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The pentagram map takes a planar polygon $$P$$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $$P$$ . This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $$P$$ is a Poncelet polygon, then the image of $$P$$ under the pentagram map is projectively equivalent to $$P$$ . In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.
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The Limit Point of the Pentagram Map and Infinitesimal Monodromy
Abstract The pentagram map takes a planar polygon $$P$$ to a polygon $$P^{\prime }$$ 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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- Award ID(s):
- 2008021
- PAR ID:
- 10225284
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnaa258
