Introduction to smooth ergodic theory
著者
書誌事項
Introduction to smooth ergodic theory
(Graduate studies in mathematics, v. 148)
American Mathematical Society, c2013
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注記
Includes bibliographical references (p. 267-271) and index
内容説明・目次
内容説明
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 aimed at graduate students specialising 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.
目次
Preface
Part I. The core of the theory
Examples of hyperbolic dynamical systems
General theory of Lyapunov exponents
Lyapunov stability theory of nonautonomous equations
Elements of the nonuniform hyperbolicity theory
Cocycles over dynamical systems
The Multiplicative Ergodic Theorem
Local manifold theory
Absolute continuity of local manifolds
Ergodic properties of smooth hyperbolic measures
Geodesic flows on surfaces of nonpositive curvature
Part II. Selected advanced topics
Cone technics
Partially hyperbolic diffeomorphisms with nonzero exponents
More examples of dynamical systems with nonzero Lyapunov exponents
Anosov rigidity
𝐶(1) pathological behavior: Pugh's example
Bibliography
Index
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