Duke Mathematical Journal

Liouville–Arnold integrability of the pentagram map on closed polygons

Valentin Ovsienko, Richard Evan Schwartz, and Serge Tabachnikov

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The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras, and integrable systems.

Integrability of the pentagram map was conjectured by Schwartz and proved by the present authors for a larger space of twisted polygons. In this article, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.

Article information

Duke Math. J., Volume 162, Number 12 (2013), 2149-2196.

First available in Project Euclid: 9 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37J35: Completely integrable systems, topological structure of phase space, integration methods
Secondary: 51A20: Configuration theorems


Ovsienko, Valentin; Schwartz, Richard Evan; Tabachnikov, Serge. Liouville–Arnold integrability of the pentagram map on closed polygons. Duke Math. J. 162 (2013), no. 12, 2149--2196. doi:10.1215/00127094-2348219. https://projecteuclid.org/euclid.dmj/1378729686

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