Multidimensional $C^0$ transversality and the shadowing property for Axiom A diffeomorphisms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914863120384000 |
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| author | Murakami, Sogo |
| author_facet | Murakami, Sogo |
| contents | Petrov and Pilyugin (2015) generalized a notion of $C^0$ transversality of Sakai (1995) using smooth curves. Their definition involves only continuous maps from ${\mathbb R}^n$ to a manifold, which is a purely topological one. They also provided a sufficient condition for the $C^0$ transversality in terms of homological nature. In this paper, we prove that such a homological condition of Axiom A diffeomorphisms is sufficient for enjoying the shadowing property. Moreover, it is proved that the $C^0$ transversality of Axiom A diffeomorphisms with codimension one basic sets implies the homological condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multidimensional $C^0$ transversality and the shadowing property for Axiom A diffeomorphisms Murakami, Sogo Dynamical Systems 37D20, 37C50 Petrov and Pilyugin (2015) generalized a notion of $C^0$ transversality of Sakai (1995) using smooth curves. Their definition involves only continuous maps from ${\mathbb R}^n$ to a manifold, which is a purely topological one. They also provided a sufficient condition for the $C^0$ transversality in terms of homological nature. In this paper, we prove that such a homological condition of Axiom A diffeomorphisms is sufficient for enjoying the shadowing property. Moreover, it is proved that the $C^0$ transversality of Axiom A diffeomorphisms with codimension one basic sets implies the homological condition. |
| title | Multidimensional $C^0$ transversality and the shadowing property for Axiom A diffeomorphisms |
| topic | Dynamical Systems 37D20, 37C50 |
| url | https://arxiv.org/abs/2407.06588 |