Poncelet pairs and the twist map associated to the Poncelet billiard (Q541379)

For a family of indexed differentiable curves that are realized as the boundary of convex and connected domains having the same indexing, the authors exactly determine the number of \(n\)-Poncelet pairs and show that it is essentially half of Euler's totient function evaluated at \(n\). To arrive at this result, they consider a corresponding billiard problem and use area preserving twist maps. As additional technical machinery they use Aubry-Mather theory and rotation numbers of invariant curves. By using properties of continued fractions, the paper concludes with estimations on the derivative of twist maps.