Approximation of analytic functions with prescribed boundary conditions by circle packing maps

We use recent advances in circle packing theory to develop a constructive method for the approximation of an analytic function $F:\Omega \to \bold C$ by circle packing maps providing we have only been given $\Omega$, $|F'|\big|_{\Omega}$, and the set of critical points of $F$. This extends the earlier result of Ithiel Carter and Burt Rodin [CR] for $F$ with no critical points

Recurrent random walks, Liouville's theorem, and circle packings

It has been shown that univalent circle packings filling in the complex plane $\bold C$ are unique up to similarities of $\bold C$. Here we prove that bounded degree branched circle packings properly covering $\bold C$ are uniquely determined, up to similarities of $\bold C$, by their branch sets. In particular, when branch sets of the packings considered are empty we obtain the earlier result. We also establish a circle packing analogue of Liouville's theorem: if $f$ is a circle packing map whose domain packing is infinite, univalent, and has recurrent tangency graph, then the ratio map associated with $f$ is either unbounded or constant

Approximation of analytic functions with prescribed boundary conditions by circle-packing maps

Abstract. We use recent advances in circle packing theory to develop a constructive method for the approximation of an analytic function F: Ω → C by circle packing maps providing we have only been given Ω, |F ′ | ∣ ∣ Ω, and the set of critical points of F. This extends the earlier result of [CR] for F with no critical points.