Set-theoretical entropies of weighted generalized shifts

In this paper for a finite field $F$, a nonempty set $\mathrm{\Gamma }$, a self–map $\phi :\mathrm{\Gamma }\to \mathrm{\Gamma }$ and a weight vector $\mathfrak{w}\in F^{\mathrm{\Gamma }}$, we show that the set–theoretical entropy of the weighted generalized shift ${\sigma }_{\phi ,\mathfrak{w}}:F^{\mathrm{\Gamma }}\to F^{\mathrm{\Gamma }}$ is either zero or $+\mathrm{\infty }$, moreover it is equal to zero if and only if ${\sigma }_{\phi ,\mathfrak{w}}$ is quasi–periodic. On the other hand after characterizing all conditions under which ${\sigma }_{\phi ,\mathfrak{w}}:F^{\mathrm{\Gamma }}\to F^{\mathrm{\Gamma }}$ is of finite fibre, we show that the contravariant set–theoretical entropy of a finite fibre ${\sigma }_{\phi ,\mathfrak{w}}:F^{\mathrm{\Gamma }}\to F^{\mathrm{\Gamma }}$ depends only on $\phi$ and $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathfrak{w})$. In final sections we study the restriction of ${\sigma }_{\phi ,\mathfrak{w}}$ to the direct sum $\underset{\mathrm{\Gamma }}{}F$.