Measurable entire functions II

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 $T_zf(w) = f(w+z)$. Let $\mathcal{U}$ denote the set of entire functions whose orbit under $T$ is dense. Birkhoff showed, in [B], that $\mathcal{U}$ is not empty. One of the problems in the collection by T-C Dinh and N. Sibony [DS] 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.