Ergodic Theory of Random Transformations

220 Pages·1986·5.138 MB·English

by  Yuri Kifer (auth.)

Progress in Probability
and Statistics
Vol. 10
Edited by
Peter Huber
Murray Rosenblatt
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, 1948-
Ergodic 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
All rights reserved.
No part of this publication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise,
without prior permission of the copyright owner.
© 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
To my family
Table of Contents
Int rodu ction. 1
1. General analysis of random maps. 7
1.1. Markov chains as compositions of random maps. 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.
82
ill. Random bundle maps. 88
3.1. Oseledec's theorem and the
"non-random" multiplicative ergodic theorem. 88
3.2. Biggest characteristic exponent.
99
3.3. Filtration of invariant subbundles.
115
N. Further study of invariant sub bundles and characteristic
exponents. 130
4.1. Continuity of invariant subbundles. 130
4.2 Stability of lhe biggest exponent. 135
4.3. Exponential growth rales. 140
V. Smooth random transformations. 156
5.1. Random diffeomorphisms.
156
5.2. Slochastic flows.
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 fl' nF Fn c Fl ' D nf means the
C ••• C •••
differential of n f.
hp(f)- the metric entropy of f with respect to an invariant meas
ure p.
L(f)- the topological entropy of f.
I - the unit interval.
J
I1(M,'I7) - the space of functions g with ig id'17 <
00
M
= =
P m'" or p n ....
f fA j- the probability of A.
= =
XA- the indicator of a set A i.e., XA(x) 1 if x E A and 0 for oth
erwise.
(J(M)- the space of probability measures on M.
rr
m-l - the (m-l)-dimensional projective space.
IRm - the m-dimensional Eucledean space.
~ - the unit circle.
'tl'-a space of vector bundle maps.
TM - the tangent bundle of a smooth manifold M.
o = Jf- or 0 = 'tl' -.
• - the end of the proof.
Statement i.j - i denotes the section and j denotes the number of
this statement in the section. The Roman number at the begin
ning (for instance, III. 1.2) means the number of the chapter.
-1-
Introduction
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
general situation when one applies in turn different transforma
tions chosen at random. Their program was fulfilled by S. Kakutani
[23] in 1951. 'Both papers considered the case of transformations
with a common invariant measure. Recently Ohno [38] noticed
that this condition was excessive. Ergodic theorems are just the
beginning of ergodic theory. Among further major developments
are the notions of entropy and characteristic exponents.
The purpose of this book is the study of the variety of ergodic
theoretical properties of evolution processes generated by
independent applications of transformations chosen at random
from a certain class according to some probability distribution.
The book exhibits the first systematic treatment of ergodic theory
of random transformations i.e., an analysis of composed actions of
independent random maps. This set up allows a unified approach to
many problems of dynamical systems, products of random
matrices and stochastic flows generated by stochastic differential
equations.
The precise set up is the followmg. Let Jr be a space of
transformations acting in a certain space M. Suppose that Jr
possesses some measurable structure so that one can consider Jr
valued random variables f which we shall call random transforma
tions or random maps. Of course, this means that f is a Jr-valued