Liouville-Arnold integrability of the pentagram map on closed polygons
Let \(P=v_1v_2\dots v_n\) be a convex polygon (\(n\)-gon, \(n \geq 5\)) in \({\mathbb R}^2\), and let \(v_{k+n}=v_n\) for all integers \(k\). For any four consecutive points \(v_{i-1},v_{i},v_{i+1},v_{i+2}\), define the new point \(w_i=(v_{i-1},v_{i+1})\cap(v_{i},v_{i+2})\), where \((v_j,v_k)\) stands for the line through \(v_j\) and \(v_k\). Thus we obtain a new \(n\)-gon \(T(P)=w_1w_2\dots w_n\) and the so-called pentagram map \(T: {\mathcal E}_n \rightarrow {\mathcal E}_n\), where \({\mathcal E}_n\) is the moduli space of projective equivalence classes of \(n\)-gons in the projective plane \({\mathbb R}P^2\).NEWLINENEWLINEFor \(n=5\) or \(n=6\), the pentagram map is always periodic, more precisely \(T\) or \(T^2\) are equal to the identity map. It turns out that the orbit \(P,T(P),T^2(P),\dots\) of a general convex \(n\)-gon \(P\) undergoes quasi-periodic motion. The authors give a purely geometric proof of the following theorem.NEWLINENEWLINETheorem: Almost every point of \({\mathcal E}_n\) lies on a \(T\)-invariant submanifold \(M\) of dimension \(d=n-4\) for \(n\) odd or \(d=n-5\) for even \(n\). The submanifold \(M\) is defined by a locally free action of \(d\) commuting and independent at every point vector fields over \(M\).NEWLINENEWLINETo prove this discrete version of an Arnold-Liouville's type of theorem, some corner invariants \((x_{2i-1},x_{2i})\) for each point \(v_i\) have been introduced. These \(x_1,\dots, x_{2n}\) are the local coordinates in \({\mathcal E}_n\) to express \(T\) as certain rational functions, the Poisson brackets as \(\{ x_i, x_{i+2} \} = (-1)^i x_i x_{i+2}\) (otherwise vanishing), the commuting vector fields as polynomials, etc.
scientific article; zbMATH DE number 6218376
scientific article