Minimal Flows and Their Extensions

By

  • J. Auslander

This monograph presents developments in the abstract theory of topological dynamics, concentrating on the internal structure of minimal flows (actions of groups on compact Hausdorff spaces for which every orbit is dense) and their homomorphisms (continuous equivariant maps).

Various classes of minimal flows (equicontinuous, distal, point distal) are intensively studied, and a general structure theorem is obtained. Another theme is the ``universal'' approach - entire classes of minimal flows are studied, rather than flows in isolation. This leads to the consideration of disjointness of flows, which is a kind of independence condition. Among the topics unique to this book are a proof of the Ellis ``joint continuity theorem'', a characterization of the equicontinuous structure relation, and the aforementioned structure theorem for minimal flows.

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Book information

  • Published: July 1988
  • Imprint: NORTH-HOLLAND
  • ISBN: 978-0-444-70453-5

Reviews

The leisurely pace, clarity of presentation and the self consistency which is carefully preserved throughout makes it ideal for graduate students seeking an introduction to this branch of topological dynamics.
E. Glasner Bulletin of the American Mathematical Society


The book under review is a pleasant, leisurely account of the topics discussed. In places it reads like a novel. The author's approach to the subject might be termed `topological' as opposed to `algebraic'; he stresses the former and down-plays the latter wherever possible.
R. Ellis, Ergodic Theory & Dynamical Systems



Table of Contents

Flows and Minimal Sets. Equicontinuous Flows. The Enveloping Semigroup of a Transformation Group, I. Joint Continuity Theorems. Distal Flows. The Enveloping Semigroup, II. The Furstenberg Structure Theorem for Distal Minimal Flows. Universal Minimal Flows and Ambits. The Equicontinuous Structure Relation and Weakly Mixing Flows. The Algebraic Theory of Minimal Flows. Disjointness. Invariant Measures on Flows. Kakutani-Bebutov Theorems. General Structure Theorems. Appendices: Nets. Uniform Spaces.