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arXiv · 1812.03016

Area and Hausdorff dimension of Sierpi\'{n}ski carpet Julia sets

Yuming Fu·Fei Yang
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Abstract

We prove the existence of rational maps whose Julia sets are Sierpi\'{n}ski 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 Sierpi\'{n}ski 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 Sierpi\'{n}ski carpet Julia sets having Hausdorff dimension exactly $s$.

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Yuming Fu, Fei Yang. 2018-12-07. Area and Hausdorff dimension of Sierpi\'{n}ski carpet Julia sets. https://arxiv.org/abs/1812.03016

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