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We consider the quantisation of the Artin dynamical system defined on thefundamental region of the modular group. In classical regime the geodesic flowin 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 orderto investigate the influence of the classical chaotic behaviour on thequantum-mechanical properties of the Artin system we calculated thecorresponding thermal quantum-mechanical correlation functions. It wasconjectured by Maldacena, Shenker and Stanford that the classical chaos can bediagnosed in thermal quantum systems by using an out-of-time-order correlationfunction as well as the square of the commutator of operators separated intime. We demonstrated that the two- and four-point correlation functions of theLouiville-like operators decay exponentially with a temperature dependentexponent. As conjectured the square of the commutator of the Louiville-likeoperators separated in time grows exponentially, similar to the exponentialdivergency of trajectories in the classical regime. The corresponding exponentdoes not saturate the maximal growth condition.

Correlation Functions of Classical and Quantum Artin System defined on Lobachevsky Plane and Scrambling Time