Amenability of Discrete Semigroup Flows

Abstract

A discrete flow (S,X) is a semigroup S acting on a set X where both S, and X are equipped with the discrete topology. Amenability of semigroups is a topic that explores the existence of measures that are invariant under the semigroup multiplication. The goal of this thesis is to generalize these results to a semigroup acting on a set, i.e. a flow, so that the invariance is with respect to the action. We start out in Chapter 1 by giving some preliminaries that are important for the results in this thesis. Chapter 2 generalizes basic theorems characterizing amenability and gives sufficient and necessary conditions for the same. We discuss some relevant topics such as the Hahn-Banach extension theorem and an application of flow amenability - a fixed point theorem. Next, in Chapter 3, we discuss various Folner conditions - combinatorial properties that characterize aspects of amenability. Finally, in Chapter 4, we discuss the flow stucture of the Stone-Cech compactification of a flow. We then discuss the concept of density of means and apply some properties of Folner nets. In Chapter 5 we briefly get into reversible invariance - a property that is equivalent to amenability in groups (and group flows).