Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities
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1222 Anatole Katok Jean-Marie Strelcyn with the collaboration of F. Ledrappier and F. Przytycki
Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Authors Anatole Katok Mathematics 253-37, California Institute of Technology Pasadena, CA 91125, USA Jean-Marie Strelcyn Universite Paris-Nord, Centre Scientifique et Polytechnique Departernent de Mathematiques Avenue 1.-B. Clement, 93430 Villetaneuse, France Francois Ledrappier Laboratoire de Probabilites, Universite Paris VI 4 Place Jussieu, 75230 Paris, France Feliks Przytycki Mathematical Institute of the Polish Academy of Sciences ul. Sniadeckich 8, 00-950 Warsaw, Poland
Mathematics Subject Classification (1980): Primary: 28020, 34F05, 58F 11, 58F 15 Secondary: 34C35, 58F08, 58F18, 58F20, 58F22, 58F25 ISBN 3-540-17190-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17190-8 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
TABLE OF CONTENTS Introduction
V
EXISTENCE OF INVARIANT MANIFOLDS FOR SMOOTH MAPS WITH SINGULARITIES
PART I.
(by A. KATOK and J.-M. STRELCYN) 1.
Class of Transformations with Singularities
2.
Preliminaries
3.
Overcoming Influence of Singularities
10
4.
The Proof of Lemma 3.3 and Related Topics
19
5.
The Formulation of Pesin's Abstract Invariant Manifold Theorem
24
6.
Invariant Manifolds for Maps Satisfying Conditions (1.1) - (1.3)
25
7.
Some Additional Properties of Local Stable Manifolds
35
PART II.
1
5
ABSOLUTE CONTINUITY
41
(by A. KATOK and J.-M. STRELCYN) 1.
Introduction
41
2.
Preliminary Remarks and Notations
42
3.
Some Facts from Measure Theory and Linear Algebra
46
4.
Formulation of the Absolute Continuity Theorem and a Sketch of the Proof
55
5.
Start of the Proof - I
62
6.
The First Main Lemma
65 79
7.
Start of the Proof - II
8.
Projection and Covering Lemmas
9.
Comparison of the Volumes
88 107
10.
The Proof of the Absolute Continuity Theorem
117
11.
Absolute Continuity of Conditional Measures
130
12.
Infinite Dimensional Case
138
13.
Final Remarks
154
PART III.
THE ESTIMATION OF ENTROPY FROM BELOW THROUGH LYAPUNOV CHARACTERISTIC EXPONENTS
157
(by F. LEDRAPPIER and J.-M. STRELCYN) 1.
Introduction and Formulation of the Results
157
2.
Preliminaries
162
3.
Construction of the Partition
4.
Computation of Entropy
n
167
175
IV
PART IV.
THE ESTIMATION OF ENTROPY FROM ABOVE THROUGH LYAPUNOV CHARACTERISTIC EXPONENTS
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