标题： 无限维Kolmogorov定理和几乎周期呼吸子的构造 显示英文标题

标题： An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers

摘要： 在本文中，我们基于Bourgain型的非共振频率或甚至更弱的条件，提出了两个无限维的Kolmogorov定理。 更准确地说，在无限维哈密顿系统的一个非退化条件下，我们证明了具有普遍指定频率的全维KAM环面的存在性，该频率独立于任何谱渐进行为。 作为应用，我们证明了对于由\[\frac{{{{\rm d}^2}{x_n}}}{{{\rm d}{t^2}}} + V'\left( {{x_n}} \right) = \varepsilon_n {W'\left( {{x_{n + 1}} - {x_n}} \right) - \varepsilon_{n-1}W'\left( {{x_n} - {x_{n - 1}}} \right)} ,\;\;n \in \mathbb{Z},\]描述的一类扰动网络或甚至更一般的扰动网络，只要局部势$ V $和耦合势$ W $满足某些假设，就存在频率保持的准周期呼吸子。 特别是，这得到了Aubry-MacKay猜想[8,21]的第一个频率保持版本。 显示英文摘要

摘要： In this paper, we present two infinite-dimensional Kolmogorov theorems based on nonresonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a nondegenerate condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with the universally prescribed frequency that is independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by \[\frac{{{{\rm d}^2}{x_n}}}{{{\rm d}{t^2}}} + V'\left( {{x_n}} \right) = \varepsilon_n {W'\left( {{x_{n + 1}} - {x_n}} \right) - \varepsilon_{n-1}W'\left( {{x_n} - {x_{n - 1}}} \right)} ,\;\;n \in \mathbb{Z},\] or even for more general perturbed networks, almost periodic breathers with frequency-preserving do exist, provided that the local potential $ V $ and the coupling potential $ W $ satisfy certain assumptions. In particular, this yields the first frequency-preserving version of the Aubry-MacKay conjecture [8,21].