Appears in collection : Homogeneous Dynamics and Geometry in Higher-Rank Lie Groups

Let $G$ be a product of simple real algebraic groups of rank one and $\Gamma$ be a Zariski dense and Anosov subgroup with respect to a minimal parabolic subgroup $P$. Let $N$ be the unipotent radical of $P$. For each direction $u$ in the interior of Weyl chamber, we show that there exists at most one $N$-invariant measure in $\Gamma\backslash G$ which is supported on the forward recurrent subset for the $\exp(tu)$-action. This can be viewed as a generalization of the unique ergodicity result for the horospherical action due to Furstenberg, Burger, Roblin and Winter for $\Gamma$ convex cocompact. This is joint work with Or Landesberg, Elon Lindenstrauss and Hee Oh.