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 $\phi :\Gamma \to \Gamma$ we prove the set-theoretical entropy of generalized shift ${\sigma }_{\phi }:X^{\Gamma }\to X^{\Gamma }$ (${\sigma }_{\phi }((x_{\alpha }{)}_{\alpha \in \Gamma })=(x_{\phi (\alpha )}{)}_{\alpha \in \Gamma }$ (for $(x_{\alpha }{)}_{\alpha \in \Gamma }\in X^{\Gamma }$)) is either zero or infinity, moreover it is zero if and only if $\phi$ 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.

12 pages

References in corpus (1)

• Topological Entropy and Algebraic Entropy for group endomorphisms