Speiser class Julia sets with dimension near one

TL;DR

This paper constructs entire functions with three singular values whose Julia sets have Hausdorff dimension arbitrarily close to one, filling a gap in the understanding of the possible dimensions of such sets.

Contribution

It provides the first known examples of entire functions with finite singular set and Julia set dimension less than 2, specifically approaching dimension one.

Findings

Constructed entire functions with Julia set dimension near one

Demonstrated existence of functions with finite singular set and low-dimensional Julia sets

Extended understanding of the possible Hausdorff dimensions of Julia sets

Abstract

For any $ \delta >0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=\delta$. Stallard proved that the dimension must be strictly larger than 1 whenever $f$ has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.