Planar Relative Schottky Setsand Quasisymmetric Maps

Starting with a region ? in the plane and removing off from ? a countable number of open disks that are positively separate from each other and from the boundary of ?,we get a subset of ?,which is called a relative Schottky set.In the case ? is the complex plane,a set constructed as above is called a Schottky set.Therefore,a lot of questions on quasisymmetrical mappings between relative Schottky sets are more difficult than those between Schottky sets.In the present paper we focus on relative Schottky sets of measure zero starting by Jordan regions.We prove that any quasisymmetrical mapping between two of such Schottky sets must be the restriction of a conformal mapping.Also,we prove that such mappings are locally bi-Lipschitz.