Lectures On Ergodic Theory

This concise classic by a well-known master of mathematical exposition covers recurrence, ergodic theorems, ergodicity and mixing properties, and the relation between conjugacy and equivalence. 1956 edition.

These lecture notes provide a unique introduction to Pesin theory and its applications.

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.

These lectures were first given during my tenure of a Walker Ames Visiting Professorship in the Department of Astronautics and Aeronautics at the University of Washington, November 2-12, 1964. I am grateful for the interest shown there and for the tranquil hospitality of Dr. JOHN BOLLARD and Dr. ELLIS DILL, which allowed me the leisure sufficient to write the first manuscript. I thank Dean ROBERT Roy and Dr. GEORGE BENTON for the unusual honor of an invitation to deliver a series of public lectures at my own university. Apart from the footnotes on pp. 49, 50, and 85, which have been added so as to answer questions allowed by the slower pace of silence, and the obviously necessary note on p. 106, the lectures of this second series are here printed as read, February 9-25, 1965. Thus I may call these, in imitation of a famous example, " Bal timore Lectures". Acknowledgment The first lecture is based largely upon my Bingham Medal Address of 1963, part of which it reproduces verbatim. The filth lecture may be regarded as a partial summary of my course on ergodic theory at the International School of Physics, Varenna, 1960. Much of the last lecture runs parallel to my article "The Modern Spirit in Applied Mathematics", ICSU Review of World Science, Volume 6, pp. 195-205 (1964), and some paragraphs are taken from my address to the Fourth U.S. National Congress of Applied Mechanics (1961).