The Ergodic. Theorem The first major result in ergodic theory was proved in 1931 by G. D. Birkhoff [l]. Theorem l. 5: (Birkhoff Ergodic Theorem) Suppose T: (X, B, m) → (X, B, m) is measure-preserving (where we l 1 * ...i allow (X, 8, ...
Preface In 1970 I gave a graduate course in ergodic theory at the University of Maryland in College Park , and these lectures were the basis of the Springer Lecture Notes in Mathematics Volume 458 called “ Ergodic TheoryIntroductory ...
Author: Peter Walters
Publisher: Springer Science & Business Media
The first part of this introduction to ergodic theory addresses measure-preserving transformations of probability spaces and covers such topics as recurrence properties and the Birkhoff ergodic theorem. The second part focuses on the ergodic theory of continuous transformations of compact metrizable spaces. Several examples are detailed, and the final chapter outlines results and applications of ergodic theory to other branches of mathematics.
(1932d) Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. 33, 587–642. (1934) Almost periodic functions in a group, I, Trans, Amer. Math. Soc. 36, 445–92, WALTERS, PETER *(1975) Ergodic Theory: Introductory Lectures, ...
Author: Karl E. Petersen
Publisher: Cambridge University Press
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.
Author: Iakov Grigorevich SinaiPublish On: 2017-03-14
P. Billingsley, Ergodic Theory and Information. J. Wiley, 1965, 106p. 4. P. Walters, Ergodic Theory. Introductory Lectures. Lecture Notes in Mathematics No. 458, Springer-Verlag, 1975. 5. I. P. Cornfeld, S. V. Fomin, Ya.
Author: Iakov Grigorevich Sinai
Publisher: Princeton University Press
This book concerns areas of ergodic theory that are now being intensively developed. The topics include entropy theory (with emphasis on dynamical systems with multi-dimensional time), elements of the renormalization group method in the theory of dynamical systems, splitting of separatrices, and some problems related to the theory of hyperbolic dynamical systems. Originally published in 1993. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
G.  Ergodic properties of an ideal gas with an infinite number of degrees of freedom. Funct. Anal. Appl. 5, 19–21 (1971). Walters, P.  Ergodic theory. Introductory lectures. Lecture Notes in Mathematics, no. 458.
Author: I. P. Cornfeld
Publisher: Springer Science & Business Media
Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dyna mical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc. The outline of this book became clear to us nearly ten years ago but, for various reasons, its writing demanded a long period of time. The main principle, which we adhered to from the beginning, was to develop the approaches and methods or ergodic theory in the study of numerous concrete examples. Because of this, Part I of the book contains the description of various classes of dynamical systems, and their elementary analysis on the basis of the fundamental notions of ergodicity, mixing, and spectra of dynamical systems. Here, as in many other cases, the adjective" elementary" i~ not synonymous with "simple. " Part II is devoted to "abstract ergodic theory. " It includes the construc tion of direct and skew products of dynamical systems, the Rohlin-Halmos lemma, and the theory of special representations of dynamical systems with continuous time. A considerable part deals with entropy.