Advertisement

Area and Hausdorff dimension of Sierpiński carpet Julia sets

  • Yuming Fu
  • Fei YangEmail author
Article
  • 70 Downloads

Abstract

We prove the existence of rational maps whose Julia sets are Sierpiński carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpiński carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number \(s\in (1,2)\), we prove the existence of Sierpiński carpet Julia sets having Hausdorff dimension exactly s.

Keywords

Sierpiński carpet Julia set Positive area Hausdorff dimension 

Mathematics Subject Classification

Primary 37F45 Secondary 37F10 37F25 

Notes

Acknowledgements

The authors are very grateful to Huojun Ruan for providing a method to construct the Sierpiński carpets (not Julia sets) with Hausdorff dimension one (Theorem D), to Xiaoguang Wang, Yongcheng Yin and Jinsong Zeng for offering a proof of Lemma 5.2. We would also like to thank Arnaud Chéritat, Kevin Pilgrim, Feliks Przytycki, Weiyuan Qiu, Yongcheng Yin and Anna Zdunik for helpful discussions and comments. We are also very grateful to the referee for his/her helpful suggestions which largely improve the readability of this paper. This work is supported by National Natural Science Foundation of China (grant No. 11671092) and the Fundamental Research Funds for the Central Universities (grant No. 0203-14380025).

References

  1. 1.
    Avila, A., Buff, X., Chéritat, A.: Siegel disks with smooth boundaries. Acta Math. 193(1), 1–30 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Avila, A., Lyubich, M.: Lebesgue measure of Feigenbaum Julia sets, arXiv:1504.20986 (2015)
  3. 3.
    Barański, K., Wardal, M.: On the Hausdorff dimension of the Sierpiński Julia sets. Discrete Cont. Dyn. Syst. 35(8), 3293–3313 (2015)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bonk, M.: Uniformization of Sierpiński carpets in the plane. Invent. Math. 186(3), 559–665 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bonk, M., Lyubich, M., Merenkov, S.: Quasisymmetries of Sierpiński carpet Julia sets. Adv. Math. 301, 383–422 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bonk, M., Merenkov, S.: Quasisymmetric rigidity of square Sierpiński carpets. Ann. Math. 177(2), 591–643 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Buff, X., Chéritat, A.: How regular can the boundary of a quadratic Siegel disk be? Proc. Am. Math. Soc. 135(4), 1073–1080 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Buff, X., Chéritat, A.: Quadratic Julia sets with positive area. Ann. Math. 176(2), 673–746 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cheraghi, D.: Topology of irrationally indifferent attractors. arXiv:1706.02678 (2017)
  10. 10.
    Cheraghi, D., DeZotti, A., Yang, F.: Dimension paradox of irrational indifferent attractors. Manuscript in preparation (2018)Google Scholar
  11. 11.
    Devaney, R.L., Fagella, N., Garijo, A., Jarque, X.: Sierpiński curve Julia sets for quadratic rational maps. Ann. Acad. Sci. Fenn. Math. 39(1), 3–22 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Devaney, R.L., Look, D.M., Uminsky, D.: The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math. J. 54(6), 1621–1634 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Devaney, R.L., Pilgrim, K.M.: Dynamic classification of escape time Sierpiński curve Julia sets. Fund. Math. 202(2), 181–198 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Devaney, R.L., Russell, E.D.: Connectivity of Julia sets forsingularly perturbed rational maps, In: Chaos, CNN, Memristors and Beyond. World Scientific, pp. 239–245 (2013)Google Scholar
  15. 15.
    Douady, A., Hubbard, J.H.: On the dynamics of polynomials-like mappings. Ann. Sci. École Norm. Sup. 18, 287–343 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gao, Y., Haïssinsky, P., Meyer, D., Zeng, J.: Invariant Jordan curves of Sierpiński carpet rational maps. Ergod. Theory Dyn. Sys 38(2), 583–600 (2018)zbMATHCrossRefGoogle Scholar
  17. 17.
    Gao, Y., Zeng, J., Zhao, S.: A characterization of Sierpiński carpet rational maps. Discrete Cont. Dyn. Syst. 37(9), 5049–5063 (2017)zbMATHCrossRefGoogle Scholar
  18. 18.
    Graczyk, J., Jones, P.: Dimension of the boundary of quasiconformal Siegel disks. Invent. Math. 148(3), 465–493 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Haïssinsky, P., Pilgrim, K.M.: Quasisymmetrically inequivalent hyperbolic Julia sets. Revista Math. Iberoamericana 28, 1025–1034 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Inou, H., Shishikura, M.: The renormalization for parabolic fixed points and their perturbation. Preprint (2008)Google Scholar
  21. 21.
    Look, D.M.: Sierpiński carpets as Julia sets for imaginary 3-circle inversions. J. Diff. Equ. Appl. 16(5–6), 705–713 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Lyubich, M.: On the Lebesgue measure of the Julia set of a quadratic polynomial. arXiv:9201285 (1991)
  23. 23.
    Lyubich, M., Minsky, Y.: Laminations in holomorphic dynamics. J. Diff. Geom. 47(1), 17–94 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Mañé, R.: Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100(4), 495–524 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mañé, R.: Erratum: “Hyperbolicity, sinks and measure in one-dimensional dynamics”. Commun. Math. Phys. 112(4), 721–724 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Merenkov, S.: Local rigidity for hyperbolic groups with Sierpiński carpet boundaries. Compos. Math. 150(11), 1928–1938 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    McMullen, C.T.: The Classification of Conformal Dynamical Systems, Current developments in mathematics, 1995 (Cambridge, MA), 323–360. Internat Press, Cambridge, MA (1994)Google Scholar
  28. 28.
    McMullen, C.T.: Complex Dynamics and Renormalization. Ann. of Math. Studies, Vol. 135. Princeton University Press, Princeton, NJ (1994)Google Scholar
  29. 29.
    McMullen, C.T.: Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Math. 180(2), 247–292 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Milnor, J.: Geometry and dynamics of quadratic rational maps, with an appendix by J. Milnor and L. Tan. Exper. Math. 2(1), 37–83 (1993)zbMATHCrossRefGoogle Scholar
  31. 31.
    Morosawa, S.: Julia sets of subhyperbolic rational functions. Complex Var. Theory Appl. 41(2), 151–162 (2000)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Pilgrim, K.M.: Cylinders for Iterated Rational Maps. Thesis, University of California, Berkeley (1994)Google Scholar
  33. 33.
    Przytycki, F.: On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers. Bull. Pol. Acad. Sci. Math. 54(1), 41–52 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Qiu, W., Wang, X., Yin, Y.: Dynamics of McMullen maps. Adv. Math. 229(4), 2525–2577 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Qiu, W., Yang, F.: Hausdorff dimension and quasi-symmetric uniformization of Cantor circle Julia sets. arXiv:1811.10042 (2018)
  36. 36.
    Qiu, W., Yang, F., Yin, Y.: Quasisymmetric geometry of the Julia sets of McMullen maps. Sci. China. Math. 61(12), 2283–2298 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Qiu, W., Yang, F., Zeng, J.: Quasisymetric geometry of Sierpiński carpet Julia sets. Fund. Math. 244(1), 73–107 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Ruelle, D.: Repellers for real analytic maps. Ergod. Thorey. Dyn. Syst. 2, 99–107 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Shishikura, M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. 147(2), 225–267 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Shishikura, M., Yang, F.: The high type quadratic Siegel disks are Jordan domains. arXiv:1608.04106v3 (2018)
  41. 41.
    Steinmetz, N.: On the dynamics of the McMullen family \(R(z)=z^m+\lambda /z^\ell \). Conform. Geom. Dyn. 10, 159–183 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Steinmetz, N.: Sierpiński and non-Sierpiński curve Julia sets in families of rational maps. J. Lond. Math. Soc. 78(2), 290–304 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Tan, L., Yin, Y.: Local connectivity of the Julia set for geometrically finite rational maps. Sci. China Ser. A 39, 39–47 (1996)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Urbański, M.: On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point. Stud. Math. 97(3), 167–188 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Urbański, M., Zdunik, A.: Hausdorff dimension of harmonic measure for self-conformal sets. Adv. Math. 171(1), 1–58 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Whyburn, G.T.: Analytic Topology, AMS Colloquium Publications, vol. 28. American Mathematical Society, New York (1942)Google Scholar
  47. 47.
    Whyburn, G.T.: Topological characterization of the Sierpiński curves. Fund. Math. 45, 320–324 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Xiao, Y., Qiu, W., Yin, Y.: On the dynamics of generalized McMullen maps. Ergod. Theory Dyn. Syst. 34(6), 2093–2112 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Yang, F.: A criterion to generate carpet Julia sets. Proc. Am. Math. Soc. 146(5), 2129–2141 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Yang, F., Yin, Y.: Non-renormalizable quadratic Julia sets with Hausdorff dimension two. Preprint (2018)Google Scholar
  51. 51.
    Zdunik, A.: Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math. 99(3), 627–649 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Zeng, J., Su, W.: Quasisymmetric rigidity of Sierpiński carpets \(F_{n, p}\). Ergod. Theory. Dyn. Syst. 35(5), 1658–1680 (2015)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

Personalised recommendations