  A relative Schottky set in a planar domain D is a subset of D obtained by
removing from D open geometric discs whose closures are in D and are pairwise
disjoint. In this paper we study quasisymmetric and related maps between
relative Schottky sets of measure zero. We prove, in particular, that
quasisymmetric maps between such sets in Jordan domains are conformal, locally
bi-Lipschitz, and that their first derivatives are locally Lipschitz. We also
provide a locally bi-Lipschitz uniformization result for relative Schottky sets
in Jordan domains and establish rigidity with respect to local quasisymmetric
maps for relative Schottky sets in the unit disc.
