Introduction to Smooth Ergodic Theory (Graduate Studies in Mathematics)

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. Cover Introduction to Smooth Ergodic Theory Copyright © 2013 by the authors. ISBN 978-0-8218-9853-6 QA611.5.B37 2013 515'.39-dc23 LCCN 2013007773 Contents 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. Holder continuity of invariant distributions Chapter 5 Cocycles over Dynamical Systems 5.1. Cocycles and linear extensions 5.1.1. Linear multiplicative cocycles. 5.1.2. Operations with cocycles. 5.1.3. Cohomology and tempered equivalence 5.2. Lyapunov exponents and Lyapunov-Perron regularity for cocycles 5.3. Examples of measurable cocycles over dynamical systems 5.3.1. Reducible cocycles. 5.3.2. Cocycles associated with Schrodinger operators. 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 6.3.1. Lyapunov inner products. 6.3.2. The Oseledets-Pesin Reduction Theorem. 6.4. Lyapunov charts 6.4.1. A tempering kernel. 6.4.2. Construction of 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 7.3.1. Sizes of local manifolds 7.3.2. Smoothness of local manifolds 7.3.3. Graph transform property. 7.3.4. Global manifolds. 7.3.5. Stable manifold theorem for flows. 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 9.3.1. The metric entropy of a diffeomorphism. 9.3.2. Upper bound for the metric entropy 9.3.3. Lower bound for the metric entropy. Chapter 10 Geodesic Flows on Surfaces of Nonpositive Curvature 10.1. Preliminary information from Riemannian geometry 10.1.1. The canonical Riemannian metric. 10.1.2. Geodesics 10.1.3. The universal Riemannian covering 10.1.4. Curvature. 10.1.5. Fermi coordinates 10.1.6. Jacobi fields. 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 14.1.1. Transfinite hierarchy of set filtrations 14.1.1.1. Set filtrations 14.1.1.2. The hierarchy 14.1.1.3. Termination of the process 14.1.2. Proof of Theorem 14.1 14.2. The Anosov rigidity phenomenon. II Chapter 15 C1 Pathological Behavior: Pugh's Example Bibliography Index Back Cover 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. Readership: Graduate students interested in dynamical systems and ergodic theory and research mathematicians interested in smooth ergodic theory. 1. Examples Of Hyperbolic Dynamical Systems -- 2. General Theory Of Lyapunov Exponents -- 3. Lyapunov Stability Theory Of Nonautonomous Equations -- 4. Elements Of The Nonuniform Hyyperbolicity Theory -- 5. Cocycles Over Dynamical Systems -- 6. The Multiplicative Ergodic Theorem -- 7. Local Manifold Theory -- 8. Absolute Continuity Of Local Manifolds -- 9. Ergodic Properties Of Smooth Hyperbolic Measures -- 10. Geodesic Flows On Surfaces Of Nonpositive Curvature -- 11. Cone Technics -- 12. Partially Hyperbolic Diffeomorphisms With Nonzero Exponents -- 13. More Examples Of Dynamical Systems With Nonzero Lyapunov Exponents -- 14. Anosov Rigidity -- 15. C1 Pathological Behavior: Pugh's Example. Luis Barreira, Yakov Pesin. Includes Bibliographical References (pages 267-271) And Index.