A family of integrable transformations of centroaffine polygons: geometrical aspects

Arnold, Maxim; Fuchs, Dmitry; Tabachnikov, Serge

doi:10.5802/aif.3641

A family of integrable transformations of centroaffine polygons: geometrical aspects
[Sur une famille de transformations intégrables de polygones centroaffines : aspects géométriques]

Arnold, Maxim ¹ ; Fuchs, Dmitry ² ; Tabachnikov, Serge ³

¹ Department of Mathematical Sciences, University of Texas at Dallas, 800 West Campbell Road, Richardson, TX 75080 (USA)
² Department of Mathematics, University of California, Davis, CA 95616 (USA)
³ Department of Mathematics, Pennsylvania State University, University Park, PA 16802 (USA)

Annales de l'Institut Fourier, Tome 74 (2024) no. 3, pp. 1319-1363.

Résumé
Abstract

On appelle deux polygones planaires $(P_{1}, ..., P_{n})$ et $(Q_{1}, ..., Q_{n})$ $c$ -correspondants si leurs déterminants $det (P_{i}, P_{i + 1}) = det (Q_{i}, Q_{i + 1})$ et $det (P_{i}, Q_{i}) = c$ sont satisfaits pour tous $i$ .

Cette relation est un analogue équiaffine de la correspondance de bicyclette discrète étudiée par un certain nombre d’auteurs. Nous étudions la géométrie de ces relations, présentons ses intégrales, et montrons que – dans un sens approprié – ces relations commutent quand considérées pour différentes valeurs des constantes $c$ . Nous relions ce sujet à la chaîne de pansement de Veselov et Shabat.

Two polygons, $(P_{1}, ..., P_{n})$ and $(Q_{1}, ..., Q_{n})$ in $ℝ^{2}$ are $c$ -related if $det (P_{i}, P_{i + 1}) = det (Q_{i}, Q_{i + 1})$ and $det (P_{i}, Q_{i}) = c$ for all $i$ . This relation extends to twisted polygons (polygons with monodromy), and it descends to the moduli space of $SL (2, ℝ)$ -equivalent polygons. This relation is an equiaffine analog of the discrete bicycle correspondence studied by a number of authors. We study the geometry of this relations, present its integrals, and show that, in an appropriate sense, these relations, considered for different values of the constants $c$ , commute. We relate this topic with the dressing chain of Veselov and Shabat. The case of small-gons is investigated in detail.

Reçu le : 2021-12-17
Révisé le : 2022-06-12
Accepté le : 2022-09-23
Première publication : 2024-05-17
Publié le : 2024-07-03

DOI : 10.5802/aif.3641

Classification : 37J70, 37J39, 53D30
Keywords: Liouville integrability, Integrable Systems, Centroaffine geometry, Dressing chain.
Mot clés : Intégrabilité de Liouville, Systèmes intégrables, Géométrie centroaffine, Chaîne de pansement.

Affiliations des auteurs :

Arnold, Maxim ¹ ; Fuchs, Dmitry ² ; Tabachnikov, Serge ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Arnold, Maxim; Fuchs, Dmitry; Tabachnikov, Serge. A family of integrable transformations of centroaffine polygons: geometrical aspects. Annales de l'Institut Fourier, Tome 74 (2024) no. 3, pp. 1319-1363. doi : 10.5802/aif.3641. https://aif.centre-mersenne.org/articles/10.5802/aif.3641/

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