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: Con- temporary mathematics ; volume 692 | Includes bibliographical references. Identifiers: LCCN 2016052689 | ISBN 9781470425609 (alk. paper) Subjects: LCSH: Anosov, D. V. | Differentiable dynamical systems. | Hyperbolic spaces. | Bound- ary 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 sys- tems 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 er- godic 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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Photograph courtesy of the Steklov Mathematical Institute of Russian Academy of Sciences

Contents

Preface ix Dmitry Viktorovich Anosov: His life and mathematics Anatole Katok 1 D.V. Anosov and our road to partial hyperbolicity Michael Brin and Yakov Pesin 23 Escape from large holes in Anosov systems Valentin Afraimovich and Leonid Bunimovich 29 A dynamical decomposition of the torus into pseudo-circles Franc¸ois Beguin,´ Sylvain Crovisier, and Tobias Jager¨ 39 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 67 Nonlocally maximal and premaximal hyperbolic sets T. Fisher, T. Petty, and S. Tikhomirov 83 Rotation numbers for S2 diffeomorphisms John Franks 101 Path connectedness and entropy density of the space of hyperbolic ergodic measures Anton Gorodetski and Yakov Pesin 111 Around Anosov-Weil theory V. Grines and E. Zhuzhoma 123 Attractors and skew products Yu. Ilyashenko and I. Shilin 155 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 195 On a differentiable linearization theorem of Philip Hartman Sheldon E. Newhouse 209

vii viii CONTENTS

Time change invariants for measure preserving flows Marina Ratner 263 Spectral boundary value problems for Laplace-Beltrami operator: Moduli of continuity of eigenvalues under domain deformation A. Stepin and I. Zilin 275 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 dy- namics, 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 (Approxi- mation 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 contri- butions 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 re- maining 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

ix

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 SergiˇıKolyada,MartinM¨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 oenTer fDnmclSystems Dynamical of Theory Modern

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, com- plete and partial, uniform and nonuniform. • ao ta. Editors al., et Katok

ISBN 978-1-4704-2560-9

9 781470 425609

CONM/692 AMS