On generalized shift transformation semigroups

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