  We give a general review on the application of Ergodic theory to the
investigation of dynamics of the Yang-Mills gauge fields and of the
gravitational systems, as well as its application in the Monte Carlo method and
fluid dynamics. In ergodic theory the maximally chaotic dynamical systems
(MCDS) can be defined as dynamical systems that have nonzero Kolmogorov
entropy. The hyperbolic dynamical systems that fulfil the Anosov C-condition
belong to the MCDS insofar as they have exponential instability of their phase
trajectories and positive Kolmogorov entropy. It follows that the C-condition
defines a rich class of MCDS that span over an open set in the space of all
dynamical systems. The large class of Anosov-Kolmogorov MCDS is realised on
Riemannian manifolds of negative sectional curvatures and on high-dimensional
tori. The interest in MCDS is rooted in the attempts to understand the
relaxation phenomena, the foundations of the statistical mechanics, the
appearance of turbulence in fluid dynamics, the non-linear dynamics of
Yang-Mills field and gravitating N-body systems as well as black hole
thermodynamics. Our aim is to investigate classical- and quantum-mechanical
properties of MCDS and their role in the theory of fundamental interactions.
