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**An introduction to ergodic theory.**
*(English)*
Zbl 0475.28009

Graduate Texts in Mathematics, Vol. 79. New York-Heidelberg-Berlin: Springer-Verlag. ix, 250 p. (1982).

Summary: This text grew out of the author’s earlier work [Ergodic Theory – Introductory Lectures, Lect. Notes Math. 458, (1975; Zbl 0299.28012)]. It provides a useful introduction to basic results of the ergodic theory and offers a view of the current status in various topics of the field, from which the reader can pursue the literature and his own research in the subject. The book gives also a good overview of applications of ergodic theory to differentiable dynamics, differential geometry, number theory, probability theory, von Neumann algebras, and statistical mechanics. The text is divided into eleven chapters.

Chapter 0 summarizes some of the mathematical prerequisites and notations that are used in the sequel. Chapter 1 is devoted to basic properties of measure-preserving transformations, including recurrence, ergodicity, mixing properties, and Birkhoff’s ergodic theorem. In Chapter 2 the notions of isomorphism, conjugacy, and spectral isomorphism are discussed. Chapter 3 deals with ergodic measure-preserving transformations with discrete spectrum, for which the conjugacy problem is completely solved. In Chapter 4 the concept of measure-theoretic entropy is introduced, and then properties of entropy and methods for calculating entropy are presented. This chapter discusses also Bernoulli automorphisms and Kolmogorov automorphisms. Chapter 5 is concerned with dynamical properties of continuous transformations of compact metric spaces, including minimality, topological transitivity, topological conjugacy, and expansiveness. In Chapter 6 the family of invariant probability measures for a continuous transformation of a compact metric space is studied in connection with the non-wandering set, the periodic points, topological transitivity, and minimality of the transformation. Chapter 7 introduces topological entropy, and provides methods of calculating topological entropy for examples. In Chapter 8 the relationship between measure-theoretic entropy and topological entropy is investigated. In Chapter 9 the notion of topological pressure of a continuous transformation of a compact metric space is introduced, and then, on account of the variational principle, the concept of equilibrium state is discussed. Chapter 10 describes briefly topics closely related to the previous chapters, including qualitative behaviour of diffeomorphisms, the subadditive ergodic theorem, the multiplicative ergodic theorem, quasi-invariant measures, types of isomorphism, and transformations of intervals.

A remarkable feature of this book is the presence of numerous examples throughout the text, which are written out in detail, and guide the reader to a better understanding of the material. At the same time, the lack of any exercises is a drawback of the book. As for the exposition, the author takes good care to motivate situations, to explain methods, and to indicate applications. However, the book appears to be written in a somewhat disorderly manner. Thus the text contains some inconsistencies and tedious repetitions, and some general results are stated explicitly and proved after they have been used long before. For instance, definition 1.4 on p. 27 says that “A measure-preserving transformation

Chapter 0 summarizes some of the mathematical prerequisites and notations that are used in the sequel. Chapter 1 is devoted to basic properties of measure-preserving transformations, including recurrence, ergodicity, mixing properties, and Birkhoff’s ergodic theorem. In Chapter 2 the notions of isomorphism, conjugacy, and spectral isomorphism are discussed. Chapter 3 deals with ergodic measure-preserving transformations with discrete spectrum, for which the conjugacy problem is completely solved. In Chapter 4 the concept of measure-theoretic entropy is introduced, and then properties of entropy and methods for calculating entropy are presented. This chapter discusses also Bernoulli automorphisms and Kolmogorov automorphisms. Chapter 5 is concerned with dynamical properties of continuous transformations of compact metric spaces, including minimality, topological transitivity, topological conjugacy, and expansiveness. In Chapter 6 the family of invariant probability measures for a continuous transformation of a compact metric space is studied in connection with the non-wandering set, the periodic points, topological transitivity, and minimality of the transformation. Chapter 7 introduces topological entropy, and provides methods of calculating topological entropy for examples. In Chapter 8 the relationship between measure-theoretic entropy and topological entropy is investigated. In Chapter 9 the notion of topological pressure of a continuous transformation of a compact metric space is introduced, and then, on account of the variational principle, the concept of equilibrium state is discussed. Chapter 10 describes briefly topics closely related to the previous chapters, including qualitative behaviour of diffeomorphisms, the subadditive ergodic theorem, the multiplicative ergodic theorem, quasi-invariant measures, types of isomorphism, and transformations of intervals.

A remarkable feature of this book is the presence of numerous examples throughout the text, which are written out in detail, and guide the reader to a better understanding of the material. At the same time, the lack of any exercises is a drawback of the book. As for the exposition, the author takes good care to motivate situations, to explain methods, and to indicate applications. However, the book appears to be written in a somewhat disorderly manner. Thus the text contains some inconsistencies and tedious repetitions, and some general results are stated explicitly and proved after they have been used long before. For instance, definition 1.4 on p. 27 says that “A measure-preserving transformation

### MSC:

28Dxx | Measure-theoretic ergodic theory |

54H20 | Topological dynamics (MSC2010) |

37Axx | Ergodic theory |

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

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

47A35 | Ergodic theory of linear operators |

22D40 | Ergodic theory on groups |

60F15 | Strong limit theorems |

60B05 | Probability measures on topological spaces |

46L10 | General theory of von Neumann algebras |