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 $\mathrm{\Gamma }\mathrm{\backslash }G$, where $G=\mathrm{SO}(n,1{)}^{\circ }$ and $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ geometrically finite and Zariski dense that certain $U$-equivariant set-valued maps are rigid.