Strongly Rigid Flows

Abstract

We consider flows $(X,T)$, given by actions $(t, x) \to tx$, on a compact metric space $X$ with a discrete $T$ as an acting group. We study a new class of flows - the \textsc{Strongly Rigid} ($ \mathbf {SR} $) \ flows, that are properly contained in the class of distal ($ \mathbf D $) flows and properly contain the class of all equicontinuous ($ \mathbf {EQ} $) flows. Thus, $\mathbf {EQ} \ \text{flows} \subsetneqq \mathbf {SR} \ \text{flows} \subsetneqq \mathbf{ D} \ \text{flows}$. The concepts of equicontinuity, strong rigidity and distality coincide for the induced flow $(2^X,T)$. We observe that strongly rigid $(X,T)$ gives distinct properties for the induced flow $(2^X,T)$ and its enveloping semigroup $E(2^X)$. We further study strong rigidity in case of particular semiflows $(X,S)$, with $S$ being a discrete acting semigroup.