Maximally chaotic dynamical systems of Anosov-Kolmogorov
  The maximally chaotic K-systems are dynamical systems which have nonzero
Kolmogorov 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, countable Lebesgue spectrum and positive
Kolmogorov entropy. 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 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 C-K systems that are defined on Reimannian 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.
