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

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 $\ell \in \mathbb{N}$, $\ell \ge 2$. 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.