Introduction to smooth ergodic theory

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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

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