Hyperbolic Dynamical Systems

Contents Chapter 1. Introduction 7 1. Historical sketch 7 a. Homoclinic tangles 7 b. Geodesic flows 8 c. Picking up from Poincaré 8 d. Modern hyperbolic dynamics 9 e. The slowness of the initial development 9 2. Hyperbolic dynamics 10 3. Outline of this survey 10 a. Regularity 11 Chapter 2. Hyperbolic sets and stable manifolds 13 1. Definitions and examples 13 a. Hyperbolic linear maps, adapted norm 13 b. Hyperbolic sets 13 c. Basic sets, Axiom A, Anosov diffeomorphisms 14 d. Examples 14 1. The Smale horseshoe 14 2. Transverse homoclinic points and horseshoes 15 3. The Smale attractor 15 4. Toral automorphisms 16 5. Automorphisms of infranilmanifolds 16 6. Further examples 16 7. Repellers 16 e. Hyperbolic sets for flows 16 f. Examples 17 1. Geodesic flows 17 2. Suspensions 18 3. Further examples 18 g. The Banach Contraction Principle 19 h. The hyperbolic fixed point theorem 19 2. Stable manifolds 19 a. The Stable Manifold Theorem 20 b. The Perron–Irwin method 20 c. The Hadamard graph transform method 20 d. The Hadamard–Perron Theorem 21 e. Stable and unstable laminations 21 f. Fast leaves 22 g. Slow leaves 23 4 CONTENTS h. Local product structure 23 i. Stable and unstable manifolds for flows 23 3. Regularity of the invariant laminations 24 a. Definitions 24 b. Hölder regularity 25 c. Obstructions to higher regularity 26 d. Geodesic flows 26 e. Bootstrap and rigidity 27 f. Fast leaves 27 g. Absolute continuity and ergodicity of Anosov systems 28 h. Leafwise regularity 28 Chapter 3. Topological dynamics, stability, invariant measures 29 1. Expansivity and local stability 29 a. Expansivity 29 b. The Hartman–Grobman Theorem 29 c. Local topological rigidity 29 2. Shadowing 30 a. Pseudo-orbits and shadowing 30 b. The Anosov closing lemma 30 c. Shadowing Lemma 30 d. Shadowing Theorem 31 e. The specification property 31 f. Specification for flows 32 g. Closing Theorems 32 3. Transitivity 33 a. Spectral decomposition 33 b. Spectral decomposition and mixing for flows 33 c. Transitivity of Anosov systems 34 d. The Bowen–Ruelle alternative 34 4. Periodic points 35 a. Exponential growth and entropy 35 b. The #-function 35 c. Fine growth asymptotic 36 d. Periodic orbit growth for flows 36 5. Stability and classification 36 a. Strong structural stability of hyperbolic sets 37 b. Strong transversality and the Stability Theorem 37 c. The $-stability theorem 38 d. Stability of flows 39 e. Classification …