Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1435 - 1447, Volume 32, Issue 4, April 2012      doi:10.3934/dcds.2012.32.1435

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Pengfei Zhang - School of Mathematical Sciences, Peking University, Beijing 100871, China (email)

Abstract: 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.

Keywords:  Partially hyperbolic, positive volume, dynamical density basis, $acip$ measure, weak ergodicity, transitive, accessible, saturated.
Mathematics Subject Classification:  Primary 37D30, 37D10; Secondary 37C40, 37D20.

Received: June 2010;      Revised: August 2011;      Available Online: October 2011.