Introduction to the Modern Theory of Dynamical Systems

A self-contained comprehensive introduction to the mathematical theory of dynamical systems for students and researchers in mathematics, science and engineering.

Anatole Katok (Author), Boris Hasselblatt (Author)

9780521341875, Cambridge University Press

Hardback, published 28 April 1995

824 pages
24.1 x 16.5 x 5.9 cm, 1.27 kg

'This book provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline.' L'Enseignement Mathématique

This book provided the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms. The book begins with a discussion of several elementary but fundamental examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. Over 400 systematic exercises are included in the text. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate up.

Part I. Examples and Fundamental Concepts
Introduction
1. First examples
2. Equivalence, classification, and invariants
3. Principle classes of asymptotic invariants
4. Statistical behavior of the orbits and introduction to ergodic theory
5. Smooth invariant measures and more examples
Part II. Local Analysis and Orbit Growth
6. Local hyperbolic theory and its applications
7. Transversality and genericity
8. Orbit growth arising from topology
9. Variational aspects of dynamics
Part III. Low-Dimensional Phenomena
10. Introduction: What is low dimensional dynamics
11. Homeomorphisms of the circle
12. Circle diffeomorphisms
13. Twist maps
14. Flows on surfaces and related dynamical systems
15. Continuous maps of the interval
16. Smooth maps of the interval
Part IV. Hyperbolic Dynamical Systems
17. Survey of examples
18. Topological properties of hyperbolic sets
19. Metric structure of hyperbolic sets
20. Equilibrium states and smooth invariant measures
Part V. Sopplement and Appendix
21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Subject Areas: Calculus & mathematical analysis [PBK]