The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

TL;DR

This paper demonstrates that for any Hausdorff dimension between 0 and 2, there exists a meromorphic function with a Julia set of that dimension, specifically within the Speiser class with three singularities.

Contribution

It constructs meromorphic functions in the Speiser class with prescribed Julia set Hausdorff dimensions, expanding understanding of fractal geometry in complex dynamics.

Findings

Existence of meromorphic functions with Julia sets of any dimension in (0,2]

Construction within the Speiser class with three singularities

Julia set Hausdorff dimension can be precisely controlled

Abstract

We show that for each $d\in (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.