The limit point of the pentagram map and infinitesimal monodromy

The pentagram map takes a planar polygon $P$ to a polygon $P^{\mathrm{\prime }}$ 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.