On mixing and dense periodicity on spaces with a free arc
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910171175845888 |
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| author | Kwietniak, Dominik Wierzbowski, Filip |
| author_facet | Kwietniak, Dominik Wierzbowski, Filip |
| contents | We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval $(0,1)$). We provide a new proof of a result of M. Dirbák, Ľ. Snoha, V. Špitalský [Ergodic Theory Dynam. Systems, vol. 33 (2013), no. 6, pp. 1786--1812] saying that every continuous and transitive, but non-minimal map of a space with a free interval is relatively mixing, non-invertible, has positive topological entropy, and dense periodic points. The key simplification comes from short proofs of two facts. The first says that every weakly mixing map of a space with a free interval must be mixing and have positive entropy. The second says that a transitive but not minimal map of a space with a free interval has dense periodic points and is non-invertible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On mixing and dense periodicity on spaces with a free arc Kwietniak, Dominik Wierzbowski, Filip Dynamical Systems 37E05, 37E10 We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval $(0,1)$). We provide a new proof of a result of M. Dirbák, Ľ. Snoha, V. Špitalský [Ergodic Theory Dynam. Systems, vol. 33 (2013), no. 6, pp. 1786--1812] saying that every continuous and transitive, but non-minimal map of a space with a free interval is relatively mixing, non-invertible, has positive topological entropy, and dense periodic points. The key simplification comes from short proofs of two facts. The first says that every weakly mixing map of a space with a free interval must be mixing and have positive entropy. The second says that a transitive but not minimal map of a space with a free interval has dense periodic points and is non-invertible. |
| title | On mixing and dense periodicity on spaces with a free arc |
| topic | Dynamical Systems 37E05, 37E10 |
| url | https://arxiv.org/abs/2510.15073 |