Uniformization of Cantor sets with bounded geometry

Vellis, Vyron

doi:10.1090/ecgd/360

Uniformization of Cantor sets with bounded geometry
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by Vyron Vellis

Conform. Geom. Dyn. 25 (2021), 88-103

DOI: https://doi.org/10.1090/ecgd/360

Published electronically: August 10, 2021

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Abstract:

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus [Rev. Mat. Iberoamericana 15 (1999), pp. 267–277] 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 $\mathbb {R}^{n+1}$.

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