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

TL;DR

This paper studies the global analytic properties of a new class of generic real-analytic potentials in nearly-integrable Hamiltonian systems, revealing the structure of phase space and the universality of the dynamics near resonances.

Contribution

Introduces the class al Gal n_s of potentials and characterizes the phase space structure, including non-resonant, simply resonant, and non-perturbative sets, with a universal form for resonant Hamiltonians.

Findings

Most of the phase space is filled with primary KAM tori.

The simply resonant set admits a universal standard form for the averaged Hamiltonians.

The non-perturbative set's dynamics are described by a non-nearly-integrable system.

Abstract

We introduce a new class $\mathbb{G}^n_s$ of generic real analytic potentials on $\mathbb{T}^n$ and study global analytic properties of natural nearly-integrable Hamiltonians $\frac12 |y|^2+\varepsilon f(x)$, with potential $f\in \mathbb{G}^n_s$, on the phase space $\varepsilon = B \times \mathbb{T}^n$ 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 $e^{-c K}$ (where $K$ is the maximal size of resonances considered) by primary maximal KAM tori; a `simply resonant set' of measure $\sqrt{\varepsilon} K^a$ and a third set of measure $\varepsilon K^b$ 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 $k$}.