On Poncelet's maps

TL;DR

This paper explores the dynamics of Poncelet's maps for nested convex ovals, extending classical results for ellipses, providing examples with rational rotation numbers not conjugate to rotations, and offering a new proof using dynamical systems techniques.

Contribution

It generalizes Poncelet's classical results to broader convex ovals, presents new examples with unique dynamical properties, and introduces a novel dynamical proof of Poncelet's theorem.

Findings

Existence of Poncelet's maps with rational rotation numbers not conjugate to rotations.

Extension of classical Poncelet's theorem to general convex ovals.

A new proof of Poncelet's theorem using dynamical systems methods.

Abstract

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.