Set–theoretical entropies of weighted generalized shifts

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