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

Let $G$ be any connected semisimple Lie group of real rank 1 with finite center, let $\Gamma$ be any 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.