A Renormalization Proof of the KAM Theorem for Non-Analytic Perturbations

TL;DR

This paper employs a Renormalization Group approach to prove the KAM theorem for non-analytic perturbations, demonstrating convergence of solutions through analytic approximations.

Contribution

It introduces a novel RG-based proof of the KAM theorem applicable to non-analytic perturbations with high-order continuous derivatives.

Findings

Convergent solutions for non-analytic perturbations established.

Method extends KAM theorem applicability beyond analytic cases.

Provides a new framework for analyzing perturbations with limited smoothness.

Abstract

We shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non-analytic perturbation (the latter will be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which the perturbations are analytic approximations of the original one. We shall finally show that the sequence of the approximate solutions will converge to a differentiable solution of the original problem.