Ergodic Theory on Compact Spaces
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527 Manfred Denker Christian Grillenberger Karl Sigmund
Ergodic Theory on Compact Spaces
Springer-Verlag Berlin· Heidelberg· New York 1976
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
527 Manfred Denker Christian Grillenberger Karl Sigmund
Ergodic Theory on Compact Spaces
Springer-Verlag Berlin· Heidelberg· New York 1976
Authors Manfred Denker Christian Grillenberger Institut fur Mathematische Statistik LotzestraBe 13 0-3400 Gottinqen
Karl Sigmund Mathematisches Institut Strudlhofgasse 4 A-1090 Wi en
Library of Congress Cataloging in Publication Data
Denker, Manfred, 1944-
Ergodic theory on compact spaces.
(Lecture notes in mathematics ; vol. 527) Bibliography: p. Includes index. 1. Topological dynamics. 2. Ergodic theory. 3. Metric spaces. 4. Locally compact spaces. I. Grillenberger, Christian, 1941joint author. II. Sigmund, Karl, 1945joint author. III. Title. IV. Series: Lecture notes in mathematics (Berlin) ; vol. 527. QA3.I28 vol. 527 [QA6l1.5] 5l0'.8s [514'.3] 76-19105
AMS Subject Classifications (1970): 28A50, 28A65, 54H20 ISBN 3-540-07797-9 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-07797-9 Springer-Verlag New York' Heidelberg' Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
Contents
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.
Measure Theoretical Dynamical Systems.............
3
2.
Measures on Compact Metric Spaces.................
8
3.
Invariant Measures for Continuous Transformations
17
4.
Time Averages
20
5.
Ergodici ty
'. . . . . . . . . . . . . . . . . . . . . .
23
6.
Mixing and Transitivity •.•.•.•..••••..••.....••.••
30
7.
Shifts and Sub shifts • • • • . . • • . • . • • • • • . . . • . • . . . . • • • •
36
8.
Measures on the Shift Space .••••..........•..••.•
41
9.
Partitions and Generators •.••.•••.•..••••.••.•.•.•
49
10.
Information and Entropy •..•.••••••••••••.••••••.••
56
11.
Computation of Entropy ...•••.••••....••.•.....••••
62
12.
Entropy for Bernoulli- and Markov Shifts .••.•••••
68
13.
Ergodic Decompositions ••.•••••••••..••••..•.••••.•
73
14.
Topological Entropy...............................
82
15.
Topological Generators .•..•..•..••••..•••....•...•
92
16.
Expansive Homeomorphisms ...•.•..•.•••••••••..••••• 103
17.
Subshifts of Finite Type •.•.•••.•••....•.....••.•• 117
18.
Variational Principle for Topological Entropy ••••• 131
19.
Measures with Maximal Entropy Intrinsically
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