The pentagram integrals for Poncelet families

The pentagram map is now known to be a discrete integrable system. We show that the integrals for the pentagram map are constant along Poncelet families. That is, if P1P1 and P2P2 are two polygons in the same Poncelet family, and ff is a monodromy invariant for the pentagram map, then f(P1)=f(P2)f(P1)=f(P2). Our proof combines complex analysis with an analysis of the geometry of a degenerating sequence of Poncelet polygons.