Topological sensitivity for semiflow

Abstract

We give a pointwise version of sensitivity in terms of open covers for a semiflow (T, X) of a topological semigroup T on a Hausdorff space X and call it a Hausdorff sensitive point. If \((X, {\mathscr {U}})\) is a uniform space with topology \(\tau \), then the definition of Hausdorff sensitivity for \((T, (X, \tau ))\) gives a pointwise version of sensitivity in terms of uniformity and we call it a uniformly sensitive point. For a semiflow (T, X) on a compact Hausdorff space X, these notions (i.e. Hausdorff sensitive point and uniformly sensitive point) are equal and they are T-invariant if T is a C-semigroup. They are not preserved by factor maps and subsystems, but behave slightly better with respect to lifting. We give the definition of a topologically equicontinuous pair for a semiflow (T, X) on a topological space X and show that if (T, X) is a topologically equicontinuous pair in (x, y), for all \(y\in X\), then \(\overline{Tx}= D_T(x)\) where

$$\begin{aligned} D_T(x)= \bigcap \{ \overline{TU}: \text { for all open neighborhoods}\, U\, \text {of}\, x \}. \end{aligned}$$

We prove for a topologically transitive semiflow (T, X) of a C-semigroup T on a regular space X with a topologically equicontinuous point that the set of topologically equicontinuous points coincides with the set of transitive points. This implies that every minimal semiflow of C-semigroup T on a regular space X with a topologically equicontinuous point is topologically equicontinuous. Moreover, we show that if X is a regular space and (T, X) is not a topologically equicontinuous pair in (x, y), then x is a Hausdorff sensitive point for (T, X). Hence, a minimal semiflow of a C-semigroup T on a regular space X is either topologically equicontinuous or topologically sensitive.