Uniformization of Cantor sets with bounded geometry

Abstract

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus from 2 to higher dimensions. In particular, we show that a compact subset of ${\mathbb{R}}^n$ is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set $\mathcal{C}$ in ${\mathbb{R}}^{n+1}$.

13 pages. This is a small part of an older manuscript called "Extension properties of planar uniform domains"

References in corpus (2)

• Geometry of measures in real dimensions via Hölder parameterizations
• Quasiconformal and geodesic trees