Uniformization of Cantor sets with bounded geometry
Author:
Vyron Vellis
Journal:
Conform. Geom. Dyn. 25 (2021), 88-103
MSC (2020):
Primary 30C65; Secondary 30L05
DOI:
https://doi.org/10.1090/ecgd/360
Published electronically:
August 10, 2021
MathSciNet review:
4298216
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Abstract | References | Similar Articles | Additional Information
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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Additional Information
Vyron Vellis
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37916
MR Author ID:
1058764
Email:
vvellis@utk.edu
Keywords:
Quasisymmetric,
bi-Lipschitz,
Cantor set,
uniformly perfect,
uniformly disconnected
Received by editor(s):
January 24, 2021
Received by editor(s) in revised form:
May 17, 2021
Published electronically:
August 10, 2021
Additional Notes:
The author was partially supported by the Academy of Finland project 257482 and NSF DMS grant 1952510.
Article copyright:
© Copyright 2021
American Mathematical Society