Orateur
Description
We study the Euclidean asymptotics of the area-normalized pentagram map through the spectral properties of Glick’s operator. For projectively periodic orbits, we establish general criteria leading either to flattening onto a spectral line or to bounded elliptic oscillations. For pentagons, we obtain a complete projective spectral classification in terms of a single invariant and recover, with explicit spectral directions and rates, Schwartz’s long-and-thin theorem for strictly convex pentagons. We show that closely related, though richer, spectral phenomena occur for hexagons as well. We also compute the Darboux--Schwartz return spectrum for Poncelet polygons using a Jacobi-function normal form and obtain corresponding asymptotic results. Finally, exact higher-order projective returns illustrate how the same spectral mechanisms extend to larger polygons.