On uniformly disconnected Julia sets

Abstract

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in $\mathbb{S}^n$, for $n\geq 2$. Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if $E$ is a compact, uniformly perfect and uniformly disconnected set in $\mathbb{S}^n$, then it is the Julia set of a hyperbolic UQR map $f:\mathbb{S}^N \to \mathbb{S}^N$ where $N=n$ if $n=2$ and $N=n+1$ otherwise.