Shapes of Polynomial Julia Sets

TL;DR

This paper demonstrates that polynomial Julia sets can approximate any Jordan curve in the complex plane, and rational map basins can approximate finite collections of disjoint Jordan domains, highlighting the flexibility of Julia sets.

Contribution

It establishes that Julia sets of polynomials can approximate arbitrary Jordan curves, and rational maps can approximate collections of Jordan domains, expanding understanding of Julia set topology.

Findings

Polynomial Julia sets can approximate any Jordan curve.

Basins of attraction of rational maps can approximate finite collections of Jordan domains.

Julia sets exhibit versatile topological approximation properties.

Abstract

Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational maps.