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

Abstract

We introduce a new class ${\mathbb{G}}_s^n$ of generic real analytic potentials on ${\mathbb{T}}^n$ and study global analytic properties of natural nearly-integrable Hamiltonians $\frac{1}{2}|y{|}^2+\epsilon f(x)$, with potential $f\in {\mathbb{G}}_s^n$, on the phase space $\epsilon =B\times {\mathbb{T}}^n$ with $B$ a given ball in ${\mathbb{R}}^n$. The phase space $\mathcal{M}$ 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 $\sqrt{\epsilon }{K}^a$ and a third set of measure $\epsilon K^b$ which is `non perturbative', in the sense that the $H$-dynamics on it can be described by a natural system which is {\sl 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 (labelled by the resonance index $k\in {\mathbb{Z}}^n$) can be put into a universal form (which we call `Generic Standard Form'), whose main analytic properties are controlled by {\sl only one parameter, which is uniform in the resonance label $k$}.