On singularity confinement for the pentagram map

Details On singularity confinement for the pentagram map On singularity confinement for the pentagram map

2011-10-04

arxiv.org/abs/1110.0868v2

Mathematics

Combinatorics

37 pages, 25 figures. Various changes, mostly to section 8

Scientific paper

The pentagram map, introduced by R. Schwartz, is a birational map on the configuration space of polygons in the projective plane. We study the singularities of the iterates of the pentagram map. We show that a "typical" singularity disappears after a finite number of iterations, a confinement phenomenon first discovered by Schwartz. We provide a method to bypass such a singular patch by directly constructing the first subsequent iterate that is well-defined on the singular locus under consideration. The key ingredient of this construction is the notion of a decorated (twisted) polygon, and the extension of the pentagram map to the corresponding decorated configuration space.

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