Rigidity of schottky sets

We call the complement of a union of at least three disjoint (round) open balls in the unit sphere S ⁿ a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Möbius transformation on S ⁿ. In the other direction we show that every Schottky set in S ² of positive measure admits nontrivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries.