Set-theoretical entropies of generalized shifts

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.