Duke Mathematical Journal

Liouville–Arnold integrability of the pentagram map on closed polygons

Abstract

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

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

Dates
First available in Project Euclid: 9 September 2013

https://projecteuclid.org/euclid.dmj/1378729686

Digital Object Identifier
doi:10.1215/00127094-2348219

Mathematical Reviews number (MathSciNet)
MR3102478

Zentralblatt MATH identifier
1315.37035

Citation

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

References

• [1] V. I. Arnol’d, Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math. 60, Springer, New York, 1989.
• [2] V. Fock and A. Marshakov, Integrable systems, clusters, dimmers, and loop groups, in preparation.
• [3] S. Fomin and A. Zelevinsky, Cluster algebras, I: Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.
• [4] E. Frenkel, N. Reshetikhin, and M. A. Semenov-Tian-Shansky, Drinfeld-Sokolov reduction for difference operators and deformations of $W$-algebras, I: The case of Virasoro algebra, Comm. Math. Phys. 192 (1998), 605–629.
• [5] M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein, Higher pentagram maps, weighted directed networks, and cluster dynamics, Electron. Res. Announc. Math. Sci. 19 (2012), 1–17.
• [6] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster Algebras and Poisson Geometry, Amer. Math. Soc., Providence, 2010.
• [7] M. Glick, The pentagram map and Y-patterns, Adv. Math. 227 (2011), 1019–1045.
• [8] M. Glick, On singularity confinement for the pentagram map, preprint, arXiv:1110.0868v2 [math.CO].
• [9] B. Khesin and F. Soloviev, The pentagram map in higher dimensions and KdV flows, Electron. Res. Announc. Math. Sci. 19 (2012), 86–96.
• [10] B. Khesin and F. Soloviev, Integrability of higher pentagram maps, in preparation.
• [11] I. Krichever, Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem (in Russian), Uspekhi Mat. Nauk. 59, no. 6 (2004), 111–150; English translation in Russian Math. Surveys 59 (2004), 1117–1154.
• [12] S. B. Lobb and F. W. Nijhoff, Lagrangian multiform structure for the lattice Gel’fand-Dikii hierarchy, J. Phys. A 43 (2010), no. 7., 11 pp.
• [13] G. Mari-Beffa, On integrable generalizations of the pentagram map, preprint, arXiv:1303.4295v1 [math.DS].
• [14] G. Mari-Beffa, On generalizations of the pentagram map: Discretizations of AGD flows, J. Nonlinear Sci. 23 (2013), 303–334.
• [15] S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, $2$-frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier (Grenoble) 62 (2012), 937–987.
• [16] Th. Motzkin, The pentagon in the projective plane, with a comment on Napier’s rule, Bull. Amer. Math. Soc. 51 (1945), 985–989.
• [17] V. Ovsienko, R. Schwartz, and S. Tabachnikov, Quasiperiodic motion for the pentagram map, Electron. Res. Announc. Math. Sci. 16 (2009), 1–8.
• [18] V. Ovsienko, R. Schwartz, and S. Tabachnikov, The pentagram map: A discrete integrable system, Comm. Math. Phys. 299 (2010), 409–446.
• [19] V. Ovsienko and S. Tabachnikov, Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups, Cambridge Univ. Press, Cambridge, 2005.
• [20] R. Schwartz, The pentagram map, Experiment. Math. 1 (1992), 71–81.
• [21] R. Schwartz, The pentagram map is recurrent, Experiment. Math. 10 (2001), 519–528.
• [22] R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl. 3 (2008), 379–409.
• [23] R. Schwartz, S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer 32 (2010), 31–34.
• [24] R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Combin. 18 (2011), paper 171, 19 pp.
• [25] F. Soloviev, Integrability of the pentagram map, to appear in Duke Math. J., preprint, arXiv:1106.3950v3 [math.AG].
• [26] A. Tongas and F. Nijhoff, The Boussinesq integrable system: Compatible lattice and continuum structures, Glasg. Math. J. 47 (2005), 205–219.