Moser's twist theorem revisited
Yi Liu, Lin Wang
TL;DR
This work revisits Moser's twist theorem for area-preserving twist maps with invariant circles of constant-type frequency $\alpha$, seeking almost optimal regularity by adapting Katznelson–Ornstein techniques. The authors develop a framework based on Type I/II $\kappa$-vectors, three existence criteria, and a distortion-control scheme that leverages both circle-diffeomorphism methods and Mather’s action-minimizing theory. By establishing detailed bounds on distortions and their growth, they prove that an invariant circle with frequency $\alpha$ persists under perturbations of class $C^{3+\varepsilon}$ (for any $\varepsilon'>0$ with $\varepsilon'<\varepsilon$) and provide a mechanism to extend these results to all large indices via a recursive inductive argument. The approach clarifies near-optimal regularity limits in this setting and connects invariant circle persistence to both KO1 techniques and Aubry–Mather structures, with implications for two-degree-of-freedom Hamiltonian systems.
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 $α$ persists under a small perturbation (dependent on $α$) of class $C^{3+ε}$. This result was initially established independently by Herman and Rüssmann in 1983. Our method differs essentially from their approaches.