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Quasisymmetric geometry of the Julia sets of McMullen maps

Authors
  • Qiu, Weiyuan1
  • Yang, Fei2
  • Yin, Yongcheng3
  • 1 Fudan University, School of Mathematical Sciences, Shanghai, 200433, China , Shanghai (China)
  • 2 Nanjing University, Department of Mathematics, Nanjing, 210093, China , Nanjing (China)
  • 3 Zhejiang University, School of Mathematical Sciences, Hangzhou, 310027, China , Hangzhou (China)
Type
Published Article
Journal
Science China Mathematics
Publisher
Science China Press
Publication Date
Nov 19, 2018
Volume
61
Issue
12
Pages
2283–2298
Identifiers
DOI: 10.1007/s11425-016-9228-x
Source
Springer Nature
Keywords
License
Yellow

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.

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