Global properties of generic real–analytic nearly–integrable Hamiltonian systems

Abstract

We introduce a new class Gsn of generic real analytic potentials on Tn and study global analytic properties of natural nearly–integrable Hamiltonians 12|y|2+εf(x), with potential f∈Gsn, on the phase space M=B×Tn with B a given ball in Rn. The phase space M can be covered by three sets: a ‘non–resonant’ set, which is filled up to an exponentially small set of measure e−cK (where K is the maximal size of resonances considered) by primary maximal KAM tori; a ‘simply resonant set’ of measure εKa and a third set of measure εKb which is ‘non perturbative’, in the sense that the H–dynamics on it can be described by a natural system which is not nearly–integrable. We then focus on the simply resonant set – the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) – and show that on such a set the secular (averaged) 1 degree–of–freedom Hamiltonians (labeled by the resonance index k∈Zn) can be put into a universal form (which we call ‘Generic Standard Form’), whose main analytic properties are controlled by only one parameter, which is uniform in the resonance label k.