Conformal Welding of Dendrites

Abstract

We investigate the conformal welding problem, which is a way of taking quotients of Riemann surfaces by identifying points on their boundaries. The existence and uniqueness of this operation is in general difficult to determine. Our focus is on weldings which exhibit branching so that the resulting boundary interfaces are dendrites. We show that the welding relation associated to certain Julia sets in complex dynamics satisfies a regularity condition analogous to the classical quasisymmetry condition. We also show that the Brownian lamination, a random welding relation related to the continuum random tree, has a unique solution.