Set-theoretical entropies of generalized shifts

In the following text for arbitrary $X$ with at least two elements, nonempty set $\mathrm{\Gamma }$ and self-map $\phi :\mathrm{\Gamma }\to \mathrm{\Gamma }$ we prove the set-theoretical entropy of generalized shift $\sigma_\varphi:X^\Gamma\to X^\Gamma$($\sigma_\varphi((x_\alpha){\alpha\in\Gamma})=(x{\varphi(\alpha)}){\alpha\in\Gamma}$(for$(x\alpha)_{\alpha\in\Gamma}\in X^\Gamma$)) is either zero or infinity, moreover it is zero if and only if $\phi$ is quasi-periodic. We continue our study on contravariant set-theoretical entropy of generalized shift and motivate the text using counterexamples dealing with algebraic, topological, set-theoretical and contravariant set-theoretical positive entropies of generalized shifts.