This book provides a self-contained comprehensive exposition of the theory of dynamical systems. The book begins with a discussion of several elementary but crucial 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. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate and up.
Author(s): Anatole Katok, Boris Hasselblatt
Series: Encyclopedia of Mathematics and its Applications 54
Publisher: Cambridge University Press
Year: 1995
Language: English
Pages: 824
Preface xiii
0. Introduction 1
Part 1 Examples and fundamental concepts
1. First examples 15
2. Equivalence, classification, and invariants 57
3. Principal classes of asymptotic topological invariants 105
4. Statistical behavior of orbits and introduction to ergodic theory 133
5. Systems with smooth invariant measures and more examples 183
Part 2 Local analysis and orbit growth
6. Local hyperbolic theory and its applications 237
7. Transversality and genericity 287
8. Orbit growth arising from topology 307
9. Variational aspects of dynamics 335
Part 3 Low-dimensional phenomena
10. Introduction: What is low-dimensional dynamics? 381
11. HOMEOMORPHISMS OF THE CIRCLE 387
12. Circle diffeomorphisms 401
13. Twist maps 423
14. Flows on surfaces and related dynamical systems 451
15. Continuous maps of the interval 489
16. Smooth maps of the interval 519
Part 4 Hyperbolic dynamical systems
17. Survey of examples 531
18. Topological properties of hyperbolic sets 565
19. Metric structure of hyperbolic sets 597
20. Equilibrium states and smooth invariant measures 615
Supplement
Dynamical systems with nonuniformly hyperbolic behavior
by Anatole Katok and Leonardo Mendoza 659
Appendix
Background material
1. Basic topology
Topological spaces; Homotopy theory; Metric spaces
2. Functional analysis
3. Differentiable manifolds
Differentiable manifolds; Tensor bundles; Exterior calculus; Transversality
4. Differential geometry
5. Topology and geometry of surfaces
6. Measure theory
Basic notions; Measure and topology
7. Homology theory
8. Lie groups
Notes
Hints and answers to the exercises
References
Index
703
