Joint equidistribution of maximal flat cylinders and holonomies for Anosov homogeneous spaces

TL;DR

This paper proves the joint equidistribution of maximal flat cylinders and their holonomies in Anosov homogeneous spaces, extending classical rank one results to higher rank groups.

Contribution

It establishes a joint equidistribution result for maximal flat cylinders and holonomies in Anosov homogeneous spaces, generalizing rank one geodesic results.

Findings

Joint equidistribution of flat cylinders and holonomies as circumferences grow

Extension of rank one geodesic equidistribution to higher rank spaces

Description of primitive elements' conjugacy classes in Anosov subgroups

Abstract

Let $G$ be a connected semisimple real algebraic group and $P<G$ be a minimal parabolic subgroup with Langlands decomposition $P=MAN$. Let $\Gamma < G$ be a Zariski dense Anosov subgroup with respect to $P$. Since $\Gamma$ is Anosov, the set of conjugacy classes of primitive elements of $\Gamma$ is in one-to-one correspondence with the set of (positively oriented) maximal flat cylinders in $\Gamma\backslash G/M$. We describe the joint equidistribution of maximal flat cylinders and their holonomies as their circumferences tend to infinity. This result can be viewed as the Anosov analogue of the joint equidistribution result of closed geodesics and holonomies in rank one by Margulis--Mohammadi--Oh.