David J. Grabiner and Jeffrey C. Lagarias
This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2,Z) with respect to the standard fundamental domain F of PSL(2,Z). The cutting sequence for a vertical geodesic \theta+ti is related to a one-dimensional continued fraction expansion for \theta, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. We show that the additive ordinary continued fraction expansion of \theta can be computed from the cutting sequence for a vertical geodesic by a finite automaton, but not vice versa. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space {L,R,J} which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain F up to an isometry of the hyperbolic plane H.