On Poncelet's maps

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 these types 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 an n is an element of N such that P-n = Id, or in other words, Poncelet's map is conjugate 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 is not conjugate to a rotation. Finally, we also provide a new proof of Poncelet's result based on dynamical and computational tools. (c) 2010 Elsevier Ltd. All rights reserved.