In this paper, we study the prescribable conformally equivariant dilatations for orientation preserving quasiconformal homeomorphisms. The complex dilatation is a prescribable conformally equivariant dilatation in ℝ2. A Schottky set is a subset of the unit sphere Sn whose complement is the union of at least three disjoint open balls. By using the result of Bonk, Kleiner, and Merenkov that there are rigid Schottky sets of positive measure in each dimension at least 3, we prove that it is not possible to have a prescribable conformally equivariant dilatation in ℝn, where n ≥ 3.
ASJC Scopus subject areas
- Applied Mathematics