Quasi-Anosov diffeomorphisms of 3manifolds. preprint

Abstract. In 1969, Hirsch posed the following problem: given a dif-feomorphism f: N → N, and a compact invariant hyperbolic set Λ of f, describe the topology of Λ and the dynamics of f restricted to Λ. We solve the problem where Λ = M3 is a closed 3-manifold: if M3 is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions. The dynamics of the diffeomorphisms restricted to M3, called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions. 1