Gluing-orbit property, local stable/unstable sets, and induced dynamics on hyperspace
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910461654466560 |
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| author | Antunes, Mayara Carvalho, Bernardo Cordeiro, Welington Cueto, José |
| author_facet | Antunes, Mayara Carvalho, Bernardo Cordeiro, Welington Cueto, José |
| contents | We prove that local stable/unstable sets of homeomorphisms of an infinite compact metric space satisfying the gluing-orbit property always contain compact and perfect subsets of the space. As a consequence, we prove that if a positively countably expansive homeomorphism satisfies the gluing-orbit property, then the space is a single periodic orbit. We also prove that there are homeomorphisms with gluing-orbit such that its induced homeomorphism on the hyperspace of compact subsets does not have gluing-orbit, contrasting with the case of the shadowing and specification properties, proving that if the induced map has gluing-orbit, then the base map has gluing-orbit and is topologically mixing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17574 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gluing-orbit property, local stable/unstable sets, and induced dynamics on hyperspace Antunes, Mayara Carvalho, Bernardo Cordeiro, Welington Cueto, José Dynamical Systems We prove that local stable/unstable sets of homeomorphisms of an infinite compact metric space satisfying the gluing-orbit property always contain compact and perfect subsets of the space. As a consequence, we prove that if a positively countably expansive homeomorphism satisfies the gluing-orbit property, then the space is a single periodic orbit. We also prove that there are homeomorphisms with gluing-orbit such that its induced homeomorphism on the hyperspace of compact subsets does not have gluing-orbit, contrasting with the case of the shadowing and specification properties, proving that if the induced map has gluing-orbit, then the base map has gluing-orbit and is topologically mixing. |
| title | Gluing-orbit property, local stable/unstable sets, and induced dynamics on hyperspace |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2405.17574 |