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We prove the existence of invariant tori to the area-preserving maps definedon $ \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 theperturbation is ultra-differentiable in $ \theta\in \mathbb{T},$ which is veryclosed to $C^{\infty}$ regularity. Moreover, we assume that the frequency$\alpha$ is any irrational number without other arithmetic conditions and thesmallness of the perturbation does not depend on $\alpha$. Thus, both thedifficulties from the ultra-differentiability of the perturbation andLiouvillean frequency will appear in this work. The proof of the main result isbased on the Kolmogorov-Arnold-Moser (KAM) scheme about the area-preservingmaps with some new techniques.

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