Introduction to the Modern Theory of Dynamical Systems
Introduction to the Modern Theory of Dynamical Systems
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.
- Over 400 exercises, with solution hints
- Comprehensive - goes from elementary theory to recent research
- Katok is one of the world's leading researchers in dynamical systems
Reviews & endorsements
' … there is no other treatment coming close in terms of comprehensiveness and readability … it is indispensable for anybody working on dynamical systems in almost any context, and even experts will find interesting new proofs, insights and historical references throughout the book.' Monatshefte für Mathematik
'… contains detailed discussion … presents many recent results … The text is carefully written and is accompanied by many excercises.' European Mathematical Society Newsletter
'This book provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline.' L'Enseignement Mathématique
Product details
February 1997Paperback
9780521575577
824 pages
235 × 159 × 51 mm
1.16kg
Available
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.
