Abstract The interest in maximally chaotic dynamical systems is associated with the attempts to understand the relaxation phenomena, the foundation of the statistical mechanics, the appearance of turbulence in fluid dynamics, the non-linear dynamics of the Yang–Mills field, as well as the dynamical properties of gravitating N-body systems and the Black hole thermodynamics. In this respect of special interest are Anosov–Kolmogorov C-K systems that are defined on Riemannian manifolds of negative sectional curvature and on a high-dimensional tori. Here we shall review the classical- and quantum-mechanical properties of maximally chaotic dynamical systems, the application of the C-K theory to the investigation of the Yang–Mills dynamics and gravitational systems, as well as their application in the Monte Carlo method. The maximally chaotic K-systems are dynamical systems that have nonzero Kolmogorov entropy. On the other hand, the hyperbolic dynamical systems that fulfil the Anosov C-condition have exponential instability of their phase trajectories, mixing of all orders, countable Lebesgue spectrum and positive Kolmogorov entropy. The C-condition defines a rich class of maximally chaotic systems that span an open set in the space of all dynamical systems.

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最大混沌动力系统

摘要 对最大混沌动力学系统的兴趣与理解弛豫现象、统计力学的基础、流体动力学中湍流的出现、杨-米尔斯场的非线性动力学以及引力 N 体系​​统的动力学特性和黑洞热力学。在这方面特别感兴趣的是 Anosov-Kolmogorov CK 系统，它定义在负截面曲率的黎曼流形和高维环面上。在这里，我们将回顾最大混沌动力学系统的经典力学和量子力学性质，CK 理论在杨-米尔斯动力学和引力系统研究中的应用，以及它们在 Monte Carlo 方法中的应用。最大混沌 K 系统是具有非零 Kolmogorov 熵的动力系统。另一方面，满足 Anosov C 条件的双曲动力系统具有其相轨迹的指数不稳定性、所有阶的混合、可数 Lebesgue 谱和正 Kolmogorov 熵。C 条件定义了丰富的一类极大混沌系统，它们跨越所有动力系统空间中的一个开集。