DIFFERENTIABLE DYNAMICAL SYSTEMS: AN INTRODUCTION TO STRUCTURAL STABILITY AND HYPERBOLICITY. GRADUATE STUDIES IN MATHEMATICS VOLUME: 173

Preface ix
Chapter 1. Basics of dynamical systems 1
1.1. Basic concepts 4
1.2. Topological conjugacy and structural stability 10
1.3. Circle homeomorphisms 14
1.4. Conley’s fundamental theorem of dynamical systems 18
Exercises 22
Chapter 2. Hyperbolic fixed points 25
2.1. Hyperbolic linear isomorphisms 25
2.2. Persistence of hyperbolic fixed points 29
2.3. Persistence of hyperbolicity for a fixed point 34
2.4. Hartman-Grobman theorem 41
2.5. The local stable manifold for a hyperbolic fixed point 45
Exercises 54
Chapter 3. Horseshoes, toral automorphisms, and solenoids 57
3.1. Symbolic dynamics 57
3.2. Smale horseshoe 60
3.3. Anosov toral automorphisms 66
3.4. The solenoid attractor 70
Exercises 73
Chapter 4. Hyperbolic sets 75
4.1. The concept of hyperbolic set 75
4.2. Persistence of hyperbolicity for an invariant set 83
4.3. Smoothness in Lemma 2.17 and Theorem 2.18 89
4.4. Stable manifolds of hyperbolic sets 95
4.5. Structural stability of hyperbolic sets 117
4.6. The shadowing lemma 128
Exercises 134
Chapter 5. Axiom A, no-cycle condition, and O-stability 139
5.1. Spectral decomposition and Axiom A 139
5.2. Cycle and O-explosion 145
5.3. No-cycle and O-stability 147
5.4. Equivalent descriptions 150
Exercises 154
Chapter 6. Quasi-hyperbolicity and linear transversality 157
6.1. The simplest setting 157
6.2. Quasi-hyperbolicity 158
6.3. Linear transversality 166
6.4. Applications 168
6.5. A glimpse of the stability conjectures 172
Exercises 180
Bibliography 181
Index 189

This is a graduate text in differentiable dynamical systems. It focuses on structural stability and hyperbolicity, a topic that is central to the field. Starting with the basic concepts of dynamical systems, analyzing the historic systems of the Smale horseshoe, Anosov toral automorphisms, and the solenoid attractor, the book develops the hyperbolic theory first for hyperbolic fixed points and then for general hyperbolic sets. The problems of stable manifolds, structural stability, and shadowing property are investigated, which lead to a highlight of the book, the OO-stability theorem of Smale.

While the content is rather standard, a key objective of the book is to present a thorough treatment for some tough material that has remained an obstacle to teaching and learning the subject matter. The treatment is straightforward and hence could be particularly suitable for self-study.

Author
Lan Wen: Peking University, Beijing, China.