Maximally chaotic dynamical systems

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.