A family of integrable transformations of centroaffine polygons: geometrical aspects. (Sur une famille de transformations intégrables de polygones centroaffines : aspects géométriques.) (English. French summary) Zbl 07877214
Summary: 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 \(\mathrm{SL} (2,\mathbb{R})\)-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.
MSC:
| 37J70 | Completely integrable discrete dynamical systems |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 53D30 | Symplectic structures of moduli spaces |
| 53A15 | Affine differential geometry |
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