Topological Dynamics of Enveloping Semigroups

TL;DR

This paper explores the algebraic properties of enveloping semigroups in topological dynamics, extending Ellis's theory to the induced flow on the space of closed subsets, revealing new relations between these semigroups.

Contribution

It extends the theory of enveloping semigroups to the induced flow on the space of closed subsets, providing new insights into their algebraic relations.

Findings

Established relations between E(X) and E(2^X) semigroups.

Proposed a new approach to study induced flows using algebraic properties.

Extended Ellis's theory to the flow on the space of closed subsets.

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.