Sharp Nekhoroshev estimates for the three body problem around periodic orbits

We construct a Nekhoroshev-like result of stability with sharp constants for the planar three body problem, both in the planetary and in the restricted circular case, by using the periodic averaging technique. Our constructions can be generalized to any near-integrable hamiltonian system whose unperturbed hamiltonian is quasi-convex. The dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account. This allows for a deep analytical understanding of the limits of such techniques in insuring Nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases. Finally, two examples with concrete values are considered, one for the planetary case and one for the restricted one.