Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective.

*(English)*Zbl 1060.37020
Encyclopaedia of Mathematical Sciences 102. Mathematical Physics 3. Berlin: Springer (ISBN 3-540-22066-6/hbk). xviii, 384 p. (2005).

The notion of uniform hyperbolicity led to the remarkable advances in the theory of dynamical systems in the 60ies and 70ies of the 20th century but early hopes that such systems are generic in the realm of smooth dynamics were quickly disproved. First general results concerning nonuniformly hyperbolic dynamical systems appeared in the late seventies in the form of the Pesin theory and the Jacobson theorem. It is difficult to advance much further in the general Pesin’s theory framework and researchers started to look more closely at specific models such as the Hénon-type attractors which exhibit chaotic dynamics, i.e., the behavior which has a natural probabilistic description via various limit theorems.

The book starts with basic notions and proceeds to one of the main mechanisms of creation of nonuniform hyperbolic dynamics by means of homoclinic tangencies. Then, the authors discuss Hénon-like dynamics and explain recent results on the decay of correlations, central limit theorem and stochastic stability there. In the following chapters, the authors discuss such topics as robust transitivity, stable ergodicity, generic diffeomorphisms (via chain recurrence), SRB measures and Gibbs states for nonuniformly hyperbolic and expanding maps, random perturbations, decay of correlations and some related issues.

The book appears in the Springer series: Encyclopedia of Mathematical Sciences, so only main ideas of proofs are discussed with various degrees of depth and the references to the original papers are provided for readers willing to learn the details. The book is a welcome reference source for researchers and graduate students who work in or just want to get impression on this important and rapidly developing area of dynamical systems.

The book starts with basic notions and proceeds to one of the main mechanisms of creation of nonuniform hyperbolic dynamics by means of homoclinic tangencies. Then, the authors discuss Hénon-like dynamics and explain recent results on the decay of correlations, central limit theorem and stochastic stability there. In the following chapters, the authors discuss such topics as robust transitivity, stable ergodicity, generic diffeomorphisms (via chain recurrence), SRB measures and Gibbs states for nonuniformly hyperbolic and expanding maps, random perturbations, decay of correlations and some related issues.

The book appears in the Springer series: Encyclopedia of Mathematical Sciences, so only main ideas of proofs are discussed with various degrees of depth and the references to the original papers are provided for readers willing to learn the details. The book is a welcome reference source for researchers and graduate students who work in or just want to get impression on this important and rapidly developing area of dynamical systems.

Reviewer: Yuri Kifer (Jerusalem)

##### MSC:

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

37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |

37C75 | Stability theory for smooth dynamical systems |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |