Circle packing with generalized branching

Abstract

There is a fairly comprehensive theory of discrete analytic functions based on circle packing. In this theory, discrete analytic functions are represented as maps between circle packings that share combinatorial tangency patterns. Branching behavior, however, has until now been restricted by the need to place branch points at circle centers. In this paper, the authors introduce mechanisms for generalized branching which remove this restriction. Their use is illustrated by overcoming combinatorial obstructions to branching in the construction of a discrete Ahlfors function.