Set-theoretical entropies of generalized shifts

TL;DR

This paper investigates the set-theoretical entropy of generalized shifts, showing it is either zero or infinite, with zero entropy characterizing quasi-periodic maps, and explores contravariant entropy with various counterexamples.

Contribution

It establishes a dichotomy for the set-theoretical entropy of generalized shifts and introduces the study of contravariant entropy with illustrative counterexamples.

Findings

Entropy is either zero or infinity for generalized shifts.

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

Counterexamples demonstrate diverse behaviors of entropies in different contexts.

Abstract

In the following text for arbitrary $X$ with at least two elements, nonempty set $\Gamma$ and self-map $\varphi:\Gamma\to\Gamma$ 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 $\varphi$ 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.