# American Institute of Mathematical Sciences

August  2019, 39(8): 4647-4711. doi: 10.3934/dcds.2019190

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

 1 Department of Mathematics, University of Maryland, College Park, MD 20742, USA 2 Department of Mathematics, Nanjing University, Nanjing 210093, China

To the memory of Professor Isaac Namioka

Received  September 2018 Revised  March 2019 Published  May 2019

Let $T$ be any topological semigroup and $(T, X)$ with phase mapping $(t, x)\mapsto tx$ a semiflow on a compact $\text{T}_2$ space $X$. If $tX = X$ for all $t$ in $T$ then $(T, X)$ is called surjective; if $x\mapsto tx$, for each $t$ in $T$, is 1-1 onto, then $(T, X)$ is termed invertible and the latter induces a right-action semiflow $(X, T)$ with the phase mapping $(x, t)\mapsto xt: = t^{-1}x$. We show that $(T, X)$ is equicontinuous surjective iff it is uniformly distal iff $(X, T)$ is equicontinuous surjective. We then consider minimality, distality, point-distality, and sensitivity of $(X, T)$ when $(T, X)$ possesses these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of flow on a zero-dimensional space with phase group $\mathbb{Z}$.

Citation: Joseph Auslander, Xiongping Dai. Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4647-4711. doi: 10.3934/dcds.2019190
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