Measurable entire functions II
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916849149542400 |
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| author | Glücksam, Adi Weiss, Benjamin |
| author_facet | Glücksam, Adi Weiss, Benjamin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measurable entire functions II Glücksam, Adi Weiss, Benjamin Dynamical Systems Complex Variables 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. |
| title | Measurable entire functions II |
| topic | Dynamical Systems Complex Variables |
| url | https://arxiv.org/abs/2507.13182 |