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

TL;DR

This paper proves sharp bounds for the Hausdorff dimension of escaping sets of meromorphic functions in the Speiser class, extending previous results and constructing functions with prescribed dimensions.

Contribution

It extends sharp Hausdorff dimension bounds from the Eremenko-Lyubich class to the Speiser class and constructs functions with specific Julia and escaping set dimensions.

Findings

Bounds are sharp in the Speiser class.

Constructed functions with preassigned Julia and escaping set dimensions.

Extended the applicability of dimension bounds to a broader class.

Abstract

Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko-Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show that these bounds are also sharp in the Speiser class. We apply this method also to construct meromorphic functions in the Speiser class with preassigned dimensions of the Julia set and the escaping set.