Distributional chaotic generalized shifts
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866909076245446656 |
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| author | Ahmadabadi, Zahra Nili Shirazi, Fatemah Ayatollah Zadeh |
| author_facet | Ahmadabadi, Zahra Nili Shirazi, Fatemah Ayatollah Zadeh |
| contents | Suppose $X$ is a finite discrete space with at least two elements, $Γ$ is a nonempty countable set, and consider self--map $φ:Γ\toΓ$. We prove that the generalized shift $σ_φ:X^Γ\to X^Γ$ with $σ_φ((x_α)_{α\inΓ})=(x_{φ(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$) is:
$\bullet$ distributional chaotic (uniform, type 1, type 2) if and only if $φ:Γ\toΓ$ has at least a non-quasi-periodic point,
$\bullet$ dense distributional chaotic if and only if $φ:Γ\toΓ$ does not have any periodic point,
$\bullet$ transitive distributional chaotic if and only if $φ:Γ\toΓ$ is one--to--one without any periodic point.
We complete the text by counterexamples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_04832 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Distributional chaotic generalized shifts Ahmadabadi, Zahra Nili Shirazi, Fatemah Ayatollah Zadeh Dynamical Systems 54H20 Suppose $X$ is a finite discrete space with at least two elements, $Γ$ is a nonempty countable set, and consider self--map $φ:Γ\toΓ$. We prove that the generalized shift $σ_φ:X^Γ\to X^Γ$ with $σ_φ((x_α)_{α\inΓ})=(x_{φ(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$) is: $\bullet$ distributional chaotic (uniform, type 1, type 2) if and only if $φ:Γ\toΓ$ has at least a non-quasi-periodic point, $\bullet$ dense distributional chaotic if and only if $φ:Γ\toΓ$ does not have any periodic point, $\bullet$ transitive distributional chaotic if and only if $φ:Γ\toΓ$ is one--to--one without any periodic point. We complete the text by counterexamples. |
| title | Distributional chaotic generalized shifts |
| topic | Dynamical Systems 54H20 |
| url | https://arxiv.org/abs/1708.04832 |