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Entropy in Dynamical Systems

Tomasz Downarowicz
Cambridge University Press
Publication Date: 
Number of Pages: 
New Mathematical Monographs 18
[Reviewed by
Vicentiu Radulescu
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This book on entropy is part of a series intended for advanced undergraduates or beginning graduate students. The term “entropy” goes back to the French mathematician Sadi Carnot (1824) and to the German physicist Rudolf Clausius (1854), in strong relationship with the second law of thermodynamics. Austrian physicist Ludwig Boltzmann (1877) put entropy into the probabilistic setting of statistical mechanics, while John von Neumann (1932) applied the concept of entropy in the mathematical formulation of quantum mechanics.

Entropy in Dynamical Systems highlights the role of the concept of “entropy” in the theory of dynamical systems. It covers three major types of dynamics: measure preserving transformations, continuous maps on compact spaces, and operators on function spaces. The book is divided into three parts, correspondingly.

Part I deals with the role of entropy in ergodic theory. Here we find several important results, including the Shannon-McMillan-Breiman theorem, the Ornstein-Weiss Return time theorem, the Krieger generator theorem and the ergodic law of series. A central role is played by the notion of “Kolmogorov-Sinai entropy”, which is the key entropy notion in ergodic theory. The formula that provides the entropy of a process gives the following elementary interpretation of the Kolmogorov-Sinai entropy: it is the new information obtained in one step of a process given all the information from the past.

Part II offers deep insight into the theory of entropy structure and explains the role of zero-dimensional dynamics as a bridge between measurable and topological dynamics. The main notion here is that of “topological entropy”, which is a nonnegative number measuring the complexity of a system. Specifically, it measures the exponential growth rate of the number of distinguishable n-orbits as n grows to infinity.

Finally, Part III explains how both measure-theoretic and topological entropy can be extended to operators on function spaces.

This book can be used for an advanced undergraduate or a graduate course on applied mathematics. Every chapter is followed by an exercise section that contains several problems which extend some of the ideas discussed in the chapters.

Besides being a wonderful source of techniques, examples and applications, this volume is a pleasing and well-organized introduction to a subject at the interface between entropy theory and dynamical systems.

Vicentiu D. Radulescu ( ) is a Professorial Fellow at the Mathematics Institute of the Romanian Academy. He received both his Ph.D. and Habilitation at the Université Pierre et Marie Curie (Paris 6) under the coordination of Haim Brezis. His current interests are broadly in nonlinear PDEs and their applications. He wrote more than 150 papers and he published books with Oxford University Press, Cambridge University Press, Springer New York, Springer Heidelberg, Kluwer, and Hindawi.


Part I. Entropy in Ergodic Theory

1. Shannon information and entropy
2. Dynamical entropy of a process
3. Entropy theorems in processes
4. Kolmogorov–Sinai entropy
5. The ergodic law of series

Part II. Entropy in Topological Dynamics

6. Topological entropy
7. Dynamics in dimension zero
8. The entropy structure
9. Symbolic extensions
10. A touch of smooth dynamics

Part III. Entropy Theory for Operators

11. Measure theoretic entropy of stochastic operators
12. Topological entropy of a Markov operator
13. Open problems in operator entropy

Appendix A. Toolbox
Appendix B. Conditional S-M-B
List of symbols