The pentagram map: geometry, algebra, integrability (Q2895568)

This is a short and very clear review of work, mostly by the author and collaborators, on recent developments in the study of a discrete dynamical system called the pentagram map. This map was introduced by Richard Schwarz (1992) as a transformation \(T\) on the space of convex closed \(n\)-gons in the projective plane. For a given \(n\)-gon \(P\) the corresponding \(n\)-gon \(T(P)\) is the convex hull of the intersection points of consecutive shortest diagonals of \(P\). This map commutes with all projective transformations \(\phi\in \mathrm{PGL}(3,R)\), so it can be considered as a transformation on the space \({\mathcal C}_n\) of all \(n\)-gons modulo projective equivalence. The name derives from the fact that the case \(n=5\) gives the simplest nontrivial examples. Computer experiments suggest that this mapping on \({\mathcal C}_n\) may be integrable. This has not been established as yet, but complete integrability can be established for a slightly larger space that contains \({\mathcal C}_n\). Let \(v_1,v_2,\dots\) be an infinite sequence of vectors in the projective plane such that no three consecutive vectors are collinear, and there is some \(M\in \mathrm{PGL}(3,R)\), such that \(v_{i+n}=M\circ v_i\) for all \(i\). All such vector sequences, modulo projective equivalences, form the space of twisted \(n\)-gons \({\mathcal P}_n\). In the special case that \(M\) is the identity matrix we recover a closed polygon. The map \(T\) can easily be extended to \(T: {\mathcal P}_n \to {\mathcal P}_n\).NEWLINENEWLINEMuch of the paper sketches the demonstration that \(T\) is completely integrable on \({\mathcal P}_n\). This includes pointing out that this map can be posed as a difference equation \(V_i=a_iV_{i-1}-b_iV_{i-2}+V_{i-3}\) where the \(V_i\) are real numbers and \(a_i,b_i\) are \(n\)-periodic. The pentagram map can be described in terms of its action on the coordinates \(a_i,b_i\) as simple rational expressions. A Poisson bracket is introduced on this space such that the bracket is invariant with respect to the pentagram map. Then \(2[n/2]+2\) algebraically independent polynomials in the coordinates are constructed that are invariant under under \(T\) and Poisson commute. In addition there are 2 Casimirs for \(n\) odd and 4 for \(n\) even. This proves complete integrability.NEWLINENEWLINEAlso the author points out 1) that the continuous limit of this discrete dynamical system as \(n\to\infty\) is the classical Boussinesq equation and 2) that the original space \({\mathcal S}_n\) has a combinatorial interpretation as a 2-frieze pattern, an analog of the classical Coxeter-Conway frieze pattern.NEWLINENEWLINEFor the entire collection see [Zbl 1234.81013].