Poncelet Paire and the Twist Map Associated to the Poncelet Billiard

Abstract

Consider a fixed differentiable curve $K$ (which is the boundary of a convex domain) and a family indexed by $\lambda\in [0,1]$ of differentiable curves $L = L_\lambda$ such that they are the boundary of convex (connected) domains $A_\lambda$. Suppose that for $\lambda_1<\lambda_2$ we have $A_{\lambda_1}\subset A_{\lambda_2}$. Then, the number of $n$-Poncelet pairs is given by $\frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd$(m,n)=1$. The curve $K$ does not have to be part of the family. In order to show this result we consider an associated billiard transformation and a twist map which preserves area. We use Aubry-Mather theory and the rotation number of invariant curves to obtain our main result. In the last section we estimate the derivative of the rotation number of a general twist map using some properties of the continued fraction expansion.