Lyapunov指数和光滑遍历理论

本书是对光滑遍历理论的系统介绍。讨论的主题包括Lyapunov指数的一般（抽象）理论及其在微分方程稳定性理论、稳定流形理论、绝对连续性和具有非警Lyapunov指数动力系统（包括测地流）的遍历理论中的应用。 作者通过几个具有非零Lyapunov指数动力系统的非平凡实例，说明了该理论的一些基本方法和思想。本书内容自封，读者需要有实分析、测度论、微分方程和拓扑的基本知识。作者提出了光滑遍历理论的基本概念，并给出了主要结果的完整证明。作者还介绍了一些更前沿的结果，使读者对光滑遍历理论有更广泛的认识。本书可供那些希望熟悉该领域的一般读者使用。

Preface
Introduction
Chapter 1.Lyapunov Stability Theory of Diferential Equations
1.1.Lyapunov Exponents for Differential Equations
1.2.Abstract Theory of Lyapunov Exponents
1.3.Forward and Backward Regularity
1.4.Stability Theory of Nonautonomous Differential Equations
1.5.Lyapunov Regularity and the Oseledets Decomposition
Chapter 2.Elements of Nonuniform Hyperbolic Theory
2.1.Dynamical Systems with Nonzero Lyapunov Exponents
2.2.Nonuniform Hyperbolicity and Regular Sets
2.3.Holder Continuity of Invariant Distributions
2.4.Proof of the Multiplicative Ergodic Theorem
Chapter 3.Examples of Nonuniformly Hyperbolic Systems
3.1.Anosov Diffeomorphisms
3.2.Diffeomorphisms with Nonzero Lyapunov Exponents on Surfaces
3.3.A Flow with Nonzero Lyapunov Exponents
3.4.Geodesic Flows on Compact Manifolds of Nonpositive Curvature
Chapter 4.Local Manifold Theory
4.1.Existence of Local Stable Manifolds
4.2.Basic Properties of Stable and Unstable Manifolds
4.3.Absolute Continuity Property
4.4.Computing the Jacobian of the Holonomy Map
4.5.Partial Hyperbolicity
Chapter 5.Ergodic Properties of Smooth Hyperbolic Measures
5.1.Absolute Continuity and Smooth Invariant Measures
5.2.Ergodicity of Smooth Hyperbolic Measures
5.3.Local Ergodicity
5.4.The Entropy Formula
5.5.SRB-Measures and General Hyperbolic Measures
5.6.Geodesic Flows on Compact Surfaces of Nonpositive Curvature
Bibliography
Index