Explicit Examples of Coding Geodesics on Surfaces of Constant Negative Curvature

Abstract

It is well-known that there are various ways of representing geodesics on a surface M of constant negative curvature. There are two different methods on the bottom line: one is geometric coding and the other is arithmetic coding. The former is the so-called Morse method which is coding a geodesic by the cutting sequence as it passes a fixed set of curves on M. The latter, Artin method, is the construction using concatenating two sequences, obtained by using a suitable boundary expansion, of two endpoints of a lift of the geodesic (a geodesic in the unit disc). In this thesis, we investigate the more mysterious Artin method for specific examples of surfaces and show that we obtain a sofic system by using Artin method.