Escape of mass and entropy for diagonal flows in real rank one situations

TL;DR

This paper explores the relationship between entropy and escape of mass for diagonal flows in real rank one Lie groups, providing bounds on escaping mass and analyzing orbit dimensions.

Contribution

It establishes bounds on escaping mass and links entropy with escape phenomena for diagonal flows in rank one Lie groups.

Findings

Bounds on escaping mass for diagonal flows

Relation between entropy and escape of mass

Hausdorff dimension of non-escaping orbit sets is not full

Abstract

Let $G$ be a connected semisimple Lie group of real rank 1 with finite center, let $\Gamma$ be a non-uniform lattice in $G$ and $a$ any diagonalizable element in $G$. We investigate the relation between the metric entropy of $a$ acting on the homogeneous space $\Gamma\backslash G$ and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of $a$) which miss a fixed open set is not full.