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

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} $.