Rigidity of Kleinian groups via self-joinings: measure theoretic criterion

Let $n,m\ge 2$. Let $\mathrm{\Gamma }<{\text{SO}}^{\circ }(n+1,1)$ be a Zariski dense convex cocompact subgroup and $\mathrm{\Lambda }\subset {\mathbb{S}}^n$ be its limit set. Let $\rho :\mathrm{\Gamma }\to {\text{SO}}^{\circ }(m+1,1)$ be a Zariski dense convex cocompact faithful representation and $f:\mathrm{\Lambda }\to {\mathbb{S}}^m$ the $\rho$-boundary map. Let $${\mathrm{\Lambda }}_f:=\bigcup \{C\cap \mathrm{\Lambda }:\begin{array}{c}C\subset {\mathbb{S}}^n\text{ is a circle such that}\\ f(C\cap \mathrm{\Lambda })\text{ is contained in a proper sphere in }{\mathbb{S}}^m\end{array}\}\mathrm{.}$$ When there exists at least one $\mathrm{\Lambda }$-doubly stable circle in ${\mathbb{S}}^n$ (e.g., $\mathrm{\Omega }={\mathbb{S}}^n-\mathrm{\Lambda }$ is disconnected), we prove the following dichotomy: $$\text{either}{\mathrm{\Lambda }}_f=\mathrm{\Lambda }\text{ or }{\mathcal{H}}^{\delta }({\mathrm{\Lambda }}_f)=0,$$ where ${\mathcal{H}}^{\delta }$ is the Hausdorff measure of dimension $\delta ={\dim }_H\mathrm{\Lambda }$. Moreover, in the former case, we have $n=m$ and $\rho$ is a conjugation by a Möbius transformation on ${\mathbb{S}}^n$. Our proof uses ergodic theory for directional diagonal flows and conformal measure theory of discrete subgroups of higher rank semisimple Lie groups, applied to the self-joining subgroup ${\mathrm{\Gamma }}_{\rho }=(\mathrm{id}\times \rho )(\mathrm{\Gamma })<{\text{SO}}^{\circ }(n+1,1)\times {\text{SO}}^{\circ }(m+1,1)$. We also obtain an analogous theorem for any divergence-type subgroup.