Dipoles and Pixie dust

TL;DR

This paper demonstrates that any closed subset of the Riemann sphere can be approximated by Julia sets of rational maps in the Hausdorff topology, highlighting the universality of Julia sets.

Contribution

It proves a universal approximation property of Julia sets for all closed subsets of the Riemann sphere.

Findings

Any closed subset can be approximated by Julia sets

Julia sets are dense in the space of closed subsets

The approximation uses rational maps in the Hausdorff topology

Abstract

Every closed subset of the Riemann sphere can be approximated in the Hausdorff topology by the Julia set of a rational map.