光滑遍历理论导论(影印版)

本书是关于光滑遍历理 论的首次系统介绍。它由两 部分组成：第一部分介绍了 理论核心，第二部分讨论了 更高级的主题。特别是，书 中描述了Lyapunov指数的 一般理论及其在微分方程稳 定性理论中的应用、非均匀 双曲性的概念、稳定流形理 论（着重介绍不变叶状结构 的绝对连续性）以及具有非 零Lyapunov指数的动力系 统的遍历理论。作者还详细 描述了所有具有非零 Lyapunov指数的保守系统 的基本例子，包括在非正曲 率紧曲面上的测地线流。 本书是Lyapunov Exponents and Smooth Ergodic Theory的修订和大 幅扩展版，面向专攻动力系 统和遍历理论的研究生，以 及任何想要掌握光滑遍历理 论并学会使用其工具的人士 。本书设有80多个练习题， 可用作光滑遍历理论高级课 程的主要教材。本书内容自 封，读者只需要基本的实分 析、测度论、微分方程和拓 扑学的知识，即便如此，作 者仍然提供了读者所需的背 景定义和结果。

Preface 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.HSlder 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