Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems

In [Discrete Contin. Dyn. Syst. $\mathbf{\text{15}}$ (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 $\mathrm{\Lambda }$ is a strongly partially hyperbolic set with positive volume, then $\mathrm{\Lambda }$ contains the global stable manifolds over $\alpha ({\mathrm{\Lambda }}^d)$ and the global unstable manifolds over $\omega ({\mathrm{\Lambda }}^d)$. 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 $f$.