Symbolic dynamics for 4-valent lean tilings of the hyperbolic plane

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.