Pseudo-Anosov surfaces and Dynamics in 3-manifolds

We determine which closed orientable $3$-manifolds $M$ admit a self-homeomorphism restricting to a pseudo-Anosov map on an incompressible subsurface $\Sigma$, which we call a pseudo-Anosov surface. When $M$ is irreducible, we show that the self-homeomorphism of $M$ is isotopic rel $\Sigma$ to a "partially pseudo-Anosov" homeomorphism, a notion that we will introduce. This is motivated by the corresponding results for Anosov tori in irreducible $3$-manifolds, and the connection to partially hyperbolic diffeomorphisms, obtained by F. Rodriguez-Hertz, J. Rodriguez-Hertz and R. Ures.