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The general topology of dynamical systems. (English) Zbl 0781.54025
Graduate Studies in Mathematics. 1. Providence, RI: American Mathematical Society (AMS). x, 261 p. (1993).
The theory of dynamical systems was dominated in the sixties by the development of hyperbolic theory, whose construction was orchestrated by S. Smale. Notions and concepts used in this theory, such as fixed points, periodic points, nonwandering points, attractors, basic sets, recurrence are purely topological concepts. Their importance in the understanding of topological dynamics suggested to the author of the book under review a reorganization and an approach in a more general context of these basic concepts ‘what every dynamicist should know’. As the author says in the Preface of the book ‘the central theme is the role of chain recurrence in the study of dynamical systems on compact metric spaces’.
In classical topological dynamics a dynamical system is defined by a continuous mapping or a homeomorphism of a compact metric space. In the author’s book concepts and structures of topological dynamics are associated with a closed relation \(f\subset X\times X\) on the compact metric space \(X\). Closed relations are regarded as dynamical systems. In order to give the meaning and consequences of different notions of recurrence one associates to a closed relation a finite sequence of nested relations. Various types of invariant sets are introduced, Lyapunov function for a closed relation is constructed and equivalent definitions for an attractor are given. Then are examined in detail the properties of a dynamical system when the closed relation is a continuous map \(f: X\to X\), and additional featues are pointed out. A special attention is paid to the concept of topological transitivity and minimal set. An apparatus is developed which allows to construct the chain recurrent set from the limit point set.
The structures and concepts previously developed for maps are then described for semiflows. A chapter is concerned with topological perturbation. Some dynamic properties of a closed relation which are preserved under a perturbation are identified.
Notions from the hyperbolic differentiable dynamics are generalized in the topological context: expansive homeomorphism, shadowing property, Anosov homeomorphism. The book also contains some notions concerning invariant measures and gives applications to the case of the symbolic dynamics and flows on the torus.
This book is the first in a new series from AMS: “Graduate Studies in Mathematics”. 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.

54H20 Topological dynamics (MSC2010)
37B99 Topological dynamics
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
37A99 Ergodic theory
37D99 Dynamical systems with hyperbolic behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37C10 Dynamics induced by flows and semiflows