Appears in collection : The Pentagram Map and Ramifications

One of the first results from Schwartz about the pentagram map is the collapsing of convex polygons. Much later, Glick found a formula for the coordinates of the collapse point. This collapsing behavior reappears in other dyncamics, and the goal of this talk is to understand why. To do so, we'll consider the general setting of polygonal dynamics, defined for polygons in any projective spaces. We give a coordinate system on the moduli space, using generalized cross-ratios. If the dynamics admits a scaling symmetry this yields a formula for the generalized Glick operator. To do this, we use the formalism of infinitesimal monodromies, developped by Aboud and Izosimov. As a by-product, we automatically get preserved quantities (the monodromy invariants), which paves the way for integrability. Then we prove the collapsing phenomenon under a periodicity assumption on the moduli space, and present ideas to prove it in the quasi-periodic case.