Wednesday, April 03, 2024, 11:00am - 12:00pm
We study families of Poncelet maps on a pencil of ellipses, which are conjugate to a corresponding family of billiard maps. This conjugation unveils a one-parameter family of continuous symmetries that Poncelet maps inherit. We show that each pencil of ellipses has a single parameter, the pencil eccentricity, which becomes the modulus of the Jacobi elliptic functions used to construct a covering space. We find that the rotation number of the Poncelet map for any element of a pencil can then be written in terms of elliptic integrals. The resulting expression for the rotation number gives an explicit condition for Poncelet porisms, the parameters for which the rotation number is rational. For such parameters, an orbit of the corresponding Poncelet map is periodic: it forms a polygon for any initial point. These universal parameters also solve the inverse problem: given a rotation number, which member of a pencil has a Poncelet map with that rotation number? We also analyze how rotation numbers are related to Cayley's conditions.
Location: PMA 12.166