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](http://aimsciences.org/journals/pdfs.jsp?paperID=6086&mode=full). 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 $T^2$.



pseudo-Anosov surface in three manifolds
For genus greater than one, there are lots of Pseudo-Anosov mapping classes that extend over handlebodies, so you can build lots of examples that are not of the types listed above.
Here is a specific example:
D.D. Long "Pseudo-Anosov maps which extend over two handlebodies" Proceedings of Edinburgh Society(1990) 33, 181-190


There is recent work of Biringer, Johnson, and Minsky characterizing when a power of a pseudo-anosov extends over a handlebody: <http://arxiv.org/abs/1011.0021>


