Abstract
It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in \(\mathbb {S}^n\), for \(n\ge 2\). Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if E is a compact, uniformly perfect and uniformly disconnected set in \(\mathbb {S}^n\), then it is the Julia set of a hyperbolic UQR map \(f:\mathbb {S}^N \rightarrow \mathbb {S}^N\) where \(N=n\) if \(n=2\) and \(N=n+1\) otherwise.
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References
Ahlfors Lars, V.: Extension of quasiconformal mappings from two to three dimensions. Proc. Natl. Acad. Sci. USA 51, 768–771 (1964)
Antoine, L.: Sur l’homéomorphie de deux figures et de leurs voisinages, NUMDAM, [place of publication not identified] (1921)
Bergweiler, W.: Iteration of quasiregular mappings. Comput. Methods. Funct. Theorem. 10, 455–481 (2010)
Matias Carrasco Piaggio: Conformal gauge of compact metric spaces. Université de Provence - Aix-Marseille I, Theses (2011)
Daverman, R.J.: Decompositions of manifolds. AMS Chelsea Publishing, Providence, RI, Reprint of the 1986 Original (2007)
David, G., Semmes, S.: Fractured fractals and broken dreams. Oxford Lecture Series in Mathematics and its Applications, vol. 7, The Clarendon Press, Oxford University Press, New York, Self-similar geometry through metric and measure (1997)
Alastair, N.F.: Attractor sets and julia sets in low dimensions. Conf. Geom. Dyn. 23, 117–129 (2019)
Fletcher, A.N., Nicks, D.A.: Julia sets of uniformly quasiregular mappings are uniformly perfect. Math. Proc. Camb. Philos. Soc. 151(3), 541–550 (2011)
Fletcher, A.N., Stoertz, D.: Julia sets and genus \(g\) cantor sets, in preparation
Alastair, N.F., Jang-Mei, W.: Julia sets and wild Cantor sets. Geom. Ded. 174(1), 169–176 (2015)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001)
Hinkkanen, A., Gaven, J.M.: Local dynamics of uniformly quasiregular mappings. Math. Scand. 95, 80–100 (2004)
Haissinsky, P., Pilgrim, K.M.: Quasisymmetrically inequivalent hyperbolic Julia sets. Rev. Mat. Ibero. 28(4), 1025–1034 (2010)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5):713–747 (1981)
Iwaniec, T., Martin, G.: Quasiregular semigroups. Ann. Acad. Sci. Fenn. Math. 21(2), 241–254 (1996)
Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. The Clarendon Press, Oxford University Press, Oxford Mathematical Monographs (2001)
Luukkainen, J.: Assouad dimension : antifractal metrization, porous sets, and homogeneous measures. J. Kor. Math. Soc. 35(1), 23–76 (1998)
MacManus, P.: Catching sets with quasicircles. Rev. Mat. Iberoamericana 15(2), 267–277 (1999)
Gaven, J.M.: Branch sets of uniformly quasiregular maps. Conf. Geom. Dyn. 1, 24–27 (1997)
Martin, G.J.: Quasiregular mappings, curvature & dynamics. In: Proceedings of the International Congress of Mathematicians, Volume III, Hindustan Book Agency, New Delhi, pp. 1433–1449 (2010)
Mayer, V.: Uniformly quasiregular mappings of Lattès type. Conf. Geom. Dyn. 1, 104–111 (1997)
Mañé, R., Da Rocha, L.F.: Julia sets are uniformly perfect. Proc. Am. Math. Soc. 116(1), 251–257 (1992)
Gaven, J.M., Peltonen, K.: Stoïlow factorization for quasiregular mappings in all dimensions. Proc. Am. Math. Soc. 138(1), 147–151 (2010)
Munkres, J.: Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. Math. (2) 72, 521–554 (1960)
Rickman, S.: Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26. Springer, Berlin (1993)
Semmes, S.: On the nonexistence of bi-Lipschitz parameterizations and geometric problems about \(A_\infty \)-weights. Rev. Mat. Iberoamericana 12(2), 337–410 (1996)
Tukia, P., Väisälä, J.:Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 303–342 (1982)
Tyson, J.: Quasiconformality and quasisymmetry in metric measure spaces. Ann. Acad. Sci. Fenn. Math. 23(2), 525–548 (1998)
Väisälä, J.: Lectures on \(n\)-dimensional quasiconformal mappings. Lecture Notes in Mathematics, vol. 229. Springer, Berlin (1971)
Vellis, V.:Uniformization of cantor sets with bounded geometry. arXiv:1609.08763 (2016)
Yang, F.: Cantor Julia sets of Hausdorff dimension two, to appear in IMRN
Acknowledgements
We would like to thank Peter Haissinsky for helpful comments on the uniform disconnectedness of hyperbolic rational maps. We also thank the anonymous referee for their comments that improved the exposition of the paper.
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Alastair N. Fletcher was supported by a grant from the Simons Foundation (#352034, Alastair Fletcher). Vyron Vellis was partially supported by NSF DMS grant 1952510.
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Fletcher, A.N., Vellis, V. On uniformly disconnected Julia sets. Math. Z. 299, 853–866 (2021). https://doi.org/10.1007/s00209-021-02699-6
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DOI: https://doi.org/10.1007/s00209-021-02699-6