Finite intersection property and dynamical compactness
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2016
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| Acceso en línea: | |
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| _version_ | 1866916149219819520 |
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| author | Huang, Wen Khilko, Danylo Kolyada, Sergiy Peris, Alfred Zhang, Guohua |
| author_facet | Huang, Wen Khilko, Danylo Kolyada, Sergiy Peris, Alfred Zhang, Guohua |
| contents | Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in [22]. In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the $ω_{\mathcal{F}}$-limit and the $ω$-limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1605_05851 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Finite intersection property and dynamical compactness Huang, Wen Khilko, Danylo Kolyada, Sergiy Peris, Alfred Zhang, Guohua Dynamical Systems Primary 37B05, Secondary 54H20 Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in [22]. In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the $ω_{\mathcal{F}}$-limit and the $ω$-limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics. |
| title | Finite intersection property and dynamical compactness |
| topic | Dynamical Systems Primary 37B05, Secondary 54H20 |
| url | https://arxiv.org/abs/1605.05851 |