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

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 $\mathrm{HD}(\mathcal{I}(f))$ 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 $\mathrm{HD}(\mathcal{I}(f))$ 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 $\mathrm{HD}(\mathcal{I}(f))=0$.

27 pages

References in corpus (1)

• Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions