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Journal Articles Comm. Math. Phys. Year : 2010

The pentagram map: A discrete integrable system

Valentin Ovsienko
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Serge Tabachnikov
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The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from ℤ into ℝℙ2 that is periodic modulo a projective transformation called the monodromy. 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. 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 for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.
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hal-00864947 , version 1 (23-09-2013)


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Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. The pentagram map: A discrete integrable system. Comm. Math. Phys., 2010, 299 (2), pp.409--446. ⟨hal-00864947⟩
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