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 TMF, 2018, Volume 197, Number 2, Pages 230–251 (Mi tmf9549)

Artin billiard: Exponential decay of correlation functions

H. R. Poghosyana, H. M. Babujiana, G. K. Savvidib

a National Science Laboratory, Yerevan, Armenia
b Institute of Nuclear and Particle Physics, National Center for Scientific Research "Demokritos", Athens, Greece

Abstract: The hyperbolic Anosov C-systems have an exponential instability of their trajectories and as such represent the most natural chaotic dynamical systems. The C-systems defined on compact surfaces of the Lobachevsky plane of constant negative curvature are especially interesting. An example of such a system was 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 identifying points congruent with respect to the modular group, the 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. Here, we present Artin's results, calculate the correlation functions/observables defined on the phase space of the Artin billiard, and show that the correlation functions decay exponentially with time. We use the Artin symbolic dynamics, differential geometry, and the group theory methods of Gelfand and Fomin.

Keywords: Anosov C-system, hyperbolic system, Lobachevsky plane, hyperbolic geodesic flow, chaotic system, Artin billiard, correlation function, automorphic function.

 Funding Agency Grant Number European Research Council 644121 This research was supported by the European Union's Horizon 2020 research and innovation program (Marie Skĺodowska-Curie grant agreement No. 644121).

DOI: https://doi.org/10.4213/tmf9549

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English version:
Theoretical and Mathematical Physics, 2018, 197, 1592–1610

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Document Type: Article

Citation: H. R. Poghosyan, H. M. Babujian, G. K. Savvidi, “Artin billiard: Exponential decay of correlation functions”, TMF, 197:2 (2018), 230–251; Theoret. and Math. Phys., 197 (2018), 1592–1610

Citation in format AMSBIB
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