Anosov flows in dimension 3 from gluing building blocks with quasi-transverse boundary

Abstract

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. B\'eguin, 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.