论文标题
闭合引理和KAM正常形式
Closing lemma and KAM normal form
论文作者
论文摘要
在本文中，我们开发了一种基于KAM正常形式关闭引理问题的方法。新方法与现有的$ C^1 $扰动方法和光谱方法有所不同，并且可以处理高规律性，高维案例甚至Riemannian度量扰动。此外，证明是建设性有效的。我们将方法应用于庞加莱的原始几乎可集成的设置，并确认了一些较弱的构想的新旧猜想。首先，对于庞加莱的几乎可集成系统的原始设置，我们证明，对于典型的扰动，周期性的轨道在渐近的范围内渐近，因为扰动的大小趋于零。其次，我们证明，平坦的圆环上的典型平滑光滑扰动具有渐近密集的周期性轨道，这部分解决了一个空旷的问题，因为Pugh-Robinson的$ C^1 $ C^1 $ clateming-Clating Lemma。第三，我们证明，对于典型的哈密顿量或接触椭圆形流的接触扰动，在能量水平上具有渐近密集的轨道，这增强了最近对强闭合引理的研究，并且在这种情况下也证实了一部分鱼类的猜想。我们还讨论了我们的模型与最近关于物理学多体定位的研究的关系。
In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing $C^1$ perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincaré and confirm several old and new conjectures with weak formulations. First, for Poincaré's original setting of nearly integrable systems, we prove that for typical perturbations, periodic orbits are asymptotically dense as the size of perturbation tends to zero. Second, we prove that typical smooth perturbation of the geodesic flow on the flat torus has asymptotically dense periodic orbits, which partially solves an open problem since Pugh-Robinson's $C^1$-closing lemma. Third, we prove that for typical Hamiltonian or contact perturbation of the geodesic flows of the ellipsoid has asymptotically dense orbit on the energy level, which enhances the recent researches on strong closing lemma, and also confirms partially a conjecture of Fish-Hofer in this setting. We also discuss the relation of our models to the recent researches on many-body localization in physics.
