Strongly Rigid Flows

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.