Dynamics of Enveloping Semigroup of Flows

TL;DR

This paper explores the relationship between the dynamics of a flow and its enveloping semigroup, establishing key equivalences and properties related to distality, equicontinuity, sensitivity, and proximality.

Contribution

It provides new characterizations of flow properties via their enveloping semigroups, including conditions for distality, equicontinuity, and sensitivity, and shows isomorphism of iterated enveloping semigroups.

Findings

Flow is distal iff induced flow on enveloping semigroup is distal.

Induced flow on enveloping semigroup of equicontinuous flow is equicontinuous.

For any flow, the enveloping semigroup is isomorphic to its double enveloping semigroup.

Abstract

In this article, we relate the dynamics of a flow $(X, T)$ with the dynamics of the induced flow $(E(X), T)$ where $E(X)$ is the enveloping semigroup of flow $(X, T)$. We establish that a flow $(X, T)$ is distal if and only if the induced flow is also distal. We prove that while the induced flow on the enveloping semigroup of an equicontinuous flow is equicontinuous, the converse holds when $(X, T)$ is point transitive. We prove that for any flow $(X,T)$, $(E(X),T)$ is isomorphic to $(E(E(X)),T)$. We show that if a distal flow contains a minimal sensitive subsystem, the induced flow on the enveloping semigroup is also sensitive. We relate various forms of rigidity for a flow with rigidity for the induced flow. We also establish that for any proximal flow $(X, T)$, if T is abelian then the induced flow $(E(X),T)$ is also proximal.