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arXiv · 2111.01924

Conformal renormalization of compact sets

Carsten Lunde Petersen·Filip Samuelsen
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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.

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Carsten Lunde Petersen, Filip Samuelsen. 2021-11-02. Conformal renormalization of compact sets. https://arxiv.org/abs/2111.01924

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