Quasiperiodic Motion for the Pentagram Map

TL;DR

This paper demonstrates that the pentagram map on twisted polygons possesses a Poisson structure making it completely integrable, with connections to the classical Boussinesq equation in the continuous limit.

Contribution

It introduces a Poisson structure on the space of twisted polygons and proves the integrability of the pentagram map within this framework.

Findings

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

The continuous limit of the pentagram map is the classical Boussinesq equation.

Certain families of polygons exhibit quasi-periodic orbits.

Abstract

The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call {\it twisted polygons}. 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 {\it universally convex}, we translate the integrability into a statement about the quasi-periodic notion of the pentagram-map orbits. We also explain how the continuous limit of the Pentagram map is the classical Boissinesq equation, a completely integrable PDE.