Abstract

In this work, we provide an adapted version of the thermodynamic formalism for a family of one-parameter transcendental entire maps: $\flc: \mathbb C\to \mathbb C$, defined by $f_{\ell, c}(z)= c-(\ell-1)\log c+ \ell z- e^z,$ where $\ell \in \mathbb N$, with $\ell \geq 2 $ and $c$~belongs to the disk $ D(\ell, 1)$ in the complex plane. The model map for this family, is the function~$f_{2,2}(z) = 2 -\log 2 + 2z -e^z $, considered by Bergweiler's work, as an example of entire map with a Baker domain lying at a positive Euclidean distance from the post-singular set. The maps $\flc$ exhibit Baker domains of hyperbolic type and possess rich dynamics, including an attracting domain and two families of wandering domains. This study utilizes techniques and methods from the thermodynamic formalism of the one-parameter Fatou family of transcendental entire maps previously studied by Kotus and Urba\'nski, and we prove the existence and uniqueness of conformal measures, and that the Hausdorff dimension $\HD(J_r(f))$ is the unique zero of the pressure function $t\to P(t)$, for $t>1,$ where $J_r(f)$ is the radial Julia set.