Invariant Measures for Horospherical Actions and Anosov Groups

Let Γ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group G. For a maximal horospherical subgroup N of G, we show that the space of all non-trivial $NM$-invariant ergodic and A-quasi-invariant Radon measures on Γ G, up to proportionality, is homeomorphic to Rrank\,G-1, where A is a maximal real split torus and M is a maximal compact subgroup that normalizes N. One of the main ingredients is to establish the $NM$-ergodicity of all Burger–Roblin measures.