A criterion to generate carpet Julia sets

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.