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:
every unstable manifold is dense in $M$,
every stable manifold is dense in $M$
$f$ is topologically transitive,
$f$ is topologically mixing.
I want to know weather it is Ok to replace "Anosov diffeomorphism" with "Partially hyperbolic diffeomorphism" in the above theorem?