Ergodic Theory of Random Transformations
Ergodic theory of dynamical systems i.e., the qualitative analysis of iterations of a single transformation is nowadays a well developed theory. In 1945 S. Ulam and J. von Neumann in their short note [44] suggested to study ergodic theorems for the more g
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Birkhauser Boston· Basel . Stuttgart
Yuri Kifer Ergodic Theory of Random Transformations
1986
Birkhauser Boston' Basel· Stuttgart
Author: Yuri Kifer Institute of Mathematics and Computer Science Givat Ram 91904 J erusalem/Israel
Library of Congress Cataloging in Publication Data Kifer, Yuri, 1948Ergodic theory of random transformations. (Progress in probability and statistics ; vol. 10) Bibliography: p. 1. Stochastic differential equations. 2. Differentiable dynamical systems. 3. Ergodic theory. 4. Transformations (Mathematics) 1. Title. II. Series: Progress in probability and statistics ; v. 10. QA274.23.K53 1985 519.2 85-18645
CIP-Kurztitelaufnahme der Deutschen Bibliothek Kifer, Yuri: Ergodic theory of random transformations I Yuri Kifer. - Boston ; Basel ; Stuttgart Birkhauser, 1986. (Progress in probability and statistics Vol. 10) NE:GT
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© 1986 Birkhauser Boston, Inc. ISBN 978-1-4684-9177-7 ISBN 978-1-4684-9175-3 (eBook) DOI 10.1007/978-1-4684-9175-3
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Table of Contents Introdu ction.
1. General analysis of random maps. 1.1. Markov chains as compositions of random maps.
1 7 7
1.2. Invariant measures and ergodicity.
13
1. 3. Characteristic exponents in metric spaces.
26
II. Entropy characteristics of random transformations.
33
2.1. Measure theoretic entropies.
33
2.2. Topological entropy.
67
2.3. Topological pressure. ill. Random bundle maps.
82 88
3.1. Oseledec's theorem and the "non-random" multiplicative ergodic theorem. 3.2. Biggest characteristic exponent. 3.3. Filtration of invariant subbundles.
88 99
115
N. Further study of invariant sub bundles and characteristic exponents. 4.1. Continuity of invariant subbundles.
130 130
4.2 Stability of lhe biggest exponent.
135
4.3. Exponential growth rales.
140
V. Smooth random transformations. 5.1. Random diffeomorphisms.
5.2. Slochastic flows.
156 156
175
Appendix.
191
A.1. Ergodic decompositions.
191
A.2. Subadditive ergodic theorem.
200
References.
208
Frequently used notations
B(M)- the Borel a-algebra of M. [(M,N)-the space of continuous maps from M to N. [k-class - continuous together with k-derivatives. Dfthe differential of a map! lZr-the expectation of a random variable r.
Jr -a space of transformations on M. f- a random transformation with a distribution m. F'- a random bundle map with a distribution n. nf
= fn
C
•••
c fl'
nF
= Fn
C
c Fl ' D nf
•••
means
the
differential of n f.
hp(f)- the metric entropy of f with respect to an invariant measure p.
L(f)- the topological entropy of f. I -
the unit interval.
I1(M,'I7) - the space of functions g with
J
ig id'17
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