MAXIMALLY CHAOTIC DYNAMICAL SYSTEMS OF ANOSOV-KOLMOGOROV

摘要

The maximally chaotic K-systems are dynamical systems which have nonzero Kol-mogorov entropy. On the other hand, the hyperbolic dynamical systems that fulfil the Anosov C-condition have exponential instability of phase trajectories, mixing of all orders, and positive Kolmogorov entropy. Therefore, the C-condition defines a rich class of maximally chaotic systems which span an open set in the space of all dynamical systems. The interest in Anosov-Kolmogorov C-K 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 nonlinear dynamics of the Yang-Mills field, the iV-body system in Newtonian gravity and the relaxation phenomena in stellar systems, and the Black Hole thermodynamics. In this respect, of special interest are C-K systems that are defined on Reimannian manifolds of negative sectional curvatures and on high-dimensional tori. 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 will be reviewed.