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

在本工作中，我们成功地推导了高维哈密顿系统中轨道扩散(Arnold扩散)的标度律，并成功地得到了典型哈密顿系统(Froeschle模型)中不变环面和混沌界面层中的对数扩散和次扩散。此外，还证明了这些结果可以用无限遍历定理(DKA定理)在理论上表述，该定理适用于附近可积哈密顿情形下的Arnold网。从这一理论结果中，我们成功地系统地了解了哈密顿系统中慢动力学的各个方面，例如，1/f谱涨落和Nekholoshev时间的对数-威布尔分布函数，它们与标度平均涨落的大偏差性质的普适分布函数(Mittag-Leffler分布)密切相关。我们还成功地推广了DKA定理，解释了隐藏在哈密顿系统中无限遍历现象背后的详细测量理论结构，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的过渡区域的表观衰减。所有这些结果都是独立的、新的结果，它们不仅促进了哈密顿动力学遍历理论研究的未来发展，也促进了耗散动力学遍历理论研究的进一步发展。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.