Measurable entire functions II

TL;DR

This paper demonstrates the existence of invariant probability measures supported on dense orbits of entire functions, extending to several complex variables, using a modified classical construction.

Contribution

It provides a positive answer to the existence of invariant measures supported on dense orbits of entire functions and extends the construction to several complex variables.

Findings

Existence of invariant probability measures supported on dense orbits.

Construction of ergodic measures on entire functions of several variables.

Extension of classical methods to new settings.

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 $T_zf(w) = f(w+z)$. 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.