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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 theBowen-Margulis-Sullivan measure with respect to closed connected subspaces ofthe $N$ component in some Iwasawa decomposition$\mathbf{SO}^o(n+1,\mathbf{R})=KAN$. We also study the dimension of projectedPatterson-Sullivan measures along some fixed small circle.

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