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

TL;DR

This paper investigates the relationships between equicontinuity, surjectivity, and distality in semigroup actions on compact Hausdorff spaces, establishing equivalences and exploring implications for minimality, sensitivity, and recurrence.

Contribution

It proves that equicontinuous surjective semiflows are equivalent to uniformly distal ones and explores their invertible counterparts, extending understanding of their dynamical properties.

Findings

Equicontinuous surjective semiflows are equivalent to uniformly distal ones.

Invertible semiflows with these properties have specific minimality and sensitivity characteristics.

The paper analyzes recurrence and weak almost periodicity in zero-dimensional phase 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.