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

Abstract Given an exact symplectic map T of a cylinder with generating function H satisfying the so-called negative twist condition, H 12 > 0 , we study locally maximising orbits that is, configurations which are local maxima action functional $\sum_n H(q_n,q_{n+1})$?> ∑ n ( q , + 1 stretchy="false">) . We provide necessary and sufficient condition for configuration to be maximising. Using it, consider situation where has two functions respect different sets coordinates. suggest simple geometric guarantees set both these coincide. As main application show planar Birkhoff billiards satisfy this condition. apply it get following result: centrally symmetric curve γ billiard rotational invariant α four-periodic orbits. prove certain L 2 -distance between its ‘best approximating’ ellipse can bounded from above in terms measure complement filled by lying boundary phase cylinder. Moreover, estimate is sharp, giving effective version recent result on conjecture curves (Bialy Mironov 2022 Ann. Math. 196 389–413). also similar bound arbitrary relates circle.