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The Pentagram Map: A Discrete Integrable System

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Abstract

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from \({\mathbb Z}\) into \({{\mathbb{RP}}^2}\) 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. A research announcement of this work appeared in [16].

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References

  1. Adler V.E.: Cuttings of polygons. Funct. Anal. Appl. 27, 141–143 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adler V.E.: Integrable deformations of a polygon. Phys. D 87, 52–57 (1995)

    Article  MathSciNet  Google Scholar 

  3. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, 60. New York: Springer-Verlag, 1989

  4. Belov A., Chaltikian K.: Lattice analogues of W-algebras and classical integrable equations. Phys. Lett. B 309, 268–274 (1993)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Bobenko, A., Suris, Yu.: Discrete Differential Geometry: Integrable Structure. Providence, RI: Amer. Math. Soc., 2008

  6. Fock V., Goncharov A.: Moduli spaces of convex projective structures on surfaces. Adv. Math. 208, 249–273 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fomin S., Zelevinsky A.: Cluster algebras I: foundations. J. Amer. Math. Soc. 12, 497–529 (2002)

    Article  MathSciNet  Google Scholar 

  8. Falqui G., Magri F., Tondo G.: Reduction of bi-Hamiltonian systems and the separation of variables: an example from the Boussinesq hierarchy. Theoret. and Math. Phys. 122, 176–192 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  9. Frenkel E., Reshetikhin N., Semenov-Tian-Shansky M.: Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras. I. The case of Virasoro algebra. Commun. Math. Phys. 192, 605–629 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Gekhtman M., Shapiro M., Vainshtein A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3, 899–934 (2003)

    MATH  MathSciNet  Google Scholar 

  11. Henriques A.: A periodicity theorem for the octahedron recurrence. J. Alg. Combin. 26, 1–26 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hoffmann T., Kutz N.: Discrete curves in CP1 and the Toda lattice. Stud. Appl. Math. 113, 31–55 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Konopelchenko B.G., Schief W.K.: Menelaus’ theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy. J. Phys. A: Math. Gen. 35, 6125–6144 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Marshall I., Semenov-Tian-Shansky M.: Poisson groups and differential Galois theory of Schroedinger Equation on the circle. Commun. Math. Phys 284, 537–552 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Motzkin Th.: The pentagon in the projective plane, with a comment on Napiers rule. Bull. Amer. Math. Soc. 52, 985–989 (1945)

    Article  MathSciNet  Google Scholar 

  16. Ovsienko V., Schwartz R., Tabachnikov S.: Quasiperiodic motion for the pentagram map. Electron. Res. Announc. Math. Sci. 16, 1–8 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ovsienko, V., Tabachnikov, S.: Projective Differential Geometry old and New, from Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Cambridge Tracts in Mathematics, 165, Cambridge: Cambridge University Press, 2005

  18. Robbins D., Rumsey H.: Determinants and alternating sign matrices. Adv. Math. 62, 169–184 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schwartz R.: The pentagram map. Experiment. Math. 1, 71–81 (1992)

    MATH  MathSciNet  Google Scholar 

  20. Schwartz R.: The pentagram map is recurrent. Experiment. Math. 10, 519–528 (2001)

    MATH  MathSciNet  Google Scholar 

  21. Schwartz R.: Discrete monodromy, pentagrams, and the method of condensation. J. of Fixed Point Theory and Appl. 3, 379–409 (2008)

    Article  MATH  Google Scholar 

  22. Speyer D.: Perfect matchings and the octahedron recurrence. J. Alg. Combin. 25, 309–348 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Suris, Yu.: The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics, 219, Basel: Birkhauser Verlag, 2003

  24. Thurston, W.: Three-dimensional Geometry and Topology. Vol. 1, Princeton Mathematical Series, 35, Princeton, NJ: Princeton University Press, 1997

  25. Tongas A., Nijhoff F.: The Boussinesq integrable system: compatible lattice and continuum structures. Glasg. Math. J. 47, 205–219 (2005)

    Article  MathSciNet  Google Scholar 

  26. Veselov A.: Integrable mappings. Russ. Math. Surv. 46(5), 1–51 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  27. Weinstein A.: The local structure of Poisson manifolds. J. Diff. Geom. 18, 523–557 (1983)

    MATH  Google Scholar 

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Authors and Affiliations

  1. CNRS, Institut Camille Jordan, Université Lyon 1, Villeurbanne Cedex, 69622, France

    Valentin Ovsienko

  2. Department of Mathematics, Brown University, Providence, RI, 02912, USA

    Richard Schwartz

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Serge Tabachnikov

Authors
  1. Valentin Ovsienko
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  2. Richard Schwartz
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  3. Serge Tabachnikov
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Correspondence to Valentin Ovsienko.

Additional information

Communicated by N.A. Nekrasov

Dedicated to the memory of V. Arnold

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Ovsienko, V., Schwartz, R. & Tabachnikov, S. The Pentagram Map: A Discrete Integrable System. Commun. Math. Phys. 299, 409–446 (2010). https://doi.org/10.1007/s00220-010-1075-y

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  • DOI: https://doi.org/10.1007/s00220-010-1075-y

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