Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Abstract

Let $\pi\colon T\times X\rightarrow X$ with phase map $(t,x)\mapsto tx$, denoted $(\pi,T,X)$, be a \textit{semiflow} on a compact Hausdorff space $X$ with phase semigroup $T$. If each $t\in T$ is onto, $(\pi,T,X)$ is called surjective; and if each $t\in T$ is 1-1 onto $(\pi,T,X)$ is called invertible and in latter case it induces $\pi^{-1}\colon X\times T\rightarrow X$ by $(x,t)\mapsto xt:=t^{-1}x$, denoted $(\pi^{-1},X,T)$. In this paper, we show that $(\pi,T,X)$ is equicontinuous surjective iff it is uniformly distal iff $(\pi^{-1},X,T)$ is equicontinuous surjective. As applications of this theorem, we also consider the minimality, distality, and sensitivity of $(\pi^{-1},X,T)$ if $(\pi,T,X)$ is invertible with these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of $\mathbb{Z}$-flow with compact zero-dimensional phase space.