Two polygons, and in are -related if and for all . This relation extends to twisted polygons (polygons with monodromy), and it descends to the moduli space of -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 , 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 et -correspondants si leurs déterminants et sont satisfaits pour tous .
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 . Nous relions ce sujet à la chaîne de pansement de Veselov et Shabat.
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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
@article{AIF_2024__74_3_1319_0, author = {Arnold, Maxim and Fuchs, Dmitry and Tabachnikov, Serge}, title = {A family of integrable transformations of centroaffine polygons: geometrical aspects}, journal = {Annales de l'Institut Fourier}, pages = {1319--1363}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {74}, number = {3}, year = {2024}, doi = {10.5802/aif.3641}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3641/} }
TY - JOUR AU - Arnold, Maxim AU - Fuchs, Dmitry AU - Tabachnikov, Serge TI - A family of integrable transformations of centroaffine polygons: geometrical aspects JO - Annales de l'Institut Fourier PY - 2024 SP - 1319 EP - 1363 VL - 74 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3641/ DO - 10.5802/aif.3641 LA - en ID - AIF_2024__74_3_1319_0 ER -
%0 Journal Article %A Arnold, Maxim %A Fuchs, Dmitry %A Tabachnikov, Serge %T A family of integrable transformations of centroaffine polygons: geometrical aspects %J Annales de l'Institut Fourier %D 2024 %P 1319-1363 %V 74 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3641/ %R 10.5802/aif.3641 %G en %F AIF_2024__74_3_1319_0
Arnold, Maxim; Fuchs, Dmitry; Tabachnikov, Serge. A family of integrable transformations of centroaffine polygons: geometrical aspects. Annales de l'Institut Fourier, Volume 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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