On generalized shift transformation semigroups

In the following text we prove that for finite discrete $X$ with at least two elements and infinite $\mathrm{\Gamma }$, the generalized shift transformation semigroup $(\mathcal{S},X^{\mathrm{\Gamma }})$ is equicontinuous (resp. has at least an equicontinuous point, is not sensitive) if and only if for all $w\in \mathrm{\Gamma }$, $\{\phi (w):{\sigma }_{\phi }\in \mathcal{S}\}$ is finite. We continue our study regarding distality and expansivity of $(\mathcal{S},X^{\mathrm{\Gamma }})$.