Building Anosov flows on $3$–manifolds

Abstract

We prove we can build (transitive or nontransitive) Anosov flows on closed three-dimensional manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications of this result; for example:

1. We build a closed three-dimensional manifold supporting both a transitive Anosov vector field and a nontransitive Anosov vector field.

2. For any $n$, we build a closed three-dimensional manifold $M$ supporting at least $n$ pairwise different Anosov vector fields.

3. We build transitive hyperbolic attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive hyperbolic attractors.

4. We build a transitive Anosov vector field admitting infinitely many pairwise nonisotopic transverse tori.