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Area and Hausdorff dimension of Sierpi\'{n}ski carpet Julia sets
- Publication Year :
- 2018
-
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 $s$.<br />Comment: 15 pages, 3 figures; some typos corrected
- Subjects :
- Mathematics - Dynamical Systems
Mathematics - Complex Variables
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1812.03016
- Document Type :
- Working Paper