Measurable Entire Functions II
Year: 2025
Type: article
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.
DOI: 10.1093/imrn/rnaf330
Authors: Adi Glücksam, Benjamin Weiss
Institution: Hebrew University of Jerusalem
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,534
OpenAlex ID: W4415776720