Memoirs of the American Mathematical Society 2008; 152 pp; softcover Volume: 195 ISBN10: 0821841882 ISBN13: 9780821841884 List Price: US$75 Individual Members: US$45 Institutional Members: US$60 Order Code: MEMO/195/913
 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
