Topological Dynamics of Enveloping Semigroups

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.