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

Abstract

Let G be any connected semisimple Lie group of real rank 1 with finite center, let Γ 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 Γ∖G and escape of mass. Moreover, we provide bounds on the escaping 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.