HOROSPHERICAL INVARIANT MEASURES AND A RANK DICHOTOMY FOR ANOSOV GROUPS

Let G =^∏r i=1 G_i be a product of simple real algebraic groups of rank one and Γ an Anosov subgroup of G with respect to a minimal parabolic subgroup. For each v in the interior of a positive Weyl chamber, let Rv ⊂Γ\G denote the Borel subset of all points with recurrent exp(R₊v)-orbits. For a maximal horospherical subgroup N of G, we show that the N-action on Rv is uniquely ergodic if r = rank(G) ≤ 3 and v belongs to the interior of the limit cone of Γ, and that there exists no N-invariant Radon measure on Rv otherwise.