Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, Series Number 54)

"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 orbits structure. The third and fourth parts develop in depth the theories of low-dimensional dynamical systems and hyperbolic dynamical systems." "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. It contains more than four hundred systematic exercises."--BOOK JACKET 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 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. Anatole Katok, Boris Hasselblatt. Includes Bibliographical References And Index. 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.