The limit point of the pentagram map and infinitesimal monodromy

TL;DR

This paper links the limit point of the pentagram map to the infinitesimal monodromy, revealing that Glick's operator measures how a perturbed polygon fails to close, thus connecting geometric dynamics with monodromy concepts.

Contribution

It interprets Glick's operator as the infinitesimal monodromy of a polygon, providing a new geometric understanding of the pentagram map's limit behavior.

Findings

Glick's operator is the infinitesimal monodromy of a polygon.

The limit point of the pentagram map can be characterized via this monodromy.

A natural perturbation of polygons measures the non-closure via the operator.

Abstract

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. The orbit of a convex polygon under this map is a sequence of polygons which converges exponentially to a point. Furthermore, as recently proved by Glick, coordinates of that limit point can be computed as an eigenvector of a certain operator associated with the polygon. In the present paper we show that Glick's operator can be interpreted as the infinitesimal monodromy of the polygon. Namely, there exists a certain natural infinitesimal perturbation of a polygon, which is again a polygon but in general not closed; what Glick's operator measures is the extent to which this perturbed polygon does not close up.