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. Language: English