Topological Dynamics of Enveloping Semigroups

Abstract

A compact metric space $X$ and a discrete topological acting group $T$ give a flow $(X,T)$. Robert Ellis had initiated the study of dynamical properties of the flow $(X,T)$ via the algebraic properties of its "Enveloping Semigroup" $E(X)$. This concept of \emph{Enveloping Semigroups} that he defined, has turned out to be a very fundamental tool in the abstract theory of `topological dynamics'. The flow $(X,T)$ induces the flow $(2^X,T)$. Such a study was first initiated by Eli Glasner who studied the properties of this induced flow by defining and using the notion of a `circle operator' as an action of $\beta T$ on $2^X$, where $\beta T$ is the \emph{Stone-$\check{C}$ech compactification} of $T$ and also a universal enveloping semigroup. We propose that the study of properties for the induced flow $(2^X,T)$ be made using the algebraic properties of $E(2^X)$ on the lines of Ellis' \ theory, instead of looking into the action of $\beta T$ on $2^X$ via the circle operator as done by Glasner. Such a study requires extending the present theory on the flow $(E(X),T)$. In this article, we take up such a study giving some subtle relations between the semigroups $E(X)$ and $E(2^X)$ and some interesting associated consequences.