Introduction to the Modern Theory of Dynamical Systems

by Anatole Katok , Boris Hasselblatt

• ISBN13: 9780521341875
• ISBN10: 0521341876
• Format: Hardcover
• Copyright: 1995-04-28
• Publisher: Cambridge University Press

Summary

This book provides 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. Scientists and engineers working in applied dynamics, nonlinear science, and chaos will also find many fresh insights in this concrete and clear presentation.

Table of Contents

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.