Overview

This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapunov exponents and its applications to the stability theory of differential equations, the concept of nonuniform hyperbolicity, stable manifold theory (with emphasis on absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. A detailed description of all the basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature, is also presented. There are more than 80 exercises. The book is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wishes to get a working knowledge of smooth ergodic theory and to learn how to use its tools. It can also be used as a source for special topics courses on nonuniform hyperbolicity. The only prerequisite for using this book is a basic knowledge of real analysis, measure theory, differential equations, and topology, although the necessary background definitions and results are provided. In this second edition, the authors improved the exposition and added more exercises to make the book even more student-oriented. They also added new material to bring the book more in line with the current research in dynamical systems.

Full Product Details

Author:   Luis Barreira ,  Yakov Pesin
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Edition:   2nd Revised edition
Weight:   0.290kg
ISBN:  

9781470473075

ISBN 10:   1470473070
Pages:   340
Publication Date:   09 August 2023
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

The core of the theory: Examples of hyperbolic dynamical systems General theory of Lyapunov exponents Cocylces over dynamical systems The multiplicative ergodic theorem Elements of the nonuniform hyperbolicity theory Lyapunov stability theory of nonautonomous equations Local manifold theory Absolute continuity of local manifolds Ergodic properties of smooth hyperbolic measures Geodesic flows on surfaces of nonpositive curvature Topological and ergodic properties of hyperbolic measures Selected advanced topics: Cone techniques Partially hyperbolic diffeomorphisms with nonzero exponents More examples of dynamical systems with nonzero Lyapunov exponents Anosov rigidity $C^1$ pathological behavior: Pugh's example Bibliography Index

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Luis Barreira, Universidade de Lisboa, Lisbon, Portugal. Yakov Pesin, Pennsylvania State University, University Park, PA.

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