Herman's converse KAM mechanism revisited

TL;DR

This paper revisits Herman's counterexample to the KAM theorem, demonstrating the essential role of the bump function through advanced estimates and renormalization techniques, clarifying the conditions under which the theorem holds.

Contribution

It proves the necessity of the bump function in Herman's counterexample using improved Siegel--Brjuno estimates and a parameter-dependent resonance renormalization.

Findings

The bump function is necessary for certain hyperbolic perturbations.

Improved Siegel--Brjuno estimates are crucial for the proof.

Resonance renormalization clarifies the role of the bump function.

Abstract

In his celebrated counterexample to the KAM theorem, Herman introduced a perturbation of an integrable system consisting of two components: a hyperbolic term and a bump function. He also remarked that it was unclear whether the bump function was truly necessary. In this note, we prove that the bump function is indeed necessary when more natural hyperbolic perturbations are considered. The proof of this necessity relies on an improved Siegel--Brjuno estimate and a parameter-dependent renormalization of resonances within the direct KAM method.