Deformations on Symbolic Cantor Sets and Ultrametric Spaces

  • Qingshan Zhou
  • Xining Li
  • Yaxiang LiEmail author


By introducing new deformations on symbolic Cantor sets and ultrametric spaces, we prove that doubling ultrametric spaces admit bilipschitz embedding into Cantor sets. If in addition the spaces are uniformly perfect, we show that they are quasisymmetrically equivalent to Cantor sets. Moreover, we provide a new proof for a recent work of Heer regarding quasimöbius uniformization of Cantor set.


Symbolic Cantor set Ultrametric space Bilipschitz map Quasisymmetric map Quasimöbius map 

Mathematics Subject Classification

Primary: 30C65 30F45 Secondary: 30C20 



The authors are indebted to the referee for the valuable suggestions.


  1. 1.
    Bonk, M., Foertsch, T.: Asymptotic upper curvature bounds in coarse geometry. Math. Z. 253, 753–785 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bonk, M., Kleiner, B.: Rigidity for quasimöbius group actions. J. Differ. Geom. 61, 81–106 (2002)CrossRefGoogle Scholar
  3. 3.
    Brodskiy, N., Dydak, J., Higes, J., Mitra, A.: Dimension zero at all scales. Topol. Appl. 154(14), 2729–2740 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Buckley, S.M., Herron, D., Xie, X.: Metric space inversions, quasihyperbolic distance, and uniform spaces. Indiana Univ. Math. J. 57, 837–890 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    David, G., Semmes, S.: Fractured Fractals and Broken Dreams: Self-Similar Geometry Through Metric and Measure. Oxford Lecture Series in Mathematics and Its Applifications, vol. 7. Clarendon Press, Oxford (1997)zbMATHGoogle Scholar
  6. 6.
    Freeman, D., Donne, E. L.: Toward a quasi-Möbius characterization of invertible homogeneous metric spaces. arXiv:1812.03313 [math.MG] (2018)
  7. 7.
    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. Cambridge University Press, Cambridge (2015)CrossRefGoogle Scholar
  8. 8.
    Heer, L.: Some invariant properties of Quasimöbius maps. Anal. Geom. Metr. Spaces 5, 69–77 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ibragimov, Z.: Möbius maps between ultrametric spaces are local similarities. Ann. Acad. Sci. Fenn. Math. 37, 309–317 (2012) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Koblitz, N.: p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, 2nd edn. Springer, New York (1984)CrossRefGoogle Scholar
  11. 11.
    Lapidus, M., van Frankenhuijsen, M.: Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. Springer, New York (2006)CrossRefGoogle Scholar
  12. 12.
    Lapidus, M., Lu, H.: Nonarchimedean Cantor set and string. J. Fixed Point Theory Appl. 3, 181–190 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Martínez-Pérez, A.: Zig–zag chains and metric equivalences between ultrametric spaces. Topol. Appl. 158, 1595–1606 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ryoo, C.S., Kim, T.: An analogue of the zeta function and its applications. Appl. Math. Lett. 19, 1068–1072 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Schikhof, W.H.: Ultrametric Calculus: An Introduction to p-Adic Analysis. Cambridge University Press, Cambridge (1984)zbMATHGoogle Scholar
  16. 16.
    Väisälä, J.: Quasimöbius maps. J. Anal. Math. 44, 218–234 (1984/1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wang, Y., Yang, J.: The pointwise convergence of \(p\)-adic Möbius maps. Sci. China Math. 57, 1–8 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang, X., Zhou, Q.: Quasimöbius maps, weakly quasimöbius maps and uniform perfectness in quasi-metric spaces. Ann. Acad. Sci. Fenn. Math. 42(1), 257–284 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhou, Q., Li, Y., Li, X.: Sphericalization and flattening with their applications in quasimetric measure spaces. arXiv:1911.01760 [math.CV] (2019)

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.School of Mathematics and Big DataFoshan UniversityFoshanPeople’s Republic of China
  2. 2.Department of MathematicsSun Yat-sen UniversityGuangzhouPeople’s Republic of China
  3. 3.School of Mathematics (Zhuhai)Sun Yat-sen UniversityZhuhaiPeople’s Republic of China
  4. 4.Department of MathematicsHunan First Normal UniversityChangshaPeople’s Republic of China

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