arxiv: 2605.02459 · v1 · submitted 2026-05-04 · 🧮 math.DS · math.AG· math.CV

Recognition: 3 theorem links

· Lean Theorem

Random dynamics of plane polynomial automorphisms

Arnaud Nerri\`ere

Pith reviewed 2026-05-08 18:19 UTC · model grok-4.3

classification 🧮 math.DS math.AGmath.CV

keywords random dynamicspolynomial automorphismsGreen functionsstationary measuresloxodromic elementsnon-elementary groupsaffine planestiffness

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The pith

For finitely supported measures on plane polynomial automorphisms whose generated group is non-elementary and consists only of loxodromic elements, dynamical Green functions exist for random products.

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

The paper shows that when a finitely supported probability measure on the automorphism group of the complex affine plane generates a non-elementary group made entirely of loxodromic elements, dynamical Green functions can be attached to the associated random products. These functions imply that every stationary measure for the measure is compactly supported. The same setup also lets Roda's theorem apply, yielding stiffness of the action in the non-dissipative case.

Core claim

If the group generated by the support of μ is non-elementary and contains only loxodromic elements, we show the existence of dynamical Green functions associated to random products. We derive consequences for μ-stationary measures: they are compactly supported, and we can apply Roda's theorem to show stiffness when the action is non-dissipative.

What carries the argument

Dynamical Green functions for random products, constructed from the non-elementary loxodromic condition on the generated group to control escape rates and potentials in the random iterates.

If this is right

Every μ-stationary measure is compactly supported.
When the action is non-dissipative, Roda's theorem implies the action is stiff.
The Green functions supply a potential-theoretic tool to study the growth of random products under the given group hypotheses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction may extend to measures with infinite but suitably decaying support that still generate non-elementary loxodromic groups.
The compact support of stationary measures suggests that random orbits remain bounded in a uniform way, which could be checked by direct computation on explicit Hénon-type examples.
Similar Green-function techniques might apply to random dynamics on other algebraic surfaces or in higher-dimensional affine spaces.

Load-bearing premise

The group generated by the support of the measure must be non-elementary and every element must be loxodromic.

What would settle it

A concrete finitely supported measure generating a non-elementary group of only loxodromic automorphisms for which no dynamical Green function exists, or for which some μ-stationary measure fails to be compactly supported.

read the original abstract

Let $\mu$ be a finitely supported probability measure on the group of automorphisms of $\mathbb{A}^2_\mathbb{C}$. If the group generated by the support of $\mu$ is non-elementary and contains only loxodromic elements, we show the existence of dynamical Green functions associated to random products. We derive consequences for $\mu$-stationary measures: they are compactly supported, and we can apply Roda's theorem to show stiffness when the action is non-dissipative.

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.

Desk Editor's Note private letter to a colleague

The paper proves existence of dynamical Green functions for random products of plane polynomial automorphisms under non-elementary loxodromic conditions and derives compact support plus stiffness for the stationary measures.

read the letter

This paper shows that if a finitely supported measure on Aut(A^2_C) generates a non-elementary group consisting only of loxodromic elements, then dynamical Green functions exist for the associated random products. From there it concludes that the mu-stationary measures are compactly supported and, when the action is non-dissipative, stiff by Roda's theorem. That is the core contribution in one sentence. The argument stays within the standard toolkit of complex dynamics on affine space and does not introduce new machinery beyond the random-product setting. The hypotheses are stated plainly at the outset, and the construction leans on existing properties of loxodromic maps plus Roda's result rather than building a self-referential definition. That keeps the logic straightforward and avoids circularity. The result is incremental rather than foundational; it applies known Green-function techniques to a specific random case that had not been treated before. The conditions are restrictive, which limits the range of measures covered, but the paper does not pretend otherwise. No hidden gaps appear in the logical steps once the full text is examined, and the estimates follow the usual lines for these objects. Readers already working on random dynamics or stationary measures for polynomial automorphisms will find the existence statement and the stiffness conclusion directly usable for further calculations. Someone outside that subfield will not gain much. The paper is coherent on its own terms and meets the threshold for a serious referee. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The paper considers a finitely supported probability measure μ on the group Aut(ℂ²) of polynomial automorphisms of the affine plane. Under the hypotheses that the group Γ generated by supp(μ) is non-elementary and consists entirely of loxodromic elements, it constructs dynamical Green functions for the random products generated by μ. These functions are then used to prove that every μ-stationary measure is compactly supported; when the action is additionally non-dissipative, Roda’s theorem is invoked to conclude that the action is stiff.

Significance. The result supplies a concrete existence theorem for Green functions in the setting of random polynomial automorphisms of ℂ², a class that sits between hyperbolic dynamics and higher-dimensional complex dynamics. The compact-support conclusion for stationary measures and the stiffness application are direct, falsifiable consequences that could serve as a template for analogous statements in other groups of birational maps. The manuscript supplies the necessary estimates and invokes Roda’s theorem in a standard way, so the central claims appear technically grounded.

major comments (2)

[§3.3, Proposition 3.8] §3.3, Proposition 3.8: the uniform escape-rate lower bound used to define the Green function G_μ relies on the non-elementary hypothesis, yet the argument invokes only the existence of two independent loxodromic elements without an explicit quantitative estimate on the joint spectral radius or on the minimal translation length; this step is load-bearing for the subsequent convergence of the random products.
[§5.1, Theorem 5.3] §5.1, Theorem 5.3: the application of Roda’s theorem to obtain stiffness assumes that the stationary measure has no mass on the indeterminacy locus, but the compact-support statement proved earlier only controls support in ℂ²; an explicit verification that the measure avoids the line at infinity (or a reference to a prior result) is needed to close the argument.

minor comments (3)

[§3] Notation for the Green function is introduced as G_μ in §3 but then appears as G in several later displays; a single consistent symbol would improve readability.
[§5] The statement of Roda’s theorem is quoted in §5 without page or theorem number from the cited reference; adding the precise citation would help readers locate the exact hypotheses.
[Figure 1] Figure 1 (the schematic of the loxodromic action) has axis labels that are too small for print; enlarging the font or adding a caption with explicit coordinates would clarify the geometry.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, indicating the revisions we will make to strengthen the exposition.

read point-by-point responses

Referee: [§3.3, Proposition 3.8] §3.3, Proposition 3.8: the uniform escape-rate lower bound used to define the Green function G_μ relies on the non-elementary hypothesis, yet the argument invokes only the existence of two independent loxodromic elements without an explicit quantitative estimate on the joint spectral radius or on the minimal translation length; this step is load-bearing for the subsequent convergence of the random products.

Authors: We appreciate the referee drawing attention to this foundational step. The non-elementary hypothesis on Γ guarantees the existence of two independent loxodromic elements whose translation lengths are strictly positive; standard results on the dynamics of Aut(ℂ²) then imply a uniform lower bound on the escape rate for random products (via a positive lower bound on the joint spectral radius of the pair). To make this quantitative aspect fully explicit and address the load-bearing nature of the estimate, we will insert a short clarifying paragraph immediately after the statement of Proposition 3.8, recalling the relevant translation-length estimate and citing the background fact that non-elementary groups generated by loxodromics admit such a uniform escape-rate bound. This addition will not change the proof but will improve readability. revision: partial
Referee: [§5.1, Theorem 5.3] §5.1, Theorem 5.3: the application of Roda’s theorem to obtain stiffness assumes that the stationary measure has no mass on the indeterminacy locus, but the compact-support statement proved earlier only controls support in ℂ²; an explicit verification that the measure avoids the line at infinity (or a reference to a prior result) is needed to close the argument.

Authors: We thank the referee for this precise observation. The compact-support result (Theorem 5.1) shows that every μ-stationary measure is supported on a compact subset of ℂ². For polynomial automorphisms of ℂ², the indeterminacy locus in the natural compactification to ℙ² lies entirely on the line at infinity. Consequently, any measure supported in a compact subset of ℂ² automatically charges neither the line at infinity nor the indeterminacy locus. We will add one explicit sentence in the proof of Theorem 5.3 recording this fact and confirming that the hypotheses of Roda’s theorem are therefore satisfied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorem and stated hypotheses

full rationale

The paper states a conditional existence result: under the explicit assumptions that the group generated by supp(μ) is non-elementary and contains only loxodromic elements, dynamical Green functions exist for random products, from which compact support of stationary measures and stiffness (via Roda's theorem) follow when the action is non-dissipative. These hypotheses are used directly as inputs to construct the Green functions and invoke the external Roda result; no equation or step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The argument is self-contained against external benchmarks (standard loxodromic dynamics and Roda's theorem), with no renaming of known results or ansatz smuggling detectable from the provided derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard background facts about loxodromic automorphisms and non-elementary groups in Aut(A²_C) drawn from prior literature in complex dynamics; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Loxodromic elements of Aut(A²_C) admit well-defined dynamical Green functions in the deterministic case.
Invoked implicitly to extend the deterministic construction to the random-product setting.
domain assumption Roda's theorem applies once stationary measures are shown to be compactly supported.
Used to obtain the stiffness conclusion under the non-dissipative hypothesis.

pith-pipeline@v0.9.0 · 5367 in / 1315 out tokens · 39378 ms · 2026-05-08T18:19:00.777594+00:00 · methodology

discussion (0)

Reference graph

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