692

Modern Theoryof Dynamical Systems

A Tribute toDmitry Victorovich Anosov

Anatole KatokYakov Pesin

Federico Rodriguez HertzEditors

American Mathematical Society

Modern Theoryof Dynamical Systems

A Tribute toDmitry Victorovich Anosov

Anatole KatokYakov Pesin

Federico Rodriguez HertzEditors

692

Modern Theoryof Dynamical Systems

A Tribute toDmitry Victorovich Anosov

Anatole KatokYakov Pesin

Federico Rodriguez HertzEditors

American Mathematical SocietyProvidence, 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 dynamicalsystems: general theory – Smooth dynamical systems: general theory. msc | Dynamical sys-tems and ergodic theory – Dynamical systems with hyperbolic behavior – Dynamical systemswith hyperbolic behavior. msc | Dynamical systems and ergodic theory – Low-dimensionaldynamical 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–dc23LC record available at https://lccn.loc.gov/2016052689

DOI: http://dx.doi.org/10.1090/conm/692

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Contents

Preface ix

Dmitry Viktorovich Anosov: His life and mathematicsAnatole Katok 1

D.V. Anosov and our road to partial hyperbolicityMichael Brin and Yakov Pesin 23

Escape from large holes in Anosov systemsValentin Afraimovich and Leonid Bunimovich 29

A dynamical decomposition of the torus into pseudo-circlesFrancois Beguin, Sylvain Crovisier, and Tobias Jager 39

On irreducibility and disjointness of Koopman and quasi-regular representationsof weakly branch groups

Artem Dudko and Rostislav Grigorchuk 51

Isolated elliptic fixed points for smooth HamiltoniansBassam Fayad and Maria Saprikina 67

Nonlocally maximal and premaximal hyperbolic setsT. Fisher, T. Petty, and S. Tikhomirov 83

Rotation numbers for S2 diffeomorphismsJohn Franks 101

Path connectedness and entropy density of the space of hyperbolic ergodicmeasures

Anton Gorodetski and Yakov Pesin 111

Around Anosov-Weil theoryV. Grines and E. Zhuzhoma 123

Attractors and skew productsYu. Ilyashenko and I. Shilin 155

Thermodynamic formalism for some systems with countable Markov structuresMichael Jakobson 177

Non-uniform measure rigidity for Zk actions of symplectic typeAnatole Katok and Federico Rodriguez Hertz 195

On a differentiable linearization theorem of Philip HartmanSheldon E. Newhouse 209

vii

viii CONTENTS

Time change invariants for measure preserving flowsMarina Ratner 263

Spectral boundary value problems for Laplace-Beltrami operator: Moduli ofcontinuity of eigenvalues under domain deformation

A. Stepin and I. Zilin 275

Measure-theoretical properties of center foliationsMarcelo Viana and Jiagang Yang 291

Preface

This volume of the “Contemporary mathematics ” series is dedicated to theachievements and memory of Dmitry Viktorovich Anosov (1936–2014), one of thefounders of the modern dynamical systems theory . While Anosov lived and workedall his life in the Soviet Union and Russia, his work beginning from 1960s, had greatinternational resonance. Anosov’s name is forever connected with hyperbolic dy-namics, the area where he made his most important contributions. S. Smale namedone 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 featurescaptured by that notion are so striking that various derivative and related objectswere given names that still refer to Anosov. Another important contribution ofAnosov is the discovery of a very flexible and rather paradoxical AbC (Approxi-mation by Conjugation) method of constructing smooth dynamical systems withinteresting, often unexpected, properties. In the literature this method, that is stillwidely 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 themare fully or partially surveys dedicated primarily to various aspects of Anosov’swork. Thematically hyperbolic dynamics in a broad sense appears as the subjectin nine of those papers. Four of those are fairly directly connected with the themesand contents of Anosov’s work. Two more papers include new applications of theAbC 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 theUnited States; (ii) senior Western mathematicians directly influenced by Anosov’swork, and (iii) mathematicians of younger generation who did not know Anosovpersonally but have been influenced by his work or by the developments directlybased on that work.

The editors hope that this volume will serve as a fitting memorial to one of theoutstanding mathematicians of the second half of the twentieth century.

Anatole KatokYakov Pesin

Federico Rodriguez Hertz

ix

Published Titles in This Series

692 Anatole Katok, Yakov Pesin, and Federico Rodriguez Hertz, Editors, ModernTheory 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 HigherRepresentation Theory, 2017

682 Gregory Arone, Brenda Johnson, Pascal Lambrechts, Brian A. Munson,and Ismar Volic, Editors, Manifolds and K-Theory, 2017

681 Shiferaw Berhanu, Nordine Mir, and Emil J. Straube, Editors, Analysis andGeometry in Several Complex Variables, 2017

680 Sergei Gukov, Mikhail Khovanov, and Johannes Walcher, Editors, Physics andMathematics of Link Homology, 2016

679 Catherine Beneteau, Alberto A. Condori, Constanze Liaw, William T. Ross,and Alan A. Sola, Editors, Recent Progress on Operator Theory and Approximation inSpaces 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 andComputer Science, 2016

676 Pierre Martinetti and Jean-Christophe Wallet, Editors, NoncommutativeGeometry and Optimal Transport, 2016

675 Ana Claudia Nabarro, Juan J. Nuno-Ballesteros, Raul Oset Sinha, and MariaAparecida Soares Ruas, Editors, Real and Complex Singularities, 2016

674 Bogdan D. Suceava, Alfonso Carriazo, Yun Myung Oh, and Joeri Van derVeken, Editors, Recent Advances in the Geometry of Submanifolds, 2016

673 Alex Martsinkovsky, Gordana Todorov, and Kiyoshi Igusa, Editors, RecentDevelopments in Representation Theory, 2016

672 Bernard Russo, Asuman Guven Aksoy, Ravshan Ashurov, and ShavkatAyupov, Editors, Topics in Functional Analysis and Algebra, 2016

671 Robert S. Doran and Efton Park, Editors, Operator Algebras and TheirApplications, 2016

670 Krishnendu Gongopadhyay and Rama Mishra, Editors, Knot Theory and ItsApplications, 2016

669 Sergiı Kolyada, Martin Moller, 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, DmitryKhavinson, Simeon Reich, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems VI: Part 2: Complex Analysis, QuasiconformalMappings, Complex Dynamics, 2016

666 Vicentiu D. Radulescu, Adelia Sequeira, and Vsevolod A. Solonnikov, Editors,Recent Advances in Partial Differential Equations and Applications, 2016

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.

ISBN978-1-4704-2560-9

9 781470 425609

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