Integrability of the Pentagram Map

TL;DR

This paper proves algebraic-geometric integrability of the pentagram map for all polygons, extending previous results by analyzing spectral curves and symplectic structures, thus establishing a comprehensive integrability framework.

Contribution

It demonstrates algebraic-geometric integrability of the pentagram map for both twisted and closed polygons, using spectral curve analysis and zero-curvature equations.

Findings

Proves algebraic-geometric integrability for all monodromies.

Shows the pentagram map as a discrete zero-curvature equation.

Establishes the equivalence of Poisson brackets with universal symplectic structures.

Abstract

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 generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show 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. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever-Phong's universal formula.