Hyperbolic sub-dynamics: compact invariant 3-manifolds

In 1970, Hirsch asked what kind of compact invariant sets could be part of a hyperbolic set. Here we obtain that, in case such an invariant set is a 3D manifold, it is a connected sum of tori with handles quotiented by involutions. Moreover, if the manifold is orientable, the involutions are all trivial. In 1975, Ma{~n}{'e} characterized hyperbolic dynamics restricted to manifolds and called them quasi Anosov. We also classify here quasi-Anosov dynamics in 3D-manifolds.