Dynamics of Enveloping Semigroup of Flows

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.