Circle packings have many applications in geometry, analysis and dynamics. For a circle packing P, one can associate a plane graph called the nerve of P. It is natural and important to understand
1. Given a plane graph G, when is it isomorphic to the nerve of a circle packing?
2. Is the circle packing rigid? Or more generally, what is the moduli space of circle packings with nerve isomorphic to G?
3, How are different circle packings with isomorphic nerves related?
For finite graphs, Kobe-Andreev-Thurston’s circle packing theorem give a complete answer to the above questions. The situation is much more complicated for infinite graphs, and has been extensively studied for locally finite triangulations.
In this talk, I will describe a new perspective of using skinning map and renormalization theory to study these questions for infinite graphs. In particular, I will explain how it gives complete answers to the above questions for graphs with subdivision rules.
I will discuss some similarities and differences between this and the renormalization theory for quadratic like maps and mapping class group action on quasi-Fuchsian groups. I will also discuss some applications on quasiconformal geometries for gasket Julia set and limit set.
This is a joint work with Y. Zhang.