The General Topology of Dynamical Systems (Graduate Studies in Mathematics)

Topology, the foundation of modern analysis, arose historically as a way to organize ideas like compactness and connectedness which had emerged from analysis. Similarly, recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results (such as attractors, chain recurrence, and basic sets). This book collects these results, both old and new, and organizes them into a natural foundation for all aspects of dynamical systems theory. No existing book is comparable in content or scope. Requiring background in point-set topology and some degree of "mathematical sophistication", Akin's book serves as an excellent textbook for a graduate course in dynamical systems theory. In addition, Akin's reorganization of previously scattered results makes this book of interest to mathematicians and other researchers who use dynamical systems in their work. Readership: Graduate students and research mathematicians interested in dynamical systems. It contains a wealth of information concerning topological dynamics, most of which has not appeared before in such an organization and presentation. It offers to a graduate-level student a very comprehensive overview on the basic concepts in the theory of dynamical systems. —Zentralblatt MATH No other single text has heretofore presented such a unified treatment of these topological ideas at this level of generality. —Mathematical Reviews Topology, the foundation of modern analysis, arose historically as a way to organize ideas like compactness and connectedness which had emerged from analysis. Similarly, recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results (such as attractors, chain recurrence, and basic sets). This book collects these results, both old and new, and organizes them into a natural foundation for all aspects of dynamical systems theory. No existing book is comparable in content or scope. Requiring background in point-set topology and some degree of “mathematical sophistication”, Akin's book serves as an excellent textbook for a graduate course in dynamical systems theory. In addition, Akin's reorganization of previously scattered results makes this book of interest to mathematicians and other researchers who use dynamical systems in their work. Includes a wealth of information concerning topological dynamics, most of which has not appeared before in such an organization and presentation. It offers to a graduate-level student a very comprehensive overview on the basic concepts in the theory of dynamical systems. --Zentralblatt MATH No other single text has heretofore presented such a unified treatment of these topological ideas at this level of generality. --Mathematical Reviews Topology, the foundation of modern analysis, arose historically as a way to organize ideas like compactness and connectedness which had emerged from analysis. Similarly, recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results (such as attractors, chain recurrence, and basic sets). This book collects these results, both old and new, and organizes them into a natural foundation for all aspects of dynamical systems theory. No existing book is comparable in content or scope. Requiring background in point-set topology and some degree of ``mathematical sophistication'', Akin's book serves as an excellent textbook for a graduate course in dynamical systems theory. In addition, Akin's reorganization of previously scattered results makes this book of interest to mathematicians and other researchers who use dynamical systems in their work Preface Chapter 0. Introduction: Gradient Systems Chapter 1. Closed Relations and Their Dynamic Extensions Supplementary exercises Chapter 2. Invariant Sets and Lyapunov Functions Supplementary exercises Chapter 3. Attractors and Basic Sets Supplementary exercises Chapter 4. Mappings-Invariant Subsets and Transitivity Concepts Supplementary exercises Chapter 5. Computation of the Chain Recurrent Set Supplementary exercises Chapter 6. Chain Recurrence and Lyapunov Functions for Flows Supplementary exercises Chapter 7. Topologically Robust Properties of Dynamical Systems Supplementary exercises Chapter 8. Invariant Measures for Mappings Supplementary exercises Chapter 9. Examples-Circles; Simplex; and Symbols Supplementary exercises Chapter 10. Fixed Points Supplementary exercises Chapter 11. Hyperbolic Sets and Axiom A Homeomorphisms Supplementary exercises Historical Remarks References Subject Index Recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results. This book collects these results, both old and new, and organises them into a natural foundation for all aspects of dynamical systems theory.