Separating Orbits by Entire Functions

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.