Synopses & Reviews
This major volume presents a comprehensive introduction to the study of topological transformation groups with respect to topological problems which can be traced back to the qualitative theory of differential equations, and provides a systematic exposition of the fundamental methods and techniques of abstract topological dynamics. The contents can be divided into two parts. The first part is devoted to a broad overview of the topological aspects of the theory of dynamical systems (including shift systems and geodesic and horocycle flows). Part Two is more specialized and presents in a systematic way the fundamental techniques and methods for the study of compact minima flows and their morphisms. It brings together many results which are scattered throughout the literature, and, in addition, many examples are worked out in detail. The primary purpose of this book is to bridge the gap between the `beginner' and the specialist in the field of topological dynamics. All proofs are therefore given in detail. The book will, however, also be useful to the specialist and each chapter concludes with additional results (without proofs) and references to sources and related material. The prerequisites for studying the book are a background in general toplogy and (classical and functional) analysis. For graduates and researchers wishing to have a good, comprehensive introduction to topological dynamics, it will also be of great interest to specialists. This volume is recommended as a supplementary text.
Synopsis
A comprehensive introduction to topological transformation groups, with respect to topological problems which can be traced back to the qualitative theory of differential equations. It provides a systematic exposition of the fundamental methods and techniques of abstract topological dynamics.
Synopsis
This book is designed as an introduction into what I call 'abstract' Topological Dynamics (TO): the study of topological transformation groups with respect to problems that can be traced back to the qualitative theory of differential equa- is in the tradition of the books GH] and EW. The title tions. So this book (, Elements . . . ' rather than 'Introduction . . . ') does not mean that this book should be compared, either in scope or in (intended) impact, with the 'Ele- ments' of Euclid or Bourbaki. Instead, it reflects the choice and organisation of the material in this book: elementary and basic (but sufficient to understand recent research papers in this field). There are still many challenging prob- lems waiting for a solution, and especially among general topologists there is a growing interest in this direction. However, the technical inaccessability of many research papers makes it almost impossible for an outsider to under- stand what is going on. To a large extent, this inaccessability is caused by the lack of a good and systematic exposition of the fundamental methods and techniques of abstract TO. This book is an attempt to fill this gap. The guiding principle for the organization of the material in this book has been the exposition of methods and techniques rather than a discussion of the leading problems and their solutions. though the latter are certainly not neglected: they are used as a motivation wherever possible.
Table of Contents
Preface. Notation.
I: Various aspects of the theory of dynamical systems.
II: Continuous and discrete flows.
III: Important examples.
IV: The general framework.
V: Equicontinuity and distality.
VI: Structure of extensions.
Appendices: A: Topology.
B: Compact right semitopological semigroups.
C: Integration.
D: Enveloping semigroups and compactifications.
E: Topological transformation groups. References. Index of authors. Index of symbols. Index of terms.