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 and here

  • 1
    $\begingroup$ The time one map of an Anosov flow is partially hyperbolic. And an Anosov flow can be topologically transitive without being mixing. Typically, a constant suspension flow above an Anosov diffeomorphism will not be mixing. This is very similar to the answer below. $\endgroup$ – Barbara Schapira Jan 19 '14 at 21:12

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 $g(M \times U) = M \times U$


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