Set-theoretical entropies of weighted generalized shifts

Abstract

In this paper for a finite field $F$, a nonempty set $\Gamma$, a self--map $\phi :\Gamma \to \Gamma$ and a weight vector $\mathfrak{w}\in F^{\Gamma }$, we show that the set--theoretical entropy of the weighted generalized shift ${\sigma }_{\phi ,\mathfrak{w}}:F^{\Gamma }\to F^{\Gamma }$ is either zero or $+\infty$, 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 ${\sigma }_{\phi ,\mathfrak{w}}:F^{\Gamma }\to F^{\Gamma }$ is of finite fibre, we show that the contravariant set--theoretical entropy of a finite fibre ${\sigma }_{\phi ,\mathfrak{w}}:F^{\Gamma }\to F^{\Gamma }$ depends only on $\phi$ and $\mathrm{supp}(\mathfrak{w})$. In final sections we study the restriction of ${\sigma }_{\phi ,\mathfrak{w}}$ to the direct sum $\underset{\Gamma }{⨁}F$.

28 pages