Geometrically Markov geodesics on the modular surface

The Morse method of coding geodesics on a surface of constant negative curvature consists of recording the sides of a given fundamental region cut by the geodesic. For the modular surface with the standard fundamental region each geodesic (which does not go to the cusp in either direction) is represented by a bi-infinite sequence of non-zero integers called its geometric code.
In this paper we show that the set of all geometric codes is not a finite-step Markov chain, and identify a maximal 1-step topological Markov chain of admissible geometric codes which we call, as well as the corresponding geodesics, geometrically Markov. We also show that the set of geometrically Markov codes is the maximal symmetric 1-step topological Markov chain of admissible geometric codes, and obtain an estimate from below for the topological entropy of the geodesic flow restricted to this set.