Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets

Abstract

We prove that for any infinite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasiconformal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the quasiconformal surgery to obtain the desired transcendental entire function. In addition, the transcendental entire function has order zero.