Sierpiński carpet Julia sets of transcendental entire functions

Abstract

We prove that Sierpiński carpet parameters are dense in the bifurcation loci of some natural families of finite type transcendental entire functions having exactly one active critical orbit. Moreover, some specific transcendental entire functions with Sierpiński carpet Julia sets are constructed such that they contain either an attracting basin, a parabolic basin, a Siegel disk or wandering domains. The proofs are mainly based on applying the polynomial-like renormalization and some results on the local connectivity of transcendental Julia sets obtained before.

arXiv comment20 pages, 4 figures