Simultaneous Approximation by Attracting Basins

Abstract

We show that any $d\ge 3$ pairwise-disjoint open sets $A_1$, ..., $A_d\subset \hat{\mathbb{C}}$ 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 $\mathcal{F}(r)={\mathcal{A}}_1\bigsqcup ...\bigsqcup {\mathcal{A}}_d$ and so that the Julia set $\mathcal{J}(r)$ approximates $J$.