The general topology of dynamical systems / Ethan AkinType de document : MonographieCollection : Graduate studies in mathematics, 1Langue : anglais.Pays: Etats Unis.Éditeur : Providence : American Mathematical Society, 1993Description : 1 vol. (x-261 p.) : ill. ; 26 cmISBN: 0821838008.ISSN: 1065-7339.Bibliographie : Bibliographie p. 255-258. Index.Sujet MSC : 37Bxx, Dynamical systems and ergodic theory - Topological dynamics
37Axx, Dynamical systems and ergodic theory - Ergodic theory
37Dxx, Dynamical systems and ergodic theory - Dynamical systems with hyperbolic behavior
37C70, Smooth dynamical systems: general theory, Attractors and repellers and their topological structureEn-ligne : Zentralblatt | MathSciNet Item type: Monographie
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Bibliographie p. 255-258. Index
... The book begins by defining recurrent sets commonly associated with iteration of closed relations on compact metric spaces. Various forms of recurrent behaviors are considered, including fixed points, periodic points, nonwandering points, and chain recurrent points. A tower of relationships between these sets is constructed with chain recurrence the most general. Most of the general topological properties presented thereafter are stated in terms of chain recurrence.
Invariant sets, in particular attractors and repellors, are investigated next, as are Lyapunov functions to determine their stability. Special results are derived for maps, including the decomposability of the metric space into invariant sets, minimal sets, and topological transitivity. In this case the chain recurrent set is estimated with the use of covers of increasingly finer mesh size. For homeomorphisms the chain recurrent set and the attractor-repellor structure are determined from the set of limit points.
Analogous results for flows and semi-flows are developed by the use of the time-one map. Special results are derived for Lyapunov functions and recurrent sets in this case. A topological version of structural stability is developed next, and is followed by a discussion of invariant measures, which allows the introduction of topological notions of ergodicity and mixing. Results of the previous chapters are then applied to several standard examples of dynamical systems, e.g., circle maps, shift maps, and flows on a torus.
The work concludes with an investigation into hyperbolicity, first for fixed points and then for Axiom A homeomorphisms. ... The style of the book is primarily of a graduate-level text with many exercises included. While nearly all the results are stated and proven formally, a number of lesser proofs are left as exercises. The book's principal use, however, may be as a reference for dynamical systems specialists. (MathSciNet)