Non-degenerate Liouville tori are KAM stable

TL;DR

This paper proves that non-degenerate, non-resonant quasi-periodic tori in Hamiltonian systems are KAM stable regardless of whether their frequencies are Diophantine or Liouville, extending stability results.

Contribution

It establishes KAM stability for non-degenerate, non-resonant tori with Liouville frequencies, answering a recent open question and broadening the class of stable invariant tori.

Findings

Non-resonant tori are KAM stable if Kolmogorov non-degenerate.

Stability holds for Diophantine and Liouville frequencies.

Regularity of tori matches the Hamiltonian's smoothness class.

Abstract

In this short note, we prove that a quasi-periodic torus, with a non-resonant frequency (that can be Diophantine or Liouville) and which is invariant by a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov non-degenerate. When the Hamiltonian is smooth (respectively Gevrey-smooth, respectively real-analytic), the in-variant tori are smooth (respectively Gevrey-smooth, respectively real-analytic). This answers a question raised in a recent work by Eliasson, Fayad and Krikorian ([EFK]). We also take the opportunity to ask other questions concerning the stability of non-resonant invariant quasi-periodic tori in (analytic or smooth) Hamiltonian systems.