Rational maps whose Julia sets are generalized Sierpi\'{n}ski gaskets
Dynamical Systems
2025-09-16 v2
Abstract
It has been shown that the Sierpi\'nski gasket-like sets can appear as the Julia sets of some geometrically finite rational maps. In this paper we prove that such type of Julia sets can also appear in the rational maps containing Siegel disks, Cremer points or which are infinitely renormalizable. Based on this, we prove the existence of gasket Julia sets with positive area. Moreover, we present a criterion which guarantees the existence of gasket Julia sets in some rational maps having exactly one fixed attracting or parabolic basin.
Keywords
Cite
@article{arxiv.2503.18264,
title = {Rational maps whose Julia sets are generalized Sierpi\'{n}ski gaskets},
author = {Xiaole He and Yingqing Xiao and Fei Yang},
journal= {arXiv preprint arXiv:2503.18264},
year = {2025}
}
Comments
18 pages, 4 figures; to appear in Indiana Univ. Math. J