Methods in thermodynamic formalism for the Bergweiler family of transcendental entire maps
Abstract
We study the family of transcendental entire functions $ f_{\ell , c}: \mathbb C\to \mathbb C $ defined by $$ f_{\ell , c}(z) = c - (\ell - 1)\log c + \ell z - e^z, \quad \ell \geq 2, \quad |c-\ell | \lt 1, $$ which exhibits a rich dynamical behavior including attracting domains, wandering domains, and Baker domains of hyperbolic type that are positively separated from the post-singular set.
We show that the core techniques of thermodynamic formalism, such as the construction of conformal measures, the definition of pressure, and Bowen’s formula, persist in this more intricate setting. In particular, we establish the existence and uniqueness of conformal measures for the associated map on the infinite cylinder. We also verify that the Hausdorff dimension of the radial Julia set is the unique zero of the pressure function. This case illustrates how thermodynamic methods remain robust even in the presence of multiple Fatou components and a more complex post-singular geometry.
