The pentagram integrals for Poncelet families (Q478641)

The pentagram map is a projectively natural map defined on the space of \(n\)-gons. It is now known to be a discrete integrable system. In this work, it is shown that the integrals of the pentagram map are constant along Poncelet families. That is, if \(P_1\) and \(P_2\) are two polygons in the same Poncelet family and \(f\) is a monodromy invariant for the pentagram map, then \(f(P_1) = f(P_2)\). The proof combines complex analysis with an analysis of the geometry of a degenerating sequence of Poncelet polygons.