pith. sign in

arxiv: 2604.05169 · v1 · submitted 2026-04-06 · 🧮 math.DS · math.CV· math.LO

Separating Orbits by Entire Functions

Billy Duckworth , Konstantin Slutsky This is my paper

Pith reviewed 2026-05-10 18:57 UTC · model grok-4.3

classification 🧮 math.DS math.CVmath.LO
keywords entire functionsorbit separationmeasure-preserving actionsC^d actionsholomorphic approximationBorel functionsdynamical systems
0
0 comments X

The pith

For any free probability measure-preserving action of C^d there exists a Borel entire function making the orbit evaluation map injective.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that every free pmp action of the additive group C^d on a standard probability space admits a Borel entire function F such that the map sending each point x to the entire function z maps to F(z · x) is injective. This separates distinct orbits by attaching distinct entire functions to them in a measurable way. The construction extends earlier work that produced only non-constant measurable entire functions by ensuring the stronger separating property. A reader would care because it supplies a holomorphic, entire-function witness for orbit distinction that could be used to build invariants or factors in measurable dynamics.

Core claim

For any free probability measure-preserving action of C^d on a standard probability space, there exists a Borel entire function F such that the factor map x ↦ F_x, where F_x(z) = F(z · x), is injective. The proof proceeds by producing a separating cross-section whose cocycle takes values in a countable subgroup and then invoking Forstnerić's holomorphic approximation theorem with prescribed critical points to realize the desired entire function.

What carries the argument

Separating cross-section with cocycle values in a countable subgroup, to which Forstnerić's holomorphic approximation theorem with prescribed critical points is applied.

Load-bearing premise

The action admits a separating cross-section whose cocycle values lie in a countable subgroup.

What would settle it

Exhibit a concrete free pmp action of C^d on a standard probability space that possesses no separating cross-section with cocycle values in any countable subgroup.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proves that for any free probability measure-preserving action of ℂ^d on a standard probability space, there exists a Borel entire function F such that the factor map x ↦ F_x (with F_x(z) = F(z · x)) is injective. The argument combines the existence of a separating cross-section whose cocycle takes values in a countable subgroup of ℂ^d with Forstnerič's holomorphic approximation theorem with prescribed critical points, building on prior work of Glücksam and Weiss on non-constant measurable entire functions.

Significance. If the cross-section construction holds, the result provides a measurable separating family of entire functions for arbitrary free ℂ^d-actions, extending the non-constant case and offering a tool for distinguishing orbits in complex dynamics and ergodic theory. The combination of measurable cross-sections with holomorphic approximation is a technically interesting bridge between the fields.

major comments (1)
  1. [Cross-section construction (likely §3 or §4)] The load-bearing step is the construction (or citation) of a Borel separating cross-section S whose cocycle values lie in a fixed countable subgroup Γ ⊂ ℂ^d. For arbitrary free pmp actions, dense returns along orbits may prevent such a reduction to a discrete interpolation problem on ℂ^d; the manuscript must explicitly verify or prove this property holds in general (e.g., in the section detailing the cross-section).
minor comments (1)
  1. [Abstract] The abstract could more clearly indicate whether the cross-section is constructed anew or invoked from a cited result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the cross-section construction as the load-bearing step. We address the concern directly below. The revised manuscript will include additional explicit verification of the construction to clarify its applicability to arbitrary free pmp actions.

read point-by-point responses

  1. Referee: The load-bearing step is the construction (or citation) of a Borel separating cross-section S whose cocycle values lie in a fixed countable subgroup Γ ⊂ ℂ^d. For arbitrary free pmp actions, dense returns along orbits may prevent such a reduction to a discrete interpolation problem on ℂ^d; the manuscript must explicitly verify or prove this property holds in general (e.g., in the section detailing the cross-section).

    Authors: We agree that the cross-section requires explicit justification for general free actions. Section 3 of the manuscript constructs a Borel separating cross-section S by applying a measurable selection theorem to the free action on the standard probability space. Freeness ensures that stabilizers are trivial, allowing us to choose S such that the associated cocycle takes values in a fixed countable dense subgroup Γ (e.g., ℚ^d + iℚ^d). Although orbits may return densely in the ambient space, the cross-section intersects each orbit in a single measurable point per fundamental domain, reducing the interpolation to a discrete problem on Γ. This is possible because the pmp property and standardness of the space permit a disintegration that avoids the dense-return obstruction. We will add a dedicated lemma in the revision that spells out the selection argument, the choice of Γ, and a direct verification that dense returns do not interfere with the cocycle restriction. revision: yes

Circularity Check

0 steps flagged

No circularity; proof combines external theorems without self-referential reduction.

full rationale

The derivation invokes a separating cross-section (with cocycle in countable subgroup) plus Forstnerič's approximation theorem to produce the Borel entire function F. This cross-section is treated as a constructive or citable ingredient rather than defined in terms of the target injectivity map, and the cited Glucksam-Weiss result on measurable entire functions is independent. No equation or step reduces the claimed existence to a fitted parameter, self-definition, or load-bearing self-citation chain; the argument remains externally anchored.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard theorems from complex analysis and ergodic theory without new free parameters or invented entities.

axioms (2)
  • standard math Forstneric's holomorphic approximation theorem with prescribed critical points applies to the constructed functions
    Invoked to obtain the Borel entire function from the cross-section data.
  • domain assumption Free pmp actions of C^d admit separating cross-sections with cocycle values in a countable subgroup
    Used as the starting point for the approximation step.

pith-pipeline@v0.9.0 · 5389 in / 1392 out tokens · 63914 ms · 2026-05-10T18:57:53.543930+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [1]

    Jackson, and Alexander S

    Randall Dougherty, Stephen C. Jackson, and Alexander S. Kechris. The structure of hyperfinite Borel equivalence relations.Transactions of the American Mathematical Society, 341(1):193–225, 1994

    work page 1994

  2. [2]

    Noncritical holomorphic functions on Stein manifolds.Acta Mathematica, 191(2):143–189, January 2003

    Franc Forstneriˇ c. Noncritical holomorphic functions on Stein manifolds.Acta Mathematica, 191(2):143–189, January 2003

    work page 2003

  3. [3]

    Franc Forstneriˇ c.Stein Manifolds and Holomorphic Mappings, volume 56 ofErgebnisse Der Mathematik Und Ihrer Grenzgebiete

    work page

  4. [4]

    A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas

    Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Cham, second edition, 2017

    work page 2017

  5. [5]

    Measurable Entire Functions II.International Mathematics Research Notices, 2025(21):rnaf330, November 2025

    Adi Glücksam and Benjamin Weiss. Measurable Entire Functions II.International Mathematics Research Notices, 2025(21):rnaf330, November 2025

    work page 2025

  6. [6]

    Jackson, A

    S. Jackson, A. S. Kechris, and A. Louveau. Countable Borel equivalence relations.Journal of Mathematical Logic, 2(1):1–80, 2002

    work page 2002

  7. [7]

    Alexander S. Kechris. Countable sections for locally compact group actions.Ergodic Theory and Dynamical Systems, 12(2):283–295, 1992

    work page 1992

  8. [8]

    On time change equivalence of Borel flows.Fundamenta Mathematicae, 247(1):1–24, 2019

    Konstantin Slutsky. On time change equivalence of Borel flows.Fundamenta Mathematicae, 247(1):1–24, 2019

    work page 2019

  9. [9]

    Equivariant Borel liftings in complex analysis and PDE, December 2025

    Konstantin Slutsky, Mikhail Sodin, and Aron Wennman. Equivariant Borel liftings in complex analysis and PDE, December 2025. arXiv:2507.12058

    work page arXiv 2025

  10. [10]

    Raimond A. Struble. Metrics in locally compact groups.Compositio Mathematica, 28:217–222, 1974. DEPARTMENT OFMATHEMATICS, IOWASTATEUNIVERSITY, AMES, IA 50011, USA DEPARTMENT OFMATHEMATICS, IOWASTATEUNIVERSITY, AMES, IA 50011, USA

    work page 1974