Symmetry Reduction and Rotation Numbers for Poncelet maps
Poncelet maps are circle maps constructed geometrically for a pair of nested ellipses; they are related to the classic billiard map on an elliptical domain when the orbit has an elliptical caustic. Here we show how the rotation number of the elliptical billiard map can be obtained from a symmetry generated from the flow of a pendulum Hamiltonian system. When such a symmetry flow has a global cross section, we previously showed that there are coordinates in which the map takes a reduced, skew-product form on a covering space. In particular, for elliptic billiard map this gives an explicit form for the rotation number of each orbit. We show that the family Poncelet maps on a pencil of ellipses is conjugate to a corresponding family of billiard maps, and thus the Poncelet maps inherit the one-parameter family of continuous symmetries. Such a pencil has a single parameter, the pencil eccentricity, which becomes the modulus of the Jacobi elliptic functions used to construct a covering space that simultaneously simplifies all of the Poncelet maps. The rotation number of the Poncelet map for any element of a pencil can then be written in terms of elliptic functions as well. An implication is that the rotation number of the pencil has a monotonicity property: it is monotone increasing as the caustic ellipse shrinks. 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? Explicit conditions are given for a general rotation numbers and we see how they are related to Cayley's classic porism theorem.