TY - JOUR

T1 - The Limit Point of the Pentagram Map and Infinitesimal Monodromy

AU - Aboud, Quinton

AU - Izosimov, Anton

N1 - Publisher Copyright:
© 2020 The Author(s). Published by Oxford University Press. All rights reserved.

PY - 2022/4/1

Y1 - 2022/4/1

N2 - 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.

AB - 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.

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U2 - 10.1093/imrn/rnaa258

DO - 10.1093/imrn/rnaa258

M3 - Article

AN - SCOPUS:85127958935

VL - 2022

SP - 5383

EP - 5397

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 7

ER -