Ergodic decompositions of Burger-Roblin measures on Anosov homogeneous spaces

Minju Lee, Hee Oh

Published 2020-10-21Version 1

Let $G$ be a connected semisimple real algebraic group and $\Gamma$ a Zariski dense Anosov subgroup of $G$. Let $N$ be a maximal horospherical subgroup of $G$ and $P$ its normalizer with a fixed Langlands decomposition $P=MAN$. We show that each $N$-invariant Burger-Roblin measure on $\Gamma\backslash G$ decomposes into at most $[M:M^\circ]$-number of $N$-ergodic components, and deduce the following refinement of the main result of the previous paper by the authors: the space of all non-trivial $N$-invariant ergodic and $M^\circ A$-quasi-invariant Radon measures on $\Gamma\backslash G$, modulo the translations by $M$, is homeomorphic to ${\mathbb R}^{\text{rank }G-1}$.