Entropy in dynamical systems.

*(English)*Zbl 1220.37001
New Mathematical Monographs 18. Cambridge: Cambridge University Press (ISBN 978-0-521-88885-1/hbk; 978-1-139-06618-1/ebook). xii, 391 p. (2011).

The main aim of this very interesting book is to offer an answer to the following question: When and how is it possible to digitalize a dynamical system such that no information is lost, i.e., in such a way that after viewing the entire sequence of symbols we can completely reconstruct the evolution of the system? The reader will find answers to the above question at two major levels: measure-theoretic and topological. In the first case the digitalization is governed by the Kolmogorov-Sinai entropy of the dynamical system, the first major subject of this book. In the topological setup, the situation is more complicated since the topological entropy, the next important notion, turns out to be insufficient to decide about digitalization that preserves the topological structure. Thus, another parameter, called symbolic extension entropy, emerges as the third main object discussed in the book.

In consequence, the text is structured in three large parts: Entropy in ergodic theory; Entropy in topological dynamics; Entropy theory for operators. In each case, the study is restricted to the most classical case of the action of iterates of a single transformation (or operator) on either a standard probability space or on a compact metric space. So, the book appears as a self-contained course, from the basics through more advanced material to the newest developments. Very few theorems are quoted without a proof, mainly in the sections or paragraphs marked with an asterisk. A very important remark is that several chapters contain very recent results for which this is the textbook debut.

In consequence, the text is structured in three large parts: Entropy in ergodic theory; Entropy in topological dynamics; Entropy theory for operators. In each case, the study is restricted to the most classical case of the action of iterates of a single transformation (or operator) on either a standard probability space or on a compact metric space. So, the book appears as a self-contained course, from the basics through more advanced material to the newest developments. Very few theorems are quoted without a proof, mainly in the sections or paragraphs marked with an asterisk. A very important remark is that several chapters contain very recent results for which this is the textbook debut.

Reviewer: Mircea Crâşmăreanu (Iaşi)

##### MSC:

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |