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arxiv: 1907.05116 · v1 · pith:BBM6TKJDnew · submitted 2019-07-11 · 🧮 math.CV · math.DS

Further remarks on rigidity of H\'{e}non maps

Ratna Pal This is my paper

Pith reviewed 2026-05-24 22:55 UTC · model grok-4.3

classification 🧮 math.CV math.DS
keywords Hénon mapsGreen functionpolynomial automorphismsShort C^2rigiditylevel setscomplex dynamics
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The pith

For Hénon maps in C^2, polynomial automorphisms preserving any fixed Green function level set are characterized as affine ones.

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

The paper characterizes the polynomial automorphisms of C^2 that keep any fixed level set of the Green function of a Hénon map completely invariant. It shows that the interior of any non-zero sublevel set of this Green function is a Short C^2 space, and concludes there are no polynomial automorphisms other than possibly affine ones that act as automorphisms on these spaces. It further proves that if level sets of the Green functions for two Hénon maps coincide, then the maps almost commute. These results refine how dynamical invariants limit the symmetries available to such maps.

Core claim

For a Hénon map H in C^2, the polynomial automorphisms keeping any fixed level set of its Green function completely invariant are characterized; as a consequence the interiors of non-zero sublevel sets, which form Short C^2 spaces, admit no polynomial automorphisms apart from possibly the affine ones, and if any two level sets of the Green functions of a pair of Hénon maps coincide then the maps almost commute.

What carries the argument

The level sets of the Green function of the Hénon map, whose complete invariance under a polynomial automorphism is used to derive the rigidity statements for the associated Short C^2 spaces.

Load-bearing premise

The standard definition and invariance properties of the Green function for polynomial Hénon maps in C^2 suffice to support the stated characterizations and consequences for Short C^2 spaces.

What would settle it

An explicit non-affine polynomial automorphism of C^2 that keeps some level set of the Green function of a Hénon map completely invariant would falsify the characterization.

read the original abstract

For a H\'{e}non map $H$ in $\mathbb{C}^2$, we characterize the polynomial automorphisms of $\mathbb{C}^2$ which keep any fixed level set of the Green function of $H$ completely invariant. The interior of any non-zero sublevel set of the Green function of a H\'{e}non map turns out to be a Short $\mathbb{C}^2$ and as a consequence of our characterization, it follows that there exists no polynomial automorphism apart from possibly the affine automorphisms which acts as an automorphism on any of these Short $\mathbb{C}^2$'s. Further, we prove that if any two level sets of the Green functions of a pair of H\'{e}non maps coincide, then they almost commute.

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

0 major / 1 minor

Summary. For a Hénon map H in C^2, the manuscript characterizes the polynomial automorphisms of C^2 which keep any fixed level set of the Green function of H completely invariant. The interior of any non-zero sublevel set of the Green function of a Hénon map is shown to be a Short C^2, with the consequence that no polynomial automorphism apart from possibly the affine ones acts as an automorphism on any of these Short C^2's. It is further proved that if any two level sets of the Green functions of a pair of Hénon maps coincide, then the maps almost commute.

Significance. If the results hold, they add to the body of rigidity theorems for polynomial automorphisms and Green functions on C^2, with direct implications for the automorphism groups of certain Short C^2 domains. The approach via complete invariance of level sets under the functional equation G(H(z)) = deg(H) G(z) is a natural extension of existing techniques in several complex variables.

minor comments (1)
  1. [Abstract] The abstract states three distinct theorems without numbering or section references, which makes it harder to locate the corresponding statements and proofs in the body of the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and positive assessment of its significance in the context of rigidity results for polynomial automorphisms and Green functions. The recommendation is listed as uncertain, but the report contains no specific major comments to address. We are happy to respond to any additional concerns if provided.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its characterizations of polynomial automorphisms preserving level sets of the Green function G for a Hénon map H, the Short C² nature of non-zero sublevel interiors, and the almost-commuting property when level sets coincide, directly from the standard invariance G(H(z)) = deg(H)·G(z) and complete invariance of level sets. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the logical chain is self-contained against the external benchmark of established Green function properties in complex dynamics of Hénon maps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure mathematics contribution relying on standard background from complex analysis and dynamics; no free parameters or invented entities are indicated.

axioms (1)
  • domain assumption Standard properties of the Green function for Hénon maps, including its plurisubharmonicity and invariance under the map.
    Invoked throughout the abstract to define level sets and support the characterizations.

pith-pipeline@v0.9.0 · 5652 in / 1287 out tokens · 35080 ms · 2026-05-24T22:55:03.813139+00:00 · methodology

discussion (0)

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Reference graph

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