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.