  We consider the quantisation of the Artin dynamical system defined on the
fundamental region of the modular group. In classical regime the geodesic flow
in the fundamental region represents one of the most chaotic dynamical systems,
it has mixing of all orders, Lebesgue spectrum and non-zero Kolmogorov entropy.
As a result, the classical correlation functions decay exponentially. In order
to investigate the influence of the classical chaotic behaviour on the
quantum-mechanical properties of the Artin system we calculated the
corresponding thermal quantum-mechanical correlation functions. It was
conjectured by Maldacena, Shenker and Stanford that the classical chaos can be
diagnosed in thermal quantum systems by using an out-of-time-order correlation
function as well as the square of the commutator of operators separated in
time. We demonstrated that the two- and four-point correlation functions of the
Louiville-like operators decay exponentially with a temperature dependent
exponent. As conjectured the square of the commutator of the Louiville-like
operators separated in time grows exponentially, similar to the exponential
divergency of trajectories in the classical regime. The corresponding exponent
does not saturate the maximal growth condition.
