Ergodic Theory
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 questio
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Editors
M. Artin S. E. Heinz F. ·W. Magnus W. Schmidt
S. Chern 1. L. Doob A. Grothendieck Hirzebruch L. Hormander S. Mac Lane C. C. Moore 1. K. Moser M. Nagata D. S. Scott 1. Tits B. L. van der Waerden
Managing Editors
B. Eckmann
S. R. S. Varadhan
I. P. Cornfeld S. V. Fomin Ya. G. Sinai
Ergodic Theory
Springer-Verlag New York Heidelberg Berlin
I. P. Cornfeld S. V. Fomin Ya. G. Sinai Landau Institute of Theoretical Physics Academy of Sciences Vorobiew Chasse-2 Moscow V-334 USSR
Translator
A. B. Sossinskii
AMS Subject Classification (1980): 47A35, 54H20, 58Fll
Library of Congress Cataroging in Publication Data Fomin, S. V. (SergeI Vasil'evich). Ergodic theory. (Grundlehren der mathematischen Wissenschaften; 245) Bibliography: p. Includes index. 1. Ergudic theory. 2. Differentiable dynamical systems. I. Cornfeld, I. P. II. Sinai, lAkov III. Title. IV. Series. Grigor'evich, 193581-5355 QA611.5.F65 515.4'2 AACR2
© 1982 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Ave., New York, NY 10010, USA.
9 8 7 6 543 2 1 ISBN 978-1-4615-6929-9 ISBN 978-1-4615-6927-5 (eBook) DOI 10.1007/978-1-4615-6927-5
Preface
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 dynamical 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 construction 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. We have included here the proof of Ornstein's theorem on the isomorphism of Bernoulli automorphisms with the same entropy due to Keane and Smorodinski; this proof is nearer to information theory than Ornstein's original proof. Before the appearance of the entropy theory of dynamical systems, the principal invariant of a dynamical system was thought to be its spectrum. Problems
