Area-Normalized Pentagram Map Dynamics: Spectral Flattening and Elliptic Asymptotics

Abstract

The pentagram map sends a polygon to the intersections of consecutive short diagonals. We study its shape dynamics after translating and positively rescaling each iterate to restore unsigned area and barycenter. For polygons whose dynamics is generated, after passing to a finite iterate and cyclic relabelling, by one projectivity, we prove a spectral dichotomy. A dominant real projective line yields flattening and unbounded diameter, whereas a dominant real eigenvalue with a subdominant non-real pair produces asymptotic motion on concentric homothetic ellipses. We apply this framework to pentagons and hexagons via Glick's operator and to Poncelet polygons via the Darboux--Schwartz projectivity. We obtain spectral diagrams and a Jacobi-function formula for the Poncelet return spectrum. Consequently, we recover Schwartz's long-and-thin result for strictly convex non-projectively-regular pentagons, prove a line-or-ellipse dichotomy for convex hexagons under explicit nondegeneracy assumptions, and establish flattening for strictly convex non-projectively-regular Poncelet polygons.