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

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 $\mathbb {R}^{n+1}$.

References

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