A family of integrable transformations of centroaffine polygons: geometrical aspects
Annales de l'Institut Fourier, Online first, 45 p.

Two polygons, (P 1 ,...,P n ) and (Q 1 ,...,Q n ) in 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,)-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.

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.

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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.
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)
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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, Online first, 45 p.

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