The Pentagram Map: A Discrete Integrable System

By Quang Nhat Le

November 20, 2013

Abstract

I'll define the pentagram map, a simple and projectively natural map on the space of polygons. it can be extended to act on the space P_n of twisted polygons. The beauty of this map is that it acts like translations along tori. What can be simpler than that? Indeed, P_n can be foliated by submanifolds, which can be given Euclidean structures such that the pentagram map preserves the foliation and is simply a translation on the leaves. This fact is obtained by constructing a Poisson structure on P_n and enough smooth functions on P_n which are invariant under the pentagram map.