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 $\mathbb{R}^2\times\mathbb{T}$ \begin{equation*} \bar{x}=F(x,θ), \qquad \barθ=θ+α\, \,(α\in \mathbb{R}\setminus\mathbb{Q}), \end{equation*} where $F$ is closed to a linear rotation, and the perturbation is ultra-differentiable in $θ\in \mathbb{T},$ which is very closed to $C^{\infty}$ regularity. Moreover, we assume that the frequency $α$ is any irrational number without other arithmetic conditions and the smallness of the perturbation does not depend on $α$. 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.

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