A family of integrable transformations of centroaffine polygons: geometrical aspects

Abstract

Two polygons, $(P_1,\dots ,P_n)$ and $(Q_1,\dots ,Q_n)$ 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.