The Limit Point of the Pentagram Map and Infinitesimal Monodromy

Quinton Aboud, Anton Izosimov

Research output: Contribution to journalArticlepeer-review

Abstract

The pentagram map takes a planar polygon P to a polygon P ′ whose vertices are the intersection points of the consecutive shortest diagonals of P. The orbit of a convex polygon under this map is a sequence of polygons that 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.

Original languageEnglish (US)
Pages (from-to)5383-5397
Number of pages15
JournalInternational Mathematics Research Notices
Volume2022
Issue number7
DOIs
StatePublished - Apr 1 2022

ASJC Scopus subject areas

  • Mathematics(all)

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