论文标题
Arnold扩散中的拓扑阴影方法：较弱的扭转和多个时间尺度
Topological Shadowing Methods in Arnold Diffusion: Weak Torsion and Multiple Time Scales
论文作者
论文摘要
考虑一个具有横向同型通道的任何偶数尺寸的正常双曲线歧管的符号图。我们开发了一个拓扑阴影的论点，以证明沿不变的流形的阿诺德扩散的存在，从而阴影不变的歧管带来的内部动力学以及由稳定且不稳定的叶子引起的外部动力学。在这样做的过程中，我们基于正确对齐窗口的方法和所谓的横向交流参数，在[26]中概括了Gidea和de la llave的想法。我们的证明允许不变流形的动力学仅满足不均匀的扭曲条件，最重要的是，对于应用，分离的分裂在某些方向上很小，因此相关的动作中相关的漂移非常慢。扩散发生在具有非微小分裂的歧管的方向上。此外，我们还提供扩散时间的估计值。
Consider a symplectic map which possesses a normally hyperbolic invariant manifold of any even dimension with transverse homoclinic channels. We develop a topological shadowing argument to prove the existence of Arnold diffusion along the invariant manifold, shadowing some iterations of the inner dynamics carried by the invariant manifold and the outer dynamics induced by the stable and unstable foliations. In doing so, we generalise an idea of Gidea and de la Llave in [26], based on the method of correctly aligned windows and a so-called transversality-torsion argument. Our proof permits that the dynamics on the invariant manifold satisfy only a non-uniform twist condition, and, most importantly for applications, that the splitting of separatrices be small in certain directions and thus the associated drift in actions very slow; diffusion occurs in the directions of the manifold having non-small splitting. Furthermore we provide estimates for the diffusion time.
