The problem of coding geodesics by symbolic dynamics has been of interest for many years. The classic examples come from Morse [12] and Artin [3] stretching as far back as the 1920's. Morse studied the geodesics on very specific surfaces and Artin came up with a symbolic dynamics for geodesics on the modular surface. Even before that, as early as 1898, Hadamard [9] studied geodesics on negatively curved spaces. In the late 1960's Adler and Weiss [2] found a symbolic dynamics for automorphisms of the 2-torus. In the early 1970's, Bowen [4] studied Axiom A flows and found a symbolic dynamic coding for these flows
We look at a compact orbifold quotient of the hyperbolic plane by a group of symmetries of a certain tiling. For a 4-valent tiling by lean polygons, we make a correspondence between geodesics and certain sequences of tiles. This is used to find a symbolic dynamic coding for geodesics of the compact orbifold
Bowen's work includes geodesics on compact orbifolds with negative curvature but does not give an explicit coding. Our methods are similar to those used by Fried [5], [6] for noncompact quotients of the hyperbolic plane
