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.