Introduction to Smooth Ergodic Theory 1st Edition by Luís Barreira, Yakov Pesin ISBN 978-0821898536 0821898531
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ISBN 10: 0821898531
ISBN 13: 978-0821898536
Author: Luís Barreira, Yakov Pesin
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.
Table of contents:
Part 1. The Core of the Theory
Chapter 1. Examples of Hyperbolic Dynamical Systems
§1.1. Anosov diffeomorphisms
§1.2. Anosov flows
§1.3. The Katok map of the 2-torus
§1.4. Diffeomorphisms with nonzero Lyapunov exponents on surfaces
§1.5. A flow with nonzero Lyapunov exponents
Chapter 2. General Theory of Lyapunov Exponents
§2.1. Lyapunov exponents and their basic properties
§2.2. The Lyapunov and Perron regularity coefficients
§2.3. Lyapunov exponents for linear differential equations
§2.4. Forward and backward regularity. The Lyapunov-Perron regularity
§2.5. Lyapunov exponents for sequences of matrices
Chapter 3. Lyapunov Stability Theory of Nonautonomous Equations
§3.1. Stability of solutions of ordinary differential equations
§3.2. Lyapunov absolute stability theorem
§3.3. Lyapunov conditional stability theorem
Chapter 4. Elements of the Nonuniform Hyperbolicity Theory
§4.1. Dynamical systems with nonzero Lyapunov exponents
§4.2. Nonuniform complete hyperbolicity
§4.3. Regular sets
§4.4. Nonuniform partial hyperbolicity
§4.5. Hölder continuity of invariant distributions
Chapter 5. Cocycles over Dynamical Systems
§5.1. Cocycles and linear extensions
§5.2. Lyapunov exponents and Lyapunov-Perron regularity for cocycles
§5.3. Examples of measurable cocycles over dynamical systems
Chapter 6. The Multiplicative Ergodic Theorem
§6.1. Lyapunov-Perron regularity for sequences of triangular matrices
§6.2. Proof of the Multiplicative Ergodic Theorem
§6.3. Normal forms of measurable cocycles
§6.4. Lyapunov charts
Chapter 7. Local Manifold Theory
§7.1. Local stable manifolds
§7.2. An abstract version of the Stable Manifold Theorem
§7.3. Basic properties of stable and unstable manifolds
Chapter 8. Absolute Continuity of Local Manifolds
§8.1. Absolute continuity of the holonomy map
§8.2. A proof of the absolute continuity theorem
§8.3. Computing the Jacobian of the holonomy map
§8.4. An invariant foliation that is not absolutely continuous
Chapter 9. Ergodic Properties of Smooth Hyperbolic Measures
§9.1. Ergodicity of smooth hyperbolic measures
§9.2. Local ergodicity
§9.3. The entropy formula
Chapter 10. Geodesic Flows on Surfaces of Nonpositive Curvature
§10.1. Preliminary information from Riemannian geometry
§10.2. Definition and local properties of geodesic flows
§10.3. Hyperbolic properties and Lyapunov exponents
§10.4. Ergodic properties
§10.5. The entropy formula for geodesic flows
Part 2. Selected Advanced Topics
Chapter 11. Cone Technics
§11.1. Introduction
§11.2. Lyapunov functions
§11.3. Cocycles with values in the symplectic group
Chapter 12. Partially Hyperbolic Diffeomorphisms with Nonzero Exponents
§12.1. Partial hyperbolicity
§12.2. Systems with negative central exponents
§12.3. Foliations that are not absolutely continuous
Chapter 13. More Examples of Dynamical Systems with Nonzero Lyapunov Exponents
§13.1. Hyperbolic diffeomorphisms with countably many ergodic components
§13.2. The Shub-Wilkinson map
Chapter 14. Anosov Rigidity
§14.1. The Anosov rigidity phenomenon. I
§14.2. The Anosov rigidity phenomenon. II
Chapter 15. C¹ Pathological Behavior: Pugh’s Example
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Tags: Luís Barreira, Yakov Pesin, Introduction to Smooth, Ergodic Theory