KAM below $\mathbf C^n$

TL;DR

This paper extends KAM theory to rotational flows on n-dimensional tori with lower regularity than previously thought, showing that $L^2$ derivatives suffice and challenging longstanding regularity conjectures.

Contribution

It demonstrates that KAM theory applies under weaker regularity conditions, specifically with derivatives in $L^2$, thus disproving the minimal $C^n$ regularity conjecture.

Findings

KAM theory applies with $L^2$ derivatives of order $n$

Disproves the conjecture that $C^n$ regularity is necessary for KAM

Shows applicability to flows with weaker regularity than $C^n$

Abstract

We consider the KAM theory for rotational flows on an $n$-dimensional torus. We show that if its frequencies are diophantine of type $n-1$, then Moser's KAM theory with parameters applies to small perturbations of weaker regularity than $C^n$. Derivatives of order $n$ need not be continuous, but rather $L^2$ in a certain strong sense. This disproves the long standing conjecture that $C^n$ is the minimal regularity assumption for KAM to apply in this setting while still allowing for Herman's $C^{n-\epsilon}\!$-counterexamples.