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DCDS

In [23] 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 $acip$ at all.

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 $acip$ at all.

keywords:
Partially hyperbolic
,
$acip$ measure
,
positive volume
,
saturated.
,
weak ergodicity
,
transitive
,
accessible
,
dynamical density
basis

## Year of publication

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