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arXiv · 2604.05169

Separating Orbits by Entire Functions

Billy Duckworth·Konstantin Slutsky
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Abstract

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 Gl\"ucksam 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 Forstneri\v{c}'s holomorphic approximation theorem with prescribed critical points.

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Billy Duckworth, Konstantin Slutsky. 2026-04-06. Separating Orbits by Entire Functions. https://arxiv.org/abs/2604.05169

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