### Abstract

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from Z into 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].

Original language | English (US) |
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Pages (from-to) | 409-446 |

Number of pages | 38 |

Journal | Communications In Mathematical Physics |

Volume | 299 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*299*(2), 409-446. https://doi.org/10.1007/s00220-010-1075-y

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*Communications In Mathematical Physics*, vol. 299, no. 2, pp. 409-446. https://doi.org/10.1007/s00220-010-1075-y

**The Pentagram Map : A Discrete Integrable System.** / Ovsienko, Valentin; Schwartz, Richard; Tabachnikov, Sergei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Pentagram Map

T2 - A Discrete Integrable System

AU - Ovsienko, Valentin

AU - Schwartz, Richard

AU - Tabachnikov, Sergei

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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. A research announcement of this work appeared in [16].

AB - 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. A research announcement of this work appeared in [16].

UR - http://www.scopus.com/inward/record.url?scp=77955846727&partnerID=8YFLogxK

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U2 - 10.1007/s00220-010-1075-y

DO - 10.1007/s00220-010-1075-y

M3 - Article

VL - 299

SP - 409

EP - 446

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -