Transcendental Julia Sets of Minimal Hausdorff Dimension

TL;DR

This paper constructs transcendental entire functions with Julia sets of minimal Hausdorff dimension 1, featuring infinitely connected Fatou components and singleton complementary components, advancing understanding of complex dynamics.

Contribution

It introduces a quasiconformal-surgery method to produce transcendental functions with Julia sets of Hausdorff dimension 1 and specific Fatou component properties.

Findings

Existence of transcendental entire functions with Julia sets of Hausdorff dimension 1

All Fatou components have infinite inner connectivity

Presence of singleton complementary components of Fatou components

Abstract

We show the existence of transcendental entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of $f$ has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of $f$, answering a question of Rippon and Stallard (arXiv:1703.11001). Our proof relies on a quasiconformal-surgery approach developed in arXiv:2101.04219.