# American Institute of Mathematical Sciences

2009, 16: 1-8. doi: 10.3934/era.2009.16.1

## Quasiperiodic motion for the pentagram map

 1 CNRS, Institut Camille Jordan, Université Lyon 1, Villeurbanne Cedex 69622, France 2 Department of Mathematics, Brown University, Providence, RI 02912, United States 3 Department of Mathematics, Penn State University, University Park, PA 16802

Received  January 2009 Revised  January 2009 Published  March 2009

The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call 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 universally convex, we translate the integrability into a statement about the quasi-periodic motion of the pentagram-map orbits. We also explain how the continuous limit of the pentagram map is the classical Boussinesq equation, a completely integrable P.D.E.
Citation: Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1
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