KAM \`{a} la R

TL;DR

This paper simplifies R"ussmann's new KAM theory variant for analytic perturbations of constant vector fields on a torus, providing a potentially shortest complete proof for such perturbations.

Contribution

It adapts R"ussmann's KAM scheme to a simpler setting, making the proof more accessible and concise for perturbations of integrable vector fields.

Findings

Simplified KAM proof for perturbations of constant vector fields

Accessible iteration scheme for analytic perturbations

Shortest known complete proof for this class of problems

Abstract

Recently R\"ussmann proposed a new new variant of KAM theory based on a slowly converging iteration scheme. It is the purpose of this note to make this scheme accessible in an even simpler setting, namely for analytic perturbations of constant vector fields on a torus. As a side effect the result may be the shortest complete KAM proof for perturbations of integrable vector fields available so far.