# The Topological Dynamics of Ellis Actions

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*Ethan Akin; Joseph Auslander; Eli Glasner*

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

# Table of Contents

## The Topological Dynamics of Ellis Actions

- Contents v6 free
- Introduction 18 free
- Chapter 1. Semigroups, Monoids and Their Actions 916 free
- Chapter 2. Ellis Semigroups and Ellis Actions 1926
- Chapter 3. Continuity Conditions 3542
- Chapter 4. Applications Using Ideals 4552
- Chapter 5. Classical Dynamical Systems 5966
- Chapter 6. Classical Actions: The Group Case 8794
- Chapter 7. Classical Actions: The Abelian Case 121128
- Chapter 8. Iterations of Continuous Maps 135142
- Table 143150
- Bibliography 145152
- Index 149156