The Pentagram map: a discrete integrable system

TL;DR

The paper demonstrates that the pentagram map, a geometric iteration on polygons, is a completely integrable system with a Poisson structure, connecting discrete geometry to the classical Boussinesq equation.

Contribution

It introduces a Poisson structure for the pentagram map and proves its complete integrability, linking discrete polygons to continuous integrable systems.

Findings

The pentagram map is completely integrable in the Arnold-Liouville sense.

A Poisson structure on twisted polygons is constructed.

The continuous limit of the pentagram map relates to the Boussinesq equation.

Abstract

The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation). 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 universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.