Construction of Anosov flows in dimension 3 by gluing blocks

Paulet, Neige

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

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 ${\bigcap }_{t\in ℝ}{X}^t\left(P\right)$ 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.

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

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