Ergodic decompositions of geometric measures on Anosov homogeneous spaces

Abstract

Let $G$ be a connected semisimple real algebraic group and $\hat{I}“$ a Zariski dense Anosov subgroup of $G$ with respect to a minimal parabolic subgroup $P$. Let $N$ be the maximal horospherical subgroup of $G$ given by the unipotent radical of $P$. We describe the $N$-ergodic decompositions of all Burger-Roblin measures as well as the $A$-ergodic decompositions of all Bowen-Margulis-Sullivan measures on $\hat{I}“\backslash G$. As a consequence, we obtain the following refinement of the main result of [LO]: the space of all {\it non-trivial} $N$-invariant ergodic and $P^{\circ }$-quasi-invariant Radon measures on $\hat{I}“\backslash G$, up to constant multiples, is homeomorphic to ${\mathbb{R}}^{\text{rank}G-1}\times \{1,\cdots ,k\}$ where $k$ is the number of $P^{\circ }$-minimal subsets in $\hat{I}“\backslash G$.

32 pages, To appear in Israel J. Math