Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.

中文翻译：

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KAM 曲线的唯一性

经典 KAM 理论保证了对于足够光滑的非退化近可积系统的不变环的正测度集的存在。当被视为频率的函数时，这个不变的环面集合称为系统的 KAM 曲线。受限于解析规律，我们为这些对象获得了很强的唯一性。特别是，我们证明了 KAM 曲线完全表征了底层系统。我们还展示了对 KAM 曲线具有某些共同特征的系统的一些动态影响。