Circle packings, renormalizations and subdivision rules

TL;DR

This paper develops a renormalization framework for circle packings with subdivision rules using skinning maps on Teichmüller spaces, establishing conditions for boundedness, contraction, and rigidity.

Contribution

It introduces a new renormalization operator for circle packings with subdivision rules and proves its uniform contraction under certain conditions.

Findings

The skinning map has bounded image under specific conditions.

The renormalization operator is uniformly contracting.

Geometrically finite Kleinian circle packings are combinatorially rigid.

Abstract

In this paper, we use iterations of skinning maps on Teichm\"uller spaces to study circle packings and develop a renormalization theory for circle packings whose nerves satisfy certain subdivision rules. We characterize when the skinning map has bounded image. Under the corresponding condition, we prove that the renormalization operator $\mathfrak{R}$ is uniformly contracting. This allows us to give complete answers for the existence and moduli problems for such circle packings. The exponential contraction of $\mathfrak{R}^n$ means that despite the non-rigidity of such circle packings, they are geometrically inflexible. As an application, we show that any geometrically finite Kleinian circle packing is combinatorially rigid.