Quasiperiodic Motion for the Pentagram Map

Details Quasiperiodic Motion for the Pentagram Map Quasiperiodic Motion for the Pentagram Map

2009-01-12

arxiv.org/abs/0901.1585v1

Mathematics

Dynamical Systems

This note is a short announcement of arXiv:0810.5605

Scientific paper

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.

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