A generalized family of transcendental functions with one dimensional Julia sets

Abstract

A generalized family of non-polynomial entire functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to 1, there exist multiply connected wandering domains, and the dynamics can be completed described. For any s ∈ (0, +∞], there is a function taken from this family with the order of growth s. This is the first example, where the Julia set has packing dimension 1, and arbitrarily positive or even infinite order of growth. This gives an answer to an open problem in [6, Problem 4].