This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapunov exponents and its applications to the stability theory of differential equations, the concept of nonuniform hyperbolicity, stable manifold theory (with emphasis on the absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. The authors also present a detailed description of all basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature. This book is a revised and considerably expanded version of the previous book by the same authors Lyapunov Exponents and Smooth Ergodic Theory (University Lecture Series, Vol. 23, AMS, 2002). It is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wants to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools. With more than 80 exercises, the book can be used as a primary textbook for an advanced course in smooth ergodic theory. The book is self-contained and only a basic knowledge of real analysis, measure theory, differential equations, and topology is required and, even so, the authors provide the reader with the necessary background definitions and results.
Author(s): Luis Barreira, Yakov Pesin
Series: Graduate Studies in Mathematics 148
Publisher: American Mathematical Society
Year: 2013
Language: English
Pages: 289
Preface VII
Part 1. The Core of the Theory
Chapter 1. Examples of Hyperbolic Dynamical Systems 3
1.1. Anosov diffeomorphisms 4
1.2. Anosov flows 8
1.3. The Katok map of the 2-torus 13
1.4. Diffeomorphisms with nonzero Lyapunov exponents on surfaces 23
1.5. A flow with nonzero Lyapunov exponents 27
Chapter 2. General Theory of Lyapunov Exponents 33
2.1. Lyapunov exponents and their basic properties 33
2.2. The Lyapunov and Perron regularity coefficients 38
2.3. Lyapunov exponents for linear differential equations 41
2.4. Forward and backward regularity. The Lyapunov-Perron regularity 51
2.5. Lyapunov exponents for sequences of matrices 56
Chapter 3. Lyapl1nov Stability Theory of Nonautonomous Equations 61
3.1. Stability of solutions of ordinary differential equations 62
3.2. Lyapunov absolute stability theorem 68
3.3. Lyapunov conditional stability theorem 72
Chapter 4. Elements of the Nonuniform Hyperbolicity Theory 77
4.1. Dynamical systems with nonzero Lyapunov exponents 78
4.2. Nonuniform complete hyperbolicity 88
4.3. Regular sets 91
4.4. Nonuniform partial hyperbolicity 93
4.5. Holder continuity of invariant distributions 94
Chapter 5. Co cycles over Dynamical Systems 99
5.1. Cocycles and linear extensions 100
5.2. Lyapunov exponents and Lyapunov-Perron regularity for cocycles 105
5.3. Examples of measurable co cycles over dynamical systems 109
Chapter 6. The Multiplicative Ergodic Theorem 113
6.1. Lyapunov-Perron regularity for sequences of triangular matrices 114
6.2. Proof of the Multiplicative Ergodic Theorem 120
6.3. Normal forms of measurable co cycles 124
6.4. Lyapunov charts 128
Chapter 7. Local Manifold Theory 133
7.1. Local stable manifolds 134
7.2. An abstract version of the Stable Manifold Theorem 137
7.3. Basic properties of stable and unstable manifolds 147
Chapter 8. Absolute Continuity of Local Manifolds 155
8.1. Absolute continuity of the holonomy map 157
8.2. A proof of the absolute continuity theorem 161
8.3. Computing the Jacobian of the holonomy map 167
8.4. An invariant foliation that is not absolutely continuous 168
Chapter 9. Ergodic Properties of Smooth Hyperbolic Measures 171
9.1. Ergodicity of smooth hyperbolic measures 171
9.2. Local ergodicity 176
9.3. The entropy formula 183
Chapter 10. Geodesic Flows on Surfaces of Nonpositive Curvature 195
10.1. Preliminary information from Riemannian geometry 196
10.2. Definition and local properties of geodesic flows 198
10.3. Hyperbolic properties and Lyapunov exponents 200
10.4. Ergodic properties 205
10.5. The entropy formula for geodesic flows 210
Part 2. Selected Advanced Topics
Chapter 11. Cone Technics 215
11.1. Introduction 215
11.2. Lyapunov functions 217
11.3. Co cycles with values in the symplectic group 221
Chapter 12. Partially Hyperbolic Diffeomorphisms with Nonzero Exponents 223
12.1. Partial hyperbolicity 224
12.2. Systems with negative central exponents 227
12.3. Foliations that are not absolutely continuous 229
Chapter 13. More Examples of Dynamical Systems with Nonzero Lyapunov Exponents 235
13.1. Hyperbolic diffeomorphisms with countably many ergodic components 235
13.2. The Shub-Wilkinson map 246
Chapter 14. Anosov Rigidity 247
14.1. The Anosov rigidity phenomenon. I 247
14.2. The Anosov rigidity phenomenon. II 255
Chapter 15. C 1 Pathological Behavior: Pugh's Example
Bibliography
Index
