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 \(n\ge 2\). 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 \(f:\mathbb {S}^N \rightarrow \mathbb {S}^N\) where \(N=n\) if \(n=2\) and \(N=n+1\) otherwise.