
Book DetailsGraduate Studies in MathematicsVolume: 148; 2013; 277 ppMSC: Primary 37
Now available in Second Edition: GSM/231
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 the absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. The authors also present a detailed description of all basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature.
This book is a revised and considerably expanded version of the previous book by the same authors Lyapunov Exponents and Smooth Ergodic Theory (University Lecture Series, Vol. 23, AMS, 2002). It is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wants to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools. With more than 80 exercises, the book can be used as a primary textbook for an advanced course in smooth ergodic theory. The book is selfcontained and only a basic knowledge of real analysis, measure theory, differential equations, and topology is required and, even so, the authors provide the reader with the necessary background definitions and results.
ReadershipGraduate students interested in dynamical systems and ergodic theory and research mathematicians interested in smooth ergodic theory.

Table of Contents

Part 1. The core of the theory

Chapter 1. Examples of hyperbolic dynamical systems

Chapter 2. General theory of Lyapunov exponents

Chapter 3. Lyapunov stability theory of nonautonomous equations

Chapter 4. Elements of the nonuniform hyperbolicity theory

Chapter 5. Cocycles over dynamical systems

Chapter 6. The Multiplicative Ergodic Theorem

Chapter 7. Local manifold theory

Chapter 8. Absolute continuity of local manifolds

Chapter 9. Ergodic properties of smooth hyperbolic measures

Chapter 10. Geodesic flows on surfaces of nonpositive curvature

Part 2. Selected advanced topics

Chapter 11. Cone technics

Chapter 12. Partially hyperbolic diffeomorphisms with nonzero exponents

Chapter 13. More examples of dynamical systems with nonzero Lyapunov exponents

Chapter 14. Anosov rigidity

Chapter 15. $C^1$ pathological behavior: Pugh’s example


Additional Material

RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
Now available in Second Edition: GSM/231
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 the absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. The authors also present a detailed description of all basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature.
This book is a revised and considerably expanded version of the previous book by the same authors Lyapunov Exponents and Smooth Ergodic Theory (University Lecture Series, Vol. 23, AMS, 2002). It is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wants to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools. With more than 80 exercises, the book can be used as a primary textbook for an advanced course in smooth ergodic theory. The book is selfcontained and only a basic knowledge of real analysis, measure theory, differential equations, and topology is required and, even so, the authors provide the reader with the necessary background definitions and results.
Graduate students interested in dynamical systems and ergodic theory and research mathematicians interested in smooth ergodic theory.

Part 1. The core of the theory

Chapter 1. Examples of hyperbolic dynamical systems

Chapter 2. General theory of Lyapunov exponents

Chapter 3. Lyapunov stability theory of nonautonomous equations

Chapter 4. Elements of the nonuniform hyperbolicity theory

Chapter 5. Cocycles over dynamical systems

Chapter 6. The Multiplicative Ergodic Theorem

Chapter 7. Local manifold theory

Chapter 8. Absolute continuity of local manifolds

Chapter 9. Ergodic properties of smooth hyperbolic measures

Chapter 10. Geodesic flows on surfaces of nonpositive curvature

Part 2. Selected advanced topics

Chapter 11. Cone technics

Chapter 12. Partially hyperbolic diffeomorphisms with nonzero exponents

Chapter 13. More examples of dynamical systems with nonzero Lyapunov exponents

Chapter 14. Anosov rigidity

Chapter 15. $C^1$ pathological behavior: Pugh’s example