Area and Hausdorff dimension of Sierpiński carpet Julia sets

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 $s$.

15 pages, 3 figures; some typos corrected