On the length spectrum of analytic convex billiard tables

Billiard maps are a type of area-preserving twist maps and, thus, they inherit a vast num-ber of properties from them, such as the Lagrangian formulation, the study of rotational invariant curves, the types of periodic orbits, etc. For strictly convex billiards, there exist at least two (p, q)-periodic orbits. We study the billiard properties and the results found up to now on measuring the lengths of all the (p, q)-trajectories on a billiard. By using a standard Melnikov method, we find that the first order term of the difference on the lengths among all the (p, q)-trajectories orbits is exponentially small in certain perturba-tive settings. Finally, we conjecture that the difference itself has to be exponentially small and also that these exponentially small phenomena must be present in many more cases of perturbed billiards than those we have presented on this work