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arXiv · 1709.01579

Set-theoretical entropies of generalized shifts

Zahra Nili Ahmadabadi·Fatemah Ayatollah Zadeh Shirazi
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Abstract

In the following text for arbitrary $X$ with at least two elements, nonempty set $Γ$ and self-map $φ:Γ\toΓ$ we prove the set-theoretical entropy of generalized shift $σ_φ:X^Γ\to X^Γ$ ($σ_φ((x_α)_{α\inΓ})=(x_{φ(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$)) is either zero or infinity, moreover it is zero if and only if $φ$ 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.

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BibTeXRIS

Zahra Nili Ahmadabadi, Fatemah Ayatollah Zadeh Shirazi. 2018-02-01. Set-theoretical entropies of generalized shifts. https://arxiv.org/abs/1709.01579

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