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Ergodic measures of high entropy for Hénon-Sibony maps are supported on the Julia set.

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Every ergodic f-invariant measure with entropy > log max{d_+^{p-1}, d_-^{k-p-1}} is supported on the Julia set J.

T0 review reviewed 2026-06-27 challenge →

load-bearing objection The paper gives a precise entropy threshold forcing ergodic measures onto the Julia set for Hénon-Sibony maps.

arxiv 2606.08754 v1 pith:XFJI3RYK submitted 2026-06-07 math.DS math.CV

On the support of measures of large entropy for H\'enon-Sibony maps

Sobir Boymurodov This is my paper
classification math.DS math.CV
keywords Hénon-Sibony mapsJulia setergodic measurestopological entropyinvariant measuresalgebraic degreecomplex dynamics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper establishes that for Hénon-Sibony maps of complex space of algebraic degree at least 2, every ergodic invariant measure whose entropy exceeds log of the maximum of two lower-degree terms must be supported on the Julia set. A sympathetic reader would care because the Julia set is the locus of chaotic expansion where the map behaves like its algebraic degree, while measures outside it are constrained to lower entropy by the inverse degree. The result therefore locates all sufficiently chaotic invariant measures and rules out their placement on proper invariant subsets. It applies under the assumption that topological entropy exactly equals log of the degree to the power p.

Core claim

For a Hénon-Sibony map f of C^k of algebraic degree d+ ≥ 2 whose inverse has algebraic degree d-, with topological entropy equal to log d_+^p = log d_-^{k-p}, every ergodic f-invariant measure ν satisfying h_ν(f) > log max{d_+^{p-1}, d_-^{k-p-1}} is supported on the Julia set J of f.

What carries the argument

The entropy threshold log max{d_+^{p-1}, d_-^{k-p-1}} that upper-bounds entropy for measures whose support avoids the Julia set.

Load-bearing premise

The map must be a Hénon-Sibony map whose topological entropy exactly equals log d_+^p.

What would settle it

An explicit ergodic f-invariant probability measure whose entropy exceeds the threshold yet whose support lies strictly outside the Julia set.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The measure of maximal entropy, whose entropy equals log d_+^p, must be supported on the Julia set.
  • Any ergodic invariant measure achieving entropy strictly between the threshold and the topological entropy is confined to the Julia set.
  • Invariant measures supported away from the Julia set cannot exceed the entropy bound given by the lower of the two degree powers.
  • The support statement applies uniformly to all ergodic components of any invariant measure satisfying the entropy inequality.

Where Pith is reading between the lines

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

  • The same entropy bound may serve as a test for whether a candidate measure is an equilibrium state for a suitable potential.
  • Decomposing non-ergodic measures into ergodic components would extend the support conclusion to all invariant measures above the threshold.
  • The result supplies a dynamical criterion that could be checked numerically by approximating entropy and support for concrete polynomial maps.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for a Hénon-Sibony map f of C^k of algebraic degree d_+ ≥ 2 whose inverse has algebraic degree d_-, with topological entropy equal to log d_+^p = log d_-^{k-p}, every ergodic f-invariant measure ν satisfying h_ν(f) > log max{d_+^{p-1}, d_-^{k-p-1}} is supported on the Julia set J of f.

Significance. If the result holds, it supplies a concrete entropy threshold guaranteeing that high-entropy ergodic measures lie on the Julia set rather than in Fatou components. This extends classical entropy-support statements from one-dimensional rational maps and polynomial automorphisms to the Hénon-Sibony setting in several variables and may be useful for locating measures of maximal entropy or studying their Lyapunov exponents.

minor comments (2)
  1. The statement of the topological entropy hypothesis (log d_+^p = log d_-^{k-p}) should be accompanied by a brief reference or short argument confirming that this equality holds for the maps under consideration, to make the hypotheses self-contained.
  2. Notation for the Julia set is introduced as Σ in the abstract but rendered as Σ in the title; consistent use of script J or a single symbol throughout would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a theorem on the support of high-entropy ergodic measures for Hénon-Sibony maps, with the topological entropy hypothesis given explicitly as part of the setup and the entropy threshold derived from the algebraic degrees in the standard manner for such maps. No equations, definitions, or self-citations in the abstract or stated claim reduce the support conclusion to a fitted parameter, a self-referential construction, or a load-bearing prior result by the same authors. The derivation chain is therefore self-contained against external benchmarks in complex dynamics and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the definition of Hénon-Sibony maps, the given entropy formula, and standard facts about ergodic measures and Julia sets in complex dynamics; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Topological entropy of f equals log d_+^p = log d_-^{k-p}
    Explicitly stated in the abstract as a property of the maps under consideration.
  • standard math Standard properties of ergodic invariant measures, topological entropy, and Julia sets for polynomial maps on C^k
    Invoked implicitly as background for the support statement.

reviewed 2026-06-27 · how reviews work

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Cite this review

Pith. "Pith review of On the support of measures of large entropy for H\'enon-Sibony maps." pith.science (2026). https://pith.science/paper/XFJI3RYK

@misc{pith2026260608754,
  author       = {Pith},
  title        = {Pith review of: On the support of measures of large entropy for H\'enon-Sibony maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XFJI3RYK}},
  note         = {Machine review of arXiv:2606.08754}
}
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abstract

Let $f$ be a H\'enon-Sibony map of $\mathbb{C}^k$ of algebraic degree $d_+\geq 2$, whose inverse $f^{-1}$ has algebraic degree $d_-$. The topological entropy of $f$ is equal to $\log d_+^{p} = \log d_-^{k-p}$. We show that every ergodic $f$-invariant measure $\nu$ satisfying $h_\nu(f)>\log \max\{ d_+^{p-1},d_-^{k-p-1}\}$ is supported on the Julia set $\mathcal{J}$ of $f$.

discussion (0)

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Reference graph

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This paper was first reviewed by grok-4.3 on June 27, 2026.