On the Hausdorff dimension of the Julia set of a regularly growing entire function

TL;DR

This paper proves that for transcendental entire functions with sufficiently regular growth, both the Julia set and the escaping set have Hausdorff dimension 2, indicating their maximal fractal complexity.

Contribution

It establishes a new link between regular growth conditions of entire functions and the Hausdorff dimension of their Julia and escaping sets.

Findings

Julia set has Hausdorff dimension 2 under regular growth

Escaping set also has Hausdorff dimension 2 under the same conditions

Provides a criterion connecting growth regularity to fractal dimension

Abstract

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.