This book is a comprehensive account of modern smooth ergodic theory and a representative survey on the theory of dynamical systems with nonzero Lyapunov exponents. In popular terms, this is a book on the mathematical theory of “deterministic chaos”. The theory of nonuniformly hyperbolic systems emerged as an independent discipline at the beginning of the 70s. It has a lot of applications in physics, biology, etc. Despite an enormous amount of research in the last few decades, there have been relatively few comprehensible texts covering the whole field until now. This extensive monograph fills the gap. Basic concepts (Lyapunov exponents, cocycles, multiplicative ergodic theorems and methods of estimating the exponents) are defined in the first part. Part II starts with classical, important examples (such as horseshoes) and it then develops nonlinear theory (such as stable and unstable manifolds). Part III contains a discussion of the ergodic theory of smooth and SRB (Sinai-Ruelle-Bowen) measures. Part IV deals with entropy, dimension and other topological properties of hyperbolic measures. The book covers many developments in this important branch of the theory of dynamical systems, mostly with complete proofs and starting from the very beginning. Of course, even in a book of such scope, some important subjects had to be omitted (such as random dynamical systems and chaotic billiards). Historical remarks are added in each section and the bibliography has 250 items. The presented encyclopedia of (nonuniformly) hyperbolic systems will be indispensable for any mathematically inclined reader with a serious interest in the subject.

Reviewer:

mzah