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 $Γ$ 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 $Γ\backslash G$ 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.

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