CUTTING SEQUENCES FOR GEODESIC FLOW ON THE MODULAR SURFACE AND CONTINUED FRACTIONS

DAVID J. GRABINER AND JEFFREY C. LAGARIAS

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 : −\frac{1}{2} ≤ x ≤ \frac{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 one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction expansion, 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 θ can be computed from the cutting sequence for a vertical geodesic by a finite automaton. In the reverse direction, the cutting sequence for a vertical geodesic cannot be computed from the additive continued fraction expansion using any finite automaton, but can be computed by a Turing machine in quadratic time using linear space. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space {L̄, R̄, J̄} 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.

1. Introduction

This paper describes the symbolic dynamics of cutting sequences of geodesics for the standard fundamental domain F = \{z = x + iy : −\frac{1}{2} ≤ x < \frac{1}{2}, |z| ≥ 1\} of the modular group P SL(2, ℤ) acting on the upper half plane ℌ = {z ∈ ℂ : Im(z) > 0}.

We study in particular the cutting sequences of vertical geodesics, {θ + it : t > 0} for θ ∈ ℝ, which are related to a continued fraction expansion introduced in 1850 by Hermite [18] in terms of quadratic forms, and studied by Humbert [20, 21]. Our motivation for studying these cutting sequences arose from the problem of determining the legal continued fraction expansions for the one-dimensional case of a multidimensional continued fraction studied in Lagarias [28]. This expansion, called the Minkowski geodesic continued fraction expansion (MGCF expansion), is based on following a parametrized family of lattice bases in GL(d + 1, ℤ)\GL(d + 1, ℝ) as the parameter t varies; in the one-dimensional case the family associated to θ

is [ \begin{matrix}1 & 0 \\ −θ & t \end{matrix} ]. The MGCF expansion finds good Diophantine approximations in all

dimensions; however, the one-dimensional case does not coincide with the ordinary

Some of this work was done while both authors were visiting the Mathematical Sciences Research Institute; MSRI is supported by NSF Grant DMS-9022140. The first author is supported by an NSF Postdoctoral Fellowship.

continued fraction expansion. We show in §3 that the one-dimensional MGCF expansion of θ is essentially the same as the cutting sequence expansion of the vertical geodesic {θ + it : 0 < t < ∞}. We study the relation of these cutting sequences to the additive continued fraction of θ, which is the variant of the ordinary continued fraction which includes all intermediate convergents. We show that the additive continued fraction expansion of θ can be computed from the MGCF expansion of {θ + it : t > 0} by a finite automaton, but not vice-versa. However the MGCF expansion of θ can be computed from the additive continued fraction expansion of θ by a Turing machine using quadratic time and linear space.

Another motivation for studying cutting sequences on ℌ/P SL(2, ℤ) arises from the study of geodesic flows on constant negative curvature surfaces of finite volume. If Γ is a finitely generated discrete group of conformal isometries of the hyperbolic plane ℌ, such that ℌ/Γ has finite volume, then ℌ/Γ is a compact Riemann surface^{1} minus a finite number of punctures, and geodesic flow on ℌ/Γ contains both topological and conformal information about this Riemann surface. Symbolic codings of geodesics were introduced by Hadamard [14] in 1898 as a way of understanding the complicated motions on geodesics on such surfaces. E. Artin [5] showed in 1924 that there exist dense geodesics on the modular surface ℌ/P SL(2, ℤ), using symbolic encodings with continued fractions, and in 1935 G. Hedlund [16] used Artin’s coding to show that geodesic flow on this surface was ergodic. Some other symbolic encodings introduced were “boundary expansions” by Nielsen [38] and cutting sequences with respect to a fundamental domain of Γ by Koebe [25] and Morse [36]. Cutting sequence expansions can be viewed as a generalization of the reduction theory of indefinite binary quadratic forms to arbitrary finitely-generated Fuchsian groups, see Katok [22]. More recently Bowen and Series [8] and Series [43], [44] used “modified boundary expansions” to obtain a particularly simple symbolic expansion from which strong forms of ergodicity could be deduced. Adler and Flatto [2], [3] used “rectilinear map” codings of geodesic flow on ℌ/Γ to explain why certain specific maps of the interval, such as the continued fraction map and backwards continued fraction map, have invariant measures of a simple form. They showed that these invariant measures are inherited from the invariant measure on geodesic flow (Liouville measure) using a cross-section map followed by a factor map. The codings of Bowen and Series [8], [44] and Adler and Flatto [3] have a particularly simple structure – they are shifts of finite type – and they preserve topological information about the Riemann surface, but they lose conformal information, for they are identical for all Riemann surfaces of the same genus. (See Theorem 8.4 of [3].) Finally we remark that other encodings of geodesic flow on the modular surface ℌ/P SL(2, ℤ) appear in Arnoux [4] and Lagarias and Pollington [29].

CUTTING SEQUENCES ON THE MODULAR SURFACE AND CONTINUED FRACTIONS 3

Cutting sequence encodings are of special interest because they are related to good Diophantine approximations and also apparently encode more information about geodesic flow than some of these other encodings. Adler and Flatto [3, §10] raise the question whether cutting sequence encodings preserve conformal information about the Riemann surface and about the fundamental domain. Consider any hyperbolic polygon P in ℌ which is a fundamental domain for a properly discontinuous group Γ acting on ℌ, such that ℌ/Γ has finite volume, and suppose that P has |P| sides. Let Σ_{P} denote the closed subshift of the shift on |P| letters generated by the cutting sequences for P for geodesic flow on general ℌ/Γ, as defined in §2.2. Adler and Flatto^{2} ask: does Σ_{P} determine P up to conformal isometry and Γ up to hyperbolic conjugacy? We show here that this is the case for the standard fundamental domain F of P SL(2, ℤ).

1There are some mild extra conditions needed on Γ for compactness, see Lehner [30, p. 203–205] for sufficient conditions. It suffices for Γ to have a fundamental region F that is a hyperbolic polygon with a finite number of sides, with only parabolic vertices at infinity.

Our main results are as follows.

1. The additive continued fraction expansion of θ can be computed from either the cutting sequence expansion or the Minkowski geodesic continued fraction expansion of the vertical geodesic θ + it by a finite automaton, but not viceversa. (Theorems 4.1, 4.2 and 6.4.) The Minkowski geodesic continued fraction expansion can be computed from the additive continued fraction expansion by a Turing machine using quadratic time and linear space. (Theorem 5.2.)

2. The set of cutting sequence expansions for (irrational) vertical geodesics Π_{F}^{0} is characterized in terms of forbidden blocks (Theorem 6.1). A generating set of forbidden blocks is enumerated. (Theorems 5.1 and 6.2.). The number of minimal forbidden blocks of length at most k grows exponentially in k. (Theorem 6.3).

3. The cutting sequence expansions for all geodesics comprise (essentially) a closed two-sided subshift Σ_{F} of the full shift on three symbols {L̄, R̄, J̄}. The set of forbidden blocks of Σ_{F} is the same as that for Π_{F}^{0}. (Theorem 7.1). Each symbol sequence in ∑_{F} corresponds to a unique oriented geodesic on ℌ/P SL(2, ℤ), with the exception of the two sequences L̄^{∞} and R̄^{∞}. In the converse direction, each oriented geodesic gives rise to a finite number of shift-equivalence classes of symbol sequences in Σ_{F}. This number is at most eight if it is not a periodic geodesic. (Theorem 7.2). The shift Σ_{F} is not a sofic system. (Theorem 7.3)

4. The shift Σ_{F} characterizes F up to an isometry of ℌ. If P is a hyperbolic polygon of finite area (possibly with some ideal vertices) which is a fundamental domain of a discrete subgroup Γ of P SL(2, ℝ) and if the subshift Σ_{P} is isomorphic to Σ_{F} by a permutation of symbols, then there is a hyperbolic isometry g ∈ P SL(2, ℝ) such that F = gP and P SL(2, ℤ) = gΓg^{−1}. (Theorem 8.1).

Result 1 gives a specific sense in which cutting sequences contain information coded in a more concise form than the ordinary continued fraction expansion. To obtain result 2 we show in §5 that vertical geodesics have special features (not shared by general geodesics) that facilitate characterizing forbidden blocks: any vertical geodesic that hits a P SL(2, ℤ)-translate of a corner of the fundamental domain must have rational θ, and each such geodesic hits at most one such corner, with the exception of those θ ≡ \frac{1}{2}(mod 1), which hits two corners. Furthermore the continued fraction expansion of any rational θ such that {θ + it : t > 0} hits a corner has a symmetry which can be described in terms of the linear fractional transformation

2They raise the question only for the (8g − 4)-sided fundamental polygons discussed in [3], for genus g ≥ 2.

z → Nz with N = [ \begin{matrix}1 & 2 \\ 2 & 1 \end{matrix} ], see Theorem 5.1. In formulating result 3 we define

cutting sequence expansions for geodesics that hit a corner of the fundamental domain F of ℌ/P SL(2, ℤ) to be limits of general-position geodesics that hit no corner. This procedure assigns infinitely many different cutting sequences to certain periodic geodesics. Finally, result 4 shows that the shift Σ_{F} encodes conformal information about F. Nearly all the proofs use the fact that the group is P SL(2, ℤ), and do not appear to easily generalize to other finitely generated Fuchsian groups Γ.

For terminology in symbolic dynamics we follow Adler and Flatto [3] and Lind and Marcus [31].

2. Cutting Sequences

We describe cutting sequences for finite-sided hyperbolic polygons that are the fundamental domain of a given discrete subgroup Γ of P SL(2, ℝ) that acts discontinuously on ℌ, and then specialize to standard fundamental domain F of the modular group P SL(2, ℤ). General references for geodesic flow and for cutting sequences are Adler and Flatto [3] and Series [46].

2.1. Cutting Sequence Shifts. Let ℌ denote the hyperbolic plane, represented as the upper half-plane ℌ = {z = x + iy : y > 0} with hyperbolic line element ds^{2} = \frac{1}{y^{2}} (dx^{2} + dy^{2}) and volume \frac{dxdy}{y^{2}}. An oriented geodesic γ = ⟨θ_{1}, θ_{2}⟩ in ℌ is uniquely specified by its two ideal endpoints θ_{1}, θ_{2} ∈ ℝ ∪ {∞}, with θ_{1} ≠ θ_{2}. It is oriented from its head θ_{1} to its foot θ_{2}. A vertical geodesic ⟨∞, θ⟩ is the vertical line θ + it, with t decreasing from ∞ to 0. Other geodesics are semicircles perpendicular to the real axis with endpoints θ_{1} and θ_{2}, centered at \frac{θ_{1}+θ_{2}}{2} , and oriented to start at θ_{1} and travel to θ_{2}.

Let Γ be a finitely generated discrete subgroup of P SL(2, ℝ) which acts properly discontinuously on the upper half-plane ℌ = {x + iy : y > 0} as a group of linear fractional transformations. We suppose that ℌ/Γ has finite volume with respect to the hyperbolic volume on ℌ. Let P be any finite-sided convex hyperbolic polygon which is a fundamental domain for Γ, i.e. P is such that the set of Γ-translates {gP : g ∈ Γ} tessellates ℌ, and no element g ∈ Γ sends P to itself except the identity. We allow polygons P to have one or more ideal vertices (“cusps”), and they may also have extra vertices, called elliptic vertices, which have an angle of π and which fall in the middle of a side of P regarded as a geometric object, as discussed below. Fundamental domains exist for each such finitely generated Γ, cf. [4, Theorem 10.1.4]. Such polygons P necessarily have an even number of “sides,” where by convention each elliptic vertex divides a side of the geometric object P into two “sides.” If P is to be a fundamental domain for Γ, the “sides” of P must be identified under the action of Γ. Specifically, there is a pairing of the “sides” of P which assigns to each “side” s of P a “side” s′ and an isometry g(s, s′) ∈ Γ such that g(s, s′) maps s to s′. Furthermore (s′)′ = s and

g(s′, s) = (g(s, s′))^{−1},

while if s = s′ then g(s, s) is the identity on s. The pairing indicates how P tessellates ℌ. For g = g(s, s′), the domain of P is adjacent to P in such a way that the “side” labeled s of gP coincides with the “side” labeled s′ of P, while g(Int(P))∩Int(P) = ∅. If s = s′ then necessarily g^{2} is the identity on ℌ, and g is a hyperbolic reflection. The neighborhood set of P

(2.1) N_{Γ}(P) := {g ∈ Γ : g = g(s, s′) for some side s of P} .

is a set of generators of the group Γ. It will comprise the set of symbols used in the cutting sequence encodings of geodesics described below.

The number of elements of N_{Γ}(P) is generally equal to the number of “sides” of P, but can be less if there are two or more elements g(s, s′) that are equal. For convex polygons P this situation can only arise when there are hyperbolic reflections g = g(s, s′) with s ≠ s′, in which case g(s, s′) = g(s′, s). In such a case the “sides” s and s′ must lie on a common geodesic of ℌ and share a common endpoint,^{3} which is then a vertex of P having an angle of π. We call any such vertex an elliptic vertex. These two “sides” s and s′ are then assigned the same generator, so if we erase the elliptic vertex and glue them together, we get a single geometric side of the polygon P regarded as a geometric object. We therefore have: For a convex fundamental domain P of a finitely generated group Γ, the number of elements of N_{Γ}(P) is equal to the number of geometric sides of P. In particular, the number of elements of N_{Γ}(P) may be odd.

As an example, the standard fundamental domain F of Γ = P SL(2, ℤ), is geometrically a triangle with one ideal vertex, but as a fundamental domain for Γ is a quadrilateral having an elliptic vertex at z = i. In this case N_{Γ}(F) has three elements.

We associate to geodesic flow on ℌ/Γ the (two-sided) cutting sequence subshift Σ_{P,Γ} which is a closed subshift of the full shift on |N_{Γ}(P)| letters.

Definition 2.1. The set G_{P,Γ}^{0} of general position geodesics for P consists of those geodesics γ on ℌ which intersect the interior of P, and which are transverse to all {g(P) : g ∈ Γ} in the sense that γ contain no vertex nor part of any side of positive length of any translated polygon {gP : g ∈ Γ}. If P contains ideal vertices (“cusps”), then the endpoints of γ must not coincide with any Γ-translate of any such vertex.

To each γ ∈ G_{P,Γ}^{0}, we assign the doubly-infinite cutting sequence

C(γ) := (…, g_{−2}, g_{−1}, g_{0}, g_{1}, g_{2}, …) (2.2)

in which all g_{i} ∈ N_{Γ}(P), as follows. We first label each “side” s of the fundamental domain P with the label g(s, s′) on its “inside” edge in P. We similarly label all the sides of the translated domain {hP : h ∈ Γ}, so that the edge of hP which is hs has the label g(s, s′). Every “side” is now assigned two labels, on its “inside” edge and “outside” edge, because it abuts two copies of a fundamental domain. (Figure 2.1 below indicates such labels.) The geodesic γ = ⟨θ_{1}, θ_{2}⟩ visits in order a sequence of translates {h_{i}P; i ∈ ℤ} in which h_{i} ∈ Γ and h_{0} is the identity. At a crossing of the edge from h_{i−1}P to h_{i}P we assign the symbol g_{i} on the side of the domain that the

3For non-convex P this may fail to hold.

6 DAVID J. GRABINER AND JEFFREY C. LAGARIAS

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tured geodesic has cutting sequence …, g_{0}, g_{1}, g_{2}, …, starting with g_{1}
from P.

geodesic enters. (This is the convention of Katok [24]; and opposite to that of Adler and Flatto [3, p. 243], who use the symbol g_{i} on the exit edge.) This convention yields:

Lemma 2.1. The cutting sequence (g_{1}, …, g_{j}) follows a geodesic from P to the translated domain P_{j} = h_{j}P with

h_{j} = g_{1}g_{2} ⋯ g_{j}. (2.3)

Proof. The edge h_{i}s of h_{i}P that borders h_{i−1}P is assigned the edge label inside h_{i}P of g_{i} = g(s, s′). Now we have (h_{i−1}g(s, s′)h_{i−1}^{−1})h_{i−1}P = h_{i}P, hence

h_{i−1}g_{i} = h_{i} , (2.4)

see Figure 2.1.

The transversality hypothesis says that if γ intersects some translate g(P) for g ∈ Γ, then γ intersects the interior of g(P), which implies that the expansion (2.2) is well-defined and unique. If γ^{−1} = ⟨θ_{2}, θ_{1}⟩ denotes the reversed geodesic, then γ^{−1} ∈ G_{P,Γ}^{0} has the cutting sequence

C(γ^{−1}) = (…, g′_{−1}, g′_{0}, g′_{1}, …) (2.5)

in which g′_{i} := (g_{−i})^{−1}.

Let Σ_{P,Γ}^{0} denote the set of all cutting sequences C(γ) for γ ∈ G_{P,Γ}^{0}. This set is invariant under the (forward) shift operator σ({g_{i}}) = {g_{i+1}} since

σ(C(γ)) = C(g_{0}γ) ,

and the geodesic g_{0}γ is in G_{P,Γ}^{0}.

Definition 2.2. The orbit or shift-equivalence class [C] of a cutting sequence C is the union of all its forward and backward shifts

[C] := ⋃_{k∈ℤ}σ^{k}(C); (2.6)

i.e., it is the smallest shift-invariant set containing C.

Thus [C(γ)] is contained in Σ_{P}^{0}, but is generally not a closed set. The geodesics on the surface ℌ/Γ are projections of geodesics on ℌ. Since the shift operator on cutting sequences corresponds to a motion of the geodesic by an element of Γ, the orbit [C] is an invariant of the projected geodesic.

Definition 2.3. The cutting sequence shift Σ_{P,Γ} is the closure of Σ_{P,Γ}^{0} in the symbol topology. That is, Σ_{P,Γ} consists of all symbol sequences

(…, g_{−1}, g_{0}, g_{1}, …)

such that every finite block (g_{i}, g_{i+1}, …, g_{i+k}) occurs in some C(γ) for a general position geodesic γ ∈ G_{P,Γ}^{0}.

We now extend the definition of cutting sequences to apply to all geodesics γ that hit the interior of P. The set of cutting sequences C(γ) for a general geodesic γ = ⟨θ′, θ⟩ consists of all symbol sequences that can be obtained as a limit point (in the sequence topology) of a sequence of C(γ_{j}) having γ_{j} ∈ G_{P,Γ}^{0} such that γ_{j} = ⟨θ′_{j}, θ_{j}⟩ with θ′_{j} → θ′ and θ_{j} → θ as j → ∞. There always exist such convergent sequences γ_{j} : j ≥ 1}, because Σ_{P,Γ} is compact in the symbol topology; hence C(γ) ≠ ∅. The set C(γ) may be infinite for some geodesics γ. All cutting sequences in C(γ) are contained in Σ_{P,Γ}; however, the closure operation used in defining Σ_{P,Γ} allows the possibility that Σ_{P,Γ} contains some symbol sequences not coming from any geodesic.

Definition 2.4. The shift-equivalence class [C(γ)] is the union of all orbits of cutting sequences in C(γ).

We show in the specific case Γ = P SL(2, ℤ) and fundamental domain F that each [C(γ)] is a union of a finite number of orbits of individual cutting sequences (Theorem 7.2).

The totality of all possible convex polygons P that are fundamental domains of some group Γ has an explicit characterization. A general treatment appears in Maskit [32], which covers non-convex fundamental domains, and also covers groups of isometries of ℌ, which may include orientation-reversing isometries. Maskit [32, section 2] gives a sufficient condition for P to be a fundamental domain, but for finitely-generated groups where ℌ/Γ has finite volume and P is hyperbolically convex it is a necessary condition as well.

2.2. Cutting Sequences for the Modular Surface. We now specialize to cutting sequence expansions for geodesics on the modular surface ℌ/P SL(2, ℤ). The modular group P SL(2, ℤ) = SL(2, ℤ)/ ± I has many different convex fundamental domains P that are hyperbolic quadrilaterals; see Beardon [6, Example 9.4.4]. The standard fundamental domain for the modular group acting on ℌ is the hyperbolic triangle

F = \{z : |z| ≥ 1 and −\frac{1}{2} ≤ ℝe(z) ≤ \frac{1}{2}\} , (2.7)

which has one ideal vertex (“cusp”) at i∞. When regarded as a fundamental domain, it is a quadrilateral having an elliptic vertex added at z = i, see Figure 2.2.

[figure: −1; −\frac{1}{2}; 0; \frac{1}{2}; 1; ^{.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................}; F; L̄F; R̄F; J̄F; s_{1}; s_{2}; s_{3}; s_{4}; −\frac{1}{2} + \frac{\sqrt{3}}{2}i; \frac{1}{2} + \frac{\sqrt{3}}{2}i; i; g(s_{1}, s_{2}) = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] = L̄; g(s_{2}, s_{1}) = [ \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ] = R̄; g(s_{3}, s_{4}) = [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] = J̄; g(s_{4}, s_{3}) = [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] = J̄]

Figure 2.2. The fundamental domain F with side-pairings. The four “sides” of F are indicated in Figure 2.2 along with the two side pairings

of s_{1} with s_{2} and s_{3} with s_{4}, and with elements g(s, s′) ∈ P SL(2, ℤ), which comprise

N(F) := N_{P SL(2,ℤ)}(F) = \{[ \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] , [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ]\} . (2.8)

The “sides” s_{3} and s_{4} containing the elliptic vertex together make up one side of F, and the three sides are labeled inside the fundamental domain as follows:

R̄ = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ]  ⇔ s_{1} := \{−\frac{1}{2} + it : t > \frac{\sqrt{3}}{2}\}

L̄ = [ \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ]  ⇔ s_{2} := \{\frac{1}{2} + it : t > \frac{\sqrt{3}}{2}\}

J̄ = [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] ⇔ s_{3} ∪ s_{4} := \{|z| = 1 : −\frac{1}{2} < ℝe(z) < \frac{1}{2}\} .

These matrices represent the P SL(2, ℤ)-motions needed to move a neighboring fundamental domain F′ in ℌ to F. These neighboring fundamental domains are labeled with the appropriate symbol in Figure 2.2.

The action of P SL(2, ℤ) tiles ℌ with copies of F, and the boundaries of all tiles fit together to cover a countable collection of geodesics of ℌ, which are exactly those geodesics having two rational endpoints ⟨\frac{a}{b}, \frac{c}{d}⟩ such that

det [ \begin{matrix}a & c \\ b & d \end{matrix} ] = ±2 , (2.9)

where we adopt the convention that the cusp i∞ is the rational \frac{1}{0}. The resulting labeling of the tiles with the labels {L̄, R̄, J̄} is indicated in Figure 2.3.

The cutting sequence shift Σ_{F} := Σ_{F,P SL(2,ℤ)} is defined by the method of section 2.1, and is a closed subshift of the shift on three letters {L̄, R̄, J̄}. For this special case, we present some additional cutting sequence labelings that keep track of corners, which we call generalized cutting sequences. We label the two finite corners of F symbolically by

(2.10)  C̄_{1} = J̄R̄J̄ = L̄J̄L̄ = [ \begin{matrix}−1 & 0 \\ 1 & −1 \end{matrix} ] ⇔ SC^{−} := z = \frac{1}{2} + i\frac{\sqrt{3}}{2}
(2.11)  C̄_{2} = J̄L̄J̄ = R̄J̄R̄ = [ \begin{matrix}−1 & 0 \\ −1 & −1 \end{matrix} ] ⇔ SC^{+} := z = −\frac{1}{2} + i\frac{\sqrt{3}}{2} .

General geodesics on the modular surface ℌ/P SL(2, ℤ) are given by geodesics on ℌ that pass through the fundamental domain. A representative set of lifts covering every geodesic in the modular surface consists of all vertical geodesics with −\frac{1}{2} < θ < \frac{1}{2}, plus all semicircular geodesics in ℌ whose maximum point lies in the interior of the fundamental domain. The latter must necessarily hit the vertical line {it : t > 0} and have endpoints (θ_{1}, θ_{2}) satisfying

|θ_{2} − θ_{1}| > \sqrt{3}, and − 1 ≤ θ_{1} + θ_{2} < 1.

These conditions imply that θ_{1}θ_{2} < 0.

A generalized cutting sequence for a general geodesic γ is the sequence of symbols from {L̄, R̄, J̄, C̄_{1}, C̄_{2}} that describe the successive sides of the images of the fundamental domains that it hits.

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desic, starting from x, has cutting sequence R̄, R̄, J̄, L̄.

Finally, for the domain F we define a set of vertical cutting sequences which are one-sided cutting sequences associated to the set of oriented geodesics of ℌ that emanate from the cusp {i∞} of F and pass through F, which are exactly the set of vertical geodesics \{(∞, θ) : −\frac{1}{2} < θ < \frac{1}{2}\}.

Definition 2.5. The irrational vertical cutting sequence set Π_{F}^{0} consists of cutting sequences of those vertical geodesics ⟨∞, θ⟩ for irrational θ with −\frac{1}{2} < θ < \frac{1}{2}.

An important result for our analysis is the following simple fact.

Lemma 2.2. An irrational vertical geodesic cannot hit any finite corner of a P SL(2, ℤ)-translate of F.

Proof. The generators of P SL(2, ℤ) acting on the upper half-plane are z → z + 1 and z → −1/z. Both generators preserve the field ℚ(\sqrt{−3}), in which every element has rational real part. The finite corners ±1/2+\sqrt{−3}/2, and thus all of their images, are in this field.

Thus the cutting sequences in Π_{F}^{0} are one-sided infinite sequences which contain no symbol C̄_{1} or C̄_{2}, hence they are all of the form:

C^{+}(θ) := (g_{0}, g_{1}, g_{2}, ⋯) all g_{i} ∈ N(F) = {L̄, R̄, J̄} . (2.12)

The expansion C^{+}(θ) is a kind of continued fraction of θ, and we call it the cutting sequence expansion of θ. Note that, for a given point θ + it corresponding to a fundamental domain h_{i}F, that there is some point z ∈ F such that

g_{0}g_{1} ⋯ g_{i−2}g_{i−1}(z) = θ + it, (2.13)

according to Lemma 2.1.

Definition 2.6. The vertical cutting sequence space Π_{F} is the closure of Π_{F}^{0} in the one-sided sequence topology on the one-sided shift space on three symbols {L̄, R̄, J̄}.

Note that the set Π_{F}^{0} is not closed under the one-sided shift operator σ^{+}, because the first symbol of every element of Π_{F}^{0} is J,

We can now define cutting sequences C(θ) for rational θ with −\frac{1}{2} < θ < \frac{1}{2} in Π_{F} as limits of cutting sequences C(θ_{i}) of irrational θ_{i} → θ, as described earlier; in this case C(θ) is a set of one-sided infinite cutting sequences.

We define generalized cutting sequence expansions for vertical geodesics, drawn from the alphabet {L̄, R̄, J̄, C̄_{1}, C̄_{2}}, similarly to the case of general geodesics.

3. Continued Fraction Expansions

This section describes symbolic dynamics for vertical cutting sequences associated to additive continued fraction (ACF) expansions, and for Minkowski geodesic continued fraction expansions (MGCF) for vertical geodesics. It shows that the cutting sequence expansion and MGCF expansions for a geodesic can each be computed from the other using a finite automaton.

3.1. Additive Continued Fraction Expansions. The ordinary continued fraction (OCF) expansion for a real θ, written θ = [a_{0}, a_{1}, a_{2}, ⋯], can be represented using 2 × 2 matrices. The use of such matrices to represent continued fractions appears in Frame [12] and Kolden [26], and is described in Stark [48]. Namely, one has

[ \begin{matrix}a_{0} & 1 \\ 1 & 0 \end{matrix} ] … [ \begin{matrix}a_{n} & 1 \\ 1 & 0 \end{matrix} ] = [ \begin{matrix}p_{n} & p_{n−1} \\ q_{n} & q_{n−1} \end{matrix} ] , (3.1)

where \frac{p_{n}}{q_{n}} = [a_{0}, a_{1}, …,a_{n}] is the n-th convergent of θ. The ordinary continued fraction expansion for θ > 0 has a symbolic dynamics which is a one-sided shift on infinitely many symbols, using the alphabet

L_{k} = [ \begin{matrix}k & 1 \\ 1 & 0 \end{matrix} ] , k ≥ 0. (3.2)

The additive continued fraction (ACF) expansion of θ > 0 is a symbolic expansion using the two symbols

R = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] , F = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] , (3.3)

which is obtained from (3.2) by expanding the symbol L_{k} as

[ \begin{matrix}k & 1 \\ 1 & 0 \end{matrix} ] = [ \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ]^{k} [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] . (3.4)

The additive continued fraction expansion of θ > 0 is thus

FR^{a_{1}}FR^{a_{2}}FR^{a_{3}} ⋯, (3.5)

regarded as a string of letters in the alphabet {R, F}. The finite truncations of this expression are 2 × 2 matrices that encode the convergents and intermediate convergents of θ.

We next construct a related continued fraction for real θ > 0 which uses the symbols

R := [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] , D = [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ] , (3.6)

The Farey tree expansion of a real number θ > 0 that has OCF expansion θ = [a_{0}, a_{1}, a_{2}, …] is:

R^{a_{0}}D^{a_{1}}R^{a_{2}}D^{a_{3}} ⋯ (3.7)

regarded as a sequence of letters in the alphabet {R, D}. This sequence encodes the steps of subtraction needed in the division process to encode the ordinary continued fraction of θ; it also essentially gives all the intermediate convergents to θ, cf. Richards [41], Theorem 2.1. For θ > 0 the set of allowable symbol sequences for the additive expansion is the full one-sided shift on two letters {R, D}. The association to the Farey tree is described in Lagarias [28].

The symbolic expansions (3.5) and (3.7) are equivalent in the sense that each is convertible to the other using a finite automaton; see section 3.4.

3.2. Minkowski Geodesic Continued Fraction Expansions. The Minkowski geodesic multidimensional continued fraction (MGCF) expansion is described in Lagarias [27]. We consider here the one-dimensional case. If −\frac{1}{2} < θ < \frac{1}{2} it follows a parametrized series of lattice bases

B_{t}(θ) = [ \begin{matrix}1 & 0 \\ −θ & t \end{matrix} ] (3.8)

for the parametrized family of lattices Λ_{t} = ℤ[(1, 0), (−θ, t)]. The bases B_{t}(θ) correspond in a natural way to a vertical geodesic γ = {θ + it : t > 0} in ℌ. There exists a sequence of critical values

∞ = t_{0} > t_{1} > t_{2} > ⋯

with lim_{n→∞} t_{n} = 0, and an associated sequence of convergent matrices {P^{(n)} : n = 0, 1, 2, ⋯ } in GL(2, ℤ) such that P^{(n)}B_{t}(θ) is a Minkowski-reduced lattice basis of the lattice when t_{n} > t > t_{n+1}. We write

P^{(n)} := [ \begin{matrix}p_{1}^{(n)} & q_{1}^{(n)} \\ p_{2}^{(n)} & q_{2}^{(n)} \end{matrix} ] , (3.9)

so that

P^{(n)}B_{t}(θ) = [ \begin{matrix}p_{1}^{(n)} − q_{1}^{(n)}θ & q_{1}^{(n)}t \\ p_{2}^{(n)} − q_{2}^{(n)}θ & q_{2}^{(n)}t \end{matrix} ] . (3.10)

We require that q_{i}^{(n)} ≥ 0 for all n ≥ 1, and if q_{i}^{(n)} = 0 then p_{i}^{(n)} > 0.

The associated partial quotient matrices A^{(n)} are defined by

P^{(n)} = A^{(n)}P^{(n−1)} ,

so that

P^{(n)} = A^{(n)}A^{(n−1)} ⋯ A^{(1)}P^{(0)} . (3.11)

Here P^{(0)} = I is the identity matrix if −\frac{1}{2} ≤ θ < \frac{1}{2}. We now show that the partial quotient matrices A^{(n)} are drawn from a finite set.

Lemma 3.1 (Minkowski partial quotient set). . The allowed partial quotients for the one-dimensional Minkowski geodesic continued fraction algorithm are

\{[ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] , [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 \\ 1 & −1 \end{matrix} ] , [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ]\} . (3.12)

Proof. Let M = [ \begin{matrix}v_{1} \\ v_{2} \end{matrix} ] be a 2 × 2 matrix with det(M) > 0, and associate to it the

positive definite quadratic form with coefficient matrix

(3.13) MM^{T} = [ \begin{matrix}∥v_{1}∥^{2} & ⟨v_{1}, v_{2}⟩ \\ ⟨v_{1}, v_{2}⟩ & ∥v_{2}∥^{2} \end{matrix} ] .

The Minkowski-reduction conditions for the quadratic form Q(x) = x^{T} (MM^{T})x are

(3.14)  ∥v_{1}∥ ≤ ∥v_{2}∥
(3.15)  ∥v_{2}∥ ≤ ∥v_{1} + v_{2}∥
(3.16)  ∥v_{2}∥ ≤ ∥v_{1} − v_{2}∥,

see Cassels [9, p. 257].

We use the fact that all vectors in the lattice L_{t}(θ) have the special form

v_{i} = v_{i}(t) := (p_{i} − q_{i}θ, q_{i}t) with q_{i} ≥ 0. (3.17)

Whenever a critical value t_{c} occurs with ∥(p − qθ, qt_{c})∥ = ∥(p′ − q′θ, q′t_{c})∥, then the inequality

∥(p′ − q′θ, q′t)∥ > ∥(p − qθ, qt)∥ for 0 < t < t_{c}

holds exactly when |q′| > |q|. Thus when the shortest vector in the basis P^{(n−1)} is replaced by a vector in P^{(n)} the associated denominator must increase.

We first consider “generic” convergents which occur when exactly two of ∥v_{1}(t)∥, ∥v_{2}(t)∥, ∥v_{1}(t) + v_{2}(t)∥ and ∥v_{1}(t) − v_{2}(t)∥ become equal at t = t_{c}.

Suppose first that q_{1} < q_{2}. If at t = t_{c} we have ∥v_{1}(t)∥ = ∥v_{2}(t)∥ then the partial

quotient is [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ]; it exchanges v_{1} and v_{2}. If ∥v_{2}(t)∥ = ∥v_{1}(t) + v_{2}(t)∥ then

the partial quotient is [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ]; it replaces v_{2} with v_{1} + v_{2}. The case ∥v_{2}(t)∥ =

∥v_{1}(t) − v_{2}(t)∥ cannot occur.

Suppose next that q_{1} ≥ q_{2}. At t = t_{c} we cannot have ∥v_{1}(t)∥ = ∥v_{2}(t)∥ since an exchange of v_{1} and v_{2} would not increase the denominator. We may have ∥v_{1}(t) + v_{2}(t)∥ = ∥v_{2}(t)∥, giving the partial quotient [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ]. If q_{1} > 2q_{2} then q′ = q_{1} −q_{2} >

q_{2} hence the partial quotient [ \begin{matrix}1 & 0 \\ 1 & −1 \end{matrix} ] is possible; it replaces v_{2} with v_{1} − v_{2}.

There remain situations in which three or more of ∥v_{1}(t)∥, ∥v_{2}(t)∥, ∥v_{1}(t) + v_{2}(t)∥ and ∥v_{1}(t) − v_{2}(t)∥ simultaneously becoming equal at t = t_{c}. Only one case is possible; it is

∥v_{1}(t)∥ = ∥v_{2}(t)∥ = ∥v_{1}(t) + v_{2}(t)∥ ,

which can only occur when q_{1} < q_{2}. In this case the partial quotient is [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ]. The

other case consistent with an increasing denominator is ∥v_{2}(t)∥ = ∥v_{1}(t) + v_{2}(t)∥ = ∥v_{1}(t) − v_{2}(t)∥, but this implies ∥v_{1}(t)∥ = 0, which is impossible unless t = 0.

We define the Minkowski geodesic continued fraction expansion of any real θ satisfying −\frac{1}{2} < θ < \frac{1}{2} to be the symbol sequence

(A^{(1)}, A^{(2)}, A^{(3)}, …) . (3.18)

We write this sequence right-to-left, although the matrix product in (3.11) runs leftto-right. The associated Minkowski geodesic symbol set is

L = [ \begin{matrix}1 & 0 \\ 1 & −1 \end{matrix} ] , R = [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ] , J = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] , C = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] . (3.19)

The symbol C occurs only for “exceptional” geodesics, and we will usually be concerned with symbolic expansions drawn from the smaller symbol set {L, R, J}.

The Minkowski geodesic continued fraction expansion corresponding to a general geodesic on F is the expansion attached to Minkowski reduction of the parametrized lattice bases

B_{t}(θ_{1}, θ_{2}) = [ \begin{matrix}1 & (θ_{1})^{−1}t \\ −θ_{2} & t \end{matrix} ] .

The associated quadratic form is

Q_{t}(x, y) = (x − θ_{2}y)^{2} + t^{2} (\frac{1}{θ_{1}} x − y)^{2} . (3.20)

3.3. Cutting Sequences and Minkowski Geodesic Continued Fractions. The cutting sequence expansion of θ for −\frac{1}{2} < θ < \frac{1}{2} is essentially the same as the Minkowski geodesic continued fraction expansion of ⟨∞, θ⟩, as formulated in Theorem 3.1 below. To demonstrate the equivalence, we first transform lattice bases in GL(2, ℝ) to the cone of positive definite symmetric matrices P_{2} (equivalently, to positive definite quadratic forms), and then we relate these to elements of ℌ.

A basis B ∈ GL(2, ℝ) corresponds to the positive definite symmetric matrix M = BB^{T} ∈ P_{2} with associated quadratic form

Q(x_{1}, x_{2}) = [x_{1}x_{2}]M[ \begin{matrix}x_{1} \\ x_{2} \end{matrix} ] = m_{11}x_{1}^{2} + (m_{12} + m_{21})x_{1}x_{2} + m_{22}x_{2}^{2} . (3.21)

For the basis matrix B_{t}(θ) in (3.8) this gives

M := M_{t}(θ) = [ \begin{matrix}1 & −θ \\ −θ & θ^{2} + t^{2} \end{matrix} ] . (3.22)

with associated quadratic form

Q_{t}(x_{1}, x_{2}) = (x_{1} − θx_{2})^{2} + t^{2}x_{2}^{2} = (x_{1} − (θ + it)y_{1})(x_{1} − (θ − it)x_{2}) . (3.23)

The action of P ∈ GL(2, ℤ) which sends the matrix B_{t}(θ) to PB_{t}(θ) is transformed to the GL(2, ℤ)-action on P_{2} that takes M to PMP^{T}, and this takes the associated quadratic form Q(x_{1}, x_{2}) to

Q′(x_{1}, x_{2}) = Q(p_{11}x_{1} + p_{21}x_{2}, p_{21}x_{1} + p_{22}x_{2}) . (3.24)

The Minkowski reduction domain M̄ for GL(2, ℤ)\P_{2} (see Cassels [9]) is the set of quadratic forms (3.21) such that

Q(1, 0) ≤ Q(0, 1) ,
(3.25)  Q(0, 1) ≤ Q(1, 1) ,
Q(0, 1) ≤ Q(−1, 1) .

This is equivalent to the conditions

|m_{12} + m_{21}| ≤ m_{11} ≤ m_{22} . (3.26)

on the form Q in (3.21). The subgroup

H_{2} := \{[ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] , [ \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] , [ \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ]\}

of GL(2, ℤ) maps the domain M̄ into itself, hence we actually let H_{2}\GL(2, ℤ) act on P_{2}. By definition the Minkowski reduction domain M of GL(2, ℤ)\GL(2, ℝ), is the preimage of M̄ under the mapping B̄ ↦ BB^{T}, and the group actions by GL(2, ℤ) coincide. The Minkowski geodesic continued fraction algorithm picks a unique element of GL(2, ℤ) out of a four-element coset of H_{2}\GL(2, ℤ) by requiring that p_{12} ≥ 0 and p_{22} ≥ 0, with p_{11} > 0 if p_{12} = 0 and with p_{21} > 0 if p_{22} = 0.

For the second transformation from P to ℌ, note that a positive definite quadratic form can be uniquely written

Q(x, y) = a(x_{1} − θx_{2})(x_{1} − θ̄x_{2}) , (3.27)

with θ and θ̄ complex conjugates and Im(θ) > 0. We map P to ℌ by sending the quadratic form Q to the unordered pair of roots {θ, θ̄} and then identify this with θ ∈ ℌ = {z : Im(z) > 0}. If P ∈ GL(2, ℤ) sends Q to Q′ given by (3.26), then a calculation yields

Q′ = a′(x_{1} − ϕx_{2})(x_{1} − ϕ̄x_{2}) ,

with a′ = Q(p_{11}, p_{12}) and with a root ϕ given by

ϕ = (P^{T})^{−1}θ = [ \begin{matrix}p_{22} & −p_{21} \\ −p_{12} & p_{11} \end{matrix} ] (θ) , (3.28)

where the action of the matrix is a linear fractional transformation on ℂ. If det(P) = 1, then ϕ ∈ ℌ, while if det(P) = −1 then Im(ϕ) < 0 is the complex conjugate of the root we want, i.e. it reverses the ordering of the roots. Now each four element coset of H_{2}\GL(2, ℤ) contains two elements in SL(2, ℤ) which form a coset of P SL(2, ℤ) = {±I}\SL(2, ℤ), so the action (3.28) of H_{2}\GL(2, ℤ) on unordered pairs {θ, θ̄} is equivalent to the standard P SL(2, ℤ)-action on ℌ. Finally, the conditions (3.25) for a form Q to be Minkowski reduced translate via (3.27) to

θθ̄ ≥ 1, −1 ≥ θ + θ̄ ≥ 1 ,

which is exactly the fundamental domain F of P SL(2, ℤ).

The mapping from GL(2, ℝ) to ℌ obtained by composing these two transformations sends the set of matrices {B_{t}(θ) : t > 0} in GL(2, ℝ) to the geodesic {θ + it : t > 0} in ℌ. Since the H_{2}\GL(2, ℤ) action on GL(2, ℝ) is equivalent to the P SL(2, ℤ) action on ℌ and the reduction domains correspond, the Minkowski geodesic continued function expansion for θ is essentially the same as the cutting sequence expansion for θ. In the reverse direction, θ ∈ ℌ determines a positive ray {aQ : a > 0} in the cone P_{2}, i.e. an element of P_{2}/ℝ^{+}, and this in term determines a family {aBQ : a ∈ ℝ^{+}, Q ∈ O(n, ℝ)} in GL(2, ℝ), i.e. an element of GL(2, ℝ)/ℝ^{+}O(2, ℝ). This does not affect the symbolic dynamics because the Minkowski domain M̄ is invariant under the ℝ^{+}-action, and because the Minkowski reduction domain M is invariant under the R^{∗}O(2, ℝ)-action.

The precise correspondence between the symbol sequences, involves specifying the relation among the four elements of the H_{2}-coset in H_{2}\GL(2, ℤ) for the Minkowski geodesic continued fraction expansion and the two elements in the {±I}-coset in P SL(2, ℤ) for the cutting sequence expansion.

Definition 3.1. The parity of a word W = S_{1} … S_{n} in the alphabet {L, R, J} is even or odd according to whether there are an even or odd number of L and J. That is,

(3.29) det(W) = (−1)parity(W).

We obtain:

Theorem 3.1. For irrational θ with −\frac{1}{2} < θ < \frac{1}{2}, the one-sided cutting sequence expansion of the vertical geodesic ⟨∞, θ⟩ = {θ + it : t > 0} in Π_{F}^{0} is obtained from the Minkowski geodesic continued fraction expansion of θ by the following procedure: If the current initial word of the MGCF expansion has even parity, on the next symbol make the letter replacement L → L̄, R → R̄ and J → J̄; if it has odd parity, make the letter replacement L → R̄, R → L̄ and J → J̄. The MGCF can be obtained from the cutting sequence by the reverse process, in which the parity is determined by the current symbols of the MGCF expansion.

Proof. By Lemma 2.2, the assumption of irrational θ ensures that the MGCF expansions and cutting sequence expansions of θ are both infinite and never use a symbol C. The MGCF expansion ⋯ A^{(3)}A^{(2)}A^{(1)} uses the symbols

L = [ \begin{matrix}1 & 0 \\ 1 & −1 \end{matrix} ] , R = [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ] , J = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] ,

drawn from GL(2, ℤ) and moves right to left, while the cutting sequence expansion uses the symbols

L̄ = [ \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ] , R̄ = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] , J̄ = [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ]

drawn from SL(2, ℤ) and moves left to right by Lemma 2.1. For notational convenience, we introduce the matrix

K:= [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] ,

so H_{2} is {I, K, −K, −I}. We first arrange for matrices to multiply in the same direction. The matrices in the MGCF are chosen so that P^{(n)}B_{t}(θ) ∈ F for a given t, while the matrices in the cutting sequence are chosen so that the cutting sequence product h_{n} = S_{1} … S_{n} has γ(t) ∈ h_{n}F. Therefore, we have h_{n} = (P^{(n)})^{−1}. We expand (P^{(n)})^{−1} in terms of symbols L, R, J, K using the relations L^{−1} = L, R^{−1} = KRK and J^{−1} = J to obtain an expansion which proceeds left to right. To convert this expansion to the cutting sequence expansion, we must convert to symbols L̄, R̄, J̄ and remove the symbols K, which encode the parity. We proceed by an intermediate conversion to symbols

R̃ := R, L̃ := [ \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ] , J̃ := [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ]

in SL(2, ℤ). To convert a product of matrices from the form L, R, J to L̃, R̃, J̃, starting from the right end of the product, we use the relations LK = R̃, RK = KL̃, and JK = −J̃. In doing this we pick up or lose a factor of K whenever we encounter a matrix L or J and this multiplies the determinant by −1; meanwhile, every R in the MGCF becomes R^{−1} = KRK when inverted, and thus is itself encoded with the opposite parity but leaves the parity unchanged for the next matrix. (We sometimes pick up a matrix factor of −I, but this commutes with everything and may be ignored.) Similarly, to reverse this, we use L̃ = KL and J̃ = KJ.

To convert from L̃, R̃, J̃ to L̄, R̄, J̄, we use the relations (L̃^{T})^{−1} = R̄, (R̃^{T})^{−1} = L̄ and (J̃^{T})^{−1} = J̄. Thus the conversion from L̃, R̃, J̃ to L̄, R̄, J̄ interchanges R with L.

3.4. Finite Automata. By a finite automaton we mean a deterministic finite-state automaton, as defined in Hopcroft and Ullman [19]. We consider finite automata used as transducers, as in Raney [40], in which each transition edge of the automaton specifies the printing of a specific finite string of output letters, possibly empty. The automaton starts in an initial state, and the input sequence is presented as a sequence of letters, and each letter determines the transition edge to the next state of the automaton.

In this paper we present a number of results asserting the existence of finite automata to convert one-sided infinite symbol sequences of one form to another form. In the proofs we only indicate the “finite-state” character of the conversion process, and generally omit details of the (routine but sometimes involved) construction of the automaton.

[figure: +1; −1; ^{.........................................................................................................................................................................................................................................................................................................................................................}; ^{.........................................................................................................................................................................................................................................................................................................................................................}; ^{............................................................................................................................................................................................................................................................................................................}; ^{............................................................................................................................................................................................................................................................................................................}; F : ∅; F : ∅; R : D; R : R]

(a) Additive continued fraction to Farey tree

[figure: +1; −1; ^{.........................................................................................................................................................................................................................................................................................................................................................}; ^{.........................................................................................................................................................................................................................................................................................................................................................}; ^{.........................................................................................................................................................................................................................................................................................................................................................................................................}; ^{.........................................................................................................................................................................................................................................................................................................................................................................................................}; ^{............................................................................................................................................................................................................................................................................................................}; ^{............................................................................................................................................................................................................................................................................................................}; J : J̄; L : R̄; L : L̄; J : J̄; R : L̄; R : R̄]

(b) MGCF to cutting sequence

Figure 3.1. Finite Automata–Transducers (initial state for both automata is +1).

The discussion in section 3.1 yields:

Theorem 3.2. For real θ > 1, the additive continued fraction expansion of θ can be converted to the Farey tree expansion of θ by a finite automaton, and vice versa.

Proof. To convert from the Farey shift expansion (3.7) to the additive continued fraction expansion (3.5), we use D = FRF and F^{2} = I.

A finite automaton which converts the additive continued fraction expansion (3.5) to the Farey tree expansion (3.7) must keep two states to keep track of the sign ±1 of det(S_{1} ⋯ S_{m}) of the symbols S_{i} = F or R examined so far. The initial state is +1.

The discussion in section 3.3 yields:

Theorem 3.3. For irrational θ with −\frac{1}{2} < θ < \frac{1}{2} there exists a finite automaton to convert the Minkowski geodesic continued fraction expansion of θ to that of the vertical cutting sequence expansion of ⟨∞, θ⟩ and vice-versa.

Proof. This follows from Theorem 3.1. For each direction, the finite automaton constructed needs two states, to keep track of whether there a factor of K present; i.e., to keep track of the sign of det(S_{1} ⋯ S_{n}). The initial state is +1.

We illustrate two of these automata in Figure 3.1. The edge symbol S : W specifies that this edge is taken if the current input symbol is S, and W denotes a symbol sequence to be output. The initial states are the vertices labeled +1.

4. Vertical Cutting Sequences to Additive Continued Fractions

The ordinary continued fraction expansion of an irrational real number −\frac{1}{2} < θ < \frac{1}{2} is easily determined from either its cutting sequence expansion or its Minkowski geodesic continued fraction expansion. We begin with the latter case.

Theorem 4.1. Given θ with −\frac{1}{2} < θ < \frac{1}{2}, with ordinary continued fraction expansion θ = [a_{0}; a_{1}, a_{2}, a_{3}, ⋯], let S_{0}S_{1}S_{2} ⋯ be the Minkowski geodesic continued fraction expansion of θ in the alphabet {L, R, J, C}. The MGCF expansion can be uniquely parsed into segments B_{0}B_{1} ⋯ where B_{0} = J or JL and each B_{i} for i ≥ 1 is R^{k}J for some k ≥ 1, R^{k+1}JL for some k ≥ 1, or R^{k}C for some k ≥ 1. Each segment encodes exactly one or two partial quotients of the OCF expansion. For general segments, R^{k}J encodes a partial quotient a_{n} = k, while R^{k+1}JL or R^{k}C encode two partial quotients a_{n} = k, a_{n+1} = 1. For the segment B_{0}, the symbol J encodes a_{0} = 0, while JL encodes the partial quotients a_{0} = −1, a_{1} = 1. The case R^{k}C can only occur when θ is rational.

Proof. We say that p/q is a best approximation to θ if |qθ − p| < |q′θ − p′| for 0 < q′ < q. Using this definition, it is a basic result about ordinary continued fractions that the convergents are the complete set of best approximations; see Hardy and Wright [15, Theorem 182]. We say that p/q is a better approximation to θ than p′/q′ if |qθ − p| < |q′θ − p′|. The best approximation property thus proves that p_{n}/q_{n} is a better approximation than any fraction with denominator less than q_{n+1}.

We start with P^{(0)} = [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ], the identity matrix. Thus P^{(0)}B_{t}(θ) is [ \begin{matrix}1 & 0 \\ −θ & t \end{matrix} ].

Neither of the Minkowski inequalities (3.15) and (3.16) can hold with equality for any t, since |θ| < \frac{1}{2}. We get t_{1}^{2} = 1−(⌊θ + \frac{1}{2}⌋ − θ)^{2} as the value at which (3.14) holds with

equality. The first partial quotient matrix is thus J. That makes P^{(1)} = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ].

Now P^{(1)}B_{t}(θ) = [ \begin{matrix}−θ & t \\ 1 & 0 \end{matrix} ].

We check the Minkowski inequalities. Inequality (3.14) is satisfied, with equality only at t = t_{1}. Inequality (3.15) is an equality if 1 = (1 − θ)^{2} + t_{2}^{2}, and (3.16) is an equality if 1 = (−1 − θ)^{2} + t_{2}^{2}. If θ = 0, neither one holds with equality for any t > 0, and J is thus the whole MGCF, encoding θ = [0]. If 0 < θ < \frac{1}{2}, then only (3.15) can hold with equality, so the next partial quotient matrix will be R. In this case, we have

P^{(1)} = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] = [ \begin{matrix}p_{0} & q_{0} \\ p_{−1} & q_{−1} \end{matrix} ] ,

and we let B_{0} = J. If −\frac{1}{2} < θ < 0, then only (3.16) can hold with equality, so the next partial quotient matrix will be L. For −\frac{1}{2} < θ < 0, the continued fraction for θ begins [−1, 1, …], and thus p_{1} = 0, q_{1} = 1. This gives

P^{(2)} = [ \begin{matrix}0 & 1 \\ −1 & 1 \end{matrix} ] = [ \begin{matrix}p_{1} & q_{1} \\ p_{0} & q_{0} \end{matrix} ] ,

and we let B_{0} = J, L.

In either case, after the first step, we have encoded all coefficients through a_{n−1}, and our current matrix is

P^{(j)} = [ \begin{matrix}p_{n−1} & q_{n−1} \\ p_{n−2} & q_{n−2} \end{matrix} ] . (4.1)

The rest of the theorem is proved by induction, with the induction hypothesis that after each segment B_{k}, the matrix P^{(j)} is in the form (4.1), with the coefficients through a_{n−1} encoded.

First, assume that the current P^{(j)} is in the form (4.1). Then

P^{(j)}B_{t}(θ) = [ \begin{matrix}p_{n−1} − q_{n−1}θ & tq_{n−1} \\ p_{n−2} − q_{n−2}θ & tq_{n−2} \end{matrix} ] .

Since q_{n−1} > q_{n−2}, the requirement of an increasing denominator makes J impossible for the next partial quotient. The Minkowski inequality for R is

((p_{n−1} − q_{n−1}θ) + (p_{n−2} − q_{n−2}θ))^{2} − (p_{n−2} − q_{n−2}θ)^{2}
(4.2)  + t^{2}((q_{n−1} + q_{n−2})^{2} − q_{n−2}^{2}) ≥ 0,

and for L, it is

((p_{n−1} − q_{n−1}θ) − (p_{n−2} − q_{n−2}θ))^{2} − (p_{n−2} − q_{n−2}θ)^{2}
(4.3)  + t^{2}((q_{n−1} − q_{n−2})^{2} − q_{n−2}^{2}) ≥ 0.

By the best approximation condition, p_{n−1}/q_{n−1} is a better approximation to θ than (p_{n−1} − p_{n−2})/(q_{n−1} − q_{n−2}). Thus (4.3) is satisfied for small t, and thus for all t < t_{j}, so we cannot have L next. If θ is rational and a_{n−1} is the last term, then (4.2) is satisfied for all t; thus the MGCF terminates. Otherwise, since every even convergent is less than θ, and every odd convergent is greater, p_{n−1} −q_{n−1}θ and p_{n−2} −q_{n−2}θ have opposite signs, and the absolute value of their sum is less than the absolute value of p_{n−2} − q_{n−2}θ. Thus (4.2) is not satisfied for sufficiently small t, and t_{j+1} is the value of t at which it holds with equality. Thus the next matrix is R; hence

P^{(j+1)} = RP^{(j)} = [ \begin{matrix}p_{n−1} & q_{n−1} \\ p_{n−2} + p_{n−1} & q_{n−2} + q_{n−1} \end{matrix} ] .

The remainder of the proof considers the case where P^{(j)} is of the form

P^{(j)} = [ \begin{matrix}p_{n−1} & q_{n−1} \\ p_{n−2} + mp_{n−1} & q_{n−2} + mq_{n−1} \end{matrix} ] , (4.4)

which can be reached after m applications of R to the form (4.1). Now we have

P^{(j)}B_{t}(θ) = [ \begin{matrix}p_{n−1} − q_{n−1}θ & tq_{n−1} \\ (p_{n−2} − q_{n−2}θ) + m (p_{n−1} − q_{n−1}θ) & t(q_{n−2} + mq_{n−1}) \end{matrix} ] .

Since q_{n−2} + mq_{n−1} > q_{n−1}, the next symbol A^{(j+1)} cannot be L. The other two Minkowski inequalities, for J and R, are

((p_{n−2} − q_{n−2}θ) + m (p_{n−1} − q_{n−1}θ))^{2} − (p_{n−1} − q_{n−1}θ)^{2}
(4.5)  + t^{2}((q_{n−2} + mq_{n−1})^{2} − q_{n−1}^{2}) ≥ 0,
((m + 1) (p_{n−1} − q_{n−1}θ) + (p_{n−2} − q_{n−2}θ))^{2}
− (m (p_{n−1} − q_{n−1}θ) + (p_{n−2} − q_{n−2}θ))^{2}
(4.6)  + t^{2}(((m + 1)q_{n−1} + q_{n−2})^{2} − (mq_{n−1} + q_{n−2})^{2}) ≥ 0.

Thus the next symbol is A^{(j+1)} = J if (4.5) holds with equality for a larger t < t_{j} than (4.6), A^{(j+1)} = R if (4.6) holds with equality for a larger t < t_{j}, and A^{(j+1)} = C if both hold with equality at the same t. Recall that we have, for 0 ≤ m ≤ a_{n}, and n even,

\frac{p_{n−1} + p_{n}}{q_{n−1} + q_{n}} = \frac{(a_{n} + 1)p_{n−1} + p_{n−2}}{(a_{n} + 1)q_{n−1} + q_{n−2}} > θ ≥ \frac{mp_{n−1} + p_{n−2}}{mq_{n−1} + q_{n−2}} , (4.7)

with the reverse inequalities holding for n odd. The last inequality holds with equality only if θ = p_{n}/q_{n} and m = a_{n}. If m < a_{n}, then mq_{n−1} + q_{n−2} < q_{n}, so we must have

(p_{n−1} − q_{n−1}θ)^{2} < ((p_{n−2} − q_{n−2}θ) + m (p_{n−1} − q_{n−1}θ))^{2},

and thus the left-hand side of (4.5) is positive for small t, so A^{(j+1)} cannot be J or C. Thus, if m < a_{n}, the only possible case is R, and the matrix P^{(j+1)} is still of the form (4.4), with m replaced by m + 1.

We next consider the case when m = a_{n} in (4.4). In this case, the next symbol A^{(j+1)} may be any of R, J, and C. First suppose A^{(j+1)} = J. If so, we have finished our segment B_{k} = R^{a_{n}}J, and P^{(j+1)} is now in the correct form (4.1), since it is

[ \begin{matrix}p_{n} & q_{n} \\ p_{n−1} & q_{n−1} \end{matrix} ] .

Next suppose that m = a_{n} and A^{(j+1)} = R. For this to happen, we need (4.6) to be true only for t ≥ t_{c}. Since m = a_{n}, we have mq_{n−1} + q_{n−2} = q_{n}, and similarly for p_{n}. For (4.6) to be true only for sufficiently large t, (p_{n−1} + p_{n})/(q_{n−1} + q_{n}) must be a better approximation to θ than p_{n}/q_{n}. This cannot happen if θ = p_{n}/q_{n}, hence a_{n+1} is defined. By the best approximation property, we must have q_{n−1} + q_{n} ≥ q_{n+1}, and it follows that a_{n+1} = 1, p_{n+1} = p_{n} + p_{n−1}, and q_{n+1} = q_{n} + q_{n−1}, so that in this case

P^{(j+1)} = RP^{(j)} = [ \begin{matrix}p_{n−1} & q_{n−1} \\ p_{n+1} & q_{n+1} \end{matrix} ] .

This matrix is in the form (4.4) but with m = a_{n} + 1. Now A^{(j+2)} cannot be R or C, because the left-hand side of left-hand side of (4.6) is positive for small t, so A^{(j+2)} must be J, which gives

P^{(j+2)} = JRP^{(j)} = [ \begin{matrix}p_{n+1} & q_{n+1} \\ p_{n−1} & q_{n−1} \end{matrix} ] . (4.8)

This gives

P^{(j+2)}B_{t}(θ) = [ \begin{matrix}p_{n+1} − q_{n+1}θ & tq_{n+1} \\ p_{n−1} − q_{n−1}θ & tq_{n−1} \end{matrix} ] ,

The following symbol A^{(j+3)} cannot be J, because q_{n+1} > q_{n−1} and we must have an increasing denominator. The Minkowski inequality for R is

((p_{n+1} − q_{n+1}θ) + (p_{n−1} − q_{n−1}θ))^{2} − (p_{n−1} − q_{n−1}θ)^{2}
(4.9)  + t^{2}((q_{n+1} + q_{n−1})^{2} − q_{n−1}^{2}) ≥ 0.

The constant term here is positive because p_{n+1} − q_{n+1}θ and p_{n−1} − q_{n−1}θ have the same sign, so this inequality holds for all t. The Minkowski inequality for L is

((p_{n+1} − q_{n+1}θ) − (p_{n−1} − q_{n−1}θ))^{2} − (p_{n−1} − q_{n−1}θ)^{2}
(4.10)  + t^{2}((q_{n+1} − q_{n−1})^{2} − q_{n−1}^{2}) ≥ 0,

which is not satisfied for small enough t because (p_{n+1} − q_{n+1}θ) − (p_{n−1} − q_{n−1}θ) = p_{n} − q_{n}θ, and thus its constant term is negative. Thus A^{(j+3)} = L, so that

P^{(j+3)} = LJRP^{(j)} = [ \begin{matrix}p_{n+1} & q_{n+1} \\ p_{n} & q_{n} \end{matrix} ] . (4.11)

This matrix has the form (4.1), and we have encoded the two coefficients, a_{n} and a_{n+1} = 1, with a segment B_{k} = R^{a_{n}+1}JL.

Finally, suppose that m = a_{n} and A^{(j+1)} = C. Again, (4.6) must hold only for sufficiently large t, and it follows that a_{n+1} = 1. Thus, since q_{n+1} = q_{n} + q_{n−1} and p_{n+1} = p_{n} + p_{n−1}, we have

P^{(j+1)} = CP^{(j)} = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] [ \begin{matrix}p_{n−1} & q_{n−1} \\ p_{n} & q_{n} \end{matrix} ] = [ \begin{matrix}p_{n+1} & q_{n+1} \\ p_{n} & q_{n} \end{matrix} ] . (4.12)

Here P^{(j+1)} is in the form (4.1), and we have again encoded the two coefficients, a_{n} and a_{n+1} = 1 as R^{a_{n}}C. This completes the induction step.

Lemma 2.2 shows that the last case R^{k}C can occur only for rational θ.

There is an analogous conversion method from the cutting sequence expansion to the additive ordinary continued fraction expansion, as follows.

Theorem 4.2. Given θ with ordinary continued function expansion θ = [a_{0}; a_{1}, a_{2}, a_{3}, ⋯], let S_{0}^{∗}S_{1}^{∗}S_{2}^{∗} ⋯ be the cutting sequence expansion for the geodesic ⟨∞, θ⟩ = {θ + it : t > 0} in the alphabet {L̄, R̄, J̄, C̄_{1}, C̄_{2}}. It can be uniquely parsed into segments B̄_{0}B̄_{1}B̄_{2} ⋯ where B̄_{0} = J̄ or J̄R̄, encoding a_{0} = 0 or a_{0} = −1, a_{1} = 1, respectively, and each succeeding segment is R̄^{k}J̄ or L̄^{k}J̄ encoding a_{n} for n even and odd, respectively, or is R̄^{k+1}J̄ R̄, or R̄^{k}C̄_{1} encoding a_{n} = k, a_{n} + 1 = 1 for n even, or is L̄^{k+1}J̄ L̄ or L̄^{k}C̄_{2} encoding a_{n} = k, a_{n+1} = 1 for n odd. The symbols C̄_{1}, C̄_{2} can only occur in expansions of rational θ.

Proof. This follows from Theorem 4.1 by noticing that the parity of the initial word W_{i} = B_{0} … B_{i} changes after each segment B_{i} encoding one term, and does not change after any segment B_{i} encoding two terms, while the segment B_{0} has odd parity if it encodes no terms, and even parity if it encodes a_{1}.

An immediate consequence of Theorem 4.1 and 4.2 is the following result.

Theorem 4.3. There exists a finite automaton which converts the Minkowski geodesic continued fraction expansion of each irrational θ with −\frac{1}{2} < θ < \frac{1}{2} to the additive continued fraction expansion of θ. There exists a finite automaton that converts the vertical cutting sequence expansion of ⟨∞, θ⟩ to the additive ordinary continued fraction expansion of θ.

Proof. The segment-partition of the MGCF expansion given in Theorem 4.1 permits the Farey tree expansion (3.7) to be computed by a finite automaton, because the necessary information to decide on the symbol R versus L depends only on the determinant ±1 of the product of the MGCF matrices scanned plus the values of the last four MGCF symbols in the expansion. Next, Theorem 4.2 guarantees that a finite automaton exists to convert the cutting sequence expansion as well. Finally, Theorem 3.2 applies to give the additive continued fraction from the Farey tree expansion.

Theorem 4.1 implies that the Minkowski geodesic continued fraction expansion of θ can represented in an abbreviated form

θ := [ã_{0}, ã_{1}, ã_{2}, ã_{3}, ...]

similar to its ordinary continued fraction expansion

θ = [a_{0}; a_{1}, a_{2}, a_{3}, ...] ,

with the change that each symbol 1 in the OCF expansion is to be replaced by one of three possible symbols 1_{h}, 1_{m} and 1_{c}. Here 1_{h} means that the continued fraction partial quotient a_{n} = 1 begins a new segment RJ or RRJL in the MGCF expansion (so that the previous convergent p_{n−1}/q_{n−1} was “hit” at the end of a segment), 1_{m} means that it combines with the previous partial quotient in a block R^{k+1}JL (so that p_{n−1}/q_{n−1} was “missed”), and 1_{c} means that it combines with the previous partial quotient in a block with a C-symbol (a “corner”). However it does not specify which sequence of symbols actually occur as legal expansions. We study this next.

5. Additive Continued Fractions to Vertical Cutting Sequences

In this section, we show how to construct the Minkowski geodesic continued fraction expansion of θ given the additive continued fraction expansion of θ. Let θ have the ordinary continued fraction expansion

θ = [a_{0}; a_{1}, a_{2}, a_{3}, ⋯],

in which a_{0} = 0 or −1. In view of the results of section 4, it suffices to determine for each a_{n} = 1 whether or not it has label 1_{h}, 1_{m} or 1_{c} in the partition of the MGCF expansion given in Theorem 4.1.

Theorem 5.1. Given θ = [a_{0}; a_{1}, a_{2}, a_{3}, ⋯] with −\frac{1}{2} < θ < \frac{1}{2} and suppose a_{n+1} = 1.

Set α_{n} = [0, a_{n}, a_{n−1}, …,a_{1}] = \frac{q_{n−1}}{q_{n}}, and β_{n} = [a_{n+1}, a_{n+2}, ⋯], so that α_{n} ∈ [0, 1]

and β_{n} ∈ [1, 2]. In terms of the linear fractional transformation

N(z) := [ \begin{matrix}1 & 2 \\ 2 & 1 \end{matrix} ] (z) = \frac{z + 2}{2z + 1} , (5.1)

the Minkowski geodesic continued fraction expansion of θ has

(5.2)  ã_{n+1} = \{ \begin{matrix}1_{h} & if β_{n} > N(α_{n}) , \\ 1_{c} & if β_{n} = N(α_{n}) , \\ 1_{m} & if β_{n} < N(α_{n}) . \end{matrix}

Remark. The matrix N acting as a linear fractional transformation maps [0, 1] to

[1, 2] while reversing orientation. Its inverse N^{−1} = \frac{1}{3} [ \begin{matrix}−1 & 2 \\ 2 & −1 \end{matrix} ] is not integral. The

theorem could also be formulated in terms of the linear fractional transformation −N which maps [0, 1] to [−2, −1] and is an involution, i.e. −N(−N(z)) ≡ z. In particular, since N sends ℝ^{+} into itself,

β < N(α) ⟺ α > −N(−β) . (5.3)

The symbol 1_{c} can occur only when θ is rational, because N(α_{n}) is always rational, while β_{n} is rational if and only if the expansion of θ terminates.

Proof of Theorem 5.1. Note that if a_{1} = 1, it always becomes 1_{m}, and we have α_{n} = 0 and thus N(α_{n}) = 2 > β_{n} as required. The following discussion assumes that a_{n+1} = 1, with n ≥ 1, so that we are not in the segment B_{0}.

Whether a_{n+1} becomes 1_{h}, 1_{c}, or 1_{m} depends on which of (4.5) and (4.6) holds with equality for a larger value of t; we have 1_{m} if it is (4.6), and 1_{c} if both hold with equality for the same t. Since a_{n+1} = 1, we have

p_{n+1} = p_{n} + p_{n−1}.
q_{n+1} = q_{n} + q_{n−1}.
p_{n+1} − q_{n+1}θ = (p_{n} − q_{n}θ) + (p_{n−1} − q_{n−1}θ) ,

The condition for 1_{m} is thus

\frac{(p_{n} − q_{n}θ)^{2} − (p_{n+1} − q_{n+1}θ)^{2}}{q_{n+1}^{2} − q_{n}^{2}} > \frac{(p_{n−1} − q_{n−1}θ)^{2} − (p_{n} − q_{n}θ)^{2}}{q_{n}^{2} − q_{n−1}^{2}} , (5.4)

or, equivalently,

\frac{(p_{n} − q_{n}θ)^{2} − (p_{n+1} − q_{n+1}θ)^{2}}{(p_{n−1} − q_{n−1}θ)^{2} − (p_{n} − q_{n}θ)^{2}} > \frac{q_{n}^{2} − q_{n+1}^{2}}{q_{n−1}^{2} − q_{n}^{2}} , (5.5)

and the condition for 1_{c} is equality. Putting everything in terms of p_{n−1}, p_{n}, q_{n−1}, and q_{n} gives

\frac{(p_{n−1} − q_{n−1}θ)^{2} + 2 (p_{n−1} − q_{n−1}θ) (p_{n} − q_{n}θ)}{(p_{n−1} − q_{n−1}θ)^{2} − (p_{n} − q_{n}θ)^{2}} > \frac{q_{n−1}^{2} + 2q_{n−1}q_{n}}{q_{n−1}^{2} − q_{n}^{2}} . (5.6)

Now let

(5.7)  α_{n} = \frac{q_{n−1}}{q_{n}} ,
(5.8)  β_{n} = −\frac{p_{n−1} − q_{n−1}θ}{p_{n} − q_{n}θ}.

These numbers have ordinary continued-fraction expansions α_{n} = [0, a_{n}, a_{n−1}, …,a_{1}], and β_{n} = [a_{n+1}, a_{n+2}, ⋯]; see Venkov [49, section 2.4]. Clearly 0 < α_{n} < 1 and β_{n} > 1, and, since a_{n+1} = 1, β_{n} < 2. Now (5.6) becomes

\frac{β_{n}^{2} − 2β_{n}}{β_{n}^{2} − 1} > \frac{α_{n}^{2} + 2α_{n}}{α_{n}^{2} − 1} , (5.9)

or, by subtracting 1 from each side,

\frac{−2β_{n} + 1}{β_{n}^{2} − 1} > \frac{2α_{n} + 1}{α_{n}^{2} − 1}. (5.10)

Let C = (2α_{n} + 1)/(α_{n}^{2} − 1). Then (5.10) holds if and only if −β_{n} is between the roots of

Cx^{2} − 2x − 1 − C = 0. (5.11)

The sum of the roots of (5.11) is 2/C, and the larger root is α_{n}, so the other root is

x = \frac{2α_{n}^{2} − 2}{2α_{n} + 1} − α_{n} = \frac{−α_{n} − 2}{2α_{n} + 1} = −N(α_{n}). (5.12)

This gives our desired condition; we have ã_{n+1} = 1_{m} if β_{n} < N(α_{n}). Since −N is an involution and is increasing on [−2, −1] and on [0, 1], we can also write this condition as α_{n} > −N(−β_{n}).

Theorem 5.1 suffices to classify all θ containing the symbol 1_{c}, i.e. all vertical geodesics that hit a corner of a translate of a fundamental domain.

Corollary 5.1. Let −\frac{1}{2} < θ < \frac{1}{2} and suppose that the MGCF expansion of θ contains a symbol 1_{c}. Then θ is rational and has ordinary continued fraction expansion of the form

θ = \begin{cases}[0, a_{1}, a_{2}, , …,a_{n}, 1_{c}, b_{1}, …,b_{m}], & if 0 ≤ θ < \frac{1}{2}, \\ [−1, 1, a_{1} − 1, a_{2}, , …,a_{n}, 1_{c}, b_{1}, …,b_{m}], & if −\frac{1}{2} < θ < 0. \end{cases} (5.13)

in which the additive continued fraction expansion of α_{n}^{∗} := [1, b_{1}, …,b_{m}] is computable from the additive continued fraction expansion of α_{n} = [0, a_{n}, …,a_{1}] by a finite automaton, and vice-versa. Furthermore there is an absolute constant c_{0} such that

(5.14) ACF-length (α_{n}^{∗}) ≤ c_{0} ( ACF-length (α_{n})) .

Proof. Raney [40] proves that given any fixed linear fractional transformation M̃(z) := \frac{az+b}{cz+d} with integer coefficients a, b, c, d and ad − bc ≠ 0, the ACF expansion of M̃(θ) can be computed from the ACF expansion of θ using a finite automaton. Furthermore the conversion process inflates the ACF-length by at most a multiplicative constant (depending on M̃). Apply this to β_{n} = N(α_{n}). The specific automaton for N(z) = \frac{z+2}{2z+1} is given in Raney [40, pp. 274–275], and the constant in this case is 3.

These results give a bound on the computational complexity of computing the MGCF expansion from the additive continued fraction expansion.

Theorem 5.2. The Minkowski geodesic continued fraction expansion of θ can be computed from the additive continued fraction expansion by a Turing machine in quadratic time using linear space. That is, there are absolute constants c_{1}, c_{2}, c_{3} such that for any θ with −\frac{1}{2} < θ < \frac{1}{2}, the first ℓ symbols of the MGCF expansion of θ can be computed using the first c_{1}ℓ symbols of the additive continued fraction expansion of θ using at most c_{2}ℓ^{2} time steps and c_{3}ℓ space locations.

Proof. Theorem 4.1 shows that the main problem is to resolve whether a given symbol a_{n+1} = 1 which appears as the ℓth symbol in the ordinary continued fraction expansion of θ is to be 1_{m}, 1_{h}, or 1_{c} in the MGCF expansion. For this, we use Theorem 5.1. At worst, we must look all the way back to the beginning of the MGCF expansion. Corollary 5.1 shows that the ACF expansion for N(α_{n}) is of length at most c_{0}ℓ, and thus comparing it to β_{n} requires looking at no more than c_{0}ℓ+1 symbols, and thus O(ℓ) time and O(ℓ) space. Since there may be O(ℓ) different 1’s to be resolved, the total time required is O(ℓ^{2}).

There are examples which do require Ω(ℓ^{2}) time steps; for example,

θ = (\sqrt{3} − 1)/2 = [0, \overline{2, 1_{h}}],

has this property. For this θ, it is necessary to backtrack all the way to the first symbol to determine that each 1 is 1_{h}, because the sequence [0, (2, 1_{h})^{j}, 2, 1_{c}, (2, l_{m})^{j}, 4] is a 1_{c}-sequence for each j ≥ 1.

In the next section, we will show that the MGCF expansion cannot be computed from the additive continued fraction expansion using a finite automaton, or even using a pushdown automaton with one stack, as defined in Hopcroft and Ullman [19].

6. Vertical Geodesics: Forbidden Blocks

In this section, we characterize the allowable cutting sequences of Π_{F}^{0}.

Definition 6.1. A finite word W in the symbol set {L̄, R̄, J̄} is a forbidden block of Π_{F}^{0} if it occurs in no cutting sequence of Π_{F}^{0}; otherwise, it is an admissible block. It is an excluded initial block of Π_{F}^{0} if does not occur as an initial segment of any cutting sequence of Π_{F}^{0}; otherwise, it is an included initial block.

It is easy to see that excluded initial blocks alone can be used to characterize any subset of the one-sided shift on {L̄, R̄, J̄}. We show that Π_{F}^{0} has the stronger property that it is determined by its set of forbidden blocks, as follows.

Theorem 6.1. A finite word W in the symbols L̄, R̄, J̄ is an excluded initial block in Π_{F}^{0} if and only if at least one of the blocks L̄W and R̄W is a forbidden block of Π_{F}^{0}.

Proof. We prove the contrapositive. First, we show that for every included initial block W, both R̄^{n}Wand L̄^{n}Ware admissible blocks of Π_{F}^{0} for all n ≥ 1. Geometrically, this encodes the fact that the geodesic ⟨∞, θ⟩ is a limit of geodesics ⟨θ′, θ⟩ as |θ′| → ∞, where θ′ → −∞ is associated to R̄^{n}W̄ and θ → +∞ is associated to L̄^{n}W̄. The block W corresponds to a finite initial segment of the cutting sequence of θ + it for some irrational θ, say for t_{0} ≤ t ≤ ∞ and it has first symbol J̄. By Lemma 2.2, this geodesic cannot hit a corner of any translate of F, hence there is some positive ϵ such that it is at distance at least ϵ from any corner for t_{0} ≤ t ≤ θ. Pick p/q a large negative rational number, and observe that the geodesic ⟨θ′, θ⟩ contains the word R̄^{n}W in its one-sided infinite cutting sequence, in which it enters the domain F just after the R̄^{n}. Here we require that the radius r = θ − p/q is at least n + 2 so that it produces the sequence R̄^{n}, and satisfies r − \sqrt{r^{2} − 1} < ϵ so that the geodesic is within ϵ of ⟨∞, θ⟩ over the range t_{0} ≤ t < 1 and thus produces the sequence W. Now take a matrix

M = [ \begin{matrix}q′ & p′ \\ q & −p \end{matrix} ] ∈ SL(2, ℤ), so that M(γ) = ⟨∞, θ′⟩ with θ′ = M(θ), and note that

θ′ is necessarily irrational. The P SL(2, ℤ)-action doesn’t affect cutting sequences, so the cutting sequence of M(γ) still contains the word R̄^{n}W. There is a unique choice of q′ and p′ such that −\frac{1}{2} < θ′ < \frac{1}{2}. This vertical geodesic has the same cutting sequence as γ. Thus R̄^{n}W and likewise L̄^{n}W are admissible blocks of Π_{F}^{0} for all n ≥ 1.

Second, we prove that if W = S_{1} ⋯ S_{n} is a word in L̄, R̄, J̄ for which R̄Wand L̄Ware both admissible blocks of Π_{F}^{0}, there is a vertical geodesic whose cutting sequence has W as an initial segment. To show this, note that the first symbol in W is necessarily J̄, since L̄R̄ and R̄L̄ are forbidden blocks. Let γ_{1} and γ_{2} be irrational vertical geodesics whose cutting sequences contain the words R̄W and L̄W, respectively. Translate each of them by the appropriate elements of P SL(2, ℤ) to geodesics γ′_{1} = ⟨p_{1}/q_{1}, θ′_{1}⟩ and γ′_{2} = ⟨p_{2}/q_{2}, θ′_{2}⟩ in such a way that the translated geodesics enter the fundamental domain F at the symbol immediately preceding W in their cutting sequences. Since the cutting sequence of γ′_{1} has the letter R̄ as it enters F, it is oriented in the direction of increasing real part and has p_{1}/q_{1} < −1/2, while γ′_{2} is oriented in the direction of decreasing real part and has p_{2}/q_{2} > 1/2. The first symbol in W is J̄, hence each geodesic exits F through its bottom edge at a point with real part between −1/2 and 1/2. If γ′_{1} and γ′_{2} do not intersect, that is if θ′_{1} < θ′_{2}, then pick any irrational θ between θ′_{1} and θ′_{2}, say θ = θ′_{1}, and let γ = ⟨∞, θ⟩. If γ′_{1} and γ′_{2} do intersect, let θ be the real part of their intersection and let γ = ⟨∞, θ⟩. In either case, γ passes through F since −\frac{1}{2} < θ′_{1} ≤ θ ≤ θ′_{2} < \frac{1}{2}. We show that γ has initial word W in its cutting sequence. Since γ′_{1} hits the vertical line Re(z) = −1/2 and then the unit circle, it is outside the unit circle at all points with Re(z) ≤ −1/2 and inside with Re(z) ≥ 1/2; likewise, γ′_{2} is inside the unit circle at all points Re(z) ≤ −1/2 and outside with Re(z) > 1/2. Thus the intersection must have real part θ with −\frac{1}{2} < θ < \frac{1}{2}. In either case, after leaving F, both γ′_{1} and γ′_{2} pass through the same sequence of translated fundamental domains F_{1}, …, F_{n}, entering along the same edge of each, corresponding to the symbols in W. The vertical geodesic γ must then enter F_{j} on the same edge at a point z_{j} between the points z_{1,j} and z_{2,j} where γ′_{1} and γ′_{2} hit it, because the domains F_{j} are hyperbolically convex; see Figure 6.1. Also, it cannot hit any other fundamental domain between z_{j−1} and z_{j}, because every point in that interval on γ is on a geodesic between points of γ′_{1} and γ′_{2} which are in F.

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\frac{p_{2}}{q_{2}}; θ; \frac{p_{1}}{q_{1}}; γ; γ′_{1}; γ′_{2}; F_{j}; z_{1,j}; z_{2,j}]

Figure 6.1. Intersecting geodesics: γ is trapped between γ′_{1} and γ′_{2} within F_{j}.

Thus the initial cutting sequence of γ is W. Finally, if the finite endpoint θ of γ is rational, then since γ hits no corners up to hitting the edge W_{n}, it is a Euclidean distance at least ϵ away from every corner on the W-edges. For every irrational θ′ with |θ′ − θ| < ϵ, the irrational vertical geodesic γ′ = {θ′ + it : t > 0} has the same initial segment W.

We next obtain from Theorem 5.1 a characterization of the cutting sequences in Π_{F}^{0} in terms of certain forbidden blocks plus a subset of excluded initial blocks. These conditions involve a critical symbol 1_{h} or 1_{m} which is mislabeled in a cutting sequence expansion.

Definition 6.2. A finite sequence [d_{1}, …,d_{n}, 1, b_{1}, …,b_{m}] of positive integers is ambiguous if it does not suffice to determine the MGCF-label on the critical symbol 1. More precisely, if

δ_{0} := [0, d_{n}, …,d_{1}], and δ_{1} := [0, d_{n}, …,d_{1} + 1],
β_{0} := [1, b_{1}, …,b_{m}], and β_{1} := [1, b_{1}, …,b_{m} + 1] ,

then [d_{1}, …,d_{n}, 1, b_{1}, …,b_{n}] is ambiguous if and only if

(6.1) int[β_{0}, β_{1}] ∩ int[N(δ_{0}), N(δ_{1})] ≠ ∅, Here int[δ_{0}, δ_{1}] denotes the closed interval determined by δ_{0} and δ_{1} with any ordering of the endpoints, i.e. δ_{1} < δ_{0} may occur.

Note that the interval int[N(δ_{0}), N(δ_{1})] contains all real numbers whose continuedfraction expansions begin with the same partial quotients as δ_{0}.

Definition 6.3. A central sequence [d_{1}, …, d_{n}, 1, b_{1}, …, b_{n}] is an ambiguous sequence such that p/q = [0 or − 1, d_{1}, …,d_{n}, 1, b_{1}, …,b_{m}] is a 1_{c} sequence. We say that p/q is the rational number associated to this central sequence.

We designate initial words of Π_{F}^{0} using the initial symbol d_{1} = ∞.

Definition 6.4. An initial sequence [∞, d_{2}, …, d_{n}, 1, b_{1}, …, b_{1}] denotes an initial word, which if −\frac{1}{2} < θ < 0 has d_{2} = 1, d_{3} = a_{1} − 1, and d_{i} = a_{i−2} for i ≥ 4, while if 0 < θ < \frac{1}{2}, d_{i} = a_{i−1} for all i ≥ 2.

We define ambiguous initial sequences as in Definition 6.2. If we set

δ_{0} = δ_{1} = [0, d_{n}, …, d_{2}] = [0, d_{n}, …, d_{2}, ∞], (6.2)

then the initial sequence is ambiguous if

(6.3) int[β_{0}, β_{1}] ∩ int[N(δ_{0}), N(δ_{1})] ≠ ∅,

where int[N(δ_{0}), N(δ_{1})] is the single point N(δ_{0}).

Theorem 6.2 (Characterization of Π_{F}^{0}). The set Π_{F}^{0} of cutting sequences of irrational θ in −\frac{1}{2} < θ < \frac{1}{2} uses the alphabet \{R̄, L̄, J̄\}. It consists of all sequences that factorize in segments B_{0}B_{1}B_{2} … as in Theorem 4.2, with the additional property that no sequence of consecutive segments corresponds to a Minkowski geodesic continued fraction expansion

[d_{1}, …, d_{n}, 1_{∗}, b_{1}, …, b_{m}]

with d_{1} = ∞ allowed, with associated δ_{0}, δ_{1}, β_{0}, β_{1}, such that:

(i) Both of [d_{2}, …, d_{n}, 1, b_{1}, …, b_{m}] and [d_{1}, d_{2}, …, d_{n}, 1, b_{1}, …, b_{m−1}] are ambiguous.

(ii) Either

(6.4) 1_{∗} = 1_{h} and int[β_{0}, β_{1}] < int[N(δ_{0}), N(δ_{1})],

or

(6.5) 1_{∗} = 1_{m} and int[β_{0}, β_{1}] > int[N(δ_{0}), N(δ_{1})] .

Proof. Immediate from Theorem 5.1 using Theorems 4.1 and 4.2.

Theorem 6.2 characterizes a large set of forbidden blocks which are given by conditions (i) and (ii) when d_{1} is finite. It also gives an additional set of excluded initial blocks when d_{1} = ∞. These are sufficient to determine Π_{F}^{0} because they identify each 1 as 1_{h} or 1_{m} as appropriate. However, Theorem 6.2 does not give a complete set of minimal forbidden blocks, because there are extra forbidden blocks in which several symbols d_{i} = 1 and b_{i} = 1 are replaced by 1_{h} and 1_{m}, which if left as indeterminate 1’s would not be forbidden.

The complete set of minimal forbidden blocks of Π_{F}^{0} seems harder to characterize. Consider a given block of symbols L̄, R̄, J̄; this can be parsed by Theorem 4.2 into a block of complete segments [a_{1}, …, a_{m}] in which each symbol a_{i} is either 1_{h}, 1_{m}, or an integer at least 2, together with some conditions on the adjacent incomplete segments; for example, an incomplete segment L̄^{4} can be part of a complete segment encoding either a_{m+1} ≥ 4, or a_{m+1} = 3, a_{m+2} = 1_{m}. Then [a_{1}, …, a_{m}] is a forbidden block if and only if a certain finite set of linear fractional conditions on two real numbers α, β of the form

α < \frac{aβ + b}{cβ + d} (6.6)

with ad − bc = ±1 are inconsistent. There is one such inequality for each symbol 1_{h} or 1_{m} in the block, which encodes the condition that [α, a_{1}, …, a_{m}, β] produces the correct symbol 1_{h} or 1_{m}. There may also be one or two inequalities on α alone (or on β alone) if there is an incomplete segment at that end, and one more condition (6.6) if the possible encoding of that segment includes another symbol 1_{h} or 1_{m}. The total number of inequalities is linear in m.

Tables 6.1, 6.2, and 6.3 below list some central sequences, ambiguous sequences and forbidden blocks, respectively; these were obtained by applying the transformation N to simple δ_{i}. Table 6.3 illustrates the computation of forbidden sequences from a central sequence; any change to the terms in a central sequence which increases either α_{n} (and thus decreases N(α_{n})) or β_{n} forces the 1 to be 1_{h}, and conversely for 1_{m}.

Theorem 6.2 implies that Π_{F}^{0} is complicated in the sense that its set of minimal forbidden blocks is very large.

Theorem 6.3. The number n(k) of minimal forbidden blocks of Π_{F}^{0} of length at most k grows exponentially in k; in fact

lim inf_{k→∞} n(k)^{1/k} ≥ 2^{1/12}. (6.7)

Proof. We will show that each central sequence associated to a rational p/q ≠ 1/2 yields two minimal forbidden blocks. The forbidden blocks are produced by adding one symbol to each end of the central sequence, and by replacing the central 1_{c} with a three-symbol word. All these forbidden blocks are distinct. Assuming these facts are proved, consider the central sequences

[d_{1}, d_{2}, …, d_{n}, 1_{c}, b_{1}, …, b_{m}]. (6.8)

in which each d_{i} = 1 or 2, and in which [b_{1}, …, b_{m}] is determined from [d_{1}, …, d_{n}]. The number of symbols in the cutting sequence encoding of [d_{1}, …, d_{n}] is at most 3n, and by Corollary 5.1, the number of symbols in the cutting sequence encoding of [b_{1}, …, b_{m}] is at most 9n. In obtaining forbidden blocks, the 1_{c} term is encoded by three symbols, and one symbol is added at each end, hence all resulting forbidden blocks contain at most 12n + 5 symbols. We conclude that there are at least 2^{n+1} minimal forbidden blocks of length at most 12n + 5, which proves (6.7).

\begin{matrix}[2, & 1_{c}, 4] & [2, & 1_{c}, 3, 1] & [1, 1, & 1_{c}, 4] & [1, 1, & 1_{c}, 3, 1] \\ [3, & 1_{c}, 2, 2] & [3, & 1_{c}, 2, 1, 1] & [1, 2, & 1_{c}, 2, 2] & [1, 2, & 1_{c}, 2, 1, 1] \\ [4, & 1_{c}, 2] & [4, & 1_{c}, 1, 1] & [1, 3, & 1_{c}, 2] & [1, 3, & 1_{c}, 1, 1] \\ [3j + 1, & 1_{c}, 1, j] & [3j + 1, & 1_{c}, 1, j − 1, 1] & [1, 3j, & 1_{c}, 1, j] & [1, 3j, & 1_{c}, 1, j − 1, 1] \\ [3j + 2, & 1_{c}, 1, j, 3] & [3j + 2, & 1_{c}, 1, j, 2, 1] & [1, 3j + 1, & 1_{c}, 1, j, 3] & [1, 3j + 1, & 1_{c}, 1, j, 2, 1] \\ [2, 2, & 1_{c}, 3] & [2, 2, & 1_{c}, 2, 1] & [1, 1, 2, & 1_{c}, 3] & [1, 1, 2, & 1_{c}, 2, 1] \\ [2, 3, & 1_{c}, 2, 5] & [2, 3, & 1_{c}, 2, 4, 1] & [1, 1, 3, & 1_{c}, 2, 5] & [1, 1, 3, & 1_{c}, 2, 4, 1] \\ [3, 2, & 1_{c}, 3, 4] & [3, 2, & 1_{c}, 3, 3, 1] & [1, 2, 2, & 1_{c}, 3, 4] & [1, 2, 2, & 1_{c}, 3, 3, 1] \\ [4, 3, & 1_{c}, 2, 3] & [4, 3, & 1_{c}, 2, 2, 1] & [1, 3, 3, & 1_{c}, 2, 3] & [1, 3, 3, & 1_{c}, 2, 2, 1] \\ [5, 2, & 1_{c}, 3, 2] & [5, 2, & 1_{c}, 3, 1, 1] & [1, 4, 2, & 1_{c}, 3, 2] & [1, 4, 2, & 1_{c}, 3, 1, 1] \\ [j, 1, & 1_{c}, 3j + 1] & [j, 1, & 1_{c}, 3j, 1] & [1, j − 1, 1, & 1_{c}, 3j + 1] & [1, j − 1, 1, & 1_{c}, 3j, 1] \\ [3, j, 1, & 1_{c}, 3j + 2] & [3, j, 1, & 1_{c}, 3j + 1, 1] & [1, 2, j, 1, & 1_{c}, 3j + 2] & [1, 2, j, 1, & 1_{c}, 3j + 1, 1] \end{matrix}

Table 6.1. Some central sequences, including all with at most five terms.

[ …, j, 1, \underline{1}, 3j + 1, any, …] [ …, any, 3j + 1, \underline{1}, 1, j, …]
[ …, j, 1, \underline{1}, 3j + 2, any, …] [ …, any, 3j + 2, \underline{1}, 1, j, …]
[ …, j, 1, \underline{1}, 3j + 3, any, …] [ …, any, 3j + 3, \underline{1}, 1, j, …]
[ …, 1, 2, \underline{1}, 2, 1, …]
[ …, 2, 2, \underline{1}, 3, ≥ 4, …]  [ …, ≥ 4, 3, \underline{1}, 2, 2, …]
[ …, 3, 2, \underline{1}, 3, 2, …]  [ …, 2, 3, \underline{1}, 2, 3, …]
[ …, 3, 2, \underline{1}, 3, 3, …]  [ …, 3, 3, \underline{1}, 2, 3, …]
[ …, 4, 2, \underline{1}, 3, 2, …]  [ …, 2, 3, \underline{1}, 2, 4, …]
[ …, ≥ 5, 2, \underline{1}, 3, 1, …]  [ …, 1, 3, \underline{1}, 2, ≥ 5, …]

Table 6.2. Non-central ambiguous sequences which go two terms forward and two terms back from the underlined 1.

\begin{matrix}[≥ 3, & 1_{h}, & ≥ 3] \\ [≥ 3, & 1_{h}, & 2, 1] \\ [1, & 1_{m}, & 1] \\ [≥ 1, 2, & 1_{m}, & 1] \\ [1, & 1_{m}, & 2, ≥ 2] \\ [≥ 1, 2, & 1_{m}, & 2, ≥ 2] \end{matrix}

Table 6.3. Forbidden sequences obtained from the central sequence [1, 2, 1_{c}, 2, 2]; the reverses of these sequences are also forbidden.

We now construct the minimal forbidden blocks. We use the result of Appendix A, which shows that any vertical geodesic for θ = p/q with −\frac{1}{2} < θ < \frac{1}{2} hits at most one corner of a fundamental domain. This fact implies that each of the central sequences (6.8) contains only one symbol 1_{c}, and thus each p/q comes from at most one central sequence. Associated to the central sequence is a word W_{1}SW_{2} in which S = C̄_{1} or C̄_{2} is a corner symbol. For the corner symbol, there are two choices of three symbols in {R̄, L̄, J̄}, which replace the 1_{c} by 1_{h} or 1_{m}; for C̄_{1}, the choices are J̄R̄J̄ or L̄J̄L̄. We consider the eight words obtained from W_{1}SW_{2} by replacing S by either choice of a three-symbol block, and by adding a prefix symbol and a suffix symbol, each of which may be either L̄ or R̄. The continued fraction for p/q can be recovered from any one of these eight blocks, and since it has only a single 1_{c}, the critical 1_{c} in the central sequence is also uniquely determined.

We claim that six of these eight blocks are admissible blocks for Π_{F}^{0} and the other two are forbidden blocks. The vertical geodesic γ = ⟨∞, p/q⟩ corresponding to

\frac{p}{q} = [0 or − 1, d_{1}, …, d_{m}, 1_{c}, b_{1}, …, b_{m}]

can be approximated by geodesics in the symbol topology in six different ways. Two of these consist of approximating geodesics which do not cross γ at all but approach it from the left and right, respectively. The other four consist of approximating geodesics which cross γ, either crossing above or below the corner that γ hits at the 1_{c}, and initially approaching γ either from the left or from the right. For sufficiently good approximations, these produce the admissible blocks. (Here we again use the fact that the geodesic γ hits exactly one corner, which implies that all geodesics sufficiently close to γ have the same sequences W_{1} and W_{2}.) The other two blocks are forbidden by condition (ii) of Theorem 6.2. They encode cutting sequences for a geodesic that would have to approach γ from the left, pass to the right of the corner, and then return to the left of γ, or vice versa; such a geodesic would have to cross γ twice, which is impossible.

Let C̄_{a} and C̄_{b} denote the two possible three-symbol encodings of S. Of the four blocks

R̄W_{1}C̄_{a}W_{2}R̄ , R̄W_{1}C̄_{a}W_{2}L̄ , L̄W_{1}C̄_{a}W_{2}R̄ and L̄W_{1}C̄_{a}W_{2}L̄ ,

exactly three are admissible and one is forbidden. Every sub-block of the forbidden block appears in one of the three admissible blocks, hence the forbidden block is minimal. The same argument applies to the other four blocks containing C̄_{b}, and produces a second minimal forbidden block.

Theorem 6.4. There does not exist a finite automaton which when given the additive continued fraction expansion of an irrational number θ with 0 < θ < \frac{1}{2} computes the Minkowski geodesic continued fraction expansion of θ, or, equivalently, the cutting sequence expansion of the geodesic ⟨∞, θ⟩.

Proof. Any finite automaton that would compute the Minkowski geodesic expansion of θ must output the n-th term of this expansion after seeing at most a bounded number of symbols following the n-th symbol of the ACF expansion of θ. However the forbidden block criteria of Theorem 6.2 show that it is sometimes necessary to see an arbitrarily large string of symbols after the n-th symbol, to decide if 1_{h} or 1_{m} should be used. For example, for each j ≥ 1, the sequence

[2^{4j+2}, 1_{c}, 3, (8, 4)^{j}] (6.9)

is a central sequence. Adding some later terms decreases β_{4j+2}, and thus changes the 1_{c} to 1_{m}, while decreasing the final 4 in (8, 4)^{j} to a 3 and then adding some further terms increases β_{4j+2} and thus changes the 1_{c} to 1_{h}. A finite automaton cannot look ahead through the 12j + 4 terms which are necessary, since j can be any integer. In fact, even a pushdown automaton (with one stack) cannot correctly compute all such cutting sequences, because it must look 12j +4 steps ahead, but also 8j +4 steps back to distinguish [a_{0}; 2^{4j}, 1_{m}, 3, (8, 4)^{j}, 8, 3, …] from [a_{0}; 2^{4j+2}, 1_{h}, 3, (8, 4)^{j}, 8, 3, …].

7. Two-sided Cutting Sequences: Structure of Σ_{F}

We now use the information on vertical cutting sequences Π_{F}^{0} to characterize the two-sided cutting sequences Σ_{F}.

Theorem 7.1. The cutting sequence shift Σ_{F} for the fundamental domain F of P SL(2, ℤ) is the closed subshift whose forbidden blocks coincide with the set of forbidden blocks of Π_{F}^{0}.

Proof. This result follows from the fact that the set of images of the set of irrational vertical geodesics under P SL(2, ℤ) is dense in the space of all geodesics of ℌ/P SL(2, ℤ).

Let S_{Π}^{0} and S_{Σ}^{0} and S_{Σ} denote the complete sets of forbidden blocks of Π_{F}^{0}, Σ_{F}^{0}, and Σ_{F}, respectively. Since Σ_{F} is the closure of Σ_{F}^{0}, we have S_{Σ}^{0} = S_{Σ}.

We first show that

S_{Σ}^{0} ⊆ S_{Π}^{0}. (7.1)

For this it suffices to show that every admissible word W in Π_{F}^{0} is an admissible word in Σ_{F}^{0}. Suppose that the word W appears in the cutting sequence of the irrational vertical geodesic ⟨∞, θ⟩ = {θ + it : t > 0}. This geodesic is a limit of geodesics ⟨ϕ_{i}, θ⟩ where ϕ_{i} → ∞ through a sequence of values such that ⟨ϕ_{i}, θ⟩ hits no corner of an image of F. The word W = S_{1} … S_{r} corresponds to a specific set of edges of translated fundamental domains {g_{j}F : 0 ≤ j ≤ r} with g_{j} ∈ SL(2, ℤ). For all sufficiently large ϕ_{i}, the geodesic ⟨ϕ_{i}, θ⟩ passes through the same sequence of fundamental domains {g_{j}F : 0 ≤ j ≤ r}, hitting the same sequence of edges in the same order. Thus W occurs in the two-sided cutting sequence of ⟨ϕ_{i}, θ⟩, so it is an admissible word of Σ_{F}^{0}.

The reverse inclusion

S_{Π}^{0} ⊆ S_{Σ}^{0}. (7.2)

is proved similarly. Let W be an admissible word in some general position geodesic γ = ⟨ϕ, θ⟩. There is a sequence of translated fundamental domains {F_{j} : 0 ≤ j ≤ r} which ϕ passes through that corresponds to W. Since γ hits no corners, we can choose a rational number p/q sufficiently close to ϕ such that the geodesic γ′ = ⟨p/q, θ⟩ passes through the same set of fundamental domains {F_{j} : 0 ≤ j ≤ r}, hitting the same sequence of edges in the same order, hence its cutting sequence (which is only one-sided infinite) contains the word W. Now apply to γ′ the transformation M in P SL(2, ℤ) which takes p/q to ∞, and θ to some θ′ in (−\frac{1}{2}, \frac{1}{2}). For this choice, the cutting sequence of M(γ′) is in Π_{F}^{0}, and (7.2) follows.

Theorem 7.2. Every element in Σ_{F} is a cutting sequence for a unique oriented geodesic on ℌ/P SL(2, ℤ) which hits F, except for the two sequences R̄^{∞} or L̄^{∞}. Every oriented geodesic γ on ℌ/P SL(2, ℤ) has at least one and at most finitely many shift-equivalence classes of cutting sequences in Σ_{F}. If γ is not periodic then it has at most eight shift-equivalence classes of cutting sequences in Σ_{F}.

To establish this result, we first prove a preliminary lemma.

Lemma 7.1. Let γ_{j} = ⟨θ′_{j}, θ_{j}⟩ for j = 1, 2, … be a sequence of general position geodesics that intersect F which have cutting sequences {S_{i}^{(j)} : i ∈ ℤ} such that the symbol S_{0}^{(j)} corresponds to the geodesic γ_{j} entering the fundamental domain F. If the cutting sequences {S_{i}^{(j)}} converge in the symbol topology as j → ∞ to a limit sequence {S_{i}} then the endpoints θ′_{j} and θ_{j} converge to unequal limiting values

(7.3) θ′ = lim_{j→∞} θ′_{j} and θ = lim_{j→∞} θ_{j}.

in ℝ ∪ {−∞, ∞}. The geodesics γ_{j} converge to a limiting geodesic γ = ⟨θ′, θ⟩ if at least one of θ, θ′ is finite. The exceptional cases ⟨−∞, ∞⟩ and ⟨∞, −∞⟩ correspond to the limiting symbol sequences R̄^{∞} and L̄^{∞}, respectively.

Proof. Let S_{+} = {S_{i} : i > 0} and S_{−} = {S_{i} : i ≤ 0} denote the positive and negative part of the limit sequence, respectively. If S_{+} is R̄^{∞}, then since any geodesic γ_{j} which enters F after S_{0} and has S_{i}^{(j)} = R̄ for 1 ≤ i ≤ n must have θ_{j} ≥ n + 1/2, we conclude that lim_{j→∞} θ_{j} = ∞ in this case. Similarly, if S_{+} is L̄^{∞}, then lim_{j→∞} θ_{j} = −∞; if S_{−} is R̄^{∞}, then lim_{j→∞} θ′_{j} = −∞ if S_{−} is L̄^{∞}, then lim_{j→∞} θ′_{j} = −∞. In particular, this shows that the two-sided limit sequence R̄^{∞} corresponds to ⟨−∞, ∞⟩, and L̄^{∞} corresponds to ⟨∞, −∞⟩

We next observe that since R̄L̄ and L̄R̄ are forbidden blocks, if S_{+} is not R̄^{∞} or L̄^{∞} it must contain a symbol J̄. Let the initial segment of S_{+} up to the first such symbol be R̄^{n}J̄ (resp. L̄^{n}J̄) Any geodesic γ_{j} which matches these symbols necessarily has θ_{j} ≤ n + 3/2 (resp. θ_{j} ≥ n − 3/2) and is positively oriented (resp. negatively oriented), hence θ_{j} ≥ −1/2 (resp. θ_{j} ≤ 1/2). In either case, if a limit (7.3) exists it must be finite. Similar reasoning applies to S_{i} to show that lim_{j→∞} θ′_{j} is infinite if and only if S_{−} does not contain a symbol J̄.

We now suppose that S_{+} contains a symbol J̄, and claim that lim_{j→∞} θ_{j} exists and is finite. The argument above shows that {θ_{j}} is bounded, so to prove this claim it suffices to show that if {θ_{j}} is not a Cauchy sequence then the one-sided sequences {S_{i}^{(j)} : i > 0} cannot converge in the symbol topology. If it is not a Cauchy sequence, there is some ϵ > 0 such that for any N, we have j, k ≥ N with |θ_{j} − θ_{k}| > ϵ. The general-position geodesics γ_{j} and γ_{k} enter F and thus have radius at least \sqrt{3}/2. Consequently, any point z = x + iy on the geodesic γ_{j} with |x − θ_{j}| < 1/4 and 0 < y < ϵ′ for ϵ′ < ϵ^{2}/36 will actually have |x − θ_{j}| < ϵ/3, and thus |x − θ_{k}| > 2ϵ/3; similarly, points on γ_{k} with 0 < y < ϵ′ and |x − θ_{k}| < 1/4 must have |x − θ_{k}| > 2ϵ/3. This implies an upper bound N_{ϵ} on the number of fundamental domains which γ_{j} and γ_{k} both hit somewhere inside the box |Re(z) − (θ_{j} + θ_{k})/2| ≤ 2ϵ, Im(z) ≤ ϵ. We may assume ϵ < 1/2, hence any such fundamental domain has a cusp at a finite rational point p/q. The rosette R(p/q) of all translated fundamental domains that touch p/q has Euclidean diameter less than 4/q^{2}. Thus any translated fundamental domain F′ which is hit by both γ_{j} and γ_{k} inside the box must have two points whose real parts differ by at least ϵ/3, hence q < \sqrt{12/ϵ}. In addition, p/q must lie within Euclidean distance 4/q^{2} of both θ_{j} and θ_{k}, hence there are only finitely many such rational p/q. Finally, there is a finite bound depending only on q and ϵ on the number of fundamental domains in each rosette R(p/q) which have Euclidean diameter exceeding ϵ/3. Together this yields a finite upper bound N_{ϵ} on the number of common fundamental domains F′ between γ_{j} and γ_{k} in the box. Outside the box, after exiting F the geodesics γ_{j} and γ_{k} traverse the strip ϵ′ ≤ Im(z) ≤ 1, and thus can hit at most N′_{ϵ} translated fundamental domains F′ in that strip, because there are only finitely many geodesics with Euclidean radius at least ϵ′ which intersect the region −\frac{1}{2} < Re(z) < \frac{1}{2}. We conclude that the positive cutting sequences γ_{j} and γ_{k} cannot agree on all of their first N_{ϵ} + N′_{ϵ} + n + 1 symbols. Thus the sequences S_{+}^{(j)} = {S_{i}^{(j)} : j > 0} cannot converge in the symbol topology, which proves the claim.

A similar argument applies to the limit sequence S_{−} if it contains the symbol J̄, to prove that lim_{j→∞} θ′_{j} = θ′ exists and is finite.

The geodesic γ = ⟨θ′, θ⟩ is encoded as a limit of the sequence of geodesics γ_{i}. If all of the γ_{i} for i ≥ N intersect a particular domain F′, then since the endpoints θ′_{i} and θ_{i} converge to θ′ and θ and F is closed, γ also intersects F, at least on the boundary. If no γ_{i} for i ≥ N intersect a domain F′, then γ does not intersect the interior of F′. Thus, if we consider γ to hit a domain if either it passes through the interior, or it intersects the boundary and all γ_{i} for sufficiently large i intersect the interior, we can define a cutting sequence as the set of edges which separate these domains, and this sequence is the limit in the symbol topology of the cutting sequences of γ_{i}.

Proof of Theorem 7.2. Since Σ_{F} is a compact set, every oriented geodesic γ = ⟨θ′, θ⟩ which hits the interior of F has at least one cutting sequence in C(γ) by taking the limit point of cutting sequences from a family γ_{j} = ⟨θ′_{j}, θ_{j}⟩ with θ′_{j} → θ′ and θ_{j} → θ as j → ∞.

It remains to show that each oriented geodesic in ℌ which hits F has only a finite number of shift-equivalence classes of cutting sequences in [C(γ)]. The simplest cases are general position geodesics, for which C(γ) is a single cutting sequence, and [C(γ)] consists of a single shift-equivalence class.

We determine how many shift-equivalence classes of cutting sequences are possible for a limiting geodesic. First, consider the limiting geodesics with exactly one rational endpoint (considering ∞ as rational). These correspond to vertical geodesics under the P SL(2, ℤ) action, so by Lemma 2.2, such geodesics cannot hit any internal corners, and thus have two possible encodings; either can be obtained by using approximating geodesics which approach the rational endpoint from either side. Thus [C(γ)] contains two shift-equivalence classes.

Next, consider those geodesics which have two rational endpoints. Without loss of generality, we can move one endpoint to ∞ and so have a vertical geodesic γ = ⟨∞, p/q⟩. Such a geodesic hits only finitely many corners of fundamental domains. If it hits n corners at finite values of t, then C(γ) contains exactly 2n + 4 cutting sequences. Two of them come from geodesics approximating γ from either side without crossing it, and the other 2n + 2 result from approximating geodesics which cross γ between the kth and (k +1)st corners with 0 ≤ k ≤ n, either from left to right or from right to left. (Here the 0th corner is ∞, and the (n + 1)st corner is p/q.) In Appendix A, we prove that n ≤ 1 for all rational p/q, except those p/q ≡ \frac{1}{2}(mod1), which have n = 2. Thus [C(γ)] contains at most eight shift-equivalence classes.

It remains to bound the number of shift-equivalence classes of cutting sequences for geodesics which have two irrational endpoints. If such a geodesic hits no corners, then it has only one cutting sequence in C(γ). If it hits exactly one corner, then it has exactly two cutting sequences, which are obtained using geodesics which approach it while staying on opposite sides of the corner. The difficult case occurs with geodesics that hit at least two corners. We show that all such geodesics hit infinitely many corners and are periodic. In this case, C(γ) will be an infinite set. To show periodicity, we observe that a corner in the upper half-plane is the intersection of two circles with centers at rational points on the x-axis and rational radii, so its x-coordinate is rational and its y-coordinate is the square root of a rational number. Thus, if the circle with radius r and center (x_{0}, 0) passes through two such points (x_{1}, y_{1}) and (x_{2}, y_{2}), then it satisfies the equations

(x_{1} − x_{0})^{2} + y_{1}^{2} = r^{2},
(x_{2} − x_{0})^{2} + y_{2}^{2} = r^{2}.

Since y_{1}^{2} and y_{2}^{2} are rational, so is r^{2}; also, equating the left sides gives a linear equation for x_{0} with rational coefficients. Thus the circle intersects the x-axis in two algebraically conjugate real quadratic surds. Pell’s equation allows us to find M ∈ SL(2, ℤ) which preserves one endpoint (and thus its conjugate). Applying this transformation to our geodesic re-scales it while preserving the orientation; hence the geodesic necessarily is a periodic geodesic on ℌ/P SL(2, ℤ). The set C(γ) is then infinite because we can choose a set of approximating geodesics which cross between any pair of consecutive corners of γ, and the resulting limit cutting sequences are all distinct.

We complete this case by showing that if a periodic geodesic γ on ℌ/P SL(2, ℤ) hits exactly n corners of translates of fundamental domains in its period, then there are exactly 2n + 2 shift-equivalence classes in [C(γ)]. Label the corners which are hit in one period c_{1}, …, c_{n}. The approximating geodesics γ_{i} can only be close in the symbol topology if they all cross γ between the same pair of corners, or if none cross γ at all. If they cross, then we can use the periodicity of γ’s cutting sequence to shift the crossing point between c_{1} and c_{n+1}. With n possible crossing regions, and crossing possible either from inside to outside or vice versa, there are 2n encodings; we get two more from approximating geodesics which are completely inside or outside γ, for a total of exactly 2n + 2 shift-equivalence classes.

Theorem 7.2 gives no uniform upper bound on the number of shift-equivalence classes of cutting sequences that correspond to periodic geodesics. There are periodic geodesics which correspond to 10 cutting sequence shift-equivalence classes because they hit four corners in one period, such as ⟨−\sqrt{13}, \sqrt{13}⟩ and ⟨−\sqrt{133}, \sqrt{133}⟩. It may well be that this is the maximal number that occurs.

Theorem 7.1 implies that the shift Σ_{F} is a relatively complicated set. Indeed Theorem 6.3 showed that the set of minimal forbidden blocks is very large. We now show that Σ_{F} is not a sofic shift. A sofic shift is any shift that is a factor of a shift of finite type (see Marcus and Lind [31].) Alternatively, it is the set of possible bi-infinite walks on an edge-labeled finite graph.

Theorem 7.3. The cutting sequence shift Σ_{F} for the fundamental domain F of P SL(2, ℤ) is not a sofic shift.

Proof. If Σ is a shift, then the follower set F_{Σ}(W) of a one-sided word W = (…, S_{−2}, S_{−1}) is the set of all one-sided words W^{+} = (S_{0}, S_{1}, …) such that WW^{+} is an element of Σ. Sofic shifts are characterized by the property that the totality of different follower sets {F_{Σ}(W) : all W} is finite; see Marcus and Lind [31, Theorem 3.2.10].

We show that Σ_{F} has infinitely many follower sets. Both of the sequences [3, 2^{4j+2}, 1_{c}, 3, (8, 4)^{j}, 10] and [4, 2^{4j+2}, 1_{c}, (8, 4)^{j}, 13] are central. Therefore, in […, 3, 2^{4j+2}, 1_{∗}, 3, (8, 4)^{k}, 13, …], the 1_{∗} is 1_{h} if j ≥ k but 1_{m} if j < k, regardless of the symbols on either side. Thus the follower sets of one-sided words which end [3, 2^{4j+2}] are different for all j.

8. The Shift Σ_{F} Determines F up to Isometry

Our object is to prove the following result.

Theorem 8.1. Let Γ be a finitely generated discrete subgroup of P SL(2, ℝ) that acts properly discontinuously on ℌ, and which has a polygonal fundamental domain P which is hyperbolically convex. Suppose that the cutting sequence shift Σ_{P} is isomorphic to Σ_{F,P SL(2,ℤ)} by a permutation of symbol alphabets. Then there is an element g ∈ P SL(2, ℝ) such that F = gP and

P SL(2, ℝ) = gΓg^{−1} .

Proof. Since the cutting sequence contains only three symbols, the polygon must be a triangle.

Since a geodesic can hit the L̄edge an unlimited number of consecutive times, the L̄ and L̄^{−1} edges of the triangle must intersect at an angle of zero. The L̄^{−1} edge cannot be the same as the L̄ edge because a geodesic can hit it twice consecutively, nor can it be the J̄ edge because no geodesic can hit that edge twice consecutively; thus it must be the R̄ edge. Since a geodesic cannot hit the J̄ edge twice consecutively, the J̄ edge must equal the J̄^{−1} edge. Thus the generator at the J̄ edge is an involution, and to have determinant 1, it must be an inversion.

Let the angle between J̄ and L̄ be α, and the angle between J̄ and R̄ be β. If 2α + β < π, then a geodesic approaching the JR-corner could hit edges J̄, L̄, J̄, L̄ in sequence; however, this cutting sequence is not possible for our fundamental domain. Likewise, 2β + α ≥ π. Also, for any hyperbolic triangle, the sum of the angles is less than π, so α + β < π.

The conditions for a given polygon to be a fundamental domain are given in Maskit [32, section 2]. In our case, it is necessary that either n(α + β) = 2π for some n, or nα = π and mβ = π for some m and n. The only cases consistent with the conditions on α and β are α = β = π/3, the desired fundamental domain; and α = π/3, β = π/2, which is the half of our fundamental domain with x > 0. But that domain is only a fundamental domain for a group including the reflection in the line x = 0, which has determinant −1. (That domain also has different symbolic dynamics; a geodesic can hit L̄ and R̄ consecutively.)

9. Open Problems

(1). We have shown in one special case that the polygon P can be recovered from the data Σ_{P} up to isometry. The example F comes from a Riemann surface of genus 0. Does the result persist for higher-genus Riemann surfaces? Can any Σ_{P} be explicitly determined for a Riemann surface of genus at least one?

(2). Since Σ_{F} determines F up to isometry, in principle it determines the volume of (ℌ/P SL(2, ℤ)). Can vol(ℌ/P SL(2, ℤ)) = \frac{π}{3} be easily computed directly from Σ_{F}?

(3). The zeta function ζ_{Σ}(z) of a shift Σ is defined by

ζ_{Σ}(z) = exp(∑_{k=1}^{∞} N_{k} \frac{z^{k}}{k}),

in which N_{k} counts the number of periodic words in Σ of period k. Is there a simple formula for the (dynamical) zeta function of Σ_{F}? What is the topological entropy of Σ_{F}?

(4). Cutting sequence shifts Σ_{P} can be constructed in higher-dimensional cases along the lines considered in [28], if one restricts to a suitable subclass of geodesics, called “flat” geodesics in [28]. Can any such Σ_{P} be determined explicitly?

Acknowledgment. We are indebted to L. Flatto and M. Sheingorn for helpful comments and references.

Appendix A. A Bound on the Number of Corners on a Vertical Geodesic

Lemma A.1. For a rational θ, the vertical geodesic γ = {θ + it : t > 0} has at most one value of t such that θ +it is a corner of a P SL(2, ℤ)-translate of F, unless θ ≡ \frac{1}{2} (mod 1), in which case it has exactly two such values, which are t = \sqrt{3}/2 and \sqrt{3}/6.

Proof. Since the corner −1/2 + \sqrt{−3}/2 of F is obtained from the corner 1/2 + \sqrt{−3}/2 by the transformation z → z − 1, every corner is a P SL(2, ℤ)-translate of

1/2 + \sqrt{−3}/2. Let the element of SL(2, ℤ) be [ \begin{matrix}a & b \\ c & d \end{matrix} ], with ad − bc = 1. We then

have

[ \begin{matrix}a & b \\ c & d \end{matrix} ] (\frac{1}{2} + \frac{\sqrt{−3}}{2}) = \frac{a (\frac{1}{2} + \frac{\sqrt{−3}}{2}) + b}{c (\frac{1}{2} + \frac{\sqrt{−3}}{2}) + d}

= \frac{(4ac + 2ad + 2bc + 4bd) + (2ad − 2bc)\sqrt{−3}}{4(c^{2} + cd + d^{2})}
(A.1)  = \frac{(2ac + ad + bc + 2bd) + \sqrt{−3}}{2(c^{2} + cd + d^{2})}.

Let N = 2ac + ad + bc + 2bd and D = c^{2} + cd + d^{2}, so that the real part of (A.1) is N/2D. We will show that N/2D is either in lowest terms or can be reduced to lowest terms by dividing both N and D by 3. (The factor 3 can occur; for example, take a = 2, b = 1, c = 1, d = 1.) Since ad − bc = 1, ad + bc is odd and thus N is not divisible by 2. Now substitute a = (1 + bc)/d in (A.1); this gives

\frac{N}{2D} = \frac{2\frac{1+bc}{d}c + 2bd + \frac{1+bc}{d}d + bc}{2D}
= \frac{2c + 2bc^{2} + 2bd^{2} + d + 2bcd}{2d(c^{2} + cd + d^{2})}
(A.2)  = \frac{b(c^{2} + cd + d^{2}) + 2c + d}{2d(c^{2} + cd + d^{2})}.

Since (c, d) = 1, we have c^{2} + cd + d^{2} ≡ 1 (mod 2), and also (d, c^{2} + cd + d^{2}) = (d, c^{2}) = 1. It follows that

(N, D) = (dN, D) = (bD + 2c + d, D) = (2c + D, D) = (2c + d, c^{2} + cd + d^{2})
= (2c + d, c^{2} + cd + d^{2} − d(2c + d)) = (2c + d, c(c − d)).

Now, (c, 2c + d) = (c, d) = 1, and (2c + d, c − d) = (3c, c − d) ≤ (3, c − d)(c, c − d) = 1 or 3. Therefore (N, D) = 1 or 3.

Suppose the geodesic is θ + it with θ = v/w. There are only two possible values of (A.1) on this geodesic; N/2D can only equal v/w in lowest terms if w = 2D or w = 2D/3. The imaginary part of (A.1) is either \sqrt{−3}/w or \sqrt{−3}/3w. However, these two points are separated by a hyperbolic distance of ln 3, the length of the finite side of the domain F; therefore, they can both be P SL(2, ℤ)-translates of corners only if they are connected by a P SL(2, ℤ)-translate of an edge. This edge is the geodesic θ + it itself, and the only vertical edges of translates gF occur when θ = (n + 1/2) for n ∈ ℤ.

For θ = n + 1/2, there are two SL(2, ℤ)-translates of corners. If we take a = 1, b = n, c = 0, d = 1, we get the corner (n + 1/2) + \sqrt{−3}/2; if we take a = n + 1, b = n, c = 1, d = 1, we get the corner (n + 1/2) + \sqrt{−3}/6.

References

[1] R. Adler, Geodesic flows, interval maps, and symbolic dynamics, in: Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces, T. Bedford, M. Keane, and C. Series (eds.), Oxford: Oxford University Press, 1991, 93–123.

[2] R. Adler and L. Flatto, Cross Section Maps for Geodesic Flows I. The Modular Surface, in: A. Katok, Ed., Ergodic Theory and Dynamical Systems II, Proceedings, Special Year, Maryland 1979–80, Boston: Birkhäuser, 1982, pp. 103–161.

[3] R. Adler and L. Flatto, Geodesic flows, interval maps and symbolic dynamics, Bull. Amer. Math. Soc. 25 (1991), 229–334.

[4] P. Arnoux, Le codage des flot geódeśique sur la surface modulaire, Enseign. Math. 40 (1994), 29–48.

[5] E. Artin, Eine mechanische System mit quasiergodischen bahnen, Hamburger Math. Abhandlungen 3 (1924) 170–175. (Collected Papers, pp. 499-505.)

[6] A. F. Beardon, The Geometry of Discrete Groups, New York: Springer-Verlag 1983.

[7] G. D. Birkhoff, Dynamical Systems, AMS Colloquium Publ. No. 9, Providence: American Math. Society. 1927. (Revision: 1966, J. Moser).

[8] R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Publ. Math. I.H.E.S. 50 (1979), 153–170.

[9] J. W. S. Cassels, Rational Quadratic Forms, New York: Academic Press 1978.

[10] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, New York: Springer-Verlag 1982.

[11] T. W. Cusick and M. Flahive, The Markoff and Lagrange Spectra, Providence: American Math. Society, 1990.

[12] J. S. Frame, Continued Fractions and Matrices, Amer. Math. Monthly 56 (1949), 98–103.

[13] J. R. Goldman, Hurwitz sequences, the Farey process, and generalized continued fractions, Adv. in Math. 72 (1988), 239–260.

[14] J. Hadamard, Les surfaces a’ corbures opposseés et leurs lignes geódésiques, J. de Math. 5 seris IV (1898), 27–73.

[15] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fourth Edition), Oxford: Oxford University Press, 1960.

[16] G. A. Hedlund, A metrically transitive group defined by the modular group, Amer. J. Math. 52 (1935), 668–678.

[17] G. A. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc. 45 (1939), 241–260.

[18] C. Hermite, Sur l’introduction des variables continues dans la theorie des nombres, J. reine Angew. Math. 41 (1851), 191–216.

[19] J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages and Computation, Reading, Mass: Addison-Wesley 1979.

[20] G. Humbert, Sur la méthode d’approximation d’Hermite, J. Math Pures Appl. (7th Series) 2 (1916), 70–103.

[21] G. Humbert, Sur les fractiones continues ordinaires et les formes quadatique binaires indéfinies, J. Math Pures Appl. (7th Series) 2 (1916), 104–154.

[22] S. Katok, Reduction theory for Fuchsian groups, Math. Annalen 273 (1986), 461–470.

[23] S. Katok, Fuchsian Groups, University of Chicago Press: Chicago 1992.

[24] S. Katok, Coding of closed geodesics after Gauss and Morse, Geometriae Dedicata 63 (1996), 123–145.

[25] P. Köebe, Riemannsche Manningfaltigkeiten und nichteucklidische Raumfornen (Vierte Mitteilungen: Verlanf Geodatischer Linien), Sitzb. Preuss. Akad. Wiss. 1929, 414–457.

[26] K. Kolden, Continued fractions and linear substitutions, Arch. Math og Naturvid. 50 (1949), 141–196.

[27] J. C. Lagarias, Number theory and dynamical systems, in The Unreasonable Effectiveness of Number Theory (S. A. Burr, ed.), Proc. Symp. Appl. Math. No. 46, American Mathematical Society, 1992, pp. 35–72.

[28] J. C. Lagarias, Geodesic multidimensional continued fractions, Proc. London Math. Soc., 69 (1994), 464–488.

[29] J. C. Lagarias and A. M. Pollington, The continuous Diophantine approximation mapping of Szekeres, J. Australian Math. Soc., Series A, 59 (1995), 148–172.

[30] J. Lehner, Discontinuous Groups and Automorphic Functions, AMS: Providence, RI 1964.

[31] J. D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge: Cambridge University Press, 1995.

[32] B. Maskit, On Poincaré’s theorem for fundamental polygons, Advances in Math. 7 (1971), 219–230.

[33] R. Moeckel, Geodesics on modular surfaces and continued fractions, Ergodic Theory Dyn. Sys. 2 (1982), 69–83.

[34] M. Morse, A one-to-one representation of geodesics on a surface of negative curvature, Amer. J. Math. 43 (1921), 33–51.

[35] M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84–100.

[36] M. Morse, Symbolic Dynamics, Institute for Advanced Study Lecture Notes 1966 (unpublished).

[37] M. Morse and G. A. Hedlund, Symbolic dynamics, Amer. J. Math. 60, (1938), 815–866.

[38] J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächer, Acta. Math. 50 (1927), 189–358.

[39] P. J. Nicholls, Diophantine approximation via the modular group, J. London Math. Soc. 17 (1978), 11–17.

[40] G. N. Raney, On continued fractions and finite automata, Math. Annalen 206 (1973), 265–283.

[41] I. Richards, Continued fractions without tears, Math. Mag. 54 (1981), 163–171.

[42] C. Series, Symbolic dynamics for geodesic flows, Acta Math. 146(1981), 103–128.

[43] C. Series, The modular surface and continued fractions, J. London Math. Soc. 31 (1985), 69–80. (Correction: see [46, p. 148].)

[44] C. Series, The geometry of Markoff numbers, Math. Intelligencer 7, No. 3 (1985) 20–29.

[45] C. Series, Geometric Markov coding on surfaces of constant negative curvature, Ergod. Th. Dynam. Sys. 6 (1986), 601–625.

[46] C. Series, Geometrical methods of symbolic coding, in: Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane, C. Series (Eds.), Oxford: Oxford U. Press, 1991, pp. 125–151.

[47] H. J. S. Smith, Memoire sur les equations modulaires, Accad. de. Lincei 1877, in Collected Papers, New York: Chelsea, pp. 224–241.

[48] H. M. Stark, An Introduction to Number Theory, Chicago: Markham, 1971 (Reprint: M.I.T. Press).

[49] B. A. Venkov, Elementary Number Theory, Groningen, Netherlands: Walters-Noordhoff Pub-

lishing (1970).

  E-mail address: grabiner@math.lsa.umich.edu

David J. Grabiner, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109

  E-mail address: jcl@research.att.com

Jeffrey C. Lagarias, AT&T Laboratories–Research, Florham Park, NJ 07932-0971