Anosov flows in dimension 3 from gluing building blocks with quasi-transverse boundary
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929732914774016 |
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| author | Paulet, Neige |
| author_facet | Paulet, Neige |
| contents | We prove a new result allowing to construct Anosov flows in dimension 3 by gluing building blocks. By a building block, we mean a compact 3-manifold with boundary $P$, equipped with a $C^1$ vector field $X$, such that the maximal invariant set $\cap_{t \in \mathbb{R}} X^t (P)$ is a saddle hyperbolic set, and the boundary $\partial P$ is quasi-transverse to $X$, i.e. transverse except for a finite number of periodic orbits contained in $\partial P$. Our gluing theorem is a generalization of a recent result of F. Béguin, C. Bonatti, and B. Yu who only considered the case where the block does not contain attractors nor repellers, and the boundary $\partial P$ is transverse to $X$. The quasi-transverse setting is much more natural. Indeed, our result can be seen as a counterpart of a theorem by Barbot and Fenley which roughly states that every 3-dimensional Anosov flow admits a canonical decomposition into building blocks (with quasi-transverse boundary). We will also show a number of applications of our theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13054 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Anosov flows in dimension 3 from gluing building blocks with quasi-transverse boundary Paulet, Neige Dynamical Systems 37D20, 37D05, 37C10 (Primary) 57K30, 57R30 (Secondary) We prove a new result allowing to construct Anosov flows in dimension 3 by gluing building blocks. By a building block, we mean a compact 3-manifold with boundary $P$, equipped with a $C^1$ vector field $X$, such that the maximal invariant set $\cap_{t \in \mathbb{R}} X^t (P)$ is a saddle hyperbolic set, and the boundary $\partial P$ is quasi-transverse to $X$, i.e. transverse except for a finite number of periodic orbits contained in $\partial P$. Our gluing theorem is a generalization of a recent result of F. Béguin, C. Bonatti, and B. Yu who only considered the case where the block does not contain attractors nor repellers, and the boundary $\partial P$ is transverse to $X$. The quasi-transverse setting is much more natural. Indeed, our result can be seen as a counterpart of a theorem by Barbot and Fenley which roughly states that every 3-dimensional Anosov flow admits a canonical decomposition into building blocks (with quasi-transverse boundary). We will also show a number of applications of our theorem. |
| title | Anosov flows in dimension 3 from gluing building blocks with quasi-transverse boundary |
| topic | Dynamical Systems 37D20, 37D05, 37C10 (Primary) 57K30, 57R30 (Secondary) |
| url | https://arxiv.org/abs/2312.13054 |