Semiflows: Prolongation sets, point-transitive, topologically transitive

Abstract

Let φ:T×X→X, or simply (T,X), be any topological semiflow on a space X with a topological semigroup T. We give some new results about prolongation sets for the semiflow (T,X). Also, we recall notions of point-transitivity and topological transitivity for (T,X) and give some examples to study the relations between them. Let X be a locally compact Hausdorff space and let (T,X) be point-transitive. Then Mγ={x∈X:γ(x)=min⁡(γ)} is dense in X for every T-invariant upper semicontinuous function γ:X→[0,∞). The converse does hold if (T,X) is a flow on the compact metric space (X,d). We show that the semiflow (T,X) on a Hausdorff space X with TX‾=X is topologically transitive if and only if every T-invariant function γ:X→[0,∞), that is continuous on a comeagre set, is constant on a comeagre set. We introduce a TP-point in a topological space X, which is weaker than an isolated point, and show that if X is a regular topological space with a TP-point p∈X and (T,X) is syndetically transitive, then p is a periodic point and X=Kp for some compact set K⊆T. Finally, we show that every expansive semiflow (T,[0,1]) is point-transitive if T is a right C-semigroup.