The Pentagram Map

A Discrete Integrable System

Valentin Ovsienko, Richard Schwartz, Sergei Tabachnikov

Research output: Contribution to journalArticle

50 Citations (Scopus)

Abstract

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].

Original languageEnglish (US)
Pages (from-to)409-446
Number of pages38
JournalCommunications In Mathematical Physics
Volume299
Issue number2
DOIs
StatePublished - Jan 1 2010

Fingerprint

Pentagram
Integrable Systems
Discrete Systems
Poisson Structure
polygons
Polygon
Boussinesq Equations
Projective Transformation
Quasi-periodic Motion
Monodromy
Integrability
Modulo

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Ovsienko, Valentin ; Schwartz, Richard ; Tabachnikov, Sergei. / The Pentagram Map : A Discrete Integrable System. In: Communications In Mathematical Physics. 2010 ; Vol. 299, No. 2. pp. 409-446.
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The Pentagram Map : A Discrete Integrable System. / Ovsienko, Valentin; Schwartz, Richard; Tabachnikov, Sergei.

In: Communications In Mathematical Physics, Vol. 299, No. 2, 01.01.2010, p. 409-446.

Research output: Contribution to journalArticle

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