Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity

Description

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 OMEGA-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. Contents Preface 1. Basics of dynamical systems 1.1. Basic concepts 1.2. Topological conjugacy and structural stability 1.3. Circle homeomorphisms 1.4. Conley's fundamental theorem of dynamical systems Exercises 2. Hyperbolic fixed points 2.1. Hyperbolic linear isomorphisms 2.2. Persistence of hyperbolic fixed points 2.3. Persistence of hyperbolicity for a fixed point 2.4. Hartman-Grobman theorem 2.5. The local stable manifold for a hyperbolic fixed point Exercises 3. Horseshoes, toral automorphisms, and solenoids 3.1. Symbolic dynamics 3.2. Smale horseshoe 3.3. Anosov toral automorphisms 3.4. The solenoid attractor Exercises 4. Hyperbolic sets 4.1. The concept of hyperbolic set 4.2. Persistence of hyperbolicity for an invariant set 4.3. Smoothness in Lemma 2.17 and Theorem 2.18 4.4. Stable manifolds of hyperbolic sets 4.5. Structural stability of hyperbolic sets 4.6. The shadowing lemma Exercises 5. Axiom A, no-cycle condition, and $\Omega$-stability 5.1. Spectral decomposition and Axiom A 5.2. Cycle and $\Omega$-explosion 5.3. No-cycle and $\Omega$-stability 5.4. Equivalent descriptions Exercises 6. Quasi-hyperbolicity and linear transversality 6.1. The simplest setting 6.2. Quasi-hyperbolicity 6.3. Linear transversality 6.4. Applications 6.5. A glimpse of the stability conjecctures Exercises Bibliography Index Selected Published Titles in This Series