Measurable Entire Functions II

Let ɛ denote the space of entire functions with the topology of uniform convergence on compact sets. The action of C by translations on ɛ is defined by T_zf(w) = f(w + z). Let U denote the set of entire functions whose orbit under T is dense. Birkhoff showed, in 1929, that 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 ɛ whose support is contained in 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.