Quasiperiodic motion for the pentagram map

Valentin Ovsienko; Richard Schwartz; Serge Tabachnikov

doi:10.3934/era.2009.16.1

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

This volume Previous Article Next Article A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems

Quasiperiodic motion for the pentagram map

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

Received: December 31, 2008

Revised: December 31, 2008

Published: December 2008

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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 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.

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Mathematics Subject Classification: Primary: 37J35; secondary: 51A99.

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