Compact non-uniformizable Li-Yorke chaotic dynamical systems via an example
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918260412252160 |
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| author | Pourattar, Mehrnaz Shirazi, Fatemah Ayatollah Zadeh |
| author_facet | Pourattar, Mehrnaz Shirazi, Fatemah Ayatollah Zadeh |
| contents | The main aim of this paper is extending the concept of scambled pair and Li--Yorke chaos to non--uniform compact dynamical systems. We show for finite (compact Alexandroff) topological space $X$ with at least two elements the following statements are equivalent:
$\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ is Li--Yorke chaotic,
$\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ has at least one scrambled pair,
$\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ has at least one non--asymptotic pair,
$\bullet$ there exists $a,b\in X$ such that $\overline{\{a\}}\cap\overline{\{b\}}=\varnothing$,
$\bullet$ $\bigcap\{\overline{\{a\}}:a\in X\}=\varnothing$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19881 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Compact non-uniformizable Li-Yorke chaotic dynamical systems via an example Pourattar, Mehrnaz Shirazi, Fatemah Ayatollah Zadeh Dynamical Systems General Topology 37B02, 54C05 The main aim of this paper is extending the concept of scambled pair and Li--Yorke chaos to non--uniform compact dynamical systems. We show for finite (compact Alexandroff) topological space $X$ with at least two elements the following statements are equivalent: $\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ is Li--Yorke chaotic, $\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ has at least one scrambled pair, $\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ has at least one non--asymptotic pair, $\bullet$ there exists $a,b\in X$ such that $\overline{\{a\}}\cap\overline{\{b\}}=\varnothing$, $\bullet$ $\bigcap\{\overline{\{a\}}:a\in X\}=\varnothing$. |
| title | Compact non-uniformizable Li-Yorke chaotic dynamical systems via an example |
| topic | Dynamical Systems General Topology 37B02, 54C05 |
| url | https://arxiv.org/abs/2512.19881 |