On Ergodic Properties of the Burger-roblin Measure

In this note we intend to describe some dynamical properties of oneparameter unipotent flows on the frame bundle of a convex cocompact hyperbolic 3-manifold. Much effort and study have been done in the case of manifolds with finite volume, and quite a rich theory is developed in this case. The case of infinite volume manifolds, however, is far less understood. The goal here is to highlight some of the difficulties one faces, and possible modifications, in extending techniques developed in the finite volume case to the case of infinite volume manifolds. Throughout, G = PSL2(C), the group of orientation preserving isometries of the hyperbolic space H3. We let Γ be a Zariski dense discrete subgroup of G which is convex cocompact, that is, the convex hull of the limit set of Γ is compact modulo Γ. Equivalently, Γ\H3 admits a finite sided fundamental domain with no cusps. The frame bundle of the manifold Γ\H3 is identified with the homogeneous space X = Γ\G. Certain subgroups of G will be of particular importance in the sequel. Let K = PSU2, A = {as : s ∈ R}, and N = {nz : z ∈ C}, where