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Conformal Geometry and Dynamics

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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: 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