The limit point of the pentagram map

TL;DR

This paper investigates the limit point of the pentagram map, providing an explicit algebraic description of its coordinates as roots of specific cubic polynomials, advancing understanding of its asymptotic behavior.

Contribution

It offers a precise algebraic characterization of the pentagram map's limit point, which was previously only known to exist and be unique.

Findings

The limit point can be explicitly described using roots of degree three polynomials.

The pentagram map's convergence to the limit point is exponential.

Provides a new algebraic perspective on the asymptotic behavior of the pentagram map.

Abstract

The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the first paper on the subject, R. Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials.