Separating Orbits by Entire Functions
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910107306033152 |
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| author | Duckworth, Billy Slutsky, Konstantin |
| author_facet | Duckworth, Billy Slutsky, Konstantin |
| contents | We show that for any free probability measure-preserving action of $\mathbb{C}^{d}$ on a standard probability space, there exists a Borel entire function $F$ such that the factor map $x \mapsto F_{x}$, where $F_{x}(z) = F(z \cdot x)$, is injective. This work builds on a result of Glücksam and Weiss, who constructed non-constant measurable entire functions for such actions. The proof combines a separating cross-section whose cocycle values lie in a countable subgroup with Forstnerič's holomorphic approximation theorem with prescribed critical points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05169 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Separating Orbits by Entire Functions Duckworth, Billy Slutsky, Konstantin Dynamical Systems Complex Variables Logic We show that for any free probability measure-preserving action of $\mathbb{C}^{d}$ on a standard probability space, there exists a Borel entire function $F$ such that the factor map $x \mapsto F_{x}$, where $F_{x}(z) = F(z \cdot x)$, is injective. This work builds on a result of Glücksam and Weiss, who constructed non-constant measurable entire functions for such actions. The proof combines a separating cross-section whose cocycle values lie in a countable subgroup with Forstnerič's holomorphic approximation theorem with prescribed critical points. |
| title | Separating Orbits by Entire Functions |
| topic | Dynamical Systems Complex Variables Logic |
| url | https://arxiv.org/abs/2604.05169 |