The Pentagram Map: A Discrete Integrable System

By Valentin Ovsienko, Richard Evan Schwartz, Serge Tabachnikov. 2008

146 citations recorded by the source corpus.

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].

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/432021303bf78ff0047bfacda3ca82693988b334

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Citations in the corpus

Polynomial cubic differentials and convex polygons in the projective plane (2014, background): “Ovsienko, Schwartz, and Tabachnikov introduced in [OST10] a natural”
2-frieze patterns and the cluster structure of the space of polygons (2010, methodology): “In [25, 20] two different systems of coordinates were introduced and used to study the space of twisted n-gons and the transformation under the pentagram map.”
Liouville–Arnold integrability of the pentagram map on closed polygons (2011, methodology): “In this section, we explain the proof of the main result in our paper [15], the Liouville-Arnold integrability of the pentagram map on the space of twisted n-gons.”
Higher pentagram maps, weighted directed networks, and cluster dynamics (2011, background): “Recently, it has attracted a considerable attention: see [11, 16, 17, 20, 21, 22, 26, 27, 28, 29, 30] for various aspects of the pentagram map and related topics.”
Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics (2015, background): “e above theorem is stated in [MGOT12], and was proved in a more general form in [MGOST14]. We explain below in details the explicit construction of the isomorphism. The construction also uses idea of [OST10]. Construction of the ismorphisms of Theorem 1.15 and 1.16. We explain here how from an eleme”
Introduction to Cluster Algebras (2018, background): “ex of B lies at the intersection of two consecutive “shortest” diagonals of A. The pentagram map was introduced by R. Schwartz [48] and extensively studied by V. Ovsienko, Schwartz and S. Tabachnikov [42, 43]. In particular, they demonstrated that the pentagram map is a completely integrable system.”
INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS (2014, unknown): “It was shown in [38] that the pentagram map was intimately related to the so-called octahedral recurrence (also known as the discrete Hirota equation), and it was conjectured in [31,32] that the pentagram map was related to cluster transformations.”
The pentagram map and Y-patterns (2010, methodology): “ly, if Ais indexed by Zthen the x-coordinates of T(A) are x′′ j = ( xj+1 1−xj+3xj+2 1−xj−1xj−2 , jeven xj−1 1−xj−3xj−2 1−xj+1xj+2 , jodd As observed by V. Ovsienko, R. Schwartz, and S. Tabachnikov in [2], the products xjxj+1 are themselves cross ratios. In fact, xjxj+1 equals the cross ratios used i”