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arXiv · 0901.1585

Quasiperiodic Motion for the Pentagram Map

Valentin Ovsienko·Richard Schwartz·Serge Tabachnikov
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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.

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Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. 2009-01-12. Quasiperiodic Motion for the Pentagram Map. https://arxiv.org/abs/0901.1585

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