Ergodic Theory (Cambridge Studies in Advanced Mathematics Book 2) by Karl E. Petersen (PDF)

Description

The study of dynamical systems forms a vast and rapidly developing field even when one considers only activity whose methods derive mainly from measure theory and functional analysis. Karl Petersen has written a book which presents the fundamentals of the ergodic theory of point transformations and then several advanced topics which are currently undergoing intense research. By selecting one or more of these topics to focus on, the reader can quickly approach the specialized literature and indeed the frontier of the area of interest. Each of the four basic aspects of ergodic theory – examples, convergence theorems, recurrence properties, and entropy – receives first a basic and then a more advanced, particularized treatment. At the introductory level, the book provides clear and complete discussions of the standard examples, the mean and pointwise ergodic theorems, recurrence, ergodicity, weak mixing, strong mixing, and the fundamentals of entropy. Among the advanced topics are a thorough treatment of maximal functions and their usefulness in ergodic theory, analysis, and probability, an introduction to almost-periodic functions and topological dynamics, a proof of the Jewett-Krieger Theorem, an introduction to multiple recurrence and the Szemeredi-Furstenberg Theorem, and the Keane-Smorodinsky proof of Ornstein’s Isomorphism Theorem for Bernoulli shifts. The author’s easily-readable style combined with the profusion of exercises and references, summaries, historical remarks, and heuristic discussions make this book useful either as a text for graduate students or self-study, or as a reference work for the initiated.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This text is a must have for anyone studying measure theoretic ergodic theory. The introductory chapter is broad (and difficult) but it does show the vast breadth of the subject. I especially liked the historical references and background material there. The treatment of the mean ergodic theorem is also very nice and worked well with my MSc. students. The inclusion of very new material, like section 4.5, was very helpful for me as a Ph.D. student struggling to understand the research papers.

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