Projections of Patterson-Sullivan measures and the Mohammadi-Oh dichotomy

Abstract: Let $\Gamma$ be some discrete subgroup of $\mathbf{SO}^o(n+1,\mathbf{R})$ with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the $N$ component in some Iwasawa decomposition $\mathbf{SO}^o(n+1,\mathbf{R})=KAN$. We also study the dimension of projected Patterson-Sullivan measures along some fixed small circle.