Smooth Ergodic Theory of Random Dynamical Systems
This book studies ergodic-theoretic aspects of random dynam- ical systems, i.e. of deterministic systems with noise. It aims to present a systematic treatment of a series of recent results concerning invariant measures, entropy and Lyapunov exponents of s
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1606
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1606
Springer
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Pei-Dong Liu Min Qian
Smooth Ergodic Theory of Random Dynamical Systems
Springer
Authors Pei-Dong Liu Min Qian Department of Mathematics and Institute of Mathematics Peking University Beijing 100871, P. R. China E-mail: [email protected]
Library of Congress Cataloging-In-Publication Data
Llu, Pei-Dong, 1964Smooth ergodtc theory of random dynamtcal systems I Pet-Dong Liu, Mtn Qlan. p. cm. -- (Lecture notes in mathematics; 1606) Includes bibl10graphtcal references and 1ndex. ISBN 3-540-60004-3 (Berl1n : acid-free). -- ISBN 0-387-60004-3 (New York: acld-free) 1. Ergodic theory. 2. Different1able dynam1cal systems. 3. Stochastic d1fferential equat1ons. I. Ch' ten, Min. II. Title. III. Serles: Lecture notes 1n mathematlcs (Springer-Verlag) ; 1606. QA3.L28 no. 1605 [QA641.5] 510 s--dc20 [514' .74] 95-21996 CIP
Mathematics Subject Classification (1991): 58Fll, 34F05, 34D08, 34C35, 60HlO
ISBN 3-540-60004-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions ofthe German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Great Britain
SPIN: 10130336
46/3142-543210 - Printed on acid-free paper
Table of Contents
Introduction
Vll
Chapter O. Preliminaries §1. Measure Theory §2. Measurable Partitions §3. Conditional Entropies of Measurable Partitions §4. Conditional Entropies of Measure-Preserving Transformations: I §5. Conditional Entropies of Measure-Preserving Transformations: II
1 1 5 7 9 16
Chapter I. Entropy and Lyapunov Exponents of Random Diffeomorphisms . 22 §1. The Basic Measure Spaces and Invariant Measures 22 §2. Measure-Theoretic Entropies of Random Diffeomorphisms 31 §3. Lyapunov Exponents of Random Diffeomorphisms 37 Chapter II. Estimation of Entropy from Above Through Lyapunov Exponents §1. Preliminaries §2. Proof of Theorem 0.1
45 45 51
Chapter III. Stable Invariant Manifolds of Random Diffeomorphisms §1. Some Preliminary Lemmas §2. Some Technical Facts About Contracting Maps §3. Local and Global Stable Manifolds §4. Holder Continuity of Sub bundles §5. Absolute Continuity of Families of Submanifolds §6. Absolute Continuity of Conditional Measures
55 55 61 63 73 84 86
Chapter IV. Estimation of Entropy from Below Through Lyapunov Exponents §1. Introduction and Formulation of the Main Result §2. Construction of A Measurable Partition §3. Est
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