Introduction to ergodic theory. (Introdução à teoria ergódica).

*(Portuguese)*Zbl 0581.28010
Projeto Euclides, 14. Rio de Janeiro: Instituto de Matemática Pura e Aplicada. Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). VIII, 389 p. (1983).

The monograph under review is an interesting introduction to the ergodic theory with special emphasis in its interrelations with the theory of differentiable dynamical systems.

In the Chapter 0 the author gives the basic notions of measure theory necessary for the subsequent four chapters, presenting complete proofs of some theorems concerning derivation with respect to a sequence of partitions of a probability space which are not founded in classical textbooks of the subject. Chapter 1 is concerned with measure preserving transformations, where the author discusses a number of interesting examples of dynamical systems motivating recurrence and ergodic theorem. This latter is fully presented in Chapter 2 which is concerned with classical concepts and theorems on ergodicity, except of a recent result of M. I. Brin, J. Feldman and A. Katok [Ann. Math., II. Ser. 113, 159-179 (1981; Zbl 0477.58021)] on the existence of a Bernoulli diffeomorphism on a \(C^{\infty}\) compact connected manifold (possibly with boundary) of dimension greater than one; in the monograph under review is given a sketch of the main ideas of the proof for the two- dimensional case. Chapter 3 is concerned with expanding endomorphisms and Anosov diffeomorphisms of a flat manifold which are typical objects of the differential ergodic theory. The last Chapter 4 presents the entropy and, after some classical preliminaries, the author introduces the topological entropy, the variational principle of the entropy and the construction of the measure with maximal entropy for hyperbolic homeomorphisms. Finally the Lyapunov exponents are presented and the formula of Ya. B. Pesin [Russ. Math. Surv. 32, No.4, 54-114 (1977); translation from Usp. Mat. Nauk 32, No.4(196), 55-112 (1977; Zbl 0359.58010)] is proved and applied to calculate the entropy of an Anosov diffeomorphism.

Almost every section of Chapter 1 to 4 ends with a few exercises, carefully selected for a better understanding of the different subjects. The monography is well-written, concise and gives an excellent introduction to the modern ergodic theory.

In the Chapter 0 the author gives the basic notions of measure theory necessary for the subsequent four chapters, presenting complete proofs of some theorems concerning derivation with respect to a sequence of partitions of a probability space which are not founded in classical textbooks of the subject. Chapter 1 is concerned with measure preserving transformations, where the author discusses a number of interesting examples of dynamical systems motivating recurrence and ergodic theorem. This latter is fully presented in Chapter 2 which is concerned with classical concepts and theorems on ergodicity, except of a recent result of M. I. Brin, J. Feldman and A. Katok [Ann. Math., II. Ser. 113, 159-179 (1981; Zbl 0477.58021)] on the existence of a Bernoulli diffeomorphism on a \(C^{\infty}\) compact connected manifold (possibly with boundary) of dimension greater than one; in the monograph under review is given a sketch of the main ideas of the proof for the two- dimensional case. Chapter 3 is concerned with expanding endomorphisms and Anosov diffeomorphisms of a flat manifold which are typical objects of the differential ergodic theory. The last Chapter 4 presents the entropy and, after some classical preliminaries, the author introduces the topological entropy, the variational principle of the entropy and the construction of the measure with maximal entropy for hyperbolic homeomorphisms. Finally the Lyapunov exponents are presented and the formula of Ya. B. Pesin [Russ. Math. Surv. 32, No.4, 54-114 (1977); translation from Usp. Mat. Nauk 32, No.4(196), 55-112 (1977; Zbl 0359.58010)] is proved and applied to calculate the entropy of an Anosov diffeomorphism.

Almost every section of Chapter 1 to 4 ends with a few exercises, carefully selected for a better understanding of the different subjects. The monography is well-written, concise and gives an excellent introduction to the modern ergodic theory.

Reviewer: P.Morales

##### MSC:

28D05 | Measure-preserving transformations |

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

37D99 | Dynamical systems with hyperbolic behavior |

28D20 | Entropy and other invariants |

57R50 | Differential topological aspects of diffeomorphisms |