Circle packings, renormalizations, and subdivision rules

Abstract In this paper, we use iterations of skinning maps on Teichmüller 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 is uniformly contracting. This allows us to give complete answers for the existence and moduli problems for such circle packings. The exponential contraction of means that despite the nonrigidity of such circle packings, they are geometrically inflexible. As an application, we show that any geometrically finite Kleinian circle packing is combinatorially rigid.