TITLE: Is there a similar theorem in the partially hyperbolic case?
QUESTION [6 upvotes]: 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:

$NW(f)=M$,
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?
You can find the definitions of hyperbolicity and partial hyperbolicity here and here

REPLY [3 votes]: 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 $g(M \times U) = M \times U$