A criterion to generate carpet Julia sets

Abstract

It has been known that the Sierpiński 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 Sierpiński carpets.