On factor rigidity and joining classification for infinite volume rank one homogeneous spaces

We classify locally finite joinings with respect to the Burger-Roblin measure for the action of a horospherical subgroup $U$ on $\Gamma \backslash G$, where $G = \operatorname{SO}(n,1)^\circ$ and $\Gamma$ is a convex cocompact and Zariski dense subgroup of $G$, or geometrically finite with restrictions on critical exponent and rank of cusps. We also prove in the more general case of $\Gamma$ geometrically finite and Zariski dense that certain $U$-equivariant set-valued maps are rigid.