Singular KAM Theory

TL;DR

This paper introduces a new singular KAM theory to analyze the measure of invariant tori in nearly-integrable Hamiltonian systems, providing bounds that support longstanding conjectures about their size relative to perturbation strength.

Contribution

It develops a novel singular KAM approach to estimate the measure of invariant tori near resonances, confirming conjectures for generic Hamiltonian systems.

Findings

Lower bounds on measure of invariant tori match conjectures

Supports exponential smallness of non-torus set in 2D systems

Provides measure estimates for higher-dimensional systems

Abstract

The question of the total measure of invariant tori in analytic, nearly--integrable Hamiltonian systems is considered. In 1985, Arnol'd, Kozlov and Neishtadt, in the Encyclopaedia of Mathematical Sciences \cite{AKN1}, and in subsequent editions, conjectured that in $n=2$ degrees of freedom the measure of the non torus set of general analytic nearly--integrable systems away from critical points is exponentially small with the size $\e$ of the perturbation, and that for $n\ge 3$ the measure is, in general, of order $\e$ (rather than $\sqrt\e$ as predicted by classical KAM Theory). In the case of generic natural Hamiltonian systems, we prove lower bounds on the measure of primary and secondary invariant tori, which are in agreement, up to a logarithmic correction, with the above conjectures. The proof is based on a new {\sl singular} KAM theory, particularly designed to study analytic properties in neighborhoods of the secular separatrices generated by the perturbation at simple resonances.