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

Let . Let be a Zariski dense convex cocompact subgroup and be its limit set. Let be a Zariski dense convex cocompact faithful representation and the ‐boundary map. Let When there exists at least one ‐doubly stable circle in (e.g., is disconnected), we prove the following dichotomy: where is the Hausdorff measure of dimension . Moreover, in the former case, we have and is a conjugation by a Möbius transformation on . 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 . We also obtain an analogous theorem for any divergence‐type subgroup.