Science China Mathematics

, Volume 61, Issue 12, pp 2283–2298 | Cite as

Quasisymmetric geometry of the Julia sets of McMullen maps

  • Weiyuan Qiu
  • Fei YangEmail author
  • Yongcheng Yin


We study the quasisymmetric geometry of the Julia sets of McMullen maps fλ(z) = zm + λ/z, where λ ∈ ℂ {0} and ℓ and m are positive integers satisfying 1/ℓ+1/m < 1. If the free critical points of fλ are escaped to the infinity, we prove that the Julia set Jλ of fλ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet (which is also standard in some sense). If the free critical points are not escaped, we give a suffcient condition on λ such that Jλ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.


Julia sets Sierpiński carpet quasisymmetrically equivalent 


37F45 37F10 37F25 



This work was supported by National Natural Science Foundation of China (Grant Nos. 11671091, 11731003, 11771387 and 11671092). The authors thank the referees for their careful reading and helpful comments.


  1. 1.
    Bonk M. Uniformization of Sierpiński carpets in the plane. Invent Math, 2011, 186: 559–665MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bonk M, Lyubich M, Merenkov S. Quasisymmetries of Sierpiński carpet Julia sets. Adv Math, 2016, 301: 383–422MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    David G, Semmes S. Fractured Fractals and Broken Dreams. Oxford Lecture Series in Mathematics and Its Applications, vol. 7. New York: Oxford University Press, 1997zbMATHGoogle Scholar
  4. 4.
    Devaney R L. Baby Mandelbrot sets adorned with halos in families of rational maps. In: Complex Dynamics. Contemporary Mathematics, vol. 396. Providence: Amer Math Soc, 2006, 37–50CrossRefzbMATHGoogle Scholar
  5. 5.
    Devaney R L. Singular perturbations of complex polynomials. Bull Amer Math Soc (NS), 2013, 50: 391–429MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Devaney R L, Look D. Buried Sierpiński curve Julia sets. Discrete Contin Dyn Syst, 2005, 13: 1035–1046MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Devaney R L, Look D, Uminsky D. The escape trichotomy for singularly perturbed rational maps. Indiana Univ Math J, 2005, 54: 1621–1634MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Devaney R L, Russell E D. Connectivity of Julia sets for singularly perturbed rational maps. In: Chaos, CNN, Memristors and Beyond. Singapore: World Scientific, 2013, 239–245CrossRefGoogle Scholar
  9. 9.
    Douady A, Hubbard J H. On the dynamics of polynomial-like mappings. Ann Sci École Norm Sup (4), 1985, 18: 287–343MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Haïssinsky P, Pilgrim K. Quasisymmetrically inequivalent hyperbolic Julia sets. Rev Mat Iberoam, 2012, 28: 1025–1034MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Haïssinsky P, Pilgrim K. Examples of coarse expanding conformal maps. Discrete Contin Dyn Syst, 2012, 32: 2403–2416MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Heinonen J. Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag, 2001CrossRefzbMATHGoogle Scholar
  13. 13.
    McMullen C T. Automorphisms of rational maps. In: Holomorphic Functions and Moduli I. Mathematical Sciences Research Institute Publications, vol. 10. New York: Springer, 1988, 31–60CrossRefzbMATHGoogle Scholar
  14. 14.
    McMullen C T. Complex Dynamics and Renormalization. Annals of Mathematics Studies, vol. 135. Princeton: Princeton University Press, 1994zbMATHGoogle Scholar
  15. 15.
    Milnor J. Periodic orbits, external rays and the Mandelbrot set: An expository account. Astérisque, 2000, 261: 277–333MathSciNetzbMATHGoogle Scholar
  16. 16.
    Qiu W, Wang X, Yin Y. Dynamics of McMullen maps. Adv Math, 2012, 229: 2525–2577MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Qiu W, Xie L, Yin Y. Fatou components and Julia sets of singularly perturbed rational maps with positive parameter. Acta Math Sin (Engl Ser), 2012, 28: 1937–1954MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Qiu W, Yang F, Yin Y. Rational maps whose Julia sets are Cantor circles. Ergodic Theory Dynam Systems, 2015, 35: 499–529MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Qiu W, Yang F, Yin Y. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete Contin Dyn Syst, 2016, 36: 3375–3416MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Qiu W, Yang F, Zeng J. Quasisymmetric geometry of the carpet Julia sets. Fund Math, 2018, in pressGoogle Scholar
  21. 21.
    Steinmetz N. On the dynamics of the McMullen family R(z) = zm + λ/zℓ. Conform Geom Dyn, 2006, 10: 159–183MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Whyburn G T. Topological characterization of the Sierpiński curve. Fund Math, 1958, 45: 320–324MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xie L. The dynamics of McMullen maps with real positive parameter (in Chinese). PhD Thesis. Shanghai: Fudan University, 2011Google Scholar
  24. 24.
    Yin Y. On the Julia set of semi-hyperbolic rational maps. Chinese J Contemp Math, 1999, 20: 469–476MathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Department of MathematicsNanjing UniversityNanjingChina
  3. 3.School of Mathematical SciencesZhejiang UniversityHangzhouChina

Personalised recommendations