The algebraic dynamics of the pentagram map

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 dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev's proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J. 162(15) (2013), 2815-2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.

Automated summary

This paper studies the properties of the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2 and proves that the pentagram map on twisted polygons is a discrete integrable system. In the course of the proof, the moduli space of twisted n-gons is constructed, formulas for the pentagram map are derived, and the Lax representation is calculated using characteristic-independent methods.