KAM below $\mathbf C^n$

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.