Theorem 5.10.3 from *Introduction to dynamical systems*, by Brin & Stuck:


Let $f:M\rightarrow M$ be an Anosov diffeomorphism. Then the following are equivalent:


1. $NW(f)=M$,
2. every unstable manifold is dense in $M$,
3. every stable manifold is dense in $M$
4. $f$ is topologically transitive,
5. $f$ is topologically mixing.


I want to know weather it is Ok to replace "Anosov diffeomorphism" with "Partially hyperbolic diffeomorphism" in the above theorem?


You can find the definitions of hyperbolicity and partial hyperbolicity [here](http://www.scholarpedia.org/article/Partial_hyperbolicity) and [here](http://en.wikipedia.org/wiki/Hyperbolic_set)



Is there a similar theorem in the partially hyperbolic case?
I think this runs into trouble at condition 1. For example, let $f : M \to M$ be an Anosov diffeomorphism that satisfies these conditions. The map $g: M \times S^1 \to M \times S^1$ defined by $g = f \times identity$ is partially hyperbolic (center direction is along the $S^1$ fibers) and the non-wandering set is the whole product manifold. But none of conditions 2-5 hold, since for any open set $U \subset S^1$ we have $g(M \times U) = M \times U$


