Set-theoretical entropies of weighted generalized shifts
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| Format: | Preprint |
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2022
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| _version_ | 1866917820614311936 |
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| author | Shirazi, Fatemah Ayatollah Zadeh Hosseini, Arezoo Mousavi, Lida Rezavand, Reza |
| author_facet | Shirazi, Fatemah Ayatollah Zadeh Hosseini, Arezoo Mousavi, Lida Rezavand, Reza |
| contents | In this paper for a finite field $F$, a nonempty set $Γ$, a self--map $φ:Γ\toΓ$ and a weight vector $\mathfrak{w}\in F^Γ$, we show that the set--theoretical entropy of the weighted generalized shift $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ is either zero or $+\infty$, moreover it is equal to zero if and only if $σ_{φ,\mathfrak{w}}$ is quasi--periodic. On the other hand after characterizing all conditions under which $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ is of finite fibre, we show that the contravariant set--theoretical entropy of a finite fibre $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ depends only on $φ$ and $\rm{supp}(\mathfrak{w})$. In final sections we study the restriction of $σ_{φ,\mathfrak{w}}$ to the direct sum $\mathop{\bigoplus}\limits_ΓF$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_09118 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Set-theoretical entropies of weighted generalized shifts Shirazi, Fatemah Ayatollah Zadeh Hosseini, Arezoo Mousavi, Lida Rezavand, Reza General Mathematics 03E20, 37B99 In this paper for a finite field $F$, a nonempty set $Γ$, a self--map $φ:Γ\toΓ$ and a weight vector $\mathfrak{w}\in F^Γ$, we show that the set--theoretical entropy of the weighted generalized shift $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ is either zero or $+\infty$, moreover it is equal to zero if and only if $σ_{φ,\mathfrak{w}}$ is quasi--periodic. On the other hand after characterizing all conditions under which $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ is of finite fibre, we show that the contravariant set--theoretical entropy of a finite fibre $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ depends only on $φ$ and $\rm{supp}(\mathfrak{w})$. In final sections we study the restriction of $σ_{φ,\mathfrak{w}}$ to the direct sum $\mathop{\bigoplus}\limits_ΓF$. |
| title | Set-theoretical entropies of weighted generalized shifts |
| topic | General Mathematics 03E20, 37B99 |
| url | https://arxiv.org/abs/2206.09118 |