Simultaneous Approximation by Attracting Basins

TL;DR

The paper demonstrates that for any collection of disjoint open sets sharing a boundary, one can approximate these sets with attracting basins of a rational map, with the Julia set closely matching the boundary.

Contribution

It introduces a method to approximate multiple disjoint open sets by attracting basins of a rational map sharing a common boundary.

Findings

Attracting basins can approximate arbitrary disjoint open sets.

The Julia set of the rational map approximates the shared boundary.

The construction works for any number of sets with dimension at least 3.

Abstract

We show that any $d\geq3$ pairwise-disjoint open sets $A_1$, ..., $A_d\subset\widehat{\mathbb{C}}$ sharing a common boundary $J$ can be simultaneously approximated by the $d$ attracting basins $\mathcal{A}_1$, ..., $\mathcal{A}_d$ of a rational map $r$ having Fatou set $\mathcal{F}(r)=\mathcal{A}_1\sqcup...\sqcup\mathcal{A}_d$ and so that the Julia set $\mathcal{J}(r)$ approximates $J$.