Poncelet pairs and the Twist Map associated to the Poncelet Billiard

Details Poncelet pairs and the Twist Map associated to the Poncelet Billiard Poncelet pairs and the Twist Map associated to the Poncelet Billiard

2007-05-28

arxiv.org/abs/0705.4057v2

Mathematics

Dynamical Systems

Scientific paper

We show that for a fixed curve $K$ and for a family of variables curves $L$, the number of $n$-Poncelet pairs is $\frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd $ (m,n)=1$. The curvee $K$ do 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 .

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