On \epsilon -escaping trajectories in homogeneous spaces

Abstract

Let G/Γ be a finite volume homogeneous space of a semisimple Lie group G, and {exp(tD)} be a one-parameter Ad-diagonalizable subgroup inside a simple Lie subgroup G 0 of G. Denote by Z ,D the set of points x ∈ G/Γ whose {exp(tD)}-trajectory has an escape for at least an -portion of mass along some subsequence. We prove that the Hausdorff codimension of Z ,D is at least c , where c depends only on G, G 0 and Γ.