##
**Ergodic theory and differentiable dynamics. Transl. from the Portuguese by Silvio Levy.**
*(English)*
Zbl 0616.28007

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 8. A Series of Modern Surveys in Mathematics. Berlin etc.: Springer-Verlag. XII, 317 p.; DM 148.00 (1987).

[The Portuguese original was published in 1983, cf. Zbl 0581.28010.]

In the last decade the literature about ergodic theory has been enriched by the appearance of such books as those by I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, ”Ergodic theory” (1980 in Russian, 1982 in English, see this Zbl 0493.28007) and by P. Walters, ”An introduction to ergodic theory” (1982; Zbl 0475.28009).

Mañé’s book under review is an important and welcome contribution to this list of books; it is an excellent introduction to ergodic theory, with emphasis on its relationship with the theory of differentiable dynamical systems. Once again we are fortunate that one of the major researchers of a subject has presented a scholarly treatise which synthetizes a background and some of the greatest results of the theory.

The book consists of five chapters. Chapter 0, a quick review of measure theory is included as a reference. Chapter I starts with a quick and superficial introduction, then presents the main kinds of dynamical systems around which ergodic theory has been developed. In modern, concise form there are developed relevant elements of the theory of volume-preserving diffeomorphisms and flows, Hamiltonians, continued fractions, topological and Lie groups, and shifts. The chapter is closed by a paragraph on equivalent maps.

Chapter II is devoted to classical concepts and theorems of ergodic theory. It begins with Birkhoff’s theorem, then presents a large number of various examples of ergodic maps. In the second part of this chapter are paragraphs devoted to spectral theory, ergodic decomposition of invariant measures, mixing and ergodic Markov shifts and Kolmogorov automorphisms.

Chapter III - Expanding maps and Anosov diffeomorphisms - is a typical example of differentiable ergodic theory. In the first paragraph an expanding map f on a metric space X is given together with a probability measure \(\lambda\) on the Borel \(\sigma\)-algebra of X. Existence and properties of f-invariant probability measures on the Borel \(\sigma\)- algebra of X which are also absolutely continuous with respect to \(\lambda\) are studied. In the second paragraph Sinai’s theorem on existence of the unique ergodic measure for transitive Anosov diffeomorphism is proved. In the third paragraph of this chapter absolute continuity of the stable foliation is proved for a Hölder \(C^ 1\) Anosov diffeomorphism.

Entropy is the subject of Chapter IV; it starts with the basic formalism and the calculation of simple examples, then discusses topological entropy, the variational principle of entropy and the construction of the unique entropy-maximizing measure for hyperbolic homeomorphisms. It is concluded with Lyapunov exponents, the Pesin formula for the entropy of volume-preserving diffeomorphisms, and the Brin-Katok local entropy formula.

A good part of the information in this book is contained in exercises which follow nearly all paragraphs and encourage the reader in better understanding of the subject.

In order not to deprive the reader of a comprehensive and up-to-date panorama of the subject, many advanced results are included without proof. This is especially true for the results for which good and readily accessible expositions are available or for the results which are outside the main stream of ideas presented in this book.

The only defect of the book is a large number of misprints which slightly hamper otherwise excellent readability of the text.

In the last decade the literature about ergodic theory has been enriched by the appearance of such books as those by I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, ”Ergodic theory” (1980 in Russian, 1982 in English, see this Zbl 0493.28007) and by P. Walters, ”An introduction to ergodic theory” (1982; Zbl 0475.28009).

Mañé’s book under review is an important and welcome contribution to this list of books; it is an excellent introduction to ergodic theory, with emphasis on its relationship with the theory of differentiable dynamical systems. Once again we are fortunate that one of the major researchers of a subject has presented a scholarly treatise which synthetizes a background and some of the greatest results of the theory.

The book consists of five chapters. Chapter 0, a quick review of measure theory is included as a reference. Chapter I starts with a quick and superficial introduction, then presents the main kinds of dynamical systems around which ergodic theory has been developed. In modern, concise form there are developed relevant elements of the theory of volume-preserving diffeomorphisms and flows, Hamiltonians, continued fractions, topological and Lie groups, and shifts. The chapter is closed by a paragraph on equivalent maps.

Chapter II is devoted to classical concepts and theorems of ergodic theory. It begins with Birkhoff’s theorem, then presents a large number of various examples of ergodic maps. In the second part of this chapter are paragraphs devoted to spectral theory, ergodic decomposition of invariant measures, mixing and ergodic Markov shifts and Kolmogorov automorphisms.

Chapter III - Expanding maps and Anosov diffeomorphisms - is a typical example of differentiable ergodic theory. In the first paragraph an expanding map f on a metric space X is given together with a probability measure \(\lambda\) on the Borel \(\sigma\)-algebra of X. Existence and properties of f-invariant probability measures on the Borel \(\sigma\)- algebra of X which are also absolutely continuous with respect to \(\lambda\) are studied. In the second paragraph Sinai’s theorem on existence of the unique ergodic measure for transitive Anosov diffeomorphism is proved. In the third paragraph of this chapter absolute continuity of the stable foliation is proved for a Hölder \(C^ 1\) Anosov diffeomorphism.

Entropy is the subject of Chapter IV; it starts with the basic formalism and the calculation of simple examples, then discusses topological entropy, the variational principle of entropy and the construction of the unique entropy-maximizing measure for hyperbolic homeomorphisms. It is concluded with Lyapunov exponents, the Pesin formula for the entropy of volume-preserving diffeomorphisms, and the Brin-Katok local entropy formula.

A good part of the information in this book is contained in exercises which follow nearly all paragraphs and encourage the reader in better understanding of the subject.

In order not to deprive the reader of a comprehensive and up-to-date panorama of the subject, many advanced results are included without proof. This is especially true for the results for which good and readily accessible expositions are available or for the results which are outside the main stream of ideas presented in this book.

The only defect of the book is a large number of misprints which slightly hamper otherwise excellent readability of the text.

Reviewer: J.Šiška

### MSC:

28D05 | Measure-preserving transformations |

37A99 | Ergodic theory |

37D99 | Dynamical systems with hyperbolic behavior |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

28D20 | Entropy and other invariants |

57R50 | Differential topological aspects of diffeomorphisms |

37C10 | Dynamics induced by flows and semiflows |