On a class between Devaney chaotic and Li-Yorke chaotic generalized shift dynamical systems
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2017
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| _version_ | 1866913199075360768 |
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| author | Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh Jooyan, Maryam Hagh Hosseini, Arezoo |
| author_facet | Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh Jooyan, Maryam Hagh Hosseini, Arezoo |
| contents | In the following text, for finite discrete $X$ with at least two elements, nonempty countable $Γ$, and $φ:Γ\toΓ$ we prove the generalized shift dynamical system $(X^Γ,σ_φ)$ is densely chaotic if and only if $φ:Γ\toΓ$ does not have any (quasi-)periodic point. Hence the class of all densely chaotic generalized shifts on $X^Γ$ is intermediate between the class of all Devaney chaotic generalized shifts on $X^Γ$ and the class of all Li-Yorke chaotic generalized shifts on $X^Γ$. In addition, these inclusions are proper for infinite countable $Γ$.
Moreover we prove $(X^Γ,σ_φ)$ is Li-Yorke sensitive (resp. sensitive, strongly sensitive, asymptotic sensitive, syndetically sensitive, cofinitely sensitive, multi-sensitive, ergodically sensitive, spatiotemporally chaotic, Li-Yorke chaotic) if and only if $φ:Γ\toΓ$ has at least one non-quasi-periodic point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_04868 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On a class between Devaney chaotic and Li-Yorke chaotic generalized shift dynamical systems Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh Jooyan, Maryam Hagh Hosseini, Arezoo Dynamical Systems 37B99 In the following text, for finite discrete $X$ with at least two elements, nonempty countable $Γ$, and $φ:Γ\toΓ$ we prove the generalized shift dynamical system $(X^Γ,σ_φ)$ is densely chaotic if and only if $φ:Γ\toΓ$ does not have any (quasi-)periodic point. Hence the class of all densely chaotic generalized shifts on $X^Γ$ is intermediate between the class of all Devaney chaotic generalized shifts on $X^Γ$ and the class of all Li-Yorke chaotic generalized shifts on $X^Γ$. In addition, these inclusions are proper for infinite countable $Γ$. Moreover we prove $(X^Γ,σ_φ)$ is Li-Yorke sensitive (resp. sensitive, strongly sensitive, asymptotic sensitive, syndetically sensitive, cofinitely sensitive, multi-sensitive, ergodically sensitive, spatiotemporally chaotic, Li-Yorke chaotic) if and only if $φ:Γ\toΓ$ has at least one non-quasi-periodic point. |
| title | On a class between Devaney chaotic and Li-Yorke chaotic generalized shift dynamical systems |
| topic | Dynamical Systems 37B99 |
| url | https://arxiv.org/abs/1708.04868 |