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

## TL;DR

This paper demonstrates the existence of rational maps with Sierpiński carpet Julia sets that have positive or zero area, and precisely controlled Hausdorff dimensions between 1 and 2.

## Contribution

It constructs examples of rational maps with Sierpiński carpet Julia sets exhibiting various area and dimension properties, including positive area, zero area, and exact Hausdorff dimension.

## Key findings

- Existence of rational maps with positive area Sierpiński carpet Julia sets.
- Construction of zero-area Sierpiński carpet Julia sets with Hausdorff dimension two.
- Existence of Sierpiński carpet Julia sets with Hausdorff dimension exactly s for any s in (1,2).

## 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 $s$.