  In [Discrete Contin. Dyn. Syst. \textbf{15} (2006), no. 3, 811--818.] Xia
introduced a simple dynamical density basis for partially hyperbolic sets of
volume preserving diffeomorphisms. We apply the density basis to the study of
the topological structure of partially hyperbolic sets. We show that if
$\Lambda$ is a strongly partially hyperbolic set with positive volume, then
$\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$ and
the global unstable manifolds over ${\omega}(\Lambda^d)$.
  We give several applications of the dynamical density to partially hyperbolic
maps that preserve some $acip$. We show that if $f$ is essentially accessible
and $\mu$ is an $acip$ of $f$, then $\text{supp}(\mu)=M$, the map $f$ is
transitive, and $\mu$-a.e. $x\in M$ has a dense orbit in $M$. Moreover if $f$
is accessible and center bunched, then either $f$ preserves a smooth measure or
there is no $acip$ of $f$.
