Locally maximizing orbits for the non-standard generating function of convex billiards and applications

Abstract

Given an exact symplectic map $T$ of a cylinder with a generating function $H$ satisfying the so-called negative twist condition, $H_{12}>0$, we study the locally maximizing orbits of $T$, that is, configurations which are local maxima of the action functional $\sum_n H(q_n,q_{n+1})$. We provide a necessary and sufficient condition for a configuration to be locally maximizing. Using it, we consider a situation where $T$ has two generating functions with respect to two different sets of symplectic coordinates. We suggest a simple geometric condition which guarantees that the set of locally maximizing orbits with respect to both of these generating functions coincide. As the main application we show that the two generating functions for planar Birkhoff billiards satisfy this geometric condition. We apply it to get the following result: consider a centrally symmetric curve $\gamma$, for which the Birkhoff billiard map has a rotational invariant curve $\alpha$ of $4$-periodic orbits. We prove that a certain $L^2$-distance between $\gamma$ and its "best approximating" ellipse can be bounded from above in terms of the measure of the complement of the set filled by locally maximizing orbits lying between $\alpha$ and the boundary of the phase cylinder. Moreover, this estimate is sharp, giving an effective version of a recent result on Birkhoff conjecture for centrally symmetric curves. We also get a similar bound for arbitrary curves $\gamma$, that relates the measure of the complement of the set of locally maximizing orbits with the $L^2$-distance between $\gamma$ and its "best approximating" circle.