论文标题
具有超差异扰动和liouvillean频率
Invariant tori for area-preserving maps with ultra-differentiable perturbation and Liouvillean frequency
论文作者
论文摘要
我们证明了在$ \ mathbb {r}^2 \ times \ mathbb {t} $上定义的区域保护地图不变的托里的存在 \ begin {equation*} \ bar {x} = f（x，θ）， \ qquad \barθ=θ+α\，\，（in \ in \ mathbb {r} \ setMinus \ mathbb {q}）， \ end {equation*}其中$ f $在线性旋转上封闭，并且在\ m athbb {t}中，扰动在$θ\中是非常差异的，它非常接闭，非常接近$ c^{\ infty} $。此外，我们假设频率$α$是没有其他算术条件的任何非理性数字，并且扰动的较小性不取决于$α$。因此，在这项工作中，扰动和liouvillean频率的超差异性的困难都将出现。主要结果的证明是基于Kolmogorov-Arnold-Moser（KAM）方案，内容涉及采用一些新技术的区域保护地图。
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.
