifolds def u {W (m) : m ∈ Ω} = W u (Ω) form a local C 1 lamination that consists of nontrivially recurrent leaves. Each leaf of W u (Ω) is everywhere dense in W u (Ω). This, combined with Theorem 4.8, implies the following proposition. Theorem 5.17. Let f : M 2 → M 2 be an A-diffeomorphism of a closed orientable hyperbolic surface M 2 , and Ω a one-dimensional widely disposed (in particular, orientable) expanding attractor of f . Then (1) the geodesic framework G(W u (Ω)) of W u (Ω) is a weakly irrational geodesic lamination; (2) any geodesic of G(W u (Ω)) is a co-asymptotic geodesic of a leaf belonging to W u (Ω). Similar to Section 5.2, we introduce the concept of a distinguished geodesic as a geodesic that is co-asymptotic for more than one leaf of the lamination W u (Ω). The family of distinguished geodesics forms the distinguished set. The next theorem follows from results obtained by Grines [37], and R. Plykin [61] (see also [22], [42], [43]) and give necessary and sufficient conditions for the conjugacy of one-dimensional basic sets via a homotopically trivial homeomorphism. Theorem 5.18. Let f1 , f2 : M 2 → M 2 be two A-diffeomorphisms of a closed orientable hyperbolic surface M 2 , and let Ω1 and Ω2 be two one-dimensional widely disposed (in particular, orientable) expanding attractors of these diffeomorphisms, respectively. Then f1 and f2 are conjugate on Ω1 and Ω2 via a homotopically trivial homeomorphism M 2 → M 2 if and only if the geodesic frameworks G(W u (Ω1 )) and G(W u (Ω2 )) are equal (without regard to the orientation on the geodesics), and they have the same family of distinguished geodesics, and there exist lifts f 1 , f 2 : Δ → Δ of these diffeomorphisms whose extensions to S∞ coincide, f 1 |S∞ = f 2 |S∞ . Two homeomorphisms of a hyperbolic surface are homotopic if and only if they have lifts with identical extensions to the absolute. Therefore, Theorem 5.18 can be reformulated as follows.

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Theorem 5.19. Let f1 , f2 : M 2 → M 2 be two homotopic (to each other) Adiffeomorphisms of a closed orientable hyperbolic surface M, and let Ω1 and Ω2 be their one-dimensional widely disposed (in particular, orientable) expanding attractors, respectively. Then f1 and f2 are conjugate on Ω1 and Ω2 via a homotopically trivial homeomorphism M 2 → M 2 if and only if the geodesic frameworks G(W u (Ω1 )) and G(W u (Ω2 )) are equal (without regard to the orientation on the geodesics) and have the same family of distinguished geodesics. Note that it is possible to get a generalization of the last two theorems for nonorientable closed surfaces using results from [45].

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[33] T. M. Cherry, Topological Properties of the Solutions of Ordinary Differential Equations, Amer. J. Math. 59 (1937), no. 4, 957–982, DOI 10.2307/2371361. MR1507295 [34] P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. MR0336648 [35] Albert Fathi, Fran¸cois Laudenbach, and Valentin Po´ enaru, Thurston’s work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. MR3053012 [36] A. A. Glutsyuk, Limit sets at infinity for liftings of non-self-intersecting curves on a torus to the plane (Russian, with Russian summary), Mat. Zametki 64 (1998), no. 5, 667–679, DOI 10.1007/BF02316282; English transl., Math. Notes 64 (1998), no. 5-6, 579–589 (1999). MR1691209 [37] V. Z. Grines, The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. I (Russian), Trudy Moskov. Mat. Obˇsˇ c. 32 (1975), 35–60. MR0418161 [38] V. Z. Grines, The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. II (Russian), Trudy Moskov. Mat. Obˇsˇ c. 34 (1977), 243–252. MR0474417 [39] V. Z. Grines, Diffeomorphisms of Two-Dimensional Manifolds with Transitive Foliations. Methods of the Qualitative Theory of Differential Equations, AMS Transl., Ser. 2, 149(1991), 193-199. [40] V. Z. Grines, On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers (Russian, with Russian summary), Mat. Sb. 188 (1997), no. 4, 57–94, DOI 10.1070/SM1997v188n04ABEH000216; English transl., Sb. Math. 188 (1997), no. 4, 537–569. MR1462029 [41] V. Z. Grines, Structural stability and asymptotic behavior of invariant manifolds of Adiffeomorphisms of surfaces, J. Dynam. Control Systems 3 (1997), no. 1, 91–110, DOI 10.1007/BF02471763. MR1436551 [42] V. Z. Grines, Topological classification of one-dimensional attractors and repellers of Adiffeomorphisms of surfaces by means of automorphisms of fundamental groups of supports, J. Math. Sci. (New York) 95 (1999), no. 5, 2523–2545, DOI 10.1007/BF02169053. Dynamical systems. 7. MR1712741 [43] V. Z. Grines, On topological classification of A-diffeomorphisms of surfaces, J. Dynam. Control Systems 6 (2000), no. 1, 97–126, DOI 10.1023/A:1009573706584. MR1738742 [44] V. Z. Grines, V. S. Medvedev, and E. V. Zhuzhoma, On surface attractors and repellers in 3-manifolds (Russian, with Russian summary), Mat. Zametki 78 (2005), no. 6, 813–826, DOI 10.1007/s11006-005-0181-1; English transl., Math. Notes 78 (2005), no. 5-6, 757–767. MR2249032 [45] V. Z. Grines and R. V. Plykin, Topological classification of amply situated attractors of A-diffeomorphisms of surfaces, Methods of qualitative theory of differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, vol. 200, Amer. Math. Soc., Providence, RI, 2000, pp. 135–148, DOI 10.1090/trans2/200/11. MR1769568 [46] Michael Handel and William P. Thurston, New proofs of some results of Nielsen, Adv. in Math. 56 (1985), no. 2, 173–191, DOI 10.1016/0001-8708(85)90028-3. MR788938 [47] P. Koebe, Riemannische Manigfaltigkeiten und nichteuklidiche Raumformen, IY. Sitzung. der Preuss. Akad. der Wissenchaften, 1929, 414-457. [48] A. Mayer, De trajectoires sur les surfaces orient´ ees (French), C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 673–675. MR0002240 [49] A. Mayer, Trajectories on the closed orientable surfaces (Russian, with English summary), Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 71–84. MR0009485 [50] Nelson Groh Markley, THE STRUCTURE OF FLOWS ON TWO-DIMENSIONAL MANIFOLDS, ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)–Yale University. MR2615823 [51] Nelson G. Markley, The Poincar´ e-Bendixson theorem for the Klein bottle, Trans. Amer. Math. Soc. 135 (1969), 159–165. MR0234442 [52] Nelson G. Markley, Invariant simple closed curves on the torus, Michigan Math. J. 25 (1978), no. 1, 45–52. MR497881 [53] N. G. Markley and M. H. Vanderschoot, An exotic flow on a compact surface. part 1, Colloq. Math. 84/85 (2000), no. part 1, 235–243. Dedicated to the memory of Anzelm Iwanik. MR1778853

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[54] N. G. Markley and M. H. Vanderschoot, Remote limit points on surfaces, J. Differential Equations 188 (2003), no. 1, 221–241, DOI 10.1016/S0022-0396(02)00065-7. MR1954514 [55] Harold Marston Morse, A One-to-One Representation of Geodesics on a Surface of Negative Curvature, Amer. J. Math. 43 (1921), no. 1, 33–51, DOI 10.2307/2370306. MR1506428 [56] Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Fl¨ achen (German), Acta Math. 50 (1927), no. 1, 189–358, DOI 10.1007/BF02421324. MR1555256 [57] Dmitri Panov, Foliations with unbounded deviation on T2 , J. Mod. Dyn. 3 (2009), no. 4, 589–594, DOI 10.3934/jmd.2009.3.589. MR2587087 [58] R. V. Plykin, The topology of basic sets of Smale diffeomorphisms (Russian), Mat. Sb. (N.S.) 84 (126) (1971), 301–312. MR0286134 [59] R. V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces (Russian), Mat. Sb. (N.S.) 94(136) (1974), 243–264, 336. MR0356137 [60] R. V. Plykin, Hyperbolic attractors of diffeomorphisms (Russian), Uspekhi Mat. Nauk 35 (1980), no. 3(213), 94–104. International Topology Conference (Moscow State Univ., Moscow, 1979). MR580625 [61] R. V. Plykin, The geometry of hyperbolic attractors of smooth cascades (Russian), Uspekhi Mat. Nauk 39 (1984), no. 6(240), 75–113. MR771099 [62] H. Poincar´ e, Sur les courbes d´ efinies par les equations differentielles. J. Math. Pures Appl. 2(1886), 151-217. [63] V. I. Pupko, Non-Self-Intersecting Curves on Closed Surfaces. Sov. Math. Dokl., 8(1967), 1405-1407. [64] R. Clark Robinson and R. F. Williams, Finite stability is not generic, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, pp. 451–462. MR0331430 [65] A. Weil, On systems of curves on a ring-shaped surface. J. Indian Math. Soc., 19(1932), 5, 109-112. [66] A. Weil, Les familles de curbes sur le tore. Mat. Sbornik, 43(1936), 5, 779-781. ´ [67] R. F. Williams, Expanding attractors, Inst. Hautes Etudes Sci. Publ. Math. 43 (1974), 169– 203. MR0348794 National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya, 603005, Nizhni Novgorod, Russia E-mail address: [email protected] National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya, 603005, Nizhni Novgorod, Russia E-mail address: [email protected]

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13926

Attractors and skew products Yu. Ilyashenko and I. Shilin To the memory of Dmitry Anosov, a great mathematician and a generous person Abstract. Different notions of attractors and relations between them are considered. The major new result claims that Lyapunov unstable Milnor attractors are topologically generic in a space of diffeomorphisms of any manifold of dimension greater than one. This result is due to the second author. A sketch of the proof is given. New robust properties of diffeomorphisms obtained with the help of the so called Ilyashenko-Gordetski strategy are described.

There are zillions of papers dedicated to attractors and skew products. This one is a survey of a small part of these investigations proceeded during the last twenty years in the seminar on dynamical systems supervised by the first author and cosupervised in different time periods by A. Gorodetski, A. Fishkin and currently by the second author. Main unpublished results of this survey are due to the second author and presented in Section 1.3 that he wrote; the rest of the paper is written by the first author. The investigations presented here are strongly based on the partial hyperbolic theory, a daughter of the theory of Anosov maps, initiated by M. Brin and Ya. Pesin, students of Anosov and Katok, and independently by M. Hursh, C. Pugh and M. Shub. The first author appreciates the chance to present this survey in a book dedicated to the memory of D. V. Anosov, with whom he shared so much in his life. 1. Attractors 1.1. Various definitions of attractors. In general, by attractor we understand an attracting set (whatever it means) in a phase space of a dynamical system. The word attractor was first mentioned by Auslander et al in [ABS], 1964. There is a great variety of rigorous definitions of an attractor, serving different needs. Let us fix the notations: let F be a diffeomorphism of some manifold X with boundary or without it. In general, we will denote by A with subscripts an attractor of the system. 2010 Mathematics Subject Classification. Primary 37C70. The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economic (HSE) in (2016-17) (grant # 16-05-0066) and supported within the framework of a subsidy granted to the HSE by the Government of Russian Federation for the implementation of the Global Competitiveness program. c 2017 American Mathematical Society

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Figure 1. The construction of Amax 1.1.1. Maximal attractor. Historically the first rigorous definition of attractor was a definition of a maximal attractor. Definition 1. Suppose there exists an open set B ⊂ X such that F (B) ⊂ B — then we can define so-called maximal attractor in the following way (see Figure 1): ∞ % Amax = F n (B) n=1

Remark 1. The general definition of Amax requires only a topological structure on X, which is defined since X is a manifold. We say that Amax is a global attractor corresponding to a pair (X, F ) iff for a.e. point one of its iterations eventually arrives into B. Note that here we need X to be a set with a measure. In the case when X is a Riemannian manifold, the measure is defined by the metric. Let us note that in some sense the system chooses the attractor for itself: to understand what points belong to attractors and what points do not, we have just to put in work a mechanism of F , wait, and then intersect. Note also that the global maximal attractor is not unique. Yet any such attractor contains the (defined below, and defined uniquely) Milnor attractor. Example 1. A simplest example of a map demonstrating a maximal attractor is a north-south map of a circle (Fig. 2): any open domain which doesn’t contain a neighborhood of the repeller S is mapped in a domain close to an attractor N. Amax = N . Example 2. Let us extend the previous example, i.e., let’s take an annulus X containing a circle with a north-south map f on it. The coordinates on an annulus are chosen in such a way that ϕ is polar angle of a point of a ring, x is its shifted radial coordinate: x = 0 on the circle. F is now given by x (1) F : (ϕ, x) → f (ϕ), 2 where f is the north-south map mentioned above. Here the south becomes a saddle and the north becomes a node, see Figure 3.

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Figure 2. A north-south map of a circle

Figure 3. An excessive maximal attractor In this case Amax = S 1 (if we choose B = X), and that seems to be not an appropriate result, because the answer Amax = N is expected intuitively. Although if B is chosen properly (not containing S and its stable separatrices), then Amax (B) = N as it should be. Example 3. Consider now a diffeomorphism of a circle with a unique parabolic fixed point P (Figure 4). For such f , there is no nonempty proper open subset which is mapped strictly into itself, so in this case Amax = S 1 . But, as before, our intuition requires the answer Amax = P . However, this example can not be saved by the choice of B. The definition of Amax has the disadvantage illustrated in the examples above: first, a natural Amax = X doesn’t exist for all the systems. Second, if we require the existence of Amax , we end up with having too many extra points in the attractor. Nevertheless, almost twenty years the definition of a maximal attractor was the only one in a world of attractors.

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Figure 4. No proper maximal attractors Remark 2. One can also consider a related definition of a local attractor: a closed transitive set is called (local) topological attractor if it is a maximal attractor in some its neighborhood. An attracting periodic orbit is a natural example. The transitivity requirement guarantees the absence of redundant points. However, in the relatively recent work [BLY] C. Bonatti, M. Li, and D. Yang constructed a nonempty open domain in the space of diffeomorphisms of arbitrary manifold of dimension at least three where topologically generic diffeomorphisms have no topological attractors. In their example each set that is an intersection of a sequence of dissipative neighborhoods is generically accumulated by a sequence of periodic orbits which prevents it from being a topological attractor. 1.1.2. Milnor attractor. In 1985 in his paper [M85] John Milnor introduced a new concept of attractor. Let us give two equivalent definitions of Milnor attractor: Definition 2. AM is the smallest by inclusion closed set containing ω-limit sets of a.e. point. Definition 3. AM is the smallest closed set such that a.e.

dn (x) = d (f n (x), AM ) −−−−→ 0 n→∞

Here we assume that some measure is fixed on the phase space X. For diffeomorphisms this is usually a Riemannian volume or any other measure equivalent to the Lebesgue measure when restricted to any chart. In Examples 1 and 2 above AM = N . In the case of f with a parabolic point, Example 3, AM = P . Note that in this example the attractor is Lyapunov unstable in sense of Definition 7. 1.1.3. Minimal and statistical attractors. The following definitions were given in [AAIS] and [GI96] (see also [I91]that contains an error corrected in [GI96]). Definition 4. Let X be a metric measure space, and f : X → X be a homeomorphism. For this map Astat is the smallest closed set with the following property: almost all points spend almost all time in any neighborhood of Astat . This definition is suggested by what may be seen in the numeric experiments as described in [AAIS]. The first author presented this definition to Arnold in 1985,

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and he discussed it with Kolmogorov, whom he visited in a hospital. Later Arnold told that Kolmogorov was satisfied by this definition, and considered that the orbits may approach the attractor, then eventually quit its neighborhood, then come back for a longer time, and so on. An equivalent definition of the statistical attractor was suggested by Milnor in his article Attractors in Scholarpedia. For the future study of these attractors see [Ca] and references therein. Definition 5. Let X be a metric measure space and f : X → X be a homeomorphism. For this map Astat is the smallest closed set with the following property. Let the function dn be the same as in the definition of the Milnor attractor. Then the Chesaro series (the series of arithmetic averages) of dn (x) converges to zero almost everywhere (with respect to the measure on X). Another natural definition provides a notion of even smaller attractor. Definition 6. Consider a homeomorphism of a smooth manifold with a finite Riemannian volume. Consider an averaging procedure for this volume, like in the Krylov-Bogolyubov theorem. Take all the limit measures that occur in this procedure. The minimal attractor (with respect to the Riemannian volume) is the closure of the union of supports of all these measures. Another definition of the minimal attractor repeats almost verbatim Definition 5 of a statistical attractor for the case of the Lebesgue measure. The only difference is that the convergence dn → 0 is required not almost everywhere, but in measure. Milnor, statistical, and minimal attractor exist for any homeomorphism of a compact metric space endowed with a Borel measure. This is an easy consequence of the definitions proved for the Milnor attractor in [M85] and for statistical and minimal attractors in [GI96]. Note that ordinary convergence implies Chesaro convergence, and convergence almost everywhere implies convergence in measure. Hence, Amin ⊆ Astat ⊆ AM ⊆ Amax . It turns out that all the inclusions above may be strict. 1.2. Non-coincidence of attractors. Let us mention the examples of noncoincidence of the attractors defined above, for the case of the Lebesgue measure. Amax = AM : a diffeomorphism of a circle with a unique parabolic point P , see Figure 4, Amax = S 1 , AM = P . Astat = AM : let us take a separatrix loop of a saddle S of a planar vector field v, and let X be an interior of the separatrix loop. Suppose the trajectories tend to the separatrix loop (see Figure 5). If we take F = gv1 , the time one phase flow transformation of v, then AM (F ) is the separatrix loop. However, Astat = S. Indeed, the time spent in a neighborhood of a saddle is growing while the time spent near the other parts of the loop is approximately the same. Amin = Astat : let us consider a vector field with one saddle S and one saddlenode SN with common separatrices as in Figure 6. Let F be the time one phase flow transformation of this field. Then Astat (F ) = S ∪ SN , Amin (F ) = SN . This is a delicate result due to V. Kleptsyn [Kl]. All the attractors coincide in the case of C 2 Axiom A diffeomorphisms, e.g. see [G96].

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Figure 5. Non-coincidence of the Milnor and statistical attractors

Figure 6. Non-coincidence of the statistical and minimal attractors One of the main problems in the domain is the following. Problem 1. Is non-coincidence of attractors of diffeomorphisms a generic phenomenon? In the previous examples, non-coincidence of maximal and Milnor attractors is a degeneracy of codimension one; non-coincidence of Milnor, statistical, and minimal attractors is shown in examples of infinite codimension. Indeed, these maps are defined as phase flow transformations of vector fields. For flows, the definitions of Milnor, statistical, and minimal attractors are parallel to the case of diffeomorphisms. In the previous examples, non-coincidence of the Milnor and statistical attractors occurs on a set of flows of codimension 1. Non-coincidence of the minimal and statistical attractors occurs on a set of flows of codimension 3. This motivates the following problem. Problem 2. Is non-coincidence of attractors of flows a generic phenomenon? It was noticed by A. Okunev that, if a diffeomorphism has infinitely many periodic sinks, its Milnor attractor is not asymptotically stable and, therefore, cannot be a maximal attractor in some open set.1 Let us recall relevant definitions. 1 The

proof of this is included in [Sh] with permission from A. Okunev.

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Definition 7. An invariant set A of a homeomorphism F is Lyapunov stable provided that for any neighborhood U of A there exists a neighborhood V of A such that for any n > 0, F n (V ) ⊂ U . Definition 8. An invariant set A of a homeomorphism F is asymptotically stable if it is Lyapunov stable and contains ω-limit sets of all points of some its neighborhood. Since coexistence of infinitely many sinks is a locally topologically generic phenomenon (see, e.g., [N74]), so is the non-coincidence of Milnor and maximal attractors. Moreover, in the relatively recent preprint [B] by P. Berger it has been shown that there exist locally topologically generic finite-parameter families of diffeomorphisms where the unit ball in the parameter space corresponds to diffeomorphisms which have infinitely many sinks. Thus, in this setting we also have Milnor attractors which are not maximal attractors. The second author showed recently (see [Sh]) that Lyapunov instability of Milnor attractors is also rather abundant and the same is true for statistical and minimal attractors. We discuss this result in the next subsection. 1.3. Lyapunov instability of Milnor, statistical, and minimal attractors. 1.3.1. The instability theorem. Theorem 1. [[Sh]] For any r ≥ 2 and any compact manifold M of dimension at least 2, with or without boundary, there exists an open set U ⊂ Diff r (M ) with the following property: for a topologically generic diffeomorphism F ∈ U , the Milnor, the statistical, and the minimal attractors of F are Lyapunov unstable. Remark 3. The same is true for r = 1 if M has dimension greater than two. The reason is that at the heart of the proof, as we will see shortly, lie persistent homoclinic tangencies associated with sectionally dissipative saddles, and such tangencies have been shown to exist in C 1 for any manifold of dimension at least three. Below we discuss only Milnor attractors; instability of statistical and minimal attractors is proved in the same way. 1.3.2. Description of the set U from Theorem 1. First we need to recall several definitions. Definition 9. A hyperbolic periodic orbit of a diffeomorphism with the multipliers from both sides of the unit circle is called a periodic saddle. It is sectionally dissipative if it has a unique “expanding” multiplier λ1 : |λ1 | > 1, and for any two multipliers λi , λj (i = j) one has |λi · λj | < 1. Definition 10. Let p be a sectionally dissipative periodic saddle of some diffeomorphism F ∈ U ⊂ Diff r (M ). There is a U -persistent tangency associated with p provided that: i) any diffeomorphism G ∈ U has a sectionally dissipative periodic saddle p(G), continuous in G, and p(F ) = p; ii) there exists a dense set W ⊂ U such that for any G ∈ W the manifolds W u (p(G)) and W s (p(G)) have a quadratic tangency at some point q(G).

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Note that the periodic orbit p(G) depends continuously on G, but the point of homoclinic tangency q(G) does not. The set U in Theorem 1 is characterized by the following properties: a) There is a U -persistent tangency associated with a sectionally dissipative periodic saddle p; b) For any G ∈ U , at least one orbit in W u (p(G)) belongs to a basin of some sink a(G). The existence of the set U described above is not obvious, and we will justify it now. Open domains with persistent tangencies in Diff r (M ), r ≥ 2, dimM ≥ 2, were constructed by S. Newhouse in the paper [N74]. He also indicated in [N80] how to construct such domains in C 1 when the dimension of M is at least three; this construction was later rediscovered by M. Asaoka and was presented in the paper [A]. Another tool of constructing such domains is the so-called blender introduced in [BD] by C. Bonatti and L. D´ıaz. In the case r ≥ 2 there is an even more powerful theorem that says that domains with persistent tangencies are to be found near any diffeomorphism with a homoclinic tangency for a sectionally dissipative saddle. This has been shown in dimension two by S. Newhouse in the paper [N79] and generalized for higher dimensions by J. Palis and M. Viana in [PV]. Let us now construct a set U with the properties a) and b). Consider an open set V ⊂ Diff r (M ) with a persistent tangency associated with (a continuation of) some sectionally dissipative saddle p. The set V itself and any its open subset have the property a). To achieve the property b), we will use the following lemma. Lemma 1 (capture lemma, [Sh]). Let F ∈ Diff r (M ), 1 ≤ r ≤ ∞, have a sectionally dissipative periodic hyperbolic saddle p = p(F ) with a homoclinic tangency. Then arbitrary C r -close to F there is a diffeomorphism G for which W u (p(G)) intersects the basin of some periodic sink. Let us now construct a dense subset U ⊂ V having the property b). By definition of V -persistent tangency, there exists a dense subset W ⊂ V such that any F ∈ W satisfies the assumptions of the capture lemma. By this lemma, arbitrary close to F there exists a map G that satisfies property b). But this property is open. Hence, there exists an open set U (F ) that satisfies the property b) and contains F in its closure. The union U = ∪F ∈W U (F ) is dense in V and satisfies properties a) and b). The construction of U is over. The proof of the capture lemma is rather technical, so we do not present it here and refer the reader to [Sh] instead. It must be mentioned that, as J.C. Tatjer kindly informed the second author, in dimension two the capture lemma follows from the main result of the paper [TS]. 1.3.3. A sufficient condition for the Lyapunov instability of the Milnor attractor. Proposition 1. Suppose that a diffeomorphism F ∈ Diff 1 (M ) satisfies the following conditions: • F has a hyperbolic saddle p whose unstable manifold W u (p) intersects the basin of attraction of some sink a(F ). • F has a sequence of periodic sinks γj , j ∈ N, that accumulate to p, i.e., dist (γj , p) → 0 as j → ∞.

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Then AM (F ) is Lyapunov unstable. Proof. First note that the Milnor attractor AM (F ) lies in the non-wandering set Ω(F ). This follows easily from the definition of the Milnor attractor: ω-limit points are dense in AM (F ), these points are non-wandering, and the non-wandering set of F is closed, therefore AM (F ) ⊂ Ω(F ). By the same definition, every sink γj belongs to the Milnor attractor since the basin of attraction of γj contains a neighbourhood of this sink and, therefore, has positive measure. As these sinks accumulate to the saddle p and the Milnor attractor is closed, we get: p ∈ AM (F ). By the first assumption of the proposition, there is a point that is attracted to the sink a(F ) in the future and to p in the past. This point is wandering and, therefore, separated from the attractor AM (F ), but its preimages under the iterates of F come arbitrarily close to p ∈ AM (F ), which contradicts Lyapunov stability of the attractor.  We can now finish the sketch of the proof of Theorem 1. As we discussed earlier, there exists an open domain U ⊂ Diff r (M ) with the properties a) and b). To prove the theorem it suffices to show that a topologically generic diffeomorphism G ∈ U has a sequence of sinks accumulating to the saddle p(G) and then apply Proposition 1 to such diffeomorphisms. Let W ⊂ U be a dense subset from Definition 10. For G ∈ W , denote by q(G) the point of the quadratic homoclinic tangency. According to Prop. 1 of [N74], when unfolding a homoclinic tangency by a C r -small perturbation, we can create a hyperbolic periodic sink that passes arbitrarily close to the point q(G). Since finite segments of orbits depend continuously on the initial point and the mapping, these sinks pass arbitrary close to the saddle p(G) as well. Therefore, for any n there exists a sequence Hn,k → G as k → ∞ with the following property. For any k, the map Hn,k has a sink passing at a distance smaller than n1 from the saddle p(Hn,k ). The latter property is open, so it holds for some neighborhood Un,k of Hn,k . Let Un (G) = ∪k Un,k , and let Un = ∪G∈W Un (G). r r r The  sets Un are C -open and C -dense in U ⊂ Diff (M ). Hence, the set R = n Un is a residual subset of U . For any map G ∈ R, arbitrarily close to p(G) there is a sink. Therefore G has a sequence of sinks that approach the saddle p(G). Any diffeomorphism G ∈ R satisfies the assumptions of Proposition 1: it has sinks accumulating to a saddle; the unstable manifold of this saddle intersects the basin of the sink a(G) by the property b) of the domain U . Then, by Proposition 1, G has an unstable Milnor attractor. The sketch of the proof of Theorem 1 is complete.2

1.4. Problems. Problem 3. Is Lyapunov instability of the Milnor attractor locally metrically generic? 2 The second author anticipates that the same arguments with minor modifications can be applied to the construction presented in [B] by P. Berger, which would prove the existence of locally topologically generic finite-parameter families where the unit ball in the parameter space corresponds to diffeomorphisms with Lyapunov unstable Milnor (statistical, minimal) attractors.

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There is a drastic difference between topological and metric genericity in the theory of dynamical systems. The Newhouse phenomenon is the coexistence of an infinite number of sinks for topologically generic diffeomorphisms in some open set in Diff(M ) for a compact manifold M . Like persistent tangencies, it is observed in C 1 for any manifold of dimension at least three and in C 2 for any manifold of dimension at least two. But the Palis conjecture [P05] implies that for a metrically generic set in Diff(M ) only a finite number of sinks may coexist. A partial result that confirms this conjecture is proved in [GK07]. The recent result by P. Berger [B] is a progress in an opposite direction. This suggests the following Conjecture 1. The set of diffeomorphisms with an unstable Milnor attractor is not shy (negligible from the point of view of metric genericity). Problem 4. Could the inequalities Amin = Astat = AM be obtained for diffeomorphisms in a finite codimension? 2. Gorodetski-Ilyashenko strategy 2.1. Three steps. The strategy named in the title is oriented for finding new robust properties of dynamical systems. It is based on the following heuristic principle. All effects observed in a finitely generated free semigroup action by diffeomorphisms of a manifold, may be observed for a single diffeomorphism of a manifold of higher dimension. This heuristic principle is realized in three steps. Step 1. Step skew products. A free semigroup action on a manifold M may be realized as an action of one homeomorphism on a larger space X. Let f1 , . . . , fN ∈ Diff(M ) be the generators of the semigroup, and ΣN be the set of bi-infinite sequences of N symbols with the Bernouli measure and the standard metric on it. Let X = ΣN × M . In this context the space ΣN will be called the base, and M will be called the fiber. The homeomorphism that realizes the action of the group has the form: (2)

F : X → X, (ω, x) → (σω, fω0 (x));

here σ is the Bernoulli shift, ω0 is an element of the sequence ω at position zero. All the effects observed for the action of the semigroup generated by the maps fj may be also observed in the cascade3 generated by the skew product F , see (2). The map (2) is called a step skew product. The reason is the following. Consider a skew product whose fiber maps fω depend on the whole sequence ω, and not only its zero element ω0 : (3)

F : X → X, (ω, x) → (σω, fω (x)).

Such skew products are called mild. Step skew products are distinguished from the mild ones by the assumption: fω = fω0 . The fiber maps for these products are the same and equal to fj on the cylinder {ω0 = j} and thus resemble step functions. This motivates the name for the map (2). Step 2. Smooth realization of the step skew product. Consider N disjoint horizontal rectangles Dj in the plane and a Smale horseshoe map h of the union D of these rectangles to the union D of N vertical disjoint rectangles, as in Fig. 7. The diffeomorphism h has a hyperbolic invariant set Λ. 3 The term cascade, parallel to the term flow, was suggested by D. Anosov for the action of the iterates of one map.

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Figure 7. Smale horseshoe map The restriction h|Λ is topologically conjugate to the Bernoulli shift σ : ΣN → ΣN . Consider a diffeomorphism (4)

F : D × M → D × M, (b, x) → (h(b), fb (x)), fb = fj for b ∈ Dj .

The map F has an invariant set Λ × M . The restriction of F to this set is topologically conjugate to the map (2). Hence, all the effects found for step skew products may be observed for diffeomorphisms restricted to their invariant sets. Step 3. The last and the most difficult step in the program is to prove that these properties persist under small perturbation of the skew product diffeomorphism in the space of all diffeomorphisms of a neighborhood of D × M to D × M . This is done by the use of the Hirsh-Pugh-Shub theory [HPS] and by some H¨older continuity theorems going back to Anosov and described below. 2.2. Skew products over a solenoid and Anosov maps. A disadvantage of the above strategy is that the dynamics is studied on a partially hyperbolic invariant set. Almost all points of a small neighborhood of this set spend but a finite time in this neighborhood . It is more interesting to replace the invariant partially hyperbolic set by an attractor, or by the whole phase space. To do that, one may consider skew products over a solenoid map, or an Anosov map of a torus. But in this case the invariant set of the base is connected, and there are no step functions continuous on these sets. Skew products over a solenoid and Anosov maps can not be step, but can only be mild ones. Studying the iterates of a mild skew products meets the following difficulty: no finite segment w of a sequence ω allows us to determine precisely the fiber map fω . It now depends not only on the zero element ω0 , and not on a finite set of elements, but on the whole sequence ω. This difficulty is bypassed by considering so called H¨ older skew products . 2.3. H¨ older skew products and their iterates. Definition 11. A skew product (3) is called H¨ older, provided that the fiber maps are H¨older continuous with respect to the point of the base. More precisely, for some r ∈ N, α ∈ (0, 1), C > 0 the following holds: (5)

dC r (fω , fω ) ≤ Cdα (ω, ω  ).

The iterates of skew products are skew products again. Denote by fm,ω the fiber map of the iterate F m over the point ω, where F is a mild skew product (3).

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We have: F m (ω, x) = (σ m ω, fm,ω (x)). For any positive integer m, (6)

fm,ω = fσm−1 ω ◦ · · · ◦ fσω ◦ fω ,

(7)

−1 f−m,ω = fσ−1 −m ω ◦ · · · ◦ fσ −1 ω .

If two sequences ω, ω  have the same finite subsequence corresponding to a word, say, w = ω−2m . . . ω0 . . . ω2m , that is (8)

  ω−2m . . . ω2m = ω−2m . . . ω2m ,

then the sequences σ k ω, σ k ω  for 2|k| ≤ m are close to each other exponentially with respect to m. The corresponding fiber maps of F are also exponentially close, by (5). Therefore, the difference between the compositions fk,ω and fk,ω , as well as f−k,ω and f−k,ω , is small exponentially in m. The fiber maps fk,ω and fk,ω , are close uniformly in m, as well as f−k,ω and f−k,ω . These arguments are formalized in lemmas 3.1, 3.2 of [GI00]. 2.4. Perturbations of skew product diffeomorphisms. It is shown below that H¨older skew products naturally occur in perturbations. In this section B is a smooth manifold with or without boundary, and h : B → B is either an Anosov diffeomorphism onto B, or a diffeomorphism onto its image, which may be a proper subset of B, having a hyperbolic maximal attractor Δ. For simplicity, we will consider the first case only. Parallel results, with B replaced by Δ, hold in the second case. Consider a skew product diffeomorphism (9)

F : B × M, (b, x) → (h(b), fb (x)).

Suppose that h is an Anosov diffeomorphism and has hyperbolicity constants λ < 1 < μ. Definition 12. A map (9) satisfies an r-dominated splitting condition provided that λ < L−r < 1 < Lr < μ, where L is the maximal Lipschitz constant of the fiber maps: L = max(Lip(fb ), Lip(fb−1 )). b∈B

Theorem 2. [HPS] Let M be a compact manifold and F : X = B × M → X be a diffeomorphism that satisfies the r-dominated splitting condition. Then any sufficiently C r -small perturbation G of F is topologically conjugate to a skew product G˜ over the same map h : B → B with the same fiber M (10)

G˜ : X → X, (b, x) → (h(b), gb (x)),

where gb are C r -smooth in x and continuous in b. Moreover, there exists a semiconjugacy p : X → B that makes the following diagram commutative:

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G

X −→ X p↓ ↓p B

h

−→

B

The fibers Mb = p−1 (b) are C r -smooth, diffeomorphic to M , and continuous in b. This is a so called foliation stability theorem. It implies that the map h permutes new invariant central leaves of the perturbed map. Therefore, the skew product structure of the diffeomorphism F persists. The perturbed diffeomorphism, a priori, has no skew product structure, but it may be restored by the use of the foliation stability theorem. The only loss is that the fibers and the fiber maps of the unperturbed diffeomorphism F depended smoothly on the base and for a perturbed map they are only continuous with respect to the base point. It occurs that, loosely speaking, the perturbed fibers and fiber maps are not only continuous, but rather H¨ older continuous with respect to the base point. Precise statements follow. 2.5. H¨ older property by Gorodetski. Let M be a closed manifold, X = B × M . Consider a skew product diffeomorphism with the fiber maps the identity: (11)

F = h × id.

Theorem 3 ([G06]). A C r+1 -small perturbation G of F is topologically conolder jugate to a skew product (10), where the fiber maps are C r -smooth and H¨ continuous in b ∈ B. Namely, (12)

dC r (gb , gb ) ≤ Cdα (b, b )

for some α ∈ (0, 1), C > 0. The H¨ older exponent α and the constant C may be taken the same for all maps G from a C r+1 -small neighborhood of F . The advantage of this theorem is that the H¨older property (12) is proved for any r > 0. The disadvantage is that the unperturbed fiber maps are supposed to be identity. A particular case of Theorem 3 was proved in [NT]. The proof of Theorem 3 is based on a theorem by Anosov. This theorem claims that the homeomorphism that conjugates two Anosov diffeomorphisms is H¨older continuous. This theorem remained unpublished for a long time, its proof first appeared in [KH]. Gorodetski transferred this theorem to so called intrinsic hyperbolic maps, that is, maps of the sets considered by themselves, independently of any ambient space. Such maps were introduced by V. Alexeev and M. Yakobson [AYa]. Gorodetski considers the central fibers of the perturbed map as points in the functional space, and the map G that permutes them as an “intrinsic hyperbolic” map. This map is topologically conjugate to h : B → B. Hence, the linking map is H¨older. The following two theorems discuss the property (12) in case r = 0.

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2.6. Generalizations and upper estimates of the H¨ older exponent. The main result of [IN2] improves the Gorodetski theorem in two ways: the fiber maps of the unperturbed diffeomorphism are no more identity; the H¨older exponent is explicitely estimated from above. But the estimate (12) is proved for r = 0 only. Yet it is sufficient for various applications. The most general result in the field for r = 0 is obtained in [PSW], where partially hyperbolic maps are considered independently of skew products. Theorem 4 ([PSW]). Consider a partially hyperbolic diffeomorphism with compact central fibers. Under mild additional assumptions, the central fibers are H¨ older continuous in the C-norm with respect to the initial conditions. The H¨ older exponent may be expressed through the hyperbolicity constants of the map, that is, through the constants that characterize strong contraction and strong expansion. This theorem implies the previous result. An interesting problem is whether the C-norm in this theorem may be replaced by the C r -norm, r > 0. Skew products over the Anosov maps are studied by many authors. One of the most striking results is the so called Fubini nightmare discovered in [SW] and studied in more detail in [RW]. 3. New robust properties of diffeomorphisms of closed manifolds and manifolds with boundary 3.1. Dense orbits with zero intermediate Lyapunov exponents. The first result obtained in frames of the strategy outlined above was the following Theorem 5. [[GI99], [GI00], [G06]] Consider a closed manifold M, dimM ≥ 4, and an arbitrary open interval I ⊂ R, 0 ∈ I. There exists an open set U ⊂ Diff2 (M ) such that every f ∈ U has a partially hyperbolic invariant set Δ which is a maximal attractor in its neighborhood and has the following properties. (i) There are numbers l1 and l2 = l1 +1 such that the hyperbolic periodic orbits with the index (dimension of the stable manifold) lj , j = 1, 2, are dense in Δ. (ii) For every λ ∈ I there is an orbit dense in Δ with one intermediate Lyapunov exponent equal to λ. Property (i) was discovered in [BD] with the use of so called blender, a special partially hyperbolic set, which is another powerful tool of building locally generic diffeomorphisms with new robust properties. 3.2. Non hyperbolic invariant measures. To what extent is the behaviour of a generic dynamical system hyperbolic ? A number of problems in the modern theory of dynamical systems may be viewed as some forms of this question. As an example, let us mention the following problem. Problem 5. Does a generic diffeomorphism of a compact Riemannian manifold have all Lyapunov exponents nonzero for any good measure? (An invariant measure is good if it can be obtained from the Lebesgue measure by the Krylov-Bogolyubov procedure). This problem stated by M. Shub and A. Wilkinson in [SW] in a slightly different form is still open. The following results suggest an idea that the answer to the previous question may be negative.

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Currently there are two methods of construction ergodic non-hyperbolic measures in smooth dynamics. First, the method based on periodic approximations was used in [KN] to construct open sets of diffeomorphisms with such measures. Theorem 6. [KN] In the space of C 2 diffeomorphisms of the three-dimensional torus, there exists an open set of mappings having an ergodic nonatomic measure with one of the Lyapunov exponents equal to zero. This result is obtained in frames of the GI strategy, and based on the following theorem. 2

Theorem 7. [GIKN] In the space (Diff1 (S 1 )) of pairs of diffeomorphisms of a circle equipped with the C 1 -topology, there exists an open set U such that for each pair in U the corresponding step skew product (2) has a non-atomic ergodic invariant measure with zero Lyapunov exponent along the fiber. The open set in the Kleptsyn-Nalski theorem 6 is a neighborhood of a skew product diffeomorphism with the fiber a circle. This approach was applied to generic non-hyperbolic homoclinic classes of diffeomorphisms [DG,BDG]. It provides conditions for a sequence of atomic measures to converge to a non-trivial non-hyperbolic ergodic measure. Second, in [BBD], some C 1 -open conditions sufficient for a diffeomorphism to possess a nonhyperbolic ergodic measure with positive entropy were stated. These conditions are satisfied for a large class of non-hyperbolic C 1 diffeomorphisms, and they imply existence of a partially hyperbolic compact sets with one-dimensional center direction and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The method uses a construction of a blender and allows one to construct a C 1 -open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with non-hyperbolic ergodic measures with positive entropy. A wide class of step skew products with the fiber a circle over the Bernoulli shift was constructed in [DGR]. The maps of this class are transitive and have nonhyperbolic invariant measures. 3.3. Attractors with intermingled basins. Skew product diffeomorphisms of manifolds with boundary are another source of new effects. A famous example by Ittai Kan provides attractors with so called intermingled basins [K94]. The map analyzed in [K94] is a particular endomorphism of an annulus onto itself that preserves the boundary. This endomorphism is a skew product over a circle doubling. Milnor and Bonifant [BM] suggested a more general construction. Consider a skew product F : S 1 × [0, 1] → S 1 × [0, 1], (b, x) → (kb, fb (x)), k > 2, with the following properties. Let the Swartzian derivative of the fiber map be negative almost everywhere. Let the following monotonicity property hold: the fiber map f0 over a fixed point 0 is a north-south map with a hyperbolic attractor 0 and hyperbolic repeller 1; the 1 , has the similar property, with fiber map fx+ over another fixed point x+ = k−1 the only difference that 0 is a repeller and 1 is an attractor.

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Theorem 8. [BM] Under the assumptions above, the Milnor attractor of the map F is the union: (13)

AM = A0 ∪ A1 , A0 = S 1 × {0}, A1 = S 1 × {1}.

The basins B0 of A0 and B1 of A1 (that is, the sets of points whose ω-limit sets belong to A0 and A1 respectively) are dense and metrically dense: their intersection with any disc has positive Lebesgue measure. 3.4. Local genericity of attractors with intermingled basins. Previous results provide a large set of examples of attractors with intermingled basins. It is important to justify that these examples illustrate an effect, that is, the phenomenon that they show is at least locally generic. Theorem 9. [IKS], [KS] There exists an open set in the space of the endomorphisms of the cylinder onto itself whose Milnor attractor AM is the union (13), and the basins B0 and B1 of A0 and A1 are both metrically dense. Note that this result may be also proved by the arguments developed in [BDV], Section 11.1.1. By definition of the Milnor attractor, its basin may be not all, but almost all phase space. The reminder set that is not attracted to AM has, by definition, measure zero. In case of the intermingled basins, this set is not empty. What else may be said about this set? Theorem 10. [KS] The open set in the previous theorem has an additional property. For any map from this set, the neutral set N = X \ (B0 ∪ B1 ), consisting of points that are not attracted to AM , has Hausdorff dimension smaller than 2. Attractors of diffeomorphisms with intermingled basins on the product of a disc to a two torus were considered in [I08]. But the intermingled basins in the examples do not persist under small perturbations. Recently, R. Ures and C. V´ asquez proved that on a three torus there are no diffeomorphisms with intermingled basins of attraction that persist under small perturbations [UV]. 3.5. Thick attractors. Boundary preserving maps may have thick attractors, that is, attractors of positive Lebesgue measure that do not coincide with the whole phase space. Again, this is done with the use of the same strategy. First, an open set of maps with a thick Milnor attractor is constructed in a space of step skew products. 2

Theorem 11. [I10] In the space (Diff2f ix (I)) of pairs of C 2 -diffeomorphisms of an interval with endpoints fixed, there exists an open set U such that for each pair in U the corresponding step skew product (2) has a thick Milnor attractor.4 This set of pairs (f0 , f1 ) mentioned in this theorem is described as follows. Both maps preserve 0 and 1. The map f0 has attractor 0, repeller 1 and no other fixed points; the map f1 has exactly three fixed points: two repelling fixed points 0 and 1, and one attractor a ∈ (0, 1). Finally, f0 (0) · f1 (0) > 1. 4 The measure on the phase space of a skew product is obtained as a product of the Bernoulli measure on the base and the Lebesgue measure on the fiber.

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A more delicate theorem, which does not imply the previous one, but heuristically is much stronger, was proved in [I11]. Below it is stated in the form obtained in [MO]. A set in a functional space is quasiopen if it may be obtained from an open set by removing a countable number of hypersurfaces. Theorem 12. There exists a quasiopen set in the space of boundary preserving diffeomorphisms of a product of a two torus to a segment that consists of maps with thick Milnor attractors. 3.6. Other developments. Other results are obtained in frames of the same strategy, but we have not enough space to describe them. So we will only name the topics, and give references. Skew products with one-dimensional fibers. These are step and mild skew products as well as skew product diffeomorphisms, with the fiber a segment or a circle. This is a rich domain sometimes called a mini Universe of dynamical systems. It is thoroughly investigated up to now in the papers [VY] (independent on the strategy described above), [KKO], [KV2], [Kud2], [Oku], [OSh]. Nondensity of orbital shadowing property in C 1 -topology, a new effect in the shadowing theory, was found in [Os] with the use of skew products with the fiber a circle. Invisible attractors. It occurred that some large parts of hyperbolic attractors may be practically invisible in numeric experiments [IN]. The corresponding domains in the functional space are neighborhoods of skew products. Thurston suggested that the rate of invisibility (see [IN] for the precise definition) increases fast when the dimension of the fiber in the unperturbed skew product grows. This was confirmed in [IV]. Bony attractors. A new kind of attractors, so called bony ones, was introduced in [K10]. These attractors have an unexpected shape, and they are studied in [K10], [I12], [DGe], [DGe1]. Special ergodic theorems. Consider an ergodic map. Let a large deviation set be a set of points such that the time average of some continuous function along the orbits of the points of this set differs more than by a fixed positive constant from the space average of the same function. It occurs that, for a large class of ergodic diffeomorphisms, the large deviation set has the Hausdorff dimension smaller than that of the ambient space. Precise statements may be found in [IN], [S11], [KMR]. It seems that the potential of the IG strategy is far from being exhausted. Acknowledgments The authors are grateful to Anton Gorodetski and Federico Rodrigez Hertz for many valuable comments that improved the text and enlarged the bibliography. The authors also thank all the participants of the Summer School “Dynamical systems 2012” who prepared the lecture notes of the minicourse given by Yu. Ilyashenko, and expecially to Olga Romaskevich. References 

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V. Kleptsyn and D. Volk, Physical measures for nonlinear random walks on interval (English, with English and Russian summaries), Mosc. Math. J. 14 (2014), no. 2, 339– 365, 428. MR3236497 [KV2] V. Kleptsyn and D. Volk, Nonwandering sets of interval skew products, Nonlinearity 27 (2014), no. 7, 1595–1601, DOI 10.1088/0951-7715/27/7/1595. MR3225874 [K10] Yu. G. Kudryashov, Bony attractors (Russian), Funktsional. Anal. i Prilozhen. 44 (2010), no. 3, 73–76, DOI 10.1007/s10688-010-0028-8; English transl., Funct. Anal. Appl. 44 (2010), no. 3, 219–222. MR2760517 [Kud2] Yu. Kudryashov, Des orbites p´ eriodiques et des attracteurs des syst´ emes dynamiques, PhD thesis, ENS Lyon, December 2010. [M85] John Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985), no. 2, 177–195. MR790735 [MO] S. S. Minkov, A. V. Okunev, Omega-limit sets of generic points of partially hyperbolic diffeomorphisms, Funct. Anal. and Its Appl., 2016, 50:1, 48-53. MR3526973. [N74] Sheldon E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18. MR0339291 [N79] Sheldon E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets ´ for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math. 50 (1979), 101–151. MR556584 [N80] Sheldon E. Newhouse, Lectures on dynamical systems, Dynamical systems (Bressanone, 1978), Liguori, Naples, 1980, pp. 209–312. MR660646 [NT] Viorel Nit¸ic˘ a and Andrei T¨ or¨ ok, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. J. 79 (1995), no. 3, 751–810, DOI 10.1215/S0012-7094-95-07920-4. MR1355183 [Oku] A. Okunev, Milnor Attractors of Skew Products with the Fiber a Circle, J. Dyn. Control Syst., 2017, Volume 23, Issue 2, 421-433. MR3625005. [OSh] A. Okunev, I. Shilin, On the attractors of step skew products over the Bernoulli shift, to appear in the Proceedings of the Steklov Institute, v. 297, 2017, arXiv:1703.01763 [Os] A. V. Osipov, Nondensity of the orbital shadowing property in C 1 -topology (Russian, with Russian summary), Algebra i Analiz 22 (2010), no. 2, 127–163, DOI 10.1090/S10610022-2011-01140-9; English transl., St. Petersburg Math. J. 22 (2011), no. 2, 267–292. MR2668126 [P05] J. Palis, A global perspective for non-conservative dynamics (English, with English and French summaries), Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 22 (2005), no. 4, 485–507, DOI 10.1016/j.anihpc.2005.01.001. MR2145722 [PV] J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math. (2) 140 (1994), no. 1, 207–250, DOI 10.2307/2118546. MR1289496 [PSW] Charles Pugh, Michael Shub, and Amie Wilkinson, H¨ older foliations, revisited, J. Mod. Dyn. 6 (2012), no. 1, 79–120, DOI 10.3934/jmd.2012.6.79. MR2929131 [RW] David Ruelle and Amie Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys. 219 (2001), no. 3, 481–487, DOI 10.1007/s002200100420. MR1838747 [S11] P. S. Saltykov, A special ergodic theorem for Anosov diffeomorphisms on the 2-torus (Russian, with Russian summary), Funktsional. Anal. i Prilozhen. 45 (2011), no. 1, 69– 78, DOI 10.1007/s10688-011-0006-9; English transl., Funct. Anal. Appl. 45 (2011), no. 1, 56–63. MR2848741 [Sh] I. Shilin, Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors, arXiv:1604.02437. [SW] Michael Shub and Amie Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508, DOI 10.1007/s002229900035. MR1738057 [TS] J. C. Tatjer and C. Sim´ o, Basins of attraction near homoclinic tangencies, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 351–390, DOI 10.1017/S0143385700007914. MR1279475 [UV] R. Ures, C. H. V´ asquez, On the non-robustness of intermingled basins, arXiv:1503.07155 [math.DS]. [VY] Marcelo Viana and Jiagang Yang, Physical measures and absolute continuity for onedimensional center direction, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 30 (2013), no. 5, 845–877, DOI 10.1016/j.anihpc.2012.11.002. MR3103173 [KV]

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National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000 Russia – and – Deparment of Mathematics, Cornell University, Ithaca, New York 14853 Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, Moscow 119002 Russia

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13919

Thermodynamic formalism for some systems with countable Markov structures Michael Jakobson To the memory of Dmitry Viktorovich Anosov Abstract. We study ergodic properties of certain piecewise smooth twodimensional systems by constructing countable Markov partitions. Using thermodynamic formalism we prove exponential decay of correlations for H¨ older functions. That extends previous results of M. Jakobson and S. Newhouse (2000), where Bernoulli property was proved for such systems. Our approach is motivated by the original method of D.V. Anosov and Ya.G. Sinai (1967).

1. Motivation: Folklore Theorem in dimension 1 A well-known Folklore Theorem in one-dimensional dynamics can be formulated as follows. Folklore Theorem. Let I = [0, 1] be the unit interval, and suppose {I1 , I2 , . . .} is a countable collection of disjoint open subintervals of I such that i Ii has the full Lebesgue measure in I. Suppose there are constants K0 > 1 and K1 > 0 and mappings fi : Ii → I satisfying the following conditions. (1) fi extends to a C 2 diffeomorphism from the closure of Ii onto [0, 1], and inf z∈Ii | Dfi (z) | > K0 for all i. | D2 fi (z) | (2) supz∈Ii | I | < K1 for all i. | Dfi (z) | i Then, the mapping F (z) defined by F (z) = fi (z) for z ∈ Ii , has a unique invariant ergodic probability measure μ equivalent to Lebesgue measure on I. For the proof of the Folklore theorem , the ergodic properties of μ and the history of the question see for example [4] and [18]. In [9] , [10] the Folklore Theorem was generalized to two-dimensional maps F which piecewise coincide with certain hyperbolic diffeomorphisms fi . As in the one-dimensional situation there is an essential difference between a finite and an infinite number of fi . In the case of an infinite number of fi , their derivatives grow with i and relations between first and second derivatives become crucial. Models with infinitely many fi appear when we study non-hyperbolic systems, such as quadratic-like maps in dimension 1, and Henon-like maps in dimension 2. 2010 Mathematics Subject Classification. Primary 37Dxx. c 2017 American Mathematical Society

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2. Model under consideration. Geometric and hyperbolicity conditions (1) As in [9] , [10] we consider the following 2-d model. Let Q be the unit square. Let ξ = {E1 , E2 , . . . , } be a countable collection of closed curvilinear rectangles in Q. Assume that each Ei lies inside a domain of definition of a C 2 diffeomorphism fi which maps Ei onto its image Si ⊂ Q. We assume each Ei connects the top and the bottom of Q. Thus each Ei is bounded from above and from below by two subintervals of the line segments {(x, y) : y = 1, 0 ≤ x ≤ 1} and {(x, y) : y = 0, 0 ≤ x ≤ 1}. Hyperbolicity conditions that we formulate below imply that the left and right  (i)  boundaries of Ei are graphs of smooth functions x(i) (y) with  dxdy  ≤ α where α is a real number satisfying 0 < α < 1. The images fi (Ei ) = Si are narrow strips connecting the left and right sides of Q and that they are bounded on the left and right by the two subintervals of the line segments {(x, y) : x = 0, 0 ≤ y ≤ 1} and {(x, y) : x = 1, 0 ≤ y ≤ 1} and above and below by the graphs of (i) smooth functions Y i (X), | dY dX | ≤ α.  We are saying that Ei s are full height in Q while the Si s are full width in Q. (2) For z ∈  Q, let z be the horizontal line through z. We define δ z (Ei ) = diam(z Ei ), δ i,max = maxz∈Q δ z (Ei ), δ i,min = minz∈Q δ z (Ei ). We assume the following Geometric conditions. G1. For i = j holds int Ei ∩ int Ej = ∅ and int Si ∩ int Sj = ∅ . G2. mes(Q \ ∪i int Ei ) = 0 where mes stands for Lebesgue measure. G3. − i δ i,max log δ i,min < ∞. (3) In the standard coordinate system for a map F : (x, y) → (F1 (x, y), F2 (x, y)) we use DF (x, y) to denote the differential of F at some point (x, y) and Fjx , Fjy , Fjxx , Fjxy , etc., for partial derivatives of Fj , j = 1, 2 . Let JF (z) =| F1x (z)F2y (z) − F1y (z)F2x (z) | be the absolute value of the Jacobian determinant of F at z. Hyperbolicity conditions. There exist constants 0 < α < 1 and K0 > 1 such that for each i the map F (z) = fi (z) for z ∈ Ei satisfies H1. | F2x (z) | + α| F2y (z) | + α2 | F1y (z) | ≤ α| F1x (z) | H2. | F1x (z) | − α| F1y (z) | ≥ K0 . H3. | F1y (z) | + α| F2y (z) | + α2 | F2x (z) | ≤ α| F1x (z) | H4. | F1x (z) | − α| F2x (z) | ≥ JF (z)K0 . For a real number 0 < α < 1, we define the cones u Kα = {(v1 , v2 ) : | v2 | ≤ α| v1 |} s = {(v1 , v2 ) : | v1 | ≤ α| v2 |} Kα u s and the corresponding cone fields Kα (z), Kα (z) in the tangent spaces at 2 points z ∈ R .

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The following proposition proved in [10] relates conditions H1-H4 above with the usual definition of hyperbolicity in terms of cone conu ditions. It shows that conditions H1 and H2 imply that the Kα cone is mapped into itself by DF and expanded by a factor no smaller than K0 s while H3 and H4 imply that the Kα cone is mapped into itself by DF −1 and expanded by a factor no smaller than K0 . Unless otherwise stated, we use the max norm on R2 , | (v1 , v2 ) | = max(| v1 |, | v2 |). Proposition 2.1. Under conditions H1-H4 above, we have (1)

u u ) ⊆ Kα DF (Kα

(2)

u v ∈ Kα ⇒ | DF v | ≥ K0 | v |

(3)

s s DF −1 (Kα ) ⊆ Kα

(4)

s v ∈ Kα ⇒ | DF −1 v | ≥ K0 | v |

Remark 2.2. The first version of hyperbolicity conditions appeared in [17]. It was developed in particular in [5] and [8] . Here we use hyperbolicity conditions from [10]. In [9] we used hyperolicity conditions from [5] which implied the invariance of cones and uniform expansion with respect to the sum norm | v | = | v1 | + | v2 |. (4) The map F (z) = fi (z) for z ∈ int Ei ˜ 0 =  int Ei , and, define is defined almost everywhere on Q. Let Q i  ˜ n , n > 0, inductively by Q ˜ n−1 . Let Q ˜ ˜n = Q ˜ 0 F −1 Q ˜ =  Q n≥0 Qn be  ˜ the set of points whose forward orbits always stay in i int Ei . Then, Q ˜ has full Lebesgue measure in Q, and F maps Q into itself. The hyperbolicity conditions H1–H4 imply the estimates on the derivatives of the boundary curves of Ei and Si which  we described earlier. They also imply that any intersection fi Ei Ej is full width in  Ej . Further, Eij = Ei fi−1 Ej is a full height subrectangle of Ei and Sij = fj fi Eij is a full width substrip in Q. Given a finite string i0 . . . in−1 , we define inductively % Ei1 i2 ...in−1 . Ei0 ...in−1 = Ei0 fi−1 0 Then, each set Ei0 ...in−1 is a full height subrectangle of Ei0 . Analogously, for a string i−m . . . i−1 we define % Si−m ...i−1 = fi−1 (Si−m ...i−2 Ei−1 ) and get that Si−m ...i−1 is a full width strip in Q. It is easy to see that Si−m ...i−1 = fi−1 ◦ fi−2 ◦ . . . ◦ fi−m (Ei−m ...i−1 ) and that fi−1 (Si−m ...i−1 ) is 0 a full-width substrip of Ei0 . We also define curvilinear rectangles Ri−m ...i−1 ,i0 ...in−1 by % Ri−m ...i−1 ,i0 ...in−1 = Si−m ...i−1 Ei0 ...in−1

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If there are no negative indices then respective rectangle is full height in Q. For infinite strings, we have the following Proposition. Proposition 2.3. Any C 1 map F satisfying the above geometric conditions G1–G3 and hyperbolicity conditions H1–H4 has a ”topological attractor” in the sense of [10] %  Si−k ...i−1 Λ= ...i−n ...i−1 k≥1

 1 The infinite intersections ∞ k=1 Si−k ...i−1 define C curves y(x), |dy/dx| ≤ α which are the unstable ∞ manifolds for the points of the attractor. The infinite intersections k=1 Ei0 ...ik−1 define C 1 curves x(y), |dx/dy| ≤ α which are the stable manifolds for the points of the attractor. The infinite intersections ∞ % ∞ % Ri−m ...i−1 ,i0 ...in−1 m=1 n=1

define points of the attractor.

(5)

Proposition 2.3 is a well known fact in hyperbolic theory. For example it follows from Theorem 1 in [5]. See also [12]. The union of the stable manifolds has full measure in Q. The trajectories of all points in this set converge to Λ. That is the reason to call Λ a topological attractor. (5) An F −invariant Borel probablility measure μ on Q is called a Sinai − Ruelle − Bowen measure (or SRB-measure) for F if μ is ergodic and there is a set A ⊂ Q of positive Lebesgue measure such that for x ∈ A and any continuous real-valued function φ : Q → R, we have n−1 1 k φ(F x) = φdμ. lim n→∞ n k=0

Existence of an SRB measure is a much stronger result, than 2.3. It allows to describe statistical properties of trajectories in a set of positive phase volume. It requires some additional assumptions. 3. Distortion conditions As we have a countable number of domains the derivatives of fi grow. We formulate certain assumptions on the second derivatives. We use the distance function d((x, y), (x1 , y1 )) = max(| x − x1 |, | y − y1 |) associated with the norm | v | = max(| v1 |, | v2 |) on vectors v = (v1 , v2 ). As above, for a point z ∈ Q, let lz denote the horizontal line through z, and  if E ⊆ Q, let δ z (E) denote the diameter of the horizontal section lz E. We call δ z (E) the z − width of E. In given coordinate systems we write fi (x, y) = (fi1 (x, y), fi2 (x, y)). We use fijx , fijy , fijxx , fijxy , etc. for partial derivatives of fij , j = 1, 2. We define | D2 fi (z) | =

max

j=1,2,(k,l)=(x,x),(x,y),(y,y)

| fijkl (z) |.

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Next we formulate distortion conditions which are used to control the fluctuation of the derivatives of iterates of F along unstable manifolds, and to construct Sinai local measures. Suppose there is a constant C0 > 0 such that the following distortion condition holds | D2 fi (z) | δ (E ) < C0 . D1. supz∈Ei ,i≥1 | fi1x (z) | z i Our conditions imply the following theorem proved in [9], [10]. Theorem 3.1. Let F be a piecewise smooth mapping as above satisfying the geometric conditions G1–G3, the hyperbolicity conditions H1–H4 and the distortion condition D1. Then, F has an SRB measure μ supported on Λ whose basin has full Lebesgue measure in Q. Dynamical system (F, μ) satisfies the following properties.

(6)

(1) (F, μ) is measure-theoretically isomorphic to a Bernoulli shift. (2) F has finite entropy with respect to the measure μ, and the entropy formula holds hμ (F ) = log |Du F |dμ where Du F (z) is the norm of the derivative of F in the unstable direction at z. (3) 1 log | DF n (z) | n where the latter limit exists for Lebesgue almost all z and is independent of such z.

(7)

hμ (F ) = lim

n→∞

4. Additional hyperbolicity and distortion conditions and statement of the main theorem When applying thermodynamic formalism to hyperbolic attractors one considers the function φ(z) = − log(Du F (z)). Thermodynamic formalism is based on the fact that the pullback of φ(z) into a symbolic space determined by some Markov partition is a locally H¨ older function. We prove H¨ older property of φ(z) assuming an extra hyperbolicity condition, and a distortion condition D2 stronger than D1. Hyperbolicity condition H5. H5.

1 K02

+ α2 < 1.

Distortion condition D2. D2.

supz∈Ei ,i≥1

| D2 fi (z) | < C0 . | fi1x (z) |

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Remark 4.1. Condition D2 is too strong to be useful for systems with critical | < c instead of | FFixx | < c. However inpoints. In dimension 1 it reads as | FFixx 2 ix ix stead of D2 one can assume additional hyperbolicity conditions, which can be vaguely formulated as ”contraction of fi grows faster than expansion” . That approach will be discussed in a forthcoming paper. Assuming additionally H5 and D2 we prove that H¨ older functions have exponential decay of correlations. Let Hγ be the space of functions on Q satisfying H¨older property with exponent γ | φ(x) − φ(y) | ≤ c| x − y |γ Then the following theorem holds. Theorem 4.2. Let F be a piecewise smooth mapping as above satisfying the geometric conditions G1–G3, the hyperbolicity conditions H1–H5 and the distortion condition D2. Then (F, μ) has exponential decay of correlations for φ, ψ ∈ Hγ . Namely there exist η(γ) < 1 and C = C(φ, ψ) such that    (8) | φ(ψ ◦ F n )dμ − φdμ ψdμ | < Cη n 5. H¨ older properties of log(Du F (z)) (1) Although Markov partitions are partitions of the attractor, we need to check H¨ older property on actual two-dimensional curvilinear rectangles Ri−m ...i−1 ,i0 ...in−1 . We call respective partition Markov as well. In our model Markov partition consists of initial full height rectangles Ei . We consider rectangles Ri−m ...i−1 ,i0 ...in−1 with m ≥ 0, n ≥ 1. We use notation m = 0 if there are no negative coordinates, which means Ri−m ...i−1 ,i0 ...in−1 = Ri0 ,...,in−1 is a full height rectangle. For any function a(x, y) the variation of a(x, y) over a rectangle R is defined as (9)

var(a(x, y))|R =

sup

| a(x1 , y1 ) − a(x2 , y2 ) |

(x1 ,y1 )∈R,(x2 ,y2 )∈R

By definition the function log Du F is locally H¨ older if for m ≥ 0, n ≥ 1 the variation of log Du F on Ri−m ...i−1 ,i0 ...in−1 satisfies (10)

min(m,n)

var(log Du F )|Ri−m ...i−1 ,i0 ...in−1 < Cθ0

for some C > 0, θ0 < 1. The assumption n ≥ 1, means that variations are measured between points which belong to the same full height rectangle. older function. Proposition 5.1. log Du F is a locally H¨ We prove Proposition 5.1 with some θ0 and C determined by hyperbolicity and distortion conditions. (a) The sets Ri−m ...i−1 ,i0 ...in−1 are bounded from above and below by some arcs of two unstable curves Γui−m ...i−1 , which are images of some ˜ and from left and right by some pieces of the top and bottom of Q, arcs of two stable curves Γsi0 ...in−1 , which are preimages of some pieces

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˜. the left and right boundaries of Q Let Z1 , Z2 ∈ Ri−m ...i−1 ,i0 ...in−1 be two points on the attractor. We connect Z1 , Z2 by two pieces of their unstable manifolds to two points Z3 , Z4 which belong to the same stable manifold. Let γ1 = γ(Z1 , Z3 ) ⊂ W u (Z1 ), γ2 = γ(Z2 , Z4 ) ⊂ W u (Z2 ), γ3 = γ(Z3 , Z4 ) ⊂ W s (Z3 ) be respective curves all located inside Ri−m ...i−1 ,i0 ...in−1 . We estimate | log Du F (Z1 ) − log Du F (Z2 ) | ≤ | log Du F (Z1 ) − log Du F (Z3 ) |+ | log Du F (Z3 ) − log Du F (Z4 ) | + | log Du F (Z4 ) − log Du F (Z2 ) | (b) First we estimate | log Du F (Z1 ) − log Du F (Z3 ) |. We cover γ1 by a chain of small rectangles with sides parallel to the standard axes R = Δx×Δy ⊂ Ri−m ...i−1 ,i0 ...in−1 . Because of cone conditions we can choose rectangles R = Δx × Δy satisfying | Δy | ≤ α| Δx |. Then | log Du F (Z1 ) − log Du F (Z3 ) | is majorated by the sum of similar differences for points z1 , z2 ∈ W u (Z1 ) ∩ R. Here z1 , z2 are points on the vertical boundaries of R. Inside R we can use the mean value theorem. Hyperbolicity conditions imply the following properties, see [10]. u at a point z ∈ Ei , in particular a tangent (i) Any unit vector in Kα u vector to W (z), has coordinates (1, az ) with | az | < α. (ii) | Du F (z) | = | F1x (z) + az F1y (z) |

(11) (iii)

| F1y | 0, 0 < θ0 < 1 such that | aZ3 − aZ4 | < c0 θ0m

(31) Proof.

We assume by induction that for any rectangle Ri−m ...i−1 ,i0 ...in−1 , and for any points Z3 , Z4 ∈ Ri−m ...i−1 ,i0 ...in−1 of intersection of two unstable manifolds W1u , W2u with the same stable manifold W0s , the inequality 31 holds. Then we prove | aF (Z3 ) − aF (Z4 ) | < c0 θ0m+1

(32)

DF maps a unit vector v = (1, a) into (F1x +F1y a, F2x +F2y a). Then the normalized vector DF v has second coordinate 

a =

(33)

F2x F1x

+

1+

F2y F1x a

F1y F1x a

We denote Z3 = z, Z4 = w and estimate (34)

(35)

(36)

F2x F1x (z)

1+

+

F2y F1x (z)a(z)

F1y F1x (z)a(z)

−

F2x F1x (w)

1+

+

F2y F1x (w)a(w)

F1y F1x (w)a(w)

After cross multiplying we get denominator bounded away from 0. Therefore it is enough to estimate two terms

 F2x

 F2x F1y F1y (w) 1 + (z)a(z) − (z) 1 + (w)a(w) F1x F1x F1x F1x and

 F2y

 F1y F1y F2y (w)a(w) 1 + (z)a(z) − (z)a(z) 1 + (w)a(w) F1x F1x F1x F1x

186

(37)

(38)

(39)

(40)

(41)

MICHAEL JAKOBSON

Both expressions are estimated similarly. To estimate 36 we split it into F2y F2y (w)a(w) − (z)a(z) F1x F1x and

F2y  F1y F2y F1y a(z)a(w) (w) (z) − (z) (w) F1x F1x F1x F1x As above we use elementary algebra and get expressions of the type F1x (w) − F1x (z) F1x (z) and F2y (w) − F2y (z) F1x (z) We split γ3 into small intervals, and apply the mean value theorem. F1x (θ) or equivalently the differences log F1x (θ)−log F1x (z) The ratios F 1x (z) for close points θ, z on the same stable manifold are estimated (using again the mean value theorem and D2) as log F1x (θ) − log F1x (z) < C0 (1 + α)Δy Thus for any two points z and θ on the same stable manifold

(42)

log F1x (θ) − log F1x (z) < C| z − θ | In particular for all points z and θ on the same stable manifold the 1x (θ) ratios F F1x (z) are uniformly bounded. Thus estimate 38 contributes

(43)

(44)

(45)

(46)

(47)

(48)

C| γ3 | When estimating 37 we get similar terms estimated as 43, and  F2y

(z) a(z) − a(w) F1x After we combine all terms except 44 we get an estimate 1 M0 C0 m K0 where M0 is a uniform constant, which depends on the number of similar terms that we added above, and C0 is the distortion constant from condition D2. For 44 we use inductive assumption 31 and get a total estimate 

1 1 + α2 c0 θ0m | aF (Z3 ) − aF (Z4 ) | < M0 C0 m + K0 K02 As K0 > 1 we can choose θ0 < 1 satisfying 1 θ0 > K0 Also H5 implies that we can choose θ0 < 1 satisfying simulteneously 1 + α 2 < θ0 K02

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Then if (49)

c0 >

M0 C0 θ0 − ( K12 + α2 ) 0

we get the left side of 46 less than c0 θ0m+1 . Q.E.D. Remark 5.3. A related result for classical systems was proved by A. Pinto and D. Rand in [11]. They prove that if Λ is an invariant hyperbolic set with local product structure for a C 1+γ diffeomorphism with one-dimensional unstable leaves, then holonomies between unstable leaves are C 1+α for some α > 0. From Lemma 5.2 and 30 we get (50)

| log Du F (z3 ) − log Du F (z4 ) | < C3 θ0m Combining 27, 28, 50 we conclude the proof of Proposition 5.1. (2) We combine several corollaries from Proposition 5.1 and from the arguments used in its proof.

(51)

Corollary 5.4. There exists c independent of i such that for any z1 , z2 ∈ Ei holds | fi1x (z1 ) | 0 such that L∗φ ν = λν, Lφ h = λh, ν(h) = 1, and for every uniformly continuous function f such that ||f h−1 ||∞ < ∞ holds λ−n Lnφ f → ν(f )h uniformly on compacts. Positive recurrence and convergence result in Theorem 6.1 hold for matrices A satisfying the following Big Images and Preimages property . BIP There is a finite set of states i1 , i2 , . . . iN such that for every state j in the alphabet there are k, l such that aik j ajil = 1. The following result of Sarig from [15] extends the results of [14] and works of Aaronson, Denker , Mauldin, Urbanski and Yuri, see [2], [3], [19]. Consider the space of functions L with bounded norm | f |L which is the older norm. sum of ||f ||∞ and some fixed H¨ Theorem 6.2. Suppose (XA , σ) is topologically mixing, φ is locally H¨ older continuous, P (φ) < ∞ and BIP property holds. Then (a) φ is positive recurrent, and there exist λ, h, ν as in theorem 6.1. (b) h is bounded away from zero and infinity and ν(X) < ∞. (c) There exist K > 0, θ ∈ (0, 1) such that for f ∈ L holds (66)

| λ−n Lnφ f − hν(f ) |L < Kθ n | f |L (2) As a corollary from Theorem 6.2 we get Proposition 6.3. Suppose there is a Markov partition of the attractor satisfying the following properties. (a) The matrix A of admissible transitions is topologically mixing and satisfies BIP property. older on the space of admissible sequences. (b) Φ(x, y) = −log| Du F | is H¨ (c) For some φ(x) cohomologous to Φ(x, y) holds P (φ(x)) < ∞. then 66 holds. Proposition 6.3 gives sufficient conditions for exponential decay of correlations for H¨ older (in particular smooth) functions restricted to the attractor.

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7. Proof of the exponential decay of correlations We check properties (a) - (c). (1) Recall that in our model we consider the partition of the square into full height rectangles Ei . Our shift is Bernoulli, all rows (and columns) are the same row of 1-s, so it is topologically mixing and property (a) is satisfied. (2) Property (b) follows from 10. (3) Next we prove property (c). As in the case of attractors for Axiom A systems we prove Proposition 7.1. For φ(x) = φu (x) topological pressure P (φu (x)) equals zero. Proof. u We fix some symbol a, respective rectangle Ea , and W0a = W0u ∩ Ea . When evaluating Zn (φ, a) in 63 we consider respective sum over all periodic orbits of period n starting in Ea . Each cylinder set Eai1 ...in−1 contains one periodic orbit of period n. When evaluating φu (x) we use formula 60. We can evaluate that expression at a point z of intersection between the stable manifold of a periodic point u . Then each term in 63 is a product of two expresin Eai1 ...in−1 and W0a sions. The first expression equals Du F1n (z) , which coincides up to a uniformly bounded factor with the length of W u (z, Eai1 ...in−1 ). The second expression eu(z)−u(F (z) is uniformly bounded away from zero and infinity. Therefore up to a uniformly bounded factor the sum 63 equals to the u . That implies P (φ) = 0. Q.E.D. length of W0a So all properties of Proposition 6.3 are satisfied, and we get exponential decay of correlations for one-sided shift. As in [7] it implies exponential decay of correlations for two-sided shift and therefore for H¨ older functions on Q. That proves Theorem 4.2. Remark 7.2. Under conditions of Theorem 4.2 the central limit theorem holds for H¨ older functions on Q which are not cohomologous to constants . Remark 7.3. Corollary 4 from [15] implies that under conditions of Theorem 4.2 there are no ”phase transitions” in the sense that the function t → P (−t log Du f ) is real analytic in a neighborhood of t = 1. Let us denote μS the invariant measure on Λ constructed in [9], [10] following Sinai method, and let μRB be the invariant measure on Λ constructed above following Ruelle-Bowen method, see [13], [7]. Let μ1 be the projection of μS onto one-sided sequences, and let μ be the measure on one-sided sequences constructed above by Ruelle-Bowen method. In both constructions measures of cylinder sets [i0 i1 . . . in−1 ] of any rank equal up to a uniform constant to the length of the crossections of Ei0 ...in−1 by W0u . So μ1 and μ are equivalent and therefore they coincide. That is a particular case of the characterization of Gibbs measures proved in [14]. As in the classical case that implies

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Corollary 7.4. Measures μS and μRB coincide. Acknowledgements. I want to thank Sheldon Newhouse, David Ruelle and Omri Sarig for useful discussions during the preparation of this paper. Special thanks to Omri Sarig for many valuable remarks. References [1] Jon Aaronson and Manfred Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn. 1 (2001), no. 2, 193–237, DOI 10.1142/S0219493701000114. MR1840194 [2] R. Daniel Mauldin and Mariusz Urba´ nski, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math. 125 (2001), 93–130, DOI 10.1007/BF02773377. MR1853808 [3] Jon Aaronson, Manfred Denker, and Mariusz Urba´ nski, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), no. 2, 495–548, DOI 10.2307/2154231. MR1107025 [4] Rufus Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys. 69 (1979), no. 1, 1–17. With an afterword by Roy L. Adler and additional comments by Caroline Series. MR547523 [5] V. M. Alekseev. Quasi-random dynamical systems, I. Math. of the USSR, Sbornik, 5(1):73– 128, 1968. [6] D. V. Anosov and Ya. G. Sinai. Some smooth ergodic systems. Russian Math. Surveys, 22: 103–167, 1967. [7] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR0442989 [8] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR0271991 [9] Michael Jakobson and Sheldon Newhouse, A two-dimensional version of the folklore theorem, Sina˘ı’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 89–105, DOI 10.1090/trans2/171/09. MR1359096 [10] Michael Jakobson and Sheldon Newhouse, Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle (English, with English and French summaries), Ast´erisque 261 (2000), xii, 103–159. G´eom´ etrie complexe et syst` emes dynamiques (Orsay, 1995). MR1755439 [11] A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics, Bull. London Math. Soc. 34 (2002), no. 3, 341–352, DOI 10.1112/S0024609301008670. MR1887706 [12] Charles Pugh and Michael Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), no. 1, 1–54, DOI 10.2307/2001206. MR983869 [13] David Ruelle, A measure associated with axiom-A attractors, Amer. J. Math. 98 (1976), no. 3, 619–654. MR0415683 [14] Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565–1593, DOI 10.1017/S0143385799146820. MR1738951 [15] Omri Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1751–1758 (electronic), DOI 10.1090/S0002-9939-03-06927-2. MR1955261 [16] Ya. G. Sina˘ı, Topics in ergodic theory, Princeton Mathematical Series, vol. 44, Princeton University Press, Princeton, NJ, 1994. MR1258087 [17] Stephen Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR0182020 [18] Peter Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121–153. MR0466493 [19] Michiko Yuri, Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indag. Math. (N.S.) 6 (1995), no. 3, 355–383, DOI 10.1016/0019-3577(95)93202-L. MR1351153

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Department of Mathematics, University of Maryland, College Park, Maryland 20742

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13923

Non-uniform measure rigidity for Zk actions of symplectic type Anatole Katok and Federico Rodriguez Hertz Abstract. We make a modest progress in the nonuniform measure rigidity program started in 2007 and its applications to the Zimmer program. The principal innovation is in establishing rigidity of large measures for actions of Zk , k ≥ 2 with pairs of negatively proportional Lyapunov exponents which translates to applicability of our results to actions of lattices in higher rank semisimple Lie groups other than SL(n, R), namely, Sp(2n, Z) and SO(n, n; Z).

1. Introduction This paper is a part of the non-uniform measure rigidity program that started in [6] and continued in [8, 10, 11]. While we refer the reader to those papers for the motivation and the general outline of the program, several comments are in order. In the present paper, for the fist time in the non-uniform setting, we are able to deal with the situation where negatively proportional Lyapunov exponents appear. Presence of those exponents constitutes a fundamental difficulty in carrying out the central recurrence argument. In the previous work on the algebraic actions this was handled either by the methods that essentially rely on non-commutativity of stable and unstable foliations (first mentioned in [13] and developed in [2] and in later papers) that are not applicable in the torus situation, or by using specific Diophantine properties of eigenspaces for algebraic actions on the torus [3] that do not extend to the non-algebraic situation. In [12] non-uniform measure rigidity was applied to the study of actions of “large” groups. Specifically, we considered actions of finite index subgroups of SL(n, Z) on the torus Tn with the standard homotopy data, i.e. inducing the same action on the first homology group Zn as the standard action by automorphisms. The nonuniform measure rigidity results needed for that concerned actions of Cartan (i.e maximal rank semisimple abelian) subgroups of SL(n, Z) and were taken from [6, 10]. The method is to first consider the restriction of the action to a maximal split Cartan subgroup and to use our results for that action. However the conditions that appeared in our previous papers are too restrictive and essentially applicable only to certain actions of SL(n, Z) and its finite index subgroups. Results of this paper significantly extend applicability 2010 Mathematics Subject Classification. Primary 37C40, 37D25, 37A35, 37C85. The work of the first author was based on research supported by NSF grants DMS-1002554. The work of the second author was partially supported by the Center for Dynamics and Geometry at Penn State and NSF grants DMS-1201326. c 2017 Copyright A. Katok and F. Rodriguez Hertz

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of our methods to actions of other lattices in higher rank semisimple Lie groups, e.g. SO(n, n; Z) and Sp(2n, Z). We extensively use terminology and notations from [6, 8, 11] with proper reminders, and refer to various results form those papers. The reader should keep in mind that some of those results may nominally refer to more restrictive situations than that of the present paper but if slightest modifications of the arguments are needed we provide appropriate explanations. Otherwise the arguments we refer to are directly applicable to the situations at hand. 2. Preliminary results on behavior of semi-conjugacies 2.1. Semi-conjugacies for maps homotopic to infranilmanifold Anosov diffeomorphisms. For reader’s convenience we recall the setting from [11, Section 2.2]. Let N be a simply connected nilpotent Lie group and A a group of affine transformations of N acting freely that contains a finite index subgroup Γ of translations that is a lattice in N . Then the orbit space M = N/A is a compact manifold that is called an infranilmanifold. An automorphism of N that maps orbits of A onto orbits of A generates a diffeomorphism of N/A that is called an infranilmanifold automorphism. If N is abelian i.e. N = Rm , the infranilmanifold N/A is called an infratorus. An action α0 of Zk by automorphisms of an infranilmanifold M is an Anosov action if induced linear action on the Lie algebra N of N has all Lyapunov exponents non-zero, or equivalently there is one element of the linear action that is an Anosov automorphism. Now let α be an action of Zk by diffeomorphisms of M such that its elements are homotopic to elements of an Anosov action by automorphisms. We will say that α has homotopy data α0 . There may exist affine actions with homotopy data α0 that are not isomorphic to α0 . This happens when α0 has more than one fixed point and affine action interchanges those fixed points. Notice any affine action with homotopy data α0 coincides with α0 on a finite index subgroup A ⊂ Zk . There exists an affine action α ˜ with homotopy data α0 and a continuous map h : M → M homotopic to identity such that (2.1)

h◦α=α ˜ ◦ h.

and hence for γ ∈ A (2.2)

h ◦ α(γ) = α0 (γ) ◦ h.

See [4, 11]. The map h is customarily called a semi-conjugacy between α and α. ˜ There are finitely many semi-conjugacies that differ by some translations by elements of the fixed point group of α. ˜ 2.2. Ledrappier-Young entropy formula and its extensions. Let g : N → N be a C 1+H¨older diffeomorphism preserving an ergodic measure ν. Let χ1 > · · · > χs > 0 be the positive Lyapunov exponents of g w.r.t. ν with associated Oseledets splitting of the unstable distribution Egu = E1 ⊕ · · · ⊕ Es . For 1 ≤ i ≤ u let us define $ #

 1 V i (x) = y ∈ M : lim sup log d g −n (x), g −n (y) ≤ −χi . n→∞ n 1 i For ν-a.e. x, V (x) is a smooth manifold tangent to j≤i Ej and we thus have the flag V 1 ⊂ V 2 ⊂ · · · ⊂ V s with V s = W u , the unstable manifold. In [16, Section 9] a

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class of increasing partitions ξ i subordinate to V i is constructed, i.e. ξ i (x) ⊂ V i (x) and ξ i (x) is a bounded neighborhood of x in V i (x) for ν-a.e. x. Consider conditional measures νxi associated with those partitions. Let B i (x, ε) be the ε ball in V i (x) centered in x with respect to the induced Riemannian metric. Then log νxi B i (x, ε) ε→0 log ε exists a.e. and does not depend on x. Moreover, writing γi = γi (g) = δi − δi−1 we have the Ledrappier–Young entropy formula (see [16, Theorem C])  (2.3) hν (g) = γj χ j . δi = δi (g) = lim

1≤j≤s

If hν (g, ξ) denotes the entropy w.r.t. the partition ξ, then [16] also gives the formula  hν (g, ξ i ) = γj χ j . 1≤j≤i

Thus the entropy does not depend of a particular choice of partition as long as it is increasing and subordinated to V i . The following corollary of the result by F. Ledrappier and Jian-Sheng Xie [14] provides the following consequence of the vanishing of the leading coefficient γs . Proposition 2.1. If γs = 0, i.e. hν (g) = hν (g, ξ s−1 ) for some partition ξ s−1 subordinated to V s−1 , then the conditional measure of ν on almost every leaf of V s = W u is supported on a single leaf of V s−1 . 2.3. Non-collapsing of Lyapunov directions under semiconjugacies. Let f : M → M be a C 1+H¨older diffeomorphism. Assume that g is a factor of f via a continuous surjective map h : M → N such that h ◦ f = g ◦ h. Let μ be an ergodic invariant measure for f such that h∗ μ = ν. It is important that no assumption on hyperbolicity of the measures be made since we will apply the results below in the setting without any a priori information on the measure μ. Proposition 2.2. If h(Wfu (x)) ⊂ Vgs−1 (h(x)) for μ-a.e. x then γsg = 0 and hence the conditional measure ν u of ν along Wgu , is supported on a single leaf of Vgs−1 . As a corollary we get the following Corollary 2.3. Let f , g and h be as above. Assume that conditional measures of ν along Wgu are absolutely continuous w.r.t. Lebesgue on Wgu then for μ-a.e. x, h(Wfu (x)) is not contained in Vgs−1 (h(x)). In particular this holds whenever ν is a measure absolutely continuous w.r.t. Lebesgue measure. To prove Proposition 2.2 we will use Proposition 2.1 and the 2 following fact from abstract ergodic theory. Here for a partition ξ, we denote ξS = n∈Z S n ξ. Proposition 2.4. [11, Proposition 6.2] Let T : (X, μ) → (X, μ) and S : (Y, ν) → (Y, ν), be measure preserving transformations and assume that S is a factor of T via a measure-preserving map p : (X, μ) → (Y, ν). Let η be a fullentropy partition for T (i.e. hμ (T ) = hμ (T, η)) and ξ a generating partition for S, i.e. ξS = ε the partition into points and p−1 ξ < η. Then ξ is a full-entropy partition for S, i.e. hν (S) = hν (S, ξ).

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Proof of Proposition 2.2. We shall proof that for some partition ξ s−1 subordinated to V s−1 , hν (g, ξ s−1 ) = hν (g) and then the results will follow from Proposition 2.1. We can build as in [16] an increasing partition ξ s−1 subordinated to Vgs−1 . Moreover, we can build again as in [16] a partition η subordinated to Wfu such that h−1 ξ u−1 < η. This can be done since h−1 (V u−1 (h(x))) ⊃ W u (x) for μ-a.e. x and h is continuous. Since ξ s−1 is subordinated to V s−1 we have that ξgs−1 = ε. Since η is an increasing partition subordinated to Wfu we have that  hμ (f ) = hμ (f, η). Hence Proposition 2.4 implies that hν (g) = hν (g, ξ s−1 ). 3. Formulation of results 3.1. Actions of higher rank abelian groups. We consider an action of Zk , k ≥ 2 by diffeomorphisms of a compact manifold. We say that an action has simple coarse Lyapunov spectrum if Lyapunov exponents are simple and no pair of Lyapunov exponents are positively proportional. Let α0 be an Anosov action of Zk by automorphisms of an infranilmanifold with simple coarse Lyapunov spectrum and let α be a smooth action with homotopy data α0 . Let us call an α-invariant Borel probability measure μ large if the push-forward h∗ μ is Haar measure. Our first result is a generalization of Theorem 2.5 from [11]. Here we assume the same regularity as in the previous section, i.e. C 1+β , β > 0. Theorem 3.1. Let α0 be a Zk , k ≥ 2, Anosov action on an infranilmanifold M with simple coarse Lyapunov spectrum. Let μ be an ergodic large invariant measure for an action α with homotopy data α0 . Then (1) μ is absolutely continuous; (2) Lyapunov characteristic exponents of the action α with respect to μ are equal to the Lyapunov characteristic exponents of the action α0 . Recall that a resonance is a relation between different Lyapunov exponents l χ1 , . . . , χl of a Zk action of the form χ1 = i=2 mi χi where m2 , . . . ml are positive integers. The following theorem generalizes to the case of simple coarse Lyapunov spectrum Theorems 2.6 and 2.7 and Proposition 7.2 from [11]. Theorem 3.2. Let α0 be a hyperbolic linear Zk action with simple coarse Lyapunov spectrum and without resonances on an infratorus N . Let α be a C ∞ action with homotopy data α0 . Then (1) α has a unique large invariant measure μ; (2) the semiconjugacy p is bijective μ-a.e. and effects a measurable isomorphism between (α, μ) and an affine action α ˜ with homotopy data α0 with Haar measure; (3) the semiconjugacy is smooth and bijective on μ a.e. stable manifold of any action element; (4) The semiconjugacy is smooth in Whitney sense on sets of μ measure arbitrary close to one. Once absolute continuity of μ and smoothness of the semiconjugacy along the Lyapunov foliations is established, all statements of Theorem 3.2 are deduced exactly as in [11, Section 7]. In particular, in order to get uniqueness of the large

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invariant measure we will use smoothness of the semiconjugacy along stable manifolds. In [11, Section 7.1] smoothness is deduced from the existence of invariant affine structures and to get those we need to assume high regularity (although not necessarily C ∞ ) and absence of resonances. Remark 1. We think that using higher order normal forms it is also possible to get smoothness of the semiconjugacy and hence allow resonances. Next we will show existence of proper periodic orbits generalizing to our setting [10, Theorem 3.1]. Theorem 3.3. Let α be an action as in Theorem 3.2 with μ the large measure. Then there is a periodic point p ∈ supp(μ) such that the semiconjugacy is a diffeomorphism when restricted to the stable and unstable manifolds of p for some element of the action. Moreover, the stable and unstable manifold for that element of the action are in the support of μ, i.e. W s (p) ∪ W u (p) ⊂ supp(μ). Corollary 3.4. Derivative of α at p is conjugate to the linear part of α0 . 3.2. Actions of lattices in higher rank Lie groups on the torus. Let G be a simply connected semisimple Lie group with no compact factors, finite center of R-rank greater than one, and let Γ be a lattice in G. Let ρ be an action of Γ on a torus Tm . Induced action ρ∗ on the first homology group can be viewed as an embedding Γ → GL(n, Z) that determines an action ρ0 by automorphisms of Tm that, similarly to the abelian case, we call the homotopy data of ρ. By Margulis Normal Subgroup Theorem [17] either the image of ρ∗ if finite or ρ∗ has finite kernel. We will assume that the second alternative takes place. By Margulis Superrigidity Theorem [17] the restriction of ρ∗ to a finite index subgroup Γ0 ⊂ Γ can be extended to a homomorphism ρ˜∗ : G → SL(n, R) with discrete kernel. Recall that weights of ρ˜∗ are eigenvalues of the restriction of ρ˜ to a maximal split Cartan subgroup of G. Since there is a maximal split Cartan subgroup of G that intersects ρ∗ (Γ0 ) by a lattice L we may speak about weights of ρ∗ that are essentially the exponentials of the Lyapunov characteristic exponents of the action ρ0 of L by automorphisms of the torus Tm . Thus we can define resonances between weights. Notice that while individual elements of actions ρ0 and ρ are homotopic, the actions may not be. Let us call a compact set K ⊂ Tm large if there is no continuous map p on the torus homotopic to the identity such that p(K) is a point. Theorem 3.5. Let ρ be a C ω (real-analytic) action of a lattice Γ in a simply connected semisimple Lie group with no compact factors finite center and of R-rank greater than one on the torus T2n such that induced action ρ∗ on the first homology group has finite kernel. Assume that (1) ρ preserves an ergodic measure μ whose support is large; (2) ρ∗ has no zero weights, no multiple weights, no positively proportional weights and no resonances between weights. Then there is a finite index subgroup Γ ⊂ Γ, a finite ρ0 -invariant set F and a bijective real-analytic map H : Tm \ F → D where D is a dense subset of supp μ, such that for every γ ∈ Γ , H ◦ ρ0 (γ) = ρ(γ) ◦ H.

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Here are two cases where Theorem 3.5 applies. Consider the standard inclusion of Sp(2n, Z) ⊂ SL(2n, Z) and the standard inclusion of SO(n, n; Z) ⊂ SL(2n, Z) Let Γ be any finite index subgroup of Sp(2n, Z) or SO(n, n; Z) and consider the standard action ρ0 of Γ on T2n . Corollary 3.6. Let Γ ⊂ Sp(2n, Z), n ≥ 2 or Γ ⊂ SO(n, n; Z), n ≥ 2 be as above. Let ρ be a real-analytic action of Γ on T2n with homotopy data ρ0 preserving an ergodic measure μ whose support is a large set. Then there is a finite index subgroup Γ ⊂ Γ, a finite ρ0 -invariant set F and a bijective real-analytic map H : Tm \ F → D where D is a dense subset of supp μ, such that for every γ ∈ Γ , H ◦ ρ0 (γ) = ρ(γ) ◦ H. Notice that since Weyl groups of Sp(2n) and SO(n, n) contain the central symmetry, for every finite-dimensional representation of those groups weights come in pairs of opposite sign. Thus the present extension of nonuniform measure rigidity results that allows pairs of simple negatively proportional Lyapunov exponents is crucial for applications to any actions of lattices in those groups. In the standard cases of Corollary 3.6 the weights are simple, there are no positively proportional ones but they still appear in pairs of opposite sign. Notice however that those may be the only new examples compared to the action of finite index subgroups. SL(n, Z) on Tn with standard homotopy data considered in [12]. For example, no new examples appear for SL(3, Z) actions.1 The proof of Theorem 3.5 repeats almost verbatim the proof of [12, Theorem 1.1] with Theorem 3.2 replacing [10, Corollary 2.2] and Theorem 3.3 taking place of [10, Theorem 3.1] (that is restated as Theorem 2.4 in [12]). The only difference is that in our setting we cannot assert that pre-image of any point under the semiconjugacy is the intersection of boxes ([10, Theorem 2.1]). For reader’s convenience in Section 6 we describe all steps of the proof and explain in detail how the above minor difference is handled. The crucial step where analyticity is used is in the application os the CairnsGhys local linearization [1]. For smooth actions we have a weaker result parallel to Theorem 3.2 that follows from the initial steps of the proof of Theorem 3.5. We formulate it for the C ∞ case. Theorem 3.7. Let ρ be a C ∞ action satisfying the rest of the assumptions of Theorem 3.5, then there exists a finite index subgroup Γ ⊂ Γ and a continuous map h : supp μ → Tm such that (1) for γ ∈ Γ , ρ0 (γ) ◦ h = h ◦ ρ(γ); (2) the map h is bijective μ-a.e. and effects a measurable isomorphism between (ρ, μ) and ρ0 with Lebesgue measure; (3) the map h is smooth and bijective on μ a.e. stable manifold of any action element; (4) The map h is smooth in Whitney sense on sets of μ measure arbitrary close to one. 1 We

thank A. Gorodnik for showing that.

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Remark 2. In [12] we actually prove a stronger statement in the case of an action of a finite index subgroup Γ of SL(n, Z) on Tn with standard homotopy data. Namely we show that h above extends to a continuous map Tn → Tn homotopic to identity; in fact it coincides with a semiconjugacy for restriction of ρ to any any Cartan subgroup of Γ; see Step 4 in the proof of [12, Proposition 2.1]. Notice that in the proof of Theorems 3.5 and 3.7 we restrict the action of Γ to a maximal rank semisimple abelian subgroup and apply to this action our results using only the fact that its homotopy data has simple coarse Lyapunov spectrum and no resonances. However, the fact that this abelian action appears as a restriction of a lattice action implies stronger properties that follow from cocycle super-rigidity. Systematic use of this additional information allows to obtain nonuniform measure rigidity for abelian actions that appear as restrictions of lattice actions under considerably more general assumptions on Lyapunov exponents than those of Theorem 3.2 and hence to extend the assertions of that theorem for those situations. 4. Absolute continuity of μ While we shall follow here the scheme of proof of previous papers c.f. [6,11] this section contains new arguments that allow us to deal with negatively proportional Lyapunov exponents. 4.1. Matching of Lyapunov half-spaces. The first step is to prove matching of Lyapunov half-spaces for α and α0 . We need to prove that the semiconjugacy does not collapse the unstable manifold. This a general result that does not need any assumption on the action α0 : Lemma 4.1. [11, Lemma 6.3] If L is a Lyapunov hyperplane (the kernel of a Lyapunov exponent) for α0 then L is a Lyapunov hyperplane for α and Lyapunov half-spaces match. Since the proof of the matching of Lyapunov half-spaces was not completely written in [11] we will add it here. Proof. Take L a Lyapunov hyperplane for α0 . We have, as proven in [11, Lemma 6.3], that it is a Lyapunov hyperplane for α also. We will prove that if χ is a Lyapunov exponent for α0 such that ker χ = L then there is a Lyapunov exponent for α positively proportional to χ. Assume this is not the case. Then any Lyapunov exponent χ ˜ for α with ker χ ˜ = L is negatively proportional to χ. Assume χ is such that all α0 Lyapunov exponents proportional to χ has rate of proportionality smaller than one (i.e. χ is the largest Lyapunov exponent among the positively proportional to it). Take t ∈ L such that t is not in ker χ ˆ for any Lyapunov exponent χ ˆ (for α or α0 ) not proportional to χ. Take n a regular element for α0 close to t such that χ(n) > 0 but smaller that any other non-proportional ˜ < 0 for any α-Lyapunov exponent χ ˜ positive α0 -Lyapunov exponent. Then χ(n) u u (x) = Wα(t) (x) for μ a.e. x but proportional to χ. Hence we have that Wα(n) Wαu0 (t) (y)  Wαu0 (n) (y) for every y. Moreover Wαu0 (t) is inside the fast expanding direction of α0 (n) (Vαu−1 in the notation of Section 2) by the choice of n. So we 0 (n) have that u u (x)) = h(Wα(t) (x)) ⊂ Wαu0 (t) (h(x)) ⊂ Vαu−1 (h(x)) h(Wα(n) 0 (n)

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and Corollary 2.3 gives a contradiction with the fact that λ is Lebesgue measure.



To end with the matching of Lyapunov half-spaces we need to prove the following Proposition 4.2. If α0 has simple coarse Lyapunov spectrum then dim M ≥ dim N and if dim M = dim N then Lyapunov hyperplanes and half-spaces for α and α0 coincide. In particular (α, μ) has also simple Lyapunov spectrum. Proof. The Proposition is true by counting Lyapunov exponents. Since α0 has simple coarse Lyapunov spectrum we can list the Lyapunov exponents of α0 as χ1 , . . . , χn , n = dim N , in such a way that for some l ≥ n/2, χl+i is negatively proportional to χi for i = 1, . . . , n − l (here we use the convention that l > n if there are no negatively proportional exponents). Lemma 4.1 says that for every ˜i positively proportional to χi , this implies χi there is an α-Lyapunov exponent χ that dim M ≥ n. If dimension of M is n, then there cannot be other α Lyapunov exponents and we get the Proposition.  4.2. A dichotomy for the conditionals along Lyapunov foliations. Let us fix a Lyapunov hyperplane L = ker χ for α with associated Lyapunov exponent χ. Take a generic singular element t ∈ ker χ of the action i.e. such that χ(t) = 0 and χj (t) = 0 for any non-proportional Lyapunov exponent and call f = α(t). Let W be the Lyapunov foliation associated to χ, put Eχ (x) = Tx W(x) and call E the α0 -Lyapunov direction corresponding to the α0 -Lyapunov exponent proportional to χ. In this subsection we shall proof that μW x is absolutely continuous w.r.t. Lebesgue for μ-a.e. x. In the sequel η will denote an element of the ergodic decomposition of μ w.r.t. f and ηxW its conditional measures w.r.t. to W. The proof of the next proposition follows, with minor changes, exactly along the lines of section 3 in [6]. Observe that π-partition argument is used in [6] to get recurrence to the W leaf (i.e. W is inside an element of the ergodic partition), instead here we use the ergodicity of η w.r.t. f . Proposition 4.3. One of the following holds: (1) ηxW is absolutely continuous w.r.t. Lebesgue for almost every ergodic component η and for η-a.e. x or, (2) ηxW is atomic for almost every ergodic component η and for η-a.e. x. For the proof of Proposition 4.3 let us recall Proposition 3.1 in [6]. An affine structure on a manifold is an atlas whose change of variables are affine maps. Proposition 4.4. [6, Proposition 3.1] There exists a unique measurable family of C 1+ε smooth α-invariant affine structures on the leaves W(x). Moreover, within a given Pesin set they depend H¨ older continuously in the C 1+ε topology. Once we have affine structure we need to freeze the the dynamics along W when we iterate f (remember that f = α(t) where t is in the Weyl chamber wall L = ker χ). We will prove as in [6], the following Lemma 4.5. For μ a.e. ergodic component η, for every Pesin set Λ and for every ε > 0 there is a set Λε ⊂ Λ and K > 0 such that η(Λ \ Λε ) < ε and K −1 ≤ Df k |Eχ (x) ≤ K if both x and f k (x) are in Λε .

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Denote with BrW (x) the ball inside W(x) centered in x and of radius r and denote by l(A) the length of an interval in E or any translate of E. To prove Lemma 4.5 we need to prove first the following: Lemma 4.6. For μ a.e. ergodic component η, for every Pesin set Λ and for every ε > 0 and r > 0 there is a set Λε ⊂ Λ and m > 0 such that η(Λ \ Λε ) < ε and l(h(BrW (x))) ≥ m. Proof. Let us show first that for μ-a.e. x, length of h(W(x)) is different from 0, i.e. h(W(x)) = h(x). By ergodicity of μ w.r.t. α we get that either for μ-a.e. x, h(W(x)) = h(x) or for μ-a.e. x, h(W(x)) = h(x). Let us assume by contradiction this latter is the case. Take an element n close to t ∈ ker χ such that χ(n) > 0 but it is still the smallest positive Lyapunov exponent. We have u u = Eα(t) ⊕ Eχ and Eαu0 (n) = Eαu0 (t) ⊕ E. On the other on one hand that Eα(n) hand, since  h(W(x)) = h(x) for μ-a.e. x by our contradiction assumption, and u u Wα(n) (x) = z∈W(x) Wα(t) (z) we get that   u u h(Wα(n) (x)) = h(Wα(t) (z)) ⊂ Wαu0 (t) (h(z)) = Wαu0 (t) (h(x)) z∈W(x)

z∈W(x)

By the choice of n we have that Wαu0 (t) = Vαu−1 (recall that Vαu−1 is the 0 (n) 0 (n) direction of fast expansion as defined in Section 2). Then Corollary 2.3 gives a contradiction with the fact that λ is Lebesgue measure. So we get that for μ-a.e. x, length of h(W(x)) is different from 0. Taking again n such that χ(n) > 0 we have that the corresponding α0 -Lyapunov exponent for the linear is also positive and hence α0 (n) expands the E direction. Hence, α0 -invariance of h(W(x)) implies that ones the length is nontrivial it has to be ∞. Finally, take r > 0, then if h(BrW (x)) = h(x) for a set of positive μ-measure then ergodicity of μ, expansion of W by α(n) and expansion of the E direction by α0 (n) would imply that h(W(x)) = h(x) for μ-a.e. x, which is a contradiction. Then, we have that for every r > 0, l(h(BrW (x))) > 0 for μ-a.e. x. Then for μ a.e. ergodic component η and for η a.e. point x, l(h(BrW (x))) > 0. So, given η, a Pesin set Λ and ε > 0, for m small enough, the set, Λε of x ∈ Λ such that l(h(BrW (x))) ≥ m  has measure η(Λ \ Λε ) < ε. Once we have that h does not collapse the W ”foliation” the proof of Lemma 4.5 follows as in [6]. Proof of Lemma 4.5. We shall proof the first inequality, the second one follows taking the inverse. Take r small and Λε as in Lemma 4.6. Take x ∈ Λε and k such that f k (x) ∈ Λε . Then we have that l(h(f k (BrW (x)))) = l(α0 (kt)(h(BrW (x)))) = l(h(BrW (x))) ≥ m since α0 (kt) is an isometry along the E direction. Hence, since h is continuous there is δ > 0 that only depends on m and there is y ∈ f k (BrW (x)) such that d(y, f k (x)) ≥ δ, so, if z = f −k (y) we have that d(x, z) < r and d(f k (z), f k (x)) ≥ δ. Finally, using the affine coordinates and that they vary continuously over Pesin sets we get that Df k |Eχ (x) ≥ δC −2 /r = K −1 , where C is a bound for the derivative of the affine structure over the Pesin set Λ.  Finally, to end the proof of Proposition 4.3, let us define for η a.e. x the group Gx = {L : Eχ (x) → Eχ (x) affine, s.t. L∗ ηx = cηx for some c > 0}.

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Here we identify Eχ (x) with W(x) using the affine structure. Gx is a closed subgroup of the affine maps of the line. If Gx is not a discrete group the ηx is absolutely continuous w.r.t. Lebesgue. Using Lemma 4.5 and ergodicity of η we get that ηx (Gx (x)) > 0 (Gx (x) = {L(x) : L ∈ Gx }). So if ηx is not atomic, then Gx is not discrete and hence ηx is absolutely continuous w.r.t. Lebesgue. 4.3. Absolute continuity of conditionals along Lyapunov foliations. In case (1) of Proposition 4.3 we get that μW x is absolutely continuous w.r.t. Lebesgue for μ-a.e. x. So let us assume by contradiction that we are in case (2). Let us take s an element of the action close to t such that χ(s) < 0 but still χ(s) is the closest to zero among the negative Lyapunov exponents of α(s) and no s s ⊂ Wα(s) and in other Lyapunov exponent change sign. We have then that Wα(t) s s fact Eα(s) = Eα(t) ⊕ Eχ . The next proposition says that conditional measures of η s s alongWα(s) sits inside Wα(t) . Proposition 4.7. η Wα(s) = η Wα(t) for almost every ergodic component η. s

s

Proof. Here we follow the proof of Proposition 4.2. in [11] but instead of using time change to freeze the dynamics along W we use Lemma 4.5.  Let us reach to a contradiction then. Recall that λ(=Lebesgue measure) is an ergodic measure for α0 (t), hence, since μ projects into Lebesgue, i.e. h∗ μ = λ we have that almost every measure η in the ergodic decomposition of μ also projects into Lebesgue, that is h∗ η = λ. Now we shall argue as in Lemma 2.3. in [6]. Let x0 be a point in a Pesinset and let Λ be a neighborhood of x0 in this Pesin set. Let us consider s s s (x) ∩ Λ. Since η Wα(s) = η Wα(t) , we have that η(R) = R = x∈W u (x0 ) Wα(t) α(s)  s (x) ∩ Λ) > 0. Hence, since h∗ η = λ we get that λ(h(R)) ≥ η( x∈W u (x0 ) Wα(s) α(s)

η(R) > 0. On the other hand, h(R) ⊂ h(x0 )(Wαu0 (s) ⊕ Wαs0 (t) ) which has dimension less than n and hence has zero Lebesgue measure. It has less dimension since Lyapunov hyperplanes match. Hence we get a contradiction and μW is absolutely continuous w.r.t. Lebesgue measure. Once conditional measures of μ along all Lyapunov foliations are absolutely continuous w.r.t. Lebesgue, absolute continuity of μ follows by the following theorem: Theorem 4.8. [11, Theorem 5.2] Let f : M → M be a C 1+θ diffeomorphism preserving an ergodic measure μ. Let T M = E u ⊕ E c ⊕ E s be the Oseledets splitting associated to μ. Let us assume that: (1) E c is tangent to a smooth foliation O, Df |E c is an isometry with respect to to the standard metric in M , and conditional measures along O are Lebesgue measure; (2) E u = E1 ⊕ · · · ⊕ Eu and E s = Es ⊕ · · · ⊕ Er , where χi < χj if i < j; (3) each Ei is tangent to an absolutely continuous Lyapunov foliation W i and the conditional measures along W i are absolutely continuous with respect to Lebesgue measure for a.e. point. As a corollary of the proof we get the smoothness of the semiconjugacy along the one dimensional W and the matching of the Lyapunov exponents.

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Proposition 4.9. For every Lyapunov foliation W, the semiconjugacy h restricted to W(x) is a smooth diffeomorphism for μ-a.e. x and the Lyapunov exponents for the linear and the nonlinear actions match. Proof. As in [6] we have here that h intertwines the affine structure along W  for α and the one of α0 . 4.4. Uniqueness of large invariant measure. The proof of Theorem 3.2 follows from the proof in Section 7 in [11] which works without any change allowing existence of negatively proportional exponents. Remark 3. It is in the use of Lemma 7.5. in [11] (see also Lemma 5.1 in the next section) where we use that the universal covering of M is contractible, i.e. M is a K(π, 1) manifold. This Lemma is false without the K(π, 1) assumption and examples can be constructed along the lines in Section 4 in [10] gluing two or more tori and presenting more than one large measure. 5. Existence of proper periodic orbits Proof of Theorem 3.3. The first issue is to get smoothness of h along stable and unstable manifolds of different elements of the action. To this end we will use that there are no resonances in the linear action. This follows as in [11, Section 7]. Take a hyperbolic element α(n) of the action. Consider a Pesin set Λ. It follows from the proof of the Main Lemma in [7] that for a.e. x ∈ Λ there is a periodic point p for α(n) such that the local stable and unstable manifolds of p are approached by admissible (w.r.t. Λ) local stable and unstable manifolds of a sequence of points zk that may be taken in a fixed Pesin set Λ ⊃ Λ. Moreover this points zk lie in the intersection of (global) stable and unstable manifolds of the orbit of x and hence are in the support of μ since conditional measures of μ along stable and unstable manifolds are equivalent to Lebesgue measure. Hence taking x s u (x) and Wα(n) (x) be affine such that the restriction of the semiconjugacy h to Wα(n) w.r.t. the corresponding affine structures and using the fact that affine structures varies H¨ older continuously along Pesin sets we get that the semiconjugacy h is a local smooth diffeomorphism along the local stable and unstable manifolds of p. By invariance we get that the semiconjugacy is a smooth diffeomorphism along the global stable and unstable manifold of p for α(n). Let us see that p has finite α-orbit. Indeed we will see that α(m)(p) = p if and only if α0 (m)(h(p)) = h(p). Of course one implication is obvious, let us see the other one. To this end we shall make use of the following Lemma that appeared as Lemma 7.5 in [11], Lemma 5.1. Let E i ⊂ Rn , i = 1, 2 be two subspaces such that E 1 ⊕ E 2 = Rn . Let pi : E i → Rn , i = 1, 2 be two proper embeddings at a bounded distance from inclusion. Call pi (E i ) = Wi , i = 1, 2. Then W1 ∩ W2 = ∅. Assume α0 (m)(h(p)) = h(p), then α(m)(p) is a periodic point for α(n) s s (α(m)(p)) = α(m)(Wα(n) (p)). But then, by Lemma 5.1 we have that and Wα(n) s u Wα(n) (α(m)(p)) ∩ Wα(n) (p) = ∅, call z a point in this intersection. Since stable manifolds are sent into stable manifolds by h and unstable manifolds are sent into unstable manifolds, it follows that h(z) = h(p) and hence p = z since h restricted to the unstable manifold of p is a diffeomorphism, but this clearly implies that α(m)(p) = p. This and the fact that h is a diffeomorphism when restricted to

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s u Wα(n) (p) and Wα(n) (p) implies that Dp α(m) is conjugated to α0 (m) whenever α(m)(p) = p. 

Again is in the use of Lemma 5.1 that we need the ambient manifold to be a K(π, 1) space. 6. Differentiable rigidity of real analytic actions. For the proof of Theorem 3.5 we will use the following consequence of Zimmer’s cocycle superrigidity as we did in [12], see also [5, 18]. Proposition 6.1. Let M, N, Γ, α, α0 , po , pˆ and μ be as above. Then there is a finite index subgroup Γ ⊂ Γ and a measurable map η : M → N defined μ a.e. such that if we define the μ-a.e. defined map p : M → N , p(x) = η(x)ˆ p then p ◦ α(γ)(x) = α0 (γ) ◦ p(x) for μ-a.e. x ∈ M and for every γ ∈ Γ . Moreover, if α0 is weakly-hyperbolic as defined in [18] (in particular if there is an Anosov element for α0 ) then η and hence p extends to a continuous map p : M → N which is a semiconjugacy on supp(μ) but may fail to be a semiconjugacy for the action outside supp(μ). Proof of Theorem 3.5. Take Γ, α, α0 and μ in the hypothesis of Theorem 3.5. By Proposition 6.1 we have a continuous map P : T2n → T2n homotopic to identity such that P ◦ α(γ)(x) = α0 (γ) ◦ P (x) for x ∈ supp(μ) and for γ ∈ Γ ⊂ Γ a finite index subgroup. Hence P (supp(μ)) is compact and α0 -invariant and so equals either T2n or a point. Assuming the support of the measure μ is large, P (supp(μ)) = T2n . If we take C a Cartan subgroup of Γ then this Cartan subgroup is a full abelian symplectic subgroup. Hence its action on T2n has simple coarse Lyapunov spectrum and is without resonances and so it is in the hypothesis of Theorems 3.1 and 3.2. And hence μ is absolutely continuous w.r.t. Lebesgue measure. Moreover, by Theorem 3.3 there is a periodic point q ∈ supp(μ) for the restriction to the Cartan action C such that W s (q) ∪ W u (q) ⊂ supp(μ) and P restricted to W s (q) and W u (q) is a smooth diffeomorphism onto P (q) + E s and P (q) + E u respectively. Let us prove that q is a periodic point for the complete action of α restricted to Γ . P (q) is a rational point since it is a periodic point for the action α0 restricted to C. Hence P (q) is a periodic orbit for the action α0 restricted to Γ . Take any γ ∈ Γ , then α(γ)(q) is a periodic orbit for α restricted to γCγ −1 . Take n ∈ C, we have that α0 (γ)Eαs 0 (n) ∩ Eαu0 (n) = {0} and hence α0 (γ)Eαs 0 (n) ⊕ Eαu0 (n) = Rn . s s s Moreover, Wα(γnγ −1 ) (α(γ)(q)) = α(γ)(Wα(n) (q)) and hence P |Wα(γnγ −1 ) (α(γ)(q)) s s is a diffeomorphism onto α0 (γ)Eα0 (n) = Eα0 (γnγ −1 ) . Using Lemma 5.1 we have that s u Wα(γnγ −1 ) (α(γ)(q)) ∩ Wα(n) (q) = ∅. Now, P (α(γ)(q)) = α0 (γ)(P (q)) = P (q) since γ ∈ Γ and P (q) is periodic for α0 restricted to Γ . Then we have that s u P (Wα(γnγ −1 ) (α(γ)(q)) ∩ Wα(n) (q)) = P (q).

So, if z is in s u Wα(γnγ −1 ) (α(γ)(q)) ∩ Wα(n) (q) u then P (z) = P (q) and hence z = q since P is injective when restricted to Wα(n) (q) s and also z = α(γ)(q) because P is injective when restricted to Wα(γ −1 nγ) (α(γ)(q)).

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So α(γ)(q) = q for any γ ∈ Γ and hence is periodic. Also, by smoothness of P along the invariant manifolds through q we have that Dq α is conjugated to α0 . Hence the rest of the proof follows as in [12] using local linearization near fix points of real analytic actions proved by Cairns and Ghys [1].  References [1]

[2]

[3]

[4] [5] [6] [7] [8] [9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

´ Grant Cairns and Etienne Ghys, The local linearization problem for smooth SL(n)-actions (English, with English and French summaries), Enseign. Math. (2) 43 (1997), no. 1-2, 133– 171. MR1460126 Manfred Einsiedler and Anatole Katok, Invariant measures on G/Γ for split simple Lie groups G, Comm. Pure Appl. Math. 56 (2003), no. 8, 1184–1221, DOI 10.1002/cpa.10092. Dedicated to the memory of J¨ urgen K. Moser. MR1989231 Manfred Einsiedler and Elon Lindenstrauss, Rigidity properties of Zd -actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 99–110 (electronic), DOI 10.1090/S1079-6762-03-00117-3. MR2029471 John Franks, Anosov diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 61–93. MR0271990 David Fisher and Kevin Whyte, Continuous quotients for lattice actions on compact spaces, Geom. Dedicata 87 (2001), no. 1-3, 181–189, DOI 10.1023/A:1012041230518. MR1866848 Boris Kalinin and Anatole Katok, Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori, J. Mod. Dyn. 1 (2007), no. 1, 123–146. MR2261075 A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes ´ Etudes Sci. Publ. Math. 51 (1980), 137–173. MR573822 Boris Kalinin, Anatole Katok, and Federico Rodriguez Hertz, Nonuniform measure rigidity, Ann. of Math. (2) 174 (2011), no. 1, 361–400, DOI 10.4007/annals.2011.174.1.10. MR2811602 Boris Kalinin, Anatole Katok, and Federico Rodriguez Hertz, Errata to “Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori” and “Uniqueness of large invariant measures for Zk actions with Cartan homotopy data” [MR 2261075; MR2285730], J. Mod. Dyn. 4 (2010), no. 1, 207–209, DOI 10.3934/jmd.2010.4.207. MR2643892 Anatole Katok and Federico Rodriguez Hertz, Uniqueness of large invariant measures for Zk actions with Cartan homotopy data, J. Mod. Dyn. 1 (2007), no. 2, 287–300, DOI 10.3934/jmd.2007.1.287. MR2285730 Anatole Katok and Federico Rodriguez Hertz, Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups, J. Mod. Dyn. 4 (2010), no. 3, 487–515, DOI 10.3934/jmd.2010.4.487. MR2729332 Anatole Katok and Federico Rodriguez Hertz, Rigidity of real-analytic actions of SL(n, Z) on Tn : a case of realization of Zimmer program, Discrete Contin. Dyn. Syst. 27 (2010), no. 2, 609–615, DOI 10.3934/dcds.2010.27.609. MR2600682 A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 751–778, DOI 10.1017/S0143385700009081. MR1406432 Fran¸cois Ledrappier and Jian-Sheng Xie, Vanishing transverse entropy in smooth ergodic theory, Ergodic Theory Dynam. Systems 31 (2011), no. 4, 1229–1235, DOI 10.1017/S0143385710000416. MR2818693 F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539, DOI 10.2307/1971328. MR819556 F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (1985), no. 3, 540–574, DOI 10.2307/1971329. MR819557 G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, SpringerVerlag, Berlin, 1991. MR1090825 Gregory A. Margulis and Nantian Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices, Ergodic Theory Dynam. Systems 21 (2001), no. 1, 121–164, DOI 10.1017/S0143385701001109. MR1826664

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[19] Daniel J. Rudolph, ×2 and ×3 invariant measures and entropy, Ergodic Theory Dynam. Systems 10 (1990), no. 2, 395–406, DOI 10.1017/S0143385700005629. MR1062766 Department of mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802 E-mail address: katok [email protected] Department of mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802 E-mail address: [email protected]

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13920

On a differentiable linearization theorem of Philip Hartman Sheldon E. Newhouse Dedicated to the memory of Dmitri Anosov and Philip Hartman Abstract. Given a linear automorphism L of a Banach space E, let ρ(L) denote its spectral radius and let c(L) = ρ(L)ρ(L−1 ) denote its spectral condition number. Given the direct sum decompostion L = A ⊕ B let ch = max(c(A), c(B)), ρh = max(ρ(A−1 ), ρ(B)). For 0 < α < 1, the map L is called α-hyperbolic if L can be written as L = A ⊕ B so that ch ρα h is less than one. A fixed point of a C 1 diffeomorphism T is called α − hyperbolic if its derivative DT (p) is α−hyperbolic. A well-known theorem of Philip Hartman states that if E is finite dimensional with an α−hyperbolic fixed point and, in addition, the derivative DT is uniformly Lipschitz near p, then the map T is locally C 1,β linearizable near p for some β > 0. We obtain the same result under the weaker assumption that DT is uniformly α−H¨ older near p. We also extend the result to Banach spaces with C 1,α bump functions and obtain continuous dependence of the linearization on parameters. The results apply to give simpler proofs under weaker regularity assumptions of classical theorems of L. P. Shilnikov and others on the existence of horseshoe type dynamics and bifurcations near homoclinic curves.

1. Introduction Linearization theorems are of fundamental importance in Dynamical Systems. On the one hand they provide simple descriptions of the behavior near critical points and periodic motions, and on the other, they can often be applied with nonlocal techniques to give substantial information about global structures, e.g. as in [12],[18], [34], [35], [26], [36]. The so-called Grobman-Hartman Theorem states that a C r (with r a positive integer) diffeomorphism or flow can be (locally) C 0 linearized near a hyperbolic fixed point1 . While this theorem gives topological information about orbits which remain near the fixed point, it is inadequate for the study of orbits which recur near the fixed point after traveling a relatively large distance away from it. The analysis of such orbits often requires estimates of derivatives, and hence, makes good use of linearizations with various amounts of smoothness. 2010 Mathematics Subject Classification. Primary 37B40, 37C05, 37C29, 37D10; Secondary 37B10. Key words and phrases. Linearization, hyperbolic, H¨ older continuous, Philip Hartman. 1 This theorem, first proved in Euclidean space independently by Hartman and Grobman, was extended to Banach spaces, independently by Palis [25] and Pugh [7]. c 2017 Copyright is retained by the author: Sheldon E. Newhouse. Duplication in whole or in part is permitted by the Author.

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Standard smooth (local) linearization theorems near a hyperbolic fixed point in a finite dimensional manifold have the following form. First, by a suitable choice of coordinates, we may assume that the fixed point is the origin in Euclidean space. Next, given a desired smoothness k for the linearization, there is an integer N (k) such that if the N (k)−jet of the system at 0 is linear (the non-linear terms up to order N (k) vanish), then there is a linearization of order k near the fixed point. This vanishing of the nonlinear terms is most often guaranteed via a preliminary change of coordinates under so-called diophantine inequalities or non-resonance conditions. These are polynomial inequalities involving the eigenvalues of the derivative of the map or flow at the fixed point. See Hartman [28], Sternberg [37], [38], Bronstein and Kopanskii [5], and the references contained therein for more details. Several papers of Chaperon [20], [21] describe a beautiful development of Invariant Manifold Theory and its applications to linearization theorems. In this paper we deal with local linearizations under rather weak smoothness assumptions and, in the bi-circular case defined below, without diophantine inequalities. Since the results and arguments hold in Banach spaces with sufficiently smooth bump functions, let us set up the notation in that case. Let E and F be real Banach spaces and let L(E, F ) denote the Banach space of bounded linear maps from E to F with the usual supremum norm, for L ∈ L(E, F ), | L | = sup | Lx |. |x|=1

Let Aut(E) denote the linear automorphisms of E; i.e., the invertible elements of L(E, E). Let U be a non-empty open, connected subset of E and let α > 0 be a positive real number2 . The map f : U → F is called α−H¨older continuous at a point x ∈ U (or simply α−H¨older at x) if def

Hol(f, x, U ) =

(1)

sup y∈U,y=x

| f (y) − f (x) | < ∞. α |y−x|

The map f is called α−H¨ older in U if def

(2)

Hol(f, U ) =

sup x,y∈U,y=x

| f (y) − f (x) | < ∞. α |y−x|

The latter condition is sometimes called uniformly α−H¨ older in U . If α = 1 in the above inequalities, we say that f is, respectively, Lipschitz at x or Lipschitz in U . We refer to the constant Hol(f, U ) in (1) (or (2)) as the H¨ older constant of f at x (in U ). When α = 1, we use the term Lipschitz constant of f , and we write this as Lip(f, U ). The map f is differentiable at a point x ∈ U if there is a bounded linear map Df (x) ∈ L(E, F ) such that | f (x + h) − f (x) − Df (x)h | = 0. |h| |h|→0 lim

2 In

this paper, unless otherwise stated, all open sets will be assumed to be connected.

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Sometimes this concept is referred by saying that f is Fr´echet differentiable at x. For notational convenience, we sometimes write Df (x) as Dfx . If f is differentiable at each x ∈ U , and, in addition, the map x → Df (x) is continuous from U into L(E, F ), then we say that f is continuously differentiable or C 1 in U . If the C 1 map f : U → F is such that y → Df (y) is α−H¨older at x, (resp., in U ), then we say that f is C 1,α at x (resp., in U ). Unless otherwise stated, we will typically use this for 0 < α < 1. When α = 1, we will say that Df is Lipschitz at x (resp., in U ). older constant For a C 1,α map f : U → E, it is convenient to define its D−H¨ Hol(Df, U ) by | Df (x) − Df (y) | . Hol(Df, U ) = sup α |x−y | x=y∈U Recall that two norms | · | and || · || on a linear space E are equivalent if there is a positive constant C > 0 such that C −1 ≤

|x| ≤C || x ||

for every x = 0 in E. Note that differentiability of functions f : E → F is independent of the choice of equivalent norms on E or F . Throughout this paper, we denote the ball of radius δ about 0 in E by Bδ = Bδ (0). Let 0 < α < 1. We will call a Banach space (E, | · |) a C 1,α Banach space if there is a C 1,α bump function on E. This is a C 1,α function φ : E → R such that φ(E) = [0, 1], and there are positive numbers c1 < c2 such that φ(x) = 1 for x ∈ Bc1 and φ(x) = 0 for x ∈ / Bc2 . Replacing φ(x) by x → φ(c2 x), we may assume that c2 = 1. This is a somewhat restrictive condition on Banach spaces. For instance, it holds for the Lp spaces with p > 1 but fails for L1 . If there is an equivalent norm || · || on E which is C 1,α on open sets which do not contain 0, then it is easy to see that such bump functions exist. In particular, this is true in Hilbert Spaces since their norms are C ∞ away from 0. . The study of the existence or non-existence of various smooth norms or bump functions in Banach spaces is a central part of the geometric theory of Banach spaces. We refer the reader to [11] and [19] for more information. Given a linear automorphism L : E → E, the spectral radius of L, denoted ρ(L), is defined to be (3)

1

ρ(L) = lim | Ln | n n→∞

To see that this limit exists, observe that the sequence an = log(| Ln |) is subadditive (i.e. an+m ≤ an + am ) and the numbers ann are bounded below (in fact, by the number −log(| L−1 |)), so Lemma 1.18 in [4] implies that the quantity an an def h0 = lim = inf n→∞ n n≥1 n exists. Then, ρ(L) = exp(h0 ). If E is n−dimensional Euclidean space and L is a linear automorphism, then ρ(L) is, of course, the maximum of the absolute values of the eigenvalues of L.

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A linear automorphism L is called contracting if ρ(L) < 1. It is expanding if ρ(L−1 ) < 1. A linear automorphism L is called hyperbolic if it satisfies exactly one of the three conditions (1) L is contracting, (2) L is expanding, or (3) L can be written as a direct sum L = A ⊕ B in which A is expanding and B is contracting. In the last case, we say that L is hyperbolic of saddle type or, simply of saddle type, and we call A the expanding part of L and B the contracting part of L. They are unique. The spectral condition number of the linear automorphism L is the quantity c(L) = ρ(L)ρ(L−1 ). It is an easy consequence of the spectral radius formula (3) that c(L) ≥ 1. Indeed, for each positive integer m, we have I = L−m Lm , so 1 ≤ | L−m || Lm |. Now, take m−th roots and the limit as m approaches infinity. A linear automorphism L is called circular if c(L) = 1. In the finite dimensional case, this means that all of its eigenvalues lie in a single circle in the complex plane. If L is circular and contracting, then we call it c−contracting. Analogously, we say that L is c−expanding if L−1 is c−contracting. We say that L is bi-circular if it is hyperbolic of saddle type, and its expanding and contracting parts are both circular. Trivially, hyperbolic linear automorphisms of saddle type on the plane are bicircular as are hyperbolic linear automorphisms of saddle type in R3 which have a pair of non-real complex conjugate eigenvalues. An obvious, but important remark, is that every hyperbolic linear automorphism of saddle type on a finite dimensional space is bi-circular on the direct sum of its leading eigenspaces. These are the subspaces on which the eigenvalues are closest to 1 in absolute value (see Section 7 for details on the analogous conditions for vector fields and applications). Given the real number α with 0 < α < 1, we say that L is α−contracting if c(L)ρ(L)α < 1. Since c(L) is always at least 1, this implies that ρ(L) < 1. It is wellknown (see Lemma 3.4 below) that this implies the existence of an equivalent norm on E whose induced norm on L is less than 1 (whence the name contracting). In the finite dimensional case, this implies that the absolute values of the eigenvalues lie in a complex annulus whose bounding circles have radii in the open-closed interval (ρ(L)1+α , ρ(L)]. We say that L is α-expanding if L−1 is α-contracting. Let L be a hyperbolic linear automorphism of saddle type, and let L = A ⊕ B with A expanding and B contracting. Let (4)

ch = ch (L) = ch (A, B) = max(c(A), c(B)). We call this the hyperbolic condition number of L. Similarly, we define the hyperbolic spectral radius of L to be

(5)

ρh = ρh (L) = ρh (A, B) = max(ρ(A−1 ), ρ(B)).

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We say that L is α − hyperbolic if it can be written as a direct sum L = A ⊕ B so that (6)

α c h ρα h = ch (L)ρh (L) < 1.

Remark 1.1. Since definition of ρh in (5) requires that the inverse operator appearing is the expanding part in a hyperbolic decomposition of L, it follows that ρh (L−1 ) = ρh (L). Remark 1.2. Since the condition (6) implies that c(A)ρ(A−1 )α < 1 and c(B)ρ(B)α < 1, we see that the contracting and expanding parts of an α−hyperbolic automorphism must, in fact, be α−contracting and α−expanding, respectively. Note that if L is bi-circular, then it is α−hyperbolic for every α ∈ (0, 1). Let Hypα (E) be the set of α−hyperbolic linear automorphisms of E. Using the formula (3) on the contracting and expanding parts of an element of Hypα (E) and the techniques in Section 4 of Hirsch and Pugh [16], it can be shown that, for each α ∈ (0, 1), Hypα (E) is an open subset of L(E, E). Let r ≥ 1, k ≥ 1 be real numbers (not-necessarily integers). Let [r] denote the greatest integer less than or equal to r. When r is not an integer, we say that a map older. Setting r − [r] = β, T is C r if it is C [r] and its [r]−th derivative is r − [r]-H¨ [r],β we will also write this as C . Given the Banach space E and a point p ∈ E, let T be a C r diffeomorphism from an open neighborhood U of p onto its image such that T (p) = p. We say that p is an α−contracting fixed point of (or for) T if the derivative DT (p) is α−contracting. In a similar way we define α−expanding, α−hyperbolic, and bi-circular fixed points. A C k linearization of T at p (or near p) is a C k diffeomorphism R from a neighborhood of p onto a neighborhood of 0 such that RT R−1 = L on some neighborhood of 0. An equivalent condition is that LR = RT on some neighborhood of p. When such a diffeomorphism R exists, we say that T is C k linearizable at (or near) p. Sometimes the term locally C k linearizable at (or near) p is used and the map R is called a local linearization at (or near) p. At times it will be convenient to use the statement p is C k linearizable to mean that p is a fixed point of a C r diffeomorphism T (for some r ≥ k) which has a C k linearization at p. Since the neighborhoods in the domains of definition of the various maps considered often change, the concept of germ of a map at a point is often used (as in [20], [18]). As far as we know, S. Sternberg was the first to obtain linearization results for finitely smooth systems without explicit diophantine inequality assumptions. In the paper [37], published in 1957, he proved that, for k ≥ 2, C k maps of an interval with contracting or expanding fixed points are locally C k−1 linearizable near the fixed points. In [27], Philip Hartman extended the C 2 case of Sternberg’s one dimensional results in the following significant ways.

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Theorem 1.3. (Hartman). Let E denote the Euclidean space Rn , and let U be an open neighborhood of 0 in E. Let T be a C 1,1 diffeomorphism from U to its image such that T (0) = 0 and DT (0) = L where either (a) L is contracting3 , or (b) L is α−hyperbolic for some α ∈ (0, 1). Then, T is C 1 linearizable at 0. As is well-known, the more general case in which T (p) = p and p is not necessarily 0 can be reduced to the case of the theorem by replacing T by S −1 T S where S(x) = x + p is the translation by p. Remark 1.4. In the contracting case, Hartman proved that the linearization R was C 1,β for some β > 0. He stated that this was true in the α−hyperbolic case, but did not present the proof. It is natural to ask if one can reduce the regularity assumption on T and still get a C 1 linearization. On page 101 in [37] Sternberg shows that, for any a ∈ (0, 1), the C 1 map T on the real line defined by  1 ax(1 − log(|x|) ) if x = 0 T (x) = 0 if x = 0 has no C 1 linearization at 0. Hence, the most natural assumption is that we consider the case the DT is uniformly H¨ older instead of uniformly Lipschitz. That is, we assume that T is C 1,α (on some neighborhood of 0) for some positive α with 0 < α < 1. It is known (even in arbitrary Banach spaces) that a C 1,α diffeomorphism T with an α−contracting fixed point at 0 has a C 1,α linearization near 0. This is a consequence of Corollary 1.3.3 in Chaperon [20]. Since this result is fundamental for our work here, we will include an elementary proof (see Theorem 3.1) below. In the finite dimensional case, this result was stated (under the stronger assumption that T was C 1,1 ) with different notation in the last sentence of the first paragraph in section 8 on page 235 in [27]. There is also a statement on page 222 of [27] that the regularity assumption on T could be weakened to some C 1,γ with γ > α0 where α0 depends on the eigenvalues of L. We suspect that this α0 equals our α, but, since it is not given explicitly, we cannot be sure of this. In the infinite dimensional case, the weaker result that there is a C 1 linearization whose derivative is H¨older at 0 was proved by Mora and Sol` a-Morales in Theorem 3.1 in [22]. Since any circular linear contraction is α−contracting for any α > 0, it follows that any C 1,α map T with a c-contracting fixed point p has a C 1,α linearization at p. The next case to consider is contractions which are not circular. Here, in [39], Zhang and Zhang study such C 1,α contractions in the plane. They obtain C 1,β linearizations for various β depending on α. They also show that any linear noncircular contraction can be perturbed to give a C 1,α diffeomorphism fixing the origin which has no C 1,β linearizations for any β > 0. It is not known in these examples if there is a C 1 linearization. 3 Since

(RT R−1 )−1 = RT −1 R−1 , the theorem also applies when L is expanding.

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In [15], Rodrigues and Sol`a-Morales give examples of C 1,1 diffeomorphisms in infinite dimensional Banach spaces with contracting fixed points which are not C 1 linearizable. An alternative approach to linearizations of C 2 flows via the Lie Derivative is in Chicone-Swanson [6]. The main result in the present paper can be paraphrased as follows. In a C 1,α Banach space with 0 < α < 1, every α−hyperbolic fixed point of a 1,α diffeomorphism is C 1,β linearizable for some β > 0. C After translating the fixed point to the origin as above, we have the more precise statement. Theorem 1.5. Let 0 < α < 1 and let E be a C 1,α real Banach space. Let U be a neighborhood of 0 in E and let T : U → E be a C 1,α map such that T (0) = 0 and L = DT (0) is an α−hyperbolic automorphism of E. Then, there are a subneighborhood V ⊂ U of 0, a real number β ∈ (0, α), and a C 1,β diffeomorphism R from V onto its image such that R(0) = 0 and % (7) L(R(x)) = R(T (x)) for all x ∈ V T −1 V Remarks. (1) In the contracting C 1,1 case (i.e., part (a) of Theorem 1.3), Hartman proved that the linearization could be made C 1,β for some β > 0. In the α−hyperbolic case, he stated that his proof could be modified to give a linearization for some β > 0, but he did not present the proof. Thus, our proof, even in the finite dimensional case, may be the first available proof that such a C 1,β linearization exists. (2) Given real numbers a > b > 1 > c > 0, and = 0, Hartman shows in [27] that, if b = ac, then the quadratic polynomial map (8)

Ta,b,c (x, y, z) = (ax, b(y + xz), cz) 1

has no C linearization at (0, 0, 0). It is interesting to note that Hartman’s example is sharp in the followb xz, z), then LR = RT so ing sense. If b = ac, and R(x, y, z) = (x, y + b−ac the map R gives even a quadratic polynomial linearization of T at (0, 0, 0). (3) In [14], Rodrigues and Sol`a-Morales consider C 1,1 diffeomorphisms having a hyperbolic saddle fixed point at 0 in a C 1,1 Banach space. They prove that, under a spectral condition which is equivalent to our notion of α-hyperbolicity, the diffeomorphisms are C 1 linearizable provided α is sufficiently close to 1. See Theorem 2 in [14]. (4) Given real numbers r ≥ 1 and r ≥ k ≥ 0, let us say that a fixed point p of a C r diffeomorphism T is robustly (r, k)- linearizable if, for any S sufficiently C r close to T , there is a pair (pS , RS ) with p = pT , depending continuously on S, such that S(pS ) = pS and RS is a C k linearization of S at pS . More precisely, let U be an open set in the Banach space E. Consider the space Dr (U, E) of C r diffeomorphisms T from U to T (U ) with the uniform C r topology. For k = 0, let Dk (U, E) denote the space of homeomorphisms T from U onto T (U ) with the uniform C 0 topology. The fixed point p of the diffeomorphism T ∈ Dr (U, E) is said to be robustly

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(5)

(6)

(7)

(8)

(r, k)-linearizable if there are neighborhoods W of T in Dr (U, E) and V of p in E such that, for any S ∈ W , there is a pair (pS , RS ) such that (a) p = pT , (b) S(pS ) = pS , (c) pS ∈ V , (d) RS ∈ Dk (V, E), (e) RS (pS ) = 0,  (f) DSpS (x) = (RS ◦ S ◦ RS−1 )(x) for x ∈ RS (U S −1 V ), and (g) the map S → (pS , RS ) from W into E × Dk (V, E) is continuous. The typical linearization results that we are aware of (e.g. those given by the Grobman-Hartman Theorem, Sternberg’s Theorems, Hartman’s Theorem 1.3 above, etc) yield fixed points which are robustly (r, k)−linearizable for some (r, k). As a consequence of Theorem 6.2 below we also obtain that, for r = 1 + α, the α−hyperbolic and bi-circular fixed points considered in this paper, are, in fact, robustly (r, 1 + β) linearizable for some β > 0. This paper owes much to Hartman’s paper [27]. The scheme of the proof of Theorem 1.5 is similar to that indicated in pages 235-238 in [27] for C 1,1 maps T . In actuality, Hartman only gave the proof that the linearization was C 0 . He stated that similar methods could be used to prove that it was C 1 , and, as we have already mentioned, he also stated that the proof could be modified to give a linearization whose derivative was uniformly H¨ older continuous. In the fifty-five years since the paper [27] appeared, two significant developments have occurred that make our arguments possible. (a) We now know that no smoothness is lost for stable and unstable manifolds at a hyperbolic fixed point. In particular, if T is C 1,α with a hyperbolic fixed point at p, then the stable and unstable manifolds at p are locally the graphs of C 1,α maps. (b) A C 1,α map T with fixed point p such that DT (p) is α−contracting has a C 1,α linearization. See Theorem 3.1 below. There are several additional papers in the literature which are relevant to the work presented here. In particular, the papers by Samovol [29], [30], [31], Belitskii [1], [2], [3], and Stowe [9] all contain interesting results on smooth linearizations with finite class of differentiability. Many of these are discussed and generalized in the book of Bronstein and Kopanskii [5]. The contents of this paper are as follows. The proof of Theorem 1.5 will be carried out in Sections (2)-(5). Section 6 considers the dependence of the linearization R in Theorem 1.5 on external parameters, and Section 7 presents some motivation and applications of the results.

Acknowledgements. (1) Being relatively unaware of the recent literature on the subject, we gave a lecture at Penn State in October, 2014, in which we discussed the generalization to the C 1,α case of part (b) in Hartman’s Theorem 1.3 above for bi-circular fixed points.The result was still for finite dimensional systems, including the finite dimensional version of Theorem 3.1 below, and only considered the existence of C 1 linearizations. We were not aware that the general result in Theorem 3.1 was known. We wish to thank Misha

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Guysinsky for subsequently providing many references to the recent literature, including, in particular, [13] and [39]. His remarks and references provided substantial impetus for us to study the recent literature, and eventually extend our results to the infinite dimensional case as presented here. (2) We have already indicated the importance of Hartman’s paper [27] in connection with this work. This was a special time for him in that, on May 16, 2015, he reached his 100th birthday. We recently learned, sadly, that he passed away on August 28, 2015. We dedicate this paper to both Anosov and Hartman–two icons in Dynamical Systems whose mathematical achievements will have a long lasting legacy. 2. The Associated Algebraic Problem The standard way to approach the functional equation (7) is to convert it into a related algebraic equation on a suitable Banach space E as follows. Let I denote the identity map on E. Writing T = L + f , we try to find R of the form R = I + φ with φ ∈ E so that the equation LR = RT becomes L(I + φ) = (I + φ)T = (I + φ)(L + f ) which leads to L + Lφ Lφ φ

= L + f + φ(L + f ) = f + φ(L + f ) = L−1 f + L−1 φ(L + f ),

or (9)

φ − L−1 φ(L + f ) = L−1 f. Defining the operator H on functions φ by

(10)

H(φ) = L−1 φ(L + f ) = L−1 φ ◦ T,

we see that H is linear, and equation (9) becomes (11)

(I − H)(φ) = L−1 f.

The problem, then, is reduced to finding a Banach space E so that the operator H is a well-defined bounded linear map such that I − H has a bounded linear inverse, in which case we obtain (12)

φ = (I − H)−1 L−1 f.

Theorem 1.5 will be proved by using the map H defined by (10) and the ensuing equations (11) and (12) in several different function spaces and then combining the results. A rough outline (assuming the hypotheses of Theorem 1.5) is as follows. Step 1: We show that if L is contracting; i.e., ρ(L) < 1, then T = L + f can be C 1,α linearized near 0. That is, there is a local C 1,α diffeomorphism R near 0 satisfying RT R−1 = L near 0. It follows, of course, that RT −1 R−1 = L−1 near 0.

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Step 2: Assuming L is hyperbolic of saddle type, let E = E u ⊕ E s be the associated unstable and stable subspaces of E. We choose C 1,α coordinates near 0 so that the local stable and unstable manifolds are flattened; i.e., are contained in E s and E u , respectively. Applying the result in Step 1, we can locally C 1,α conjugate T to a map S1 which is locally preserves E s and E u and is linear when restricted to those subspaces. Then, using a bump function, we extend the map S1 (actually its restriction to a small neighborhood of 0) to a C 1,α diffeomorphism S from the whole Banach space E onto itself such that S(0) = 0, DS(0) = L, Lip(S − L) is small, and S = L on the union of E u , E s , and the complement of a small neighborhood V of 0. Step 3: For V and Lip(S − L) small enough, we define an appropriate Banach space E of C 1 functions φ defined on all of E and solve the equation (12) to obtain a global C 1 conjugacy I + φ with φ ∈ E from S to L defined on all of E. Step 4: Choosing V small enough, and making use of (12) in two separate Banach spaces of functions, we show that, for appropriately chosen small β > 0, the map φ is uniformly β−H¨older on bounded subsets of the orbit of V . This gives that S is locally C 1,β linearizable at 0, and, hence, so is T . The operator H occurs often enough in this kind of problem that it deserves a name. We call it the associated linearization operator for T or the T −linearization operator.

3. The α−Contracting Case Let E be a Banach space, and let U be an open neighborhood of 0 in E. Let 1,α 0 < α < 1 and consider the set CU1,α = C0,U of C 1 functions g : U → E such that (13)

g(0) = 0, Dg(0) = 0,

and (14)

Hol(Dg, U ) =

sup x,y∈U,x=y

| Dg(x) − Dg(y) | < ∞. α |x−y |

Using the techniques for statement (8.6.3) in [10] one can verify that the DH¨older constant of g, | g |U = | g |U,α = Hol(Dg, U ), defines a norm on CU1,α making it into a Banach space. Observe that the usual way of defining a norm on a space of C r functions into a Banach space involves taking the supremum of the C 0 size of the function as well as that of its derivatives. Otherwise one only gets a semi-norm in that the vanishing of this semi-norm does not imply the vanishing of the functions. However, our functions already vanish at 0 (and U is connected), so this issue does not occur. Note that if V ⊂ U and φ ∈ CU1,α , then the restriction map from φ → φ|V induces a continuous map from (CU1,α , | · |U,α ) into (CV1,α , | · |V,α ). We will use the same notation for the maps φ and φ | V letting the context determine which space we are using at a given moment. When U is the open ball Bδ (0) we will denote CU1,α by Cδ1,α . Let I denote the identity map on E.

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The goal of this section is to prove the following theorem: Theorem 3.1. Let U be an open neighborhood of 0, let α be a positive real number with 0 < α < 1 and let T be a map of the form T (x) = Lx + f (x) where L : E → E is a linear α−contracting automorphism and f ∈ CU1,α . 1,α such that, setting Then, for sufficiently small δ, there is a unique map φ ∈ C0,δ R = I + φ, we have % (15) L(R(x)) = R(T (x)) for all x ∈ Bδ (0) T −1 Bδ (0). Remark 3.2. For δ small, the map R will be Lipschitz close to the identity. Therefore, the Inverse Function Theorem implies that R is a C 1,α diffeomorphism onto its image with inverse which is also C 1,α . Also, L = RT R−1 implies that L−1 = RT −1 R−1 , so Theorem 3.1 also holds when L is a linear α−expansion. Remark 3.3. Consider a C 1,α diffeomorphism T : RN → RN on the Euclidean space RN with a hyperbolic fixed point at 0 and derivative L = DT (0). Then, we have the direct sum decomposition RN = E 1 ⊕ E 2 ⊕ . . . ⊕ E s into L−invariant subspaces such that L | Ei is circular for each 1 ≤ i ≤ s. From invariant manifold theory, [17], [8], for some 0 < α < 1, there are T locally invariant C 1,α submanifolds Wi tangent at 0 to Ei for each i. These submanifolds are generally not unique, of course, and, even if T is linear, they are not generally C 1,1 . Nevertheless, using Theorem 3.1, Remark 3.2, and the techniques in Section 4.4 below, one can find an α ∈ (0, 1) and a local C 1,α coordinate chart (U, ζ) near 0 such that ζ(0) = 0, ζ(Wi ) ⊂ Ei for each i, and ζT ζ −1 restricted to each Ei is linear near 0. As we mentioned in the introduction, Theorem 3.1 is not new. It is a consequence of a more general result proved by Chaperon (see Corollary 1.3.3 in [20]). Since the result is needed in an essential way in the present paper and it is relatively simple to prove, we give an alternate proof here for completeness. We begin with a standard result which shows that, given a Banach space (E, | · |), and a bounded linear operator L : E → E, we can approximate the spectral radius ρ(L) by the supremum norm | L |1 induced by a norm | · |1 on E which is equivalent to | · |. Lemma 3.4. Let (E, | · |) be a Banach space and let L : E → E be a bounded linear map with spectral radius ρ(L). Then, given > 0 there is a norm | · |1 on E which is equivalent to | · | such that the induced norm | L |1 defined by | L |1 =

sup | Lv |1 | v |1 =1

satisfies | L |1 ≤ ρ(L) + Proof. Setting ρ1 = ρ(L) + and using formula (3), we can choose a positive integer q such that, for n ≥ q, we have | Ln | < ρn1 .

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This gives, for any v and n ≥ q, | Ln v | ≤ | Ln || v | ≤ ρn1 | v |, which implies that n ρ−n 1 | L v | ≤ | v |.

As usual, we define L0 = I. Then, setting n K = max ρ−n 1 | L |, 0≤n 0 sufficiently small, we take the space E alluded to in section 2 to be Cδ1,α , and we use the norm | φ |δ = | φ |δ,α for elements φ ∈ E. Let L, , f, T be as in the hypotheses of Theorem 3.1. All numbers δ below will be chosen in (0, ) so that f is defined and C 1,α in Bδ (0). Recalling the procedure in section 2, we consider the functional equation L(I + φ) = (I + φ)(L + f ) and the resulting algebraic equation (I − H)(φ) = L−1 f. where H(φ) = L−1 φ(L + f ). We will show that, for δ small enough, the operator H has the properties that (16)

H is defined as a function on Cδ1,α ,

(17)

H maps Cδ1,α into itself

and, for some positive integer m, | H m | < 1.

(18)

In then follows that ρ(H) < 1, and, as we have already noted, the required solution is obtained as φ = (I − H)−1 L−1 f . Before going to the proof of (16)-(18), we need some easy estimates obtained from the Mean Value Theorem and our other assumptions. As we have already noted, the α−contracting condition on L implies that ρ(L) < 1, so we may use Lemma 3.4 to renorm E so that | L | < 1,

(19) which implies that

n

| Ln | < | L | < 1

(20)

for every positive integer n. From the fact that L is α-contracting, formula (3) gives us a positive integer m such that

m1 | L−m || Lm |1+α < 1, which, in turn, implies | L−m || Lm |1+α < 1. Given a linear automorphism L, we recall that its minimum norm, denoted by m(L) (or mL ) is defined by (21)

m(L) = mL = inf | Lx |. |x|=1

It is easily checked that m(L) = | L−1 | Let 1 > 0 be such that

−1

.

(22)

| L | + 1 < 1,

(23)

m(L) − 1 > 0,

(24)

| Lm | + 1 < 1,

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and m(L−m ) − 1 > 0,

(25) and

| L−m |(| Lm | + 1 )1+α < 1.

(26)

Using Df0 = 0, choose δ > 0 small enough so that, for x ∈ Bδ (0), | Dfx | < 1 .

(27) In particular,

Lip(f, Bδ (0)) ≤ 1 .

(28) Now, for | x | < δ, (29)

| T (x) | = | Lx + f (x) | ≤ (| L | + 1 )| x | ≤ | x |,

and, for x = y ∈ Bδ , we have | T (x) − T (y) | = | (L + f )x − (L + f )y | ≥ | L(x − y) | − 1 | x − y | > (mL − 1 )| x − y | > 0. In particular, since T (0) = 0, it follows that | T (x) | > 0 for | x | > 0. Further, | T (x) − T (y) |

= | Lx + f (x) − Ly − f (y) | ≤ (| L | + Lip(f ))| x − y | ≤ (| L | + 1 )| x − y |,

so, (30)

Lip(T, Bδ (0)) ≤ | L | + 1 < 1.

This implies that both T and T m map Bδ into itself. An easy induction shows that DT m (0) = Lm . So, since the composition of 1,α maps is C 1,α , if we set fm = T m − Lm , then we have that fm ∈ Cδ1,α . C Now, shrinking δ, if necessary, we may assume that Lip(fm , Bδ ) < 1 ,

(31) and

Lip(T m , Bδ ) < | Lm | + 1 .

(32) For n = 1 or n = m, let

Kn,1 = | L−n |(| Ln | + 1 )α δ α | fn |δ , and Kn,2 = | L−n |(| Ln | + 1 )1+α , and τ n = Kn,1 + Kn,2 . From (26), we have Km,2 < 1, so we can choose δ > 0 small enough so that τ m < 1.

ON A DIFFERENTIABLE LINEARIZATION THEOREM

223

Let us now, proceed to the proof of (16)-(18). Let φ ∈ Cδ1,α . From (30), it follows that T (Bδ (0)) ⊂ Bδ (0), so H(φ) is defined on Bδ (0), and this is (16). Again using that the composition of C 1,α maps is again C 1,α , it is clear that H(φ) is C 1,α . It is also evident that it vanishes at 0 together with its derivative. For x, y ∈ Bδ with | x − y | > 0 and | x | > 0, and n = 1 or n = m, we have | D(H n (φ))(x) − D(H n (φ))(y) | = | L−n DφT n (x) DTxn − L−n DφT n (y) DTyn | ≤ | L−n || DφT n (x) DTxn − DφT n (x) DTyn | +| L−n || DφT n (x) DTyn − DφT n (y) DTyn | = (A) + (B) where (A) = | L−n || DφT n (x) DTxn − DφT n (x) DTyn | ≤ | L−n || DφT n (x) | | Ln + Dfn,x − (Ln + Dfn,y ) | = | L−n || DφT n (x) | | Dfn,x − Dfn,y | and (B)

= | L−n || DφT n (x) DTyn − DφT n (y) DTyn | ≤ | L−n |(| Ln | + 1 )| DφT n (x) − DφT n (y) |.

For x = 0, we have (A) ≤ | L−n | ≤ | L−n

| DφT n (x) |

α α n α | T (x) | | fn |δ | x − y | | T n (x) | | | φ |δ (| Ln | + 1 )α | x |α | fn |δ | x − y |α

≤ | L−n | | φ |δ (| Ln | + 1 )α δ α | fn |δ | x − y |

α

α

= Kn,1 | φ |δ | x − y | . and, for x = y, (B) ≤ | L−n |(| Ln | + 1 )| DφT n (x) − DφT n (y) | | DφT n (x) − DφT n (y) | n α | T (x) − T n (y) | | T n (x) − T n (y) |α α |(| Ln | + 1 ) | φ |δ | T n (x) − T n (y) | |(| Ln | + 1 ) | φ |δ Lip(T n )α | x − y |α

≤ | L−n |(| Ln | + 1 ) ≤ | L−n ≤ | L−n

≤ | L−n |(| Ln | + 1 )1+α | φ |δ | x − y |α α

= Kn,2 | φ |δ | x − y | . Hence, we have | D(H n (φ))(x) − D(H n (φ))(y) | ≤ (A) + (B) ≤ (Kn,1 | φ |δ + Kn,2 | φ |δ ) | x − y |α (33)

= τ n | φ |δ | x − y |α . Dividing both sides by | x − y |α and taking the supremum over x = y, gives | H n (φ) |δ ≤ τ n | φ |δ .

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SHELDON E. NEWHOUSE

For n = 1, this gives (17), and for n = m, this gives (18), completing the proof of Theorem 3.1. 4. Preliminary Constructions Let us begin with an outline of Hartman’s method in the proof of part (b) of Theorem 1.3. He first changes coordinates so that the stable and unstable manifolds are flattened (i.e., are contained in the stable and unstable subspaces, respectively) near the origin. Next, using his earlier result for C 1,1 contracting T , he chooses C 1,β coordinates (where β is related to the contracting and expanding eigenvalues) so that, near the origin, T preserves and becomes linear when restricted to the stable and unstable subspaces. Finally, he globalizes the mapping T , using a bump function to replace T by a map T1 which equals T on a ball B 0 (0) about zero, equals L off a slightly larger ball B 1 (0), and is globally Lipschitz close to L. This globalization preserves the earlier properties that the stable and unstable manifolds are flattened and T1 is linear when restricted to the entire stable and unstable subspaces. This allows him to get sufficiently good estimates on the partial derivatives of the non-linear part of T1 to use the method of successive approximations to obtain the desired local linearization. In fact, he only proves that the linearization is continuous. He states that similar methods will give that it is C 1 and that the proof can be modified to show that it has H¨ older continuous derivatives. We extend and modify his techniques to obtain our result. In addition, we write our solution in a way that it makes use of certain linear contracting maps in suitable Banach spaces. This provides the additional benefit that we can get continuous dependence of the linearization on parameters as in Theorem 6.2 below. 4.1. Estimates of the Lipschitz and D-H¨ older contants of inverses and compositions. The proof of Theorem 1.5 requires estimates of the non-linear parts of the diffeomorphisms T, T m , T −1 and T −m for a certain positive integer m in a sufficiently small ball Bδ (0) about 0 in the Banach space E. The results in this section can be used to give some information about these estimates in terms of the non-linear part of the original map T . Strictly speaking, they are not needed if one only wants the linearization R to exist on some small neighborhood of 0 and one does not need to estimate the size of that neighborhood. Let us begin with a simple lemma relating the Lipschitz and D-H¨older constants of a composition S ◦ T of C 1,α maps in terms of those of the maps S and T . Lemma 4.1. Let U and V be open subsets of the Banach space E, and consider maps S ∈ C 1,α (U, E) and T ∈ C 1,α (V, E). Then, % (34) Lip(S ◦ T, U T (V )) ≤ Lip(S, U )Lip(T, V ), and (35) % Hol(D(S ◦ T ), U T (V )) ≤ Lip(S, U )Hol(DT, V ) + Lip(T, V )1+α Hol(DS, U ). Proof. Letting x, y ∈ V , and leaving out the obvious domains of the maps involved, statement (34) is immediate from | S(T (x)) − S(T (y)) | ≤ Lip(S)| T (x) − T (y) | ≤ Lip(S)Lip(T )| x − y |.

ON A DIFFERENTIABLE LINEARIZATION THEOREM

225

For statement (35), we have | D(S ◦ T )(x) − D(S ◦ T )(y) | = | DST x DTx − DST y DTy | ≤ | DST x DTx − DST y DTx | + | DST y DTx − DST y DTy | α α ≤ | DTx |Hol(DS)| T x − T y | + | DST y |Hol(DT )| x − y | α α ≤ Lip(T )1+α Hol(DS)| x − y | + Lip(S)Hol(DT )| x − y | . Now, divide both sides by | x − y |α and take the supremum over x, y to complete the proof of Lemma 4.1.  Remark 4.2. Observe that, if T has the form T = L + f where L is bounded, f (0) = 0, and Df (0) = 0, then, since the derivative of a bounded linear map L is just L at every point, it cancels in the calculation of DT . Hence, (36)

Hol(DT ) = Hol(Df ).

That is, the D-H¨ older constant is determined by the nonlinear part of T . In particular, the addition of another bounded linear map to T does not change Hol(DT ). Corollary 4.3. Let S, T, U, V be as in the hypotheses of Lemma 4.1, and assume that S = L1 + f1 and T = L2 + f2 where L1 , L2 are bounded linear maps on E and f1 , f2 vanish at 0 together with their derivatives. Let f3 = S ◦ T − L1 L2 . Then, (37)

Lip(f3 ) ≤ | L1 |Lip(f2 ) + Lip(f1 )Lip(T ),

and (38)

Hol(Df3 ) ≤ Lip(S)Hol(DT ) + Lip(T )1+α Hol(Df1 ). Proof. We have ST

= (L1 + f1 )(L2 + f2 ) = L1 (L2 + f2 ) + f1 (T ) = L1 L2 + L1 f2 + f1 (T ),

giving f3 = L1 f2 + f1 (T ), and (37) immediately follows using the fact that the function Lip(·) is subadditive and submultiplicative. Statement (38) follows immediately from (35) since Hol(Df3 ) = Hol(D(S ◦ T )).  We will need the following estimate which we first saw in Hirsch and Pugh [16]. Lemma 4.4. For any linear automorphisms h1 , h2 , we have (39)

−1 −1 −1 | h−1 1 − h2 | ≤ | h2 || h1 − h2 || h1 |.

Proof. We have −1 −1 −1 −1 −1 | h−1 1 − h2 | = | h2 h2 h1 − h2 h1 h1 | −1 = | h−1 2 (h2 − h1 )h1 | −1 ≤ | h−1 2 || h2 − h1 || h1 | −1 = | h−1 2 || h1 − h2 || h1 |



226

SHELDON E. NEWHOUSE

Let diam(U ) denote the diameter of a subset U of a Banach space E. In Lemma 4.5, we again use the notation m(L) = mL = inf | Lx | = | L−1 |

−1

|x|=1

for a linear automorphism L. Lemma 4.5. Let (E, | · |) be a real Banach space, and let L : E → E be a linear automorphism. Let 0 < α < 1, and let U and V be neighborhoods of 0 in E such that (40)

max(diam(U ), diam(V )) < 1.

Let T : U → V be a C 1+α diffeomorphism from U onto V such that T (0) = 0 and DT (0) = L, and let f = T − L and g = T −1 − L−1 be the nonlinear parts of T and T −1 , respectively. Assume that Lip(f, U ) = sup | Df (z) | < mL .

(41)

z∈U

Then, we have (42)

Lip(T −1 , V ) ≤ (mL − Lip(f, U ))−1 ,

(43)

Lip(g, V ) ≤ | L−1 |(mL − Lip(f, U ))−1 Lip(f, U ),

(44)

Hol(DT −1 , V ) ≤ (mL − Lip(f, U ))−(2+α) Hol(Df, U ) and Hol(Dg, V ) ≤ (mL − Lip(f, U ))−(2+α) Hol(Df, U ).

(45)

Proof. For notational convenience, we will use the following notation. lf = Lip(f, U ), hf = Hol(Df, U ). From (41), we have that, for any z ∈ U and non-zero vector v, | DTz v | = | Lv + Df (z)v | ≥ mL | v | − lf | v | = (mL − lf )| v |. Now, taking any non-zero u and setting v = DTT−1 (z) u and w = T (z), we get | u | = | DTz DTw−1 u | ≥ (mL − lf )| DTw−1 u |, or | DTw−1 u | ≤ (mL − lf )−1 | u |. Thus, for every w ∈ V , we have | DTw−1 | ≤ (mL − lf )−1 , and (42) follows.

ON A DIFFERENTIABLE LINEARIZATION THEOREM

Again, writing w = T (z) = Lz + f (z), and solving for z = z(w), we get z(w) = T −1 (w) = L−1 w − L−1 f (z(w)).

(46) Thus,

g(w) = −L−1 f (z(w)), Dg(w) = −L−1 Dfz(w) DTw−1 , and | Dg(w) | ≤ | L−1 || Dfz(w) || DTw−1 | = | L−1 || Dfz(w) |(mL − lf )−1 ≤ | L−1 |lf (mL − lf )−1 This implies (43). Proceding to the proof of (44), from (46), we have DTw−1 = L−1 − L−1 Dfz(w) DTw−1 (I + L−1 Dfz(w) )DTw−1 = L−1 , DTw−1 = (I + L−1 Dfz(w) )−1 L−1 . From (41), we have, for each w ∈ V , | (I + L−1 Dfz(w) )−1 | ≤ (1 − | L−1 |lf )−1 = (| L−1 |mL − | L−1 |lf )−1 = | L−1 |

−1

(ml − lf )−1

= mL (ml − lf )−1 This, together with (39) gives − DTw−1 | ≤ | DTw−1 1 2

which is (44).

m2L (mL − lf )−2 | L−1 | | Dfz(w2 ) − Dfz(w1 ) | 2

=

(mL − lf )−2 | Dfz(w2 ) − Dfz(w1 ) |

≤

(mL − lf )−2 hf | z(w2 ) − z(w1 ) |

≤

(mL − lf )−2 hf l(T −1 )α | w2 − w1 |

≤

(mL − lf )−2 hf (mL − lf )−α | w2 − w1 |α

≤

(mL − lf )−2−α hf | w2 − w1 |

α

α α

227

228

SHELDON E. NEWHOUSE

Also, (45) holds since Hol(DT −1 , W ) = Hol(Dg, W ). This completes the proof of Lemma 4.5.  T

The next lemma gives analogous estimates of the non-linear parts of T m and for integers m > 1.

−m

Lemma 4.6. Let T, U, V, f, g be as in Lemma 4.5, let m be a positive integer, and assume that Lip(T ) ≥ 1. Consider the maps T m , T −m and fm , gm , given by fm = T m − Lm , gm = T −m − L−m . with corresponding domains Um , Vm given by m %

Um =

T

−j

U and Vm =

j=0

m %

T j V,

j=0

respectively. Then, T m and T −m are defined and C 1,α on Um and Vm , respectively, and (47)

Lip(T m ) ≤ Lip(T )m ,

(48)

Lip(fm ) ≤ mLip(f )Lip(T )m−1 ,

(49)

Hol(Dfm ) ≤ mHol(Df )Lip(T )(1+α)(m−1) ,

(50)

Lip(gm ) ≤ mLip(g)Lip(T −1 )m−1 ,

and (51)

Hol(Dgm ) ≤ mHol(Dg)Lip(T −1 )(1+α)(m−1) .

Proof. Since the composition of C 1,α maps is again C 1,α , it is clear that T m and T −m are defined and C 1,α on Um and Vm , respectively. Also, the Lipschitz composition formula (34) gives (47) immediately by induction. Using that DT n (0) = Ln , we get T n+1

= (L + f )(T n ) = LT n + f (T n ) = L(Ln + fn ) + f (T n ) = Ln+1 + Lfn + f (T n ).

This implies that, for each n ≥ 1, (52)

fn+1 = Lfn + f (T n ).

We will show that, for different choices of positive numbers An , B, C, the numbers Lip(fm ), Lip(gm ), Hol(Dfm ) and Hol(Dgm ) all satisfy the recursion relation (53)

An+1 ≤ BAn + B n C, A1 = C

for each 1 ≤ n ≤ m − 1. This recursion relation has the solution (54)

An+1 ≤ (n + 1)B n C

ON A DIFFERENTIABLE LINEARIZATION THEOREM

229

as can be seen from An+1

≤ BAn + B n C ≤ B(BAn−1 + B n−1 C) + B n C = = .. . ≤ .. . = =

(55)

B 2 An−1 + B n C + B n C B 2 An−1 + 2B n C B n A1 + nB n C B n A1 + nB n C (n + 1)B n C.

Now, consider Lip(fm ). Since | L | ≤ Lip(T ), we have Lip(fn+1 ) ≤ | L |Lip(fn ) + Lip(f )Lip(T n ) ≤ Lip(T )Lip(fn ) + Lip(f )Lip(T n ) and we can take An = Lip(fn ), B = Lip(T ) and C = Lip(f ). Next, observe that Hol(Dfm ) = Hol(DT m ) since | DT m x − DT m y | = | Lm + Dfm (x) − (Lm + Dfm (y)) | = | Df m x − Df m y |. The assumption that Lip(T ) ≥ 1 implies that Lip(T ) ≤ Lip(T )1+α . Using this and (38), we have Hol(DT n+1 )

= Hol(D(T ◦ T n )) ≤ Lip(T )Hol(DT n ) + Lip(T n )1+α Hol(DT ) ≤ Lip(T )1+α Hol(DT n ) + Lip(T )n(1+α) Hol(DT )

and we can take An = Hol(DT n ), B = Lip(T )1+α and C = Hol(DT ). Similar recursions hold for Lip(gm ) and Hol(Dgm ). Thus, formula (54) applied, in turn, to each of Lip(fm ), Hol(Dfm ), Lip(gm ) and Hol(Dgm ) gives (48)-(51) to complete the proof of Lemma 4.6.  4.2. Global Extension Via Bump Functions. Let (E, | · |) be a C 1,α Banach space. Thus, there is a C 1,α real valued function λ defined on E and a real number c ∈ (0, 1) such that λ(E) = [0, 1], λ(x) = 1 for | x | ≤ c, λ(x) = 0 for | x | ≥ 1, Lip(λ, E) = sup | Dλ(x) | < ∞, x∈E

and Hol(Dλ, E) = sup x=y∈E

| Dλ(x) − Dλ(y) | < ∞. α |x−y |

230

SHELDON E. NEWHOUSE

We call λ a unit bump function on E. Given such a bump function λ and a positive real number δ, we define the associated δ − scaled version of λ to be x (56) λδ (x) = λ( ). δ Obviously, the function λδ vanishes off Bδ and is 1 on Bcδ . The Lipschitz and H¨ older constants of λδ are easily computed as follows. We have x 1 Dλδ (x) = Dλ( ), δ δ and | Dλδ (x) − Dλδ (y) | = | 1δ (Dλ( xδ ) − Dλ( yδ )) | 1 α Hol(Dλ, E)| xδ − yδ | ≤ δ 1 = Hol(Dλ, E)| x − y |α 1+α δ for any x = y ∈ E. Thus, (57)

Lip(λδ ) =

Lip(λ) Hol(Dλ) and Hol(Dλδ ) = δ δ 1+α

Recall that, for a C 1,α function f : U → E, we have defined Lip(f, U ) = sup | Df (x) | x∈U

and Hol(Df, U ) = sup x=y∈U

| Df (x) − Df (y) | . α |x−y |

For functions g defined on all of E, we leave out the domain E and simply write Lip(g) = Lip(g, E) and Hol(Dg) = Hol(Dg, E). Lemma 4.7. Let U be an open neighborhood of 0 in the C 1,α Banach space E and let f : U → E be a C 1,α map such that f (0) = 0 and both Lip(f, U ) and Hol(Df, U ) are finite. Let λ : E → R be the unit bump function defined above, and define the functions C1 (λ) and C2 (λ) by C1 (λ) = 1 + Lip(λ) and C2 (λ) = 1 + 2Lip(λ) + 3Hol(Dλ). For any δ > 0 be such that Bδ (0) ∈ U , consider the function g defined by the product g(x) = λδ (x)f (x) where λδ is the δ−scaled version of λ, as defined in (56).

ON A DIFFERENTIABLE LINEARIZATION THEOREM

231

Then, for c as in the definition of λ(·), the function g is defined and C 1,α on all of E and satisfies the following. (58)

g(x) = f (x) for | x | ≤ c δ,

(59)

g(x) = 0 for | x | ≥ δ,

(60)

Lip(g) ≤ C1 (λ)Lip(f, U )

and Hol(Dg) ≤ C2 (λ)Hol(Df, U )

(61)

Proof. Statements (58) and (59) are obvious. For statement (60), using | x | ≤ δ and f (0) = 0, we have | D(λδ (x)f (x)) | = | Dλδ (x)f (x) + λδ (x)Df (x) | Lip(λ) ≤ | f (x) | + | λδ (x) || Df (x) | δ Lip(λ) Lip(f, U )| x | + Lip(f, U ) ≤ δ ≤ Lip(λ)Lip(f, U ) + Lip(f, U ) = Lip(f, U )(Lip(λ) + 1) as needed. We now proceed to verify (61). For ease of notation, let us denote λδ (x) as γ(x). It suffices to prove that, for x = y ∈ U , α

| D(γf )(x) − D(γf )(y) | ≤ C2 (λ)Hol(Df, U )| x − y | .

(62) We have

| D(γf )(x)−D(γf )y | = | Dγ(x)f (x)+γ(x)Df (x)−(Dγ(y)f (y)+γ(y)Df (y)) | ≤ | Dγ(x)f (x)−Dγ(y)f (y) |+| γ(x)Df (x)−γ(y)Df (y) | Let us estimate the two expressions R1 = | Dγ(x)f (x) − Dγ(y)f (y) | and R2 = | γ(x)Df (x) − γ(y)Df (y) | separately. Let H0 (Df ) represent the H¨ older constant of Df at 0.

232

SHELDON E. NEWHOUSE

We have R1

= | Dγ(x)f (x) − Dγ(y)f (y) | ≤ | Dγ(x)f (x) − Dγ(y)f (x) | + | Dγ(y)f (x) − Dγ(y)f (y) | ≤ Hol(Dγ)| x − y |α H0 (Df )| x |1+α +H0 (Dγ)| y |α sup | Df ((1 − τ )x + τ y) || x − y | 0≤τ ≤1 Hol(Dλ) ≤ H0 (Df )| x − y |α δ 1+α δ 1+α Hol(Dλ) + | δ |α H0 (Df ) max(| x |, | y |)α | x − y | δ 1+α Hol(Dλ) α 1−α α ≤ Hol(Dλ)H0 (Df )| x − y | + |x−y | H0 (Df )δ α | x − y | δ Hol(Dλ) α α H0 (Df )δ α (2δ)1−α | x − y | ≤ Hol(Dλ)H0 (Df )| x − y | + δ α α ≤ Hol(Dλ)H0 (Df )| x − y | + Hol(Dλ)H0 (Df )2| x − y | α ≤ 3Hol(Dλ)H0 (Df )| x − y | α ≤ 3Hol(Dλ)Hol(Df, U )| x − y |

and R2

= ≤ ≤ ≤ ≤ ≤ ≤

| γ(x)Df (x) − γ(y)Df (y) | | γ(x)Df (x) − γ(y)Df (x) | + | γ(y)Df (x) − γ(y)Df (y) | Lip(γ)| x − y || Df (x) | + | Df (x) − Df (y) | Lip(λ) | x − y |α | x − y |1−α Hol(Df, U )| x |α + Hol(Df, U )| x − y |α δ Lip(λ) α α | x−y | 21−α max(| x |, | y |)1−α Hol(Df, U )δ α +Hol(Df, U )| x−y | δ Lip(λ) α α | x − y | 2δ 1−α Hol(Df, U )δ α + Hol(Df, U )| x − y | δ α (2Lip(λ) + 1)Hol(Df, U )| x − y |

Hence, | D(γf )(x) − D(γf )(y) | = R1 + R2 α ≤ (3Hol(Dλ) + 2Lip(λ) + 1)Hol(Df, U )| x − y | α ≤ C2 (λ)Hol(Df, U )| x − y | This completes the proof of Lemma 4.7. 4.3. Convention when using direct sum decompositions. When the Banach space E is written as a direct sum decomposition E u ⊕E s , it is often convenient to identify T : E → E with the map T˜ from E u × E s to E u × E s defined by taking the unique representations of z = x+y, T (z) = x1 +y1 , with x, x1 ∈ E u , y, y1 ∈ E s , and defining T˜ (x, y) = (x1 , y1 ). The map T˜ is the conjugate RT R−1 where R(z) = (x, y). Thus, T˜ is simply the map obtained using R as a linear change of coordinates.

ON A DIFFERENTIABLE LINEARIZATION THEOREM

233

Some statements have a more elegant formulation using T , but their proofs are best given in terms of the map T˜. We call the map T˜ the product representation of T . Letting π u : E u ⊕E s → E u and π s : E u ⊕ E s → E s denote the natural projections π u (z) = x, π u (z) = y, and writing (π u ◦ T )(x, y) = f1 (x, y), and (π s ◦ T )(x, y) = f2 (x, y), we will identify T with the map T˜ and simply say we may write T as T (x, y) = (f1 (x, y), f2 (x, y)). Similarly, the origin will be written as 0 or (0, 0), and balls centered at 0 in E will be written as Bδ = Bδ (0) or Bδ (0, 0). 4.4. Flattening and Linearizing on Invariant Manifolds. Let E be a real Banach space. For a positive real number δ we use the notation Bδ = Bδ (0) for the open ball of radius δ centered at 0; i.e., the set of points x ∈ E such that | x | < δ. We will say that a property holds near 0 if it holds in Bδ (0) for some small δ > 0. In this section, all neighborhoods of 0 in E will be assumed to be open and convex. If U is such a neighborhood and f : U → E, then the Mean Value Theorem can be applied to show that Lip(f, U ) = sup | Dfx |, x∈U

and we will often use this fact. Given a neighborhood U of 0 in E and an injective map T : U → E, define % (63) WUs = T −n (U ) n≥0

and (64)

WUu =

%

T n (U ).

n≥0

These sets are called the U -stable and U -unstable sets of T , respectively. From the definitions, it is immediate that WUs (T ) = WUu (T −1 ), WUu (T ) = WUs (T −1 ) T (WUs (T )) ⊂ WUs (T ), and T −1 (WUu (T )) ⊂ WUu (T ). When U is a ball Bδ (0), we write WUs (T ) = Wδs (0, T ), WUu (T ) = Wδu (0, T ). Let r ≥ 1 be a real number. A local C r diffeomorphism at 0 is a pair (T, U ) such that U is an open neighborhood of 0 in E and T is a C r diffeomorphism from U onto its image such that T (0) = 0. Note that T (U ) is an open set by the Inverse Function Theorem. As usual, we often ignore the domain of the local diffeomorphism, and simply say that T is a local diffeomorphism at 0. When considering compositions and inverses of local C r diffeomorphisms, we simply shrink domains as needed to make the definitions correct. When U = E, we sometimes say that T is a global

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SHELDON E. NEWHOUSE

diffeomorphism to emphasize that we are considering a bijection from E onto E. Since we only consider diffeomorphisms with a fixed point at 0 in this and the next two sections, we sometimes drop the at 0 and simply use the terms local diffeomorphism or global diffeomorphism. Two local C r diffeomorphisms T and S are C r conjugate if there is a local C r diffeomorphism R such that RT R−1 = S near 0. If S happens to be linear, then, setting L = DT0 , M = DR0 , and P = M −1 R, we have P T P −1 = M −1 SM, DP (0) = I, and D(P T P −1 ) = L. Hence, T is C r conjugate to some linear map at 0 if and only if it is C r conjugate to its derivative at 0, and it may be assumed that the conjugacy R has the properties that R(0) = 0 and DR0 = I. To emphasize this concept, we will say that T is strongly C r conjugate to S if it is C r conjugate to S and DS(0) = DT (0). We call the local C r diffeomorphism T hyperbolic if 0 is a hyperbolic fixed point of T . Let (T, U ) be a C r local hyperbolic diffeomorphism, let L = DT (0), and let E = E u ⊕ E s be the associated L−invariant splitting; i.e., L | E u is expanding and L | E s is contracting. We will say that T is flat in U if % % (65) T (E u U ) ⊂ E u and T (E s U ) ⊂ E s . We will say that T is locally flat at (or near) 0 if there is some open neighborhood U of 0 such that T is flat in U . It is well-known (see [8], [16]) that, for δ > 0 small, the sets Wδs (0, T ) and u Wδ (0, T ) are C r embedded submanifolds of E which are tangent at 0 to E s and E u , respectively. The following lemma is well-known. Lemma 4.8. Every hyperbolic local C r diffeomorphism (T, U ) is strongly C r conjugate to a locally flat one.  More precisely, one can find a ball Bδ (0) ⊂ U T (U ) and a C r diffeomorphism R from Bδ (0) onto its image such that R(0) = 0, DR(0) = I, and the map T1 = RT R−1 is flat on R(T −1 Bδ (0)). We call an R as in Lemma 4.8, a flattening map (or diffeomorphism) for T . Proof. We begin by identifying E with E u × E s and take the product representation (66)

T (x, y) = (Ax + X(x, y), By + Y (x, y))

where A ∈ Aut(E ) is expanding, B ∈ Aut(E s ) is contracting, and X : E u × E s → E u and Y : E u × E s → E s are C r maps which vanish at (0, 0) and have their partial derivatives also vanishing at (0, 0). Letting π u : E u × E s → E u and π s : E u × E s → E s be the natural projections, and writing Eδu = π u (Bδ (0, 0)) and Eδs = π s (Bδ (0, 0)), it is proved in [8] and [16]) that, for small δ, there are C r functions gu : Eδu → E s and gs : Eδs → E u such that u

ON A DIFFERENTIABLE LINEARIZATION THEOREM

(67)

gu (0) = (0), Dgu (0) = 0,

(68)

gs (0) = (0), Dgs (0) = 0,

235

and Wδu (0, T ) and Wδs (0, T ) are the graphs of gu and gs , respectively. Setting R(x, y) = (x − gs (y), y − gu (x)), it is readily verified that, for δ > 0 small enough, the map R is C r diffeomorphism from Bδ onto its image, and, hence, is a local flattening map for T . Remark 4.9. In section 6, we consider continuous dependence of our linearizations on parameters. It will be necessary to have the maps gs , gu depend continuously on the parameters as well. The most elegant proof of this result is given in [8] where the Irwin method for proving the existence of invariant manifolds is generalized and simplified. Our discussion so far shows that, in moving toward the proof of Theorem 1.5, we may assume the T is locally flat after a C 1,α coordinate change. It will be convenient to have a stronger condition which requires another definition. Definition 4.10. Let U be an open neighborhood of 0 in the Banach space E, and let T : U → V be a C 1,α diffeomorphism from U onto its image with a hyperbolic fixed point at 0. Let L = DT (0), and let E = E u ⊕ E s be the hyperbolic splitting associated to L. We say that T is hyperbolically linear (or h-linear ) in U if   % U. (69) T (x) = Lx for x ∈ E u E s Observe that if T is h−linear on U and U is small enough, then it is locally flat m in U T −1 is h-linear on T (U ). Also, m forj positive integers m, T is h-linear on m , and −j −m U , and T is h-linear on j=0 T U , j=0 T In general, a C r local diffeomorphism with a hyperbolic fixed point at 0 will not be conjugate to even a C 1 h−linear one. However, the following lemma shows that a C 1,α local diffeomorphism with an α−hyperbolic fixed point at 0 is, in fact, strongly C 1,α conjugate to an h-linear one.. As in section 3, given an open neighborhood U of 0, consider the space CU1,α of C 1,α maps f : U → E such that f (0) = 0 and Df (0) = 0 with finite D-H¨ older norm | f |U = Hol(Df, U ) < ∞. Lemma 4.11. For 0 < α < 1, let E be a real Banach space, and let L ∈ Aut(E) be an α−hyperbolic linear automorphism with hyperbolic splitting E = E u ⊕ E s . Let U be an open neighborhood of 0 in E, let f ∈ CU1,α , and let T = L + f . Then, for any > 0, there are neighborhoods V and W of 0 in E and a C 1,α diffeomorphism R from V onto W such that  % % (70) V W ⊂ B (0) ⊂ U T (U ) T −1 (U ), (71)

R(0) = 0, DR(0) = I,

and the following conditions are satisfied. Letting T1 = RT R−1 and f1 = T1 − L,

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SHELDON E. NEWHOUSE

and defining   % % V1 = R T (V ) V T −1 (V ) , we have (72)

T1 is defined and h-linear on V1

(73)

Lip(f1 , V1 ) < , and

(74)

Hol(Df1 , V1 ) < ∞. def

Further, the map T1 is a C 1,α diffeomorphism from V1 onto its image T (V1 ) = W1 , and, setting g1 = T1−1 − L−1 , we have (75)

T1−1 is defined and h-linear on W1

(76)

Lip(g1 , W1 ) < ,

and (77)

Hol(Dg1 , W1 ) < ∞.

Proof. Taking > 0 sufficiently small, we may assume that T is locally flat in the ball B = B (0). We use the product representation for T and write T (x, y) = (Ax + X(x, y), By + Y (x, y)). with A α−expanding, B α−contracting, and X, Y C 1,α maps vanishing at (0, 0) together with their partial derivatives. In these coordinates, of course, L(x, y) = (Ax, By). It suffices to find a small neighborhoods V, W of (0, 0) contained in B (0, 0) and a local C 1,α map R from V onto W such that (78)

R(0, 0) = (0, 0), DR(0, 0) = I,

and, setting T1 = RT R−1 , we have (79)

T1 (x, 0) = (Ax, 0) and T1 (0, y) = (0, By)

for (x, y) ∈ W . Indeed, it is clear that, for V and W small enough, the conditions (73), (74), (76), and (77) will all be satisfied. Let us proceed to find the small neighborhoods V, W and the appropriate map R. Let E u = π u (B ) and E s = π s (B ), and consider the maps T u : E u → E u and s T : E s → E s defined by T u (x) = Ax + X(x, 0), T s (y) = By + Y (0, y). Clearly, T u has 0 as an α−expanding fixed point, and T s has 0 as an α−contracting fixed point.

ON A DIFFERENTIABLE LINEARIZATION THEOREM

237

By Theorem 3.1, and the remark following it, there are small neighborhoods V u of 0 in E u , V s of 0 in E s , and C 1,α diffeomorphisms R1 : V u → R1 (V u ) and R2 : V s → R2 (V s ) such that R1 (0) = 0, DR1 (0) = I, R2 (0) = 0, DR2 (0) = I, R1 T u R1−1 = A on R1 (V u ), and

R2 T s R2−1 = B on R2 (V s ). Now, let V = V u × V s , W = R1 (V u ) × R2 (V s ), and let R : V → R(V ) be the product map R(x, y) = (R1 (x), R2 (y)). The map R satisfies (78), and the map T1 = RT R−1 satisfies T1 (x, 0) = R−1 T R(x, 0) = (Ax, 0) and

T1 (0, y) = R−1 T R(0, y) = (0, By). This completes the proof of Lemma 4.11.  We will need the analog of Lemma 4.11 for powers of T .

Lemma 4.12. Let > 0, n be a positive integer, and let R, T1 , f1 , g1 , V1 , W1 be as in the statement of Lemma 4.11. Then, there are neighborhoods Vn ⊂ V1 and Wn ⊂ W1 such that, T1n is an h-linear diffeomorphism from Vn onto Wn , and, defining fn = T1n − Ln and gn = T1−n − L−n , we have (80)

Lip(fn , Vn ) < ,

(81)

Hol(Dfn , Vn ) < ∞,

(82)

Lip(gn , Wn ) < , and

(83)

Hol(Dgn , Wn ) < ∞.

Proof. As usual, writing Bδ = Bδ (0) for any positive δ, first pick δ ∈ (0, ) so that n % % T1k (V1 W1 ). Bδ ⊂ k=−n

Next, choose Vn = Bδn (0) where δn is small enough so that T1j (Vn ) ⊂ Bδ for | j | ≤ n. It is then clear that, setting Wn = T1n (Vn ), we have Vn ⊂ V1 , Wn ⊂ W1 , that n T1 is an h-linear diffeomorphism taking Vn onto Wn . The only thing remaining is to get the Lipschitz estimates (80) and (82). But, an easy induction shows that DT n (0) = Ln , which implies that Dfn (0) = 0 = Dgn (0).

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Then, it is clear that we can shrink δn enough so that (80) and (82) hold. In fact, using (43), (48), and (50), one can get explicit estimates of δn in terms of Lip(f1 ).  4.5. Extension of local h-linear diffeomorphisms to global diffeomorphisms. Let T1 be the diffeomorphism obtained in Lemma 4.11. We now wish to use Lemma 4.7 to find an h-linear C 1,α diffeomorphism S defined on all of E which agrees with T1 on a small neighborhood of 0 and is linear off a slightly larger ball Bδ where δ is small relative to Hol(DS, E). Let C01,α denote the space of C 1,α maps from E into E such that f (0) = 0 and Df (0) = 0 with the norm

(84)

Hol(Df ) = sup x=y

| Dfx − Dfy | < ∞. | x − y |α

Recall that Aut(E) is the set of linear automorphisms of E. Let D01,α denote the set of maps T = L + f with L ∈ Aut(E) and f ∈ C01,α such that (85)

Lip(f, E) ≤

1 . 2| L−1 |

From the Inverse Function Theorem, one sees that the maps T ∈ D01,α are bijective C 1,α maps with C 1,α inverses such that T (0) = 0 and DT (0) = L. By a C 1,α diffeomorphism of E, we will mean an element of D01,α . More general diffeomorphisms can, of course, be defined, but we don’t need them in this paper. Definition 4.13. Given a Banach space E, let T = L+f where L is a bounded linear map, and let f ∈ C01,α (E). We define the non-linear support of T to be the set of points x ∈ E such that T (x) = L(x). We denote this by nsupp(T ). The non-linear size of T , denoted Ns (T ), is defined by (86)

Ns (T ) = inf{ξ ∈ R : nsupp(T ) ⊂ Bξ (0)}.

It is clear that Ns (T ) = 0 if and only if T = L and that, in general, Ns (T ) can range from 0 to ∞. We will be interested in h-linear C 1,α maps T for which Ns (T ) is small compared to Hol(DT, E). Now, let C1 (λ) < C2 (λ) be the constants given in Lemma 4.7, and consider the map T1 = L + f1 defined on the neighborhood W1 obtained in Lemma 4.11. For δ > 0, let (87)

f2 = λδ f1 , T2 = L + f2 where λδ is the δ-scaled version of the unit bump function λ defined in (56).

ON A DIFFERENTIABLE LINEARIZATION THEOREM

239

Let (88)

0<
0, we may choose δ > 0 small enough so that Bδ (0) ⊂ W1 and Lip(f2 , Bδ (0)) ≤ δ α Hol(Df2 , Bδ (0)) < δ α C2 (λ)Hol(Df1 , Bδ (0)) < .

(89)

From (88) and (89) it is easily seen that T2 is a C 1,α global diffeomorphism on E. Let Bδc = E \ Bδ denote the set-theoretic complement of Bδ in E. Since f1 vanishes in %  Bδ (0, 0) [(E u × {0}) ({0} × E s )], and λδ vanishes in Bδc , we have that f2 vanishes on (E u × {0})

  ({0} × E s ) Bδc .

Thus, T2 is a C 1,α h-linear global diffeomorphism from E onto E with nonlinear support in Bδ . Next, let m be an integer greater than 1, and consider the powers T2m and T2−m . Let m %

+ Bm =

T2−k Bδc ,

k=0

and − Bm

=

m %

T2k Bδc ,

k=0 + In view of (52), we see that for each 1 ≤ j ≤ m − 1, and x ∈ Bm , if fj (x) = 0, then fj+1 (x) = 0 as well. In particular, + fm (x) = 0 for x ∈ Bm .

(90)

A similar argument shows that − , gm (x) = 0 for x ∈ Bm

(91) which yields

+ c ) = nsupp(T m ) ⊂ (Bm

(92)

m 

T −k Bδ ,

k=0

and

(93)

− c nsupp(T −m ) ⊂ (Bm ) =

m  k=0

T k Bδ .

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SHELDON E. NEWHOUSE

Now, defining δm by δm = δ max(Lip(T )m , Lip(T −1 )m ),

(94)

we see that (92) and (93) give nsupp(T m )

(95)



nsupp(T −m ) ⊂ Bδm .

Obviously, for any > 0, we may choose δ small enough so that δm ∈ (0, ). With the above definitions, let us summarize the results obtained above in the following proposition. Proposition 4.14. Let 0 < α < 1, let E be a C 1,α Banach space, and let L be an α−hyperbolic linear automorphism. Let U be an open neighborhood of 0, let f : U → E be a function in CU1,α , and let T = L + f . Then, for any > 0 and any non-zero integer n, there are a δn = δn (L, f, ) < and a global h-linear C 1,α diffeomorphism S ∈ D01,α which is strongly C 1,α conjugate to T on Bδn , and setting fn = S n − Ln , we have Lip(fn , E) <

(96) and

nsupp(S n ) ⊂ Bδn .

(97)

We will need the following simple consequence of Proposition 4.14. Lemma 4.15. Let S, n, δn , fn be as in Proposition 4.14, and, using the product representation E = E u × E s , write fn (x, y) = (Xn (x, y), Yn (x, y)). Then, for every (x, y) ∈ E u × E s we have (98)

α

max(| Xn,x (x, y) |, | Yn,x (x, y) |) ≤ Hol(Dfn ) min(δnα , | y | )

and (99)

α

max(| Xn,y (x, y) |, | Yn,y (x, y) |) ≤ Hol(Dfn ) min(δnα , | x | ).

Proof. We will only give the arguments for Xn since those needed for Yn are similar. Because we are using the maximum norm on E u × E s , we have Hol(Dfn ) = max(Hol(DXn ), Hol(DYn )). Since Xn (x, 0) = 0 for each (x, 0) we also have that Xn,x (x, 0) = 0 for each (x, 0). Similarly, Xn (0, y) = 0 implies that Xn,y (0, y) = 0 for each (y, 0). Hence, (100)

| Xn,x (x, y) | = | Xn,x (x, y) − Xn,x (0, 0) | ≤ Hol(DXn )| (x, y) |α ,

(101)

| Xn,y (x, y) | = | Xn,y (x, y) − Xn,y (0, 0) | ≤ Hol(DXn )| (x, y) | ,

(102)

| Xn,x (x, y) | = | Xn,x (x, y) − Xn,x (x, 0) | ≤ Hol(DXn )| y |α

α

and (103)

| Xn,y (x, y) | = | Xn,y (x, y) − Xn,y (0, y) | ≤ Hol(DXn )| x |α

ON A DIFFERENTIABLE LINEARIZATION THEOREM

241

The non-linear support condition (97) for Xn implies that Xn (x, y) and its partial derivatives vanish at points (x, y) such that | (x, y) | > δn . So, (100) and (101) imply (104)

max(| Xn,x (x, y) |, | Xn,y (x, y) |) ≤ Hol(DXn )δnα

This, together with (102), (103) and the fact that Hol(DXn ) ≤ Hol(Dfn ), gives formulas (98) and (99).  4.6. Norm Conditions induced by α-hyperbolicity. Assuming L is αhyperbolic, and using the product representation E = E u × E s , the map L has the form L(x, y) = (Ax, By). with (105)

max(ρ(A−1 ), ρ(B)) < 1.

By Lemma 3.4, we may assume the maps A and B satisfy (106)

| A−1 | < 1 and | B | < 1.

Next, using (6) and formula (3), we choose a positive integer m such that 1

(| A−m || Am || B m | ) m < 1, α

and

−m α 1 |B || B m || A−m | m < 1. Taking the m-th power, these imply that

(107)

| A−m || Am || B m |α < 1

and (108)

α

| B −m || B m || A−m | < 1.

Following this, we choose > 0 small enough so that (109)

min(m(Am ) − , m(B −m ) − ) > 1,

(110)

max(| A−1 | + , | B | + ) < 1,

(111)

| A−m |(| Am | + )(| B m | + )α < 1,

and (112)

| B m |(| B −m | + )(| A−m | + )α < 1.

The estimates (109) and (110) of course imply that (113)

m(A) − > 1

and (114)

| B m | + < 1.

Finally, we choose 0 < η < 1 close enough to 1 so that (115)

| A−m |(| Am | + )(| B m | + )αη (| Am | + )α(1−η) < 1,

and (116)

| B m |(| B −m | + )(| A−m | + )αη (| B −m | + )α(1−η) < 1.

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SHELDON E. NEWHOUSE

5. Proof of Theorem 1.5 Let E, T, L be as in the hypotheses of Theorem 1.5, and let E = E u ⊕ E s be the hyperbolic splitting associated to L. Applying Proposition 4.14 for n = 1 and arbitrary > 0, there are a δ = δ1 ∈ (0, ) and an h-linear map S ∈ D01,α of the form S = L + f1 with f1 ∈ C01,α such that S is strongly C 1,α conjugate to T on Bδ with nsupp(S) ⊂ Bδ . In subsection 5.1 below, we will show that for satisfying conditions (109)(112) and δ > 0 sufficiently small, the map S is globally strongly C 1 conjugate to L. That is, there is a C 1 diffeomorphism R mapping E onto itself such that R(0) = 0, DR0 = I, and RSR−1 = L. Then, in subsection 5.2, we will show that, adding conditions (115) and (116) and shrinking δ further, the map R is C 1,β on bounded subsets of the orbit of Bδ (0). Once these things are done, the restriction to a small neighborhood of 0 of the composition of R with the local C 1,α conjugacy from S to T provides a local C 1,β conjugacy from T to L and completes the proof of Theorem 1.5. 5.1. Global C 1 linearization of the map S. To C 1 linearize S = L + f1 on E, we will employ the method described in section 2. The map R will have the form R = I + φ, where (117)

φ = (I − H)−1 L−1 f1

where H is an automorphism on a suitable Banach space of functions E = E s ⊕ E u which we now define. Given a Banach space Z and a C 1 function ψ : E u ⊕ E s → Z, consider the following two real-valued functions. (118)

γ 1 (ψ, x, y, α) =

| ψx (x, y) | α |y|

(119)

γ 2 (ψ, x, y, α) =

| ψy (x, y) | α |x|

Each of the preceding functions is defined to have value zero if its denominator vanishes. Otherwise, it is given by the indicated expression. Consider the corresponding suprema: | γ i (ψ, α) | =

sup γ i (ψ, x, y, α) for i = 1, 2,

x=0,y=0

and let C01 (E, Z) be the set of C 1 functions from ψ : E u × E s to Z such that (120)

ψ(x, 0) = ψ(0, y) = 0 ∀ (x, y),

(121)

ψx (0, 0) = ψy (0, 0) = 0,

and (122)

| γ i (ψ) | = | γ i (ψ, α) | < ∞ for i = 1, 2.

ON A DIFFERENTIABLE LINEARIZATION THEOREM

243

For such ψ, set (123)

|| ψ || = || ψ ||α = max | γ i (ψ, α) |. i=1,2

C01 (E, Z)

is a linear space with the usual pointOne can check that the set wise operations of addition and scalar multiplication, and that the function || ψ || provides a norm on it so that the pair (C01 (E, Z), || · ||) becomes a Banach space. Setting Z to be E u and E s , in turn, gives the two Banach spaces E s = 1 C0 (E, E u ) and E u = C01 (E, E s ). For φ = (φs , φu ) ∈ E = E s ⊕ E u , we define the norm || φ || = || (φs , φu ) || = max(|| φs ||, || φu ||). For φ = (φs , φu ) ∈ E s ⊕ E u , the map H(φ)(x, y) = L−1 φ(S(x, y)) is expressed as H(φ)(x, y) = (Hs (φs )(x, y), Hu (φu )(x, y)) where (124)

Hs (φs )(x, y) = A−1 φs (Ax + X1 (x, y), By + Y1 (x, y))

and (125)

Hu (φu )(x, y) = B −1 φu (Ax + X1 (x, y), By + Y1 (x, y)).

Since S is h-linear on E and the composition of C 1,α maps is again C 1,α , we see that, at least as an operator on the function space E s × E u without consideration of norms, H is represented as the direct sum of operators H = Hs ⊕ Hu . Proposition 5.1. Let E, T, L, f be as in the statement of Theorem 1.5, and assume that L(x, y) = (Ax, By) and m > 0, are such that ( 109)-( 114) are valid. There is a δ ∈ (0, ) such that if S as in Proposition 4.14, then, the associated maps Hs and Hu satisfy (126)

Hs ∈ Aut(E s ) and | Hsm | < 1

and (127)

Hu ∈ Aut(E u ) and | Hu−m | < 1.

Hence, ρ(Hs ) < 1 and ρ(Hu−1 ) < 1, so H is hyperbolic. Assuming Proposition 5.1, we use Lemma 3.4 to find norms | · |u and | · |s on E u and E s , respectively so that (128)

| Hs | < 1

and (129)

| Hu−1 | < 1.

Letting Is : E s → E s , Iu : E u → E u denote the identity maps on E s , E u , respectively, we then have  (Is − Hs )−1 = Hsn n≥0

244

SHELDON E. NEWHOUSE

and (Iu − Hu )−1

= Hu−1 Hu (Iu − Hu )−1 = Hu−1 (Hu−1 )−1 (Iu − Hu )−1 = Hu−1 ((Iu − Hu )Hu−1 )−1 = Hu−1 (Hu−1 − Iu )−1 = −Hu−1 (Iu − Hu−1 )  = −Hu−1 Hu−n = −



n≥0

Hu−n

n≥1

giving (130)

(I − H)−1 = (Is − Hs )−1 ⊕ (Iu − Hu )−1 = (



Hsn , −

n≥0



Hu−n ).

n≥1 1

Then, formula (12) in Section 2 shows that a (global) C linearization of S is given by R = I + ψ where (131)

ψ = (I − H)−1 L−1 f = (ψs , ψu )

with (132)

ψs = (Is − Hs )−1 A−1 X1 =



Hsn (A−1 X1 )

n≥0

and (133)

ψu = (Iu − Hu )−1 B −1 Y1 = −



Hu−n (B −1 Y1 ).

n≥1

Let us proceed to the proof of Proposition 5.1. We first show that the statement (126) is implied by assumptions (110), (111), and (113). Following this and using the expression (134)

Hu−n (φu ) = B n φu (A−n x + X−n (x, y), B −n y + Y−n (x, y)),

it is easy to see that the same method works for (127) by replacing S, A, B by S −1 , B −1 , A−1 , respectively. using the assumptions (110), (112), and (113). Thus, it suffices to prove (126). In the sequel, all real numbers δn will be assumed to be in (0, ) where satisfies (106)-(112). Also, once a condition is specified by a choice of δn , it will continue to hold for smaller δn , so this will be assumed without further mention. Moving to (126), consider the following real numbers Kn,1 = | A−n |(| An | + )(| B n | + )α , Kn,2 = | A−n |(| An | + )α δnα Hol(Dfn ), Kn,3 = | A−n |(| B n | + )α δnα Hol(Dfn ), Kn,4 = | A−n |(| An | + )(| B n | + ), and τ n = max(Kn,1 + Kn,2 , Kn,3 + Kn,4 ).

ON A DIFFERENTIABLE LINEARIZATION THEOREM

245

By (110), we see that | B n | + < 1, so Kn,4 < Kn,1 . Letting n = m with m as in (111) and (112), we have Km,4 < Km,1 < 1. So, we may choose δm small enough so that τ m < 1. Further, for n = 1 or n = m, and δn small enough, we can guarantee that (135)

max(Lip(X1 ), Lip(Y1 )) < ,

(136)

max(Lip(Xm ), Lip(Ym )) < ,

and (137)

max(Ns (Xm ), Ns (Ym )) < δ α Hol(Dfm ).

In addition, since we are using the δ-scaled version of the bump function λ for S, we have that the functions X1 , Y1 , Xm , Ym all vanish on

  (138) E u × {0} {0} × E s Bδc . Now, the main step in the proof of (126) is Proposition 5.2. Assume that conditions ( 109)-( 112) on are satisfied. Then, for any ψ ∈ E s , and n = 1 or n = m, we have (139)

γ 1 (Hsn (ψ)) ≤ (Kn,1 + Kn,2 )|| ψ ||

and (140)

γ 2 (Hsn (ψ)) ≤ (Kn,3 + Kn,4 )|| ψ ||.

Since || ψ || = max(γ 1 (ψ), γ 2 (ψ)), conditions (139) and (140) imply that, for n = 1 or n = m, we have || Hsn (ψ) || < τ n || ψ ||. Thus, for n = 1 and n = m, the maps Hsn are bounded linear maps and | Hsm | < 1. They are also bijective with inverses Hs−n (ψ) = An ψ(S −n ). By the open mapping theorem, the maps Hsn are automorphisms (i.e., have bounded inverses) and this proves (126). Let us proceed to the proof of Propostion 5.2; i.e., to the proofs of (139) and (140). Define the functions un (x, y), vn (x, y) by (141)

un = un (x, y) = An x + Xn (x, y),

and (142)

vn = vn (x, y) = B n y + Yn (x, y).

Lemma 5.3. For every (x, y) ∈ E u × E s , we have (143)

| un (x, y) | ≤ (| An | + )| x |,

(144)

| vn (x, y) | ≤ (| B n | + )| y |,

(145)

| un,x | ≤ | An | +

246

SHELDON E. NEWHOUSE

(146)

| un,y | ≤ Hol(Dfn ) min(δnα , | x |α ),

(147)

| vn,x | ≤ Hol(Dfn ) min(δnα , | y | ),

α

and | vn,y | ≤ | B n | + .

(148)

Proof. The inequalities (145) and (148) follow from (136), and inequalities (146) and (147) are a restatement of (98) and (99). From (136), the Mean Value Theorem, and the vanishing of Xn , Yn on  E u × {0} {0} × E s , we get | un (x, y) | = ≤ ≤

An x + Xn (x, y) | An || x | + | Xn (x, y) − Xn (0, y) |   n | A || x | + sup | Xn,x (sx, y) | | x |

≤

| An || x | + | x |,

0 0 small enough, the map R is C 1,β on bounded subsets of O(δ). To prove Proposition 5.4, we first show that (154) holds. Then, as in the proof of (127), we obtain (155) with the same method by replacing S, A, B with S −1 , B −1 , A−1 , respectively. Let us use ψ | Z to denote the restriction of the function ψ to a set Z. Our proof of (132) implied that the series ψs =



Hsn (A−1 X1 )

n≥0 1

converges uniformly in the C topology, but, of course, it gives no information about D-H¨ older convergence. However, observe that the restriction A−1 X1 | O+ (δ) 1,α is C on O+ (δ) since it vanishes off Bδ (0). Moreover,  ψs | O+ (δ) = Hsn (A−1 X1 | O+ (δ)). n≥0

Of course, we also have that A and 0 < β < α.

−1

X1 | O+ (δ) is C 1,β on O+ (δ) since 0 < δ < 1

ON A DIFFERENTIABLE LINEARIZATION THEOREM

249

We proceed to introduce a Banach space E + = E + (δ) of C 1,β functions on O+ (δ) with the property that if δ > 0 is sufficiently small, then function Hs (ψ) = A−1 ψ ◦S is a well-defined bounded linear self-map of E + satisfying (168) below. It will follow from this that the series  Hsn (A−1 X1 | O+ (δ)) n≥0

converges uniformly in E . This will show that ψs |O+ (δ) is C 1,β , thus proving (154). +

Let us proceed to define the space E + . For (x, y), (h, k) ∈ E u × E s , let (156)

 = max(| x |, | h |), Mx,h

(157)

Ny,k = max(| y |, | k |),

(158)

 Mx,h = min(1, Mx,y ),

and, let α(1−η)

(159)

αη U (x, h, k) = Mx,h | (h, k) |

(160)

αη V (y, h, k) = Ny,k | (h, k) |

α(1−η)

.

Given a C 1 function ψ : O+ (δ) → E u , and points (x, y) ∈ O+ (δ), (h, k) ∈ E × E s , with (x + h, y + k) ∈ O+ (δ) and | (h, k) | > 0, consider the real-valued functions | ψx (x + h, y + k) − ψx (x, y) | (161) γ 3 (ψ, x, y, h, k) = V (y, h, k) u

and (162)

γ 4 (ψ, x, y, h, k) =

| ψy (x + h, y + k) − ψy (x, y) | U (x, h, k)

and their suprema (163)

| γ 3 (ψ) | =

sup

γ 3 (ψ, x, y, h, k).

0 1, then | ΔXn | ≤ .

 But, if Mx,h > 1, then, since | (x, y) | < δ < 12 , we have

| (x + h, y + k) | ≥ | x + h | > | h | − δ > 1 − δ > δ,

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SHELDON E. NEWHOUSE

giving Xn (x + h, y + k) = 0, and | ΔXn | = | Xn (x + h, y + k) − Xn (x, y) | = | Xn (x, y) | ≤ 1 | (x, y) | ≤ 1 δ ≤ . Moving to (177), we proceed as follows. Frst, since Hol(DXn ) ≤ Hol(Dfn ), we have α

| ΔXn,x | ≤ Hol(Dfn )| (h, k) | .

(182)

Next, using (98) at the points (x + h, y) and (x, y), we have | Xn,x (x + h, y) − Xn,x (x, y) | ≤ 2Hol(Dfn )| y |α

(183) for each x, y, h. Hence,

| ΔXn,x | = | Xn,x (x + h, y + k) − Xn,x (x, y) | ≤ | Xn,x (x + h, y + k) − Xn,x (x + h, y) |

(184)

+| Xn,x (x + h, y) − Xn,x (x, y) | α α ≤ Hol(Dfn )| k | + 2Hol(Dfn )| y | α ≤ 3Hol(Dfn )Ny,k

Putting this together with (182) gives | ΔXn,x | = | ΔXn,x |η+1−η   α (1−η) α η ≤ 3Hol(Dfn )Ny,k [Hol(Dfn )| (h, k) | ]

α η (1−η) ≤ Hol(Dfn ) 3Ny,k (| (h, k) |α ) αη = 3η Hol(Dfn )Ny,k | (h, k) |α(1−η)

(185)

≤ 3Hol(Dfn ))V (y, h, k),

and this proves (177). The proof of (178) is similar to that for (177), except we have to use Mx,h =  min(1, Mx,h ) instead of Ny,k . Working with Xn,y we have (186)

| ΔXn,y | ≤ Hol(Dfn )| (h, k) |α , | Xn,y (x, y + k) − Xn,y (x, y) | ≤ 2Hol(Dfn )| x |α ,

and | ΔXn,y | = | Xn,y (x + h, y + k) − Xn,y (x, y) | ≤ | Xn,y (x + h, y + k) − Xn,y (x, y + k) | +| Xn,y (x, y + k) − Xn,y (x, y) | ≤ Hol(Dfn )| h |α + 2Hol(Dfn )| x |α  ≤ 3Hol(Dfn )(Mx,h )α .  As in the proof of (181), if Mx,h > 1, then | Xn,y (x + h, y + k) | = 0, Mx,h = 1,

and | ΔXn,y | = ≤ ≤

| Xn,y (x, y) | ≤ Hol(Dfn )δ α 3Hol(Dfn ) α 3Hol(Dfn )Mx,h .

ON A DIFFERENTIABLE LINEARIZATION THEOREM

253

Now, replacing Ny,k by Mx,h in the technique used to get (185), we obtain the statement involving | ΔXn,y | in (178), and this completes the proof of Lemma 5.7. Now, set (187)

˜ n,1 = | A−n |(| An | + )(| B n | + )αη (| An | + )α(1−η) , K

(188)

˜ n,2 = | A−n |(| An | + )δ α(1−η) Hol(Dfn ), K

(189)

˜ n,3 = | A−n |(| B n | + )αη (| An | + )α(1−η) Hol(Dfn )δ α(1−η) , K ˜ n,4 = | A−n |(| An | + )(| B n | + ), K

(190) and

˜ n,1 + K ˜ n,2 , K ˜ n,3 + K ˜ n,4 ). τ˜n = max(K

(191)

Observe that we can choose η close enough to 1 so that ˜ m,4 < K ˜ m,1 < 1. K Then, we can choose δ small enough, depending on 1 − η, such that ˜ m,1 + K ˜ m,2 < 1, (192) K and ˜ m,4 < 1. ˜ m,3 + K K

(193) This gives

τ˜m < 1.

(194)

Next, we define some new functions and make some more estimates. Write u = un = An x + Xn (x, y), v = vn = B n y + Yn (x, y), u1,n = An (x + h) + Xn (x + h, y + k), and v1,n = B n (y + k) + Yn (x + h, y + k) so that T n (x, y) = (un , vn ) and T n (x + h, y + k) = (u1,n , v1,n ). Then, setting Δu = u1,n − un = An h + ΔXn and Δv = v1,n − vn = B n k + ΔYn , we have, for ψ ∈ E + , (195) Hsn (ψ)(x + h, y + k) − Hsn (ψ)(x, y) = A−n ψ(u + Δu, v + Δv) − A−n ψ(u, v). The proof of Lemma 5.6 involves estimating the x and y partial derivatives of the left side of (195) in terms of those of ψ and will be concluded in the inequalities (207) and (208) below. First, we have the estimates | Δu |

= | An h + ΔXn | ≤ (| An || h | + min(Mx,h , Ny,k , | (h, k) |)  ≤ (| An || h | + min(Mx,h , Ny,k , | (h, k) |)  ≤ (| An | + ) min(Mx,h , | (h, k) |),

254

SHELDON E. NEWHOUSE

and | Δv | = | B n k + ΔYn | ≤ | B n || k | + min(Mx,h , Ny,k , | (h, k) |) ≤ (| B n | + ) min(Ny,k , | (h, k) |). Putting these together with  | u | = | An x + Xn (x, y) | ≤ (| An | + 1 )| x | ≤ (| An | + 1 )Mx,h

and | v | = | B n y + Yn (x, y) | ≤ (| B n | + 1 )| y | ≤ (| B n | + 1 )Ny,k , we have (196)

  n Mu, Δu ≤ (| A | + )Mx,h ,

(197)

Nv,Δv ≤ (| B n | + )Ny,k ,

and | (Δu, Δv) | ≤ (| An | + )| (h, k) |.

(198)

From (196), we have (199)

  n min(1, Mu, Δu ) ≤ min(1, (| A | + )Mx,h ).

Observe that, if a, b, c are non-negative real numbers and c ≥ 1, then min(a, bc) ≤ c · min(a, b).

(200)

 and c = | An | + , we get Applying this with a = 1, b = Mx,h   n min(1, Mu, Δu ) ≤ (| A | + ) min(1, Mx,h ),

which, by definition, is (201)

Mu,Δu ≤ (| An | + )Mx,h .

Consider the compositions of U (x, h, k) and V (y, h, k) with u, v, Δu, Δv: (202) (203)

αη U1 = U1 (x, h, k) = U (u, Δu, Δv) = Mu, Δu | (Δu, Δv) |

α(1−η)

,

α(1−η) αη V1 = V1 (y, h, k) = V (v, Δu, Δv) = Nv, . Δv | (Δu, Δv) |

Using (| An | + ) > 1, we have (204)

U1 ≤ (| An | + )αη (| An | + )α(1−η) U = (| An | + )α )U ≤ (| An | + )U

and (205)

V1 ≤ (| B n | + )αη (| An | + )α(1−η) V.

Let us observe that, with 0 < δ ≤ 12 , we have (206)

max(Mx,h , Ny,k ) ≤ 1.

Indeed, the definition of Mx,h makes it no larger than 1. If Ny,k > 1, then, since both (x, y) and (x + h, y + k) are in O+ (δ), we would get that | y | < δ ≤ 12 , which, in turn, would imply that Ny,k = | k | > 1. This would give δ > | y + k | ≥ | k | − | y | > 1 − δ, contradicting the condition that δ ≤ 12 .

ON A DIFFERENTIABLE LINEARIZATION THEOREM

255

Now, we have vx = Yn,x and uy = Xn,y . Since max(| Xn,y |, | Yn,x |) = 0 when | (x, y) | > δ, we have that U | vx | > 0 or V | uy | > 0 ⇒ | (x, y) | ≤ δ. This, together with (206), gives | A−n |U1 | vx | = | A−n |U1 | Yn,x | ≤ | A−n |(| An | + )U | Yn,x | αη ≤ | A−n |(| An | + )Mx,h | (h, k) |

α(1−η)

α

Hol(Dfn )| y |

≤ | A−n |(| An | + )Hol(Dfn )| y | | (h, k) | α

α(1−η)

= | A−n |(| An | + )Hol(Dfn )| y |αη | y |α(1−η) | (h, k) |α(1−η) ≤ | A−n |(| An | + )Hol(Dfn )| y | ˜ n,2 | y |αη | (h, k) |α(1−η) = K

αη α(1−η)

δ

α(1−η)

| (h, k) |

˜ n,2 N αη | (h, k) |α(1−η) ≤ K y,k ˜ n,2 V. ≤ K In addition, V | uy | = V | Xn,y | α(1−η)

αη ≤ Ny,k | (h, k) |

α

Hol(Dfn )| x |

α

α(1−η)

≤ Hol(Dfn )| x | | (h, k) |

= Hol(Dfn )| x |αη | x |α(1−η) | (h, k) |α(1−η) α(1−η)

αη ≤ Hol(Dfn )δ α(1−η) Mx,h | (h, k) |

= Hol(Dfn )δ α(1−η) U which implies ˜ n,3 U. | A−n |V1 | uy | ≤ | A−n |(| B n | + )αη (| An | + )α(1−η) V | uy | ≤ K We now are in a position to prove that | γ 3 (Hsn (ψ)) | ≤ τ˜n || ψ ||

(207) and

| γ 4 (Hsn (ψ)) | ≤ τ˜n || ψ ||.

(208)

These inequalities will imply that (209)

|| Hsn (ψ) || = max (γ 3 (Hsn (ψ)), γ 4 (Hsn (ψ))) ≤ τ˜n || ψ ||.

As we mentioned above, applying this, respectively, for n = 1 and n = m, will prove Lemma 5.6, and, hence, complete the proof of Theorem 1.5. Let us proceed to the proofs of (207) and (208). For (x, y) ∈ O+ (δ), (h, k) ∈ E u × E s such that (x + h, y + k) ∈ O+ (δ) and | (h, k) | > 0, let ΔHsn (ψ)x

= ∂x (Hsn (ψ))(x + h, y + k) − ∂x (Hsn (ψ))(x, y) = ∂x (Hsn (ψ)(x + h, y + k) − Hsn (ψ)(x, y))

256

SHELDON E. NEWHOUSE

and ΔHsn (ψ)y

= ∂y (Hsn (ψ))(x + h, y + k) − ∂y (Hsn (ψ))(x, y) = ∂y (Hsn (ψ)(x + h, y + k) − Hsn (ψ)(x, y)).

Then, from (195), we have | ΔHsn (ψ)x | ≤ ≤ ≤ ≤

| A−n || ∂x (ψ(u + Δu, v + Δv) − ψ(u, v)) | | A−n || ∂u (ψ(u + Δu, v + Δv) − ψ(u, v)) || ux | +| A−n || ∂v (ψ(u + Δu, v + Δv) − ψ(u, v)) || vx | | A−n |γ 3 (ψ)V1 | ux | + | A−n |γ 4 (ψ)U1 | vx | ˜ n,2 V | A−n |γ 3 (ψ)V1 (| An | + ) + γ 4 (ψ)K

≤

| A−n |(| An | + )γ 3 (ψ)(| B n | + )αη (| An | + )α(1−η) V ˜ n,2 V +γ 4 (ψ)K

≤ ≤

˜ n,1 γ 3 (ψ)V + K ˜ n,2 γ 4 (ψ)V K τ˜n || ψ ||V.

Dividing by V and taking the supremum as (x, y) and (x + h, y + k) vary gives (207). Similarly, | ΔH n (ψ)y | ≤ | A−n || ∂y (ψ(u + Δu, v + Δv) − ψ(u, v)) | ≤ | A−n || ∂u (ψ(u + Δu, v + Δv) − ψ(u, v)) || uy | +| A−n || ∂v (ψ(u + Δu, v + Δv) − ψ(u, v)) || vy | ≤ | A−n |γ 3 (ψ)V1 | uy | + | A−n |γ 4 (ψ)U1 | vy | ˜ n,3 γ 3 (ψ)U + | A−n |γ 4 (ψ)U1 | vy | ≤ K ˜ n,3 γ 3 (ψ)U + | A−n |γ 4 (ψ)(| An | + )U (| B n | + ) ≤ K ˜ n,3 γ 3 (ψ)U + K ˜ n,4 γ 4 (ψ)U ≤ K ≤ τ˜n || ψ ||U Dividing by U and, again, taking the supremum as (x, y) and (x + h, y + k) vary gives (208). Observe that, in the above estimates, it may be assumed that | (Δu, Δv) | > 0. Otherwise, the given inequalities would trivially be satisfied, since ΔH n (ψ)x and ΔH n (ψ)y would vanish. 6. Continuous Dependence on Parameters In this section, we formulate a version of Theorem 1.5 for continuous families of maps. The proofs use more or less standard methods and will only be sketched. Let Λ be a topological space, and let X be a complete metric space. Given a map Φ : Λ × X → X and a point λ ∈ Λ, define the λ-section map Φλ : X → X to be the map given by Φλ (x) = Φ(λ, x) for x ∈ X. A map Φ : Λ × X → X is called a uniform contraction map on Λ × X if it is continuous and there is a constant 0 < μ < 1 such that each λ-section Φλ is a contraction map with Lipschitz constant less than μ.

ON A DIFFERENTIABLE LINEARIZATION THEOREM

257

The following theorem is well-known and easy to prove. Theorem 6.1. Let Φ : Λ × X → X be a uniform contraction map on Λ × X. For each λ ∈ Λ, let pλ be the unique fixed point of the λ-section map Φλ . Then, the map λ → pλ is continuous. For 0 < α < 1, and a C 1 function f : U → E, define   | Df (x) − Df (y) | | f |1,α = sup max | f (x) |, | Df (x) |, , α |x−y | x=y,x,y∈U and let C 1,α (U, E) be the set of C 1 functions from U into E such that | f |1,α < ∞. Clearly, the quantity | f |1,α defines a norm in C 1,α (U, E) making it into a real Banach space. We now state the parametrized version of Theorem 1.5 Theorem 6.2. Let 0 < α < 1, let E be a C 1,α Banach space, p ∈ E, and let U be an open neighborhood of p in E. Let Λ be a topological space, let λ0 be a point in Λ, and let F : Λ × C 1,α (U, E) be a continuous map such that p = pλ0 is an α−hyperbolic fixed point of Fλ0 . Then, there are neighborhoods L of λ0 in Λ and V ⊂ U of p in E and a real number β ∈ (0, 1) such that for each λ ∈ L, the map Fλ has a unique α−hyperbolic fixed point pλ in V and there is a C 1,β diffeomorphism Rλ from V onto a neighborhood of 0 in E such that % (210) DFλ (pλ )(Rλ (x)) = Rλ (Fλ (x)) for x ∈ V F −1 V. λ Moreover, the map λ → (pλ , Rλ ) is a continuous map from L into the product space V × C 1,β (V, E). Sketch of Proof. Replacing Fλ0 by Fλ0 (x + p) − p, we may assume that p = 0, and U is an open connected neighborhood of 0. Letting L = DFλ0 (0), we write Fλ0 (x) = Lx + fλ0 (x) with fλ0 (0) = 0 and Dfλ0 (0) = 0. The definitions and statements below require that λ be close to λ0 in Λ, and we assume this without further mention. Let fλ be the C 1,α map defined by fλ = Fλ − L. Since the operator L is hyperbolic, the operator I − L has a bounded inverse. The equation for a fixed point x of Fλ has the following forms

(211)

x = Lx + fλ (x), (I − L)x = fλ (x), or

x = (I − L)−1 fλ (x). ¯δ denote the closed ball of radius δ about 0 in E. Let B Let Gλ = (I − L)−1 fλ .

258

SHELDON E. NEWHOUSE

¯δ ⊂ U , and Given 1 ∈ (0, 1), choose δ > 0 such that, B | DGλ0 | = | (I − L)−1 Dfλ0 (x) | < 1 , ¯δ . for x ∈ B Next, let δ1 = (1 − 1 )δ and let L be a neighborhood of λ0 in Λ so that, for ¯δ , we have λ ∈ L and x ∈ B (212)

| Gλ (0) | = | (I − L)−1 fλ (0) | < δ1 ,

and (213)

| DGλ (x) | = | (I − L)−1 Dfλ (x) | < 1 .

From (213), we see that, for λ ∈ L, ¯δ ) ≤ 1 . Lip(Gλ , B ¯δ , we have Now, for each λ ∈ L and each x ∈ B

(214)

| Gλ (x) | = | Gλ (x) − Gλ (0) + Gλ (0) | ≤ 1 | x | + δ1 ≤ 1 δ + (1 − 1 )δ ≤ δ, This and (214) show that the map (λ, x) → Gλ (x) is a uniform contraction ¯δ . map on L × B ¯δ (0) which depends continHence, there is a unique fixed point pλ of Gλ in B uously on λ ∈ L. Since the set Hypα (E, E) of α−hyperbolic automorphisms of E is an open subset of the set L(E, E) of bounded linear maps on E, we may assume that the point pλ is an α−hyperbolic fixed point of Fλ . Conjugating Fλ with the translations x → x + pλ we may assume that Fλ (0) = 0 for each λ near λ0 . u Letting Lλ be the derivative of Fλ at 0, it can be shown that the subspace Eλ u u s is the graph of a bounded linear map Pλ : Eλ → Eλ of small norm. The map 0 0 λ → Pλu is continuous with the obvious topologies. A similar statement holds for s Eλ . Next, choose coordinates so that the hyperbolic splitting u s × Eλ Eλ 0 0 coincides with the coordinate subspaces

(215)

E u × {0} × {0} × E s .

For λ near 0, there is a λ−dependent family of linear coordinate changes {Aλ } preserves the splitting (215) for each λ. near the identity, such that Aλ Lλ A−1 λ Replacing Lλ with Aλ Lλ A−1 , we may assume that Lλ preserves this splitting. λ Now, all of the constructions in the proof of Theorem 1.5 vary continuously with λ. Here we use the Irwin method as in the paper of de la Llave and Wayne [8] to get the local stable and unstable manifolds via the Implicit Function Theorem. Thus, these constructions vary continuously with λ. This leads to continuously varying linearization operators λ → Hλ which are contractions on appropriate function spaces as in the proofs of (126), (127), (154), and (155).

ON A DIFFERENTIABLE LINEARIZATION THEOREM

259

This gives a family Rλ of C 1,β linearizations at p(λ) of Tλ depending continuously on λ as required for Theorem 6.2. 7. Vector Fields and Motivation Our main motivation for the results presented here is concerned with the study of vector fields. For simplicity, we consider the finite dimensional case. In this section r will denote a real number larger than 1. Let M be a C r+1 finite dimensional real manifold, and let X be a C r vector field defined in an open subset U ⊂ M . Let p be a critical point of X; i.e., X(p) = 0. Assume that the local flow φ(t, x) associated to X is defined on the product (−2, 2) × U , and let T (x) = φ(1, x) be the time one map of X. We choose an open subset U1 of U so that T and T −1 are defined and C r on U1 . For α ∈ (0, 1), and each of the properties α−contracting, α−expanding, α−hyperbolic, bi-circular, etc., we define p to have the corresponding property if it has that property as a fixed point of T . For instance, if M is the Euclidean space RN , with N a positive integer, then, using that exp(DXp ) = DTp , it can be seen that p is bi-circular if the set consisting of the real parts of the eigenvalues of DXp is a two element set {a, b} with a < 0 < b. A well-known technique of Sternberg [38], page 817, implies that a C r vector field has a local C k (here r, k ≥ 1) linearization near a critical point p if and only if its time-one map T has such a linearization at p. As a simple application of Theorem 1.5, let us consider the Shilnikov threedimensional saddle focus theorem as in [33]. One has a vector field X in R3 with a saddle type critical point at 0 such that DX0 has a pair of complex conjugate eigenvalues a ± ib with a < 0, b = 0 and a real positive eigenvalue c > 0 such that a + c > 0. It is also assumed that there is a homoclinic orbit (an orbit which is forward and backward asymptotic to 0). Shilnikov then showed that there are infinitely many saddle periodic orbits near the homoclinic orbit. In fact, the geometry shows that horseshoe type dynamics (including symbolic systems) appear. There is a geometric description of this in section 6.5 in [12]. The treatment assumes that the flow is linear in a neighborhood of the critical point. It is not difficult to see that a C 1 linearization would suffice, so our Theorem 1.5 in the bi-circular case can be applied to show that Shilnikov’s Theorem holds even if the vector field is only C 1,α . More recently, higher dimensional systems with homoclinic orbits involving saddle foci have been studied by Ovsyannikov and Shilnikov in [23] and [24]. For related material, the reader is encouraged to look at the recent books [34] and [35] which contain a beautiful and fairly complete treatment of much of the work carried out by the so-called Shilnikov School in city of Nizhny-Novogorod (formerly Gorky). See [32] for a detailed description of the work of Shilnikov and his many students and collaborators. Other treatments which include closely related work are in the books of Ilyashenko and Li [18] and Guckenheimer and Holmes [12]. Many of the proofs in works dealing with homoclinic orbits in high dimension (e.g., greater than four) are complicated. We expect that simplifications can be obtained using C 1,β (or even C 1 ) linearizations on Lyapunov center manifolds as follows.

260

SHELDON E. NEWHOUSE

Consider a C r vector field X on the Euclidean space RN with a hyperbolic critical point of saddle type at 0, with r > 1. Thus, the real parts of the eigenvalues of DX0 are non-zero and intersect both the positive and negative sets of reals. Let L = DX0 , and let am > am−1 > . . . > a1 > 0 > b1 > b2 > . . . > bn denote the distinct real parts of the eigenvalues of L. We express RN in the direct sum decomposition R N = E1 ⊕ E2 ⊕ E3 ⊕ E4 such that L(Ei ) = Ei for i = 1, 2, 3, 4, and, writing Li for the restriction L | Ei , we have (1) (2) (3) (4)

the the the the

eigenvalues eigenvalues eigenvalues eigenvalues

of of of of

L1 L2 L3 L4

have have have have

real real real real

parts parts parts parts

equal to a1 , equal to b1 , greater than a1 , and less than b1 .

Following terminology in [34] we call the eigenvalues with real parts a1 or b1 the leading eigenvalues and the corresponding subspaces E1 , E2 the leading eigenspaces. The numbers a1 and b1 are the Lyapunov exponents of L1 and L2 , respectively. That is, for each v ∈ E1 \ {0} and w ∈ E2 \ {0}, we have 1 log | etL v | = a1 t→±∞ t

χ(v) = lim and χ(w) = lim

t→±∞

1 log | etL w | = b1 . t

Accordingly, we will call the direct sum E1 ⊕E2 the Lyapunov Center Subspace, and we denote it by E cc . The theory of invariant manifolds, e.g. as in [8], [17], shows that there are submanifolds W cc , invariant by the local flow of X, tangent at 0 to the subspace E cc . We will call these Lyapunov Center Manifolds. They are not unique, but each will be C 1,α near 0 (i.e., the transition maps on local coordinate charts are C 1,α ) provided that (216)

(1 + α)a1 < a2 and (1 + α)b1 > b2 .

The derivative DX0 , restricted to E1 × E2 is bi-circular, so by Theorem 1.5 and the Sternberg technique mentioned above, the flow of X restricted to each such W cc is C 1,β linearizable at 0. Moreover, by Theorem 6.2, the linearizations can be chosen to depend continuously on external parameters. Note that even for linear differential equations, most of the manifolds W cc may not be smoother than C 1,α for some 0 < α < 1. For instance, consider the three dimensional linear system x˙ = x, y˙ = (1 + α)y, z˙ = −z for 0 < α < 1. Each surface 1+α

y = C| x |

for some non-zero constant C can be taken as one of the manifolds W cc .

ON A DIFFERENTIABLE LINEARIZATION THEOREM

261

References [1] G. R. Belicki˘ı, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class (Russian), Funkcional. Anal. i Priloˇ zen. 7 (1973), no. 4, 17–28. MR0331437 [2] G. R. Belicki˘ı, A counterexample to smooth linearization (Russian), Funkcional. Anal. i Priloˇ zen. 10 (1976), no. 2, 65–66. MR0407882 [3] G. R. Belicki˘ı, Equivalence and normal forms of germs of smooth mappings (Russian), Uspekhi Mat. Nauk 33 (1978), no. 1(199), 95–155, 263. MR490708 [4] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, volume 470 of Lecture Notes in Mathematics. Springer Verlag, 1975. [5] I. U. Bronstein and A. Ya. Kopanski˘ı, Smooth invariant manifolds and normal forms, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR1337026 [6] Carmen Chicone and Richard Swanson, Linearization via the Lie derivative, Electronic Journal of Differential Equations. Monograph, vol. 02, Southwest Texas State University, San Marcos, TX, 2000. MR1804049 [7] Charles C. Pugh, On a theorem of P. Hartman, Amer. J. Math. 91 (1969), 363–367. MR0257533 [8] Rafael de la Llave and C. Eugene Wayne, On Irwin’s proof of the pseudostable manifold theorem, Math. Z. 219 (1995), no. 2, 301–321, DOI 10.1007/BF02572367. MR1337223 [9] Dennis Stowe, Linearization in two dimensions, J. Differential Equations 63 (1986), no. 2, 183–226, DOI 10.1016/0022-0396(86)90047-1. MR848267 [10] J. Dieudonn´e, Foundations of modern analysis, Academic Press, New York-London, 1969. Enlarged and corrected printing; Pure and Applied Mathematics, Vol. 10-I. MR0349288 [11] R. Fry and S. McManus, Smooth bump functions and the geometry of Banach spaces: a brief survey, Expo. Math. 20 (2002), no. 2, 143–183, DOI 10.1016/S0723-0869(02)80017-2. MR1904712 [12] John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR709768 [13] Misha Guysinsky, Boris Hasselblatt, and Victoria Rayskin, Differentiability of the HartmanGrobman linearization, Discrete Contin. Dyn. Syst. 9 (2003), no. 4, 979–984, DOI 10.3934/dcds.2003.9.979. MR1975364 [14] Hildebrando M. Rodrigues and J. Sol` a-Morales, Smooth linearization for a saddle on Banach spaces, J. Dynam. Differential Equations 16 (2004), no. 3, 767–793, DOI 10.1007/s10884-0046116-9. MR2109165 [15] Hildebrando M. Rodrigues and J. Sol` a-Morales, Known results and open problems on C 1 linearization in Banach spaces, S˜ ao Paulo J. Math. Sci. 6 (2012), no. 2, 375–384, DOI 10.11606/issn.2316-9028.v6i2p375-384. MR3135647 [16] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR0271991 [17] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR0501173 [18] Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, Mathematical Surveys and Monographs, vol. 66, American Mathematical Society, Providence, RI, 1999. MR1650842 [19] Handbook of the geometry of Banach spaces. Vol. I, North-Holland Publishing Co., Amsterdam, 2001. Edited by W. B. Johnson and J. Lindenstrauss. MR1863688 [20] M. Chaperon, Invariant manifolds revisited, Tr. Mat. Inst. Steklova 236 (2002), no. Differ. Uravn. i Din. Sist., 428–446; English transl., Proc. Steklov Inst. Math. 1 (236) (2002), 415– 433. MR1931043 [21] Marc Chaperon, Stable manifolds and the Perron-Irwin method, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1359–1394, DOI 10.1017/S0143385703000701. MR2104589 [22] X. Mora and J. Sol` a-Morales, Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations, Dynamics of infinitedimensional systems (Lisbon, 1986), NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., vol. 37, Springer, Berlin, 1987, pp. 187–210. MR921912

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Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13939

Time change invariants for measure preserving flows Marina Ratner Abstract. We introduce an entropy-type invariant preserved by time changes of measure preserving flows. This invariant is used to show that two different cartesian powers of the horocycle flow are not Kakutani equivalent.

In this note we define an entropy type invariant preserved by time changes of measure preserving flows. Our definition was motivated by J. Feldman’s r-entropy introduced in [2] and the Kakutani equivalence theory developed in [1], [3], [4] and [7]. It was also inspired by the results in [5]. Let T = {Tt } be a measure preserving flow on a probability space (X, B, μ). For x ∈ X let xt = Tt x and let It (x) denote the orbit interval [x, xt ] = {Ts x : 0 ≤ s ≤ t}, t > 1. Let P = {P1 , . . . , Pr } be a finite measurable partition of X. If x ∈ Pj then we say that Pj is the P -name of x and write P (x) = Pj . Definition 1. For x, y ∈ X, > 0, t > 1, It (x) and It (y) are called ( , P )matchable if there exist a subset A ⊂ [0, t], l(A) > (1 − )t and an increasing absolutely continuous map h from A onto A ⊂ [0, t], l(A ) > (1 − )t such that P (Ts x) = P (Th(s) y) for all s ∈ A and the derivative h (s) satisfies (1)

|h (s) − 1| < for all s ∈ A.

Here l denotes the Lebesgue measure on [0, t]. We call h an ( , P )-match for It (x) and It (y). Define ft (x, y, P ) = inf{ > 0 : It (x) and It (y) are ( , P ) − matchable} Bt (x, , P ) = {y ∈ X : ft (x, y, P ) < }. It is clear that if y, z ∈ Bt (x, , P ) then ft (z, y, P ) < 5 . We call Bt (x, , P ) the (t, P )-ball of radius > 0 centered at x ∈ X, t > 1. A family αt ( , P ) of (t, P )-balls of radius > 0 is called an ( , t, P )-cover of X if μ(∪αt ( , P )) > 1 − . Let Kt ( , P ) = inf |αt ( , P )|, where |A| denotes the number of elements in A and inf is taken over all ( , t, P )-covers of X. Let U denote the family of all positive non-decreasing functions from R+ onto itself converging to ∞, i.e. u ∈ U iff 0 < u(t) ↑ ∞ as t → ∞. 2010 Mathematics Subject Classification. Primary 37-XX. c 2017 American Mathematical Society

263

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MARINA RATNER

For u ∈ U we denote β(u, , P ) = lim inf t→∞

log Kt ( , P ) . u(t)

It is clear that if 1 ≤ 2 , then β(u, 1 , P ) ≥ β(u, 2 , P ) and if P ≤ Q then β(u, , P ) ≤ β(u, , Q). Here the notation P ≤ Q means that every member of P is the union of some members of Q up to μ-measure zero. We define e(u, P ) = lim sup β(u, , P ) →0

e(T, u) = sup e(u, P ). P

We call e(T, u) the u-entropy of T = {Tt }. Depending on u ∈ U , e(T, u) can be zero, a positive number, or ∞. We prove the following. Theorem 1. Let {Tt } be an ergodic m.p. flow on (X, B, μ) and let P1 ≤ P2 ≤ . . . be an increasing sequence of finite measurable partitions of X such that ∨∞ n=1 Pn generates the σ-algebra B (we say that {Pi : i = 1, 2, . . . } generates B). Then e(T, u) = supm e(u, Pm ) for all u ∈ U .  Now let τ be a positive integrable function on X, X τ dμ = τ . We say that a flow {St } is obtained from {Tt } by the time change τ if St (x) = Tv(x,t) (x) for μ-almost every (a.e.) x ∈ X and all t ∈ R, where v(x, t) is defined by - v(x,t) (2) τ (Tu x)du = t. 0

The flow {St } preserves the probability measure ν defined by dν = (τ /τ )dμ. We prove the following. Theorem 2. Let S = {St } be obtained from an ergodic m.p. flow T = {Tt } by a time change τ such that τ (Tt x) is continuous in t for a.e. x ∈ X and L < τ (x) < M for some L, M > 0 and a.e. x ∈ X. Then e(S, u) = e(T, u) for all u ∈ U with limt→∞ u(at)/u(t) = 1 for all a > 0. Corollary 1. Let T = {Tt } and T˜ = {T˜t } be two ergodic Kakutani equivalent ˜ μ ˜ B, m.p. flows on (X, B, μ) and (X, ˜) respectively. Then e(T, u) = e(T˜, u) for all u ∈ U with lim u(at)/u(t) = 1 for all a > 0.

t→∞

The significance of e(T, u) is suggested by the following theorem. (n)

Theorem 3. Let h(n) = {ht = ht × · · · × ht }, n = 1, 2, . . . be the n-times Cartesian product of the horocycle flow on the unit tangent bundle of a compact surface of constant negative curvature. Let u ∈ U be u(t) = log t, t > 1. Then 3n − 3 ≤ e(h(n) , u) ≤ 3n − 2 for all n = 1, 2, . . . .

TIME CHANGE INVARIANTS FOR MEASURE PRESERVING FLOWS

265

This theorem is proved in [6, Theorem 4]. The matching maps used in the proof satisfy the conditions of Definition 1. They are constructed by intersecting twodimensional stable leaves with orbits of unstable horocycles in the -neighborhood of a point for a small > 0. These intersections induce a map h between any two segments of the horocycles in the neighborhood, so that a point x on one segment and the point h(x) on the other lie in the same stable leaf. As a map on the intervals, h is increasing and differentiable, and its derivative differs from 1 by less than a multiple of . These matching maps are applied to generating partitions consisting of sets foliated by both the horocycle segments and subsets of stable leaves of small diameter. Corollary 2. If m = n then h(n) and h(m) are not Kakutani equivalent. Proof. Let n > m. Then 3n − 3 > 3m − 2. Then by Theorem 3, one has e(h(n) , u) = e(h(m) , u) for u(t) = log t, and by Corollary 1, h(n) and h(m) are not Kakutani equivalent.  Next we shall give an equivalent definition for e(T, u), T = {Tt }.  Definition 2. For x, y ∈ X, > 0, t > 1, It (x) and It (y) are called ( , P )matchable if there exists a subset A ⊂ [0, t], l(A) > t(1 − ) and an increasing map ˜ from A onto A ⊂ [0, t], l(A ) > t(1 − ) such that P (Ts x) = P (T˜ y) for all h h(s) ˜ is measure preserving on A. s ∈ A and h Define  f˜t (x, y, P ) = inf{ > 0 : It (x) and It (y) are ( , P )-matchable}. ˜ ˜ Bt (x, , P ) = {y ∈ X : ft (x, y, P ) < }. ˜ , P ), e˜(u, P ) ˜ t ( , P ), β(u, Following the steps for the definition of e(T, u) we define K and e˜(T, u). We prove the following theorem. Theorem 4. Let T = {Tt } be ergodic. Then e˜(T, u) = e(T, u) for all u ∈ U . Finally, we mention the following Theorem proved in [6]. Theorem 5. A zero-entropy ergodic measure preserving flow T = {Tt } is loosely Bernoulli (see [1], [3], [4] and [7] for definitions) iff e(T, u) = 0 for all u ∈ U. Note. Definition 2 was first introduced in [6, Definition 1] where the require˜ being measure preserving was accidentally omitted while implicitly asment of h sumed throughout the paper. Because of this omission the proof of Theorem 3 in [6] is incomplete. Our proof of Theorem 2 here fills this gap. I am grateful to A. Katok and D. Wey for pointing out to me this omission. Proof of Theorem 1. Let > 0 and P = {P1 , . . . , Pr } be given. Let m and Q= {Q1 , . . . , Qr } be such that Q ≤ Pm and ri=1 μ(Qi !Pi ) < 0.01 . Let F = ri=1 (Pi !Qi ). Since {Tt } is ergodic, there are t0 > 0 and a set Y ⊂ X, μ(Y ) > 1 − 0.01 such that if t ≥ t0 and x ∈ Y then l(F ∩ It (x)) < 0.02 t. For t ≥ t0 let α be an (0.01 , t, Pm )-cover of X and let Z = Y ∩ ∪α, μ(Z) > 1−0.02 . Let γ = α|Z (α restricted on Z), i.e., x, y ∈ Z belong to the same element of γ iff x and y belong to the same ball of α. Clearly, |γ| ≤ |α|.

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If x and y belong to the same element of γ then ft (x, y, Pm ) < 0.05 . Let h be an (0.05 , Pm )-match between It (x) and It (y), and let A ⊂ [0, t] and A ⊂ [0, t] be as in Definition 1 for h, l(A)/t > 1 − 0.05 , l(A )/t > 1 − 0.05 . We treat A and A as subsets of It (x) and It (y) respectively. Let B = A − F , B  = A − F , l(B)/t > 1 − 0.07 , l(B  )/t > 1 − 0.07 . It follows from (1) that there are D ⊂ B, D ⊂ B  , l(D)/t > 1 − , l(D )/t > 1 − such that h maps D onto D and P (Ts x) = P (Th(s) y) for all s ∈ D. This implies that h restricted to D is an ( , P )-match between It (x) and It (y). This gives ft (x, y, P ) < . Hence |γ|-many (t, P ) balls of radius cover Z. We have just shown that given > 0 and P , there exists Pm and t0 > 0 such that if t ≥ t0 then for every (0.01 , t, Pm )-cover α there is an ( , t, P )-cover γ such that |γ| ≤ |α|. This implies that if t ≥ t0 then Kt ( , P ) ≤ Kt (0.01 , Pm ) and hence β(u, , P ) ≤ β(u, 0.01 , Pm ) ≤ e(u, Pm ) ≤ sup e(u, Pm ). m

This implies e(u, P ) ≤ sup e(u, Pm ). m

So we have e(T, u) ≤ sup e(u, Pm ) ≤ e(T, u) m



by the definition of e(T, u).

Proof of Theorem 2. Let S = {St } be obtained from T = {Tt } by a time change τ , and suppose that τ (Tt x) is continuous in t for μ-a.e. x ∈ X, and 1 < τ (x) < M M for some M > 1 and all x ∈ X. Let τ dμ = a > 0.

(3)

X

We have (4)

St (x) = Tv(x,t) (x)

where v(x, t) is defined by (5)

v(x,t)

τ (Tu x)du = t. 0

The flow {St } preserves the probability measure ν on (X, B) defined by dν = (τ /a)dμ. 1 let  Now  0 < < M be given and let {Ii : i = 1, 2, . . . , p} be a1 partition  1 of M , M into disjoint intervals Ii = [ri − r, ri + r), where ri ∈ M , M and 0 < r < 0.1 /M , i = 1, 2, . . . , p. Let Ei = {x ∈ X : τ (x) ∈ Ii }, i = 1, 2, . . . , p and let E( ) = {Ei : μ(Ei ) > 0}. Now let P = {Pi : i = 1, . . . , q} be a finite partition of X and let Q( , P ) = P ∨ E( ) = {Pi ∩ Ej : Pi ∈ P, Ej ∈ E( )}.

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Let 0 < δ0 < min{ /M, a /M } (δ0 will be chosen later) be such that if 0 < δ < δ0 then (6)

μ(A) < δ implies ν(A) < 0.1 for all A ∈ B.

Let 0 < δ < δ0 be fixed. Since {Tt } is ergodic there exists v0 > 1 and Y ⊂ X, μ(Y ) > 1 − δ/2 such that if v ≥ v0 and x ∈ Y then - v      < δv. τ (T x)du − av u   0

This implies by (2) that if v(x, t) ≥ v0 and x ∈ Y then |t − av(x, t)| < δv(x, t). It follows then from (3) that if t ≥ t0 = M v0 and x ∈ Y then (7) Now let t ≥ t0 and let

|t − av(x, t)| ≤ δtM. t˜ = (t − δtM )/a.

We have (8)

2δtM 0 < v(x, t) − t˜ < a

for all x ∈ Y , t ≥ t0 . Now let α be an (0.1δ, t˜, Q( , P ))-cover of X for T = {Tt } and let Z = Y ∩ ∪α. Then μ(Z) > 1 − δ and hence ν(Z) > 1 − 0.1 by (6). Let γ = α | Z and let x, y belong to the same element of γ. Then ft˜(x, y, Q( , P )) < 0.5δ. Let h be an (0.5δ, Q( , P ))-match for [x, Tt˜x] and [y, Tt˜y], and let A ⊂ [0, t˜], A ⊂ [0, t˜] be as in Definition 1 for h. We have 

lT (A), lT (A ) > (1 − 0.5δ)t˜ where lT (A) denotes the T -length of A. The function h maps A onto A , is increasing and absolutely continuous on A, and the derivative h (s) satisfies (9)

|h (s) − 1| < 0.5δ for all s ∈ A.

Also the Q( , P )-name of Ts x is the same as the Q( , P )-name of Th(s) y, s ∈ A. It follows from (8) that [x, St x] = [x, Tv(x,t) x] ⊃ [x, Tt˜x] and [y, St y] = [y, Tv(y,t) y] ⊃ [y, Tt˜y] and δt lS [t˜, v(x, t)], lS [t˜, v(y, t)] < 2M 2 a    1 4M  + lS (A), lS (A ) > t 1 − M δ . a a We now choose 0 < δ0 < above such that if 0 < δ < δ0 then 5M 2 δ/a < . Then we have (10)

lS (A), lS (A ) > t(1 − )

where lS (A) denotes the S-length of A. For s ∈ A ⊂ [0, v(x, t)] let ρ = ρ(s) ∈ [0, t] be such that v(x, ρ) = s and let g(ρ) ∈ [0, t] be such that v(y, g(ρ)) = h(s). We have Sρ (x) = Ts x, Sg(ρ) (y) = Th(s) (y).

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Let

B = {ρ(s) ∈ [0, t] : s ∈ A} B  = {g(ρ) : ρ ∈ B} ⊂ [0, t].

The function g is increasing and maps B onto B  . It follows from (10) that lS (B), lS (B  ) > t(1 − ).

(11) We have from (2)

-

s

τ (Tu x)du = ρ -

0 h(s)

τ (Tu y)du = g(ρ). 0

Since τ (Tt x) is continuous in t, ρ is differentiable in s and ρ (s) = τ (Ts x). Also g(ρ) is differentiable in ρ at all ρ ∈ B and g  (ρ) · ρ (s) = h (s)τ (Th(s) y). This gives g  (ρ) = h (s)

(12)

τ (Th(s) y) τ (Ts x)

for all ρ ∈ B. Since Ts x and Th(s) y have the same Q( , P )-name, there is Ei (s) ∈ E( ) such that Ts x and Th(s) y both belong to Ei (s), i = 1, . . . , q. This implies by the definition of Ei (s) that ri − 0.1 /M ≤ τ (Ts x) ≤ ri + 0.1 /M ri − 0.1 /M ≤ τ (Th(s) y) ≤ ri + 0.1 /M for some ri ∈ [1/M, M ). This gives 1 − 0.3
0 for S. It follows then from (6) that γ is an ( , t, P )-cover for S. Since |γ| ≤ |α|, this implies that Kt ( , P, S) ≤ Kt˜(0.1δ, Q( , P ), T ), t˜ = (t − δtM )/a. Now let u ∈ U be such that limt→∞ u(at)/u(t) = 1. We have (13)

log Kt˜(0.1δ, Q( , P ), T ) log Kt ( , P, S) ≤ u(t) u(t˜a)

since u is non-decreasing, and (13) implies that β( , P, S) ≤ β(0.1δ, Q( , P ), T ) ≤ e(u, Q( , P ), T ) ≤ e(T, u) since limt→∞ u(at)/u(t) = 1. This gives e(u, P, S) ≤ e(T, u) for all P

TIME CHANGE INVARIANTS FOR MEASURE PRESERVING FLOWS

269

and hence e(S, u) ≤ e(T, u). This holds for the flow S = {St } obtained from the flow T = {Tt } by the time change τ satisfying the conditions of Theorem 2. But the flow {Tt } is obtained from the flow {St } by the time change 1/τ , also satisfying the conditions of Theorem 2. Thus e(T, u) ≤ e(S, u). 

This completes the proof of the theorem.

Proof of Corollary 1. T˜ = {T˜t } is Kakutani equivalent to T = {Tt } means that {T˜ } is isomorphic to the flow S = {St } obtained from T = {Tt } by a time change τ , satisfying the conditions of Theorem 2. By this theorem we have e(T, u) = e(T˜ , u) for every u ∈ U such that limt→∞ u(at)/u(t) = 1 for all a > 0.  Next we shall prove Theorem 4. We begin with the following Lemma. Lemma 1. Let A, B be two subsets of R of positive Lebesgue measure l such    l(A)   < − 1  l(B)  for some small 0 < < 0.01. Then there exists a subset D ⊂ A with l(D)/l(A) > 1−4 and an increasing function h on D such that h maps D to B and h is measure preserving on D. that

Proof. Let I = {I1 , . . . , Ip } and J = {J1 , . . . , Jq } be two collections of disjoint open intervals such that . / p  Ii < 0.1 2 l(A) l A! i=1

⎛ (14)

l ⎝B!

q 

⎞ IJ ⎠ < 0.1 2 l(B)

j=1

l(Ii ) = l(Jj ) = r for all i, j and some small 0 < r < 0.1 2 . Let 

A =A∩

p  i=1

We have using (14)

(15)



Ii , B = B ∩

q  j=1

 p   l ( i=1 Ii )   − 1 < 0.1 2  l(A)

   q l  J   j=1 j  − 1 < 0.1 2  l(B)   l(A ) p > 1 − 0.2 2 l ( i=1 Ii ) l(B  ) 

> 1 − 0.2 2 q l j=1 Jj

Jj .

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MARINA RATNER

Now let

9  l(Ii − A ) > I = Ii : l(Ii ) 9  l(Jj − B  )  J = Jj : > . l(Jj ) 

Then we have from (15) |I  | |J  | < , < . |I| |J| Let I˜ = I − I  = {I˜i : i = 1, 2, . . . , p˜} and J˜ = J − J  = {J˜j : j = 1, 2, . . . , q˜}. We have l(A ∩ I˜i ) > (1 − )l(I˜i ) (16) l(B  ∩ J˜j ) > (1 − )l(J˜j ) ˜ J˜j ∈ J, ˜ and for all I˜i ∈ I,

.

p˜ 

l

/ I˜i

i=1

⎛ l⎝

. > (1 − )l

⎛

J˜j ⎠ > (1 − )l ⎝

j=1

Let D = A ∩ (17)

/ Ii

i=1

⎞

q˜ 

p 

⎞

q 

Jj ⎠ .

j=1

p˜

˜ Then we have D ⊂ A and . p˜ / . p /    l(D ) > (1 − )l I˜i > (1 − 2 )l Ii > (1 − 3 )l(A). i=1 Ii .

i=1

i=1

For intervals P and Q we write P < Q if x < y for every x ∈ P , y ∈ Q. We can assume that I˜1 < I˜2 < · · · < I˜p˜ J˜1 < J˜2 < · · · < J˜q˜ . For intervals P = (a, b), Q = (c, d) with b − a = d − c > 0 define a function ϕ(P, Q; •) by ϕ(P, Q; a + x) = c + x, 0 < x < b − a. Then ϕ(P, Q; •) is increasing, measure preserving and maps P onto Q. ˜ ˜ Ii by Suppose for definiteness that p˜ ≤ q˜ and define h on pi=1 h|I˜i = ϕ(I˜i , J˜i ; •) i = 1, 2, . . . , p˜. p˜ Then h is increasing on i=1 I˜i , measure preserving and maps D onto a subset p˜ F ⊂ j=1 J˜j such that ⎞ ⎛ p˜  l(F ) = l(D ) > (1 − )l ⎝ J˜j ⎠ > (1 − 3 )l(A) j=1

by (17). Also

⎛ l ⎝B  ∩

p˜  j=1

⎞

⎛

J˜j ⎠ > (1 − )l ⎝

p˜  j=1

⎞ J˜j ⎠

TIME CHANGE INVARIANTS FOR MEASURE PRESERVING FLOWS

by (16). This gives

⎛

l(F ∩ B  ) > (1 − 2 )l ⎝

p˜ 

271

⎞ J˜j ⎠ > (1 − 4 )l(A).

j=1

Now let D = {x ∈ D : h(x) ∈ F ∩ B  } ⊂ A. Then h is measure preserving on D and l(D) > (1 − 4 )l(A). This completes the proof of the Lemma.  Lemma 2. Let x, y ∈ X and let t > 1 be large. Let P = {P1 , . . . , Pk } be a finite partition of X such that for each i = 1, . . . , k the intersections Pi ∩ It (x) and Pi ∩ It (y) are disjoint unions of open intervals. Suppose that It (x) and It (y) are  ( , P )-matchable. Then It (x) and It (y) are (5 , P )-matchable. Proof. We have Pi ∩ It (x) = ∪{Pij (x) : j = 1, 2, . . . } Pi ∩ It (y) = ∪{Pij (y) : j = 1, 2, . . . } i = 1, . . . , k where Pij (x) and Pij (y) are disjoint open intervals in It (x) and It (y) respectively (we treat It (x) and It (y) as two copies of the interval [0, t]). Let h be an ( , P )-match for It (x) an It (y), and let A ⊂ [0, t], A ⊂ [0, t], l(A) > (1 − )t, l(A ) > (1 − )t be as in Definition 1. Let Aij (x) = A ∩ Pij (x), Aij (y) = A ∩ Pij (y). Let z ∈ Ai1 ,j1 (x), w ∈ Ai2 ,j2 (x) for some i1 , i2 ∈ {1, . . . , k}, j1 , j2 ∈ {1, 2, . . . } and let h(z) ∈ Ai1 ,jp (y), h(w) ∈ Ai2 ,jr (y). Suppose z < w. Then (18)

Pi1 ,j1 (x) ≤ Pi2 ,j2 (x) and Pi1 ,jp (y) ≤ Pi2 ,jr (y)

since h is increasing. (For subsets C, D ⊂ [0, t] we write C < D if c < d for every c ∈ C, d ∈ D.) Relation (18) says that h preserves the order on the intervals. Now let us fix i ∈ {1, . . . , k} and p ∈ {1, 2, . . . } and let B  = {h(s) : s ∈ Aip (x)} ⊂ A . Then B  = ∪{B  ∩ Aij (y) : j = 1, 2, . . . }. Write B  ∩ Aij (y) = Fij (y) and let Eij (x) = {s ∈ Aip (x) : h(s) ∈ Fij (y)}. We have  Eij (x) Aip (x) = j

a disjoint union. We can assume that l(Eij (x)) > 0 for all j. It follows then from (1) that    l(Eij (x))     l(Fij (x)) − 1 <

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MARINA RATNER

for all j. By Lemma 1 there is Dij (x) ⊂ Eij (x), l(Dij (x)) > (1 − 4 )l(Eij (x)) and an increasing measure preserving ϕij on Dij (x) which maps Dij (x) onto a subset of Fij (y). Let  Dij (x). Dip (x) = j

Then l(Dip (x)) > (1 − 4 )l(Aip (x)). Define ϕip on Dip (x) by ϕip = ϕij on each Dij (x). Then ϕip is increasing and measure preserving on Dip (x). Now let A˜ = ∪{Dip (x) : i = 1, . . . , k, p = 1, 2, . . . } ⊂ A ˜ on A˜ by ˜ > (1 − 4 )l(A) > (1 − 5 )t and define h l(A) ˜ D (x) = ϕip . h| ip

˜ belong to the same interval It follows from the definition of ϕip that h(s) and h(s) ˜ is ˜ Pij (y), i = 1, . . . , k, j = 1, 2, . . . for all s ∈ A. This implies via (18) that h  ˜ is an (5 , ˜ Thus h increasing and measure preserving on A. P )-match for It (x) and  It (y). This completes the proof of the Lemma. Proof of Theorem 5. It is clear that e(T, u) ≤ e˜(T, u) for every u ∈ U . To prove the converse we shall use the fact that the flow {Tt } can be viewed as a continuous flow on a metric space X preserving a Borel probability measure μ. Then there exists an increasing sequence of finite partitions P1 ≤ P2 ≤ . . . all consisting of open sets and such that {Pm : m = 1, 2, . . . } generate the Borel σ-algebra B. Repeating the argument from the proof of Theorem 1 we have e˜(T, u) = sup e˜(u, Pm ). m

Now let x, y ∈ X and suppose that It (x) and It (y) are ( , Pm )-matchable, Pm = {P1 , . . . , Prm }. Since each Pi ∈ Pm is an open set, the intersections Pi ∩ It (x) and Pi ∩It (y) are disjoint unions of open intervals for all i = 1, . . . , rm . By Lemma 2,  Pm )-matchable. This implies that if y ∈ Bt (x, , Pm ) then It (x) and It (y) are (5 , ˜ y ∈ Bt (x, 5 , Pm ), and hence ˜ t (5 , Pm ) ≤ Kt ( , Pm ) and K ˜ 5 , Pm ) ≤ β(u, , Pm ) ≤ e(u, Pm ) ≤ e(T, u). β(u, for all m, and hence e˜(u, Pm ) ≤ e(T, u) e˜(T, u) = sup e˜(u, Pm ) ≤ e(T, u). m

This completes the proof of the Theorem.



References [1] J. Feldman, New K-automorphisms and a problem of Kakutani, Israel J. Math. 24 (1976), no. 1, 16–38. MR0409763 [2] Jacob Feldman, r-entropy, equipartition, and Ornstein’s isomorphism theorem in Rn , Israel J. Math. 36 (1980), no. 3-4, 321–345, DOI 10.1007/BF02762054. MR597458 [3] A. B. Katok, Monotone equivalence in ergodic theory (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 1, 104–157, 231. MR0442195

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[4] A. B. Katok and E. A. Sataev, Standardness of rearrangement automorphisms of segments and flows on surfaces (Russian), Mat. Zametki 20 (1976), no. 4, 479–488. MR0430210 [5] Marina Ratner, The Cartesian square of the horocycle flow is not loosely Bernoulli, Israel J. Math. 34 (1979), no. 1-2, 72–96 (1980), DOI 10.1007/BF02761825. MR571396 [6] Marina Ratner, Some invariants of Kakutani equivalence, Israel J. Math. 38 (1981), no. 3, 231–240, DOI 10.1007/BF02760808. MR605381 [7] B. Weiss, Equivalence of measure preserving transformations, Lecture Notes, Institute for Advanced Studies, Hebrew University of Jerusalem, 1976. Department of Mathematics, University of California Berkeley, Berkeley, California 94720

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13938

Spectral boundary value problems for Laplace–Beltrami operator: Moduli of continuity of eigenvalues under domain deformation A. M. Stepin and I. V. Tsylin Abstract. The paper is pertaining to the spectral theory of operators and boundary value problems for differential equations on manifolds. Eigenvalues of such problems are studied as functionals on the space of domains. Resolvent continuity of the corresponding operators is established under domain deformation and estimates of continuity moduli of their eigenvalues / eigenfunctions are obtained provided the boundary of nonperturbed domain is locally represented as a graph of some continuous function and domain deformation is measured with respect to the Hausdorff–Pompeiu metric.

1. Introduction. Let M be a smooth connected compact orientable Riemannian manifold (possibly with boundary), A — elliptic differential operator of second order on M . For ¯ ∩ ∂M = ∅, the following eigenvalue problem an open subset Ω  M , Ω (1)

◦

Au = λu, u ∈ H 1 (Ω),

is considered; here solutions are understood in the weak (variational) sense. Our aim is to estimate moduli of continuity for eigenvalues {λk } of (1) with multiplicities taken into account and considered as functions of rough domain Ω; among surveys on this and related topics we notice [5, 12, 15]. The problem can be considered in the context of spectral stability of inverse operator A−1 under small perturbations of Ω. In the framework of this approach it is sufficient to obtain estimate of convergence rate for corresponding inverse operators with respect to uniform topology (resolvent convergence [16]). A convenient instrument for dealing with the resolvent convergence is (equiv◦ alent to it) convergence in the sense of Mosco (see [17, 20]) of the spaces H 1 (Ω). In the case when M ⊂ Rd is compact, Frehse[11] obtained capacitive conditions equivalent to Mosco’s convergence. In nineties conditions imposed on variations of plane domains (w.r.t. Hausdorff metric) were investigated so that to ensure the uniform resolvent convergence (see [5]). However the obtained results claimed only the fact of convergence without quantitative estimates. 2010 Mathematics Subject Classification. Primary 35J25, 35R01, 35P05, 58J05. c 2017 American Mathematical Society

275

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Beginning with 2002 some progress was achieved in proving such estimates in case of plane domains, mainly, due to Burenkov and Davies’ approach [6], that was further developed in a series of papers by Burenkov, Lamberti, Lanza de Cristoforis (see a survey in [7]). Method of these authors made it possible to estimate the upper semicontinuity in Ω for eigenvalues of the problem (1) if variation of domains is restricted to some technical class. In the present paper resolvent continuity of the boundary value problem (Au = ◦ f , u ∈ H 1 (Ω)) with respect to domain perturbation (see th. 4.2) is established and using this fact estimates for moduli of continuity are obtained for eigenvalues and eigenfunctions1 of the problem (1) provided that 1) the boundary of nonperturbed domain can be locally represented as a graph of some continuous function and 2) the perturbation of domains is measured by Hausdorff–Pompeiu metric dHP (see sections 2.2, 3.1). To formulate our result about eigenvalues we write ∂Ω ∈ C 0,ω if there exists an atlas for the manifold M such that the intersection of ∂Ω with each of its charts either is empty or can be represented in local coordinates as the graph of a function with modulus of continuity not exceeding C · ω, where ω is a nondecreasing semiadditive function such that ω(0) = 0 and C is a positive constant. Theorem 1.1. Let A be a strongly elliptic operator with Lipschitz coefficients ¯ 1 ∩ ∂M = ∅, on the Riemannian manifold (M, g), Ω1 be a domain in M , Ω ∂Ω1 ∈ C 0,ω , where ω is some modulus of continuity. Then there exist positive constants Cn = Cn (Ω1 , A, M ), δ0 = δ0 (Ω1 , ω, M ) such that for any domain Ω2  M , satisfying condition = dHP (Ω1 , Ω2 ) ≤ δ0 the following estimate holds: |λn (Ω1 ) − λn (Ω2 )| ≤ Cn · (ω( ) + ). Besides, a generalization of Burenkov–Lamberti theorem [8] to the case of domains on manifold (cor. 4.3) is obtained. These results were announced in [19]. 2. Basic notions. Everywhere below we assume that the Riemannian manifold (M, g) is C 1,1 – smooth connected orientable compact (possibly with boundary); coordinate homeomorphisms map in Rd (d = dimM ) endowed with the standard Euclidean norm | · |:  (xi )2 , x = (x1 , . . . , xd ) ∈ Rd , ed = (0, . . . , 0, 1). |x|2 = i

2.1. Conditions imposed on operator A. Let A be a differential operation on (M, g) locally representable in the form 0

1 −√ ∂i aij det g ∂j u . det g Assume that coefficients aij define a continuous symmetric section A of T 2 M . Denote G the section of T 2 M associated with the Riemannian structure g; let bilinear forms Ax and Gx in Tx∗ M × Tx∗ M be values of the sections A and G. We assume that the following conditions are fulfilled. 1 in

the sense of generalized angle (see, for instance, [13], IV.2)

BOUNDARY VALUE PROBLEMS FOR LAPLACE–BELTRAMI OPERATOR

277

A1 There is a positive constant α such that ∀x ∈ M ∀ξ ∈ Tx∗ M ⇒ αGx (ξ, ξ) ≤ Ax (ξ, ξ); A2 A sections A belongs to the space C 0,1 (M ); a norm in C 0,1 (M ) is defined by fixing some finite subatlas {(U, κU )}U∈U   Ax (ξ, ξ)  def aij + AU max max , C 0,1 (M ) = max U ij x∈M ξ∈Tx∗ M,ξ=0 Gx (ξ, ξ) C 0,1 (κU (U)) U∈U

[v]C 0,1 (κU (U)) =

sup x,y∈κU (U),x=y

|v(x) − v(y)| , v : κU (U ) → R, |x − y|

where aij U are the coordinates of sections A with respect to norms  · U C 0,1 (M ) are equivalent as regards their dependence

mapping κU .

All the on the choice of finite subatlas. ◦ As the scalar products in H 1 (Ω) and L2 (Ω), Ω  M , we choose: ◦ G(∇u, ∇v)dμ, (u, v)L2 (Ω) = uvdμ, (u, v) H 1 (Ω) = Ω

Ω

where measure μ is associated with the Riemann metric g. Let Φ be continuous in ◦ H 1 (M ) bilinear form defined by the differential operation A , A(∇u, ∇v) dμ. Φ(u, v) = M ◦

Since the form Φ is positive and bounded in H 1 (Ω), operator A associated with ◦ Φ is uniquely defined (see, for example, [13]); a function u ∈ H 1 (Ω) is the weak solution of the boundary value problem Au = f, f ∈ L2 (Ω)

(2) ◦

if and only if ∀v ∈ H 1 (Ω) (3)

f vdμ.

Φ(u, v) = M

2.2. Domains with boundaries of the class C 0,ω . The ω–cusp condition. For a mapping ω : R 0+ → R+ such that ω(r) − ω(0) is nonnegative and semi-additive we set ψ(r) = r 2 + ω(r)2 , φ(r) = r + ω(r). Let Bρ (x) ⊂ Rd be the ball with center x and radius ρ in the norm | · |; we will often def

use the notion Br (X) = ∪x∈X Br (x) for a set X ⊂ Rd . Definition 2.1. We say that an open set Ω ⊂ Rd satisfies the uniform ω–cusp condition with parameter r at a point x if there exists ξx ∈ Rd , |ξx | = 1, such that   

W1 B3ψ(r) (x) ∩ Ω − Cω,r (ξx ) ∩ B2ψ(r) (x) ⊂ Ω; def

where Cω,r (ξx ) is obtained from Cω,r (e d ) = Sω,r (ed ) ∪ Fω,r (ed ) by a rotation su d d perposing ed on ξx where Fω,r (ed ) = z = (˜ z, zd ) ∈ R : |z| < ψ(r), z ≥ ω(r) , d d d d−1 z , z ) ∈ R : ω(|˜ z |) < z < ω(r), |˜ z | < r , z˜ ∈ R . Sω,r (ed ) = (˜ Note that the condition (W1) is equivalent to the following condition  

 

W2 B3ψ(r) (x)\Ω + Cω,r (ξx ) ∩ B2ψ(r) (x) ∩ Ω = ∅.

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A. M. STEPIN AND I. V. TSYLIN

In fact, by virtue of symmetric of (W1) and (W2) with respect to changing Ω by its complement it is sufficient to check the implication (W2) ⇒ (W1). Otherwise 

∃y ∈ B3ψ(r) (x) ∩ Ω : B2ψ(r) ∩ (y − Cω,r (ξx )) ⊂ Ω and hence there exists a point z ∈ ∂Ω ∩ (y − Cω,r (ξx )). Equivalently this means that y ∈ z + Cω,r (ξx ); by (W2) this inclusion leads to a contradiction: y ∈ M \Ω. Remark 2.2. In the definition above we do not assume that ω(0) = 0. All the propositions below are valid without this assumption if it is not stated explicitly. ¯ Ω ⊂ Rd define the norm: For a matrix A ∈ C 0,1 (Ω), : : : : def def 0 : AC 0,1 (Ω) , |A|2 = r˜(At A), ¯ = |A|2 L (Ω) + :max |∂j A|2 : : ∞ j L∞ (Ω)

def

where r˜ is the spectral radius. Additionally we denote B3ρ (y) = χ−1 y (B3ρ (χy (y)). We say that an atlas W = {(Wy , χy )}y∈M is (ρ, ϑ)–technical, ρ, ϑ > 0, if W3 ∀y ∈ M ⇒ B3ρ (y) ⊂ Wy , W4 For every chart (W, χ) ∈ W, C 0,1 -norm G ◦ χ−1 does not exceed ϑ, and L∞ -norm of |G ◦ χ−1 |2 can be estimated from below by ϑ−1 . W5 There exists a finite subatlas U of the atlas for M such that for all the transition functions from ∈ U into W ∈ W and their inverses C 0,1 -norm iU  ∂x of the Jacobi matrix ∂xi does not exceed ϑ. Definition 2.3. An open subset Ω ⊂ M satisfies the uniform ω–cusp condition with parameters (r, ϑ) if there exists (ψ(r), ϑ)–technical atlas W = {(Wy , χy )}y∈M such that for arbitrary y ∈ M the open set χy (Ω ∩ B3ψ(r) (y)) satisfies the ω–cusp ω condition with parameter r at the point χy (y). This class will be denoted Wr,ϑ . Definition 2.4. Boundary ∂Ω of a domain Ω ⊂ M is of class C 0,ω if there exists such a subatlas U = {(U, κU )} for M that nonempty κU (∂Ω ∩ U ) can be represented by a graph of continuous function gU with modulus of continuity not exceeding CU ω, CU ∈ R+ , ω(0) = 0, where the intersection of κU (Ω ∩ U ) and off-graph is empty. Domains Ω ⊂ M with boundaries locally representable as graphs of continuous functions we call domains of C–class. In the case of compact M by virtue of Cantor theorem one has: for any domain Ω ⊂ M there exists such a positive semi-additive function ωΩ , ωΩ (0) = 0, that ∂Ω ∈ C 0,ωΩ . Proposition 2.5. For arbitrary domain Ω ⊂ M with boundary of the class CωΩ C 0,ωΩ , ωΩ (0) = 0, there exists a class Wr,ϑ  Ω, where C ≡ const > 0. Proof. Since M is compact we may assume that for a fixed domain Ω ⊂ M with boundary of the class C 0,ωΩ , an atlas U (in the def 2.4) is finite, sets κU (U ) ⊂ ˜ ⊃U ¯ . We select a new atlas Rd are bounded, and the mappings κU are defined on U  1 minU dist(∂κU U, κU U  ) , {(U  , κU )} with the property U   U and set r = ψ −1 100 where dist(A, B) = inf |a − b|. a∈A,b∈B

For every y ∈ M we choose U  ∈ U, y ∈ U  , and set (Wy , χy ) = (U  , κU ). Then there exists such a number ϑ > 0, that W = {(Wy , χy )}y∈M is a (ψ(r), ϑ)–technical atlas. In fact, condition (W3) is satisfied according to construction while (W4)

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279

and (W5) are fulfilled in view of all the mappings χy are obtained as restrictions of finite number of coordinate diffeomorphisms. Condition (W2) is also valid for ω = CωΩ , C = maxU CU , since images of Ω under coordinate diffeomorphism χy are located no one side with respect to graph  of gy . Similar claim in the case of Lipschitz boundary can be found in [9]. It should be noted that ω ≡ 0 if ω(h) = o(h). Nevertheless in the case of manifolds the class C 0,0(·) can be nonempty. 3. Necessary background, constructions and estimates. 3.1. Hausdorff convergence. Let X, Y be an arbitrary subsets of a connected metric compact space (M, d). Set d(x, Y ) = inf y∈Y d(x, y) and consider the function e(X, Y ) = supx∈X d(x, Y ). If X\Y = ∅ then e(X, Y ) = sup d(x, ∂Y ) = e(X\Y, ∂Y ). x∈X\Y

Next introduce eˇ(X, Y ) = e(M \Y, M \X). In view of (M \Y )\(M \X) = X\Y and ∂X = ∂(M \X) one has (4)

eˇ(X, Y ) = sup d(y, M \X) = sup d(x, ∂X) = e(X\Y, ∂X). y∈M \Y

x∈X\Y

In terms of blowing X = {x ∈ M | d(x, X) < ε }, ε > 0, and contraction X −ε = {x ∈ X | ∀z ∈ M: d(x, z) < ε ⇒ z ∈ X } the basic functions e and eˇ can be described as follows   e(X, Y ) = inf {ε > 0 |X ⊂ Y ε } , eˇ(X, Y ) = inf ε > 0 X −ε ⊂ Y . ε

It follows that Hausdorff distance functions can be introduced by the formulas: (5)

dH (X, Y ) = max {e(X, Y ), e(Y, X)} ;

(6)

dH (X, Y ) = max {ˇ e(X, Y ), eˇ(Y, X)} .

These functions are metrics on the families of closed and open set respectively. It is useful to consider stronger version of Hausdorff metric, namely upper Hausdorff– Pompeiu distance  (7) dHP (X, Y ) = max dH (X, Y ), dH (X, Y ) . Remark 3.1. Notice fundamental difference between ( 7) and ( 6). The family of all open subsets in a fixed metric compact space K is compact w.r.t. ( 6) according to Blaschke theorem (see [14]) but w.r.t. ( 7) this family loses compactness property though completeness remains. To see the latter it is sufficient to consider subgraphs of the functions (2 + sin nx) considered on the closed interval [0, π]. For our purposes (see th. 4.2) the following minimum of all the distances of the Hausdorff type will be necessary:  dHS (X, Y ) = min e(XΔY, ∂Y ), e(XΔY, ∂X), dH (X, Y ), dH (X, Y ) Quantities e(X, Y ) and eˇ(X, Y ) give us four nonequivalent ways to measure distances, thus dHS is the weakest quantity defining convergence of sets among those that can be constructed by means of e(X, Y ), eˇ(X, Y ), e(Y, X), eˇ(Y, X).

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3.2. Estimates of distances between solutions. For a Hilbert space V and its closed subspaces V1 and V2 . Consider the problems: ui ∈ Vi , Φ(ui , vi ) = f, vi 

∀vi ∈ Vi ,

where f ∈ V  , and Φ is a bilinear continuous function on V possessing positive α, β ∈ R such that αu2V ≤ Φ(u, u) ≤ βu2V ∀u ∈ V. By Lax–Milgram lemma solutions ui = G(f ; Vi ) exist and are unique. All the statements of this subsections can be found in [18]. To formulate the following lemma and its corollary we need the standard notation dV (v, A) for the distance between v ∈ V and a subset A ⊂ V . Lemma 3.2. For solutions u1 and u2 the following inequality holds: ; β (dV (u1 , V1 ∩ V2 ) + dV (u2 , V1 ∩ V2 )) . (8) u1 − u2 V ≤ α Moreover, if V 1,2 is a closed subspace, containing V1 ∩ V2 then for u1,2 = G(f ; V 1,2 ) the inequality ;  β

dV (u1,2 , V1 ) + dV (u1,2 , V2 ) (9) u1 − u2 V ≤ α takes place. Corollary 3.3. The estimate ( 9) takes the form  β

dV (u1,2 , V1 ) + dV (u2,1 , V2 ) u1 − u2 V ≤ α 2,1 2,1 2,1 contains V1 ∪ V2 . where u = G(f ; V ) and V 3.3. Lemma about perturbation of eigenvalues. Let O  (M \∂M ) be def

open non-void and MO = M \O. The assumption Ω ⊂ MO will be assumed for every considered below. $Let p = pMO denote Friedrichs constant (i.e.  # domain  2 ) and inf p  uL2 (MO ) ≤ p u2◦ 1 H MO def def (10) u2VΩ = A(∇u, ∇u) dμ, u2LΩ = p |u|2 dμ, Ω

Ω

◦

be the norms in the spaces VΩ = H 1 (Ω) and LΩ = L2 (Ω) respectively with the standard norms in these spaces defined by the formulae: def def 2 2 (11) u ◦ 1 = G(∇u, ∇u)dμ; uL2 (Ω) = |u|2 dμ. H (Ω)

Ω

Ω

In addition, we denote V = VMO , L = LMO . Again by Lax-Milgram lemma it follows that problem (2) possesses unique def colution uf = G(f ; VΩ ) ∈ VΩ for arbitrary f ∈ V  ; thus G(f ; VΩ )V ≤ α−1 f V  ∀f ∈ V  .

(12) (1)

(1)

(2)

(2)

Let pairs (un , λn ), (un , λn ) be solutions to the eigenvalue problem (1) for (i) domains Ω1 and Ω2 respectively; enumeration {λn } in ascending order is meant. (1) Denote by PΩ1 : V → VΩ1 the orthogonal projection V onto VΩ1 and set Sn = (1) (1) span(u1 , . . . , un ).

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Lemma 3.4 (cf. [3]). Fix n ∈ N and assume there are positive numbers An and (1) Bn < p such that for every u ∈ Sn the inequalities PΩ1 u − u2V ≤ An u2L ,

(13)

PΩ1 u − u2L ≤ Bn u2L ,

hold; then An (2) √ . λ(1) n ≥ λn − √ ( p − Bn ) 2 p 3.4. Local estimate necessary for resolvent convergence. Arguments in this section are based on a generalization of the technique proposed in [18]. We start with the following Lemma 3.5 (cf. [10] p. 24, [1] p. 49). Let M be a compact manifold, d a metric associated with a Riemannian structure on M , F a family of opened balls with inf B∈F diamB > 0 and A the set of its centers. Then there exists a finite subfamily {Brj (aj )}Jj=1 ⊂ F such that A ⊂ ∪Jj=1 Brj (aj ), with {Brj /3 (aj )}Jj=1 being disjoint. Proposition 3.6. Let X, Y ⊂ MO be open, atlas W be (ρ, ϑ)–technical and v ∈ VY . Assume that for every y ∈ Λ = Y \X there exists a vector ν(y), such that x ∈ χy (Bρ (y)\X) ⇒ (x + ν(y)) ∈ / χy (Y ∩ B3ρ (y)) , and a function H ∈ L1 (M ) satisfies the inequality y − v2VBρ (y) ≤ HL1 (B3ρ (y)) , vν(y)

for every y ∈ Λ3ρ = ∪y∈Λ B3ρ (y), where vhy = v ◦ χ−1 y ◦ (x + h) ◦ χy . Then there exists a function w ∈ VX (independent on the choice of H) such that w − v2V ≤ CHL1 (Λ3ρ ) , where C = C(MO , ϑ, ρ, A) ≡ const. Proof. Let OR (y) = {z ∈ M | δ(z, y) < R} be the geodesic ball with respect to the metric associated with the Riemannian structure g. Applying lemma 3.5 to J ρ (y)} the family F = {O 2ϑ y∈Λ and the set A = Λ, we get the finite set {xj }j=1 such that ρ (x ) ∩ O ρ (x ) = ∅, i = j ⇒ O 6ϑ j i 6ϑ

B

ρ 6ϑ2

(xi ) ∩ B

ρ (x ) ⊂ ∪ Λ ⊂ ∪j∈J O 2ϑ (xj ), j j∈J B ρ 2 ρ 6ϑ2

(xj ) = ∅.

Hence (14)

J≤

μ(M ) μ(M ) ≤ , ρ (x )) ρ (z)) minj μ(O 6ϑ inf j z∈M μ(O 6ϑ

and by virtue of M compact and measure μ absolutely continuous there exists a point z ∈ M where (positive) inf in (14) is reached; thus J < C2 (M, ϑ, ρ). The required function w will be built by means of a certain partition of unity. To construct it we introduce the functions ϕj (x) = min{1, (3 − 3|χxj (x) − χxj (xj )|/ρ)+ }, ϕ0 (x) = min{1, 6 δ(x, Λ)/ρ}

282

A. M. STEPIN AND I. V. TSYLIN

and g(x, ξ) = Gx (ξ, ξ). Then one has 0 ≤ ϕj (x) ≤ 1 and supp(ϕj ) ⊂ Bρ (xj ), ϕj |B3ρ/4 (xj ) ≡ 1, g(x, ∇φj )1/2 ≤ C3 (M )ρ−1 1Bρ (xj ) (x), j ∈ Z+ ∩ [0, J], where 1A (x) = 1 is the indicator of A ⊂ M ; thus 1 ≤ ϕ function κj = J j ϕ possess the following properties: k=0

J 

J j=0

ϕj ≤ 1 + J. The

k

κj ≡ 1, 0 ≤ κj ≤ 1, g(x, ∇κj )1/2 ≤ C4 (M, ϑ, ρ)/ρ =

j=0

C2 (M, ϑ, ρ)C3 (M ) . ρ

x

J ˜j , j=0 κj v

j v˜j = 1B2ρ (xj ) vν(x , then j)  2  -  J J    xj 2   κj (v − v˜j ) dμ ≤ p |v − vν(x |2 dμ v − wL ≤ p  j) M  j=0 B (x )  ρ j j=0 J J   xj xj 2 2 = v − vν(x  ≤ v − v  ≤ C (M, ϑ, ρ) 2 ν(xj ) VBρ (x ) j ) LBρ (x )

If w =

j

j

j=0

H dμ,

Λ3ρ

j=0

Using the symbol a(x, ξ) = Ax (ξ, ξ) of operator A one finally has ⎛ ⎞ J  a ⎝x, ∇ κj (v − v˜j )⎠ dμ v − w2V = M

j=0

≤ 2C2 (M, ϑ, ρ)

J  j=0

+2

J  j=0

a(x, ∇κj )|v − v˜j |2 dμ

M

κj a(x, ∇v − ∇˜ vj )dμ

M

≤ 2C4 (M, ϑ, ρ)

J  j=0

B3ρ (xj )



C5 (A) H +H pρ2

 dμ ≤ CHL1 (Λ3ρ ) . 

ω . 3.5. Estimates in domains from Wr,ϑ

Lemma 3.7 (see [4]). If v ∈ H 1 (Rd ), x0 ∈ Rd , the for any h ∈ Rd , |h| < ρ, |v(x + h) − v(x)|2 dx ≤ |h|2 |∇v(x)|2 dx B2ρ (x0 )

B3ρ (x0 )

The following claim allows one to find a function H satisfying condition in Proposition 3.6. ω Theorem 3.8. Let Z ⊂ M , Z ∈ Wr,ϑ , f ∈ L2 (M ), u = G(f ; VZ ), y ∈ M , h = |h|ξy , |h| < ψ(r). Then for every positive ρ < ψ(r) there exists such a number ˜ O , r, ϑ, A) that C˜ = C(M   ˜ u − uyh 2VBρ (y) ≤ Cu VB3ρ (y) uVB3ρ (y) + f LZ∩B2ρ (y) · |h|.

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Proof. We need the function &  + < |x − y| ◦ χy , κy = min 1, 2 − ρ it possesses the following properties: 0 ≤ κy ≤ 1, g(x, ∇κy ) ≤ C3 (M )ρ−1 , κy |Bρ (y) ≡ 1. For v ∈ H 1 (B3ρ (y)) we introduce notation Ty,h v = (1 − κy )v + κy vhy , ω and remark that Ty,h v − v = κy (vhy − v). Since Z ∈ Wr,ϑ , one has

¯ v ∈ VZ ⇒ Ty,h v ∈ VZ ⊂ V, supp(v), supp(Ty,h v) ⊂ Z,

(15) and hence (16) uyh − u2VB

ψ(r) (y)

= Ty,h u − u2VB

ψ(r) (y)

≤ Φ(Ty,h u, Ty,h u) − Φ(u, u) + 2f, u − Ty,h u.

The last summand can be estimated by lemma 3.7: 2f, u − Ty,h u ≤ ϑ3 |h| uH 1 (B3ρ ) f L2 (Z∩B3ρ (y)) ≤ ϑ3 p−1 α−1 |h|uVB3ρ f LZ∩B3ρ (y) . To estimate the remaining term in (16) we use the formula ∇(Ty,h v) = κy ∇(vhy ) + (1 − κy )∇v + ∇κy (vhy − v) = Ty,h ∇v + ∇κy (vhy − v), and property (15) implies that Φ(Ty,h u, Ty,h u) − Φ(u, u) does not exceed (17) a(y, Ty,h ∇u + ∇κy (uyh − u))dμ − a(y, Ty,h ∇u)dμ+ Z Z (18) a(y, Ty,h ∇u)dμ − a(y, ∇u)dμ. Z

Z

Since for ξ, η ∈ R the following inequality d

a(x, ξ + η) − a(x, ξ) ≤ (a(x, η)a(x, 2ξ + η))1/2 ≤ a(x, η)1/2 (2a(x, ξ)1/2 + a(x, η)1/2 ), holds, we can apply it for ξ = Ty,h ∇u, η = ∇κy (uyh − u)), hence integrals in line (17) can be estimated as follows y a(y, Ty,h ∇u + ∇κy (uh − u))dμ − a(y, Ty,h ∇u)dμ Z Z ≤ a(y, Ty,h ∇u + ∇κy (uyh − u))dμ − a(y, Ty,h ∇u)dμ Z

≤

a (y, ∇κy (uyh

1/2



Z∩B2ρ (y)

(a(y, ∇κy (uyh

1/2

− u)) − u))) + 2 (a(y, Ty,h ∇u)) Z 1/2 =1/2 1/2 > ≤ a(y, η)dμ · a(y, η)dμ +2 a(y, ξ)dμ Z

≤

Z

C˜3 (M, ϑ, A)u−uyh LB2ρ (y)

Z

· C˜3 (M, ϑ, A)u−uyh LB2ρ (y) +2a(x, Ty,h ∇u)LB2ρ (y) .

1/2

 dμ

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A. M. STEPIN AND I. V. TSYLIN

It follows from definition of Ty,h that a(x, Ty,h ∇u)L2 (B2ρ (y)) ≤ 2C˜4 (M, ρ)uVB3ρ (y) . Therefore applying lemma 3.7 one has a(y, Ty,h ∇u + ∇κy (uyh − u))dμ − a(y, Ty,h ∇u)dμ Z Z

≤ 2C˜3 (M, ϑ, A) C˜3 (M, ϑ, A) + C˜4 (M, ρ) u2VB (y) |h| 3ρ

= C˜5 (M, ϑ, ρ, A)u2VB

3ρ

|h|. (y)

The integrals in (18) can be estimated with regards for convexity of a: a(x, Ty,h ∇v) − a(x, ∇v) ≤ (1 − κy )a(x, ∇v) + κy a(x,∇(vhy )) − a(x, ∇v) = κy [a(x, ∇vhy ) − a(x, ∇v)] , where x ∈ suppκy . Hence we get a(y, Ty,h ∇u)dμ − a(y, ∇u)dμ Z Z ≤ κy [a(x, ∇(vhy )) − a(x, ∇v)] dμ Z∩B2ρ (y)

=

-

Z∩χ−1 y (B2ρ (y)+h)

(κy )y−h a(x − h, ∇v)d(μy−h ) −

≤

Z∩B3ρ (y)

κy a(x, ∇v)dμ Z∩B2ρ (y)

(κy )y−h (a(x − h, ∇v) − κy a(x − h, ∇v))d(μy−h )

-

+ Z∩B3ρ (y)

κy a(x − h, ∇v))(dμy−h − dμ)

-

κy (a(x − h, ∇v) − a(x, ∇v))dμ

+ Z∩B2ρ (y)

≤ C˜6 (MO , ϑ, ρ, A)|h| · uVB3ρ (y) .  3.6. Properties of the sets satisfying uniform ω-cusp condition. 3.6.1. Local geometry. In the proposition of this subsection Ω ⊂ M ⊂ Rd deω ; notation Ωε see in subsection 3.1. notes a domain of the class Wr,1 Lemma 3.9. Let Ω satisfy uniform ω-cusp condition with parameter r at point x. Then for any postitve ε ≤ ρ = ψ(r) the set Ωε satisfies uniform ω-cusp condition with parameter r2 = ψ −1 (ψ(r)/2) at the point x. Proof. Choose h ∈ Cω,r (ξx ) and y ∈ B3ρ/2 (x); if y ∈ Ωε then there exists z ∈ Ω such that |z − y| < ε ≤ ρ. Since |z − x| < ρ + 32 ρ < 3ρ then from the local definition 2.1 one has z − h ∈ Ω, dist(y − h, Ω) ≤ |y − h − (z − h)| = |y − z| < ε, that is y − h ∈ Ωε . Remark 3.10. By simple geometric reason it is easy to see that   ψ(r) −1 0

-

f dμ , B3ρ (y)

where v = uε , ν(y) = (ϑ + 1)(φ(ε) + φ(−η))ξy . Thus, it follows from 3.14 that one can use proposition 3.6 for Y = Oε (Ω), X = Oη (Ω), v = uε , ν(y) = (ϑ + 1)(φ(ε) + φ(−η))ξy 4 3 √ 1 H = 2C˜ · (ϑ + 1)(φ(ε) + φ(−η)) (1 + κ)A(∇v, ∇v) + f , κ and get that there exists a function wη ∈ VOη (Ω) such that √ uε − wη 2V ≤ 2C˜ · (ϑ + 1)(φ(ε) + φ(−η)) · (1 + κ)uε 2V 3ψ(r) + κ−1 f 2L Λ

Choosing κ =

f L

Λ3ψ(r)

uε V

Λ3ψ(r)

we obtain the required inequality.

Λ3ψ(r)

. 

Now the following resolvent continuity will be shown with respect to domain perturbation. ω , ω(0) = 0, f ∈ L2 (M ), ui = G(f ; VΩi ), i = Theorem 4.2. Let Ω1 ∈ Wr,ϑ 1, 2, then for sufficiently small = e(Ω2 ΔΩ1 , ∂Ω1 ) there exists a constant Γ = ω , A) such that Γ(MO , Wr,ϑ

u1 − u2 2V ≤ Γ · φ( )f L f V  .

(19)

ω If in addition Ω2 ∈ Wr,ϑ and dHS (Ω1 , Ω2 ) is sufficiently small then

u1 − u2 2V ≤ Γ · φ(dHS (Ω1 , Ω2 ))f L f V  . Proof. Set ε = e(Ω2 , Ω1 ), εˇ = eˇ(Ω2 , Ω1 ), η = e(Ω1 , Ω2 ), ηˇ = eˇ(Ω1 , Ω2 ). To establish (19) we use inequality (9) of lemma 3.2, where V1 = VΩ1 , V2 = VΩ2 , V 1,2 = VOε (Ω1 ) , α = β = 1. As VO−ηˇ (Ω1 ) ⊂ V1 ∩ V2 and d(u1,2 , V2 ) ≤ d(u1,2 , VO−ηˇ (Ω1 ) ) one has that



 u1 − u2 V ≤ dV (u1,2 , VΩ1 ) + dV (u1,2 , VΩ2 ) ≤ 2dV u1,2 , VO−ηˇ (Ω1 ) . Now to get estimate (19) it is sufficient to use lemma 4.1 and inequality (12).

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A. M. STEPIN AND I. V. TSYLIN

ω If in addition Ω2 ∈ Wr,ϑ then interchanging Ω1 and Ω2 in the above arguments one obtains the estimate

u1 − u2 2V ≤ Γ · φ (min{e(Ω1 ΔΩ2 , ∂Ω1 ), e(Ω1 ΔΩ2 , ∂Ω2 )}) f L f V  . We notice incidentally that 3.12 enables one to obtain sufficient smallness of dHP (Ω1 , Ω2 ). Hence it sufficient to establish the following estimates

 u1 − u2 2V ≤ Γ · φ dH (Ω1 , Ω2 ) f L f V  , (20) (21)

u1 − u2 2V ≤ Γ · φ (dH (Ω1 , Ω2 )) f L f V  .

Now, the first one follows from lemma 4.1, inequality (12) and estimate (8) in lemma 3.2: η ) f L f V  , VO−ηˇ (Ω1 ) ⊂ V1 ∩ V2 , dV (u1 , V1 ∩ V2 )2 ≤ dV (u1 , VO−ηˇ (Ω1 ) )2 ≤ Γ · φ (ˇ ε) f L f V  . VO−ˇε (Ω2 ) ⊂ V1 ∩ V2 , dV (u2 , V1 ∩ V2 )2 ≤ dV (u2 , VO−ˇε (Ω2 ) )2 ≤ Γ · φ (ˇ As regards (21) it follows again from lemma 4.1, inequality (12) and corollary 3.3 where V 1,2 = VOε (Ω1 ) , V 2,1 = VOη (Ω2 ) .  Resolvent continuity in Ω established above enables one to estimate distance between eigenspaces of the spectral boundary value problems for A with close domains Ω1 and Ω2 . Namely, let ∂Ω1 ∈ C 0,ω , νk be k-th (in decreasing order) eigenvalue of the operator G(·, VΩ1 ) without taking multiplicity into account, Ek (Ω1 ) be the corresponding eigenspace, number r > 0 be such that B2r (νk )∩spec(G(·, VΩ1 )) = {νk }. Then there exists δ = δ(r, νk ) > 0 so that as soon as = e(Ω1 ΔΩ2 , ∂Ω1 ) < δ for the generalized angle (see, for instance, [13]) & < δ˜V (A, B) = max sup dV (u, B), sup dV (u, A) u∈A, u V =1

u∈B, u V =1

between subspaces A, B ⊂ V the following estimate takes place δ˜V (Ek (Ω1 ), Ek (Ω2 )) ≤ Γ · C(νk , r) · φ( )1/2 , where Ek (Ω2 ) is the range of the operator 1 −1 (G(·, VΩ2 ) − ξ) dξ. 2πi |νk −ξ|=r 4.2. Proof of the Theorem 1.1. By means of Proposition 2.5 for domain Ω1 Cω with the boundary of class C 0,ω one has a class Wr,ϑ (without loss of generality one (i)

can assume that C = 1). Let un ∈ VΩ1 , i = 1, 2, be eigenfunctions associated with ◦ (i) (1) ¯n ∈ H 1 (Ω2 ) be eigenvalues λn of the problem (1) considered in domains Ωi and u weak solution of the equation (1) Au = λ(1) n un ,

◦

u ∈ H 1 (Ω2 ).

Taking advantage of the Theorem 4.2 one has the estimate (1) 2 u(1) n − PΩ2 un V 2 ≤ u(1) ¯(1) n −u n V (1) (1) (1) ≤ Γ · φ(e(Ω1 ΔΩ2 , ∂Ω1 ))λ(1) n un L λn un V  ,

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besides the following inequality holds:

" (i) (i) −1 u  ≤ C(n, A, p) λn u(i) λ(i) n n H (MO ) n L2 (MO ) .

(1)

Since λn ≤ γn does not exceed the n-th eigenvalue of the problem (1), with Ω being a ball contained in Ω1 ∩ Ω2 , one has (1)

(1)

uj − PΩ2 uj 2V ≤ Cj φ(e(Ω1 ΔΩ2 , ∂Ω1 )),

j = 1, . . . , n (1)

Now applying lemma 3.4 one derives the estimate from below for λn . To obtain (2) similar estimate for λn we write (2) 2 u(2) n − PΩ1 un V 2 ≤ u(2) ¯(2) n −u n V (2) (2) (2) ≤ Γ · φ(e(Ω1 ΔΩ2 , ∂Ω1 ))λ(2) n un L λn un V  , (2)

(2) (2)

where u ¯n = G(λn un ; VΩ1 ). Carrying on the above arguments we obtain the required conclusion ω then If in addition Ω2 ∈ Wr,ϑ (1) 2 (1) (1) (1) (1) u(1) n − PΩ2 un V ≤ Γ · φ(dHS (Ω1 , Ω2 ))λn un L λn un V 

Corollary 4.3 (Manifold version of Burenkov–Lamberti theorem). Let operator A on the manifold (M, g) satisfies conditions (A1) – (A2), Ω1 , Ω2 ∈ ω , ω(0) = 0. Then there exists constants Cn = Cn (M, ω, r, ϑ, A) > 0, δ0 = Wr,ϑ δ0 (MO , ω, r, ϑ) > 0 such that conditions Ω1  MO , dHS (Ω1 , Ω2 ) ≤ δ0 imply inequality (2) |λ(1) n − λn | ≤ Cn (ω(dHS (Ω1 , Ω2 )) + dHS (Ω1 , Ω2 )), (i)

where {λn } are eigenvalues of the problem ( 1) for domains Ωi indexed in ascending order with multiplicities taken into account. References [1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR1857292 [2] Gerassimos Barbatis and Pier Domenico Lamberti, Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients, Mathematika 58 (2012), no. 2, 324–348, DOI 10.1112/S0025579311002397. MR2965976 [3] Garrett Birkhoff, C. de Boor, B. Swartz, and B. Wendroff, Rayleigh-Ritz approximation by piecewise cubic polynomials, SIAM J. Numer. Anal. 3 (1966), 188–203. MR0203926 [4] Ha¨ım Brezis, Analyse fonctionnelle (French), Collection Math´ ematiques Appliqu´ ees pour la Maˆıtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983. Th´ eorie et applications. [Theory and applications]. MR697382 [5] Dorin Bucur and Giuseppe Buttazzo, Variational methods in shape optimization problems, Progress in Nonlinear Differential Equations and their Applications, 65, Birkh¨ auser Boston, Inc., Boston, MA, 2005. MR2150214 [6] V. I. Burenkov and E. B. Davies, Spectral stability of the Neumann Laplacian, J. Differential Equations 186 (2002), no. 2, 485–508, DOI 10.1016/S0022-0396(02)00033-5. MR1942219 [7] V. I. Burenkov, P. D. Lamberti, and M. Lantsa de Kristoforis, Spectral stability of nonnegative selfadjoint operators (Russian, with Russian summary), Sovrem. Mat. Fundam. Napravl. 15 (2006), 76–111, DOI 10.1007/s10958-008-0074-4; English transl., J. Math. Sci. (N. Y.) 149 (2008), no. 4, 1417–1452. MR2336430

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[8] Victor Burenkov and Pier Domenico Lamberti, Spectral stability of higher order uniformly elliptic operators, Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 69–102, DOI 10.1007/978-0-387-85650-6 5. MR2484622 [9] Denise Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl. 52 (1975), no. 2, 189–219. MR0385666 [10] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR1158660 [11] J. Frehse, Capacity methods in the theory of partial differential equations, Jahresber. Deutsch. Math.-Verein. 84 (1982), no. 1, 1–44. MR644068 [12] Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkh¨ auser Verlag, Basel, 2006. MR2251558 [13] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR0203473 [14] Paul J. Kelly and Max L. Weiss, Geometry and convexity, John Wiley & Sons, New YorkChichester-Brisbane, 1979. A study in mathematical methods; Pure and Applied Mathematics; A Wiley-Interscience publication. MR534615 [15] Antoine Lemenant, Emmanouil Milakis, and Laura V. Spinolo, Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains, J. Funct. Anal. 264 (2013), no. 9, 2097–2135, DOI 10.1016/j.jfa.2013.02.006. MR3029148 [16] V. P. Maslov, Theory of perturbations of linear operator equations and the problem of the small parameter in differential equations (Russian), Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 531–534. MR0084752 [17] Umberto Mosco, Approximation of the solutions of some variational inequalities, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 373–394; erratum, ibid. (3) 21 (1967), 765. MR0226376 [18] Giuseppe Savar´ e and Giulio Schimperna, Domain perturbations and estimates for the solutions of second order elliptic equations, J. Math. Pures Appl. (9) 81 (2002), no. 11, 1071–1112, DOI 10.1016/S0021-7824(02)01256-4. MR1949174 [19] A. M. Stepin and I. V. Tsylin, On boundary value problems for elliptic operators in the case of domains on manifolds (Russian, with Russian summary), Dokl. Akad. Nauk 463 (2015), no. 2, 144–148; English transl., Dokl. Math. 92 (2015), no. 1, 428–432. MR3443990 [20] I. V. Tsylin, Continuity of eigenvalues of the Laplace operator according to domain, Moscow Univ. Math. Bull. 70 (2015), no. 3, 136–140, DOI 10.3103/S0027132215030079. Translation of Vestnik Moskov. Univ. Ser. I Math. Mekh. 2015, no. 3, 35–39. MR3401238 Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia RUDN University, Moscow, Russia

Contemporary Mathematics Volume 692, 2017 http://dx.doi.org/10.1090/conm/692/13922

Measure-theoretical properties of center foliations Marcelo Viana and Jiagang Yang Abstract. Center foliations of partially hyperbolic diffeomorphisms may exhibit pathological behavior from a measure-theoretical viewpoint: quite often, the disintegration of the ambient volume measure along the center leaves consists of atomic measures. We add to this theory by constructing stable examples for which the disintegration is singular without being atomic. In the context of diffeomorphisms with mostly contracting center direction, for which upper leafwise absolute continuity is known to hold, we provide examples where the center foliation is not lower leafwise absolutely continuous.

Contents 1. Introduction 2. Dimension theory for the center foliaton 3. Semiconjugacy to the linear model 4. Upper absolute continuity Appendix A. Atomic disintegration References

1. Introduction As is the case for many other developments in Dynamics over the last half century, the subject of this paper goes back to the work of Dmitry Viktorovich Anosov. Anosov’s remarkable proof [1] that the geodesic flow on any manifold with negative curvature is ergodic introduced two major ingredients in Dynamics. The first one was the observation that those geodesic flows are hyperbolic, which implies that they carry certain invariant – stable and unstable – foliations. The second one is the proof that those foliations, while not being smooth, are still regular enough (absolute continuity) that a version of the Hopf ergodicity argument can be applied. Here we consider maps exhibiting a weaker (partial) form of hyperbolicity and we want to study the properties of the invariant center foliations. Before stating our results in precise terms, let us briefly outline how this field has evolved. 2010 Mathematics Subject Classification. Primary 37D30, 37D10, 37D25. M.V. and J.Y. were partially supported by CNPq, FAPERJ, and PRONEX. c 2017 American Mathematical Society

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1.1. Partial hyperbolicity. Recall that a diffeomorphism f : M → M on a compact manifold is called an Anosov diffeomorphism (or globally hyperbolic) if there exists a decomposition T M = E s ⊕ E u of the tangent bundle T M into two continuous sub-bundles x → Exs and x → Exu such that • both E s and E u are invariant under the derivative Df and • Df | E s is a uniform contraction and Df | E u is a uniform expansion. Such systems form an open (possibly, empty) subset of the space of C r diffeomorphisms of M , for any r ≥ 1. A distinctive feature of Anosov diffeomorphisms is that they admit invariant foliations F s and F u that are tangent to the sub-bundles E s and E u at every point. Consequently, the leaves of F s are contracted by the forward iterates, whereas the leaves of F u are contracted by backward iterates of f . The leaves are smooth immersed sub-manifolds but, in general, these foliations are not differentiable, that is, they can not be “straightened” by means of C 1 local charts. However, after Anosov, Sinai [1, 2], we know that they do have a crucial differentiability property, called absolute continuity: assuming the derivative Df is H¨older continuous, the holonomy maps of both F s and F u map zero Lebesgue measure sets to zero Lebesgue sets. Indeed, this fact lies at the heart of Anosov’s proof that the geodesic flow on manifolds with negative curvature is ergodic. By the early 1970’s, Brin, Pesin [8] were proposing to extend the class of Anosov diffeomorphisms to what they called partially hyperbolic diffeomorphisms. A similar proposal was made by Pugh, Shub [37] independently and at about the same time. By partially hyperbolic, we mean in this paper1 that there exists a decomposition T M = E ss ⊕E c ⊕E uu of the tangent bundle T M into three continuous sub-bundles x → Exss and x → Exc and x → Exuu such that (i) all three sub-bundles E ss and E c and E uu are invariant under the derivative Df and (ii) Df | E ss is a uniform contraction, Df | E uu is a uniform expansion and (iii) Df | E c lies in between them: for some choice of a Riemannian metric on M (see Gourmelon [16]), one has 1 Df (x)v s  ≤ Df (x)v c  2

and

Df (x)v c  1 ≤ Df (x)v u  2

for any unit vectors v s ∈ E ss and v c ∈ E c and v u ∈ E uu and any x ∈ M (we say that E c dominates E ss and E uu dominates E c , respectively.) Again, partially hyperbolic diffeomorphisms form an open subset of the space of C r diffeomorphisms of M , for any r ≥ 1. The class of manifolds for which this set is non-empty is far from being completely understood. 1.2. Stable and unstable foliations. Part of what was said before about Anosov diffeomorphisms extends to this class. Namely, the strong-stable sub-bundle E ss and the strong-unstable sub-bundle E uu are still uniquely integrable, that is, there are unique foliations F ss and F uu whose leaves are smooth immersed submanifolds of M tangent to E ss and E uu , respectively, at every point. Moreover, 1 Brin, Pesin used a stronger definition that is sometimes called absolute partial hyperbolicity. See Hirsch, Pugh, Shub [24, pages 3–5].

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these so-called strong-stable foliation and strong-unstable foliation are still absolutely continuous. This fact plays a key role in the ergodic and geometric theory of such systems (see Pugh, Shub [38] and Burns, Wilkinson [9], for example). 1.3. Center foliations: existence and (non-)absolute continuity. The situation for the center sub-bundle E c is a lot more complicated. To begin with, E c need not be integrable, that is, there may be no foliation with smooth leaves tangent to E c at every point. The first example was probably due to Smale [43], see Wilkinson [48] and Pesin [33]; other constructions, with interesting additional features, were proposed by Hammerlindl [17] and Hertz, Hertz, Ures [21]. This later paper also shows that even when the center sub-bundle is integrable it may fail to be uniquely integrable, that is, curves tangent to E c may not be contained in a unique leaf of the integral foliation (center foliation). Notwithstanding, there are also many robust examples of partially hyperbolic diffeomorphisms with uniquely integrable center sub-bundle. The simplest construction goes as follows. Start with a hyperbolic torus automorphism A : T3 → T3 (a similar construction can be carried out in any dimension) with eigenvalues (1)

λ1 < 1 < λ2 < λ3

and corresponding eigenspaces E1 , E2 , E3 . A is an Anosov diffeomorphism, of course, and then so is any diffeomorphism f in a C 1 neighborhood. However, A may also be viewed as a partially hyperbolic diffeomorphism with invariant subbundles Exss = E 1 and Exc = E 2 and Exuu = E 3 . Then every f in a C 1 neighborhood is also partially hyperbolic. It follows from general results in [24] that the center (or “weakly expanding”) bundle E c of f is uniquely integrable in this case. Actually, a result of Potrie [36] implies that the center sub-bundle is integrable for every partially hyperbolic diffeomorphism in the isotopy class of A. Moreover, if one assumes the (stronger) absolute form of partial hyperbolicity that we alluded to before, it follows from Brin, Burago, Ivanov [7] that the center sub-bundle is uniquely integrable for any partially hyperbolic diffeomorphism of T3 . Thus, the question naturally arises whether such center foliations are still absolutely continuous. In fact, this question was first raised by A. Katok in the 1980’s, especially for Anosov diffeomorphisms in T3 as introduced in the previous paragraph. Katok also obtained the first example of a center foliation (for a non-invertible map) which is not absolutely continuous. Indeed, this foliation (see Milnor [32]) is such that some full volume subset intersects each leaf in not more than one point. Shub, Wilkinson [42] constructed partially hyperbolic, stably ergodic (with respect to volume) diffeomorphisms whose center leaves are circles and whose center Lyapunov exponent is non-zero, and they observed that for such maps the center foliation can not be absolutely continuous. Indeed, in a related setting, Ruelle, Wilkinson [40] observed that the center foliation has atomic disintegration: the Rokhlin conditional measures of the volume measure along the leaves are supported on finitely many orbits. That is the case also in Katok’s construction, as observed before, but it should be noted that in Katok’s example the center Lyapunov exponent vanishes. An extension of these results to diffeomorphisms with higher-dimensional compact center leaves was due to Hirayama, Pesin [22]. As a matter of fact, for a large class of partially hyperbolic, volume-preserving diffeomorphisms with one-dimensional center leaves one has a sharp dichotomy

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atomic disintegration vs. Lebesgue disintegration: the volume measure along center leaves are either purely Lebesgue measure; in this latter case, we also speak of This was observed by Avila, Viana, Wilkinson, in two

conditional measures of the atomic or equivalent to the leafwise absolute continuity. main situations:

• maps fixing their center leaves, including perturbations of time-one maps of hyperbolic flows [4]; • maps with circle center leaves [3], including perturbations of certain skewproducts, of the type considered in [40, 42]. Moreover, the second alternative is often very rigid: for example, for perturbations of the time-one map of a hyperbolic flow, it implies that the perturbation is itself the time-one map of a smooth flow. 1.4. Statement of main results. Partially hyperbolic diffeomorphisms that are isotopic to Anosov diffeomorphisms have center leaves that are neither compact nor fixed under the map. The measure-theoretical properties of such center foliations have also been studied by several authors, especially the intermediate foliations of Anosov diffeomorphisms which we mentioned before. Saghin, Xia [41] and Gogolev [15] exhibited conditions under which those intermediate foliations can not be absolutely continuous. Moreover, Var˜ ao [45] gave examples where the disintegration is neither atomic nor Lebesgue, thus proving that the dichotomy mentioned in the previous paragraph breaks down for such intermediate foliations of Anosov maps. On the other hand, Ponce, Tahzibi, Var˜ ao [35] prove that atomic disintegration occurs stably in the isotopy class of certain Anosov automorphisms A of the 3-torus. The rigidity phenomenon of [3, 4] also does not extend to the non-volumepreserving setting. Indeed, in [47] we exhibited stable examples of absolute continuity simultaneously for all invariant foliations (center as well as center-stable and center-unstable foliations, tangent to E ss ⊕ E c and E c ⊕ E uu , respectively). Our main result in this paper is a criterion for the disintegration of any ergodic measure μ (not just the volume measure) to have uncountable support along center leaves. By this, we mean that for some choice of a foliation box (in the sense of [4, Section 3]) for the center foliation, the supports of the conditional measures of μ along local center leaves are uncountable sets. The criterion applies to partially hyperbolic diffeomorphisms of the 3-torus in the isotopy class D(A) of an automorphism A as in (1): Theorem A. Let μ be an ergodic invariant probability measure of f ∈ D(A) with hμ (f ) > log λ3 . Then every full μ-measure set Z ⊂ M intersects almost every center leaf on an uncountable subset. Moreover, the center Lyapunov exponent along the center direction is non-negative, and even strictly positive if f is C 2 . By “almost every center leaf” we always mean “every leaf through every point in some full measure subset”. By definition, a probability measure has atomic disintegration along a foliation if there exists a full measure set that intersects almost every leaf on a countable subset (see Appendix A for a more detailed discussion of this notion). Thus the first conclusion in the theorem means precisely that the disintegration of μ along the center foliation is not atomic. Let us also point out that the bound log λ3 in Theorem A is sharp. Indeed, Ponce, Tahzibi [34] constructed an open set of volume preserving deformations f

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of a linear Anosov map A in T3 for which the volume measure has atomic disintegration; one can easily find diffeomorphisms in this open set for which the entropy with respect to the volume measure is equal to log λ3 . As an application of Theorem A, we obtain stable examples of partially hyperbolic, volume-preserving diffeomorphisms for which the disintegration of the Lebesgue measure along center leaves is neither Lebesgue nor atomic: Corollary B. Let f ∈ D(A) be a volume-preserving, partially hyperbolic C 2 diffeomorphism with hvol (f ) > log λ3 and whose integrated center Lyapunov exponent is greater than log λ2 . Then there exists a neighborhood U ⊂ D(A) of f in the space of volume-preserving C 2 diffeomorphisms such that for every g ∈ U the volume measure is ergodic and its disintegration along the center foliation (restricted to any foliation box) is neither atomic nor Lebesgue. It follows that for g ∈ U every full volume set intersects almost every center leaf on an uncountable subset. A related result was obtained by Var˜ ao [45], however, his construction is more restrictive (it applies only to certain Anosov diffeomorphisms close to the linear automorphism A) and, in particular, it is not known to be stable. More precise versions of Theorem A and Corollary B will be presented later. We also provide examples of yet another kind of measure-theoretical behavior of invariant foliations: for maps of the type constructed by Kan [26], we show that the disintegration of Lebesgue along center leaves may be absolutely continuous but not equivalent to Lebesgue measure. The precise statement is given in Theorem 4.1. Acknowledgements. We are grateful to the anonymous referee for a careful revision of the manuscript. 2. Dimension theory for the center foliaton In this section, we prove the following theorem: Theorem 2.1. Let μ be any ergodic invariant measure of the linear automorphism A. If hμ (A) > log λ3 , then every full μ-measure subset Z intersects almost every center leaf in an uncountable set. The proof of Theorem 2.1, which will be given at the end of Section 2.3, is based on the notion of partial entropy of an ergodic probability measure along an expanding foliation, that we describe in Section 2.1. We prove that if the partial entropy is positive then the invariant measure satisfies the conclusion of the theorem. The other half of the argument is to prove that the partial entropy is indeed positive under the assumptions of Theorem 2.1. This is based on an inequality for partial entropies that is stated in Proposition 2.8 and which is inspired by results of Ledrappier, Young [31]. We also get that if the partial entropy is zero then the foliation constitutes a measurable partition, in the sense of Rokhlin [39], and the conditional measures are Dirac masses. This seems to be known already, at least for extreme (strongunstable) laminations, check Ledrappier, Xie [29, Remark 1]. The construction of Ponce, Tahzibi that we mentioned before also shows that the bound log λ3 is sharp in Theorem 2.1. Indeed, by a result of Franks [14] (see also Section 3) the diffeomorphisms in the open set constructed in [34] are semiconjugate to A. Using this semiconjugacy, one finds an ergodic invariant measure μ of A whose disintegration is atomic and whose entropy is equal to log λ3 .

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2.1. Entropy along an expanding foliation. Let f : M → M be a diffeomorphism. We say that a foliation F is expanding if it is invariant and the derivative Df restricted to the tangent bundle of F is uniformly expanding. It is a classical fact (check [23]) that if f admits an invariant dominated splitting T M = E cs ⊕E uu such that Df | E uu is uniformly expanding, then E uu is uniquely integrable; in this case, the integral foliation F uu is an example of expanding foliation. In general, given an expanding foliation, its tangent bundle may not correspond to the strongest expansion and an invariant transverse sub-bundle need not exist either. Let F be an expanding foliation, μ be an invariant probability measure, and ξ be a measurable partition of M with respect to μ. We say that ξ is μ-subordinate to the foliation F if for μ-almost every x, we have (A) ξ(x) ⊂ F(x) and ξ(x) has uniformly small diameter inside F(x); (B) ξ(x) contains an open neighborhood of x inside the leaf F(x); (C) ξ is an increasing partition, meaning that f ξ ≺ ξ. Remark 2.2. Ledrappier, Strelcyn [27] proved that the Pesin unstable lamination admits some μ-subordinate measurable partition. The same is true for the strong-unstable foliation F uu of any partially hyperbolic diffeomorphism. In fact, their construction extends easily to any expanding invariant foliation F (including the center foliations of the maps we consider here), as we are going to sketch (see also [50, Lemma 3.2]). Start by choosing a finite partition A with arbitrarily small diameter such that its elements have small boundary, in the following sense: there exists c smaller than and close to 1  μ(Bck (∂A)) < ∞. (2) k≥1 A∈A F

Let A be a refinement of A whose elements are the intersections of elements of A with local plaques of F. Then the partition ∞ ?

f i (AF )

i=0

is μ-subordinate to F. In all that follows, it is assumed that μ-subordinate partitions are constructed in this way. Indeed, this construction yields the following additional property that will be useful later: (D) for any y ∈ F(x) there exists n ≥ 1 such that f −n (y) ∈ ξ(f −n (x)). Let us explain this, since it is not explicitly stated in the previous papers. Property (B) ensures that there exists a measurable function x → r(x) > 0 such that ξ(x) contains the ball of radius r(x) around x inside the leaf F(x). By recurrence, the pre-orbit f −n (x) of μ-almost point returns infinitely often to any region where r(·) is bounded from zero. On the other hand, the distance from f −n (y) to f −n (x) goes to zero as n → ∞. Thus property (D) follows. We also need some terminology from [39, § 5]. Given measurable partitions η1 and η2 , let Hμ (η1 | η2 ) denote the mean conditional entropy of η1 given η2 . The entropy of f with respect 2∞ to a measurable partition η is defined by hμ (f, η) = Hμ (η | f η + ) where η + = i=0 f i η. Thus hμ (f, η) = Hμ (η | f η) whenever η is an increasing measurable partition.

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The following result is contained in Lemma 3.1.2 of Ledrappier, Young [30]: Lemma 2.3. Given any expanding foliation F, we have hμ (f, ξ1 ) = hμ (f, ξ2 ) for any measurable partitions ξ1 and ξ2 that are μ-subordinate to F. This allows us to define the partial μ-entropy hμ (f, F) of an expanding foliation F to be hμ (f, ξ) for any μ-subordinate partition. Our next goal is to prove that the nature of the conditional probabilities of μ along the leaves of the foliation F is directly related to whether the entropy is zero or strictly positive. That is the content of Propositions 2.5 and 2.7 below. Beforehand, we must introduce a few important ingredients. Let ξ be any measurable partition μ-subordinate to F. Let {μx : x ∈ M } be the disintegration of μ with respect to ξ. By definition, μx (ξ(x)) = 1 for μ-almost every x. Keep in mind that Hμ (ξ | f ξ) = Hμ (f −1 ξ | ξ), because μ is f -invariant. Moreover, the definition gives that

 (3) Hμ (f −1 ξ | ξ) = g dμ, where g(x) = − log μx (f −1 ξ)(x) . Let dF (·, ·) denote the distance along F-leaves. Given any x ∈ M , n ≥ 0 and ε > 0, let BF (x, n, ε) = {y ∈ F(x) : dF (f i (x), f i (y)) < ε for 0 ≤ i < n}. Then define 1 log μx (BF (x, n, ε)) n 1 hμ (x, ε, ξ) = lim sup − log μx (BF (x, n, ε)). n n→∞ hμ (x, ε, ξ) = lim inf − n→∞

The following statement is contained in Ledrappier-Young [31, §§(9.2)-(9.3)]: Proposition 2.4. At μ-almost every x, lim hμ (x, ε, ξ) = lim hμ (x, ε, ξ) = Hμ (ξ|f ξ).

ε→0

ε→0

Proof. The proof that limε→0 hμ (x, ε, ξ) ≥ Hμ (ξ | f ξ) is identical to [31, § (9.2)] and so we omit it. To prove that lim hμ (x, ε, ξ) ≤ Hμ (ξ | f ξ),

ε→0

we could invoke [31, § (9.3)]. However, since we take F to be (uniformly) expanding, it is possible to give a much shorter argument, as follows. Property (A) above implies that for any ε > 0 there is kε (x) ≥ 1 such that diamF (f −m ξ(x)) < ε for any m ≥ kε (x). This ensures that, for every x, n ≥ 1 and m ≥ kε (x), ?

n+m j=0

 f −j ξ (x) ⊂ BF (x, n, ε).

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Then h(x, ε, ξ) = lim sup − n→∞

1 log μx (BF (x, n, ε)) n

n+kε (x)

?  1 f −j ξ)(x) ≤ lim sup − log μx ( n n→∞ j=0

= lim sup n→∞

1 n



n+kε (x)−1

g(f j (x)).

j=0

By ergodicity, and the Birkhoff theorem, this means that h(x, ε, ξ) ≤ g dμ = Hμ (f −1 ξ | ξ). 

This proves the claim.

2.2. Partial entropy and disintegration. We are ready to prove that vanishing partial entropy corresponds to an atomic disintegration: Proposition 2.5. The following conditions are equivalent: (a) hμ (f, F) = 0; (b) there is a full μ-measure subset that intersects each leaf on exactly one point; (c) there is a full μ-measure subset that intersects each leaf on a countable subset. Proof. Let ξ be any partition μ-subordinate to F. To prove that (c) implies (a), let Γ be a full μ-measure subset whose intersection with every leaf is countable. Replacing Γ by a suitable full μ-measure subset, we may assume that the conclusion of Proposition 2.4 holds, μx is well defined and μx (Γ ∩ ξ(x)) = 1 for any point x ∈ Γ. The latter implies that μx is an atomic measure, because Γ ∩ ξ(x) is taken to be countable. Take any y ∈ Γ ∩ ξ(x) such that μx ({y}) > 0. Since μx = μy , because ξ(x) = ξ(y), one gets that μy (BF (y, n, ε)) ≥ μy ({y}) > 0 for any ε > 0 and n ≥ 1. In view of Proposition 2.4, this implies that hμ (f, F) = H(ξ | f ξ) = 0. It remains to prove that (a) implies (b). By the relation (3), the assumption H(f −1 ξ | ξ) = hμ (f, ξ) = 0 implies that g(x) = 0 for μ-almost every x. In other words, μx (f −1 ξ(x)) = 1 for a full μ-measure subset A1 of values of x. Replacing f by f n and using the relation (Rokhlin [39, §7.2]) Hμ (f −n ξ|ξ) = nHμ (f −1 ξ|ξ) we conclude that for any n ≥ 1 there exists a full μ-measure set An such that μx (f −n ξ(x)) = 1 for every x ∈ An . Now, our assumptions ensure that the diameter of f −n ξ(x) decreases uniformly to 0. Thus, for a full μ-measure set A∞ = ∩n≥1 An of values of x, the measure μx is supported on the point x itself: μx = δx . In particular, A∞ ∩ ξ(x) = {x} for every x ∈ A∞ . Finally, consider the full μ-measure invariant set A = ∩n≥0 f n A∞ . Using property (D) above, we get from the previous paragraph that A ∩ F(x) = {x} for every x ∈ A. 

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Remark 2.6. It follows from Proposition 2.5 that if hμ (f, F) = 0 then the leaves of F define a measurable partition of M , with respect to μ. Let us also observe that hμ (f ) = 0 implies hμ (f, F) = 0 for every expanding foliation F. Thus, for example, if f : M → M is Anosov then its unstable (or stable) leaves form a measurable partition with respect to any invariant measure with zero entropy. It is well-known that such measures fill-in a generic subset of the space of all invariant probability measures μ. One way to see this is to recall that μ → hμ (f ) is upper semi-continuous (because f is expansive) and every invariant measure is approximated by measures supported on periodic orbits (by the Anosov closing lemma). These two observations imply that {μ : hμ (f ) < 1/k} is open and dense, for any k ≥ 1, and the claim follows immediately. Proposition 2.7. Let {μx : x ∈ M } be the disintegration of μ with respect to any measurable partition ξ μ-subordinate to F. The following conditions are equivalent: (a) hμ (f, F) > 0; (b) for μ-almost every point x, the measure μx is continuous, that is, it has no atoms. Moreover, any of these conditions implies that any full μ-measure subset Z intersects almost every leaf of F on an uncountable set. Proof. The fact that (b) implies (a) is a direct consequence of Proposition 2.5, so let us prove that (a) implies (b). By Proposition 2.4, there is a full μ-measure subset A of values of x for which the conditional measure μx is well defined and

1 lim inf − log μx BF (x, n, ε)) > 0. n→∞ n Clearly, the latter implies that μx ({x}) = 0 for x ∈ A. Since μy (A) = 1 for μ-almost every y and μx = μy whenever ξ(x) = ξ(y), this proves that μy is continuous for μ-almost every y, as claimed. Given any full μ-measure subset Z, let Z1 be the subset of points x ∈ Z such that μx is a continuous measure. Condition (b) ensures that Z1 has full μ-measure. Then, by the definition of a disintegration, μx (Z1 ) = 1 for every x in some full μ-measure set Z2 ⊂ Z1 . Since μx is continuous and μx (ξ(x)) = 1, this implies that Z1 ∩ ξ(x) is uncountable for every x ∈ Z2 . In particular, Z ∩ F(x) is uncountable  for every x ∈ Z2 . 2.3. Main proposition. Now we focus on the case when the dynamics is partially hyperbolic. More precisely, take f : M → M be a C 1 diffeomorphism admitting an invariant decomposition T M = E c ⊕ E wu ⊕ E uu into three continuous sub-bundles such that (i) dim E wu = dim E uu = 1 and (ii) both Df | E wu and Df | E uu are uniform expansions and (iii) E wu dominates E c and E uu dominates E wu . It is a classical fact (see Hirsch, Pugh, Shub [24]) that the sub-bundles E uu and E u = E wu ⊕ E uu are uniquely integrable: there exist unique foliations F uu and F u , respectively, whose leaves are C 1 and tangent to these sub-bundles at every point. Property (ii) implies that these foliations are expanding. Moreover, F uu sub-foliates F u .

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We also assume: (iv) there exists some invariant weak-unstable foliation F wu with C 1 leaves that are tangent to E wu at every point. Again, such a foliation is necessarily expanding. Moreover, it sub-foliates F u . We say that F wu is uniformly Lipschitz on leaves of F u if there exists K > 0 such that the F wu -holonomy map between any two segments transverse to F wu within distance 1 from each other inside any leaf of F u is K-Lipschitz. The main technical result in this paper is: Proposition 2.8. Suppose that F wu is uniformly Lipschitz on leaves of F u . Then, (4)

hμ (f, F u ) − hμ (f, F wu ) ≤ τ uu ,

where τ uu is the largest Lyapunov exponent of f with respect to μ (corresponding to the the invariant sub-bundle E uu ). Ledrappier and Young have a similar statement ([31, Theorem C’]) where the roles of F uu and F wu are exchanged and the diffeomorphism is assumed to be C 2 (C 1+ suffices, by Barreira, Pesin, Schmeling [5]). In their setting, the lamination F uu is automatically Lipschitz inside F u . That is not true, in general, for F wu . While we were writing this paper, Fran¸cois Ledrappier pointed out to us that a similar result was obtained by Jian-Sheng Xie [49, equation (2.26)] in the context of linear toral automorphisms. His result would be sufficient for our purposes, but our methods extend to non-linear maps, and so they should be useful in more generality. The arguments that follow are essentially borrowed from [31]. They can be adapted to yield a version of Proposition 2.8 where the sub-bundle E wu is assumed to be non-uniformly hyperbolic, and to admit a tangent lamination F wu satisfying a Lipschitz condition. We do not state it explicitly because it will not be necessary for our purposes. The following observation shows that, at least in this non-uniformly hyperbolic setting, the Lipschitz condition can not be omitted: Remark 2.9. Shub and Wilkinson [42] dealt with C2 volume-preserving perturbations of a skew-product map g × id : T2 × S 1 → T2 × S 1 , where g is a linear Anosov map on the 2-dimensional torus. The perturbation f is a partially hyperbolic, volume-preserving diffeomorphism with an invariant circle bundle and whose center Lyapunov exponent τ c is positive. The entropy formula (for partial entropy) gives that hμ (f, F u ) is equal to the sum τ uu + τ wu of the two positive Lyapunov exponents. On the other hand, Ruelle-Wilkinson [40] showed that every center leaf contains finitely many μ-generic points. Thus, hμ (f, F wu ) = 0 and so (4) fails in this case. The proof of Proposition 2.8 is given in Subsection 2.5. It is clear that the weak-unstable foliation of a linear Anosov diffeomorphisms is well defined and uniformly Lipschitz inside leaves of the unstable foliation F u . Thus Theorem 2.1 is an immediate corollary of Proposition 2.7 and Proposition 2.8. 2.4. Auxiliary lemmas. In the section we quote several lemmas from [31] that will be used in the proof of Proposition 2.8.

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Lemma 2.10 ([31], Lemma 4.1.3). Let (X, ν) be a Lebesgue space, π : X → Rn be a measurable map, and {νt : t ∈ Rn } be a disintegration of ν with respect to the partition {π −1 (t) : t ∈ Rn }. Let α be a countable partition of X with Hν (α) < ∞. Define  g(x) = χA (x)g A (π(x)), A∈α



g∗ (x) =

χA (x)g∗A (π(x)) and

A∈α



gδ (x) =

χA (x)gδA (π(x))

A∈α

where g (t) = νt (A) for each A ∈ α and t ∈ Rn , 1 g A d(π∗ ν) and g∗A (t) = inf gδA (t). gδA (t) = δ>0 (π∗ ν)(Bδ (t)) Bδ (t) A

Then gδ → g almost everywhere on X and − log g∗ dν ≤ Hν (α) + log c + 1 where c = c(n) is the constant that comes from Besicovitch covering lemma. Lemma 2.11 ([31], Lemma 4.1.4). Let ω be a finite Borel measure on Rn . Then lim sup ε→0

log ω(Bε (x)) ≤ n. log ε

2.5. Proof of Proposition 2.8. We are going to prove that, given any β > 0, (5)

τ uu + β ≥ (1 − β)[hμ (f, F u ) − hμ (f, F wu ) − 2β].

Proposition 2.8 follows by making β go to zero. Let β > 0 be fixed from now on. The first step is to construct two suitable μ-subordinate partitions, ξ u and ξ wu , for foliations F u and F wu , respectively. Let A be a finite partition with arbitrarily small diameter and whose elements have small boundary in the sense of (2). Denote by Au and Awu the refinements of A defined by u wu (x) ∩ A(x) and Awu (x) = Floc (x) ∩ A(x). Au (x) = Floc

Arguing as in Remark 2.2, we see that ? ? f n Au and ξ wu = f n (Awu ) ξu = n≥0

n≥0

are measurable partitions μ-subordinate to F u and F wu respectively. The next lemma states that ξ wu refines ξ u and the quotient ξ u /ξ wu is preserved by the dynamics: Lemma 2.12. Take the diameter of A to be sufficiently small. Then for μalmost every x, y ∈ M with y ∈ ξ u (x), wu (a) ξ u (x) ∩ Floc (y) = ξ wu (y) and wu −1 (b) f (ξ (f (y))) ∩ ξ u (x) = ξ wu (y).

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Proof. The relation ⊃ in (a) is clear from the definitions. To prove the conwu (y). By the definition of ξ u , the verse, let y, z ∈ ξ u (x) be such that z ∈ Floc −n −n backward iterates f (y) and f (z) belong to the same element of Au and consequently to the same element of A. By property (D) applied to the partition ξ wu , we have that f −n (z) ∈ ξ wu (f −n (y)) for every large n. In particular, f −n (y) and f −n (z) belong to the same element of Awu for every large n. Choose any such n. Since A is assumed to have small diameter, Awu (y−n ) = Awu (z−n ) also has small diameter inside the corresponding F wu -leaf. Then, by continuity, f (Awu (f −n (y))) wu −n+1 (f (y)). This proves that is contained in Floc wu −n+1 f −n+1 (z) ∈ Floc (f (y)) ∩ A(f −n+1 (y)) = Awu (f −n+1 (y)).

By (backwards) induction, this proves that f −n (y) and f −n (z) belong to the same element of Awu for every n. Thus ξ wu (y) = ξ wu (z), as we wanted to prove. The proof of part (a) is complete. wu −1 (f (y)) we immediately get that f (ξ wu (f −1 (y))) ⊂ From ξ wu (f −1 (y)) ⊂ Floc wu Floc (y). Combining this with part (a), we find that f (ξ wu (f −1 (y))) ∩ ξ u (x) ⊂ ξ wu (y). This proves the relation ⊂ in part (b) of the lemma. To prove the converse, observe that ξ wu (y) ⊂ ξ u (x), by definition, and f (ξ wu (f −1 (y))) ⊃ ξ wu (y) because the partition ξ wu is increasing.  It follows that one may identify each quotient ξ u (x)/ξ wu with a subset of the local strong-unstable leaf F uu (x). Indeed, define uu (x), πxwu : ξ u (x) → Floc

wu uu πxwu (y) = the unique point in Floc (y) ∩ Floc (x).

It is clear that this map is constant on every element of ξ wu , and part (a) of Lemma 2.12 ensures that it is injective. Thus it induces an injective map from uu (x). Using this latter map, we may transport the Riemannian ξ u (x)/ξ wu to Floc uu distance on Floc (x) to a distance dx on the quotient space ξ u (x)/ξ wu . In what follows, we define yet another distance on ξ u (x)/ξ wu which we are going to denote as d˜ and which has the advantage of being independent of x. For this, we need a kind of Pesin block construction, which is contained in the next proposition. Proposition 2.13. For any ε > 0, there is a positive measure subset Λε such that, for any x ∈ Λε and any n > 0, 1 log Df n | Exuu  ≤ τ uu + ε. n The arguments are very classical, except for the fact that here the diffeomorphism is only assumed to be C 1 , so we just outline the proof of the proposition. A similar construction appeared in [51]. Define 1 Λε = {x : log Df n | Exuu  ≤ τ uu + ε for every n > 0}. n Then Λε is a compact set, possibly empty. To prove that μ(Λε ) > 0 it suffices to show that the forward orbit Orb+ (x) of μ-almost every x intersects Λε . By the theorem of Oseledets, for μ-almost every x there exists n(x) ≥ 1 such that 1 ε log Df n | Exuu  ≤ τ uu + for every n ≥ n(x). n 2

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We also need the following variation of the Pliss lemma (see [6, Lemma 11.5]): Lemma 2.14. Given K > 0 and τ < τ¯ and any sequence {an }∞ n=1 such that an  < K for every n ≥ 1 and 1 aj < τ, n j=1 n

lim sup n→∞

there exists n0 > 0 such that 1  an +j < τ¯ for any m ∈ N. m j=1 0 m

Take K = supx∈M {Df (x)}, τ = τ uu + ε/2 and τ¯ = τ uu + ε, and define an = Df | Efuu n (x)  for n ≥ 1. From Lemma 2.14 we get that there is n(x) > 0 such that 1 uu log Df m | Efuu + ε for any m ≥ 1. n(x) (x)  ≤ τ m Thus f n(x) (x) ∈ Λε , which implies the claim that Orb+ (x) intersects Λε . This completes our outline of the proof of Proposition 2.13. From now on, let Λ = Λβ/3 . Fix r0 > 0 such that for any x, y ∈ M (6)

d(x, y) < r0 implies  log Df | Exuu − log Df | Eyuu  ≤ β/3.

Assume that the diameter of A is smaller than r0 . Then the same is true for Au and Awu , and so ξ u and ξ wu also have diameter less than r0 . Fix x0 ∈ supp(μ | Λ), that is, such that every neighborhood of x intersects Λ on a positive measure subset. Let D  x be a small codimension-1 disk transverse to F wu . Let (x1 , x2 , . . . , xd−1 ) be local smooth coordinates on D such that the x1 -axis is close to the direction of E uu . Let B be the union of the local F wu -leaves through points of D and π ˜ , from B to the x1 -axis to be the composition of the projection B → D along F wu -leaves with the projection to the x1 -axis associated with the chosen coordinates. Remark 2.15. The projections along the local coordinates are smooth maps, of course. Recall that F u is 2-dimensional and is sub-foliated by F uu and by F uu . Since we assume that the weak-unstable foliation F wu is uniformly Lipschitz inside ˜ is uniformly bi-Lipschitz restricted to each leaf of each leaf of F u , we get that π ˜ be a uniform Lipschitz constant. F uu inside B. Let K It is no restriction to suppose that B2r0 (x0 ) ⊂ B. Define Λ0 = Λ ∩ Br0 (x0 ). By further reducing r0 > 0 if necessary, we may assume that (7)

e−β(τ

uu

+β)

˜ 4μ(Λ0 ) < 1. K

We can extend the projection π ˜ from the domain B to the union of ξ u (x) over n all x ∈ ∪n≥0 f (Λ0 ), as follows. Given such an x, let n be the smallest nonnegative integer such that f −n (x) ∈ Λ0 . Since ξ u is increasing and has small diameter, f −n (y) ∈ ξ u (f −n (x)) ⊂ Br0 (Λ0 ) ⊂ B for any y ∈ ξ u (x). Just define π ˜ (y) = π ˜ (f −n (y)).

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Keep in mind that ∪n≥0 f n (Λ0 ) has full μ-measure, by ergodicity. Now we are ˜ for x ∈ ∪n≥0 f n (Λ0 ), and ready to introduce the announced transverse distance d: u y1 , y2 ∈ ξ (x), define ˜ 1 , y2 ) = |˜ d(y π (y1 ) − π ˜ (y2 )|.

(8)

˜ ·) induces a distance on the quotient space By Lemma 2.12(a), this function d(·, u wu ξ (x)/ξ which is independent of x. Let {μux : x ∈ M } and {μwu x : x ∈ M } be the disintegrations of μ with respect to the partitions ξ u and ξ wu , respectively. For μ-almost every x, consider the disk ˜ y) < ρ}. We are going to prove that BρT (x) = {y ∈ ξ u (x) : d(x, (9)

(τ uu + β) lim sup ρ→0

log μux (BρT (x)) ≥ (1 − β)[hμ (f, ξ u ) − hμ (f, ξ wu ) − 2β]. log ρ

Our definitions are such that μux (BρT (x)) coincides with the (projection) measure of an Euclidean ball of radius ρ in the x1 -axis. Since the latter is 1-dimensional, the lim sup on the left-hand side is smaller than or equal to 1 (compare Lemma 2.11). Recalling also the definition of partial entropy, we immediately conclude that (9) implies (5). Thus we have reduced the proof of Proposition 2.8 to proving (9). The rest of the argument is based on Lemma 2.10. Define g, g∗ , gδ : M → R by −1 u ξ )(y)) and g∗ (y) = inf gδ (y) with g(y) = μwu y ((f δ>0 u μy ((f −1 ξ u )(y) ∩ BδT (y)) 1 gδ (y) = u T g(z) dμuy (z) = μy (Bδ (y)) BδT (y) μuy (BδT (y))

(the last identity is a consequence of the definition of disintegration). It follows from Lemma 2.10 that − log g∗ dμ < ∞. (10) gδ → g at μ-almost everywhere and To see this, just fix x, substitute (ξ u (x), μux ) for (X, ν), let π ˜ be the projection from ξ u (x) to the x1 -axis introduced previously, and take α = f −1 ξ u |ξ u (x); finally, integrate with respect to μ. By Poincar´e recurrence, for μ-almost every x ∈ Λ0 one can find times 0 = n0 < n1 < · · · < nj < · · · < n such that f nj (x) ∈ Λ0 for any j ≥ 0. For each 0 ≤ k < n, take j ≥ 0 largest such that nj ≤ k and then define T (f k x) with δ(x, n, k) = e−(n−nj )(τ a(x, n, k) = Bδ(x,n,k)

uu

+β)

˜ 2j . K

Note that δ(x, n, k) = δ(x, n, nj ) for every k ∈ {nj , . . . , nj+1 − 1}. This will be used for proving the following invariance property: Lemma 2.16. a(x, n, k)∩(f −1 ξ u )(f k (x)) ⊂ f −1 (a(x, n, k+1)) for every x ∈ Λ0 . Proof. Suppose first that k = nj+1 − 1. Note that a(x, n, k) consists of elements ξ wu (y) of the weak-unstable partition and, of course, the same is true for a(x, n, k + 1). For each one of them, Lemma 2.12(b) ensures that f (ξ wu (y)) ∩ ξ u (f k+1 (x)) = ξ wu (f (y)) ˜ f k (x)) = for any y ∈ a(x, n, k)∩(f −1 ξ u )(f k (x)). The definition (8) ensures that d(y, k+1 ˜ (y), f (x)). Besides, as observed before, the transverse diameters δ(x, n, k) d(f

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and δ(x, n, k + 1) are the same in the present case. In this way we get that f (a(x, n, k)) ∩ ξ u (f k+1 (x)) = a(x, n, k + 1), as we wanted to prove. From now on, suppose that k = nj+1 − 1. While the transverse diameters are no longer necessary the same for k and k + 1, all we have to do is that it is still true that (11)

˜ (y), f k+1 (x)) ≤ δ(x, n, k + 1) d(f

for any y ∈ a(x, n, k) ∩ (f −1 ξ u )(f k (x)). By definition, ˜ f k (x)) ≤ e−(n−nj )(τ uu +β) K ˜ 2j . d(y, According to Remark 2.15, this implies that ˜ 2j+1 e−(n−nj )(τ uu +β) . df nj (x) (f nj −nj+1 +1 (y), f nj (x)) ≤ K Since f nj (x) ∈ Λ0 ⊂ Λβ/3 and diam(ξ u ) < r0 , Proposition 2.13 together with our choice of r0 ensure that ˜ 2j+1 e−(n−nj+1 )(τ uu +β) . df nj+1 (x) (f (y), f nj+1 (x)) ≤ K Using Remark 2.15 once again, it follows that ˜ (y), f nj+1 (x)) ≤ K ˜ 2j+2 e−(n−nj+1 )(τ uu +β) . d(f This means that f (y) ∈ a(x, n, k + 1), as we wanted to prove.



Now let us estimate the measure μux (a(x, n, 0)) for x ∈ Λ0 . Clearly, (12)

p(n)−1  μufk (x) (a(x, n, k)) μux (a(x, n, 0)) = , μufn (x) (a(x, n, p(n))) μuf(k+1) x (a(x, n, k + 1)) k=0

where p(n) = [n(1 − β)]. For each 0 ≤ k ≤ p(n) − 1 and μ-almost every x ∈ Λ0 , μufk (x) (a(x, n, k)) μuf(k+1) (x) (a(x, n, k + 1))

= μufk (x) (a(x, n, k))

μufk (x) (f −1 (ξ u (f (k+1) (x)))) μufk (x) (f −1 (a(x, n, k + 1)))

(use the essential uniqueness of the disintegration together with the fact that ξ u is an increasing partition). By Lemma 2.16, the right-hand side is bounded above by μufk (x) (a(x, n, k)) μufk (x) (f −1 ξ u (f k (x)) ∩ a(x, n, k))

μufk (x) (f −1 ξ u (f k (x))).

The quotient on the left-hand side is precisely 1/gδ(x,n,k) (f k (x)). Write the last k factor as e−I(f (x)) , where I(z) = − log μuz (f −1 ξ u (z)). Replacing these expressions in (12), we get that log μux (BeT−n(τ uu +β) (x)) = log μux (a(x, n, 0)) ≤ log

μux (a(x, n, 0)) u μf n (x) (a(x, n, p)) 

p(n)−1

≤−

k=0



p(n)−1

log gδ(x,n,k) (f k (x)) −

k=0

I(f k (x)).

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Consequently, (τ uu + β) lim sup ρ→0

log μux (B T (x, ρ)) log ρ

≥ (τ uu + β) lim inf n→∞

log μux (BeT−n(τ uu +β) (x)) log e−n(τ uu +β)

p(n)−1 1   1  k ≥ lim inf log gδ(x,n,k) (f (x)) + I(f k (x)) . n→∞ n n i=0 p(n)−1

k=0

By the Birkhoff ergodic theorem and the definition of conditional entropy 1 lim I(f k (x))) = p→∞ p i=0 p

-

I dμ = Hμ (f −1 ξ u | ξ u ).

Therefore, using also the definition of partial entropy of an expanding foliation, p(n) 1 lim I(f k (x))) = (1 − β)hμ (f, F u ). n→∞ n i=0

(13)

We are left to prove that (14)

lim sup − n→∞

p(n) 1 log gδ(x,n,k) (f k x) ≤ (1 − β)(hμ (f, F wu ) + 2β). n k=0

By (10), we may find a measurable function x → δ(x) such that − log gδ (x) ≤ − log g(x) + β for every δ < δ(x) and a constant δ0 > 0 such that {x:δ(x)≤δ0 }

− log g∗ (x) dμ(x) < β.

By ergodicity, for μ-almost all x we have #{0 ≤ i < n : f i (x) ∈ Λ0 } ≤ 2nμ(Λ0 ) for every large n. In particular, we always have j ≤ 2nμ(Λ0 ). Moreover, nj ≤ k ≤ p(n) implies that n − nj ≥ βn. Therefore, δ(x, n, k) = e−(n−nj )(τ

uu

+β)

˜ 2j ≤ e−βn(τ uu +β) K ˜ 4nμ(Λ0 ) K

for every 0 ≤ k ≤ p(n). By (7), this implies that δ(x, n, k) → 0 uniformly in 0 ≤ k ≤ p(n) when n → ∞. In particular, δ(x, n, k) < δ0 for every k ≤ p(n) if n is sufficiently large. Going back to (14), note that 

p(n)

− log gδ(x,n,k) (f k x)

k=0

≤

 k:δ(f k (x))>δ0

− log g(f k (x)) + β +

 k:δ(f k (x))≤δ0

− log g∗ (f k (x))

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and, by the Birkhoff ergodic theorem, this leads to lim sup − n

p(n) 1 log gδ(x,n,k) (f k x) n k=0

≤ (1 − β) ≤ (1 − β)

 

-

− log g dμ + β +

-

 − log g dμ + 2β .

{x:δ(x)≤δ0 }

− log g∗ dμ



−1 wu ξ (x)) and so To conclude, note that g(x) = μwu x (f − log g dμ = Hμ (f −1 ξ wu | ξ wu ) = hμ (f, F wu ).

This completes the proof of (9) and thus of Proposition 2.8. 3. Semiconjugacy to the linear model Let f : T → T be a C 1 diffeomorphism in the isotopy class D(A) of a linear automorphism A as described in the previous section. By Potrie [36], such a diffeomorphism is dynamically coherent: there exist invariant foliations F cs and F cu tangent to the center-stable and center-unstable sub-bundles, respectively. Intersecting their leaves, one obtains an invariant center foliation tangent to E c . By Franks [14], there exists a continuous surjective map φ : T3 → T3 that semiconjugates f to A, that is, such that φ ◦ f = A ◦ φ. Moreover, by construction, φ lifts to a map φ˜ : R3 → R3 that is at bounded distance from the identity: there exists C > 0 such that ˜ x) − x (15) φ(˜ ˜ ≤ C for every x ˜ ∈ R3 . 3

3

3.1. Geometry of the center foliation. Proposition 3.1. For all z ∈ T3 , the pre-image φ−1 (z) is a compact connected subset of some center leaf of f (that is, either a point or an arc) with uniformly bounded length. This was proven by Ures [44, Proposition 3.1], assuming absolute partial hyperbolicity, and by Fisher, Potrie, Sambarino [13], in the present context. We outline the arguments, to highlight where the uniform bound comes from. Sketch of proof of Proposition 3.1: Let f˜ and A˜ be the lifts of f and c A, respectively, to the universal cover R3 of T3 . The center foliations Ffc and FA c ˜ also lift to foliations F˜fc and F˜A in R3 and these are center foliations for f˜ and A, respectively. We need the following facts: ˜ x) = φ(˜ ˜ y ) if and only if there exists K > 0 such that f˜n (˜ (i) φ(˜ x) − f˜n (˜ y ) < K for all n ∈ Z. In fact, K may be chosen independent of x ˜ and y˜. (ii) There exists a homeomorphism h : T3 → T3 which maps each center leaf L of f to a center leaf of A so that h(f (L)) = A(h(L)) for every L. We say that h is a leaf conjugacy between f and A. (iii) The leaves of F˜fc are quasi-isometric in R3 : there exists Q > 1 such that dc (x, y) ≤ Qx − y + Q for every x, y in the same center leaf, where dc denotes the distance inside the leaf.

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Fact (i) is a direct consequence of the construction of φ in Franks [14], which is based on the shadowing lemma for the linear automorphism A. Fact (ii) was proven in Hammerlindl, Potrie [19, Corollary 1.5]. See Hammerlindl [18, Proposition 2.16] for a proof of fact (iii) in the absolute partially hyperbolic case and Hammerlindl, Potrie [19, Section 3] for an explanation on how to extend the conclusion to the present context. ˜ : R3 → R3 which is a leaf The map h in (ii) lifts to a homeomorphism h ˜ From a general property of lift maps, we get that conjugacy between f˜ and A. ˜ xn ), h(˜ ˜ yn )) → ∞ ⇒ d(˜ d(h(˜ xn , y˜n ) → ∞. c It is clear that given any distinct F˜A -leaves F1 and F2 the distance between A˜n (F1 ) n and A˜ (F2 ) goes to infinity when n → +∞ or n → −∞ or both. In view of the previous observation, the same is true for f˜: given any distinct F˜fc -leaves L1 and L2 the distance between f˜n (L1 ) and f˜n (L2 ) goes to infinity when n → +∞ or n → −∞ or both. So, by fact (ii) above, every pre-image φ−1 (z) is contained in a single F˜fc -leaf. On the other hand, the quasi-isometry property (iii) implies that if two points x ˜ and y˜ are such that f˜n (˜ x) − f˜n (˜ y ), n ∈ Z is bounded then, denoting by [˜ x, y˜]c the center segment between the two points, the length of f˜n ([˜ x, y˜]c ), n ∈ Z is also bounded. That implies that the whole segment is contained in the same pre-image  φ−1 (z).

Proposition 3.2. The image under φ : T3 → T3 of any center leaf of f is contained in some center leaf of A. Proof. We claim that the image of any center-stable leaf of f is contained in a leaf of the center-stable foliation of A; note that the center-stable leaves of A are just the translates of the center-stable subspace E cs . Analogously, one gets that the image of any center-unstable leaf of f is contained in a translate of the center-unstable subspace E cu . Taking intersections, we get that the image of every center leaf of f is contained in a center leaf of A, as stated. So, we have reduced the proof of the proposition to proving this claim. Let Ffcs be the center-stable foliation of f and F˜fcs be its leaf to the universal cover R3 . By Potrie [36] (Theorem 5.3 and Proposition A.2), there exists R > 0 such that every leaf of F˜fcs is contained in the R-neighborhood of some translate x ˜ + E cs of the plane E cs inside R3 . Then, since φ˜ − id  ≤ C, the image of every ˜ + E cs . leaf of F˜fcs under φ˜ is contained in the R + C-neighborhood of x ˜ the family of Since F˜fcs is invariant under f˜ and φ˜ semi-conjugates f˜ to A, ˜ F˜ cs (˜ ˜ The family of translates of y˜ + E cs , images φ( ˜ ∈ R3 is invariant under A. f y )), y 3 ˜ y˜ ∈ R is also invariant under A, of course. Moreover, A˜ is expanding in the direction transverse to E cs . Thus, the only way the conclusion of the previous ˜ y ) + E cs . ˜ F˜ cs (˜ paragraph may occur is if every image φ( f y )) is contained in φ(˜ Projecting this conclusion down to the torus, we get our claim.  Corollary 3.3. The pre-image under φ : T3 → T3 of any center leaf of A consists of a unique center leaf of f . c Proof. LetFA (y) be any center-leaf of A. Proposition 3.2 implies that its pre-image is a union of center leaves and Proposition 3.1 implies that the images of

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these center leaves are pairwise disjoint. So, by connectivity, it suffices to prove that c (y) then this image is an open if the image of a center leaf Ffc (x) is contained in FA c c subset of FA (y). For that, it suffices to observe that the map φ : Ffc (x) → FA (y) is monotone, which is an immediate consequence of Proposition 3.1.  Corollary 3.4. Y = {y ∈ T3 : φ−1 (y) consists of a single point} is a Borel set and the restriction φ : φ−1 (Y ) → Y is a homeomorphism with respect to the relative topologies. In particular, the inverse φ−1 is a measurable map. Proof. We already know that φ−1 (y) is always a center leaf segment. Since φ is continuous, the length of this segment is an upper semi-continuous (hence measurable) function of y. In particular, the set Y of points where the length is equal to zero is measurable, as claimed in the first part of the corollary. It is clear that the restriction φ : φ−1 (Y ) → Y is a continuous bijection. So, to prove the second part of the corollary it suffices to check that it is also a closed be written as K ∩ φ−1 (Y ) map. By definition, every closed subset of φ−1 (Y

) may  3 −1 for some compact subset of T . Observe that φ K ∩ φ (Y ) = φ(K) ∩ Y and this is a closed subset of Y , because φ(K) is a compact subset of T3 . This proves that f is indeed a closed map.  3.2. Diffeomorphisms derived from Anosov. Let φ∗ denote the pushforward map induced by φ in the space of probability measures. We have the following general result: Proposition 3.5. Consider continuous maps g : M → M and h : N → N in compact spaces and suppose there exists a continuous surjective map p : M → N such that p ◦ g = h ◦ p. Then: (a) The push-forward p∗ maps the space of g-invariant (respectively, g-ergodic) probability measures onto the space of h-invariant (respectively, h-ergodic) probability measures. (b) If ν is an h-invariant probability measure in N such that #φ−1 (y) = 1 for ν-almost every y ∈ N then there exists a unique probability measure μ in M such that φ∗ μ = ν. This measure μ is g-invariant and it is g-ergodic if and only if ν is h-ergodic. Proof. It is easy to see from the relation p ◦ g = h ◦ p that the push-forward of any g-invariant probability measure μ is an h-invariant probability measure. To prove surjectivity, we need the following consequences of the fact that p is continuous: (1) y → p−1 (y) is a map from N to the space K(M ) of compact subsets of M; (2) this map is upper semicontinuous, with respect to the Hausdorff topology on K(M ). In particular, y → p−1 (y) is measurable with respect to the Borel σ-algebras of N and K(M ). Then, [10, Theorem III.30] asserts that p admits a measurable section, that is, a measurable map σ : N → M such that σ(y) ∈ p−1 (y) for every y ∈ N or, equivalently, p ◦ σ = id. Given any h-invariant probability measure ν, let ω = σ∗ ν. Then ω is a probability measure on M , not necessarily invariant, such that p∗ ω = ν. Since p◦g = h◦p, it

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follows that every iterate f∗j ω also projects down to ν. Let μ0 be any accumulation point of the sequence n−1 1 j g ω. n j=0 ∗ It is well known that ω is g-invariant and, since the map p∗ is continuous, it follows from the previous observations that p∗ ω = ν. This proves surjectivity in the space of invariant measures. It is clear that p∗ μ is h-ergodic whenever μ is g-ergodic. Conversely, let ν be any h-ergodic probability measure. By the previous paragraph, there  exists some g-invariant probability measure μ such that p∗ μ = ν. Let μ = μP dP be the ergodic decomposition of μ (see [46, Chapter 5]). Since p∗ is a continuous linear operator, (16) ν = p∗ μ = (p∗ μP ) dP. By the previous observation, p∗ μP is h-ergodic for almost every P . Thus, by uniqueness, (16) must be the ergodic decomposition of ν. As we take ν to be ergodic, this implies that p∗ μP = ν for almost every P . That proves surjectivity in the space of ergodic measures. Now suppose that the set Z = {y ∈ N : #φ−1 (y) = 1} has full measure for ν. Let μ be any probability measure with φ∗ μ = ν. Then, in particular, μ(φ−1 (Z)) = 1. Observe also that σ ◦ φ(x) = x for every x ∈ φ−1 (Z). Then μ = σ∗ φ∗ μ = σ∗ ν = ω. This proves that μ is unique. By the surjectivity statements in the previous paragraph, μ must be g-invariant and it must be g-ergodic if ν is h-ergodic.  Theorem 3.6. The map φ∗ preserves the entropy and it is a bijection restricted to the subsets of invariant ergodic probability measures with entropy larger than log λ3 . Ures [44] proved a version of this result for measures of maximal entropy. Proof. Let μ be any invariant probability measure. Clearly, hφ∗ μ (A) ≤ hμ (f ). On the other hand, the Ledrappier-Walters formula [28] implies that hμ (f ) ≤ hφ∗ (μ) (A) + max3 h(f, φ−1 (z)), z∈T

where h(f, K) denotes the topological entropy of f on a compact set K ⊂ T3 . See Viana, Oliveira [46, Section 10.1.2], for instance. In our case, K = φ−1 (z) is a onedimensional segment whose images have bounded length. Hence, the topological entropy is zero and so we get that hμ (f ) ≤ hφ∗ (μ) (A). This proves that φ preserves the entropy. Now, in view of Proposition 3.5, we only have to show that if ν is an A-ergodic probability measure with hν (A) > log λ3 then φ−1 (y) consists of a single point for νalmost every y ∈ T3 . Suppose otherwise. Then, using the first part of Corollary 3.4, there exists a positive measure set W ⊂ T3 such that φ−1 (y) is a non-trivial arc of a center leaf of f . By ergodicity, we may suppose that W has full measure. Let L be any center leaf of A. By Corollary 3.3, the pre-image φ−1 (W ∩ L) is a subset of a unique center leaf of f . Moreover, it is the union of the non-trivial arcs φ−1 (y) with y ∈ W ∩L. Since these arcs are pairwise disjoint, there can only be countably many

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of them. Thus, W ∩ L is countable. Then, we may use Theorem 2.1 to conclude  that hν (A) ≤ log λ3 , which contradicts the hypothesis. Theorem 3.7. If μ is an ergodic measure of f with negative center exponent, then hμ (f ) ≤ log λ3 and there exists a full μ-measure subset which intersects almost every center leaf on a single point. Proof. Let us start with the following lemma: Lemma 3.8. There exists Kf > 0 depending only on f such that for any compact center segment I there exists NI ≥ 1 such the length of f −n (I) is bounded by Kf for every n ≥ NI . Proof. Let φ : T3 → T3 be the semi-conjugacy introduced previously. Then φ(I) is a segment inside a center leaf of A, and the same is true for the iterates: φ(f −n (I)) = A−n (φ(I)) for every n ≥ 1. Since A−1 contracts the center direction, because λ2 > 1, the length of A−n (φ(I)) goes to zero as n → +∞. As observed before, the map φ˜ is at bounded distance from the identity. It follows that the distance between the endpoints of f −n (I) in T3 is uniformly bounded when n is large. Since the center leaves of f are quasiisometric (property (iii) above), it follows that the length of the center segments  f −n (I) is uniformly bounded when n is large, as claimed. Let Γμ be the set of points x ∈ T3 for which the center Lyapunov exponent is well defined and coincides with the center Lyapunov exponent λc (μ) of the ergodic measure μ. Thus, 1 log |Df −n | Efc (x)| = −λc (μ) for every x ∈ Γμ . n By ergodicity, Γμ is a full μ-measure subset of the torus. lim

Lemma 3.9. There exists δμ > 0 such that for any x ∈ Γμ and any neighborhood U of x inside the center leaf of f that contains x, one has lim inf

n−1 1 length(f −i (U )) ≥ δμ . n i=0

A similar result was proven in Lemma 3.8 in our previous paper [47]. The present statement is analogous, and even easier, because here we take the center direction to be one-dimensional. Corollary 3.10. There exists Nμ ≥ 1 such that #(Γμ ∩ L) ≤ Nμ for every center leaf L of f . Proof. Take Nμ = 3Kf /δμ . Suppose that #(Γμ ∩ L) > Nμ for some center leaf L. Fix pairwise disjoint neighborhoods around each of these points, and let I ⊂ L be a compact segment containing these neighborhoods. From Lemma 3.9 , we get that n−1 δμ 1 > Kf length(f −i (I)) > Nμ n i=0 2 for every large n, which contradicts Lemma 3.8.



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We are left to prove that, up to replacing Γμ by some full measure invariant subset, we may take Nμ = 1. This can be seen as follows. Fix an orientation of the leaves of f once and for all (it is clear that the center foliation of A is orientable and then we may use the semi-conjugacy φ to define an orientation of the center leaves of f ). Let Γmin be the subset of Γ formed by the first points of Γμ on each center leaf, with respect to the chose orientation. It is clear that Γmin is invariant under f , because Γμ is. We claim that Γmin is a measurable set. Let us assume this fact for a while. If Γmin has positive measure then, by ergodicity, it has full measure. Since Γmin intersects every leaf in at most one point, this proves that we may indeed take Nμ = 1. If Γmin has zero measure, just replace Γμ with Γμ \ Γmin and start all over again. Notice that this Nμ is replaced with Nμ − 1 and so this argument must stop in less than Nμ steps. It remains to check that Γmin is indeed a measurable set. We need Lemma 3.11. There exists R > 0 such that the diameter of Γμ ∩ L inside every center leaf L is less than R. Proof. By Proposition 3.5(a), the projection φ∗ μ is an ergodic measure for A. Keep in mind that, by Corollary 3.3, φ maps center leaves to center leaves, in an one-to-one fashion. Thus, φ(Γμ ) is a full measure that intersects each center leaf of A at finitely many points. By Proposition 2.5, it follows that the intersection consists of a single point. In other words, the intersection Γμ ∩ L with each center leaf L is contained in φ−1 (z) for some z ∈ φ(L). Then the conclusion of the lemma follows directly from Proposition 3.1.  Given r > 0 and any disk D transverse to the center foliation, let D(r) denote the union of the center segments of radius r around the points of D. Consider a finite family {Di : i = 1, . . . , l} of (small) disks transverse to the center foliation, such that (i) each Di (R + 2) is homeomorphic to the product Di × [−R − 2, R + 2]; (ii) the sets Di (1), i = 1, . . . , l cover M . For each i = 1, . . . , l, denote by Γmin (i) the set formed by the first point of Γ in the center leaf through each point in Γ ∩ Di (1). Notice that Γmin (i) ⊂ Di (R + 2) and

Γmin = ∪i Γmin (i)

as a consequence of Lemma 3.11. Thus we only have to check that, up to replacing Γμ by some invariant full measure subset, each Γmin (i) is a measurable set. The latter can be seen as follows. Let i be fixed. Identify Di (R + 2) with Di × [−R − 2, R + 2] through the homeomorphism in condition (i) above. Let E ⊂ Di be the vertical projection of Γ ∩ Di (1). Theorem III.23 in [10] ensures that E is a measurable subset of Di . Moreover, Γmin (i) is the graph of a function σ : E → [−R − 2, R + 2]. Our goal is to prove that this function is measurable. If Γμ is compact then the function σ is lower semi-continuous and thus measurable. In general, by Lusin, we may find an increasing sequence of compact sets Γk ⊂ Γμ such that their union Γ (i) has full measure in Γμ ∩ Di (1). By the previous observation, the function σk : Ek → [−R − 2, R + 2] associated with each Γk is measurable. The function σ  : E  → [−R − 2, R + 2] associated with Γ (i) is given by E  = ∪k Ek

and σ  (z) = inf σk (z). k

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Thus, σ  is a measurable function as well. To get the claim, just replace Γμ with the invariant subset obtained by removing the orbits through all zero measure sets Γμ ∩ Di (1) \ Γ (i). This completes the proof of Theorem 3.7.  Remark 3.12. In the context of Theorem 3.7, the map φ∗ is not injective at μ: there is at least one more ergodic measure ν such that φ∗ μ = φ∗ ν. This can be seen as follows. The assumption that the center Lyapunov exponent of μ is negative implies that φ−1 (z) is a non-trivial segment for φ∗ μ-typical points z. Let x be an endpoint of φ−1 (z), for any such z, and ν be any accumulation point of the time average over the orbit of x. Then φ∗ ν = φ∗ μ, because z is taken to be φ∗ μ typical. Moreover, the center Lyapunov exponent of ν can not be negative, for otherwise there would a neighborhood of x inside φ−1 (z), which would contradict the choice of x. 3.3. Proof of Theorem A. By Theorem 3.6, the measure ν = φ∗ μ is ergodic and has the same entropy as μ. In particular, hν (A) > log λ3 . Let Z ⊂ M be any full μ-measure set and Z  = Z ∩ φ−1 (Y ), where Y is the as in Corollary 3.4. Then φ(Z  ) is a measurable subset of T3 . Moreover, φ(Z  ) has full ν-measure, because Y has full ν-measure (this is contained in the second part of the proof of Theorem 3.6) and so Z  has full μ-measure. Thus, by Theorem 2.1, φ(Z  ) intersects almost every center leaf of A on an uncountable subset. By Corollary 3.3, the pre-image of every center of A is a center leaf of f . It follows that Z  intersects almost every center leaf on an uncountable subset. Then the same holds for Z, of course. We have seen in Theorem 3.7 that if the center Lyapunov exponent is negative then some full measure subset intersects almost every center leaf in a single point. In view of the previous paragraph, this ensures that in the present situation the center exponent is non-negative. We are left to show that the center Lyapunov exponent is actually positive when f is a C 2 diffeomorphism. This could be deduced from the refinement of the Ruelle inequality in [25, Theorem 3.3.], as discussed by Ures [44]. Alternatively, we argue by contradiction, as follows. Assume that the center exponent of f is non-positive. Then the strong-unstable leaf F uu coincides with the Pesin unstable manifold at μ-almost every point. Define the exponential volume growth rate of any disk D contained in some strong-unstable leaf of f to be (17)

G(D) = lim inf n→∞

1 vol(f n (D)) log . n vol(D)

It was shown by Cogswell [11] that hμ (f ) ≤ G(D) whenever f is C 2 and D is a neighborhood of a μ-typical point x ∈ M inside its Pesin unstable manifold. Clearly, if the center Lyapunov exponents are non-positive then the Pesin unstable manifold coincides with the strong-unstable leaf through the point. So, to complete the proof of Theorem A it suffices to show that G(D) ≤ log λ3 for any segment D inside some strong-unstable leaf. This can be seen as follows. Let x ˜1 ˜ of the segment D to the universal cover. By and x ˜2 be the endpoints of some lift D (15), ˜ x1 )) − An (φ(˜ ˜ x2 )) = 2C + λn φ(˜ ˜ x1 ) − φ(˜ ˜ x2 ). f˜n (˜ x1 ) − f˜n (˜ x2 ) ≤ 2C + An (φ(˜ 3 It has been show by Potrie [36, Corollary 7.7] that in the present setting the lift of the unstable foliation to the universal is quasi-isometric: the distance between

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any two points along a leaf is bounded by some affine function of the distance of the two points in the ambient space. Thus, in particular, there exists a uniform constant Q > 0 such that ˜ x1 ) − φ(˜ ˜ x2 ). ˜ ≤ Q + Qf˜n (˜ x1 ) − f˜n (˜ x2 ) ≤ Q(2C + 1) + Qλn3 φ(˜ |f˜n (D)| Replacing this estimate in the definition of G(D) we get that G(D) ≤ log λ3 , as claimed. The proof of Theorem A is complete. 3.4. Proof of Corollary B. By Hammerlindl, Ures [20, Theorem 7.2], every C 2 volume-preserving partially hyperbolic diffeomorphism g ∈ D(A) whose integrated center Lyapunov exponent λc (g) is different from zero is ergodic. Thus, since the map g → λc (g) is continuous, every volume-preserving g in a neighborhood of f is ergodic. We begin by claiming that the disintegration of volume along the center foliation of such a g cannot be atomic. Indeed, suppose that the disintegration is atomic. Let B1 , . . . , Bk be a finite cover of M by foliation charts such that the conditional probabilities of each vol | Bi are purely atomic for almost every plaque. Equivalently (Appendix A), every Bi admits a full measure subset Zi whose intersection with every plaque is countable. Then Z = Z1 ∪ · · · ∪ Zk is a full measure subset of M that intersects every center leaf on a countable set. This contradicts Theorem A. Now we prove that conditional probabilities of vol along center leaves cannot be absolutely continuous respect to Lebesgue measure. Let Γ be the set of points x ∈ M such that 1 lim log Dg n | Efc (x) = λc (g), n n By ergodicity and the Birkhoff theorem, Γ has full volume. Fix ε < (λc (g) − log λ2 )/2. Then, there exists a measurable function n(x) : Γ → N such that Dg n | Egc (x) ≥ eεn λn2

for any x ∈ Γ and n ≥ n(x).

Take n0 ≥ 1 sufficiently large, such that the set Γ0 = {x ∈ Γ : n(x) ≤ n0 } has positive volume. Assuming, by contradiction, that the disintegration of the volume measure along the center foliation is absolutely continuous, we get that there exists some center plaque L such that Γ0 ∩ L has positive Lebesgue measure. By the definition of Γ, |g n (L)| ≥ volgn (L) (g n (Γ0 )) ≥ eεn λn2 volL (Γ0 ) for any n ≥ n0 , which implies that G(L) ≥ log λ2 + ε. Thus, to reach a contradiction, it suffices to show that G(L) ≤ log λ2 . For proving this latter claim, we use a variation of an argument in the proof ˜ of the segment L to of Theorem A. Let x ˜1 and x ˜2 be the endpoints of some lift L ˜ x2 ) belong to the same center ˜ x1 ) and φ(˜ the universal cover. By Proposition 3.2, φ(˜ leaf of A. Then, ˜ g n (˜ ˜ x1 )) − An (φ(˜ ˜ x1 )) = λn φ(˜ ˜ x1 ) − φ(˜ ˜ x2 ) ˜ g n (˜ x1 )) − φ(˜ x2 )) = An (φ(˜ φ(˜ 2 for every n ≥ 1. Since φ˜ is uniformly close to the identity, by (15), it follows that ˜ x1 ) − φ(˜ ˜ x2 ) + 2C. x1 ) − g˜n (˜ x2 ) ≤ λn2 φ(˜ ˜ g n (˜

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Then, using the quasi-isometry property of the center foliation (recall (iii) in the proof of Proposition 3.1), ˜ x1 ) − φ(˜ ˜ x2 ) + Q(2C + 1), |˜ g n (L)| ≤ Q˜ g n (˜ x1 ) − g˜n (˜ x2 ) + Q ≤ Qλn2 φ(˜ where Q is a uniform constant. Replacing this in (17) we get that G(L) ≤ log λ2 , as claimed. 4. Upper absolute continuity We have seen previously that the disintegration of Lebesgue measure along center leaves may be singular without being atomic. This adds to the previously known types of behaviour for the center foliation: Lebesgue disintegration (i.e. leafwise absolute continuity) and atomic disintegration. In this section we want to refine our understanding of the non-singular case. Let vol denote the Lebesgue measure in the ambient manifold and volL be the Lebesgue measure restricted to some submanifold L. Following [3, 4], we say that a foliation F is upper leafwise absolutely continuous if volL (Y ) = 0 for every leaf L through a full Lebesgue measure subset of points z ∈ M implies vol(Y ) = 0. Similarly, F is lower leafwise absolutely continuous if for every zero vol-measure set Y ⊂ M and vol-almost every z ∈ M , the leaf L through z meets Y in a zero volL -measure set. Thus, the foliation is upper leafwise absolutely continuous if the conditional measure of vol along a typical leaf L is absolutely continuous with respect to volL and it is lower leafwise absolutely continuous if volL is absolutely continuous with the respect to the conditional measure of vol along a typical leaf L. So, Lebesgue disintegration (leafwise absolute continuity) is the same as both upper and lower leafwise absolute continuity. Pesin theory may be used to show that upper leafwise absolute continuity is actually quite common (see [47, Proposition 6.2] for a precise statement). Here we describe fairly robust examples whose center foliations are upper but not lower leafwise absolutely continuous. We start from a construction due to Kan [26]. Let f0 : S 1 × [0, 1] → S 1 × [0, 1] be a C 2 map of the cylinder of the form f0 (θ, t) = (3θ, hθ (t)) with (1) (2) (3) (4)

hθ (i) = i for every i ∈ {0, 1} and every θ ∈ S 1 ;  c < 3 for some c and every θ ∈ S 1 ; |h  θ (t)| ≤  log |hθ (i)| dθ < 0 for i ∈ {0, 1}; |h0 (0)| < 1 and |h1/2 (1)| < 1 and h0 (t) < t < h1/2 (t) for t ∈ (0, 1).

The first condition means that f0 preserves the two boundary components of the cylinder S 1 × [0, 1]. The second one ensures that f0 is a partially hyperbolic endomorphism of the cylinder, with the vertical segments as center leaves. The restriction of f0 to each boundary component S 1 × {i} is uniformly expanding and preserves the Lebesgue measure mi on the boundary component. The third condition means that, for either boundary component, the transverse Lyapunov exponent is negative. Finally, the fourth condition means that 0 is an attractor for h0 and 1 is an attractor for h1/2 and their basins contain the interval (0, 1). Let K be a small neighborhood of f0 inside the space of C 2 maps of the cylinder preserving both boundary components. Every f ∈ K is partially hyperbolic, with almost vertical center foliation Ffc , and admits absolutely continuous ergodic invariant measures mi,f on the boundary components. These measures vary continuously

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with the map and so their center Lyapunov exponents log |Df | Efc (θ, i)| dmi,f (θ) ≈ log |hθ (i)| dθ (Efc denotes the center bundle, tangent to Ffc ) are still negative. This ensures that both m0,f and m1,f are physical measures for f . As observed by Kan [26], the basins B(mi,f ) are intermingled - they are both dense - and their union has full measure in the ambient cylinder. Denote by p0 (f ) and p1 (f ) the continuations of the fixed saddle-points (0, 0) and (1/2, 1), respectively. Theorem 4.1. For any f ∈ K such that ∂θ f (p0 ) = ∂θ f (p1 ), the center foliation is upper leafwise absolutely continuous but not lower leafwise absolutely continuous. Proof. Let πf be the holonomy map of the center foliation of f from the bottom boundary component S 1 × {0} to the top boundary component S 1 × {1}. Then πf is a homeomorphism and the fact that the center foliation is invariant means that it conjugates the restrictions of f to the two boundary components. It is well-known that a conjugacy between two C 2 expanding maps is either C 1 or completely singular. The assumption ∂θ f (p0 ) = ∂θ f (p1 ) prevents the former possibility, since πf maps p0,f to p1,f . Thus, πf is completely singular and so the measures m1,f and m∗1,f = (πf )∗ m0,f are mutually singular. Similarly, the measures m0,f and m∗0,f = (πf )∗ m1,f are mutually singular. In other words, for i = 0, 1 there exists a full mi,f -measure subset Λi,f of S 1 × {i} such that the sets of center leaves through Λ0,f and Λ1,f are disjoint. We claim that all four invariant measures mi,f and m∗i,f , i = 0, 1 have negative center exponents, assuming f is close enough to f0 . This can be seen as follows. First of all, that is true for f = f0 and in this case, mi,f = m∗i,f = Lebesgue measure along S 1 × {i}. Then, observe that f → mi,f is continuous (because the absolutely continuous invariant measure of a C 2 expanding map depends continuously on the map) and πf also depends continuously on f (note that f = f0 the holonomy map is just (θ, 0) → (θ, 1)). Thus, all these measures vary continuously with f . Since the center bundle is one-dimensional, so do their center Lyapunov exponents. Our claim follows. We also need the following fact: Lemma 4.2. Up to Lebesgue measure zero, for i = 0, 1, the basin of mi,f coincides with the union of the Pesin stable manifolds of the points in Λi,f , which is contained in the union W c (Λi,f ) of the center leaves through the points of Λi,f . This follows from a standard density point argument, see for instance [6, Proposition 11.1]. Similar statements appeared also in our previous papers [12, Lemma 4.6] and [47, Proposition 6.9], in somewhat different situations. Denote Yf = W c (Λ0,f ) \ B(m0,f ) ∪ W c (Λ1,f ) \ B(m1,f ). On the one hand, since the union of the basins B(mi,f ), i = 0, 1 has full Lebesgue measure (Kan [26]), the set Yf has zero Lebesgue measure in S 1 × [0, 1] and Lebesgue almost every center leaf is contained in W c (Λ0,f ) ∪ W c (Λ1,f ). On the other hand, Yf contains the Pesin stable manifold of m∗i,f -almost every point, for i = 0, 1. In particular, the intersection of Yf with Lebesgue almost every center leaf contains a non-trivial segment and, thus, has positive Lebesgue measure inside the center leaf. This proves that the center foliation is not lower leafwise absolutely continuous. On the

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other hand, by Proposition 6.2 in our previous paper [47], the center foliation is upper leafwise absolutely continuous.  Appendix A. Atomic disintegration Denote by Dk the closed unit disk in Rk . Let F be a foliation of dimension k ≥ 1 of some manifold M of dimension d > k. By this we mean that every point of M is contained in the interior of some foliation box, that is, some image B of a topological embedding Φ : Dd−k × Dk → M such that every plaque Px = Φ({x} × Dk ) is contained in a leaf of F. We say that F has C r leaves if every Φ(x, ·) : Dk → M,

y → Φ(x, y)

r

is a C embedding depending continuously on x in the C r topology. Let B be a foliation box, identified with the product Dd−k × Dk through the corresponding homeomorphism B. By Rokhlin’s disintegration theorem (see [46, Theorem 5.1.11]) there exists a probability measure νˆ on Dd−k and a family of probability measures {νx : x ∈ Dd−k } such that

 νx {y ∈ Dk : (x, y) ∈ E} dˆ ν (x) (18) ν(E) = D d−k

for every measurable set E ⊂ B. In fact, νˆ is just the projection of ν on the first coordinate and the family {νx : x ∈ Dd−k } is essentially uniquely determined. We say that ν has atomic disintegration along F if for every foliation box B and ν-almost every x ∈ B the conditional measure νx gives full weight to some countable set (equivalently, νx is a countable linear combination of Dirac masses). Lemma A.1. ν has atomic disintegration if and only if for every foliation box B there exists a full ν-measure set Z ⊂ B whose intersection with νˆ-almost every plaque Px is countable (possibly finite). Proof. Suppose that there exists some full ν-measure set Z ⊂ B whose intersection with νˆ-almost every plaque Px is countable. By (18), it follows that νx (Z ∩ Px ) = 1 (and so νx is a purely atomic measure) for νˆ-almost every x. The converse is also true: if νx is purely atomic for νˆ-almost every x then one may find a full measure subset Z of B that intersects every plaque on a countable subset. This can be deduced from the claim in Rokhlin [39, § 1.10] but, for the reader’s convenience, we provide a quick direct explanation. The idea is quite simple: we take Z = ∪x {x} × Y (x) where each {x} × Y (x) is a countable full νx -measure subset of the plaque Px . The main point is to check that Z is a measurable set (up to measure zero). Once that is done, (18) immediately gives that Z has full ν-measure. To prove measurability, start by fixing some countable basis V for the topology of B. It is also part of Rokhlin’s theorem that the map x → νx (V ) is measurable (up to measure zero) for any measurable set V ⊂ B. Thus, given any ε > 0, one may find a compact set Kε ⊂ Dd−k such that νˆ(Kε ) > 1 − ε and x → νx (V ) is continuous on Kε , for every V ∈ V. In particular, x → νx is continuous with respect to the weak∗ topology for x ∈ Kε . It is no restriction to suppose that νx is purely atomic for every x ∈ Kε , and we do so. Let ε > 0 be fixed. It is clear that, given any δ > 0, the set Γδ (x) = {y ∈ Dk : νx ({y}) ≥ δ}

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is finite, and hence compact. Moreover, the properties of Kε ensure that the function x → Γ(x, δ) is upper semi-continuous on x ∈ Kε . In other words, Λ(ε, δ) = {(x, y) : x ∈ Kε and y ∈ Γδ (x)} is a closed subset of B. Since νx is purely atomic for every x ∈ Kε , the union Λ(ε) = ∪n Λ(ε, 1/n) is a (measurable) full measure subset of Kε × Dk contained in Z. Then, ∪m Λ(1/m) is a (measurable) full measure subset of B contained in Z. This proves that Z is measurable up to measure zero, as claimed.  Notice that the statement of the lemma concerns the intersection of Z with plaques of the foliation, not entire leaves. In the special case of foliations of dimension k = 1 one can do a bit better. Indeed, consider any finite cover of the ambient manifold by foliation boxes. Since 1-dimensional manifolds have only two ends, every leaf can intersect these foliation boxes at most countably many times, that is, every leaf is covered by countably many plaques. Thus for k = 1 the condition in Lemma A.1 may be reformulated equivalently as follows: there exists a full ν-measure set Z ⊂ B such that Z ∩ Fx is countable for νˆ-almost every x. This conclusion extends to large k under the additional condition that every leaf has countably many ends. References [1] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature (Russian), Trudy Mat. Inst. Steklov. 90 (1967), 209. MR0224110 [2] D. V. Anosov and Ja. G. Sina˘ı, Certain smooth ergodic systems (Russian), Uspehi Mat. Nauk 22 (1967), no. 5 (137), 107–172. MR0224771 [3] A. Avila, M. Viana, and A. Wilkinson. Absolute continuity, Lyapunov exponents and rigidity II: compact center leaves. In preparation. [4] Artur Avila, Marcelo Viana, and Amie Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 6, 1435–1462, DOI 10.4171/JEMS/534. MR3353805 [5] Luis Barreira, Yakov Pesin, and J¨ org Schmeling, Dimension and product structure of hyperbolic measures, Ann. of Math. (2) 149 (1999), no. 3, 755–783, DOI 10.2307/121072. MR1709302 [6] Christian Bonatti, Lorenzo J. D´ıaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR2105774 [7] Michael Brin, Dmitri Burago, and Sergey Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, J. Mod. Dyn. 3 (2009), no. 1, 1–11, DOI 10.3934/jmd.2009.3.1. MR2481329 [8] M. Brin and Ya. Pesin. Partially hyperbolic dynamical systems. Izv. Acad. Nauk. SSSR, 1:177–212, 1974. [9] Keith Burns and Amie Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171 (2010), no. 1, 451–489, DOI 10.4007/annals.2010.171.451. MR2630044 [10] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR0467310 [11] Kurt Cogswell, Entropy and volume growth, Ergodic Theory Dynam. Systems 20 (2000), no. 1, 77–84, DOI 10.1017/S0143385700000055. MR1747029 [12] Dmitry Dolgopyat, Marcelo Viana, and Jiagang Yang, Geometric and measure-theoretical structures of maps with mostly contracting center, Comm. Math. Phys. 341 (2016), no. 3, 991–1014, DOI 10.1007/s00220-015-2554-y. MR3452277 [13] Todd Fisher, Rafael Potrie, and Mart´ın Sambarino, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov, Math. Z. 278 (2014), no. 1-2, 149–168, DOI 10.1007/s00209-014-1310-x. MR3267574 [14] John Franks, Anosov diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 61–93. MR0271990

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Published Titles in This Series 692 Anatole Katok, Yakov Pesin, and Federico Rodriguez Hertz, Editors, Modern Theory of Dynamical Systems, 2017 686 Alp Bassa, Alain Couvreur, and David Kohel, Editors, Arithmetic, Geometry, Cryptography and Coding Theory, 2017 685 Heather A. Harrington, Mohamed Omar, and Matthew Wright, Editors, Algebraic and Geometric Methods in Discrete Mathematics, 2017 684 Anna Beliakova and Aaron D. Lauda, Editors, Categorification in Geometry, Topology, and Physics, 2017 683 Anna Beliakova and Aaron D. Lauda, Editors, Categorification and Higher Representation Theory, 2017 682 Gregory Arone, Brenda Johnson, Pascal Lambrechts, Brian A. Munson, and Ismar Voli´ c, Editors, Manifolds and K-Theory, 2017 681 Shiferaw Berhanu, Nordine Mir, and Emil J. Straube, Editors, Analysis and Geometry in Several Complex Variables, 2017 680 Sergei Gukov, Mikhail Khovanov, and Johannes Walcher, Editors, Physics and Mathematics of Link Homology, 2016 679 Catherine B´ en´ eteau, Alberto A. Condori, Constanze Liaw, William T. Ross, and Alan A. Sola, Editors, Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions, 2016 678 Joseph Auslander, Aimee Johnson, and Cesar E. Silva, Editors, Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, 2016 677 Delaram Kahrobaei, Bren Cavallo, and David Garber, Editors, Algebra and Computer Science, 2016 676 Pierre Martinetti and Jean-Christophe Wallet, Editors, Noncommutative Geometry and Optimal Transport, 2016 675 Ana Claudia Nabarro, Juan J. Nu˜ no-Ballesteros, Ra´ ul Oset Sinha, and Maria Aparecida Soares Ruas, Editors, Real and Complex Singularities, 2016 674 Bogdan D. Suceav˘ a, Alfonso Carriazo, Yun Myung Oh, and Joeri Van der Veken, Editors, Recent Advances in the Geometry of Submanifolds, 2016 673 Alex Martsinkovsky, Gordana Todorov, and Kiyoshi Igusa, Editors, Recent Developments in Representation Theory, 2016 672 Bernard Russo, Asuman G¨ uven Aksoy, Ravshan Ashurov, and Shavkat Ayupov, Editors, Topics in Functional Analysis and Algebra, 2016 671 Robert S. Doran and Efton Park, Editors, Operator Algebras and Their Applications, 2016 670 Krishnendu Gongopadhyay and Rama Mishra, Editors, Knot Theory and Its Applications, 2016 669 Sergiˇı Kolyada, Martin M¨ oller, Pieter Moree, and Thomas Ward, Editors, Dynamics and Numbers, 2016 668 Gregory Budzban, Harry Randolph Hughes, and Henri Schurz, Editors, Probability on Algebraic and Geometric Structures, 2016 667 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, Dmitry Khavinson, Simeon Reich, Gilbert Weinstein, and Lawrence Zalcman, Editors, Complex Analysis and Dynamical Systems VI: Part 2: Complex Analysis, Quasiconformal Mappings, Complex Dynamics, 2016 666 Vicent ¸iu D. R˘ adulescu, Ad´ elia Sequeira, and Vsevolod A. Solonnikov, Editors, Recent Advances in Partial Differential Equations and Applications, 2016

CONM

692

ISBN 978-1-4704-2560-9

AMS

9 781470 425609 CONM/692

Modern Theory of Dynamical Systems • Katok et al., Editors

This volume is a tribute to one of the founders of modern theory of dynamical systems, the late Dmitry Victorovich Anosov. It contains both original papers and surveys, written by some distinguished experts in dynamics, which are related to important themes of Anosov’s work, as well as broadly interpreted further crucial developments in the theory of dynamical systems that followed Anosov’s original work. Also included is an article by A. Katok that presents Anosov’s scientific biography and a picture of the early development of hyperbolicity theory in its various incarnations, complete and partial, uniform and nonuniform.