Measurable Entire Functions II

Abstract Let $\mathcal{E}$ denote the space of entire functions with the topology of uniform convergence on compact sets. The action of $\mathbb{C}$ by translations on $\mathcal{E}$ is defined by $T_{z}f(w) = f(w+z)$. Let $\mathcal{U}$ denote the set of entire functions whose orbit under $T$ is dense. Birkhoff showed, in 1929, that $\mathcal{U}$ is not empty. One of the problems in the collection by T.-C. Dinh and N. Sibony in 2020 asks whether there exists an invariant probability measure on $\mathcal{E}$ whose support is contained in $\mathcal U$. We will show how an old construction of the second author can be modified to provide a positive answer to their question. Furthermore, we modify the construction to produce a wealth of ergodic measures on the space of entire functions of several complex variables.