Invariant Measures for Horospherical Actions and Anosov Groups

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