On the existence of a conformal and an absolutely continuous invariant measure for transcendental entire maps

TL;DR

This paper studies a class of hyperbolic transcendental entire maps, demonstrating the existence of conformal and invariant measures using thermodynamic formalism, and explores their symbolic dynamics and measure properties.

Contribution

It extends thermodynamic formalism to certain transcendental maps and characterizes their invariant measures via symbolic dynamics with countable alphabets.

Findings

Existence of conformal and invariant probability measures for the class of maps.

Dynamics conjugated to shift map over a countable alphabet.

Metric incompatibility between Euclidean and symbolic space metrics.

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 techniques 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.