Moser's twist theorem revisited

Abstract

Inspired by the work of Katznelson and Ornstein, we present a short way to achieve the almost optimal regularity in Moser's twist theorem. Specifically, for an integrable area-preserving twist map, the invariant circle with a given constant type frequency $\hat{I}\pm$ persists under a small perturbation (dependent on $\hat{I}\pm$) of class $C^{3+\hat{I}\mu }$. This result was initially established independently by Herman and Rüssmann in 1983. Our method differs essentially from their approaches.