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Area and Hausdorff dimension of Sierpiński carpet Julia sets

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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.

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Notes

  1. The definition of renormalization needs to exclude a special case, i.e. f itself is a polynomial and U is an open neighborhood of the filled Julia set of f.

  2. Lyubich proved that the Julia set of a quadratic polynomial has zero area if it has no irrational indifferent periodic points and is not infinitely renormalizable.

  3. Obviously, Cremer fixed points and Siegel disks can be defined similarly for rational maps.

  4. The infinitely renormalizable quadratic polynomials constructed by Buff and Chéritat are infinitely satellite renormalizable while Avila and Lyubich’s examples are infinitely primitive renormalizable.

  5. Buff and Chéritat’s construction of quadratic Julia sets with positive area requires that the rotation number is of high type. Moreover, the boundary of the Siegel disk of their construction does not contain the critical point. Actually, it is not known if a quadratic Julia set can have positive area and also contain the boundary of a Siegel disk passing through the critical point.

  6. For the non-renormalizable quadratic polynomials in Theorem 2.5, the post-critical sets may equal to the whole Julia sets. In order to guarantee that the post-critical set of \({\widetilde{f}}_3\) is disjoint with the \(\beta \)-fixed point, we consider the 2-renormalization \((f_3^{\circ 2},U_3,V_3)\) but not 1-renormalization.

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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).

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  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Yuming Fu & Fei Yang

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  1. Yuming Fu
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Fu, Y., Yang, F. Area and Hausdorff dimension of Sierpiński carpet Julia sets. Math. Z. 294, 1441–1456 (2020). https://doi.org/10.1007/s00209-019-02319-4

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