Set-theoretical entropies of weighted generalized shifts

TL;DR

This paper investigates the set-theoretical entropy of weighted generalized shifts over finite fields, showing it is either zero or infinite, with zero entropy characterizing quasi-periodic shifts, and explores conditions for finite fibers and restrictions to direct sums.

Contribution

It characterizes when weighted generalized shifts have zero or infinite entropy and identifies conditions for finite fibers and entropy dependence on the shift map and support.

Findings

Entropy is either zero or infinite for these shifts.

Zero entropy occurs if and only if the shift is quasi-periodic.

Contravariant entropy depends only on the shift map and support when fibers are finite.

Abstract

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