Invariant Tori for Area-Preserving Maps with Ultra-differentiable Perturbation and Liouvillean Frequency

We prove the existence of invariant tori to the area-preserving maps defined on \( \mathbb {R}^2\times \mathbb {T} \) ${\mathbb{R}}^2\times \mathbb{T}$ \(\begin{aligned} \overline{x}=F(x,\theta ), \qquad \overline{\theta }=\theta +\alpha \, \,(\alpha \in \mathbb {R} {\setminus }\mathbb {Q}), \end{aligned}\) $$\begin{array}{c}\hfill \overline{x}=F(x,\theta ),\overline{\theta }=\theta +\alpha (\alpha \in \mathbb{R}\backslash \mathbb{Q}),\end{array}$$ where F is related to a linear rotation, and the perturbation is ultra-differentiable in \( \theta \in \mathbb {T},\) $\theta \in \mathbb{T},$ which is very closed to \(C^{\infty }\) $C^{\infty }$ regularity. Moreover, we assume that the frequency \(\alpha \) $\alpha$ is any irrational number without other arithmetic conditions and the smallness of the perturbation does not depend on \(\alpha \) $\alpha$ . 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.