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

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