Thermodynamic formalism and hyperbolic Baker domains: Real-analyticity of the Hausdorff dimension

TL;DR

This paper proves that the Hausdorff dimension of the radial Julia set varies real-analytically with parameter c for a family of entire maps, using thermodynamic formalism on an infinite cylinder.

Contribution

It establishes the real-analyticity of the Hausdorff dimension function for a specific family of entire maps via thermodynamic formalism.

Findings

Hausdorff dimension of radial Julia sets depends real-analytically on parameter c

Utilizes thermodynamic formalism on an infinite cylinder model

Provides new insights into the parameter dependence of Julia set dimensions

Abstract

We consider the family of entire maps given by $f_{\ell,c}(z)=\ell+c-(\ell-1)\log c-e^z$, where $c\in D(\ell,1)$ and $\ell\in\mathbb N$, $\ell\geq2$. By using the property of $f_{\ell,c}$ to be dynamically projected to an infinite cylinder $\mathbb C/2\pi I\mathbb Z$, where the thermodynamic formalism tools are well-defined, we prove as a main result on this work, the real-analyticity of the map $c\mapsto HD(\mathcal{J}_r(f_{\ell,c}))$, here $\mathcal{J}_r(f_{\ell,c})$ is the radial Julia set.