Universal aspects of Malti-ergodic phase space structure in many body hamiltonian dynamics

一般哈密顿系统的不变性测度揭示了相空间中的许多奇异性，其中哈密顿动力学的非双曲性起着至关重要的作用。这种无限测度的出现迄今为止已经在无限遍历理论的框架中进行了研究[Aaronson，1997]。[A]我们研究了非双曲混沌的谱指数、关联函数、李雅普诺夫指数和Lempel-Ziv复杂度等特性，并成功地导出了这些特性之间的精确关系。此外，我们已经成功地制定了精确的Levy扩散方程的反常扩散在相空间的非双曲哈密顿动力学。这些结果表明，汉密尔顿动力学的测度理论结构与双曲系统中的普通SRB测度有很大的不同。[B]在团簇形成过程中发现了无限遍历性，普遍观察到Weibull型分布。我们成功地从基于Nekoroshev定理的标度理论解释了这些统计规律，并揭示了这些统计特征中的大偏差性质。这些结果表明，非常强烈的Arnold扩散可以在无限遍历理论的框架下进行表征。

The invariantt measure of generic hamiltonian systems reveals a number of singularities in phase space, where the non-hyperbolicity of hamiltonian dynamics plays an essential role. The appearance of such infinite measure has been studied so far in the framework of infinite ergodic theory [Aaronson, 1997]. [A] We have studied many characteristics of non-hyperbolic chaos, such as the spectral indeces, correlation functions, Lyapunov exponents, and the Lempel-Ziv Complexity, and we have succeeded to derive the exact interrelations among those characteristics. Furthermore, we have succeeded to formulate the exact Levy diffusion equation in terms of anomalous diffusion in the phase space of non-hyperbolic hamiltonian dynamics. These results suggest that the measure-theoretical structure of hamilton dynamics is quite different from the ordinary SRB measures in hyperbolic systems. [B] The infinite ergodic aspects are discovered in the cluster formation process, where the Weibull type distributions are universally observed. We have succeeded to explain those statistical laws from the scaling theory based on the Nekoroshev theorem, and also revealed that the large deviation properties are realized in those statistical features. These results suggest very strongly that the Arnold diffusion may be characterized in the framework of the infinite ergodic theory.