A generalized family of transcendental functions with one dimensional Julia sets

TL;DR

This paper introduces a new family of transcendental entire functions with Julia sets of Hausdorff and packing dimension one, including functions with arbitrary positive or infinite order of growth, expanding understanding of complex dynamics.

Contribution

The paper constructs a generalized family of transcendental functions with specific fractal properties and arbitrary growth orders, including multiply connected wandering domains.

Findings

Julia sets have Hausdorff and packing dimension equal to one

Existence of functions with arbitrary positive or infinite order of growth

Presence of multiply connected wandering domains in the dynamics

Abstract

A generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains, the dynamics can be completed described, and for any $s\in(0,+\infty]$, there is a function taken from this family with the order of growth $s$. Baker proved that the Hausdorff dimension of the transcendental function is no less than one in 1975, the minimum value was obtained via an elegant construction by Bishop in 2018. The order of growth is zero in Bishop's construction, the family of functions here have arbitrarily positive or even infinite order of growth.