Dimension bounds for escape on average 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 $0<\delta \le 1$.

18 pages, 1 figure