Integrability of the pentagram map. (English) Zbl 1282.14061

Soloviev, Fedor

Duke Math. J. 162, No. 15, 2815-2853 (2013).

The pentagram map was introduced by R. Schwartz in [Exp. Math. 1, No. 1, 71–81 (1992; Zbl 0765.52004)] as a map defined on convex polygons understood up to projective equivalence on the real projective plane. This map sends an \(i\)-th vertex to the intersection of 2 diagonals: \((i-1, i+1)\) and \((i, i+2)\). The definition implies that this map is invariant under projective transformations. This map stands at the intersection of many branches of mathematics: dynamical systems, integrable systems, projective geometry, and cluster algebras. This paper deals with the integrability of the pentagram map. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper the author proves algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose he shows that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresponding spectral curve, and the dynamics on its Jacobian. He also proves that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever-Phong’s universal formula.

Reviewer: Ahmed Lesfari (El Jadida)

MSC:

14H70	Relationships between algebraic curves and integrable systems
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K20	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions

Keywords:

integrable systems; pentagram map

Citations:

Zbl 0765.52004

References:

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