Partially hyperbolic sets with positivemeasure and $ACIP$ for partially hyperbolic systems

Pengfei Zhang

doi:10.3934/dcds.2012.32.1435

Article Contents

2012, Volume 32, Issue 4: 1435-1447. Doi: 10.3934/dcds.2012.32.1435

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

Pengfei Zhang^1,

1.
School of Mathematical Sciences, Peking University, Beijing 100871

Received: May 31, 2010

Revised: July 31, 2011

Published: March 2012

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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.

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Mathematics Subject Classification: Primary 37D30, 37D10; Secondary 37C40, 37D20.

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