Projections of Patterson-Sullivan measures and the Mohammadi-Oh dichotomy

TL;DR

This paper investigates the dynamics of Patterson-Sullivan measures and their projections in hyperbolic spaces, revealing new dichotomies and measure projection properties related to discrete subgroups of special orthogonal groups.

Contribution

It introduces a novel analysis of Patterson-Sullivan measures' projections and their dynamical behavior under subgroup actions, extending the understanding of measure rigidity and geometric structures.

Findings

Established a dichotomy for Patterson-Sullivan measure projections.

Analyzed the measure dynamics with respect to subgroup actions.

Provided new insights into the dimension of projected measures.

Abstract

Let $\Gamma$ be some discrete subgroup of $\mathbf{SO}^o(n+1,\mathbf{R})$ with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the $N$ component in some Iwasawa decomposition $\mathbf{SO}^o(n+1,\mathbf{R})=KAN$. We also study the dimension of projected Patterson-Sullivan measures along some fixed small circle.