Fekete polynomials and shapes of Julia sets

TL;DR

This paper characterizes which subsets of the complex plane can be approximated by polynomial filled Julia sets, providing constructive methods using Fekete polynomials and estimates on the approximation rate.

Contribution

It establishes a precise criterion for approximation by Julia sets and introduces a constructive approach using Fekete polynomials, along with approximation rate estimates.

Findings

Characterization of approximable subsets as bounded with connected complement

Constructive approximation method using Fekete polynomials

Estimate for approximation rate based on geometric and potential theoretic factors

Abstract

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation in terms of geometric and potential theoretic quantities.