Poncelet pairs and the Twist Map associated to the Poncelet Billiard

TL;DR

This paper investigates the properties of Poncelet pairs and associated twist maps, using Aubry-Mather theory and continued fractions to analyze the rotation number and count Poncelet pairs for fixed curves.

Contribution

It establishes a formula for the number of n-Poncelet pairs related to the gcd function and explores the dynamics of area-preserving twist maps using advanced mathematical theories.

Findings

Number of n-Poncelet pairs is e(n)/2, where e(n) counts coprime numbers less than n.

The study connects Poncelet pairs with twist map dynamics and rotation numbers.

Provides estimates for the derivative of the rotation number using continued fractions.

Abstract

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 .