Invariant tori for area-preserving maps with ultra-differentiable perturbation and Liouvillean frequency

TL;DR

This paper proves the existence of invariant tori in area-preserving maps with ultra-differentiable perturbations and Liouvillean frequencies, extending KAM theory to more general conditions without arithmetic restrictions.

Contribution

It establishes invariant tori existence for ultra-differentiable perturbations with Liouvillean frequencies, removing previous arithmetic restrictions and developing new KAM techniques.

Findings

Invariant tori exist under ultra-differentiable perturbations.

Results hold for any irrational frequency without arithmetic conditions.

New KAM methods handle ultra-differentiability and Liouvillean frequencies.

Abstract

We prove the existence of invariant tori to the area-preserving maps defined on $ \mathbb{R}^2\times\mathbb{T} $ \begin{equation*} \bar{x}=F(x,\theta), \qquad \bar{\theta}=\theta+\alpha\, \,(\alpha\in \mathbb{R}\setminus\mathbb{Q}), \end{equation*} where $ F $ is closed to a linear rotation, and the perturbation is ultra-differentiable in $ \theta\in \mathbb{T},$ which is very closed to $C^{\infty}$ regularity. Moreover, we assume that the frequency $\alpha$ is any irrational number without other arithmetic conditions and the smallness of the perturbation does not depend on $\alpha$. Thus, both the difficulties from the ultra-differentiability of the perturbation and Liouvillean frequency will appear in this work. The proof of the main result is based on the Kolmogorov-Arnold-Moser (KAM) scheme about the area-preserving maps with some new techniques.