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

Let $G$ be a connected semisimple Lie group of real rank 1 with finite center, let $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }\mathrm{\backslash }G$ 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.