**Proceedings of Symposia in Pure Mathematics**

Volume: 69;
2001;
881 pp;
Hardcover

MSC: Primary 11; 28; 34; 37; 53; 70;

Print ISBN: 978-0-8218-2682-9

Product Code: PSPUM/69

List Price: $190.00

Individual Member Price: $152.00

**Electronic ISBN: 978-0-8218-9374-6
Product Code: PSPUM/69.E**

List Price: $190.00

Individual Member Price: $152.00

# Smooth Ergodic Theory and Its Applications

Share this page *Edited by *
*Anatole Katok; Rafael de la Llave; Yakov Pesin; Howard Weiss*

During the past decade, there have been several major new developments in smooth ergodic theory, which have attracted substantial interest to the field from mathematicians as well as scientists using dynamics in their work. In spite of the impressive literature, it has been extremely difficult for a student—or even an established mathematician who is not an expert in the area—to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools.

Accordingly, the AMS Summer Research Institute on Smooth Ergodic Theory and Its Applications (Seattle, WA) had a strong educational component, including ten mini-courses on various aspects of the topic that were presented by leading experts in the field. This volume presents the proceedings of that conference.

Smooth ergodic theory studies the statistical properties of differentiable
dynamical systems, whose origin traces back to the seminal works of
Poincaré and later, many great mathematicians who made contributions to
the development of the theory. The main topic of this volume, smooth ergodic
theory, especially the theory of nonuniformly hyperbolic systems, provides the
principle paradigm for the rigorous study of complicated or *chaotic*
behavior in deterministic systems. This paradigm asserts that if a non-linear
dynamical system exhibits sufficiently pronounced exponential behavior, then
global properties of the system can be deduced from studying the linearized
system. One can then obtain detailed information on topological properties
(such as the growth of periodic orbits, topological entropy, and dimension of
invariant sets including attractors), as well as statistical properties (such
as the existence of invariant measures, asymptotic behavior of typical orbits,
ergodicity, mixing, decay of correlations, and measure-theoretic entropy).
Smooth ergodic theory also provides a foundation for numerous applications
throughout mathematics (e.g., Riemannian geometry, number theory, Lie groups,
and partial differential equations), as well as other sciences.

This volume serves a two-fold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a state-of-the-art report on important current aspects of the subject. The book is divided into three parts: lecture notes consisting of three long expositions with proofs aimed to serve as a comprehensive and self-contained introduction to a particular area of smooth ergodic theory; thematic sections based on mini-courses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.

#### Table of Contents

# Table of Contents

## Smooth Ergodic Theory and Its Applications

- Contents vii8 free
- Preface ix10 free
- Part I. Lecture Notes 112 free
- Lectures on Lyapunov exponents and smooth ergodic theory 314
- Appendix A: Hölder continuity of invariant distributions 91102
- Appendix B: An example of a smooth hyperbolic measure with countably Many Ergodic Components 95106
- Cocycles, cohomology, and combinatorial constructions in ergodic theory 107118
- A tutorial on KAM theory 175186

- Part II. Survey-Expository Articles 293304
- Part Ila: Systems with Hyperbolic Behavior 295306
- Decay of correlations 297308
- Recent results about stable ergodicity 327338
- Statistical properties of some almost hyperbolic systems 367378
- Random f-expansions 385396
- Dynamical zeta functions 409420
- An overview of the dimension theory of dynamical systems 429440
- Collet-Eckmann condition in one-dimensional dynamics 489500
- Monotonicity, J-algebra of Potapov and Lyapunov exponents 499510

- Part lIb: Geodesic Flows 523534
- Part llc: Algebraic Systems and Rigidity 591602
- Part lld: KAM-theory 677688

- Part III. Research Articles 747758
- Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states 749760
- Smoothness of holonomy maps derived from unstable foliation 785796
- Schrödinger operators and topological pressure on manifolds of negative curvature 791802
- Isoperimetric and ergodic properties of horospheres in symmetric spaces 797808
- Minimal but not uniquely ergodic diffeomorphisms 809820
- Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions 825836

#### Readership

Graduate students and research mathematicians interested in ergodic theory and its applications.