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The Topological Dynamics of Ellis Actions
Ethan Akin, City College, CUNY, New York, NY, Joseph Auslander, University of Maryland, College Park, MD, and Eli Glasner, Tel Aviv University, Israel
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Memoirs of the American Mathematical Society
2008; 152 pp; softcover
Volume: 195
ISBN-10: 0-8218-4188-2
ISBN-13: 978-0-8218-4188-4
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/195/913
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An Ellis semigroup is a compact space with a semigroup multiplication which is continuous in only one variable. An Ellis action is an action of an Ellis semigroup on a compact space such that for each point in the space the evaluation map from the semigroup to the space is continuous. At first the weak linkage between the topology and the algebra discourages expectations that such structures will have much utility. However, Ellis has demonstrated that these actions arise naturally from classical topological actions of locally compact groups on compact spaces and provide a useful tool for the study of such actions. In fact, via the apparatus of the enveloping semigroup the classical theory of topological dynamics is subsumed by the theory of Ellis actions. The authors' exposition describes and extends Ellis' theory and demonstrates its usefulness by unifying many recently introduced concepts related to proximality and distality. Moreover, this approach leads to several results which are new even in the classical setup.

Table of Contents

  • Introduction
  • Semigroups, monoids and their actions
  • Ellis semigroups and Ellis actions
  • Continuity conditions
  • Applications using ideals
  • Classical dynamical systems
  • Classical actions: The group case
  • Classical actions: The Abelian case
  • Iterations of continuous maps
  • Table
  • Bibliography
  • Index
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