A surface $S$ in a three manifold $M$ is pseudo-Anosov means if there exists a homeomorphism $f$ over $M$ for which $S$ is $f$ invariant and $f$ is a pseudo-Anosov on $S$. For example, $M$---- any surface bundle over circle with pseudo-Anosov monodromy map; $S$---- a fiber (surface).
Question: Which three manifolds admit a pseudo-Anosov surface?
More subtle, if $M$ is irreducible, does the example(s) above contain all cases? Moreover, is the following true?: $M$ admits a pseudo-Anosov surface iff there exists a prime factor of $M$ admits a pseudo-Anosov surface?
This question is motivated by link text. In this paper, the authors tell us: if $S$ is torus, $f$ is Anosov and $M$ is irreducible, $M$ must be one of the following 3 cases: (1) the 3-torus $T^3$; (2) the mapping torus of -id; (3) the mapping tori of hyperbolic automorphisms of $T^2$.