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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
Articles
  • 34 Downloads

Abstract

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.

Keywords

Julia sets Sierpiński carpet quasisymmetrically equivalent 

MSC(2010)

37F45 37F10 37F25 

Notes

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.

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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

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