On singularity confinement for the pentagram map

TL;DR

This paper investigates the singularity confinement phenomenon in the pentagram map, demonstrating that typical singularities resolve after finite iterations and introducing a method to bypass singularities using decorated polygons.

Contribution

It extends the pentagram map to decorated polygons and provides a systematic way to bypass and analyze singularities in its iterates.

Findings

Typical singularities disappear after finite iterations

Method to construct well-defined iterates on singular loci

Extension of the pentagram map to decorated polygons

Abstract

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.