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

Quasiperiodic motion for the pentagram map

Valentin Ovsienko 1, , Richard Schwartz 2, and  Serge Tabachnikov 3,

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.
Keywords: polygons, Poisson structre, pentagram, projective geometry, integrability, monodromy, Bouissinesq equation., iteration.
Mathematics Subject Classification: Primary: 37J35; secondary: 51A9.
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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