Twistless KAM Tori, Quasi Flat Homoclinic Intersections, and Other Cancellations in the Perturbation Series of Certain Completely Integrable Hamiltonian Systems. A Review

摘要

The author discusses the invariant tori and the splitting of their homoclinic stable and unstable manifolds for a special class of quasi integrable hamiltonian systems. The author applies and extends in the considered class, the important ideas of Melnikov and Eliasson respectively on the theory of low dimensional invariant tori and their manifolds and on the cancellations behind the convergence of the formal perturbation series for the invariant tori of maximal dimension i.e. for the KAM tori. The author also points out the analogy of the methods with those used in quantum field theory, particularly in the renormalization group approaches.