We consider a system of infinitely many penduli on an m-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasi-periodic tori with transverse heteroclinic orbits. We implement this approach in an infinite dimensional setting, both in the space of bounded ${\mathbb{Z}}^m$-sequences and in spaces of decaying ${\mathbb{Z}}^m$-sequences. Key steps in the proof are an invariant manifold theory for hyperbolic tori and a Lambda Lemma for infinite dimensional coupled map lattices with decaying interaction.

中文翻译：

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无限格子上哈密顿系统中的阿诺德扩散

我们考虑一个弱耦合的 m 维晶格上的无限多个钟摆系统。对于晶格中的任何规定路径，对于合适的耦合，我们为这个无限自由度的哈密顿系统构造轨道，该轨道沿路径在附近的摆锤之间传递能量。只要弱耦合强烈衰减，我们就允许它是近邻耦合或长程耦合。能量转移由阿诺德扩散机制给出，该机制依赖于原始的 V.I 阿诺德方法：构造一系列具有横向异宿轨道的双曲不变准周期环面。我们在无限维设置中实现这种方法，都在有界的空间中${\mathbb{Z}}^{米}$- 序列和腐烂的空间${\mathbb{Z}}^{米}$- 序列。证明的关键步骤是双曲环面的不变流形理论和具有衰减相互作用的无限维耦合映射晶格的 Lambda 引理。