Hausdorff dimension of escaping sets of meromorphic functions II

TL;DR

This paper investigates the Hausdorff dimension of escaping sets of transcendental meromorphic functions, establishing that with up to four singular values, all dimensions in [0,2] are attainable, extending previous results for functions with fewer singular values.

Contribution

It proves that the number of singular values needed to realize any Hausdorff dimension in [0,2] for escaping sets is at most four, generalizing earlier findings.

Findings

For functions with two singular values, Hausdorff dimension is either 1/2 or 2.

Speiser functions can have escaping set dimensions covering the entire interval [0,2].

Up to four singular values are sufficient to attain every Hausdorff dimension in [0,2].

Abstract

A function which is transcendental and meromorphic in the plane has at least two singular values. On one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either $2$ or $1/2$. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in $[0,2]$ (cf. \cite{ac1}). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than $4$.