A family of integrable transformations of centroaffine polygons: geometrical aspects

TL;DR

This paper explores a family of integrable transformations of centroaffine polygons, analyzing their geometric properties, integrals, and commutativity, and relating them to the dressing chain, with detailed study of small polygons.

Contribution

It introduces a new family of integrable transformations of centroaffine polygons, extending the discrete bicycle correspondence and connecting it with the dressing chain.

Findings

The transformations are integrable and commute for different parameters.

The relations extend to twisted polygons with monodromy.

Detailed analysis of small-gon cases.

Abstract

Two polygons, $(P_1,\ldots,P_n)$ and $(Q_1,\ldots,Q_n)$ in ${\mathbb R}^2$ 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 $SL(2,{\mathbb R}^2)$-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.