  The maximally chaotic dynamical systems (DS) are the systems which have
nonzero Kolmogorov entropy. The Anosov C-condition defines a reach class of
hyperbolic dynamical systems that have exponential instability of the phase
trajectories and positive Kolmogorov entropy and are therefore maximally
chaotic. The interest in Anosov-Kolmogorov 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, the N-body system in Newtonian
gravity and the relaxation phenomena in stellar systems and the Black hole
thermodynamics. 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 presented.
