Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

Abstract

This paper describes the cutting sequences of geodesic flow on the modular surface H/P SL(2, Z) with respect to the standard fundamental domain F = {z = x + iy : − 1 2 ≤ x ≤ 1 2 and |z| ≥ 1} of P SL(2, Z). The cutting sequence for a vertical geodesic {θ + it : t > 0} is related to a onedimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. 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. We give conversion methods between the cutting sequence for the vertical geodesic {θ + it : t > 0}, the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first ℓ symbols when given as input the first O(ℓ) symbols of the ACF expansion, which takes time O(ℓ 2 ) and space O(ℓ).

Symbolic dynamics for the modular surface and beyond

“…(Morse called the coding sequences admissible line elements, and some authors [S4,GL] referred to them as cutting sequences.) We assume that the geodesic intersects D and choose an initial point on it inside D. After exiting D, the geodesic enters a neighboring image of D through the side labeled, say, by g 1 (see Figure 2).…”

“…An ambiguity in assigning a Morse code occurs whenever a geodesic passes though a vertex of D: such geodesics have more than one code, and closed geodesics have nonperiodic codes along with periodic ones (see [GL,KU1] for relevant discussions).…”

“…The case when a geodesic passes through the corner ρ of F was described to a great extent in [GL,§7]. Such a geodesic has multiple codes obtained by approximating it by general position geodesics which pass near the corner ρ slightly higher or slightly lower.…”

“…However, in spite of its geometric nature and seeming simplicity, this method has two major shortcomings: if the fundamental region has vertices inside H, the geodesics passing through any of those vertices have multiple codes, and the space of all admissible codes has a complicated structure. (We believe that in general the space is not a topological Markov chain: corresponding results for the modular surface with the standard fundamental region were proved in [GL,KU1]; see Section 4. )…”

Geodesic Flow on the Quotient Space of the Action of ⟨z + 2;−1/z 〉 on the Upper Half Plane

“…Several techniques for developing codes have been introduced [4], [5] and [8] and one of the most natural codes is geometric codes appearing in [15]. In this note, we introduce geometric codes for geodesic flow on M c2 .…”