Randomness and Recurrence in Dynamical Systems aims to bridge a gap between undergraduate teaching and the research level in mathematical analysis. It makes ideas on averaging, randomness, and recurrence, which traditionally require measure theory, accessible at the undergraduate and lower graduate level. The author develops new techniques of proof and adapts known proofs to make the material accessible to students with only a background in elementary real analysis. Over 60 figures are used to explain proofs, provide alternative viewpoints and elaborate on the main text. The book explains further developments in terms of measure theory. The results are presented in the context of dynamical systems, and the quantitative results are related to the underlying qualitative phenomena—chaos, randomness, recurrence and order. The final part of the book introduces and motivates measure theory and the notion of a measurable set, and describes the relationship of Birkhoff's Individual Ergodic Theorem to the preceding ideas. Developments in other dynamical systems are indicated, in particular Lévy's result on the frequency of occurence of a given digit in the partial fractions expansion of a number. Historical notes and comments suggest possible avenues for self-study.

The book is very well written and the author clearly motivates all definitions and theorems. Excellent illustrations throughout help cement the reader's understanding of the material and proofs are given in full detail. Great attention was paid in the writing and editing of this book, as I have not come across any typos.

Peter Rabinovitch Source: MAA Online Reviews

This book, which is appropriate for first-year graduate or (more likely) upper division undergraduate study, aims to introduce standard results having dynamical underpinnings, e.g., those involving averaging, randomness, recurrence, etc., in such a way as to require little background. ... This is a unique book that has been put together with extraordinary care and attention to detail, and the topics are completely in keeping with the overall theme. Many of the sections are self-contained and read like expository articles; as such, the book could be an excellent resource for math club talks, student presentations, etc.; one can imagine a course based on this book in which advanced undergraduate mathematics majors and first year graduate students take turns giving lectures on the various sections. Useful in this regard is the fact that, in addition to exercises at the close of each chapter, a handful of 'Investigations' (like exercises, but more involved and open-ended) are suggested. A recommended book for the appropriate audience and use.

Randall McCutcheon Source: Mathematical Reviews