Strongly Rigid Flows

TL;DR

This paper introduces and studies the class of strongly rigid flows, which are a new intermediate class between equicontinuous and distal flows, revealing unique properties in their induced flows and enveloping semigroups.

Contribution

The paper defines strongly rigid flows, explores their relation to equicontinuous and distal flows, and analyzes their properties in induced flows and enveloping semigroups.

Findings

Strongly rigid flows form a proper subset of distal flows.

Equicontinuity, strong rigidity, and distality coincide for the induced flow.

Strong rigidity influences properties of the induced flow and its enveloping semigroup.

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.