Area and Hausdorff dimension of Sierpiński carpet Julia sets

  • Yuming Fu
  • Fei YangEmail author


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.


Sierpiński carpet Julia set Positive area Hausdorff dimension 

Mathematics Subject Classification

Primary 37F45 Secondary 37F10 37F25 



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).


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Authors and Affiliations

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

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