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
Full-text PDF Free Access
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 $\mathbb {R}^{n+1}$.
- Matthew Badger and Vyron Vellis, Geometry of measures in real dimensions via Hölder parameterizations, J. Geom. Anal. 29 (2019), no. 2, 1153–1192. MR 3935254, DOI 10.1007/s12220-018-0034-2
- Mario Bonk, Quasiconformal geometry of fractals, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1349–1373. MR 2275649
- Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127–183. MR 1930885, DOI 10.1007/s00222-002-0233-z
- Mario Bonk and Daniel Meyer, Quasiconformal and geodesic trees, Fund. Math. 250 (2020), no. 3, 253–299. MR 4107537, DOI 10.4064/fm749-7-2019
- Robert J. Daverman, Decompositions of manifolds, AMS Chelsea Publishing, Providence, RI, 2007. Reprint of the 1986 original. MR 2341468, DOI 10.1090/chel/362
- Guy David and Stephen Semmes, Fractured fractals and broken dreams, Oxford Lecture Series in Mathematics and its Applications, vol. 7, The Clarendon Press, Oxford University Press, New York, 1997. Self-similar geometry through metric and measure. MR 1616732
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- Alastair N. Fletcher and Vyron Vellis, Decomposing multitwists, arXiv:2106.00054, 2021.
- Alastair N. Fletcher and Vyron Vellis, On uniformly disconnected Julia sets, Math. Z., posted online March 3, 2021. DOI 10.1007/s00209-021-02699-6, to appear in print.
- Alastair Fletcher and Jang-Mei Wu, Julia sets and wild Cantor sets, Geom. Dedicata 174 (2015), 169–176. MR 3303046, DOI 10.1007/s10711-014-0010-3
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- Tadeusz Iwaniec and Gaven Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 241–254. MR 1404085
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- John M. Mackay and Jeremy T. Tyson, Conformal dimension, University Lecture Series, vol. 54, American Mathematical Society, Providence, RI, 2010. Theory and application. MR 2662522, DOI 10.1090/ulect/054
- Paul MacManus, Catching sets with quasicircles, Rev. Mat. Iberoamericana 15 (1999), no. 2, 267–277. MR 1715408, DOI 10.4171/RMI/256
- Pekka Pankka and Jang-Mei Wu, Deformation and quasiregular extension of cubical Alexander maps, preprint, arXiv:1904.09095, 2019.
- Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel) 32 (1979), no. 2, 192–199. MR 534933, DOI 10.1007/BF01238490
- Dennis Sullivan, Hyperbolic geometry and homeomorphisms, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 543–555. MR 537749
- Pekka Tukia, The planar Schönflies theorem for Lipschitz maps, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 49–72. MR 595177, DOI 10.5186/aasfm.1980.0529
- P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 97–114. MR 595180, DOI 10.5186/aasfm.1980.0531
- P. Tukia and J. Väisälä, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 303–342 (1982). MR 658932, DOI 10.5186/aasfm.1981.0626
- Jeremy Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 2, 525–548. MR 1642158
- Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, vol. 229 Springer-Verlag, Berlin, 1971.
- Vyron Sarantis Vellis, Quasisymmetric spheres constructed over quasidisks, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR 3322047
Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2020): 30C65, 30L05
Retrieve articles in all journals with MSC (2020): 30C65, 30L05
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
