Equilibrium states and the ergodic theory of Anosov diffeomorphisms.
2nd revised ed.

*(English)*Zbl 1172.37001
Lecture Notes in Mathematics 470. Berlin: Springer (ISBN 978-3-540-77605-5/pbk). viii, 74 p. (2008).

Among the impressive work of R. Bowen, the volume “Equilibrium states and the ergodic theory of Anosov diffeomorphisms” [Lecture Notes in Mathematics. 470. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0308.28010)] became a classic from the beginning.

On these notes, R. Bowen presents the ergodic theory of Anosov and Axiom A diffeomorphisms. The first chapter is devoted to the study the statistical properties of Gibbs measures associated to a subshift of finite type. In Chapter 2 the general thermodynamic formalism for continuous transformations on compact metric spaces is studied. In Chapter 3, the relation between Axiom A diffeomorphisms and symbolic dynamics is shown by using Markov partitions. Finally, in Chapter 4 the ergodic properties of Axiom A diffeomorphisms are studied.

As David Ruelle states in the preface of this 2nd revised edition “More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems”.

We celebrate the idea of Jean-RenĂ© Chazotte to make Bowen’s monograph more easily available by reprint it; in particular, the improvements are made in the bibliography.

On these notes, R. Bowen presents the ergodic theory of Anosov and Axiom A diffeomorphisms. The first chapter is devoted to the study the statistical properties of Gibbs measures associated to a subshift of finite type. In Chapter 2 the general thermodynamic formalism for continuous transformations on compact metric spaces is studied. In Chapter 3, the relation between Axiom A diffeomorphisms and symbolic dynamics is shown by using Markov partitions. Finally, in Chapter 4 the ergodic properties of Axiom A diffeomorphisms are studied.

As David Ruelle states in the preface of this 2nd revised edition “More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems”.

We celebrate the idea of Jean-RenĂ© Chazotte to make Bowen’s monograph more easily available by reprint it; in particular, the improvements are made in the bibliography.

Reviewer: Carlos Vasquez (Valparaiso)

##### MSC:

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

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

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