Conformal renormalization of compact sets

TL;DR

This paper introduces a conformal renormalization scheme for compact sets in the complex plane, enabling analysis of their equilibrium measures and providing a method to reconstruct the original set from its renormalizations.

Contribution

It develops a novel conformal renormalization framework for compact sets and proves the existence of conformal homeomorphisms relating equilibrium measures, also offering a reconstruction method.

Findings

Existence of conformal homeomorphisms for equilibrium measures

Relaxation of connectedness conditions for subsets

Reconstruction of sets from conformal renormalizations

Abstract

This paper develops a conformal renormalization scheme for compact sets $K \subset \mathbb{C}$. As one application of the conformal renormalization scheme we prove that for every isolated non-trivial connected component $E \subset K$ there exists a conformal homeomorphism $\phi$ mapping a neighbourhood of $E$ into $\mathbb{C}$ such that the equilibrium measure on $K$ restricted to $E$ equals the scaled push-forward by $\phi^{-1}$ of the equilibrium measure on $\phi(E)$. Moreover the proof shows that the condition of connectedness of $E$ can be relaxed considerably. We also introduce an inverse to the procedure of conformal renormalization, which allows one to reconstruct $K$ from its conformal renormalizations.