The Pentagram Map is Recurrent

Abstract

The pentagram map is defined on the space of convex $n$-gons (considered up to projective equivalence) by drawing the diagonals that join second-nearest-neighbors in an $n$-gon and taking the new $n$-gon formed by the intersections. We prove that this map is recurrent; thus, for almost any starting polygon, repeated application of the pentagram map will show a near copy of the starting polygon appear infinitely often under various perspectives.