# The pentagram map, Poncelet polygons, and commuting difference operators

```@article{Izosimov2019ThePM,
title={The pentagram map, Poncelet polygons, and commuting difference operators},
author={Anton Izosimov},
journal={Compositio Mathematica},
year={2019},
volume={158},
pages={1084 - 1124}
}```
• A. Izosimov
• Published 25 June 2019
• Mathematics
• Compositio Mathematica
The pentagram map takes a planar polygon \$P\$ to a polygon \$P'\$ whose vertices are the intersection points of consecutive shortest diagonals of \$P\$. This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if \$P\$ is a Poncelet polygon, then the image of \$P\$ under the pentagram map is projectively equivalent to \$P\$. In the present paper, we show that in the convex…
5 Citations
The pentagram map, introduced by Schwartz [The pentagram map. Exp. Math.1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex
We survey definitions and integrability properties of the pentagram maps on generic plane polygons and their generalizations to higher dimensions. We also describe the corresponding continuous limit
• Mathematics
• 2022
. We employ the Poisson-Lie group of pseudo-diﬀerence operators to deﬁne lattice analogs of classical W m -algebras. We then show that the so-constructed algebras coincide with the ones given by

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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
The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants \$E_1, O_1, E_2, O_2,\dots\$ By analyzing
• Mathematics
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The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for
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We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension \$d\$ there are
• Mathematics
• 2011
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
Inspired by a method from the theory of ordinary differential equations, the paper constructs roughly n algebraically independent invariants for the map, when it is defined on the space of n-gons, which strongly suggest that the pentagram map is a discrete completely integrable system.
• Mathematics
Annales de l'Institut Fourier
• 2019
In this paper we define a generalization of the pentagram map to a map on twisted polygons in the Grassmannian space Gr(n;mn). We define invariants of Grassmannian twisted polygons under the natural