Quasiperiodic motion for the pentagram map

Abstract

The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call <em>twisted polygons</em>. In this note we describe our recent work on the pentagram map, in which we find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call <em>universally convex</em>, we translate the integrability into a statement about the quasi-periodic motion of the pentagram-map orbits. We also explain how the continuous limit of the pentagram map is the classical Boussinesq equation, a completely integrable P.D.E.