Skip to main content
SpringerLink
Log in

Quasisymmetric geometry of the Julia sets of McMullen maps

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We study the quasisymmetric geometry of the Julia sets of McMullen maps fλ(z) = zm + λ/zℓ, 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bonk M. Uniformization of Sierpiński carpets in the plane. Invent Math, 2011, 186: 559–665

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonk M, Lyubich M, Merenkov S. Quasisymmetries of Sierpiński carpet Julia sets. Adv Math, 2016, 301: 383–422

    Article  MathSciNet  MATH  Google Scholar 

  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, 1997

    MATH  Google Scholar 

  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, 37–50

    Book  MATH  Google Scholar 

  5. Devaney R L. Singular perturbations of complex polynomials. Bull Amer Math Soc (NS), 2013, 50: 391–429

    Article  MathSciNet  MATH  Google Scholar 

  6. Devaney R L, Look D. Buried Sierpiński curve Julia sets. Discrete Contin Dyn Syst, 2005, 13: 1035–1046

    Article  MathSciNet  MATH  Google Scholar 

  7. Devaney R L, Look D, Uminsky D. The escape trichotomy for singularly perturbed rational maps. Indiana Univ Math J, 2005, 54: 1621–1634

    Article  MathSciNet  MATH  Google Scholar 

  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, 239–245

    Book  Google Scholar 

  9. Douady A, Hubbard J H. On the dynamics of polynomial-like mappings. Ann Sci École Norm Sup (4), 1985, 18: 287–343

    Article  MathSciNet  MATH  Google Scholar 

  10. Haïssinsky P, Pilgrim K. Quasisymmetrically inequivalent hyperbolic Julia sets. Rev Mat Iberoam, 2012, 28: 1025–1034

    Article  MathSciNet  MATH  Google Scholar 

  11. Haïssinsky P, Pilgrim K. Examples of coarse expanding conformal maps. Discrete Contin Dyn Syst, 2012, 32: 2403–2416

    Article  MathSciNet  MATH  Google Scholar 

  12. Heinonen J. Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag, 2001

    Book  MATH  Google Scholar 

  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, 31–60

    Book  MATH  Google Scholar 

  14. McMullen C T. Complex Dynamics and Renormalization. Annals of Mathematics Studies, vol. 135. Princeton: Princeton University Press, 1994

    MATH  Google Scholar 

  15. Milnor J. Periodic orbits, external rays and the Mandelbrot set: An expository account. Astérisque, 2000, 261: 277–333

    MathSciNet  MATH  Google Scholar 

  16. Qiu W, Wang X, Yin Y. Dynamics of McMullen maps. Adv Math, 2012, 229: 2525–2577

    Article  MathSciNet  MATH  Google Scholar 

  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: 1937–1954

    Article  MathSciNet  MATH  Google Scholar 

  18. Qiu W, Yang F, Yin Y. Rational maps whose Julia sets are Cantor circles. Ergodic Theory Dynam Systems, 2015, 35: 499–529

    Article  MathSciNet  MATH  Google Scholar 

  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: 3375–3416

    Article  MathSciNet  MATH  Google Scholar 

  20. Qiu W, Yang F, Zeng J. Quasisymmetric geometry of the carpet Julia sets. Fund Math, 2018, in press

    Google Scholar 

  21. Steinmetz N. On the dynamics of the McMullen family R(z) = zm + λ/zℓ. Conform Geom Dyn, 2006, 10: 159–183

    Article  MathSciNet  MATH  Google Scholar 

  22. Whyburn G T. Topological characterization of the Sierpiński curve. Fund Math, 1958, 45: 320–324

    Article  MathSciNet  MATH  Google Scholar 

  23. Xie L. The dynamics of McMullen maps with real positive parameter (in Chinese). PhD Thesis. Shanghai: Fudan University, 2011

    Google Scholar 

  24. Yin Y. On the Julia set of semi-hyperbolic rational maps. Chinese J Contemp Math, 1999, 20: 469–476

    MathSciNet  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Weiyuan Qiu

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Fei Yang

  3. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Yongcheng Yin

Authors
  1. Weiyuan Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongcheng Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Yang.

Additional information

Dedicated to the Memory of Professor Lei Tan

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiu, W., Yang, F. & Yin, Y. Quasisymmetric geometry of the Julia sets of McMullen maps. Sci. China Math. 61, 2283–2298 (2018). https://doi.org/10.1007/s11425-016-9228-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-016-9228-x

Keywords

MSC(2010)

Search

Navigation