##
**Introduction to smooth ergodic theory.**
*(English)*
Zbl 1336.37001

Graduate Studies in Mathematics 148. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9853-6/hbk). ix, 277 p. (2013).

This book, written by two leading researchers in the field of smooth dynamical systems, is a revised and considerably expanded version of the earlier book by the same authors [Lyapunov exponents and smooth ergodic theory. University Lecture Series 23. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 1195.37002)]. This book consists of two parts.

The first part (from Chapter 1 to 10) introduces the basics of smooth ergodic theory. The topics include the general theory (Lyapunov-Perron regularity, Multiplicative Ergodic theorem, etc.) of Lyapunov exponents for linear differential equations, matrix cocycles and also include its applications to the stability theory of differential equations, stable manifold and its absolute continuity, entropy formula and the ergodic theory of dynamical systems with nonzero Lyapunov exponents, etc.

The second part (from Chapter 11 to 15) introduces some selected advanced topics of smooth ergodic theory, including the introduction of cone technics; partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents, Anosov rigidity phenomenon, etc.

This book is systematic and self-contained so that it is suitable for graduate students and researchers specializing in dynamical systems and ergodic theory.

The first part (from Chapter 1 to 10) introduces the basics of smooth ergodic theory. The topics include the general theory (Lyapunov-Perron regularity, Multiplicative Ergodic theorem, etc.) of Lyapunov exponents for linear differential equations, matrix cocycles and also include its applications to the stability theory of differential equations, stable manifold and its absolute continuity, entropy formula and the ergodic theory of dynamical systems with nonzero Lyapunov exponents, etc.

The second part (from Chapter 11 to 15) introduces some selected advanced topics of smooth ergodic theory, including the introduction of cone technics; partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents, Anosov rigidity phenomenon, etc.

This book is systematic and self-contained so that it is suitable for graduate students and researchers specializing in dynamical systems and ergodic theory.

Reviewer: Xueting Tian (Shanghai)

### MSC:

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |

37C45 | Dimension theory of smooth dynamical systems |