  The hyperbolic Anosov C-systems have exponential instability of their
trajectories and as such represent the most natural chaotic dynamical systems.
Of special interest are C-systems which are defined on compact surfaces of the
Lobachevsky plane of constant negative curvature. An example of such system has
been introduced in a brilliant article published in 1924 by the mathematician
Emil Artin. The dynamical system is defined on the fundamental region of the
Lobachevsky plane which is obtained by the identification of points congruent
with respect to the modular group, a discrete subgroup of the Lobachevsky plane
isometries. The fundamental region in this case is a hyperbolic triangle. The
geodesic trajectories of the non-Euclidean billiard are bounded to propagate on
the fundamental hyperbolic triangle. In this article we shall expose his
results, will calculate the correlation functions/observables which are defined
on the phase space of the Artin billiard and demonstrate the exponential decay
of the correlation functions with time. We use Artin symbolic dynamics, the
differential geometry and group theoretical methods of Gelfand and Fomin.
