2009/01/12 by Ovsienko, Valentin, Schwartz, Richard, Tabachnikov, Serge
#37J35 (Primary) #51A99 (Secondary) #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences
paper · doi:10.48550/arxiv.0901.1585
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.