Locally Maximizing orbits for multi-dimensional Twist maps and Birkhoff billiards

Abstract

In this work we consider variational properties of exact symplectic twist maps $T$ that act on the cotangent bundle of a torus, or on a ball bundle over a sphere. An example of such a map is the well-known Birkhoff billiard map corresponding to smooth convex hypersurfaces. In this work we will focus on the important class $\mathcal{M}$ of orbits of $T$ which are locally maximizing with respect to the variational principle associated with a generating function of the symplectic twist map. Our first goal is to give geometric and variational characterization for orbits in the class $\mathcal M$. The billiard map is known to have two different generating functions, which brings forth a natural question: to compare the properties of these two generating functions. While our motivation comes from billiards, we will work in general, and assume that a general twist map $T$ has two generating functions. Thus we consider the orbits of $T$ which are locally maximizing with respect to either of the generating functions. We formulate a geometric criterion that guarantees that two generating functions of the same twist map have the same class of locally maximizing orbits, and we will show that the two generating functions for the Birkhoff billiard map do, in fact, satisfy this criterion. The proof of this last property will rely on the Sinai-Chernov formula from geometric optics and billiard dynamics.