Multi ergodicity in nearly integrable Hamiltonian systems and large deviation properties of infinite ergodic systems

在本文中，我们提出了在高维Hamilton系统中导出轨道扩散（Arnold扩散）的标度律的挑战，并成功地得到了在一个典型的Hamilton系统（Froeschle模型）中不变环面和混沌之间的界面层中的对数扩散和次扩散.进一步证明了这些结果在理论上可以用无穷遍历定理（DKA定理）来表述，该定理适用于Arnold网在可积Hamilton附近的情形.从这个理论结果，我们已经成功地系统地了解在哈密顿系统的慢动力学的各个方面，例如，1/f谱涨落和对数威布尔分布函数的Nekholoshev时间，这是密切相关的一个普遍的分布函数（米塔格-莱弗勒分布）的大偏差性质的标度平均波动。我们还成功地扩展了DKA定理，阐明了隐藏在Hamilton系统中无穷遍历现象背后的详细测度理论结构，例如，具有分形标度的相空间（蘑菇台球系统）的精细结构，以及附近可积系统中环面和混沌之间瞬态区域的明显衰减.这些结果都是独立的、新的，不仅对哈密顿动力学，而且对耗散动力学的遍历理论研究都有着重要的推动作用。事实上，奇异非混沌吸引子和异宿湍流揭示了与我们目前工作中提到的相同类型的慢动力学。

In the present works, we have challenged to derive the scaling law of the orbital diffusion(Arnold diffusion) in high dimensional Hamiltonian systems, and succeeded to obtain the log-diffusion as well as the sub-diffusion in the interface layer between the invariant torus and chaos in a typical Hamiltonian system(Froeschle's model). Furthermore, it was proved these results can be theoretically formulated by use of the infinite ergodic theorem(DKA theorem), which is applicable to the Arnold webs in the nearby integrable Hamiltonian cases. From this theoretical result, we have succeeded to understand systematically the various aspects of slow dynamics in Hamiltonian systems, for instance, 1/f spectral fluctuations and the log-Weibull distribution function for the Nekholoshev time, which are closely connected to a universal distribution function(the Mittag-Leffler distribution) for the large deviation property of scaled average fluctuations. We have also succeeded to extend the DKA theorem to elucidate the detailed measure theoretical structure hidden behind the infinite ergodic phenomena in Hamiltonian systems, for instance, the fine structure of phase space(Mushroom billiard system) with a fractal scaling and the apparent attenuation in the transient regime between torus and chaos in nearby integrable systems. All these results are independent and new ones that stimulate the future development of the ergodic theoretical studies not only in Hamiltonian dynamics but also in the dissipative dynamics. Indeed, Strange-Nonchaotic Attractors and Heteroclinic turbulence reveal the same type of slow dynamics as these mentioned in our present works.