The exact value of Hausdorff dimension of escaping sets of class B meromorphic functions

TL;DR

This paper derives an explicit formula for the Hausdorff dimension of escaping sets of certain meromorphic functions in class B, linking it to a critical exponent and demonstrating cases with zero dimension.

Contribution

It provides a closed-form expression for the Hausdorff dimension of escaping sets for class B meromorphic functions, connecting it to a natural series' critical exponent.

Findings

Exact Hausdorff dimension formula for class B meromorphic functions.

Identification of the dimension with a critical exponent of a series.

Existence of functions with infinite order and zero escaping set dimension.

Abstract

We consider the subclass of class ${\mathcal B} $ consisting of meromorphic functions $f:{\mathbb C}\to\hat{\mathbb C}$ for which infinity is not an asymptotic value and whose all poles have orders uniformly bounded from above. This class was introduced in \cite{BwKo2012} and the Hausdorff dimension ${\rm HD}({{\mathcal I} (f)})$ of the set ${{\mathcal I} (f)}$ of all points escaping to infinity under forward iteration of $f$ was estimated therein. In this paper we provide a closed formula for the exact value of ${\rm HD}({{\mathcal I} (f)})$ identifying it with the critical exponent of the natural series introduced in \cite{BwKo2012}. This exponent is very easy to calculate for many concrete functions. In particular, we construct a function from this class which is of infinite order and for which ${\rm HD}({{\mathcal I} (f)})=0$.