Measure theoretic entropy of entire maps

The entropy of rational maps has been described in classical works of Misiurewicz and Przytycki, Gromov, Lyubich, Freire, Lopes and Mane, and Bedford, Lyubich and Smillie. The topological entropy of a rational function of degree d equals log(d), and the same statement holds for Hénon maps in two complex variables. Moreover, for maps in each of these classes there exists a unique measure of maximal entropy. In an ongoing project with Leandro Arosio, Anna Miriam Benini and John Erik Fornaess, we consider the dynamics of so-called transcendental Hénon maps, combining ideas from transcendental dynamics in one variable and polynomial dynamics in two variables. It was suggested by Nessim Sibony and Romain Dujardin that the entropy of transcendental Hénon maps should always be infinite. Indeed, following the ideas of Marcus Wendt for entire maps in one variable, we previously showed that the topological entropy is always infinite.