2020/06/12 by Quinton Aboud, Aboud, Quinton, Anton Izosimov +1 · 1 citation
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Metric Geometry (math.MG) #math.DG #math.DS #math.MG #nlin.SI
paper · pdf · doi:10.48550/arxiv.2006.07413
10 pages, 4 figures; final version accepted to IMRN
arxiv created 2020/08/19 · arxiv updated 2020/08/21
The pentagram map takes a planar polygon P to a polygon P' whose vertices are the intersection points of consecutive shortest diagonals of P. The orbit of a convex polygon under this map is a sequence of polygons which converges exponentially to a point. Furthermore, as recently proved by Glick, coordinates of that limit point can be computed as an eigenvector of a certain operator associated with the polygon. In the present paper we show that Glick's operator can be interpreted as the infinitesimal monodromy of the polygon. Namely, there exists a certain natural infinitesimal perturbation of a polygon, which is again a polygon but in general not closed; what Glick's operator measures is the extent to which this perturbed polygon does not close up.