Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity
About this Title
Lan Wen, Peking University, Beijing, China
Publication: Graduate Studies in Mathematics
Publication Year: 2016; Volume 173
ISBNs: 978-1-4704-2799-3 (print); 978-1-4704-3210-2 (online)
MathSciNet review: MR3497139
MSC: Primary 37-02; Secondary 34D30, 37C20, 37C75, 37D05
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.
Selected solutions are available electronically for instructors only. Please send email to email@example.com for more information.
This book introduces the reader to some basic concepts of hyperbolic theory of dynamical systems with emphasis on structural stability. It is well written, the proofs are presented with great attention to details, and every chapter ends with a good collection of exercises. It is suitable for a semester-long course on the basics of dynamical systems.
—Yakov Pesin, Penn State University
Lan Wen's book is a thorough introduction to the “classical” theory of (uniformly) hyperbolic dynamics, updated in light of progress since Smale's seminal 1967 Bulletin article. The exposition is aimed at newcomers to the field and is clearly informed by the author's extensive experience teaching this material. A thorough discussion of some canonical examples and basic technical results culminates in the proof of the Omega-stability theorem and a discussion of structural stability. A fine basic text for an introductory dynamical systems course at the graduate level.
—Zbigniew Nitecki, Tufts University
Graduate students and research mathematicians interested in the hyperbolic theory of dynamical systems.
Table of Contents
- Chapter 1. Basics of dynamical systems
- Chapter 2. Hyperbolic fixed points
- Chapter 3. Horseshoes, toral automorphisms, and solenoids
- Chapter 4. Hyperbolic sets
- Chapter 5. Axiom A, no-cycle condition, and $\Omega $-stability
- Chapter 6. Quasi-hyperbolicity and linear transversality