Quasiperiodic Motion for the Pentagram Map

Valentin Ovsienko (1) , Richard Schwartz (2) , Serge Tabachnikov (3)

1 ICJ - Institut Camille Jordan
2 Department of Mathematics, Brown University
3 Department of Mathematics

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.

Mots clés: pentagram; iteration; polygons; projective geometry; monodromy; Poisson structre; integrability; Bouissonesq

Domaines: Systèmes dynamiques [math.DS]; Systèmes Solubles et Intégrables [nlin.SI]

Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic Motion for the Pentagram Map. 2009. ⟨hal-00351832⟩