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Entropy and generators in ergodic theory. (English) Zbl 0175.34001
Mathematics Lecture Note Series. New York-Amsterdam: W. A. Benjamin, Inc. xii, 124 p. (1969).
The book presents the abstract partition theory of dynamical systems with the aid of entropy and generators. Applications find no place in this book. The first three chapters contain the fundamental notions and theorems: the notion of a dynamical system, mean- and individual ergodic theorem (without proofs), the notion of the conditional expectation, conditional information and entropy of a countable measurable partition, martingale theorem (without proof), the basic properties of entropy and information, the McMillan’s theorem (with proof), the notion of the entropy of a transformation and that of homomorphisms and generators, resp. a-generators (i.e. automorphism-generators). The fourth chapter extends the notion of the conditional entropy to the general measurable partitions of a Lebesgue space and establishes Rohlin’s fundamental theorem of cross-sections. Chapter 5 extends the previous theorems of the book to general measurable partitions and introduces the metric spaces \(Z_\eta\). Chapter 6 is devoted to the study of special partitions. Chapter 7 deals with the existence and the properties of a-generators of ergodic automorphisms with finite entropy. In Chapter 8 it is shown that every ergodic automorphism has a countable generator. In Chapter 9, some results for zero entropy and finite generators are given, and Chapter 10 studies the \(\sigma\)-finite endomorphisms and automorphisms.
In spite of some misprints the book is well readable. The book contains no index, but it has a short bibliography.
Reviewer: J. Szücs

MSC:
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
28Dxx Measure-theoretic ergodic theory
37Axx Ergodic theory