Dimension bounds for escape on average in homogeneous spaces

TL;DR

This paper provides an upper bound on the Hausdorff dimension of points in homogeneous spaces whose trajectories escape a given open set on average, depending on the escape frequency.

Contribution

It introduces a new upper estimate for the Hausdorff dimension of escape sets in homogeneous spaces based on average escape frequency.

Findings

Derived an explicit upper bound for the Hausdorff dimension of escape sets.

Applicable to trajectories with any escape frequency between 0 and 1.

Enhances understanding of dynamical behavior in homogeneous spaces.

Abstract

Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a uniform lattice in $G$, and let $O$ be an open subset of $X$. We give an upper estimate for the Hausdorff dimension of the set of points whose trajectories escape $O$ on average with frequency $\delta$, where $0 < \delta \le 1$.