  In 1969, Hirsch posed the following problem: given a diffeomorphism, and a
compact invariant hyperbolic set, describe its topology and restricted
dynamics. We solve the problem where the hyperbolic invariant set is a closed
3-manifold: if the manifold 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 these manifolds,
called quasi-Anosov diffeomorphisms, is also classified: it is the connected
sum of DA-diffeomorphisms, quotiented by commuting involutions.
