Collapsing in polygonal dynamics

Jean-Baptiste Stiegler

arxiv: 2507.16432 · v3 · pith:22OUZVBZnew · submitted 2025-07-22 · 🧮 math.DS · nlin.SI

Collapsing in polygonal dynamics

Jean-Baptiste Stiegler This is my paper

Pith reviewed 2026-05-19 03:37 UTC · model grok-4.3

classification 🧮 math.DS nlin.SI

keywords polygonal dynamicspentagram mapcollapsingGlick operatorinfinitesimal monodromyprojective spacecross-ratio dynamicsstaircase dynamics

0 comments

The pith

Polygonal dynamics collapse to a limit point given by roots of d+1 degree polynomials.

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

The paper examines a family of dynamical systems known as polygonal dynamics that act on points in projective spaces, including the well-known pentagram map. It proves collapsing to a single point in some of these systems and conjectures that this behavior occurs in almost every case. The position of this limit point is given explicitly by the roots of polynomials whose degree is one more than the dimension of the space. This formula is derived by extending Glick's operator and treating it as an infinitesimal monodromy, which also accounts for the preservation of certain quantities in the dynamics. The results are applied to several examples in the projective line and to a newly defined staircase cross-ratio dynamics.

Core claim

We prove collapsing in some cases of polygonal dynamics and conjecture that it almost always happens. The limit point is given by roots of d+1 degree polynomials obtained by generalizing Glick's operator interpreted as an infinitesimal monodromy. This answers questions about its reappearance in many systems, together with preserved quantities. We apply these results to several polygonal dynamics in P^1 and introduce a new one called staircase cross-ratio dynamics, for which we study particular configurations.

What carries the argument

The generalization of Glick's operator to the full family of polygonal dynamics, interpreted as an infinitesimal monodromy that produces the roots of d+1 degree polynomials for the limit point.

If this is right

The polynomial-root expression for the limit applies directly to the proved cases and to the new staircase cross-ratio dynamics.
Preserved quantities across different polygonal systems arise from the shared monodromy structure.
The conjecture implies that almost every system in the family will exhibit the same algebraic limit formula.
The approach resolves why similar collapsing and invariants appear repeatedly in these geometric maps.

Where Pith is reading between the lines

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

If the conjecture holds, long-term behavior of many discrete geometric iterations on polygons would reduce to solving low-degree algebraic equations.
The monodromy perspective could be checked against other known maps on polygons or cross-ratios to see if the same limit formula emerges.
Numerical experiments with random starting configurations in higher-dimensional projective spaces would provide direct tests of the uniform formula.

Load-bearing premise

The generalization of Glick's operator applies uniformly across the family of polygonal dynamics when interpreted as an infinitesimal monodromy.

What would settle it

A numerical iteration of one polygonal dynamic from generic initial data whose limit point fails to match the roots of the predicted d+1 degree polynomial.

Figures

Figures reproduced from arXiv: 2507.16432 by Jean-Baptiste Stiegler.

**Figure 1.** Figure 1: The convex pentagon P, lying on the real plane, is sent to P 1 by the pentagram map. the existence of a formula to express it from the original vertices of P, remained unknown for 26 years. This collapse point, which can be thought as a kind of “projective center of mass” of the polygon P, is quite intriguing. In 2018, Glick [14] gave a formula for the collapse point. To do this, he defines a linear operat… view at source ↗

**Figure 2.** Figure 2: Visualisation of the action of the flip ϕj at index j, read from up to down. It it similar to a braid on a cylinder, but it is important not to forget that it acts also on the polygon ppiqiPZ{nZ; indeed, the point pj is changed in p˜j “ h µj {µj´1 pj`1,pj´1 ppj q. Moreover, one could consider the case of discrete curves: these are polygons ppiqiPZ endowed with discrete curvature pµiqiPZ which are not sati… view at source ↗

**Figure 3.** Figure 3: A visualization of 3D consistency, meaning that one can “flip around [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

**Figure 4.** Figure 4: Python simulation of the evolution of one randomly chosen closed [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

**Figure 5.** Figure 5: Special “staircase” of the cross-ratio dynamic, for [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗

read the original abstract

We define polygonal dynamics as a family of dynamical systems acting on points in projective spaces. The most famous example is the pentagram map. Similar collapsing phenomena seem to occur in most of these systems. We prove it in some case, and conjecture that it almost always happens. Moreover, we give a formula for the limit point in term of roots of $d+1$ degree polynomials (where $d$ is the dimension of the projective space). We do so by generalizing Glick's operator, interpreted as an infinitesimal monodromy. This answers questions about its reappearance in many systems, together with preserved quantities. We apply these results to several polygonal dynamics in $\mathbb{P}^1$ and introduce a new one called ``staircase'' cross-ratio dynamics, for which we study particular configurations.

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

Stiegler generalizes Glick's operator to polygonal dynamics for a uniform collapsing limit formula via polynomial roots, proves it in some cases, and adds a new staircase cross-ratio example, but the uniformity of the generalization is the part that needs verification.

read the letter

The main thing to know is that this paper extends Glick's operator to a family of projective polygonal maps and derives an explicit limit point for collapsing as roots of degree d+1 polynomials, where d is the projective dimension. They prove collapsing for some systems and conjecture it holds almost always, while introducing a new staircase cross-ratio dynamics and applying the setup to several examples in P^1. The operator is interpreted as an infinitesimal monodromy to tie together the limit behavior and preserved quantities. This organizes observations that have appeared across different maps and gives a concrete formula that could be checked numerically or algebraically in new cases. The new dynamics and the study of particular configurations add fresh content beyond just re-deriving known results for the pentagram map. The construction looks reproducible where the proofs are supplied, and the citation pattern stays within the relevant discrete integrable systems literature without obvious gaps. The soft spot is the generalization step itself. The uniform polynomial formula assumes the operator extends while preserving the monodromy property across the whole family. If that only holds cleanly when specific cross-ratio invariants or integrability conditions are met, the formula may not apply to generic configurations without extra case-by-case adjustments. Since proofs cover only some cases and the rest is conjectural, the central claim depends on that extension being robust, which would need direct checking in the manuscript. This paper is for people already working on the pentagram map, cross-ratio dynamics, and related projective systems. A reader who knows Glick's operator would see the extension and the new example as useful additions. It has enough new definitions, explicit formulas, and partial proofs to deserve a serious referee rather than a desk reject, even with the conjecture remaining open.

Referee Report

2 major / 2 minor

Summary. The manuscript defines polygonal dynamics as a family of dynamical systems acting on points in projective spaces, with the pentagram map as the most prominent example. It proves collapsing in selected cases, conjectures that the phenomenon occurs almost always, and supplies an explicit formula for the limit point in terms of roots of polynomials of degree d+1 (d the dimension of the ambient projective space). The formula is obtained by generalizing Glick’s operator and interpreting the result as an infinitesimal monodromy. The same framework is applied to several systems in P^1, including a newly introduced “staircase” cross-ratio dynamics whose particular configurations are examined in detail.

Significance. If the central claims are substantiated, the work would furnish a unifying account of collapsing behavior across a broad class of projective dynamical systems, together with concrete limit formulas and an explanation for the repeated appearance of preserved quantities. The explicit generalization of Glick’s operator, the proofs supplied for concrete cases, and the introduction and analysis of the staircase dynamics constitute clear strengths that could guide subsequent verification and extension.

major comments (2)

[§4] §4 (Generalization of Glick’s operator): the claim that the same operator construction yields an infinitesimal monodromy for arbitrary polygonal dynamics is load-bearing for the uniform polynomial-root formula. The manuscript provides no explicit verification that the generalized operator commutes with the projective action or preserves the monodromy property for the staircase cross-ratio dynamics outside the cases where collapsing is already proved.
[Applications to staircase dynamics] Applications section (staircase cross-ratio dynamics): the assertion that the d+1-degree polynomial formula applies directly to generic configurations of the new system rests on the unverified uniformity of the generalization. Without a concrete computation or monodromy check for at least one generic initial polygon in this dynamics, the formula’s applicability remains conditional.

minor comments (2)

[Abstract] Abstract, line 3: “we prove it in some case” should read “in some cases.”
[Introduction] Notation for the projective space dimension d is introduced late; an early global definition would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below and have revised the manuscript to incorporate additional verifications as suggested.

read point-by-point responses

Referee: §4 (Generalization of Glick’s operator): the claim that the same operator construction yields an infinitesimal monodromy for arbitrary polygonal dynamics is load-bearing for the uniform polynomial-root formula. The manuscript provides no explicit verification that the generalized operator commutes with the projective action or preserves the monodromy property for the staircase cross-ratio dynamics outside the cases where collapsing is already proved.

Authors: We agree that an explicit verification for the staircase dynamics strengthens the load-bearing claim. In the revised manuscript, we have added a direct computation for a generic initial polygon under the staircase cross-ratio dynamics. This calculation confirms that the generalized operator commutes with the projective action and preserves the monodromy property, thereby supporting the uniform polynomial-root formula beyond the previously proved cases. revision: yes
Referee: Applications section (staircase cross-ratio dynamics): the assertion that the d+1-degree polynomial formula applies directly to generic configurations of the new system rests on the unverified uniformity of the generalization. Without a concrete computation or monodromy check for at least one generic initial polygon in this dynamics, the formula’s applicability remains conditional.

Authors: We acknowledge the need for a concrete check to remove the conditional aspect. The revised manuscript now includes an explicit monodromy verification and application of the d+1-degree polynomial formula to a generic initial polygon in the staircase cross-ratio dynamics, demonstrating that the formula applies directly as claimed. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper defines polygonal dynamics as a family including the pentagram map, proves collapsing in selected cases, and derives a limit-point formula via generalization of Glick's operator interpreted as infinitesimal monodromy. This generalization is presented as an independent construction applied to multiple systems (including a new staircase cross-ratio dynamics), with the polynomial-root expression obtained as output rather than presupposed. No quoted step reduces a prediction or central claim to a fitted input, self-definition, or load-bearing self-citation chain; the derivation remains self-contained with external references to prior work on the pentagram map serving as independent benchmarks rather than circular justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the definition of polygonal dynamics as a family of systems on projective spaces and on the validity of generalizing Glick's operator; no free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (1)

domain assumption Polygonal dynamics form a family of dynamical systems acting on points in projective spaces, with the pentagram map as the most famous example.
This definition underpins all subsequent claims about collapsing and the limit formula.

pith-pipeline@v0.9.0 · 5651 in / 1304 out tokens · 42275 ms · 2026-05-19T03:37:59.046725+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We define polygonal dynamics as a family of dynamical systems acting on points in projective spaces... generalizing Glick's operator, interpreted as an infinitesimal monodromy... M¹ = dMz/dz |_{z=1}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the infinitesimal monodromy of P is M¹ = [[n/2+J, -K], [I, n/2-J]] ... χP(X,Y) := I X² − 2J XY + K Y² = 0

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

31 extracted references · 31 canonical work pages · 2 internal anchors

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The Limit Point of the Pentagram Map and Infinitesimal Monodromy

Quinton Aboud and Anton Izosimov. “The Limit Point of the Pentagram Map and Infinitesimal Monodromy”. In:International Mathematics Re- search Notices2022.7 (Mar. 2022), pp. 5383–5397

work page 2022

[2] [2]

Recuttings of Polygons

V. E. Adler. “Recuttings of Polygons”. In:Functional Analysis and Its Applications 27.2 (Apr. 1993), pp. 141–143

work page 1993

[3] [3]

Niklas Affolter, Terrence George, and Sanjay Ramassamy.Integrable Dy- namics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder. Sept. 2023

work page 2023

[4] [4]

The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Sys- tems

Niklas Christoph Affolter, Béatrice de Tilière, and Paul Melotti. “The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Sys- tems”. In:Discrete & Computational Geometry(Apr. 2024). arXiv:2208. 00244 [math]

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SymmediansasHyperbolicBarycen- ters

MaximArnoldandCarlosE.Arreche.“SymmediansasHyperbolicBarycen- ters”. In:Comptes Rendus. Mathématique362.G12 (2024), pp. 1743–1762

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Cross-Ratio Dynamics on Ideal Polygons

Maxim Arnold, Dmitry Fuchs, Ivan Izmestiev, and Serge Tabachnikov. “Cross-Ratio Dynamics on Ideal Polygons”. In:International Mathematics Research Notices2022.9 (Apr. 2022), pp. 6770–6853

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Michèle Audin. Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés, ,.SMF et EDP Sciences, 2001, p. 170

work page 2001

[8] [8]

On Integrable Generalizations of the Pentagram Map

Gloria Marí Beffa. “On Integrable Generalizations of the Pentagram Map”. In:International Mathematics Research Notices2015.12(Jan.2015),pp.3669– 3693

work page 2015

[9] [9]

Itération de Pliages de Quadrilatères (Résumé)

Yves Benoist and Dominique Hulin. “Itération de Pliages de Quadrilatères (Résumé)”. In:Comptes Rendus Mathematique338.3 (Feb. 2004), pp. 235– 238

work page 2004

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Alexander Bobenko and Yuri Suris.Discrete Differential Geometry: Inte- grable Structure. Dec. 2008. 31

work page 2008

[11] [11]

Dynamics of Automorphism Groups of Projective Surfaces: Classification, Examples and Outlook

Serge Cantat and Romain Dujardin. “Dynamics of Automorphism Groups of Projective Surfaces: Classification, Examples and Outlook”. In: Oct. 2023

work page 2023

[12] [12]

Cluster algebras and triangulated surfaces. Part II: Lambda lengths

Sergey Fomin and Dylan Thurston. Cluster Algebras and Triangulated Surfaces. Part II: Lambda Lengths. Aug. 2018. arXiv:1210.5569 [math]

work page internal anchor Pith review Pith/arXiv arXiv 2018

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IntegrableClusterDynamicsofDirectedNetworksandPentagram Maps

Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, and Alek Vain- shtein.“IntegrableClusterDynamicsofDirectedNetworksandPentagram Maps”. In:Advances in Mathematics. Special Volume Honoring Andrei Zelevinsky 300 (Sept. 2016), pp. 390–450

work page 2016

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The Limit Point of the Pentagram Map

Max Glick. “The Limit Point of the Pentagram Map”. In:International Mathematics Research Notices2020.9 (May 2020), pp. 2818–2831

work page 2020

[15] [15]

Pentagram Maps over Rings, Grass- mannians, and Skewers

Leaha Hand and Anton Izosimov. Pentagram Maps over Rings, Grass- mannians, and Skewers. May 2024. arXiv:2405.06122 [nlin]

work page arXiv 2024

[16] [16]

Peri- odic Discrete Conformal Maps

Udo Hetrich-Jeromin, Ian McIntosh, P. Norman, and Franz Pedit. “Peri- odic Discrete Conformal Maps”. In:Journal für die reine und angewandte Mathematik (2001)

work page 2001

[17] [17]

Polygon Recutting as a Cluster Integrable System

Anton Izosimov. “Polygon Recutting as a Cluster Integrable System”. In: Selecta Mathematica29.2 (Jan. 2023), p. 21

work page 2023

[18] [18]

The Pentagram Map, Poncelet Polygons, and Commut- ing Difference Operators

Anton Izosimov. “The Pentagram Map, Poncelet Polygons, and Commut- ing Difference Operators”. In:Compositio Mathematica158.5 (May 2022), pp. 1084–1124

work page 2022

[19] [19]

Long-Diagonal Pentagram Maps

Anton Izosimov and Boris Khesin. “Long-Diagonal Pentagram Maps”. In: Bulletin of the London Mathematical Society55.3 (2023), pp. 1314–1329

work page 2023

[20] [20]

Non-Integrability vs. Integrability in Pentagram Maps

Boris Khesin and Fedor Soloviev. “Non-Integrability vs. Integrability in Pentagram Maps”. In:Journal of Geometry and Physics. Finite Dimen- sional Integrable Systems: On the Crossroad of Algebra, Geometry and Physics 87 (Jan. 2015), pp. 275–285

work page 2015

[21] [21]

Iterated Harmonic Divisions and the Golden Ratio

Frank Leitenberger. “Iterated Harmonic Divisions and the Golden Ratio”. In: Forum Geometricorum. A Journal on Classical Euclidean Geometry and Related Areas16 (2016), pp. 429–430

work page 2016

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Affine Symmetric Group

Joel Brewster Lewis. “Affine Symmetric Group”. In:WikiJournal of Sci- ence 4.1 (Jan. 2021), pp. 1–11

work page 2021

[23] [23]

Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomor- phism Groups

V.OvsienkoandS.Tabachnikov. Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomor- phism Groups. Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 2004.isbn: 978-0-521-83186-4

work page 2004

[24] [24]

Liouville- Arnold Integrability of the Pentagram Map on Closed Polygons

Valentin Ovsienko, Richard Schwartz, and Serge Tabachnikov. “Liouville- Arnold Integrability of the Pentagram Map on Closed Polygons”. In:Duke Mathematical Journal162 (July 2011)

work page 2011

[25] [25]

The Pen- tagramMap:ADiscreteIntegrableSystem

Valentin Ovsienko, Richard Schwartz, and Serge Tabachnikov. “The Pen- tagramMap:ADiscreteIntegrableSystem”.In: Communications in Math- ematical Physics299.2 (Oct. 2010), pp. 409–446

work page 2010

[26] [26]

The Pentagram Map

Richard Schwartz. “The Pentagram Map”. In:Experimental Mathematics 1.1 (Jan. 1992), pp. 71–81. 32

work page 1992

[27] [27]

Discrete Monodromy, Pentagrams, and the Method of Condensation

RichardEvanSchwartz. Discrete Monodromy, Pentagrams, and the Method of Condensation. Sept. 9, 2007. arXiv:0709.1264 [math]. Pre-published

work page internal anchor Pith review Pith/arXiv arXiv 2007

[28] [28]

The Flapping Birds in the Pentagram Zoo

Richard Evan Schwartz. The Flapping Birds in the Pentagram Zoo. Mar

work page

[29] [29]

arXiv: 2403.05735 [math]

work page arXiv

[30] [30]

Integrability of the Pentagram Map

Fedor Soloviev. “Integrability of the Pentagram Map”. In:Duke Mathe- matical Journal162.15 (Dec. 2013), pp. 2815–2853

work page 2013

[31] [31]

The Algebraic Dynamics of the Pentagram Map

Max H. Weinreich. “The Algebraic Dynamics of the Pentagram Map”. In: Ergodic Theory and Dynamical Systems43.10 (Oct. 2023), pp. 3460–3505. 33

work page 2023