Moser's twist theorem revisited

TL;DR

This paper revisits Moser's twist theorem, providing a simplified proof that invariant circles with certain frequencies persist under small perturbations with near-optimal regularity, improving understanding of twist map stability.

Contribution

The paper offers a new, shorter proof of the persistence of invariant circles in twist maps with near-optimal regularity, differing from previous methods by Herman and Rüssmann.

Findings

Invariant circles with constant type frequency persist under small $C^{3+\epsilon}$ perturbations.

The proof simplifies existing approaches to Moser's twist theorem.

The regularity condition for persistence is nearly optimal.

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 $\alpha$ persists under a small perturbation (dependent on $\alpha$) of class $C^{3+\epsilon}$. This result was initially established independently by Herman and R\"{u}ssmann in 1983. Our method differs essentially from their approaches.