  We call a 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 Mobius transformation on S^n. In the other
direction we show that every Schottky set in S^2 of positive measure admits
non-trivial 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.
