A criterion to generate carpet Julia sets
Fei Yang

TL;DR
This paper introduces a new criterion for generating carpet Julia sets in rational maps with specific attracting or parabolic basins, expanding understanding of fractal structures in complex dynamics.
Contribution
The paper presents a novel criterion for the existence of carpet Julia sets in certain rational maps, applicable to well-known families like McMullen maps.
Findings
Criterion guarantees carpet Julia sets in specific rational maps.
Application to McMullen maps and Morosawa-Pilgrim family.
Examples of Julia sets that are Sierpiński carpets.
Abstract
It was known that the Sierpi\'{n}ski carpets can appear as the Julia sets in the families of some rational maps. In this article we present a criterion that guarantees the existence of the carpet Julia sets in some rational maps having exactly one fixed (super-) attracting or parabolic basin. We show that this criterion can be applied to some well known rational maps, such as McMullen maps and Morosawa-Pilgrim family. Moreover, we give also some special examples whose Julia sets are Sierpi\'{n}ski carpets.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Mathematics and Applications · Advanced Differential Equations and Dynamical Systems
