Artin Billiard Exponential Decay of Correlation Functions

TL;DR

This paper investigates the exponential decay of correlation functions in Artin billiard systems, a class of hyperbolic dynamical systems on the Lobachevsky plane, using symbolic dynamics and geometric methods.

Contribution

It provides a detailed calculation of correlation functions in Artin billiards and demonstrates their exponential decay over time, linking geometric and group-theoretic approaches.

Findings

Correlation functions decay exponentially with time.

Artin billiard exhibits strong chaotic properties.

Method combines symbolic dynamics and differential geometry.

Abstract

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.