Strongly Rigid Flows

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