Separating Orbits by Entire Functions

TL;DR

This paper proves the existence of a Borel entire function that uniquely encodes free probability measure-preserving actions of complex Euclidean spaces, extending previous measurable function results.

Contribution

It introduces a method to construct injective factor maps using holomorphic approximation and separating cross-sections for complex group actions.

Findings

Existence of injective factor maps via entire functions for complex actions

Extension of Gl"ucksam and Weiss's measurable entire functions to holomorphic functions

Application of Forstneric's approximation theorem with prescribed critical points

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.