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

TL;DR

This paper investigates the structure of partially hyperbolic sets with positive volume, showing they contain stable and unstable manifolds, and explores the existence and properties of absolutely continuous invariant measures in such systems.

Contribution

It introduces applications of a dynamical density basis to partially hyperbolic sets and establishes conditions for the existence and properties of acip in partially hyperbolic systems.

Findings

Partially hyperbolic sets with positive volume contain stable and unstable manifolds.

If a system is essentially accessible with an acip, it is transitive and has dense orbits.

Accessible and center bunched systems either preserve a smooth measure or lack acip.

Abstract

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 $f$.