An Outline of Ergodic Theory

By Author(s)
An Outline of Ergodic Theory
Cambridge Studies in Advanced Mathematics 122

Steven Kalikow and Randall McCutcheon

Cambridge University Press (2010)

174 pp.
Price: $59.00 (hardcover)
ISBN 978-0-521-19440-2
Reviewer: Charles H. Morgan, Jr.
Book Reviews Editor
Department of Mathematics
Lock Haven University of Pennsylvania
Lock Haven, Pennsylvania

I have never been a fan of mathematics textbooks which attempt to give intuitive, "insightful" (usually only to the author) explanations of advanced concepts, even when the author immediately follows that explanation with a rigorous definition. These attempts often result in oversimplifications which fail to present the salient features of the concepts to the reader and which defy the precision required of a mathematical discussion, particularly since many students latch onto the imprecise, intuitive "definition." Furthermore—and worst of all!—these "plain English" explanations turn the reader from an active thinker into a passive observer. Any professor who has used a 1200-page calculus textbook in his/her freshman-level calculus class will know precisely what I mean. The rigorous definitions of limit and differentiability presented to the mathematical world by Karl Weierstraß and his students have yet again been supplanted—through the writing of well-intentioned textbook authors who wish to make the material "clear and understandable"—by the intuitive "definitions" which permitted André Ampère, Joseph L. Bertrand, and Sylvestre Lacroix to "prove" that every continuous function is differentiable.[1] In too many cases, these well-intentioned, ill-performed attempts at making mathematics textbooks more accessible have begun to infect the world of graduate-level textbooks.

Let me make it clear that I am not lodging this complaint against Stephen Kalikow and Randall McCutcheon. Their book An Outline of Ergodic Theory is an excellent summary of the major results of ergodic theory. I use the word "summary" here in the most complimentary way. In only 174 pages, Kalikow and McCutcheon have written a book which presents everything which a graduate student should know about ergodic theory; yet, they have not indulged the reader in long-winded, overly simplified explanations which allow the reader to avoid having to think long and hard about the meaning of a theorem or a definition. This book is written in precisely the way that a mathematician reads a book or journal article anyway; that is, we just read the theorems, definitions, lemmas, and proofs; and we skip all the extraneous text in between as unnecessary clutter. After all, we do not need the author's opinion clouding our own understanding of the subject.

An Outline of Ergodic Theory is written more in outline form than in classic prose. There is no narrative thread to tie together the paragraphs. The paragraphs are numbered 1 through 763, and they are appropriately terse.[2] The book is very well-organized as an enumerated collection of definitions, theorems, proof sketches, exercises, and occasional comments. Yet, everything in this book is presented in a clear, logical, readable order. This book is very much like a brief version of the excellent classic text by Maurice Kendall and Alan Stuart.[3]

Kalikow and McCutcheon assume that the reader has had the standard graduate-level course in measure theory and integration. Chapter 1 begins with about four pages of reminders—definitions of algebra, measure, measurable space, event, Carathéodory's Theorem, Tychonoff's Theorem—before beginning with Lebesgue spaces, factor spaces, random variables, conditional expectation, and stochastic processes—topics which are usually not covered in a one-semester graduate course in measure theory and integration.

Chapters 2 through 7 cover all the essential topics which a dynamicist should know in ergodic theory. These include measure-preserving transformations, stationary processes, the Rokhlin Tower Theorem, Birkhoff's Ergodic Theorem, martingales, stochastic coupling, Shannon-Breiman-Macmillan Theorem, entropy, Bernoulli transformations, Ornstein's Isomorphism Theorem, and Sinai's Theorem on factors of full entropy, for example.

The seventh chapter closes the book with what is arguably its most important material. In this final chapter, Kalikow and McCutcheon discuss different types of mixing (strong mixing and weak mixing) and how these fit into K-systems and Bernoulli systems.

For more advanced students, the book could be used in a one-semester course, and there are more than enough exercises in the text to keep a graduate student busy for the entire semester. Most instructors, though, would probably use the book as a text for a two-semester course. Even then, there are certainly enough exercises for a two-semester course since the authors give only ideas or sketches of proofs and leave completion of the proofs as excellent possible exercises.

The outline structure of the book makes it an excellent reference for students who are taking a course in ergodic theory or for students who have completed one.


[1] Morris Kline, Mathematics: The Loss of Certainty, Barnes and Noble, New York, 2009, p. 193.

[2] Let the reader not be put off by my use of the word "terse," which often has a negative connotation. Rather, Merriam-Webster's dictionary defines the word well: "smoothly elegant; devoid of superfluity."

[3] Maurice Kendall and Alan Stuart, The Advanced Theory of Statistics, Macmillan Publishing Co., Inc., New York, 1977.

Categories: Magazine, Book Reviews

Please login or register to post comments.