Abstract

We identify a class of hyperbolic transcendental entire maps and we prove that some of its elements generate a class of potentials for which exhibit a conformal and invariant probability Gibbs measure. The methods and tech- niques from the thermodynamic formalism can be extended to this class of potentials. To complement this study we highlight that the dynamics of such a map on some subset of the Julia set is conjugated to the shift map over a code space with countable alphabet and the euclidean metric on the complex plane induces a metric on the symbolic space which is not compatible with the shift standard metric. From this fact, we provide a general description of the thermodynamic formalism from symbolic dynamic outlook, by studying the shift map acting on a non-compact and invariant subset of the full shift space with a countably infinite alphabet and a class of weakly H ̈older continuous potentials, to prove the existence of a conformal and absolutely continuous invariant probability measure.