Construction of Anosov flows in dimension 3 by gluing blocks

Neige Paulet

doi:10.5802/mrr.17

Neige Paulet ¹

¹ LAGA, Université Sorbonne Paris Nord, 93143 Villetaneuse FRANCE

Mathematics Research Reports, Volume 4 (2023), pp. 47-62.

Abstract

We present a new result allowing us 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 $⋂_{t \in ℝ} X^{t} (P)$ is a saddle hyperbolic set, and such that $\partial P$ 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 $\partial P$ 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 present a number of applications of our theorem.

Received: 2023-06-19
Published online: 2023-12-15

DOI: 10.5802/mrr.17

Classification: 37D20, 37D05, 37C10, 57K30, 57R30
Keywords: Anosov flows, 3-manifolds, building blocks

Author's affiliations:

Neige Paulet ¹

¹ LAGA, Université Sorbonne Paris Nord, 93143 Villetaneuse FRANCE

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Neige Paulet. Construction of Anosov flows in dimension 3 by gluing blocks. Mathematics Research Reports, Volume 4 (2023), pp. 47-62. doi : 10.5802/mrr.17. https://mrr.centre-mersenne.org/articles/10.5802/mrr.17/

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