Ergodicity of generalised Patterson-Sullivan measures in higher rank symmetric spaces

Abstract

Let X = G/K be a higher rank symmetric space of noncompact type and a discrete Zariski dense group. In a previous article, we constructed for each G-invariant subset of the regular limit set of Γ a family of measures, the so-called (b, Γ · ξ)-densities. Our main result here states that these densities are Γ-ergodic with respect to an important subset of the limit set which we choose to call the ``ray limit set''. In the particular case of uniform lattices and products of convex cocompact groups acting on the product of rank one symmetric spaces every limit point belongs to the ray limit set, hence our result is most powerful for these examples. For nonuniform lattices, however, it is a priori not clear whether the ray limit set has positive measure with respect to a (b, Γ · ξ)-density. Using a counting theorem of Eskin and McMullen, we are able to prove that the ray limit set has full measure in each G-invariant subset of the limit set.