Ergodic Theory, Entropy
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214 Meir Smorodinsky University of Warwick, Coventry Warwickshire/G.B.
Ergodic Theory, Entropy
Springer-Verlag Berlin· Heidelberg· NewYork 1971
AMS Subject Classifications (1970): 28A65, 47A35, 94A15
ISBN 3-540-05556-8 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05556-8 Springer-Verlag New York . Heidelberg· Berlin
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© by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 75-171482. Printed in Germany. Offsetdruck: ] ulius Beltz, HemsbachiBergstr.
PHEFACE
These notes are based on one term's M.Sc. course given in Warwick University, Autumn 1970.
for the course was
basic knowledge of measure theory and integration such as covered by
[1].
Espacially, the Radon Nikodym theorem is assumed.
In the history of Ergodic Theory there were three major breakthroughs.
The first was Birkhoff's individual ergodic theorem(lJ
The second was the introduction of entropy by Kollllogorov (1)[2) and Sinai (1].
The third was Sinai's weak isomorphism [2] and Ornstein's
isomorphisms theorem for
[4][5].
shifts and other results [1](2](3)
See also Friedman, Ornstein [1].
All these steps were Since my purpose was to
achieved without the use of spectral theory. introduce the stUdents
to the recent development, I chose to
omit spectral theory from the discussion.
Other important and
beautiful topics which are not mentioned are ergodic theory of operators, the problem of invariant measures and measurable transformation on spaces of infinite measure. expository works: Friedman [1] ;
We refer the reader to the following
Billingsley [1] ; Halmos [2]
Hokhlin [2] : and Sinai
Jacobs [1] ;
[4].
In the treatment of entropy theory I avoided using uncountable partitions and preferred the use of sub-a-fields.
80, these notes
are in the spirit of Billingsley [1) rather than Parry [i ], Rokhlin and Sinai [4). The symbol
invites the reader to complete a missing
arguement, or sometimes, a whole proof.
C2J
IV
I am grateful to the Mathematical Reseach Institute of Warwick University in which I was a Senior Research Fellow while writing these notes. My thanks go to Mr R. Keppler who assisted me in organizing parts of the notes and to Mrs Sue Elworthy for her patient job of them.
Mathematics Institute, Warwick University.
M.S.
fuarch
TABLE OF CONTENTS
Notation, Symbols Chapter I Chapter II Chapter III Chapter IV Chapter V
Examples
.
..
2
.. ··..···· 8 Conditional Probabilities and Expectations. Doob's Martingale Theorem 13 ...· Entropy and Conditional Entropy 19 ··.. Entropy of a Transformation . · · ·
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