Symbolic dynamics for the geodesic flow on Hecke surfaces

In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in terms of the generating map of a particular continued fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.


INTRODUCTION
Surfaces of negative curvature and their geodesics have been studied since the 1898 work of Hadamard [15] (see in particular the remark at the end of §58). Inspired by the work of Hadamard and Birkhoff [6] Morse [32] introduced a coding of geodesics essentially corresponding to what is now known as "cutting sequences" and used this coding to show the existence of a certain type of recurrent geodesics [33].
Further ergodic properties of the geodesic flow on surfaces of constant negative curvature given by Fuchsian groups were shown by e.g. Artin [5], Nielsen [37], Koebe [26], Löbell [29], Myrberg [34], Hedlund [16,18,19,17], Morse and Hedlund [31] and Hopf [21,20]. In this sequence of papers one can see the subject of symbolic dynamics emerging. For a more up-to-date account of the ergodic properties of the geodesic flow on a surface of constant negative curvature formulated in a modern language see e.g. the introduction in Series [45].
Artin's [5] approach was novel in that he used continued fractions to code geodesics on the modular surface. After Artin, coding and symbolic dynamics on the modular surface have been studied by e.g. Adler and Flatto [1,2,3] and Series [46]. For a recent review of different aspects of coding of geodesics on the modular surface see for example the expository papers by Katok and Ugarcovici [24,25].
Other important references for the theory of symbolic dynamics and coding of the geodesic flow on hyperbolic surfaces are e.g. Adler-Flatto [4], Bowen and Series [7] and Series [45].
In the present paper we study the geodesic flow on a family of hyperbolic surfaces with one cusp and two marked points, the so-called Hecke triangle surfaces, generalizing the modular surface. Symbolic dynamics for a related billiard has also been studied by Fried [13]. We now give a summary of the paper. Sections 1 and 2 contain preliminary facts about hyperbolic geometry and geodesic flows. In Section 3 we develop the theory of λ -fractions connected to the coding of the geodesic flow on the Hecke triangle surfaces. The explicit discretization of the geodesic flow in terms of a Poincaré section and Poincaré map is developed in Section 4. As an immediate application we derive invariant measures for certain interval maps in Section 5. Some rather technical lemmas are confined to the end in Section 6.

Hyperbolic geometry and Hecke triangle surfaces
Recall that any hyperbolic surface of constant negative curvature −1 is given as a quotient (orbifold) M = H /Γ. Here H = {z = x + iy | y > 0, x ∈ R} together with the metric ds = |dz| y is the hyperbolic upper half-plane and Γ ⊆ PSL 2 (R) ∼ = SL 2 (R)/ {±I 2 } is a Fuchsian group. Here SL 2 (R) is the group of real two-by-two matrices with determinant 1, I 2 = 1 0 0 1 and PSL 2 (R) is the group of orientation preserving isometries of H . The boundary of H is ∂ H = R * = R∪{∞}. If g = a b c d ∈ PSL 2 (R) then gz = az+b cz+d ∈ H for z ∈ H and we say that g is elliptic, hyperbolic or parabolic depending on whether |Tr g| = |a + d| < 2, > 2 or = 2. The same notation applies for fixed points of g. In the following we identify the elements g ∈ PSL 2 (R) with the map it defines on H . Note that the type of fixed point is preserved under conjugation z → AzA −1 by A ∈ PSL 2 (R). A parabolic fixed point is a degenerate fixed point, belongs to ∂ H and is usually called a cusp. Elliptic points z appear in pairs, one belongs to H and the other one is in the lower half-plane H and its stabilizer subgroup Γ z in Γ is cyclic of finite order m. Hyperbolic fixed points appear also in pairs with x, x * ∈ ∂ H , where x * is said to be the conjugate point of x. A geodesics γ on H is either a half-circle orthogonal to R or a line parallel to the imaginary axis and the endpoints of γ are denoted by γ ± ∈ ∂ H . We identify the set of geodesics on H with G = {(ξ , η) | ξ = η ∈ R * } and use γ (ξ , η) to denote the oriented geodesic on H with γ + = ξ and γ − = η. Unless otherwise stated all geodesics are assumed to be parametrized with hyperbolic arc length with γ (0) either at height 1 if γ is vertical or the highest point on the half-circle. It is known that z ∈ H and θ ∈ [0, 2π) ∼ = S 1 determine a unique geodesic (cf. Lemma 24) passing through z whose tangent at z makes an angle θ with the positive ξ -axis. This geodesic is denoted by γ z,θ . It is also well known that a geodesic γ (ξ , η) is closed if and only if ξ and η = ξ * are conjugate hyperbolic fixed points.
The unit tangent bundle of H , T 1 H = z∈H { v ∈ T z H | | v| = 1} is the collection of all unit vectors in the tangent planes of H with base points z ∈ H which we denote by T 1 z H . By identifying v with its angle θ with respect to the positive real axis we can view T 1 H as the collection of all pairs (z, θ ) ∈ H × S 1 . We may also view this as the set of geodesics γ z,θ on H or equivalently as G ⊆ R * 2 .
Let π : H → M be the natural projection map, i.e. π (z) = Γz and let π * : T 1 H → T 1 M be the extension of π to T 1 H . Then γ * = πγ is a closed geodesic on M if and only if γ + and γ − are fixed points of the same hyperbolic map g γ ∈ Γ. For an introduction to hyperbolic geometry and Fuchsian groups see e.g. [23,27,39].
Definition 1. For an integer q ≥ 3 the Hecke triangle group G q ⊆ PSL 2 (R) is the group generated by the maps S : z → − 1 z and T : z → z+λ where λ = λ q = 2 cos π q ∈ [1, 2). The corresponding orbifold (Riemann surface) is M q = G q \H , which we sometimes identify with the standard fundamental domain of G q F q = {z ∈ H | |ℜz| ≤ λ /2, |z| ≥ 1} with sides pairwise identified. Let ρ = ρ + = e πi q and ρ − = −ρ. We define the following oriented boundary components of F q : L 0 is the circular arc from ρ − to ρ + . L 1 is the vertical line from ρ + to i∞ and L −1 is the vertical line from i∞ to ρ − . Thus ∂ F q = L −1 ∪ L 0 ∪ L 1 is the positively oriented boundary of F q . Remark 1. The group G q is a realization of the Schwarz triangle group π ∞ , π q , π 2 and it is not hard to show (see e.g. [27,VII]) that G q for q ≥ 3 is a co-finite Fuchsian group with fundamental domain F q and the only relations (1) S 2 = (ST ) q = Id -the identity in PSL 2 (R).
Hence G q has one cusp, that is the equivalence class of parabolic points, and two elliptic equivalence classes of orders 2 and q respectively. Note that G 3 = PSL 2 (Z) -the modular group and G 4 , G 6 are conjugate to congruence subgroups of the modular group. For q = 3, 4, 6 the group G q is non-arithmetic (cf. [23, pp. 151-152]), but in the terminology of [9,44] it is semi-arithmetic, meaning that it is possible to embed G q as a subgroup of a Hilbert modular group.

THE GEODESIC FLOW ON T 1 M
We briefly recall the notion of the geodesic flow on a Riemann surface M = Γ\H with Γ ⊂ PSL 2 (R) a Fuchsian group. To any (z, θ ) ∈ T 1 H ∼ = H × S 1 we can associate a unique geodesic γ = γ z,θ on H such that γ (0) = z andγ (0) = e iθ . The geodesic flow on T 1 H can then be viewed as a map Φ t : . A more abstract and general description of the geodesic flow, which can be extended to other homogeneous spaces, is obtained by the identification T 1 H ∼ = PSL 2 (R). Under this representation the geodesic flow corresponds to right multiplication by the matrix a −1 t = e t/2 0 0 e −t/2 in PSL 2 (R) (cf. e.g. [11,Ch. 13] infinitely often, i.e. Φ t j (γ) ∈ Σ for an infinite sequence of t j → ±∞.
The corresponding first return map is the map T : Here t 0 = t 0 (z, θ ) > 0 is called the first return time.
Poincaré sections were first introduced by Poincaré [38] to show the stability of periodic orbits. For examples of cross section maps in connection with the geodesic flow on hyperbolic surfaces see e.g. [4,1].
The previous definition extend naturally to T 1 M with ϒ and Σ replaced by ϒ * = π (ϒ) and Σ * = π * (Σ). The first return map T is used to obtain a discretization of the geodesic flow, e.g. we replace Φ t (z, θ ) by {Φ t k (z, θ )} where t k (z, θ ) is a sequence of consecutive first returns. Incidentally this provides a reduction of the dynamics from three to two dimensions and it turns out that in our example the first return map also has a factor map, which allows us to study the three dimensional geodesic flow with the help of an interval map (see Sections 4.3 and 5).

Basic concepts
Continued fraction expansions connected to the groups G q , the so-called λ -fractions, were first introduced by Rosen [40] and subsequently studied by Rosen and others, cf. e.g. [42,41,43]. For the purposes of natural extensions (cf. Section 3.4) the results of Burton, Kraaikamp and Schmidt [8] are analogous to ours and we occasionally refer to their results. Our definition of λ -fractions is equivalent to Rosens definition (cf. e.g. [40, §2]).
To a sequence of integers, a 0 ∈ Z and a j ∈ Z * = Z\ {0} , j ≥ 1 (finite or infinite) we associate a λ -fraction x = a 0 ; a 1 , a 2 , . . . . This λ -fraction is identified with the point if the right hand side is convergent. When there is no risk of confusion, we sometimes write x = x. The tail of x is defined as a m+1 , a m+2 , . . . for any m ≥ 1. Note that −x = −a 0 ; −a 1 , −a 2 , . . . . If a 0 = 0, we usually omit the leading 0; . Repetitions in a sequence is denoted by a power, e.g. a, a, a = a 3 and an infinite repetition is denoted by an overline, e.g. a 1 , . . . a k , a 1 , . . . , a k , . . . = a 1 , . . . , a k . Such a λ -fraction is said to be periodic with period k, an eventually periodic λ -fraction has a periodic tail. Two λfractions x and y are said to be equivalent if they have the same tail. In this case it is easy to see that, if the fractions are convergent, then x = Ay for some A ∈ G q .
The sole purpose for introducing λ -fractions is to code geodesics by identifying the λfractions of their endpoints with elements of Z N . For reasons that will be clear later (Section 3.4), we have to consider also bi-infinite sequences Z Z and view Z N as embedded in Z Z with a zero-sequence to the left. On Z Z we always use the metrich defined byh where a i = b i for |i| < n and a n = b n or a −n = b −n . In this metric Z Z and Z N have the topological structure of a Cantor set and the leftand right shift maps σ ± : Z Z → Z Z , σ ± a j = a j±1 are continuous. We also set σ + a 1 , a 2 , . . . = a 2 , a 3 , . . . .

Regular λ -fractions
In the set of all λ -fractions we choose a "good" subset, in which almost all x ∈ R have unique λ -fractions and in which infinite λ -fractions are convergent. The first step is to choose a "fundamental region" I q for the action of T : R → R, namely I q = − λ 2 , λ 2 . Then it is possible to express one property of our "good" subset as follows: If in the fraction x = a 0 ; a 1 , a 2 , . . . the first entry a 0 = 0, then x ∈ I q . That means, we do not allow sequences with a 0 = 0 correspond to points outside I q .
A shift-invariant extension of this property leads to the following definition of regular λ -fractions: Definition 3. Let x = a 0 ; a 1 , a 2 , . . . be a finite or infinite convergent λ -fraction and let x j = σ j x = 0; a j , a j+1 , . . . , j ≥ 1, be the j-th shift of x. Let x j be the corresponding point. Then x is said to be a regular λ -fraction if and only if (*) x j ∈ I q , for all j ≥ 1.
A regular λ -fraction is denoted by a 0 ; a 1 , . . . , the space of all regular λ -fractions is denoted by A q and the subspace of infinite regular λ -fractions with a 0 = 0 is denoted by A 0,q .
For a finite fraction x = a 0 ; a 1 , . . . , a n we get x j = 0; and x j = 0 ∈ I q for j > n.
We will see later that regular λ -fractions can be regarded as nearest λ -multiple continued fractions. In the case q = 3 or λ = 1, nearest integer continued fractions were studied already by Hurwitz [22] in 1889. An account of Hurwitz reduction theory can be found in Fried [12] (cf. also the H-expansions in [24,25]). For general q this particular formulation of Rosens fractions was studied by Nakada [36].
For the remainder of the paper we let h = q−3 2 if q is odd and h = q−2 2 if q is even. The following Lemma is an immediate consequence of [8, (4)].
Lemma 1. The points ∓ λ 2 have finite regular λ -fractions given by Lemma 2. If q is odd, the point x = 1 has the finite regular λ -fraction and hence −1/a = −a. Since a > 0, this implies that a = 1.
Definition 4. Let x be the floor function defined by x = n ⇔ n < x ≤ n + 1 for x > 0, respectively n ≤ x < n + 1 for x ≤ 0, and let {x} λ = x λ + 1 2 be the corresponding nearest λ -multiple function. Then define F q : I q → I q by Lemma 3. For x ∈ R the following algorithm gives a finite or infinite regular λ -fraction c q (x) = a 0 ; a 1 , . . . corresponding to x: If x j = 0 for some j, the algorithm stops and gives a finite regular λ -fraction.
Proof. By definition we see that x j+1 = T −a j Sx j , j ≥ 1, and hence x = T a 0 ST a 1 · · · ST a n x n for any n ≥ 1. If x = a 0 ; a 1 , . . . . , then for j ≥ 1 x j = σ j x = 0; a j , . . . corresponds to the point x j and condition (*) of Definition 3 is fulfilled, since F q maps I q to itself and Remark 2. We say that F q is a generating map for the regular λ -fractions. It is also clear from Lemma 3 that F q acts as a shift map on the space A 0,q , i.e. c q F q x = σ c q (x).
An immediate consequence of Lemma 3 is the following corollary: Corollary 1. If x has an infinite regular λ -fraction, then it is unique and equal to c q (x) as given by Lemma 3.
The above choice of floor function implies that F q is an odd function and that ± λ 2 λ = 0 in agreement with Lemma 1. The ambiguity connected to the choice of floor function at integers affects only the points By Lemma 1 we conclude, that any point, which has more than one regular λ -fraction, is F q -equivalent to ± λ 2 and hence has a finite λ -fraction. We can produce in this way a regular λ -fraction c q (x) as a code for any x ∈ R. For the purpose of symbolic dynamics we prefer to have an intrinsic description of the members of the space A 0,q formulated in terms of so-called forbidden blocks, i.e. certain subsequences which are not allowed. From Definition 3 it is clear, which subsequences are forbidden and how to remove them by rewriting the sequence, using the fraction with a leading a 0 = 0 for any point outside I q instead of a 0 = 0. Lemma 4. Let q be even and set h = q−2 2 . Since by Lemma 1 one has ∓ λ 2 = (±1) h the following blocks are forbidden: (±1) h , ±m with m ≥ 1. Using (ST ) 2h+2 = 1 such blocks can be rewritten as Lemma 5. Let q be odd and set h = q−3 2 . Since by Lemma 1 one knows that ∓ λ 2 = (±1) h , ±2, (±1) h the following two different types of blocks are forbidden: (±1) h+1 and (±1) h , ±2, (±1) h , ±m , with m ≥ 1. Using (ST ) 2h+3 = 1 for q ≥ 5 these blocks can be rewritten as Remark 3. It is easy to see that Rosens λ -fractions [40] can be expressed as words in the generators T, S and JS of the group G * q =< G q , J >⊆ PGL 2 (R), where J : z → −z is the reflection in the imaginary axis. Since J is an involution of G q , e.g. JT J = T −1 and JSJ = S, it is easy to see that Rosen's and our notions of λ -fractions are equivalent: e.g. in the G q -word identified with our λ -fraction we replace any T −a by JT a J, a ≥ 1. Algorithmically this means for a λ -fraction with entries a j that the corresponding Rosen fraction has entries (ε j , |a j |) where ε 1 = −sign (a 1 ) and ε j = −sign(a j−1 a j ) for j ≥ 2.
From the definition of regular λ -fractions it is clear that Rosen's reduced λ -fractions [40, Def. 1] correspond to a fundamental interval 0, λ 2 for the action of the group T, J together with the choices made for finite fractions in [40, Def. 1 (4)- (5)]. It is easy to verify, for example using the forbidden blocks, that a finite fraction not equivalent to ± λ 2 or an infinite regular λ -fraction correspond to a reduced λ -fraction of Rosen. The main difference between our regular and Rosens reduced λ -fractions is that any λ -fraction equivalent to ± λ 2 has two valid regular λ -fractions. The root of this non-uniqueness is our choice of a closed interval I q which is in turn motivated by our Markov partitions in Section 3.5.
It is then clear, that those results of [40] and [8] pertaining to infinite reduced λ -fractions can be applied directly to our regular λ -fractions. Lemma 6. An infinite λ -fraction without forbidden blocks is convergent.
An immediate consequence of Definition 3 and Lemmas 4, 5 and 6 is the following Corollary 2. A λ -fraction is regular if and only if it does not contain any forbidden block.
It is important that rewriting forbidden blocks can be done sequentially.
Lemma 7. If the λ -fraction a 0 ; a 1 , . . . has a forbidden block beginning at a n , then the subsequence a 0 ; a 1 , . . . , a n−2 is not affected by rewriting this forbidden block.
Proof. This is easy to verify in the different cases arising in Lemmas 4 and 5 by simply noting that a n = ±1 so a n−1 = ±1. For the complete details see [30].

Dual regular λ -fractions
To encode the orbits of the geodesic flow in terms of a discrete invertible dynamical system it turns out that we still need another kind of λ -fraction, the so-called dual regular λfraction. In the case q = 3 this was already introduced by Hurwitz [22], see also [25, p. 15].
Consider the set of λ -fractions y = 0; b 1 , . . . which do not contain any reversed forbidden block, i.e. a forbidden block given in Definitions 4 or 5 read in reversed order.
Let R be the largest number in this set and define r = R − λ and I R = [−R, R].
Proof. To obtain the largest number we use the lexicographic ordering on A 0,q implied by the following sequence of easily verifiable inequalities: For the details on how this ordering extends to infinite sequences see [30]. Thus we want to take as many digits as possible close to −1 in the λ -fraction of R. The constraints set by the reversed forbidden blocks in Lemmas 4 and 5 clearly give the following expression for R: The Lemma is then clear by rewriting these numbers recursively into regular λ -fractions using Lemmas 4 and 5.
Lemma 9. For even q we have the identity R = 1.
Proof. Consider the action of S on R: Hence −1/R = −R and since R > 0 we must have R = 1.
Proof. From the explicit expansions of λ 2 and 1 in Lemmas 1 and 2 together with the lexicographic ordering mentioned in the proof of Lemma 8 it is clear that λ 2 < R < 1 (or see the proof of Lemma 3.3 in [8]). By rewriting as in Lemma 5 for q ≥ 5 we get which is identity a). A similar rewriting works for q = 3. Using the following explicit formula for the matrix (T S) n (cf. e.g. [8, p. 1279 Remark 4. Using the representation (2) one can also show that the map A r fixing r = R − 1 is given by for even q.
Definition 5. Let y = b 0 ; b 1 , . . . be a finite or infinite λ -fraction. Set y 0 = 0; b 1 , . . . , y j = σ j−1 y 0 = 0; b j , . . . and y j , j ≥ 0, the corresponding point in R. Then y is said to be a dual regular λ -fraction if and only if it has the following properties: * , the space of all dual regular λfractions by A * q and the subspace of all infinite sequences in A * q with leading 0 by A * 0,q .
Uniqueness of a subset of dual regular λ -fractions is again asserted using a generating map.
Definition 6. Let · be the floor function from Definition 4 and consider the shifted Lemma 11. For y ∈ R the following algorithm produces a finite or infinite dual regular λ -fraction c * q (y) = b 0 ; b 1 , . . . * corresponding to y: If y j = 0 for some j the algorithm stops and one obtains a finite dual regular λ -fraction.
Proof. It is easy to verify that {y} * λ = 0 ⇔ y ∈ I R and that in general It is thus clear that (D1) and (D2) are automatically fulfilled and it follows that Hence condition (D3) is also satisfied.
Remark 5. We say that F * q is a generating map for the dual regular λ -fractions and it is easily verified that F * q acts as a left shift map on A * 0,q .
It is easily seen that the points affected by the choice of floor function appearing in {·} * λ (cf. ± λ 2 in the regular case) are exactly those that are equivalent to ±r. Hence we obtain the following corollary.
Corollary 3. If y has an infinite dual regular λ -fraction expansions which is not equivalent to the expansion of ±r then it is unique and is equal to c * q (y). Lemma 12. A λ -fraction y = b 0 ; b 1 , . . . is dual regular if and only if the sequence y 0 does not contain any reversed forbidden blocks. Thereby y 0 = y if b 0 = 0 and y 0 = Proof. Consider q even and a reversed forbidden block of the form m, 1 h with m ≥ 1.
Analogous arguments work for b 0 = 0, for forbidden blocks involving −1s and for the case of odd q. For more details see [30].
Proof. The proof is similar as for the the regular λ -fractions. For details see e.g. [30].
Remark 6. Just as the regular λ -fractions are equivalent to the reduced Rosen λ -fractions, one can show that the dual regular λ -fractions are essentially equivalent to a particular instance of so-called α-Rosen λ -fractions, see [10] and [35] (in the case q = 3). Note that where T α is the generating map of the α-Rosen fractions of [10].

Symbolic dynamics and natural extensions
An introduction to symbolic dynamics and coding can be found in e.g. [28]. See also [4,45] or [4, Appendix C]. Our underlying alphabet is infinite, N = Z * = Z\ {0}. The dynamical system N Z + , σ + is called the one-sided full N −shift. Since the forbidden blocks (cf. Definitions 4 and 5) imposing the restrictions on A 0,q and A * 0,q all have finite length it follows that A 0,q , σ + and (A * 0,q , σ + ) are both one-sided subshifts of finite type (cf. [4, Thm. C7]).
One can show that c q : I q → A q and c * q : I R → A * q as given by Lemmas 3 and 11 are continuous and we call these the regular and dual regular coding map respectively. Let be the set of "G q -irrational points" and set I ∞ α = I α ∩ R ∞ for α = q, R. Since the set G q (∞) of cusps of G q is countable it is clear, that the Lebesgue measure of I ∞ α is equal to that of I α , α = q, R. From [30] we see, that the restrictions c q : I ∞ q → A 0,q and c * q : follows that the one-sided subshifts A 0,q , σ + and (A * 0,q , σ + ) are topologically conjugate to the abstract dynamical systems F q , I ∞ q and F * q , I ∞ R respectively (see [4, p. 319]). Consider the set of regular bi-infinite sequences B q ⊂ A * 0,q × A 0,q ⊂ Z Z consisting of precisely those . . . , b 2 , b 1 a 1 , a 2 , . . . which do not contain any forbidden block. Then B q , σ is a two-sided subshift of finite type extending the one-sided subshift A 0,q , σ + , . . . In the next section we will see that there exists a domain Ω ⊂ I q × I R such that C |Ω ∞ : Ω ∞ → B q is one-to-one and continuous (here we neglect points (x, y) where either x or y has a finite λ -fraction). The natural extension,F q , of F q to Ω ∞ is defined by the condition that B q , σ is topologically conjugate to F q , Ω ∞ , i.e. by the

Markov Partitions for the generating map F q
To construct a Markov partition of the interval I q with respect to F q we consider the orbits of the endpoints ± λ 2 . For x ∈ I q or y ∈ I R we define the F q -orbit and F * q -orbit of x and y respectively as It is easy to verify that the closure of the intervals form a Markov partition of I q for F q . I.e. I j covers I q , overlaps only at endpoints and F q maps endpoints to endpoints. Since the alphabet N is infinite there exist also another Markov partition of the form J n = −2 x − nλ and hence expanding and bijective unless n = 1 for q ≥ 3 or n = 2 for q = 3.
From the explicit formula ofF q −1 it is clear that we also need to consider the orbits of the endpoints of ± [r, R]. From Lemmas 9 and 10 we see that # {O * (−R)} = κ + 1. Set r 0 = −R and let 0 > r 1 > r 2 > · · · > r κ = r > −R = r 0 be an ordering of O * (−R) = r j . One can verify that (see Lemma 1 and 8). It is then easy to verify that 2 (see Lemma 1 and 8). It is then easy to verify that Hence To establish the sought correspondence between the domain Ω ∞ and B • q we first need a Lemma.
Lemma 14. r is the smallest number y in I R such that C (x, y) ∈ B q for all x ∈ I κ .
Proof. Let q be even. We know from Lemma 8 and its proof that . . * and C (φ h−1 , y) contains the forbidden block 1 h+1 .
Let q ≥ 5 be odd. Then C (φ 2h , y) contains the forbidden block 1 h , 2, 1 h , 2 . We have shown that r is the smallest number such that C (φ κ−1 , r) does not contain a forbidden block and it is easy to show that also C (x, r) ∈ B • q for any x ∈ I κ = [φ κ−1 , 0). The same argument applies to q = 3.
Proof. Just as in the proof of Lemma 14 it is not hard to verify that r j is the smallest number in I R with a dual regular expansion which can be prepended to the regular expansion of φ j and hence of all x ∈ φ j , 0 .

Definition 7.
To determine the first return map we introduce a "conjugate" region Domains Ω and Ω * for q = 7 Ω Ω * Ω * Thus Ω with Ω ∞ as a dense subset is the domain of the natural extensionF q of F q . An example of Ω and Ω * is given in Figure 1. See [36] for another choice of a "conjugate" Ω * of Ω using the maps x, y −1 and also [8] for the corresponding domain for the reduced Rosen fractions.

Reduction of λ -fractions
In a first step in our construction of a cross-section for the geodesic flow we select a set of geodesics on H which contains at least one lift of each geodesic on M q = H /G q , i.e. a set of "representative" or "reduced" geodesics modulo G q . For an overview and a discussion of different reduction procedures in the case of PSL 2 (Z) see [25,Sect. 3].
Sketch of proof. The complete proof can be found in [30]. First we find A ∈ G q such that to R ∞ × I ∞ R and applyingF −1 q repeatedly. Once the point is inside the rectangle we attach to it a bi-infinite sequence C •S (Au, Av) = . . . , b 2 , b 1 a 1 , a 2 , . . . . Using further backward shifts and rewriting of forbidden blocks we obtain C •S (Bu, Bv) ∈ B q , i.e. (Bu, Bv) ∈ Ω * . In the last step of rewriting we rely on the fact that rewriting does not propagate to the left, cf. Lemma 7. The explicit form of B is given by F * q n y for some n.
In fact, one can do slightly better than in the previous Lemma by using the important property of the number r, namely that r and −r are G q -equivalent but not orbit-equivalent, i.e. O * (r) = O * (−r) (cf. Lemmas 9 and 10). Using the explicit map identifying r and −r one can show that it is possible to reduce any geodesic to one with endpoints in Ω * without the upper horizontal boundary.
Lemma 17. If (x, y) ∈ Ω * and y has a dual regular expansion with the same tail as −r then there exists A ∈ G q such that (Ax, Ay) ∈ Ω * and Ay has the same tail as r.
Proof. ϒ is identified with the set of endpoints in Ω * , the mapS : Ω * → Ω ∞ defined in Definition 7 is a continuous bijection and by Corollary 1 and 3 respectively Lemmas 15 and 17 it is clear that C : Ω ∞ → B • q is continuous, onto and one-to-one except for points where η is equivalent to ±r where it is two-to-one. ThusC = C •S : ϒ → B • q is a homeomorphism. For the case of G q = PSL 2 (Z) see also [25, p. 19].
The set of reduced geodesics contains representatives of all geodesics on H /G q . This property is an immediate corollary to the following Lemma which additionally also provides a reduction algorithm.
Lemma 19. Let γ be a geodesic on the hyperbolic upper half-plane with endpoints in R ∞ and γ − = b 0 ; b 1 , . . . * . Then there exists an integer n ≥ 0 and A ∈ G q such that Here A is one of the maps Id, T −1 and T −1 ST −1 for even q, respectively Proof. This is an immediate consequence of Lemmas 16 and 17. Note that

Corollary 4.
If γ * is a geodesic on H /G q with all lifts having endpoints in R ∞ then ϒ contains an element of π −1 (γ * ).
Proof. It is not hard to show, that Aξ = ξ and Aη −1 = η −1 . Since η ∈ I R , with R ≤ 1 < 2 λ it is clear that ξ * = ξ and hence A is hyperbolic. The other statement in the Lemma is easy to verify by writing B in terms of generators, rewriting any forbidden blocks and going through all cases of non-allowed sequences, e.g. if B ends with an S.
Since the geodesic γ (ξ , η) is closed if and only if ξ and η are conjugate hyperbolic fixed points and since r and −r are G q -equivalent we conclude from Lemma 20 that there is a one-to-one correspondence between closed geodesics on M q = H /G q and the set of purely periodic regular λ -fractions which do not have the same tail as −r.
Remark 9. Because Ω ∞ only contains points with infinite λ -fractions, the set ϒ does not contain lifts of geodesics which disappear out to infinity, i.e. with one or both endpoints equivalent to ∞. The neglected set however corresponds to a set in T 1 M of measure zero with respect to any probability measure invariant under the geodesic flow. See e.g. the second paragraph of [24, p. 1].
The subshift of finite type B q , σ is also conjugate to the invertible dynamical system ϒ,F q . HereF q : ϒ → ϒ is the map naturally induced byF q acting on ). Using the same notation for both maps should not lead to any confusion.

Definition 9.
For an oriented geodesic arc c on H we let c denote the unique geodesic containing c and preserving the orientation, for instance L 1 = {z = λ 2 + iy y > 0} oriented upwards. Let c ± denote the forward and backward end points of c and let −c denote the geodesic arc with endpoints −c ± . Here −c should not to be confused with the geodesic c with reversed orientation, denoted by c −1 .
For z, w ∈ H ∪ ∂ H denote by [z, w] the geodesic arc oriented from z to w including the endpoints in H .

CONSTRUCTION OF THE CROSS-SECTION
As a cross section for the geodesic flow on the unit tangent bundle T 1 M of M which can be identified with F q × S 1 modulo the obvious identification of points on ∂ F q × S 1 , we will take a set of vectors with base points on the boundary ∂ F q directed inwards with respect to F q . The precise definition will be given below. For a different approach to a cross section related to a subgroup of G q see [14]. For the sake of completeness we include the case q = 3 in our exposition but it is easy to verify that our results in terms of the cross-section, first return map and return time agree with the statements in [24,25].

Strongly Reduced Geodesics
Definition 11. We define the following subsets of T 1 M : Γ r = (z, θ ) ∈ L r × S 1 |γ z,θ (s) directed inwards at z , r = 0, ±1, If q is even let Σ := ∪ 2 j=−2 Σ j and if q ≥ 5 is odd let Σ := ∪ 3 j=−3 Σ j . If q = 3 we drop the restriction on γ + in the definition of Σ ±2 and set Σ = ∪ 2 j=−2 Σ j . We will show that there is a one-to-one correspondence between Σ and a subset of reduced geodesics, which we call strongly reduced. For q = 3 these sets are identical.
If |a k | ≥ 2 then γ k + > 3λ 2 and if a k−1 a k < 0 then γ k + γ k − < 0. In both cases by definitioñ F k q γ ∈ ϒ s . Since an infinite sequence of 1's or −1's is forbidden, it is clear that there exists a k ≥ 0 such that one of these conditions apply.
Combining the above Lemma with Lemma 16 we have shown, that every reduced geodesic is G q -equivalent to a strongly reduced geodesic.
A consequence of Lemma 22 is that for any strongly reduced geodesic we can find an infinite number of strongly reduced geodesics in its forward and backwardF q -orbit (with infinite repetitions if the geodesic is closed). Furthermore, since the base-arcs L ±1 and L 0 of Σ consist of geodesics, none of whose extensions are in ϒ s , it is clear that any strongly reduced geodesic intersects Σ transversally. The set Σ thus fulfills the requirements (P1) and (P2) of Definition 2 and is a Poincaré (or cross-) section with respect to ϒ s . Since any geodesic γ * on M q which does not go into infinity has a strongly reduced lift we also have the following lemma.
Lemma 23. π * (Σ) is a Poincaré section for the part of the geodesic flow on T 1 M which does not disappear into infinity.
From the identification of Σ and Ω * s via the mapP we see that the natural extensionF q of the continued fraction map F q induces a return map for Σ, i.e. if z = (z, θ ) ∈ Σ theñ P •F n q •P −1 z ∈ Σ for an infinite number of n = 0. We give a geometric description of the first return map for Σ and we will later see that this map is in fact also induced byF q . Definition 14. The first return map T : Σ → Σ is defined as follows (cf. Figure 4): If z 0 ∈ Σ and γ = P −1 z 0 ∈ ϒ s let {w n } n∈Z be the ordered sequence of intersections in the direction from γ − to γ + between γ and the G q -translates of ∂ F with w 0 given by z 0 . Since γ + and γ − have infinite λ -fractions they are not cusps of G q and the sequence w n is bi-infinite. For each w n let A n ∈ G q be the unique map such that w n = A n w n ∈ ∂ F and γ = A n γ intersects ∂ F at w n in the inwards direction.
If γ ∈ ϒ s and Pγ = z we say that z ∈ Σ is a return of γ to Σ. If n 0 > 0 is the smallest integer such that w n 0 gives a return to Σ we say that the corresponding point PA n 0 γ = z 1 ∈ Σ is the first return and the first return map T : Σ → Σ is defined by T z 0 = z 1 where z 1 is the first return after z 0 . SometimesT : Ω * s → Ω * s given byT =P −1 • T •P is also called the first return map.
After proving some useful geometric lemmas in the next section we will show in Section 4.3 that the first return map T is given explicitly by powers ofF q .
Proof. This follows from the proof of Lemma 24 and a trivial computation.

Proof. Taking
−1 ±c−ρ 1 ∓c−ρ it is easy to verify that A j L j = iR + preserving the orientation.
Proof. It is clear, that γ intersects L 1 if and only if the intersection with L 1 = λ 2 + iR + is at a height above sin π q = ℑρ. By Lemma 6 the point of intersection is given by w (ξ , η) = We thus need to check the inequality ξ − λ 2 −η + λ 2 > sin 2 π q . With η < λ 2 < ξ it is clear that ℑw decreases as η increases for ξ fixed. Using λ 2 = cos π q we see, that Observe, that Aρ = ρ and A 2 = Id, i.e. A is elliptic of order 2. The stabilizer G q,ρ of ρ in G q is a cyclic group with q elements generated by T S. For even q = 2h + 2 one can use the explicit formula (2) to verify that A = (T S) h+1 ∈ G q . For odd q on the other hand there is no element of order 2 in G q,ρ , so A / ∈ G q .

The first return map
Our aim in this section is to obtain an explicit expression for the first return map T : Σ → Σ. The notation is as in Definition 14, see also Figure 4. The main idea is to use geometric arguments to identify possible sequences of intersections {w n } and then use arguments involving regular and dual regular λ -fractions to determine whether a particular w n corresponds to a return to Σ or not.
If it intersects this extension it has to pass first through the arc (T S) j T ρ, α j−1 . But the completion of this arc is clearly (T S) j T L 1 = β j , α j−1 and hence γ can not intersect this arc and must pass through ω j . The second case is analogous, except that we do not care about whether the next intersection is at ω j+1 or (T S) j+1 T L 1 .

FIGURE 7. Geodesics leaving the Poincaré section (q=7)
We also have to consider the return map for the second type of strongly reduced geodesics.
Proof. Consider z 0 = Pγ ∈Σ with γ = γ (ξ , η) ∈ ϒ s and assume without loss of generality that ξ > 0 with c q (ξ ) = a 0 ; 1 j , a j+1 , . . . for some j ≥ 0, a j+1 = 1 and a j+1 = ±1 if j = 0 (the case of −1's is analogous). Recall the notation in Definition 14, in particular the sequence {w n } n∈Z and the corresponding maps A n ∈ G q . It is clear, that w n gives a return if and only if A n γ ∈ ϒ s . There are two cases to consider: Either z 0 ∈ Σ −1 ∪ Σ 0 respectively z 0 ∈ Σ −1 ∪ Σ 0 ∪ Σ 3 in the case of odd q or z 0 ∈ Σ 2 . In Figure 7 these different possibilities are displayed, Pγ A ∈ Σ −1 , Pγ B ∈ Σ 0 ,Pγ C ∈ Σ 2 and Pγ D ∈ Σ 3 . It is clear, that if z 0 ∈ Σ 2 then w 0 = T Sz 0 ∈ L 2 and the sequence of {w n } is essentially different from the case z 0 ∈ Σ 2 when w 0 = z 0 .  Figure 7), then w n ∈ T n L −1 for 1 ≤ n ≤ k − 1 and k = a 0 − 1, a 0 or a 0 + 1 depending on whether z 0 ∈ Σ 0,1 or Σ 3 and whether the next intersection is on T a 0 L 0 or T a 0 −1 L 0 . Then either w k ∈ T a 0 L 0 or w k ∈ T a 0 −1 L 0 (see geodesics γ E and γ F in Figure 8). Since A n = T −n for w n ∈ T n L −1 and, as we will show in Lemma 37 T −n γ / ∈ ϒ none of the w n ∈ T n L −1 for 1 ≤ n ≤ k − 1 gives a return to Σ. There are now two possibilities: If w k ∈ T a 0 L 0 , then A k = ST −a 0 and γ k = A k γ =F q γ. If j = 0 it is clear that γ ∈ ϒ s and T z 0 = P •F q γ ∈ Σ 0 . If j ≥ 1, by Lemma 32 applied to γ we get If w k ∈ T a 0 −1 L 0 , then we will show in Lemma 39 and 42 that none of the arcs emanating from T a 0 −1 ρ gives a return (cf. Figures 8 and 6) except for the next return at T a 0 −1 ST −1 h+1 L 1 . Furthermore it follows, that T z 0 = P •F k q • P −1 z 0 ∈ Σ n with k = K (ξ ) (here h or h + 1) and n = n (ξ ).
Case 2: If z 0 ∈ Σ 2 then 3λ 2 < ξ < λ + 1 and γ must intersect T L 1 below T ρ. By the same arguments as in Case 1 we conclude that the first return is given by w q−1 ∈ T ST −1 h+1 L 1 and T z 0 = P •F k q • P −1 z 0 ∈ Σ n where k = K (ξ ) and n = n (ξ ) as in Case 1 (ii). In all cases we see, that the first return map T : Σ → Σ is given by T = P •F k q • P −1 or alternatively byT =F k q where k = 1, h or h + 1 depending on ξ .
By combining Lemma 32 and 33 it is easy to see, that the first return mapP is determined completely in terms of the coordinate ξ : ∈ Ω * s then T z =P •F q k •P −1 z ∈ Σ n where k = K (ξ ) and n = n (ξ ).
Having derived explicit expressions for the first return map, in a next step we want to get explicit formulas for the first return time, i.e. the hyperbolic length between the successive returns to Σ.
Proof. Denote by z j , θ j ∈ Σ the successive returns of γ to Σ and let w j−1 ∈ γ be the point on γ corresponding to z j . If γ is closed, the set z j j≥0 is finite with N + 1 elements for some N + 1 ≤ n, i.e. z N+1 = z 0 . It is clear that the length of γ is given by adding up the lengths of all pieces between the successive returns to Σ and a repeated application of Lemma 34 gives us Remark 11. Formula (3) can also be obtained by relating the length of γ(ξ , η) to the axis of the hyperbolic matrix fixing ξ and observing that this matrix must be given by the map F n q acting on ξ . In the case of PSL 2 (Z) and the Gauss (regular) continued fractions formula (3) is wellknown.
We are now in a position to discuss the first return time. By Lemma 26 it is clear that we need to calculate the function g j (ξ , η) = |wj−ξ| 2 v j , where w j = u j + iv j = Z j (ξ , η) for all the intersection points in Corollary 7.
Definition 18. Let B ∈ PSL 2 (R) be given by Bz = 2−λ z λ −2z . Set δ n (ξ , η) := η − T n BT −n ξ and Ξ + : Proposition 2. The first return time r for the geodesic γ (ξ , η) is given by the function K (ξ ) and n (ξ ) are defined as in Definition 17, whereas the functions g j (ξ , η) = g z j , γ for z j ∈ L j are given as in Corollary 7 and F (γ) is given as in Lemma 34.
Proof. Consider γ = γ (ξ , η) ∈ ϒ s with ξ > 0 and suppose that Pγ = z 0 ∈ Σ and T z 0 = z 1 ∈ Σ with w ∈ γ corresponding to z 1 . Since geodesics are parametrized by arc length the first return time is simply the hyperbolic length between z 0 and w, i.e.
by Lemma 34. If z 0 ∈ L j , we set g (z 0 , γ) = g j (ξ , η) as given in Corollary 7. By Corollary 8 it is easy to verify, that the sets in the definition of A (ξ , η) correspond exactly to the cases z 0 ∈ L −1 , L 0 , L 2 and L 3 respectively, where the last set is empty for even q. It is also easy to see, that B 3 4 λ − 1 λ = 3λ 2 and B (λ − 1) = λ + 1. The statement of the Proposition now follows from the explicit formula for F (γ) in Lemma 34 and the domains in Proposition 1 for whichT =F k q . That r (−ξ , −η) = r (ξ , η) follows from the invariance of the crosssection with respect to reflection in the imaginary axis.

CONSTRUCTION OF AN INVARIANT MEASURE
By Liouvilles theorem we know that the geodesic flow on T 1 H preserves the measure induced by the hyperbolic metric. This measure, the Liouville measure, is given by dm = y −2 dxdydθ in the coordinates (x + iy, θ ) ∈ H × S 1 on T 1 H . Using the coordinates (ξ , η, s) ∈ Ω * × R given by Corollary 5 we obtain the Liouville measure in these coordinates The time-discretization of the geodesic flow in terms of the cross-section and first return map thus preserves the measure dm = 2dξ dη (η−ξ ) 2 . We prefer to work with the finite domain Ω * s ⊆ Ω ∞ . So let u = Sξ and v = −η with (u, v) ∈ Ω ∞ . Hence the measure dµ (u, v) = dm (ξ , η) given by . Because dµ is equivalent to Lebesgue measure, we deduce that dµ is in fact an F is a factor map ofT . An invariant measure of f 1 : I q → I q can be obtain by integrating dµ in the v-direction. We get different alternatives depending on q being even or odd.

q = 3
In this case the set of strongly reduced and reduced geodesics are the same and

Even q ≥ 4
Here and the invariant measure of F K q for odd q is given by where χ I j is the characteristic function for the interval U j . This measure is piece-wise differentiable and finite. The finite-ness is clear since uR and ur = 1 for u ∈ I q . If I q dµ (u) = c then 1 c dµ is a probability measure on I q .

Odd q ≥ 3
Let , and the invariant measure of F K q for odd q is given by where χ I j is the characteristic function for the interval U j . This measure is piece-wise differentiable and finite. The finite-ness is clear since uR, ur and ur κ−1 = 1 for u ∈ I q . If I q dµ (u) = c then 1 c dµ is a probability measure on I q .
Remark 12. For another approach leading to an infinite invariant measure see e.g. Haas and Gröchenig [14].
it is clear that dm is invariant underF q : Ω * → Ω * and letting u = Sξ and v = −η it is easy to verify that dm (u, v) = 2dudv (1−uv) 2 is invariant underF q : Ω → Ω. We can thus obtain corresponding invariant measure dµ for F q by projecting on the first variable. Let dµ (u) = dµ j (u) for u ∈ I j ,then and the invariant measure of F q is given by where χ I j is the characteristic function for the interval I j . This measure is piece-wise differentiable and finite. If c = I q dµ then 1 c dµ is a probability measure on I q . The explicit values of c = 1 4 C where C −1 = ln 1+cos π q sin π q for even q and C −1 = ln (1 + R) for odd q (see

LEMMAS ON CONTINUED FRACTION EXPANSIONS AND REDUCED GEODESICS
This section contains a collection of rather technical lemmas necessary to show that the first return map on Σ in Lemma 33 is given by powers ofF q .
Proof. Consider Figures 6 and 8 showing the arcs around the point ρ. The picture is symmetric with respect to ℜz = λ 2 and invariant under translation, so it applies in the present case. After passing through T a 0 −1 L 0 the geodesic γ will intersect a sequence of translates of the arcs χ j and ω j which are the reflections of χ j and ω j in ℜz = λ 2 exactly as in Lemma 32, except that it now passes through every arc. Note, even the argument why γ intersects ω j and not its extension applies. Let w n and A n be as in Definition 14 except that now w 0 ∈ T a 0 −1 L 0 .
Proof. Consider once more Figure 6 showing the arcs around ρ. Analogous to the proof of Lemma 39 we have w 2 j ∈ T a 0 −1 ω j = T a 0 −1 ST −1 j SL 0 , for 0 ≤ j ≤ h + 1 and w 2 j+1 ∈ T a 0 −1 χ j = T a 0 −1 ST −1 j+1 L 1 , for 0 ≤ j ≤ h with the corresponding maps A 2 j+1 = (T S) j+1 T 1−a 0 and A 2 j = (ST ) j ST 1−a 0 . Set γ j := A j γ and ξ j := A j ξ . There are now four possibilities to produce a return to Σ:
We will see that most of these cases do not give a return. Since T 1−a 0 ξ ∈ λ 2 , 1 Lemma 36 shows that c q (ξ ) = a 0 ; 1 h , a h+1 , . . . with a h+1 ≥ 2. Suppose also, that c * q (η) = 0; b 1 , b 2 , . . . * . For the following arguments it is important to remember that the action of T S on ∂ H ∼ = R * ∼ = S 1 is monotone as a rotation around ρ.