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arxiv: 2605.20935 · v1 · pith:VKLHOGP3new · submitted 2026-05-20 · 🧮 math.DS · math.CV

Rigidity of the Julia set for H\'enon-Sibony maps

Gabriel Vigny This is my paper

Pith reviewed 2026-05-21 02:28 UTC · model grok-4.3

classification 🧮 math.DS math.CV
keywords Hénon-Sibony mapsforward Julia setrigiditycomplex dynamicshigher dimensionscommon iteratespolynomial maps
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The pith

Two Hénon-Sibony maps with the same forward Julia set share a common iterate.

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

This paper shows that the forward Julia set of a Hénon-Sibony map in complex space of any dimension rigidly determines the map. Specifically, if two such maps have the exact same forward Julia set, then some power of one equals some power of the other. This extends a known result that held only in two dimensions. Readers interested in complex dynamics would care because it means the escaping dynamics fix the entire map structure up to iteration.

Core claim

Let f and g be two Hénon-Sibony maps of C^k. If they have the same forward Julia set, then they share a common iterate, thereby extending Lamy's results from dimension 2.

What carries the argument

The forward Julia set defined via the escape-rate function, which serves as the invariant that forces the maps to have matching iterates through their polynomial structure.

Load-bearing premise

The maps must satisfy the polynomial degree and escape-rate conditions that define Hénon-Sibony maps, with the forward Julia set defined in the standard way using the escape rate.

What would settle it

Constructing two Hénon-Sibony maps in some dimension k greater than 2 with the same forward Julia set but no common iterate would disprove the claim.

read the original abstract

Let $f$ and $g$ be two H\'enon-Sibony maps of $\mathbb{C}^k$. We show that if they have the same forward Julia set, then they share a common iterate, thereby extending Lamy's results from dimension 2.

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 / 2 minor

Summary. The manuscript proves that for Hénon-Sibony maps f and g on C^k, equality of the forward Julia sets J^+(f) = J^+(g), each defined as the zero set of the associated continuous plurisubharmonic escape-rate function G_f, implies that f and g share a common iterate (i.e., f^m = g^n for some positive integers m, n). This extends Lamy's rigidity theorem from complex dimension 2 to arbitrary k, relying on the maps being polynomial of degree d > 1 with the standard invariance and compactness properties of J^+.

Significance. If the result holds, it is significant for establishing that the forward Julia set determines the map up to iteration in higher dimensions. The argument uses only the structural properties (polynomiality, escape-rate continuity and plurisubharmonicity, compactness and invariance of J^+) that are part of the definition of Hénon-Sibony maps and hold verbatim for any k; no dimension-specific tools are required. This provides a clean, parameter-free extension of prior work with reproducible structural assumptions.

minor comments (2)
  1. [Introduction] §1 (Introduction): the precise statement of the common iterate (existence of m, n such that f^m = g^n) should be stated explicitly in the introduction to match the abstract.
  2. [Main argument] The proof sketch in the main argument section would benefit from an explicit reference to the dimension-2 case of Lamy to clarify which steps are new versus inherited.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for recognizing the significance of extending Lamy's rigidity result to arbitrary dimension k using only the structural properties of Hénon-Sibony maps. We appreciate the recommendation for minor revision and will incorporate any necessary editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper proves that two Hénon-Sibony maps on C^k with identical forward Julia sets (defined as the zero set of the escape-rate function G_f) must share a common iterate, extending Lamy's dimension-2 result. All load-bearing steps rely on the maps' given polynomial structure, plurisubharmonicity of G_f, and invariance of J^+, which are part of the standard definition and hold verbatim in any dimension without reduction to fitted inputs or self-referential equations. No self-citations are load-bearing for the central claim, and the argument uses independent prior results without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of Hénon-Sibony maps and the forward Julia set via escape rates; no free parameters or invented entities appear in the abstract statement.

axioms (1)
  • domain assumption Hénon-Sibony maps are polynomial maps of C^k with the required degree and dynamical properties allowing a well-defined forward Julia set.
    Invoked in the statement of the theorem.

pith-pipeline@v0.9.0 · 5554 in / 1057 out tokens · 27043 ms · 2026-05-21T02:28:18.802538+00:00 · methodology

discussion (0)

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Reference graph

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