论文标题
无限晶格中哈密顿系统中的阿诺德扩散
Arnold diffusion in Hamiltonian systems on infinite lattices
论文作者
论文摘要
我们认为在$ m $二维的晶格中，无限多个Penduli的系统具有较弱的耦合。对于晶格中的任何规定路径，对于合适的耦合，我们为这种无限自由度系统的轨道构建轨道，该系统沿着路径沿着附近的Penduli之间传递能量。我们允许弱耦合是近代最近的邻居或远距离的近距离，只要它强烈腐烂即可。能量的转移是由依赖于原始V的Arnold扩散机制给出的。IArnold方法：构建具有横向异质轨道轨道的一系列双曲线不变的准膜状Tori。我们在无限的维度设置中实现了这种方法，包括在有限的$ \ mathbb {z}^m $ - 序列的空间和衰减的$ \ mathbb {z}^m $ seperences的空间中。证明的关键步骤是双曲线托里的不变歧管理论和无限尺寸耦合的地图晶格的Lambda引理，并具有衰减的相互作用。
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 quasiperiodic 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.
