Homoclinic orbits to invariant sets of quasi-integrable exact maps

The resonant tori of an integrable system are destroyed by a perturbation. If the Hamiltonian is convex, they give rise to hyperbolic lower-dimensional invariant tori or to Aubry-Mather invariant sets. Bolotin has proved the existence of homoclinic orbits to the hyperbolic tori but not to the Aubry-Mather invariant sets. We solve this problem and obtain, for each resonant frequency, the existence of an invariant set with homoclinic orbits.