在庞斯莱的地图上
On Poncelet's maps
摘要：
鉴于两个椭圆，一个围绕另一个，蓬斯莱介绍了一个地图$P$从外部一个本身使用的切线的内部椭圆。这个过程可以推广到任何两个光滑的，嵌套的和凸的椭圆形，我们称这种类型的地图庞斯莱的地图。我们还记得他证明了1814年左右的动力系统语言：在两个椭圆的情况下，当旋转数的P是合理的存在一个自然数，使P^n是身份，或者换句话说，蓬斯莱的地图是共轭的合理旋转。本文研究了一般的Poncelet映射，并给出了代数椭圆的几个例子，其中相应的Poncelet映射具有有理旋转数，并且不与旋转共轭。最后，我们还利用动力学工具对Poncelet的结果给出了一个新的证明。
英文摘要：
Given two ellipses, one surrounding the other one, Poncelet introduced a map $P$ from the exterior one to itself by using the tangent lines to the interior ellipse. This procedure can be extended to any two smooth, nested and convex ovals and we call this type of maps Poncelet's maps. We recall what he proved around 1814 in the dynamical systems language: In the two ellipses case and when the rotation number of P is rational there exists a natural number such that P^n is the identity, or in other words, the Poncelet's map is conjugated to a rational rotation. In this paper we study general Poncelet's maps and give several examples of algebraic ovals where the corresponding Poncelet's map has a rational rotation number and it is not conjugated to a rotation. Finally, we also provide a new proof of Poncelet's result based on dynamical tools.
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