Open Access
2009/2010 Poncelet Paire and the Twist Map Associated to the Poncelet Billiard
Artur O. Lopes, Marcos Sebastiani
Real Anal. Exchange 35(2): 355-374 (2009/2010).


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.


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Artur O. Lopes. Marcos Sebastiani. "Poncelet Paire and the Twist Map Associated to the Poncelet Billiard." Real Anal. Exchange 35 (2) 355 - 374, 2009/2010.


Published: 2009/2010
First available in Project Euclid: 22 September 2010

MathSciNet: MR2683603

Primary: 37C25 , 37D50 , 37E40 , 37E45

Keywords: Periodic points , Poncelet billiard , Poncelet pairs , rotation number , twist maps

Rights: Copyright © 2009 Michigan State University Press

Vol.35 • No. 2 • 2009/2010
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