Symbolic Dynamics for the Geodesic Flow on Two-Dimensional Hyperbolic Good Orbifolds

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2014.34.2173 DYNAMICAL SYSTEMS Volume 34, Number 5, May 2014 pp. 2173–2241

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW ON TWO-DIMENSIONAL HYPERBOLIC GOOD ORBIFOLDS

Anke D. Pohl

Mathematisches Institut, Georg-August-Universität Göttingen Bunsenstr. 3–5 37073 Göttingen, Germany

(Communicated by Sebastien Gouezel)

Abstract. We construct cross sections for the geodesic flow on the orbifolds Γ\H which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, H denotes the hyperbolic plane and Γ is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic.

1. Introduction. The construction and use of symbolic dynamics in various se- tups has a long history. It goes back to work of Hadamard [19] in 1898. Since then, symbolic dynamics found influence in several fields. One of these are transfer operator approaches to Maass cusp forms (and other modular forms and functions) and Selberg zeta functions. These transfer operator approaches have their origin in the thermodynamic formalism. They provide a link between the classical and the quantum dynamical systems of the considered orbifolds. In this article, we construct symbolic dynamics which are tailor-made for these transfer operator approaches. In the following we briefly present the initial example of such a transfer operator approach and recall the state of the field prior to the existence of the symbolic dynamics constructed here. Then we state the most important properties of the symbolic dynamics provided in this article and we list the progress made using these for transfer operator approaches. Transfer operator approaches prior to the symbolic dynamics constructed here. The modular group PSL2(Z) was the first Fuchsian lattice for which the complete transfer operator approach to both Maass cusp forms and the Selberg zeta function could be established. This approach is a combination of ground-breaking work by Artin, Series, Mayer, Chang and Mayer, and Lewis and Zagier. In [43], Series, based on work by Artin [3], provided a symbolic dynamics for the geodesic flow on the modular surface PSL (Z) H which relates this flow to the Gauss map 2 \ F : [0, 1] Q [0, 1] Q, x 1 1 , \ → \ 7→ x − x 2010 Mathematics Subject Classification. Primary: 37D40; Secondary: 37B10, 37C30. Key words and phrases. Symbolic dynamics, cross section, geodesic flow, transfer operator, orbifolds. The author is supported by the International Research Training Group 1133 “Geometry and Analysis of Symmetries” and the Volkswagen Foundation.

2173 2174 ANKE D. POHL in a way that periodic geodesics on PSL2(Z) H correspond to ﬁnite orbits of F . Here, H denotes the hyperbolic plane. We\ also refer to [1, 2] for this relation between the geodesic ﬂow and the Gauss map or classical continued fractions. Mayer [26, 27, 28] investigated the transfer operator (weighted evolution operator) with parameter β C associated to F , which here takes the form ∈ 2β 1 F,βf (x)= (x + n)− f . L x + n n N X∈ 1 He found a Banach space such that for Re β > , the transfer operator F,β B 2 L acts on , is nuclear of order 0, and has a meromorphic extension F,β to the whole complexB β-plane with values in nuclear operators of order 0 on .L He showed B that the Selberg zeta function Z is represented by the product of thee Fredholm determinants of F,β: ±L

Z(β) = det(1 F,β) det(1 + F,β). (1) e − L · L On the other side, Lewis and Zagier [24] (see Lewis [23] for even Maass cusp forms) e e proved that Maass cusp forms for PSL (Z) with eigenvalue β(1 β) are linearly 2 − isomorphic to real-analytic functions f on R>0 which satisfy the functional equation

2β x f(x)= f(x +1)+(x + 1)− f (2) x +1 and are of strong decay at 0 and . Functions of this kind are called period functions ∞ for PSL2(Z). Using this characterization of Maass cusp forms, Chang and Mayer [10] as well as Lewis and Zagier [24] deduced that even resp. odd Maass cusp forms for PSL2(Z) are linearly isomorphic to the 1-eigenspaces of Mayer’s transfer operator. Efrat [15] proved this earlier on a spectral± level, that is, the parameters β for which there exist a 1-eigenfunction of F,β are just the spectral parameters of even resp. odd Maass± cusp forms. In addition,L Bruggeman [7] provided a hyperfunction approach to the period functions fore PSL2(Z). By using representations and a hyperfunction approach, Deitmar and Hilgert [14] could induce the period functions for PSL2(Z) to ﬁnite index subgroups and show the correspondence between the period functions for these subgroups and Maass cusp forms. Also using representations, Chang and Mayer [11, 12] extended the transfer operator of the modular group to its ﬁnite index subgroups and represented the Selberg zeta function as the Fredholm determinant of the induced transfer operator family. For Hecke congruence subgroups, the combination of [20] and [16] shows a close relation between eigenfunctions of certain transfer operators and these period functions. Moreover, the following transfer operator approaches to the Selberg zeta function have been established: Pollicott [39] considered the transfer operator family associated to the Series • symbolic dynamics for cocompact lattices [42]. Here the coding sequences are not unique, so the Fredholm determinant of the arising transfer operator family is not exactly the Selberg zeta function. Instead one has a representation of the form

Z(s)= h(s) det(1 s), − L where h is a function compensating for multiple codings. SYMBOLIC DYNAMICS 2175

Fried [18] developed a symbolic dynamics for the billiard flow on Λ H for tri- • \ angle groups Λ PGL2(R), and induced it to its finite index subgroups. The Fredholm determinant⊂ of the associated transfer operator family represents the Selberg zeta function without compensation factor. Morita [31] used a modified Bowen-Series symbolic dynamics for a wide class • of non-cocompact lattices [8] and investigated the associated transfer operator families. Here the problem with multiple codings arises again. Mayer et. al. [33, 30] used a symbolic dynamics related to Rosen continued • fractions for Hecke triangle groups Γ. Also this has multiple codings. Direct transfer operator approaches to Maass cusp forms were not known. How- ever, related to this field of investigations is a characterization of Maass cusp forms in parabolic 1-cohomology for any Fuchsian lattice provided by Bruggeman, Lewis and Zagier [4] (see [5, 6, 13] for earlier relations between cohomology spaces and Maass cusp forms or automorphic forms). Main properties of the symbolic dynamics constructed here. The symbolic dynamics for the geodesic flow on the orbifolds Γ H provided in this article enjoy several properties which make them well-adapted\ for transfer operator approaches. The most important ones are the following: The symbolic dynamics arise from cross sections for the geodesic flow. The • labels (the alphabet of the symbolic dynamics) arise in a natural geometric way. They encode the location of the next intersection of the cross section and the geodesic under consideration. The alphabet used in the coding is finite, all labels are elements of Γ. • The symbolic dynamics is conjugate to a discrete dynamical system F on • subsets of R. The map F is piecewise real-analytic. The action on the pieces (intervals) are given by Möbius transformations with the labels. The boundary points of the intervals are cuspidal. In turn, the associated transfer operators have only finitely many terms. They • are finite sums of the action of principal series representation by elements formed from the labels in an algorithmic way. All periodic geodesics are coded, and their coding sequences are unique. Thus, • compensation factors for multiple codings do not occur. The construction allows a number of choices which result in different symbolic • dynamics models and transfer operator families. To some degree, this freedom can be used to control properties of the transfer operators. The first step in the construction consists of the construction of a fundamen- • tal domain of a specific type. Then the construction is a finite number of algorithmic steps. This enables constructive proofs. Moreover, it allows us to test conjectures with the help of a computer. New transfer operator approaches using the symbolic dynamics constructed here. The transfer operator families which arise from the symbolic dynamics constructed in this article can be used for direct and constructive transfer operator approaches to Maass cusp forms. For Hecke triangle groups this is shown in [32], including separate period functions for odd resp. even Maass cusp forms. In particular, the functional equation (2) is just the defining equation for 1-eigenfunctions of the transfer operators for PSL2(Z). For the Hecke congruence subgroups Γ0(p), p prime, this transfer operator approach to Maass cusp forms is performed in [38]. Finally, in [36] it is shown for all Fuchsian lattices which are admissible here, including a 2176 ANKE D. POHL discussion of the relation between period functions arising from different choices in the construction, as well as a brief comparison to period functions arising from the other methods mentioned above. In addition, transfer operator approaches to Selberg zeta functions are possible. Using a certain acceleration procedure of the symbolic dynamics, in [32], the Selberg zeta function for Hecke triangle groups is represented as the Fredholm determinant of the arising transfer operator family. Compatibility with a specific orientation-reversing Riemannian isometry on H allows a factorization of the Fred- holm determinant as in (1). For the modular group, these results reproduce Mayer’s transfer operator. In [37] it is proven, by extending the symbolic dynamics to one of a billiard flow, that this factorization for general Hecke triangle groups corresponds to the splitting into odd and even spectrum and that both Fredholm determinants are Fredholm determinants of transfer operator families belonging to the billiard flow. This extends Efrat’s results [15] to Hecke triangle groups. Further results on transfer operator approaches to Selberg zeta functions will appear in future work.

Outline of this article. In Section 2 we provide the necessary preliminaries on the geometry of the hyperbolic plane and hyperbolic orbifolds as well as on fundamental domains and symbolic dynamics. In Section 3 we present a sketch of the construction of the symbolic dynamics, of the discrete dynamical system and of the transfer operator families. Sections 4-9 contain a detailed proof of the construction, and in Section 10 we brieﬂy discuss the structure of the arising transfer operators.

2. Preliminaries.

2.1. Two-dimensional hyperbolic good orbifolds. We use the upper half plane H := z C Im z > 0 { ∈ | } 2 2 2 2 with the Riemannian metric given by the line element ds = y− (dx +dy ) as model for the two-dimensional real hyperbolic space. The associated Riemannian metric will be denoted by dH. We identify the group of orientation-preserving Riemannian isometries with PSL2(R) via the well-known left action a b az + b PSL (R) H H, g = ,z g.z = . 2 × → c d 7→ cz + d Let Γ be a Fuchsian group, that is, a discrete subgroup of PSL2(R). The orbit space Y :=Γ H \ is naturally endowed with the structure of a good Riemannian orbifold. We call Y a two-dimensional hyperbolic good orbifold. The orbifold Y inherits all geometric properties of H that are Γ-invariant. Vice versa, several geometric entities of Y can be understood as the Γ-equivalence classes of the corresponding geometric entities on H. In particular, the unit tangent bundle SY of Y can be identiﬁed with the orbit space of the induced Γ-action on the unit tangent bundle SH of H. We consider all geodesics to be parametrized by arc length. For v SH let γv ∈ denote the (unit speed) geodesic on H determined by γv0 (0) = v. The (unit speed) geodesic ﬂow on H will be denoted by Φ, hence

Φ: R SH SH, (t, v) γ0 (t). × → 7→ v SYMBOLIC DYNAMICS 2177

Let π : H Y and π : SH SY denote the canonical projection maps (there will be no danger→ in using the same→ notation for both). Then the geodesic flow on Y is given by 1 Φ := π Φ (id π− ): R SY SY. ◦ ◦ × × → 1 Here, π− is an arbitrary section of π. One easily sees that Φ does not depend on 1 b the choice of π− . Throughout, we use the convention that if e denotes an element belonging in some sense to H (like geodesics on H, subsetsb of H), then e denotes the corresponding element belonging to Y . g The one-point compactification of the closure of H in C will be denotedb by H , hence g H = z C Im z 0 . { ∈ | ≥ }∪{∞} It is homeomorphic to the geodesic compactification of H. The action of PSL2(R) g extends continuously to the boundary ∂gH = R of H in H . ∪ {∞} We let R := R denote the two-point compactification of R and extend ∪ {±∞} the ordering of R to R by the definition

If, in addition, is connected, then it is a fundamental domain for Γ in H. The Fuchsian groupF Γ is called geometrically finite if there exists a convex fundamental region for Γ in H with finitely many sides. In this article we will use isometric fundamental regions (Ford fundamental regions), whose construction and existence we recall in the following. Let Γ := g Γ g. = ∞ { ∈ | ∞ ∞} a b denote the stabilizer subgroup of in Γ. Let g = c d ΓrΓ , which implies that c = 0. Let denote the Euclidean∞ norm on C. The∈isometric∞ sphere of g is defined6 as | · |

I(g) := z H g0(z) =1 = z H cz + d =1 . { ∈ | | | } { ∈ | | | } The exterior of I(g) is the set ext I(g) := z H cz + d > 1 , { ∈ | | | } and int I(g) := z H cz + d < 1 { ∈ | | | } is its interior. If the representative of g is chosen such that c> 0, then the isometric d 1 d 1 sphere I(g) is the complete geodesic segment with endpoints c c and c + c . If z = x + iy is an element of I(g), then the geodesic segment− (z−, ) is− contained 0 0 0 0 ∞ in ext I(g), and the geodesic segment (x0,z0) belongs to int I(g). Moreover, H = ext I(g) I(g) int I(g) ∪ ∪ is a partition of H into convex subsets such that ∂ ext I(g)= I(g)= ∂ int I(g). Let := := ext I(g) K KΓ g Γ Γ∞ ∈\\ denote the common part of the exteriors of all isometric spheres of Γ. We call a point z ∂gH a cuspidal point for Γ if Γ contains a parabolic element that stabilizes z. In∈ other words, z is called cuspidal if it is a representative of a cusp of Γ or, equivalently, of Y =Γ H. If is cuspidal for Γ, then \ ∞ 1 λ Γ = ∞ 0 1 for some λ> 0. For each r R, the set ∈ (r) := ,Γ(r) := (r, r + λ)+ iR>0 F∞ F∞ is a fundamental domain for Γ in H. The following proposition states the existence of isometric fundamental domains.∞ This proposition is well-known. For proofs in various generalities we refer to, e.g., [17, 22, 40, 35, 34]. Proposition 2.1. Let Γ be a geometrically ﬁnite Fuchsian group that has as cuspidal point. Then, for any r R, the set ∞ ∈ (r) := (r) F F∞ ∩ K is a convex fundamental domain for Γ in H with ﬁnitely many sides. SYMBOLIC DYNAMICS 2179

2.3. Symbolic dynamics. As before, let Γ be a Fuchsian group and set Y =Γ H. \ Let CS (“cross section”) be a subset of SY . Suppose that γ is a geodesic on Y . If

γ0(t) CS, then we say that γ intersects CS in t. Further, γ is said to intersect CS ∈ infinitelyc often in the future if we find a sequence (tn)n N withb tn as n ∈ → ∞ → ∞ andb γ0(tcn) CS for all n N.b Analogously,cγ is said to intersectb CS infinitely oftenc ∈ ∈ in the past if we find a sequence (tn)n N with tn as n and γ0(tn) CS ∈ → −∞ → ∞ ∈ for allb n N.c Let µ be a measure on the spaceb of geodesics on cY . The set CS is ∈ called a cross section (w.r.t. µ) for the geodesic flow Φ if b c (C1) µ-almost every geodesic γ on Y intersects CS infinitely often in the pastc and in the future, b (C2) each intersection of γ andb CS is discrete inc time: if γ0(t) CS, then there is ∈ ε> 0 such that γ0((t ε,t + ε)) CS = γ0(t) . − ∩ { } c c1 We call a subset U of Yba totally geodesic suborbifold ofbY if π− (U) is a totally geodesic submanifoldb of H. Let pr: SY c Y denoteb the canonical projection on → base points. If pr(CS)b is a totally geodesic suborbifold of Y and CS doesb not contain elements tangent to pr(CS), then CS automatically satisfies (C2). Suppose that CSc is a cross section for Φ. If, in addition, CS satisfiesc the property that each geodesic intersectingc CSc at all intersects it infinitely often in the past and in the future, thenc CS will be called a strongb cross section,c otherwise a weak cross section. Clearly, every weak crossc section contains a strong cross section. The first return mapc of Φ w.r.t. the strong cross section CS is the map

R: CS CS, v γv0(t ), b → 7→ 0 c where v SH with π(v)= v, π(γv)= γv, and ∈ c c b b t0 := min t> 0 γ0 (t) CS b b v ∈ n o is the first return time ofv ˆ or γv. This definition requires that t0 = t0(v) exists b c for each v CS, which will indeed be the case in our situation. For a weak cross ∈ section CS, the first return mapb can only be defined on a subset of CS. Inb general, this subsetb isc larger than the maximal strong cross section contained in CS. Supposec that CS is a strong cross section and let Σ be an at mostc countable c set. Decompose CS into a disjoint union α Σ CSα. To each v CS we assign the ∈ Z ∈ (two-sided infinite)ccoding sequence (an)n Z Σ defined by S∈ ∈ c cn b c an := α if and only if R (v) CSα. ∈ Note that R is invertible and let Λ be the set of all sequences that arise in this way. Then Λ is invariant under the left shift b c Z Z σ : Σ Σ , σ (an)n Z := ak+1. → ∈ k Suppose that the map CS Λ is also injective, which it will be in our case. Then we → have the inverse map Cod: Λ CS which maps a coding sequence to the element → in CS it was assigned to.c Obviously, the diagram

c R CS / CS c O O Codc cCod σ Λ / Λ 2180 ANKE D. POHL commutes. The pair (Λ, σ) is called a symbolic dynamics for Φ with alphabet Σ. If CS is only a weak cross section and hence R is only partially defined, then Λ also contains one- or two-sided finite coding sequences. b cLet CS0 be a set of representatives for the cross section CS, that is, CS0 is a subset of SH such that π 0 is a bijection CS0 CS. Relative to CS0, we define |CS → the map τ : CS ∂gH ∂gH by c → × c τ(v) := (γv( ),γv( )) c ∞ −∞ 1 where v := (π 0 )− (v). For some cross sections CS it is possible to choose CS0 |CS in such a way that τ is a bijectionb between CS and some subset DS of R R. × In this case the dynamicalb system (CS, R) is conjugatec to (DS, F ) by τ, where 1 F := τ R τ − is the induced selfmap on DSc (partially defined iffCS is only a ◦ ◦ weak cross section). Moreover, to constructc a symbolic dynamicsf efor Φ, one can starte with a decomposition of DS into pairwisef disjoint subsets Dα, α cΣ. Finally, let (Λ, σ) be a symbolic dynamics with alphabet Σ. Suppose∈b that we have a map i:Λ DS for somef DS R such that i (an)n Z e depends only on → ⊆ ∈ (an)n N , a (partial) selfmap F : DS DS, and a decomposition of DS into a ∈ 0 → disjoint union α Σ Dα such that ∈ S F i((an)n Z) Dα a1 = α ∈ ∈ ⇔ for all (an)n Z Λ. Then F , more precisely the triple F, i, (Dα)α Σ , is called a generating∈ function∈ for the future part of (Λ, σ). If such a generating∈ function exists, then the future part of a coding sequence is independent of the past part. Remark 2.2. We stated a rather general definition of cross sections. In particular, we did not require any regularity of the set CS nor restricted the admissible measures µ. Any additional requirements on CS or µ should be due to applications. In the following we restrict the measures toc an admissible set and construct subsets CS of SY which are simultaneous crossc sections for all admissible measures (see Theorem 8.15). For transfer operator approaches, any regularity of CS only plays an implicitc role. For these applications, the most important properties of a cross section are that all periodic geodesics intersect it and that it is conjugatec to a discrete dynamical system (DS, F ) with a (in some weak sense) piecewise smooth map F . In all hitherto existing examples of transfer operator approaches, the geodesics which tend to a cusp in thef futuree or past direction can be neglected. For this reason,e all measures are admissible which assign zero mass to this set of geodesics. We will see that the base sets pr(CS) of the constructed cross sections CS are always a finite family of complete geodesics and hence smooth. The weak cross sections are smooth subsets of SY . In contrast,c the strong cross sections are typicallyc fractal.

3. A sketch of the construction. In this section we provide a sketch of the construction of cross sections and symbolic dynamics, illustrated by examples. For further examples we refer to [21, 32, 38, 36]. As we will see already in this sketch, the cuspidal point plays a distinguished role. Throughout let Γ be a geometrically ﬁnite Fuchsian∞ group with as cuspidal point. ∞ Additional condition on Γ. To state the additionally required condition on Γ we need a few notions. The height of a point z H is deﬁned to be ∈ ht(z) := Im z. SYMBOLIC DYNAMICS 2181

Let g = a b ΓrΓ with c chosen positive, and consider its isometric sphere c d ∈ ∞ d 1 I(g)= z H z + = . ∈ c c

Its point of maximal height

d 1 s = + i − c c is called the summit of I(g). Recall the set = ext I(g), K g ΓrΓ∞ ∈\ which is the subset of H contained in the exterior of all isometric spheres of Γ. We call an isometric sphere I of Γ relevant if I contributes nontrivially to the boundary of , that is, if I ∂ contains a submanifold of H of codimension 1. If the isometricK sphere I is∩ relevant,K then I ∂ is called its relevant part. An endpoint of the relevant part of a relevant isometric∩ K sphere is called a vertex of . From now on we impose the following condition on Γ: K For each relevant isometric sphere, its summit is contained in ∂ but not a (A) vertex of . K K π Example 3.1. (i) For n N, n 3, let λn := 2cos . The Hecke triangle group ∈ ≥ n Gn is the subgroup of PSL2(R) which is generated by 0 1 1 λ S := and T := n . 1− 0 n 0 1 The relevant isometric spheres are I(S) and its T k-translates, k Z. n ∈

3 λn λn 3 λn λn 0 λn λn − 2 − − 2 2 2

Figure 1. The set for G . K 5 (ii) Let a b PΓ (5) := PSL (Z) c 0 mod5 . 0 c d ∈ 2 ≡

The isometric spheres are the sets

d 1 Ic,d = z H z + = ∈ 5c 5c

2182 ANKE D. POHL

where c N and d Z. The set is given by (see Figure 2) ∈ ∈ K = z H z + d > 1 . K ∈ 5 5 d Z \∈

1 0 1 2 3 4 1 6 − 5 5 5 5 5 5

Figure 2. The set for PΓ (5). K 0

0 1 1 4 (iii) Let S := 1− 0 and T := [ 0 1 ], and denote by Γ the subgroup of PSL2(R) which is generated by S and T . The set is indicated in Figure 3. K

1 0 1 2 3 4 5 −

Figure 3. The set for the group Γ from Example 3.1(iii). K

The first cross section. The (first) cross section CS for the geodesic flow on Γ H will be defined with the help of a very specific tesselation of . For that we note\ K that there are two kinds of vertices of . Let v be ac vertex of . If v H, then we call v an inner vertex, otherwise v is calledK an infinite vertex.K Now we∈ construct a family of geodesic segments in the following way: Whenever v is an infinite vertex of , thenG the (complete) geodesic segment (v, ) belongs to this family. Moreover, wheneverK s is the summit of a relevant isometric∞ sphere, then the geodesic segment (s, ) belongs to this family. Let CS denote the set of unit tangent vectors on H which∞ are based on the geodesic segments in this family but which are not tangent to any of these geodesic segments.f In other words, CS consists of the unit tangent vectors whose base points are on any of these geodesic segments and which point to the right or the left, but not straight up or down.f Then CS := π(CS) is a weak cross section for the geodesic flow. By eliminating a certain set of vectors c f from it, we will also construct a strong cross section. SYMBOLIC DYNAMICS 2183

Construction of a set of representatives. The family of geodesic segments induces a tesselation of into hyperbolic triangles, quadrilateralG and strips. We call these tesselation elementsK precells in H. The union of certain finite subfamilies of these precells in H form isometric fundamental regions or even, if the union is connected, isometric fundamental domains. This relation is useful for the following two reasons. Pick such a subfamily of precells in H and let CSpre0 denote the set of unit tangent which are based on the vertical sides of these precells and point into them. Then the link to isometric fundamental regions yields that CSpre0 is a set of representatives for CS. This set, however, is not very useful for coding purposes. For this reason, we use another property of isometric fundamental regions to modify the set of representatives.c Isometric fundamental regions allow to define cycles of their sides as in the PoincaréFundamental Polyhedron Theorem. The acting elements in these cycles are just the generating elements of the relevant isometric spheres which contribute to the boundary of the fundamental region. Using these cycles we glue translates of precells in H to ideal polyhedrons in H, that is, polyhedrons all of which vertices are in ∂gH. Even more, all vertices are in Γ. . At this point, Condition (A) is essential. We call these polyhedron cells in H.∞ All these cells in H have two vertical sides, and all these non-vertical sides are Γ-translates of vertical sides of some cells in H. These elements in Γ can be determined by an algorithm from the generators of the relevant isometric spheres. Lifting this construction to the unit tangent bundle SH and using this to re- distribute CSpre0 allows us to find a set of representatives CS0 such that CS0 is the disjoint union

CS0 = CSα0 α A [∈ for some finite index set A such that for each α A, the subset CSα0 is the set of unit tangent vectors based on a vertical side (a complete∈ geodesic segment) of some cell in H and pointing into this cell. The definition of precells in H and the construction of cells in H from precells is based on ideas in [44]. Our construction differs from Vulakh’s construction in three important aspects: We define three kinds of precells in H of which only one are precells in the sense of Vulakh. Finally, contrary to Vulakh, we extend the considerations to precells and cells in the unit tangent bundle. Example 3.2. The set

λn n := z H z > 1, Re z < F ∈ | | | | 2 is an isometric fundamental domain for the Hecke triangle group Gn (see Figure 4).

Hecke triangle groups have up to equivalence under (Gn) = Tn only one precell in H. It is indicated in Figure 5. ∞ h i Figure 6 provides an idea of the lifting and redistribution procedure. Discrete dynamical system on the boundary and associated transfer operator families. The relation between isometric fundamental regions and the speciﬁc way of lifting to SH yields the following property of CS0. Whenever L is the side of some Γ-translate of some cell in H and L0 is the set of all unit tangent vectors based on that side pointing into this Γ-translate or all vectors pointing out of this Γ-translate, then there is a unique pair (α, g) A Γ such that L0 = g. CS0 . Both, ∈ × α 2184 ANKE D. POHL

n F

λn λn − 2 2

Figure 4. An isometric fundamental domain for G in H. F5 5

i λn + i

λ 0 n λn 2

Figure 5. The precell of G . A 5

Figure 6. Lifting to SH and deﬁning CS0 for G5.

α and g can be determined algorithmically. Further, Γ. CS0 is just the union of all such L0. These properties allow to read off the induced discrete dynamical system F on the boundary of H. It is given by finitely many local diffeomorphisms of the form 1 (Iα Jα ) g− .(Iα Jα ) α g.(Iα Jα ) (Iα Jα ) α , 1 × 1 ∩ 2 × 2 ×{ 1} → 1 × 1 ∩ 2 × 2 ×{ 2} (x, y, α ) (g.x, g.y, α ), (3) 1 7→ 2 SYMBOLIC DYNAMICS 2185 where α , α A, Iα , Iα ,Jα ,Jα are intervals, and g Γ depends on α , α . 1 2 ∈ 1 2 1 2 ∈ 1 2 The associated transfer operator F,β with parameter β C is then defined on spaces of functions f on the domainL of F and given by ∈ f(u) F,βf(w)= β . L − F 0(u) u F 1(w) ∈ X | | A more explicit formula is provided in Section 10.

Example 3.3. Figure 7 shows the translates of CS0 for the Hecke triangle group G5. Here, we set λ 1 U := T S = n . n n 1− 0 For general Hecke triangle groups Gn, the picture is similar. If one restricts here to

1 T5− .CS0 S.CS0 CS0 U5.CS0 T5.CS0

1 4 1 2 4 2 T5− U5 .CS0 T5− U5 .CS0 U5 .CS0 U5 .CS0 1 3 3 T5− U5 .CS0 U5 .CS0 4 2 U5 S.CS0 U5 S.CS0 3 U5 S.CS0 1 4 1 3 1 2 3 2 T − U . = λ T − U . T − U . 0 U . U . U5. = λ5 5 5 ∞ − 5 5 5 ∞ 5 5 ∞ 5 ∞ 5 ∞ ∞

Figure 7. The shaded parts are translates of CS0 (in the unit tangent bundle) as indicated. the forward-direction of the geodesic ﬂow, then the discrete dynamical system for Gn is F : R> rGn. R> rGn. , 0 ∞ → 0 ∞ given by the bijections 1 (gk.0,gk. )rGn. R> rGn. , x g− .x ∞ ∞ → 0 ∞ 7→ k for k =1,...,n 1 with − k gk := Un S.

The associated transfer operator F,β with parameter β C is then deﬁned on L ∈ functions f : R> rGn. C and given by 0 ∞ → n 1 − β F,βf(x)= g0 (x) f(gk.x). L k k=1 X For the modular group PSL2(Z)= G3 the transfer operator becomes

2β x F,βf(x)= f(x +1)+(x + 1)− f . L x +1 The 1-eigenfunctions of this transfer operator are characterized by the functional equation (2). In [32], the relation between eigenfunctions of this transfer operator and period functions, as well as the relation to Mayer’s transfer operator and the factorization (1) are discussed in detail. 2186 ANKE D. POHL

The second cross section and the reduced discrete dynamical system. For some lattices Γ it may happen that in some of the local diffeomorphisms of the form (3) we have g = id. To avoid that and at the same time to be able to eliminate the bits α and α from the domain and range of F , we will shrink the cross { 1} { 2} section CS and the set of representatives CS0 in an algorithmic way to deduce a new cross section CSred with a set of representatives CSred0 . In terms of notions to be introducedc later, we will eliminate all vectors from CS with id in the label. A group that does not satisfy (A). The previous examples as well as those c from [21, 38, 36] show that there are several geometrically finite Fuchsian groups with as cuspidal point and which satisfy the condition (A). We now provide an example∞ of a geometrically finite Fuchsian group with as cuspidal point, which does not satisfy (A). ∞ Let Γ be the subgroup of PSL2(R) which is generated by the matrices 1 17 18 5 3 1 t := 11 , g := and g := . 0 1 1 11 −3 2 10 −3 − − Set 4 3 g := g g = , 3 1 2 3 −2 − let := 2 , 19 + iR+ and (see Figure 8) F∞ 11 11 1 1 := extI(g1) ext I(g1− ) ext I(g2) ext I(g1g2) ext I (g1g2)− . F F∞ ∩ ∩ ∩ ∩ ∩

s5 F s6

s1 s8

s3 s4 s2 s7 v2v3v4 v6

v1 v5 v7

Figure 8. The fundamental domain F We indicated the points 3 i 91 √24 v0 := v3 := 10 + 10 v6 := 55 + i 55 ∞2 19 √24 19 v1 := 11 v4 := 55 + i 55 v7 := 11 14 √24 v2 := 55 + i 55 v5 :=1 and the geodesic segments

s1 := [v0, v1] s4 := [v3, v4] s7 := [v6, v7] s2 := [v1, v2] s5 := [v4, v5] s8 := [v7, v0]. s3 := [v2, v3] s6 := [v5, v6] SYMBOLIC DYNAMICS 2187

Proposition 3.4. The set is a fundamental domain for Γ in H. F Proof. The sides of are s ,s ,s s ,s ,s ,s and s . Further we have the F 1 2 3 ∪ 4 5 6 7 8 side-pairings ts1 = s8, g1s2 = s7, g2s3 = s4 and g3s5 = s6, where tv0 = v0, tv1 = v7, g1v1 = v7, g1v2 = v6, g2v2 = v4, g2v3 = v3, g3v4 = v6 and g3v5 = v5. Now Poincar´e’s Fundamental Polyhedron Theorem (see e.g. [25]) yields that is F a fundamental domain for the group generated by t,g1,g2 and g3. This group is exactly Γ.

Proposition 3.5. Γ does not satisfy (A). Proof. Proposition 3.4 states that is a fundamental domain for Γ in H. Its shape and side-pairings yield that it is anF isometric fundamental domain. Therefore, the isometric sphere I(g1) is relevant and s1 is its relevant part. The summit of I(g1) is s := 3+i . One easily calculates that s int I(g ). Thus, Γ does not satisfy (A). 11 ∈ 2

In [44], Vulakh states that each geometrically ﬁnite subgroup of PSL2(R) for which is a cuspidal point satisﬁes (A). The previous example shows that this statement∞ is not correct. This property is crucial for the results in [44]. Thus, Vulakh’s constructions do not apply to such a huge class of groups as he claims.

4. Precells in H. Let Γ be a geometric finite Fuchsian group with as cuspidal point and which satisfies Condition (A). To avoid empty statements∞ suppose that Γ =Γ , which means that there are relevant isometric spheres. 6 In this∞ section we introduce the notion of precells in H and basal families of precells in H. Moreover, we discuss their relation to fundamental regions and study some of their properties. We recall that := ext I(g). K g ΓrΓ∞ ∈\ 4.1. The structure of . We start with a short discussion of the vertex structure of . K KThe set of all isometric spheres of Γ need not be locally finite. For example, g in the case of the modular group PSL2(Z), each neighborhood of 0 in H contains infinitely many isometric spheres. However, from Γ being geometrically finite, it follows immediately that the set of relevant isometric spheres is locally finite. In turn, the set of infinite vertices of has no accumulation points. Moreover, if v is an inner vertex of , then there areK exactly two distinct relevant isometric spheres such that v is a commonK endpoint of their relevant parts. If v is an infinite vertex, then two situations can occur. If v is an endpoint of the relevant parts of two distinct relevant isometric spheres, we call v a two-sided infinite vertex. Otherwise we call v a one-sided infinite vertex. Straightforward geometric arguments prove the following statement on the local situation at one-sided infinite vertices of . For this let K g pr : H r R, pr (z) := z ht(z)=Re z ∞ {∞} → ∞ − denote the geodesic projection from to ∂gH. For a,b R we set ∞ ∈ [a,b] if a b, a,b := ≤ h i ([b,a] otherwise. 2188 ANKE D. POHL

Proposition 4.1. Let v be a one-sided infinite vertex of . Then there exists K 1 a unique one-sided infinite vertex w of such that the strip pr− ( v, w ) H is 1 K ∞ h i ∩ contained in . In particular, pr− ( v, w ) does not intersect any isometric sphere K ∞ 1 h i in H, and, of all vertices of , pr− ( v, w ) contains only v and w. K ∞ h i For any one-sided infinite vertex v of , we call the interval v, w from Propo- sition 4.1 a boundary interval, and we callK w the one-sided infiniteh vertexi adjacent to v. Boundary intervals will be needed for the definition and investigation of strip precells defined below. Example 4.2. Recall the group Γ from Example 3.1(iii). The boundary intervals of are the intervals [1 + 4m, 3+4m] for each m Z. K ∈ 4.2. Precells in H and basal families. We now define the notion of precells in H. Definition 4.3. Let v be a vertex of . Suppose first that v is an inner vertex or a two-sided infinite vertex. Then thereK are (exactly) two relevant isometric spheres I1, I2 with relevant parts [a1, v] resp. [v,b2]. Let s1 resp. s2 be the summit of I1 resp. I2. By Condition (A), the summits s1 and s2 do not coincide with v. 1 If v is a two-sided infinite vertex, then define 1 to be the hyperbolic triangle with vertices v, s and , and define to be theA hyperbolic triangle with vertices 1 ∞ A2 v, s2 and . The sets 1 and 2 are the precells in H attached to v. Precells arising in this∞ way are calledA cuspidalA . If v is an inner vertex, then let be the hyperbolic quadrilateral with vertices A s1, v, s2 and . The set is the precell in H attached to v. Precells that are constructed in∞ this way areA called non-cuspidal. Suppose now that v is a one-sided infinite vertex. Then there exist exactly one relevant isometric sphere I with relevant part [a, v] and a unique one-sided infinite 1 vertex w other than v such that pr− ( v, w ) does not contain vertices other than v and w (see Proposition 4.1). Let s∞beh thei summit of I. Define 1 to be the hyperbolic triangle with vertices v, s and , and define 2 A 1 ∞ A to be the vertical strip pr− ( v, w ) H. The sets 1 and 2 are the precells in H attached to v. The precell∞ h is calledi ∩ cuspidal, andA isA called a strip precell. A1 A2 Example 4.4. The precells in H of the congruence group PΓ0(5) from Exam- ple 3.1(ii) are indicated in Figure 9 up to PΓ0(5) -equivalence. ∞

(v ) (v ) (v ) (v ) (v ) A 0 A 1 A 2 A 3 A 4

m1 m2 m3 m4 v1 v2 v3

v0 =0 v4 =1

Figure 9. Precells in H of PΓ0(5).

1We consider the boundary of the triangle in H to belong to it. SYMBOLIC DYNAMICS 2189

The inner vertices of are K 2k +1 √3 v = + i , k =1, 2, 3, k 10 10 and their translates under PΓ0(5) . The summits of the indicated isometric spheres are ∞ k i m = + , k =1,..., 4. k 5 5 The group PΓ0(5) has cuspidal as well as non-cuspidal precells in H, but no strip precells. Example 4.5. The precells in H of the group Γ from Example 3.1(iii) are up to Γ -equivalence one strip precell 1 and two cuspidal precells 2, 3 as indicated in∞ Figure 10. Here, v = 3, v =A 1, v = 1 and m = i. A A 1 − 2 − 3

A1 A2 A3 m

v1 v2 0 v3

Figure 10. Precells in H of Γ.

Condition (A) yields that the relation between different precells and between the precells and are well-structured. K Proposition 4.6. (i) If is a precell in H, then A 1 1 = pr− pr ( ) and ◦ = pr− pr ( ◦) . A ∞ ∞ A ∩ K A ∞ ∞ A ∩ K (ii) If two precells in H have a common point, then either they are identical or they coincide exactly at a common vertical side. (iii) The set is the essentially disjoint union of all precells in H, K = precell in H , K {A|A } and contains the disjoint[ union of the interiors of all precells in H, K ◦ precell in H . {A | A } ⊆ K Proof. We use the notation[ from Definition 4.3 to discuss the relation between precells and . Suppose firstK that is a non-cuspidal precell in H attached to the inner vertex A v. Condition (A) implies that the summits s1 and s2 are contained in the relevant parts of the relevant isometric spheres I1 and I2, respectively. Therefore they are contained in ∂ . Hence is the subset of with the two vertical sides [s , ] and K A K 1 ∞ [s2, ], and the two non-vertical ones [s1, v] and [v,s2]. The geodesic projection of from∞ is A ∞ pr ( )= Re s1, Re s2 . ∞ A h i 2190 ANKE D. POHL

Suppose now that is a cuspidal precell in H attached to the infinite vertex v. Then has two verticalA sides, namely [v, ] and [s, ], and a single non-vertical side, namelyA [v,s]. As for non-cuspidal precells∞ we find∞ that [v,s] is contained in the relevant part of some relevant isometric sphere, and hence is a subset of . The geodesic projection of from is A K A ∞ pr ( )= Re s, v . ∞ A h i 1 Suppose finally that is the strip precell pr− ( v, w ) H. Then is attached to the two vertices v andA w. It has the two vertical∞ h sidesi ∩ [v, ] andA [w, ] and no non-vertical ones. The geodesic projection of from is ∞ ∞ A ∞ pr ( )= v, w . ∞ A h i By Proposition 4.1, is contained in . From these observations, the statements follow easily. A K

Before we investigate the relation between precells in H and fundamental regions in Theorem 4.8 below, we state a few properties of isometric spheres. Their proofs are straightforward. Lemma 4.7. Let g ΓrΓ . ∈ ∞ 1 (i) We have g.I(g)= I(g− ). 1 (ii) If s is the summit of I(g), then g.s is the summit of I(g− ). 1 (iii) The geodesic projection pr (s) of the summit s of I(g) is the center g− . of I(g). ∞ ∞ 1 (iv) If I(g) is relevant with relevant part [a,b], then I(g− ) is relevant with relevant part g.[a,b] = [g.a, g.b]. Let Λ be a Fuchsian group. A subset of H is called a closed fundamental F region for Λ in H if is closed and ◦ is a fundamental region for Λ in H. If, in F F addition, ◦ is connected, then is said to be a closed fundamental domain for Λ in H. NoteF that if is a non-connectedF fundamental region for Λ in H, then can happen to be a closedF fundamental domain. F Let Aj j J be a family of real submanifolds (possibly with boundary) of H g { | ∈ } or H , and let n := max dim Aj j J . We call the union { | ∈ }

Aj j J [∈ essentially disjoint if for each i, j J, i = j, the intersection Ai Aj is contained in a real submanifold (possibly with∈ boundary)6 of dimension n ∩1. − Theorem 4.8. There exists a set j j J of precells in H, indexed by J, such {A | ∈ } that the (essentially disjoint) union j is a closed fundamental region for Γ j J A in H. The set J is finite and its cardinality∈ does not depend on the choice of the S specific set of precells. The set j j J can be chosen such that j J j is a {A | ∈ } ∈ A closed fundamental domain for Γ in H. In each case, the (disjoint) union ◦ S j J Aj is a fundamental region for Γ in H. ∈ S Proof. By Proposition 4.6(ii), the union of each family of pairwise different precells in H is essentially disjoint. Let r be the center of some relevant isometric sphere I. Let r := Re s for the summit s of some relevant sphere of Γ or let r := Re v for an SYMBOLIC DYNAMICS 2191 infinite vertex v of . Let λ> 0 be the unique positive number such that K 1 λ Γ = . ∞ 0 1 From Lemma 4.7 and Proposition 4.6 one easily deduces that

(r)= (r) = ([r, r + λ]+ iR>0) F F∞ ∩ K ∩ K decomposes into a ﬁnite set A := j j J of precells in H. Clearly, (r) is a closed fundamental domain. {A | ∈ } F

Let A2 := k k K be a set of precells in H such that := k K k is a closed fundamental{A | region∈ } for Γ in H. Proposition 4.6(iii) impliesF that ∈ A S = h. h Γ , A . K { A | ∈ ∞ A ∈ } Let k A2 and pick z k◦[. Then there exists hk Γ and jk J such that A ∈ ∈ A ∈ ∞ ∈ hk.z jk . Therefore hk. k◦ jk = . The Γ -invariance of shows that hk. k ∈ A A ∩ A 6 ∅ ∞ K A is a precell in H. Then Proposition 4.6(ii) implies that hk. k = j , and in turn A A k hk and jk are unique. We will show that the map ϕ: K J, k jk, is a bijection. → 7→ To show that ϕ is injective suppose that there are l, k K such that jk = jl =: j. 1 ∈ 1 Then hl. l = j = hk. k, hence hk− hl. l = k. In particular, hk− hl. l◦ k◦ = . A A A A A A1 ∩A 6 ∅ Since h K h◦ ◦ and ◦ is a fundamental region, it follows that hk− hl = id and ∈ A ⊆F F l = k. Thus, ϕ is injective. To show surjectivity let j J and z j◦. Then there S ∈ ∈ A 1 exists g Γ and k K such that g.z k. On the other hand, k = h− . j . ∈ ∈ ∈ A A k A k Hence hkg. ◦ j = . Since j and j are convex polyhedrons, it follows that Aj ∩ A k 6 ∅ A A k hkg. ◦ j = . Since (r) is a fundamental region and ◦, ◦ (r), we find Aj ∩ A k 6 ∅ F Aj Ajk ⊆F that hkg = id and j = jk. Hence, ϕ is surjective. It follows that #K = #J. It remains to show that the disjoint union P := k K k◦ is a fundamental region ∈ A for Γ in H. Obviously, P is open and contained in ◦. This shows that P satisfies F (F1) and (F2). Since ( )= for each precell inSH and K is finite, it follows that A◦ A P = = k = . Ak◦ A F k K k K [∈ [∈ Hence, P satisfies (F3) as well, and thus it is a fundamental region for Γ in H.

Each set A := j j J of precells in H, indexed by J, with the property that {A | ∈ } := j is a closed fundamental region is called a basal family of precells in F j J A H or a family∈ of basal precells in H. If, in addition, is connected, then A is called a connectedS basal family of precells in H or a connectedF family of basal precells in H. Example 4.9. Recall the Examples 3.2, 4.4 and 4.5. The set of precells in H {A} for Gn is a connected basal family of precells in H, as well as (v0),..., (v4) for PΓ (5), and , , for Γ. {A A } 0 {A1 A2 A3} The proof of Theorem 4.8 shows the following statements. Let 1 λ t := λ 0 1 with λ> 0 be such that Γ = tλ . ∞ h i Corollary 4.10. Let A be a basal family of precells in H.

(i) For each precell in H there exists a unique pair ( 0,m) A Z such that m A A ∈ × t . 0 = . λ A A 2192 ANKE D. POHL

m( ) (ii) For each A choose an element m( ) Z. Then tλ A . A is A ∈ A ∈ mA( ) A ∈ A A a basal family of precells in H. For each , the precell tλ . is of the same type as . A ∈ A (iii) The set is theA essentially disjoint union h. h Γ , A . K { A | ∈ ∞ A ∈ } 4.3. The tesselation of H by basal familiesS of precells. The following proposition is crucial for the construction of cells in H from precells in H. Note that the element g ΓrΓ in this proposition depends not only on and b but also on the choice∈ of the basal∞ family A of precells in H. In this sectionA we will use the proposition as one ingredient for the proof that the family of Γ-translates of all precells in H is a tesselation of H. Proposition 4.11. Let A be a basal family of precells in H. Let A be a basal precell that is not a strip precell, and suppose that b is a non-verticalA side ∈ of . Then there is a unique element g ΓrΓ such that b I(g) and g.b is the non-verticalA ∈ ∞ ⊆ side of some basal precell 0 A. If is non-cuspidal, then 0 is non-cuspidal, A ∈ A A and, if is cuspidal, then 0 is cuspidal. A A Proof. Let I be the (relevant) isometric sphere with b I. We will show first that there is a generator g of I such that g.b is a side of⊆ some basal precell. Then 1 g.b g.I(g)= I(g− ), which implies that g.b is a non-vertical side. Let⊆ h ΓrΓ be any generator of I, let s be the summit of I and v the vertex of that∈ is attached∞ to. Then b = [v,s]. Further, b is contained in the relevant partK of I =AI(h). By Lemma 4.7, the set h.b = [h.v, h.s] is contained in the relevant 1 part of the relevant isometric sphere I(h− ), the point h.v is a vertex of and h.s 1 K is the summit of I(h− ). Thus, there is a unique precell h with non-vertical side A h.b. By Corollary 4.10, there is a unique basal precell 0 and a unique m Z such that A ∈ m h = t . 0 = 0 + mλ. A λ A A m m Then tλ− h.b is a non-vertical side of 0, and tλ− h.b is contained in the relevant A 1 1 m m 1 part of the relevant isometric sphere I(h− ) mλ = I(h− tλ ) = I((tλ− h)− ). m − Clearly, also g := tλ− h is a generator of I. To prove the uniqueness of g, let k be any generator of I. Then there exists a unique n Z such that k = tnh. Thus, k.b = tnh.b = h.b + nλ and therefore ∈ λ λ k = h + nλ. Then A A m+n k = 0 + mλ + nλ = t . 0, A A λ A (m+n) (m+n) and tλ− k is a generator of I such that tλ− k.b is a side of some basal precell. Moreover, (m+n) m n m tλ− k = tλ− tλ− k = tλ− h = g. This shows the uniqueness. The basal precell 0 cannot be a strip precell, since it has a non-vertical side. Finally, is cuspidalA if and only if v is an infinite vertex. This is the case if and only A if gv is an infinite vertex, which is equivalent to 0 being cuspidal. This completes the proof. A

Lemma 4.12. Let be a precell in H. Suppose that S is a vertical side of . Then A A there exists a precell 0 in H such that S is a side of 0 and 0 = . In this case, A A A 6 A S is a vertical side of 0. A SYMBOLIC DYNAMICS 2193

Proposition 4.13. Let 1, 2 be two precells in H and let g1,g2 Γ. Suppose A A ∈ 1 that g1. 1 g2. 2 = . Then we have either g1. 1 = g2. 2 and g1g2− Γ , or g . Ag .∩ isA a6 common∅ side of g . and g A. , or gA. g . is∈ a∞ point 1 A1 ∩ 2 A2 1 A1 2 A2 1 A1 ∩ 2 A2 which is the endpoint of some side of g1. 1 and some side of g2. 2. If S is a 1A A common side of g1. 1 and g2. 2, then g1− .S is a vertical side of 1 if and only if 1 A A A g− .S is a vertical side of . 2 A2

Proof. W.l.o.g. g1 = id. Let A be a basal family of precells in H. Corollary 4.10 shows that we may assume that 1 A. Let S1 = [a1, ] and S2 = [a2, ] be the vertical sides of . If hasA non-vertical∈ sides, let∞ these be S = [∞b ,b ] A1 A1 3 1 2 resp. S = [b ,b ] and S = [b ,b ]. Lemma 4.12 shows that we ﬁnd precells 0 3 1 2 4 3 4 A1 and 0 such that 0 = = 0 and S is a vertical side of 0 and S is a A2 A1 6 A1 6 A2 1 A1 2 vertical side of 0 . Proposition 4.11 shows that there exist (h , 0 ) Γ A resp. A2 3 A3 ∈ × (h , 0 ), (h , 0 ) Γ A such that h .S is a non-vertical side of 0 and h .S is 3 A3 4 A4 ∈ × 3 3 A3 4 4 a non-vertical side of 0 and h . = 0 and h . = 0 . Recall that each precell A4 3 A1 6 A3 4 A1 6 A4 in H is a convex polyhedron. Therefore 1 10 is a polyhedron with (a1, ) in its 1 A ∪ A ∞ interior, and 1 h3− . 30 is a polyhedron with (b1,b2) in its interior. Likewise for A ∪ 1 A 0 and h− . 0 . A1 ∪ A2 A1 ∪ 4 A4 Suppose ﬁrst that ◦ g . = . By Corollary 4.10 there exists a pair (h, 0) A1 ∩ 2 A2 6 ∅ A ∈ Γ A such that 2 = h. 0. Then 1◦ g2h. 0 = . Since 0 is a convex ∞ × A A A ∩ A 6 ∅ A polyhedron, ◦ g h.( 0)◦ = . Now ( )◦ A is a fundamental region A1 ∩ 2 A 6 ∅ A A ∈ for Γ in H (see Theorem 4.8). Therefore, g2h = id and, by Proposition 4.6(ii), 1 S 1 = 0. Hence g2− Γ and 1 = g2. 2. b b A A ∈ ∞ A A Suppose now that ◦ g . = . If g . (a , ) = , then g . ( 0 )◦ = A1 ∩ 2 A2 ∅ 2 A2 ∩ 1 ∞ 6 ∅ 2 A2 ∩ A1 6 ∅ and the argument from above shows that 10 = g2. 2 and g2 Γ . From this it A A ∈ ∞ follows that S1 is a vertical side of 2. A 1 If g2. 2 (b1,b2) = , then g2. 2 h3− .( 30 )◦ = . As before, g2h3 Γ and A ∩1 6 ∅ A ∩ A 6 ∅ ∈ ∞ g . = h− . 0 . Then S is a non-vertical side of . The argumentation for 2 A2 3 A3 3 A2 g2. 2 (a2, ) = and g2. 2 (b3,b4) = is analogous. AIt remains∩ ∞ the6 case∅ that gA. ∩ intersects6 ∅ is an endpoint v of some side of . 2 A2 A1 A1 By symmetry of arguments, v is an endpoint of some side of 2. This completes the proof. A

We call a family Sj j J of polyhedrons in H a tesselation of H if { | ∈ } (T1) H = j J Sj , and ∈ (T2) if Sj Sk = for some j, k J, then either Sj = Sk or Sj Sk is a common S∩ 6 ∅ ∈ ∩ side or vertex of Sj and Sk.

Corollary 4.14. Let A be a basal family of precells in H. Then

Γ.A = g. g Γ, A { A | ∈ A ∈ } is a tesselation of H which satisﬁes in addition the property that if g1. 1 = g2. 2, then g = g and = . A A 1 2 A1 A2 Proof. Let := A . Theorem 4.8 states that is a closed fundamental F {A|A∈ } F region for Γ in H, hence g Γ g. = H. This proves (T1). Property (T2) follows S ∈ F directly from Proposition 4.13. Now let (g , ), (g , ) Γ A with g . = S 1 A1 2 A2 ∈ × 1 A1 g . . Then g . ◦ = g . ◦. Recalling that ◦ is a fundamental region for Γ in H 2 A2 1 A1 2 A2 F and that ◦, ◦ ◦, we get that g = g and = . A1 A2 ⊆F 1 2 A1 A2 2194 ANKE D. POHL

5. Cells in H. Let Γ be a geometrically finite subgroup of PSL2(R) of which is a cuspidal point and which satisfies (A). Suppose that the set of relevant isometric∞ spheres is non-empty. Let A be a basal family of precells in H. To each basal precell in H we assign a cell in H, which is an essentially disjoint union of certain Γ- translates of certain basal precells. More precisely, using Proposition 4.11 we define so-called cycles in A Γ in the same way as the cycles in the PoincaréFundamental Polyhedron Theorem× are defined (see e.g. [25]). These are certain finite sequences of pairs ( ,h) A ΓrΓ , such that each cycle is determined up to a cyclic permutationA by∈ any pair× which∞ belongs to it. Moreover, if ( ,h ) is an element of some cycle, then h is an element in ΓrΓ assigned to byA PropositionA 4.11 (or h = id if is a stripA precell). Conversely,∞ if h is anA element assigned to by PropositionA A4.11, then ( ,h ) determines a cycleA in A ΓrΓ . A One of the crucial propertiesA A of each cell in H is that× it is a∞ convex polyhedron with non-empty interior of which each side is a complete geodesic segment. This fact is mainly due to the condition (A) of Γ. The other two important properties of cells in H are that each non-vertical side of a cell is a Γ-translate of some vertical side of some cell in H and that the family of Γ-translates of all cells in H is a tesselation of H. 5.1. Cycles in A Γ. Let A be a non-cuspidal precell in H. The definition of precells shows that× is attachedA ∈ to a unique (inner) vertex v of , and is the unique precell attachedA to v. Therefore we set (v) := . Further,K hasA two A A A non-vertical sides b1 and b2. Let k1( ), k2( ) be the two elements in ΓrΓ given { A A } ∞ by Proposition 4.11 such that bj I(kj ( )) and kj ( ).bj is a non-vertical side of some basal precell. Necessarily, the∈ isometricA spheresA I(k ( )) and I(k ( )) are 1 A 2 A different, therefore k1( ) = k2( ). The set k1( ), k2( ) is uniquely determined by Proposition 4.11, theA assignment6 A k ({ ) clearlyA dependsA } on the enumeration A 7→ 1 A of the non-vertical sides of . Now w := kj ( ).v is an inner vertex. Let (w) be the (unique non-cuspidal) basalA precell attachedA to w. Since one non-verticalA side 1 of (w) is kj ( ).bj , which is contained in the relevant isometric sphere I(kj ( )− ), A 1 A A and kj ( )− kj ( ).bj = bj is a non-vertical side of some basal precell, namely of , A A 1 A one of the elements in ΓrΓ assigned to (w) by Proposition 4.11 is kj ( )− . ∞ A A Construction 5.1. Let A be a non-cuspidal precell and suppose that = A ∈ A (v) is attached to the vertex v of . We assign to two sequences (hj ) of elements Ain ΓrΓ using the following algorithm:K A ∞ (step 1) Let v1 := v and let h1 be either k1( ) or k2( ). Set g1 := id, g2 := h1 and carry out (step 2). A A (step j) Set vj := gj.v and j := (vj ). Let hj be the element in ΓrΓ such 1 A A ∞ that hj ,hj− 1 = k1( j ), k2( j ) . Set gj+1 := hj gj. If gj+1 = id, then the − A A algorithm stops. If gj = id, then carry out (step j + 1). +1 6 Example 5.2. Recall the Hecke triangle group Gn and its basal family A = {A}n of precells in H from Example 3.2. The two sequences assigned to are (Un)j=1 1 n A and Un− j=1. The statements of Propositions 5.3-5.7 follow as in the PoincaréFundamental Polyhedron Theorem, using Lemma 4.7 and elementary convex geometry. For this reason we omit the detailed proofs. SYMBOLIC DYNAMICS 2195

Proposition 5.3. Let = (v) be a non-cuspidal basal precell. A A (i) The sequences from Construction 5.1 are finite. In other words, the algorithm for the construction of the sequences always terminates. (ii) Both sequences have same length, say k N. ∈ (iii) Let (aj )j=1,...,k and (bj)j=1,...,k be the two sequences assigned to . Then they are inverse to each other in the following sense: For each j =1,...,kA we have 1 aj = bk− j+1. − (iv) For j =1,...,k set cj+1 := aj aj 1 a2a1, dj+1 := bjbj 1 b2b1 and c1 := − ··· − ··· id =: d1. Then k k 1 1 := c− . cj v = d− . dj v . B j A j A j=1 j=1 [ [ Further, both unions are essentially disjoint, and is the polyhedron with the 1 B 1 1 (pairwise distinct) vertices (in this order) c1− . , c2− . , ..., ck− . resp. 1 1 1 ∞ ∞ ∞ d− . , d− . , ..., d− . . 1 ∞ 2 ∞ k ∞ Definition 5.4. Let A be a non-cuspidal precell and suppose that is attached to the vertex v of .A Let ∈ h be one of the elements in ΓrΓ assignedA to by K A ∞ A Proposition 4.11. Let(hj )j ,...,k be the sequence assigned to by Construction 5.1 =1 A with h1 = h . For j = 1,...,k set g1 := id and gj+1 := hj gj. Then the (finite) A sequence ( (gj v),hj ) is called the cycle in A Γ determined by ( ,h ). A j=1,...,k × A A Let A be a cuspidal precell. Suppose that b is the non-vertical side of A ∈ A and let h be the element in ΓrΓ assigned to by Proposition 4.11. Let 0 be the (cuspidal)A basal precell with non-vertical∞ sideAh .b. Then the (finite) sequenceA 1 A ( ,h ), ( 0,h− ) is called the cycle in A Γ determined by ( ,h ). A A A A × A A Let A be a strip precell. Set h := id. Then ( ,h ) is called the cycle in A A A Γ determinedA ∈ by ( ,h ). A × A A Example 5.5. (i) Recall Example 5.2. The cycle in A Gn determined by n × ( ,Un) is ( ,Un) . A A j=1 (ii) Recall the group PΓ (5) and the basal family A = (v ),..., (v ) from 0 {A 0 A 4 } Example 4.4. The element in PΓ0(5) r PΓ0(5) assigned to (v0) is h := 4 1 ∞ A 5 −1 . The cycle in A PΓ0(5) determined by ( (v0),h) is − × A 1 ( (v ),h), ( (v ),h− ) . A 0 A 4 3 2 2 1 Further let h1 := h, h2 := 5 −3 and h3 := 5 −2 . The cycle in A PΓ0(5) determined by ( (v ),h ) is − − × A 1 1 ( (v ),h ), ( (v ),h ), ( (v ),h ) . A 1 1 A 3 2 A 2 3 (iii) Recall the group Γ and the basal family A = 1, 2, 3 from Example 4.5. The cycle in A Γ determined by ( ,S) is {A( ,SA ),A( },S) . × A2 A2 A3 Proposition 5.6. Let A be a non-cuspidal precell in H and suppose that h is one of the elementsA ∈ in Γ r Γ assigned to by Proposition 4.11. Let A ∞ A ( j ,hj ) be the cycle in A Γ determined by ( ,h ). Let j 1,...,k A j=1,...,k × A A ∈ { } and define the sequence ( 0,al) by Al l=1,...,k hl+j 1 for l =1,...,k j +1, al := − − (hl+j 1 k for l = k j +2,...,k, − − − 2196 ANKE D. POHL and

l+j 1 for l =1,...,k j +1, l0 := A − − A ( l+j 1 k for l = k j +2,...,k. A − − − Then a1 = hj is one of the elements in ΓrΓ assigned to j by Proposition 4.11 ∞ A and ( 0,al) is the cycle in A Γ determined by ( j ,hj ). Al l=1,...,k × A Proposition 5.7. Let A be a cuspidal precell in H and let h be the element A ∈ A1 in ΓrΓ assigned to by Proposition 4.11. Let ( ,h ), ( 0,h− ) be the cycle ∞ A 1 A A A A in A Γ determined by ( ,h ). Then h− is the element in ΓrΓ assigned to × A A 1 A ∞ 0 by Proposition 4.11 and ( 0,h− ), ( ,h ) is the cycle in A Γ determined by A 1 A A A A × ( 0,h− ). A A 5.2. Cells in H and their properties. Now we can define a cell in H for each basal precell in H. Construction 5.8. Let be a basal strip precell in H. Then we set A ( ) := . B A A Let be a cuspidal basal precell in H. Suppose that g is the element in ΓrΓ A 1 ∞ assigned to by Proposition 4.11 and let ( ,g), ( 0,g− ) be the cycle in A Γ determinedA by ( ,g). Define A A × A 1 ( ) := g− . 0. B A A ∪ A The set ( ) is well-defined because g is uniquely determined. Let BbeA a non-cuspidal basal precell in H and fix an element h in Γ r Γ A A ∞ assigned to by Proposition 4.11. Let ( j ,hj ) be the cycle in A Γ A A j=1,...,k × determined by ( ,h ). For j =1,...,k set g1 := id and gj+1 := hj gj. Set A A k 1 ( ) := g− . j . B A j A j=1 [ By Proposition 5.3, the set ( ) does not depend on the choice of h . The family B := ( ) A is calledB A the family of cells in H assigned to A.A Each element of B is{B calledA | A a cell ∈ in} H. Note that the family B of cells in H depends on the choice of A. If we need to distinguish cells in H assigned to the basal family A of precells in H from those assigned to the basal family A0 of precells in H, we will call the first ones A-cells and the latter ones A0-cells.

Example 5.9. Recall the Example 5.5. For the Hecke triangle group G5, Figure 11 shows the cell assigned to the family A = from Example 3.2. For the group {A} PΓ0(5), the family of cells in H assigned to A is indicated in Figure 12. Figure 13 shows the family of cells in H assigned to the basal family A of precells of Γ. In the following series of six propositions we investigate the structure of cells and their relations to each other. This will allow to show that the family of Γ-translates of cells in H is a tesselation of H, and it will be of interest for the labeling of the cross section. Proposition 5.10. Let be a non-cuspidal basal precell in H. Suppose that h is A A an element in ΓrΓ assigned to by Proposition 4.11 and let ( j ,hj ) ∞ A A j=1,...,k SYMBOLIC DYNAMICS 2197

( ) B A

i A λ5 + i 1 4 U − U − 5 A 5 A 2 3 U − U − 5 A 5 A

4 3 2 U U U U5 5 ∞ 5 ∞ 5 ∞ ∞ =0 = λ5

Figure 11. The cell ( ) for G . B A 5

( (v )) ( (v )) ( (v )) ( (v )) ( (v )) B A 0 B A 1 B A 2 B A 3 B A 4

0 1 2 3 4 1 5 5 5 5

Figure 12. The family of cells in H assigned to A for PΓ0(5).

( ) ( ) ( ) B A1 B A2 B A3

v1 v2 0 v3

Figure 13. The family of cells in H assigned to A for Γ.

be the cycle in A Γ determined by ( ,h ). For j = 1,...,k set g1 := id and × A A gj+1 := hj gj. Then the following assertions hold true. 1 (i) The set ( ) is the convex polyhedron with vertices (in this order) g1− . , 1 B A 1 ∞ g2− . ,..., gk− . . (ii) The∞ boundary of∞ ( ) consists precisely of the union of the images of the B A 1 vertical sides of j under g− , j =1,...,k. More precisely, if sj denotes the A j summit of I(hj ) for j =1,...,k, then k k+1 1 1 ∂ ( )= gj− .[sj , ] gj− .[hj 1.sj 1, ]. B A ∞ ∪ − − ∞ j=1 j=2 [ [ (iii) For each j = 1,...,k we have gj . ( ) = ( j ). In particular, the side 1 1 B A B A 1 [gj− . ,gj−+1. ] of ( ) is the image of the vertical side [ ,hj− . ] of ( j) ∞ 1 ∞ B A ∞ ∞ B A under gj− . 2198 ANKE D. POHL

(iv) Let be a basal precell in H and let h Γ such that h. ( )◦ = . Then A ∈ A∩B1 A 6 ∅ there exists a unique j 1,...,k such that h = g− and = j . In ∈ { } j A A particular,b is non-cuspidal and h. ( )= ( ). b A B A B A b Proof. By Proposition 5.3, ( ) is the polyhedron with vertices (in this order) 1 1 b 1 B A b g1− . , g2− . ,...,gk− . . Since each of its sides is a complete geodesic segment, ( ∞) is convex.∞ This∞ shows (i). The statement (ii) follows from the proof of PropositionB A 5.3. To prove (iii), ﬁx j 1,...,k and recall from Proposition 5.6 the cycle ∈ { } ( 0,al) in A Γ determined by ( j ,hj ). For l = 1,...,k set c := id Al l=1,...,k × A 1 and c := a c . Then l+1 l l 1 gl+j 1gj− for l =1,...,k j +1 − cl = 1 − (gl+j k 1gj− for l = k j +2,...,k. − − − Hence k k 1 1 ( j)= c− . 0 = gj (clgj)− . 0 B A l Al Al l l [=1 [=1 k j+1 k − 1 1 = gj . gl−+j 1. l+j 1 gj. gl−+j k 1. l+j k 1 − A − ∪ − − A − − l=1 l=k j+2 [ [− k 1 = gj . g− . l = gj . ( ). l A B A l [=1 1 1 This immediately implies that the side [gj− . ,gj−+1. ] of ( ) maps to the side 1 1 1 ∞ ∞ B A gj.[g− . ,g− . ] = [ ,h− . ] of ( j), which is vertical. j ∞ j+1 ∞ ∞ j ∞ B A To prove (iv), ﬁx z h. ( )◦. Then there exists l 1,...,k such that 1 ∈ A∩B A 1 ∈ { } z h. gl− . l. Let b := h. gl− . l. By Proposition 4.13 there are three possibilities∈ A ∩ for Ab. b A ∩ A 1 If b =b h. = gl− . l, then glh.b = l. Since and l are both basal, it follows A1 A A A A A that h = gl− and = l. Supposeb that v Ais theA vertex ofb to which bis attached. If b is a common side 1 K A of h. and gl− . l,b then, since z ( )◦, gl.b must be a non-vertical side of l (seeA (ii)). This impliesA that v b∈. BInA turn, there is a neighborhood U of v suchA ∈ that Ub ( ) and U h.( )◦ = . Hence, h.( )◦ ( )◦ = . Thus there exists ⊆B A ∩ A 6 ∅1 A ∩B A 6 ∅ j 1,...,k such that h.( )◦ gj− . j = . Proposition 4.13 implies that = j ∈{ 1} Ab ∩ A 6 ∅ b A A and h = gj− . b 1 b If b is a point, then b = z must be the endpoint of some side of g− . l. From l A z ( )◦ it follows that z = v. Now the previous argument applies. ∈B A 1 To show the uniqueness of j 1,...,k with = j and h = gj− , suppose ∈ { } A A 1 that there is p 1,...,k with = p and h = g− . Then gj = gp. By ∈ { } A A p Proposition 5.3(iv) j = p. The remaining parts of (ivb) follow from (iii). b Proposition 5.11. Let be a cuspidal basal precell in H which is attached to the vertex v of . SupposeA that g is the element in Γ r Γ assigned to by K 1 ∞ A Proposition 4.11. Let ( ,g), ( 0,g− ) be the cycle in A Γ determined by ( ,g). Then we have the followingA properties.A × A SYMBOLIC DYNAMICS 2199

1 (i) The set ( ) is the hyperbolic triangle with vertices v, g− . , . (ii) The boundaryB A of ( ) is the union of the vertical sides of ∞with∞ the images B A 1 A of the vertical sides of 0 under g− . A (iii) The sets g. ( ) and ( 0) coincide. In particular, the non-vertical side 1 B A B A 1 [v,g− . ] of ( ) is the image of the vertical side [g.v, ] of 0 under g− . ∞ B A ∞ A (iv) Suppose that is a basal precell in H and h Γ such that h. ( )◦ = . A 1 ∈ A∩B A 6 ∅ Then either h = id and = or h = g− and = 0. In particular, is A A A A A cuspidal and h.b ( )= ( ). b B A B bA b b Proposition 5.12. Let be a basal strip precell in H. Let be a basal precell in bA A H and h Γ such that h. ( )◦ = . Then h = id and = . ∈ A∩B A 6 ∅ Ab A Corollary 5.13. The map A B, ( ) is a bijection. b → A 7→ B A b The following proposition is implied by following the glueing procedure from precells to cells. The details are easily seen by straightforward deductions. Proposition 5.14. (1) Let be a non-cuspidal basal precell in H. Suppose that A (hj )j=1,...,k is a sequence in ΓrΓ assigned to by Construction 5.1. For ∞ A j = 1,...,k set g := id and gj := hj gj. Let 0 be a basal precell in H and 1 +1 A g Γ such that ( ) g. ( 0) = . Then we have the following properties. ∈ B A ∩ B A 6 ∅ (i) Either ( )= g. ( 0), or ( ) g. ( 0) is a common side of ( ) and B A B A B A ∩ B A B A g. ( 0). B A 1 (ii) If ( )= g. ( 0), then g = g− for a unique j 1,...,k . In particu- B A B A j ∈{ } lar, 0 is non-cuspidal. A (iii) If ( ) = g. ( 0), then 0 is cuspidal or non-cuspidal. If 0 is cuspidal B A 6 B A A A and k ΓrΓ is the element assigned to 0 by Proposition 4.11, then we ∈ ∞ 1 A have ( ) g. ( 0)= g.[k− . , ]. (2) Let be aB cuspidalA ∩ B basalA precell in∞H ∞which is attached to the vertex v of . SupposeA that h ΓrΓ is the element assigned to by Proposition 4.11. LetK ∈ ∞ A 0 be a basal precell in H and g Γ such that we have ( ) g. ( 0) = . AThen the following assertions hold∈ true. B A ∩ B A 6 ∅ (i) Either ( )= g. ( 0), or ( ) g. ( 0) is a common side of ( ) and B A B A B A ∩ B A B A g. ( 0). B A 1 (ii) If ( ) = g. ( 0), then either g = id or g = h− . In particular, 0 is cuspidal.B A B A A (iii) If ( ) = g. ( 0), then 0 is cuspidal or non-cuspidal or a strip precell. B A 6 B A A 1 If 0 is a strip precell, then [h− . , ] = ( ) g. ( 0). If 0 is a cuspidalA precell attached to the vertex∞ ∞w 6 ofB Aand∩ kB AΓ r Γ Ais the K ∈ ∞ element assigned to 0 by Proposition 4.11, then ( ) g. ( 0) is either A 1 B A1 ∩ B A [v, ] = g.[w, ] or [v, ] = g.[w, k− . ] or [v,h− . ] = g.[w, ] or ∞ 1 ∞ 1 ∞ 1 ∞ 1 ∞ ∞ [v,h− . ]= g.[w, k− . ] or [h− . , ]= g.[k− . , ]. If 0 is a non- ∞ ∞ ∞ ∞ ∞1 ∞ A cuspidal precell, then we have ( ) g. ( 0) = [h− . , ]. B A ∩ B A ∞ ∞ (3) Let be a basal strip precell in H. Let 0 be a basal precell in H and g Γ A A ∈ such that ( ) g. ( 0) = . Then the following statements hold. B A ∩ B A 6 ∅ (i) Either ( )= g. ( 0), or ( ) g. ( 0) is a common side of ( ) and B A B A B A ∩ B A B A g. ( 0). B A (ii) If ( )= g. ( 0), then g = id and = 0. B A B A A A (iii) If ( ) = g. ( 0), then 0 is a cuspidal or strip precell. If 0 is cuspidal B A 6 B A A A and k ΓrΓ is the element assigned to 0 by Proposition 4.11, then we ∈ ∞ 1 A have ( ) g. ( 0) = g.[k− . , ]. B A ∩ B A 6 ∞ ∞ 2200 ANKE D. POHL

Corollary 5.15. The family of Γ-translates of all cells provides a tesselation of H. In particular, if is a cell in H and S a side of , then there exists a pair B B ( 0,g) B Γ such that S = g. 0. Moreover, ( 0,g) can be chosen such that B 1 ∈ × B ∩ B B g− .S is a vertical side of 0. B 6. The base manifold of the cross sections. Let Γ be a geometrically finite subgroup of PSL (R) of which is a cuspidal point and which satisfies (A), and 2 ∞ suppose that there are relevant isometric spheres. In this section we define a set CS which will turn out to be a cross section for the geodesic flow on Y =Γ H w.r.t. to \ certain measures µ, which will be characterized in Section 8.1 below. Here we willc already see that CS satisfies Condition (C2) by showing that pr(CS) is a totally geodesic suborbifold of Y of codimension one and that CS is the set of unit tangent vectors based onc pr(CS) but not tangent to it. To achieve this,c we start at the other end. We fix a basal family A of precells in H andc consider the family B of cells in H assigned tocA. We define BS(B) to be the set of Γ-translates of the sides of these cells. Then we show that the set BS := BS(B) is in fact independent of the choice of A. We proceed to prove that BS is a totally geodesic submanifold of H of codimension one and define CS to be the set of unit tangent vectors based on BS but not tangent to it. Then CS := π(CS) is our geometric cross section and pr(CS) = BS := π(BS). This construction shows in particular that the set CS does not depend on the choice of Ac. For future purposes we already define the sets NC(cB) andc bd(B) and show that also these are independent of the choice of A. Throughoutc let 1 λ t = λ 0 1 with λ> 0 be such that Γ = tλ . ∞ h i Definition 6.1. Let A be a basal family of precells in H and let B be the family of cells in H assigned to A. For B let Sides( ) be the set of sides of . Then set B ∈ B B Sides(B) := Sides( ) B B B∈[ and define BS(B) := Γ. Sides(B)= g.S g Γ, S Sides(B) . { | ∈ ∈ } For B define bd( [) := ∂g and let[ NC( ) be the set of geodesics on Y which haveB a ∈ representativeB on H bothB endpoints ofB which are contained in bd( ). Further set B bd(B) := g. bd( ) B g Γ B [∈ B∈[ and NC(B) := NC( ). B B B∈[ Proposition 6.2. Let A and A0 be two basal families of precells in H and suppose that B resp. B0 are the families of the corresponding cells in H assigned to A resp. A0. There exists a unique map A Z, m( ) such that → A 7→ A m( ) ψ : A precells in H , t A . →{ } A 7→ λ A SYMBOLIC DYNAMICS 2201 is a bijection from A to A0. Then m( ) χ: B B0, ( ) t A . ( ) → B A 7→ λ B A is a bijection as well. Further we have that BS(B) = BS(B0), NC(B) = NC(B0) and bd(B) = bd(B0). Proof. This follows immediately from a combination of Corollary 4.10, Theorem 4.8, Corollary 5.13, Proposition 4.11 and Proposition 5.3. We set BS := BS(B), bd := bd(B) and NC := NC(B) for the family B of cells in H assigned to an arbitrary family A of precells in H. Proposition 6.2 shows that BS, bd and NC are well-defined. Proposition 6.3. The set BS is a totally geodesic submanifold of H of codimension one. Proof. The set BS is a disjoint countable union of complete geodesic segments, which is locally finite. Let CS denote the set of unit tangent vectors in SH that are based on BS but not tangent to BS. Recall that Y denotes the orbifold Γ H and recall the canonical \ projections π : H Y , π : SH SY from Section 2. Set BS := π(BS) and CS := π(CS). → → c c Proposition 6.4. The set BS is a totally geodesic suborbifold of Y of codimension one, CS is the set of unit tangent vectors based on BS but not tangent to BS and CS satisfies (C2). c c c c 1 Proof.c Since BS is Γ-invariant by definition, we see that BS = π− (BS). Therefore, BS is a totally geodesic suborbifold of Y of codimension one. Moreover, CS = 1 π− (CS) and hence CS is indeed the set of unit tangent vectors basedc on BS but notc tangent to BS. Finally, pr(CS) = BS. Therefore, CS satisfies (C2). c c c Remark 6.5. Let NIC be the set of geodesics on Y of which at least one endpoint c cg c g c is contained in π(bd). Here, π : H Γ H denotes the extension of the canonical g → \ projection H Y to H . In Section 8.1 we will show that CS is a cross section for the geodesic→ flow on Y w.r.t. any measure µ on the space of geodesics on Y for which µ(NIC) = 0. c We end this section with a short explanation of the acronyms. Obviously, CS stands for “cross section” and BS for “base of (cross) section”. Then bd stands for “boundary” in the sense of geodesic boundary, and bd( ) is the geodesic boundary of the cell . Moreover, which will become clearer in SectionB 8.2 (see Remark 8.32), NC standsB for “not coded” and NC( ) for “not coded due to the cell ”. Finally, NIC stands for “not infinitely often coded”.B B

7. Precells and cells in SH. Let Γ be a geometrically finite subgroup of PSL2(R) which satisfies (A). Suppose that is a cuspidal point of Γ and that the set of relevant isometric spheres is non-empty.∞ In this section we define the precells and cells in SH and study their properties. The purpose of precells and cells in SH is to get very detailed information about the set CS from Section 6 and its relation to the geodesic flow on Y , see Section 8.1. c 2202 ANKE D. POHL

Definition 7.1. Let U be a subset of H and z U. A unit tangent vector v at z ∈ is said to point into U if the geodesic γv determined by v runs into U, i. e., if there exists ε > 0 such that γv((0,ε)) U. The unit tangent vector v is said to point ⊆ along the boundary of U if there exists ε > 0 such that γv((0,ε)) ∂U. It is said to point out of U if it points into H rU. ⊆ Definition 7.2. Let be a precell in H. Define to be the set of unit tangent A A vectors that are based on and point into ◦. The set is called the precell in A A A SH corresponding to . If is attached to the vertexe v of , we call a precell in SH attached to v. A A e K A e Recall the projection pr: SH H on base points. → Remark 7.3. Let be a precell in H and the corresponding precell in SH. A A Since is a convex polyhedron with non-empty interior, pr( ) is the precell in H A A to which corresponds. e A e Lemma 7.4. Let , be two different precells in H. Then the precells and e A1 A2 A1 2 in SH are disjoint. A f Proof. This is an immediate consequence of Proposition 4.6(ii) and Definition 7.2. f

Deﬁnition 7.5. Let be a precell in H and the corresponding precell in SH. A A The set vb( ) of unit tangent vectors based on ∂ and pointing along ∂ is called A A A the visual boundary of . Further, vc( ) := e vb( ) is said to be the visual A A A ∪ A closure of .e A e e e e e

Figure 14. The precell in SH and its visual boundary for a non- cuspidal precell in H.

The next lemma is clear from the deﬁnitions. Lemma 7.6. Let be a precell in H and be the corresponding precell in SH. A A Then vc( ) is the disjoint union of and vb( ). A A e A We remark that the visual boundary and the visual closure of a precell in SH e e e A is a proper subset of the topological boundary resp. closure of in SH. A e Proposition 7.7. Let j j J be a basal family of precells in H and let {A | ∈ } b j j J be the set of corresponding precells in SH. Then there is a fundamental {A | ∈ } set for Γ in SH such that fF j vc j . (4) e A ⊆ F ⊆ A j J j J [∈ [∈ f e f SYMBOLIC DYNAMICS 2203

Moreover, pr( ) = j J j . If is a fundamental set for Γ in SH such that F ∈ A F j J vc( j ), thenS satisfies (4). Conversely, if j j J is a set, F ⊆ ∈ Ae F e {A | ∈ } H indexedS by J, of precells in S such that (4) holds for some fundamental set for e f e e F Γ in SH, then the family pr( j ) j J is a basal family of precells in H. { A | ∈ } e Proof. This is a slight extension of a similar statement in [21]. e Remark 7.8. Recall from Theorem 4.8 that each basal family of precells in H contains the same finite number of precells, say m. Proposition 7.7 shows that if k k K is a set of precells in SH, indexed by K, such that (4) holds for some {A | ∈ } fundamental set for Γ in SH, then #K = m. e F Definition 7.9. Let A := j j J be a basal family of precells in H. Then e {A | ∈ } the set A := j j J of corresponding precells in SH is called a basal family of precells in {SAH or| a∈family} of basal precells in SH. If A is a connected family of basal precellse infH, then A is said to be a connected family of basal precells in SH or a connected basal family of precells in SH. e Let A := j j J be a basal family of precells in H and let A := j j J be the corresponding{A | ∈ basal} family of precells in SH. {A | ∈ } e f Definition and Remark 7.10. We call two cycles c ,c in A Γ equivalent if 1 2 × there exists a basal precell A and elements g1,g2 ΓrΓ such that ( ,g1) is A ∈ ∈ ∞ A an element of c1 and ( ,g2) is an element of c2. Obviously, equivalence of cycles is an equivalence relationA (on the set of all cycles). If [c] is an equivalence class of cycles in A Γ, then each element ( ,h ) A Γ in any representative c of [c] is called a generator× of [c]. A A ∈ × Lemma 7.11. Let be a non-cuspidal basal precell in H and suppose that h is A A an element in ΓrΓ assigned to by Proposition 4.11. Let ( j ,hj) be ∞ A A j=1,...,k the cycle in A Γ determined by ( ,h ). If = l for some l 2,...,k , then × A A A A ∈{ } hl = h . Moreover, if A q := min l 1,...,k 1 l = ∈{ − } A +1 A exists, then q does not depend on the choice of h , k is a multiple of q, and A ( l q,hl q) = ( l,hl) for l 1,...,k q . A + + A ∈{ − } Proof. This is an obvious consequence of the definition of cycles. Definition 7.12. Let be a non-cuspidal basal precell in H and let h be an A A element in ΓrΓ assigned to by Proposition 4.11. Suppose that ( j ,hj ) ∞ A A j=1,...,k is the cycle in A Γ determined by ( ,h ). We set × A A cyl( ) := min l 1,...,k 1 l = k . A ∈{ − } A +1 A ∪{ } Lemma 7.11 shows that cyl( ) is well-defined. Moreover, it implies that cyl( ) does not depend on the choiceA of the generator ( ,h ) of an equivalence classA of cycles. For a cuspidal basal precell in H we setA A A cyl( ) :=3, A and for a basal strip precell in H we define A cyl( ) :=2. A 2204 ANKE D. POHL

Example 7.13. Recall Example 5.5. For the Hecke triangle group Gn and its basal precell in H we have = and hence cyl( ) = 1. In contrast, the basal A A A2 A precell (v1) in H of the congruence group PΓ0(5) appears only once in the cycle in A PΓA (5) and therefore cyl (v ) = 3. × 0 A 1 Construction and Deﬁnition 7.14.Set := j J j . Pick a fundamental set F ∈ A for Γ in SH such that F S j vc j , e A ⊆ F ⊆ A j J j J [∈ [∈ which is possible by Propositionf7.7. Fore each basalf precell A and each z A ∈ ∈F let Ez( ) denote the set of unit tangent vectors in vc( ) based at z. Fix A F ∩ A any enumeration of the index set J of A, say J = j ,...,jk . For z and { 1 } ∈ F l e1,...,k set e e ∈{ } l 1 − z j := Ez j and z j := Ez j r Ez j . F A 1 A 1 F A l A l A m m=1 [ Further sete e e e e

( ) := z( ) F A F A z [∈A for A. Recall from Propositione 7.7 that pr(e )= . Thus, A ∈ F F z( )= , (5) F A eF z A [∈F A∈[ and the union is disjoint. For each equivalencee classe of cycles in A Γ ﬁx a generator and let S denote the set of chosen generators. Let ( ,h ) S. × Suppose that is a non-cuspidal precell in H andA letA v∈ be the vertex of to A K which is attached. Let ( j ,hj ) be the cycle in A Γ determined by A A j=1,...,k × ( ,h ). For j = 1,...,k set g1 := id and gj+1 := hj gj, and let sj be the summit A A of I(hj ). Further, for convenience, set h0 := hk and s0 := sk. In the following we partition certain ( j ) into k subsets. More precisely, we partition each element F A of the set ( j ) j =1,...,k into k subsets. Let j 1,..., cyl( ) . {F A e| } ∈{ A } For each z j◦ (hj 1.sj 1,gj.v] [gj .v, sj ) we pick any partition of z( j ) − − e ∈ A ∪ (1)∪ (k) F A into k non-empty disjoint subsets Wj,z ,...,Wj,z . (1) (2) (k) e For z [sj , ) we set Wj,z := z( j ) and Wj,z = ... = Wj,z := . ∈ ∞ F (1)A (2) ∅ (3) For z [hj 1.sj 1, ) we set Wj,z := , Wj,z := z( j ) and Wj,z = ... = ∈ − − ∞ e ∅ F A W (k) := . j,z ∅ For m 1,...,k and j 1,..., cyl( ) we set e ∈{ } ∈{ A } (m) j,m := W A j,z z j ∈A[ and e k 1 j ( ,h ) := gjgl− . l,l j+1 B A A A − l [=1 e e where the ﬁrst part (l) of the subscript of l,l j+1 is calculated modulo cyl( ) and the second part (l j + 1) is calculated moduloA − k. A − e SYMBOLIC DYNAMICS 2205

Suppose that is a cuspidal precell in H. Let ( ,h ), ( ,h ) be the cycle A A1 1 A2 2 in A Γ determined by ( ,h ). Set g := h1 = h , g1 := id and g2 := h1 = g. Suppose× that v is the vertexA ofA to which A is attached,A and let s be the summit K of I(g). Let j 1, 2 . We partition ( j ) into three subsets as follows. ∈{ } F A For z j◦ (gj .v, gj.s) we pick any partition of z( j ) into three non-empty ∈ A ∪ (1) (2) (3) e F A disjoint subsets Wj,z , Wj,z and Wj,z . (1) (2) e (3) For z (gj .v, ) we set Wj,z := z( j ) and Wj,z = Wj,z := . ∈ ∞ (1) F(2)A (3) ∅ For z [gj .s, ) we set W = W := and W := z( j ). ∈ ∞ j,z j,z ∅ j,z F A For m 1, 2, 3 and j 1, 2 wee set ∈{ } ∈{ } (m) e j,m := W . A j,z z j ∈A[ Then we define e 1 1( ,h ) := 1,1 g− . 2,2, B A A A ∪ A 2( ,h ) := g. 1,2 2,1, Be A A eA ∪ A e 1 3( ,h ) := 1,3 g− . 2,3. Be A A A e ∪ e A Suppose that is a strip precell in H. Let v1, v2 be the two (infinite) vertices of A e e e to which is attached and suppose that v1 < v2. We partition ( ) into two Ksubsets as follows.A F A For z ◦ we pick any partition of z( ) into two non-empty disjointe subsets (1) ∈ A(2) F A Wz and Wz . (1) (2) For z (v1, ) we set Wz := z( e) and Wz := . For z (v2, ) we set (1) ∈ ∞(2) F A ∅ ∈ ∞ Wz := and Wz := z( ). For m∅ 1, 2 we defineF A e ∈{ } e (1) (2) 1( ,h ) := Wz and 2( ,h ) := Wz . B A A B A A z z [∈A [∈A The set S (“selection”)e of generators of the equivalencee classes of cycles in A Γ is called a set of choices associated to A. The family ×

BS := j( ,h ) ( ,h ) S, j =1,..., cyl( ) B A A A A ∈ A is called the family of cellsn in SH associated to A and S. The elementso in this family e e are subject to the choice of the fundamental set , the enumeration of J and some choices for partitions in unit tangent bundle. However,F all further applications of BS are invariant under these choices. This justifiese to call BS the family of cells in SH associated to A and S. Each element in BS is called a cell in SH. e e Example 7.15. For the Hecke triangle group G from Example 5.5 we choose e 5 S = ( ,U ) . Here we have k = 5 and cyl( ) = 1. The first figure in Figure 15 { A 5 } A indicates a possible partition of ( ) into the sets , , , ,..., , . The unit F A A1 1 A1 2 A1 5 tangent vectors in black belong to , , those in very light grey to , , those in A1 1 A1 2 light grey to , , those in middlee grey to , , and thosee ine dark greye to , . The A1 3 A1 4 A1 4 second figure in Figure 15 shows thee cell 1( ,U5) in SH. e e Be A e Example 7.16. For the group Γ from Examplee 3.1(iii) we choose as set of choices S = ( , id), ( ,S) (cf. Example 5.5). The cells in SH which arise from ( , id) { A1 A2 } A1 2206 ANKE D. POHL

( ) ( ,U ) F A B1 A 5 e e

Figure 15. A partition of ( ) and the cell ( ,U ) in SH. F A B1 A 5 e e ( , id) ( , id) B1 A1 B2 A1 e e

v1 v2 v1 v2

Figure 16. The cells in SH arising from ( , id). A1 are shown in Figure 16. The cells in SH arising from ( 2,S) are indicated in Figure 17. A

( ,S) ( ,S) ( ,S) B1 A2 B2 A2 B3 A2 e e e

v2 0 0 v3 v2 0

Figure 17. The cells in SH arising from ( , id). A1

Let S be a set of choices associated to A.

Proposition 7.17. The union e e is disjoint and a fundamental set for Γ in BS B SH. B∈ S e Proof. Construction 7.14 picks a fundamental set for Γ in SH and chooses a F family := z( ) z , A of subsets of it. Since the union in (5) is P {F A | ∈ F A ∈ } disjoint, is a partition of . Recall the notatione from Construction 7.14. One P e F e SYMBOLIC DYNAMICS 2207 considers the family

1 := z( j ) z , ( ,h ) S, j =1,..., cyl( ) . P F A ∈F A A ∈ A The elements of nare pairwise disjoint and each element of is containedo in . P1 e P P1 Hence, is a partition of . The next step is to partition each element of into P1 F P1 a ﬁnite number of subsets. Thus, is partitioned into some family of subsets F P2 of . Then each element We of is translated by some element g(W ) in Γ to get F P2 the family 3 := g(W ).W W e2 . Since is a fundamental set for Γ in SH, thee elementsP of { are pairwise| disjoint∈ P } and F is a fundamental set for Γ in SH. P3 P3 Now 3 is partitioned into certain subsets, saye into the subsets l, l L. Each P S Q ∈ e cell in SH is the union of the elements in some l( ) such that l( 1) = l( 2) if B Q B B 6 B 1 = 2. Therefore, the union BS is disjoint and a fundamental set Bfor6 Γe inB SH. B B ∈ e e S e e e e e For each BS set b( ) := pr( ) ∂ pr( ) and let CS0( ) be the set of unit B ∈ B B ∩ B B tangent vectors in that are based on b( ) but do not point along ∂ pr( ). e e B e e B e e B Example 7.18. Recall the congruence subgroup PΓ0(5) and its cycles in A PΓ0(5) e e 1 e× from Example 5.5. We choose S := (v4),h− , (v1),h1 as set of choices associated to A and set A A 1 := (v ),h− , := (v ),h , B1 B1 A 4 B4 B1 A 1 1 1 2 := 2 (v4),h− , 5 := 2 (v1),h1, Be Be A Be Be A 1 3 := 3 (v4),h− , 6 := 3 (v1),h1, Be Be A Be Be A as well as e e e e CS0 := CS0 j j B for j =1,..., 6. Figure 18 shows the sets CSj0 . e

CS20 CS40 CS60 CS50 CS30 CS10

0 1 2 3 4 1 5 5 5 5

Figure 18. The sets CSj0 .

Lemma 7.19. Let , be two basal precells in H and let g Γ such that A1 A2 ∈ g. vc( ) vc( ) = . A1 ∩ A2 6 ∅ Suppose that 1 = 2 or g = id. Then A 6 A 6 f f g. vc vc g. vb vb . A1 ∩ A2 ⊆ A1 ∩ A2 Moreover, suppose that is cuspidal or non-cuspidal and that there is a unit A1 tangent vector w vc( f) pointingf into a non-verticalf sidef S of such that ∈ A1 1 A1 gw vc( 2). Then g.w points into a non-vertical side S2 of 2 and g.S1 = S2. ∈ A f A f 2208 ANKE D. POHL

Proof. We have pr(g. vc( )) = g. pr(vc( )) = g. and pr(g. vb( )) = g.∂ , A1 A1 A1 A1 A1 and likewise for pr(vc( 2)) = 2 and pr(vb( 2)) = ∂ 2. From A f A f A A f g. vc( ) vc( ) = f A1 ∩ Af2 6 ∅ then follows that g. 1 2 = . By Proposition 4.13, either g. 1 = 2 and g Γ A ∩A 6 ∅ f f A A ∈ ∞ or g. 1 2 g.∂ 1 ∂ 2. Assume on the contrary that g. 1 = 2 with g Γ . SinceA ∩Aand⊆ areA ∩ basal,A Corollary 4.10 shows that g = idA andA = .∈ This∞ A1 A2 A1 A2 contradicts the hypotheses of the lemma. Hence g. 1 2 g.∂ 1 ∂ 2 and therefore A ∩ A ⊆ A ∩ A g. vc( ) vc( ) g. vb( ) vb( ). A1 ∩ A2 ⊆ A1 ∩ A2 Let 1 and w fulﬁll the assumptions of the Lemma. Further let γ be the geodesic determinedA by w. Then g.γfis the geodesicf determinedf byfg.w. By deﬁnition there exists ε> 0 such that γ((0,ε)) S and g.γ((0,ε)) . Then ⊆ 1 ⊆ A2 1 1 γ (0,ε) S g− . g− . . ⊆ 1 ∩ A2 ⊆ A1 ∩ A2 1 1 Since the sets 1 and g− . 2 intersect in more than one point and 1 = g− . 2, A A 1 A 6 1 A Proposition 4.13 states that g− . is a common side of and g− . . A1 ∩ A2 A1 A2 Necessarily, this side is S1. According to Proposition 4.13, g.S1 is a non-vertical side of . Thus, g.w points along the non-vertical side g.S of . A2 1 A2 Proposition 7.20. Let ( ,h ) S and suppose that is a non-cuspidal precell A A ∈ A in H. Let ( j ,hj ) be the cycle in A Γ determined by ( ,h ). For each A j=1,...,k × A A 1 m =1,..., cyl( ) we have b( m( ,h )) = (hm− . , ) and A B A A ∞ ∞ 1 pr m( ,h ) = ( m)◦ (hm− . , ). B Ae A B A ∪ ∞ ∞ 1 Moreover, CS0( m( ,h e)) is the set of unit tangent vectors based on (hm− . , ) B A A ∞ ∞ that point into ( m)◦, and pr(CS0( m( ,h ))) = b( m( ,h )). Be A B A A B A A Proof. We use the notation from Construction 7.14. Let j 1,..., cyl( ) and e e ∈ { A } z j . First we show that z( j ) = . For each choice of we have j ∈ A F A 6 ∅ F A ⊆ vc( j ). Remark 7.3 states that pr( j ) = j . Hence Ez( j ) j = . More F ∩ A e A A A e∩ A 6 ∅ f precisely, if j z denotes the set of unit tangent vectors based on z that point into e f A f e e ◦, then j = Ez( j ) j . The set j is non-empty, since j is convex Aj A z A ∩ A A z A with non-emptyf interior. Let k J such that k = j . Then ∈ A 6 A f e f f Ez( k) Ez( j ) vc k vc j vb k vb j , A ∩ A ⊆ A ∩ A ⊆ A ∩ A where the laste inclusione follows fromf Lemmaf7.19. Sincef j vb(f j ) = by Lemma 7.6, it follows that A ∩ A ∅ f f j Ez( k)= j Ez( j ) Ez( k) j vb j = . A ∩ A A ∩ A ∩ A ⊆ A ∩ A ∅ Hence f e f e e f f j z( j ). (6) A z ⊆ F A Let j 1,..., cyl( ) set j := j( ,h ) and ∈{ A } B Bf A Ae

Tj := j◦ (hj 1.sj 1,gj .v] [gj .v, sj ). Ae ∪ e − − ∪ SYMBOLIC DYNAMICS 2209

Let m 1,...,k . Then ∈{ } (m) pr j,m = pr W A j,z z j ∈A[ e = pr W (m) pr W (m) pr W (m) j,z ∪ j,z ∪ j,z z Tj z [sj , ) z [hj− .sj− , ) [∈ ∈ [∞ ∈ 1[ 1 ∞ Tj for m / 1, 2 , ∈{ } = Tj [sj , ) for m = 1,  ∪ ∞ Tj [hj 1.sj 1, ) for m = 2. ∪ − − ∞ Note that necessarily k 3. Then  ≥ k 1 j = gjgl− . l,l j+1 B A − l [=1 j 1 k e − e 1 1 1 1 = gj gl− . l,l j+1 gj gj− . j,1 gj gj−+1. j+1,2 gj gl− . l,l j+1. A − ∪ A ∪ A ∪ A − l l j [=1 =[+2 Since l j +1 1,e2 mod k for l e1,...,j 1 e j +2,...,k , it followse that − 6≡ ∈{ − }∪{ } j 1 − 1 1 pr j = gjgl− . pr l,l j+1 pr j,1 hj− . pr j+1,2 B A − ∪ A ∪ A l=1 [ e k e e e 1 gj gl− . pr l,l j+1 ∪ A − l=j+2 [ j 1 e k − 1 1 1 1 = gjg− .Tl Tj [sj , ) h− .Tj h− .[hj .sj , ) gjg− .Tl l ∪ ∪ ∞ ∪ j +1 ∪ j ∞ ∪ l l l j [=1 =[+2 k 1 1 = gj g− .Tl (h− . , ). l ∪ j ∞ ∞ l [=1 1 For the last equality we use that pr (sj ) = hj− . by Lemma 4.7. Hence sj is ∞ 1 1 ∞ 1 contained in the geodesic segment pr− (hj− . ) H = (hj− . , ), which shows ∞ ∞ ∩ ∞ ∞1 that the union of the two geodesic segments [sj , ) and [sj ,hj− . ) is indeed 1 ∞ ∞ (h− . , ). Proposition 5.10 implies that j ∞ ∞ 1 pr j = ( j )◦ (h− . , ). B B A ∪ j ∞ ∞ 1 This shows that b( j ) = (h− . , ). The set of unit tangent vectors in j based B j e∞ ∞ B on b( j) is the disjoint union B e e (1) 1 (2) D0 := W h− . W e j j,z ∪ j j+1,z z [sj , ) z [hj .sj , ) ∈ [∞ ∈ [ ∞ 1 = z( j ) h− . z( j ). F A ∪ j F A +1 z [sj , ) z [hj .sj , ) ∈ [∞ ∈ [ ∞ e e To show that CS0( j ) is the set of unit tangent vectors based on b( j ) that point B B into ( j)◦ we have to show that Dj0 contains all unit tangent vectors based on B A 1 [sj , ) that point intoe ◦, all unit tangent vectors based on (h− . ,se j ] that point ∞ Aj j ∞ 2210 ANKE D. POHL

1 into h− . ◦ and the unit tangent vector which is based at sj and points into j Aj+1 [sj ,gjv]. If w is a unit tangent vector based on [hj .sj , ) that points into j◦+1, 1 ∞ 1 A then, clearly, hj− .w is a unit tangent vector based on [sj ,hj− . ) that points into 1 ∞ h− . ◦ . Hence, (6) shows that D0 contains all unit tangent vectors of the first j Aj+1 j two kinds mentioned above. Let w be the unit tangent vector with pr(w) = sj which points into [sj ,gj.v]. Suppose first that w . Then w vc( j ) and therefore w Es ( j ). ∈ F ∈ A ∩ F ∈ j A Let k J with k = j . Assume on the contrary that w vc( k). Lemma 7.19 ∈ A 6 A ∈ A implies that [sj ,gj .v] is a non-verticale side off k, whiche is a contradiction.e Hence A w / Es ( k). Therefore, w s ( j ) and hence w D0 . f ∈ j A ∈ F j A ∈ j Suppose now that w / . Then there exists a unique g Γr id such that ∈ F ∈ { } gw e . Let be a basal precelle in H such that g.w vc( ) . Lemma 7.19 ∈ F A ∈ A 1∩ F shows that g.[sj ,gj.v] is a non-verticale side S of . Thus, g− . ( j)◦ = . A A∩B A 16 ∅ By Propositione 5.10(iv) there is a unique l 1,...,k suche thateg = glg− and ∈ { } j = l. Now [sj ,gj.v] is mapped by hj to the non-vertical side [hj .sj,gj .v] of A A +1 j . Thus, g = hj and = j . Then hj .w vc( ]j ). As before we see that A +1 A A +1 ∈ A +1 w D0 . Moreover, pr(CS0( j)) = b( j). ∈ j B B Analogously to Proposition 7.20 one proves the following two propositions. e e Proposition 7.21. Let ( ,h ) S and suppose that is a cuspidal precell in H. A A ∈ A 1 Let v be the vertex of to which is attached and let ( ,g), ( 0,g− ) be the cycle in A Γ determinedK by ( ,h A). Then A A × A A 1 b 1( ,h ) = (v, ), b 2( ,h ) = (g.v, ), b 3( ,h ) = (g− . , ) B A A ∞ B A A ∞ B A A ∞ ∞ and e e e pr 1( ,h ) = ( )◦ (v, ), B A A B A ∪ ∞ pr 2( ,h ) = ( 0)◦ (g.v, ), Be A A B A ∪ ∞ 1 pr 3( ,h ) = ( )◦ (g− . , ). Be A A B A ∪ ∞ ∞ Moreover, CS0( m( ,h )) is the set of unit tangent vectors based on b( m( ,h )) B A A e B A A that point into pr( m( ,h ))◦, and pr(CS0( m( ,h ))) = b( m( ,h )) for m = 1, 2, 3. e B A A B A A B A Ae e e e Proposition 7.22. Let ( ,h ) S and suppose that is a strip precell. Let v1, v2 be the two (infinite) verticesA ofA ∈to which is attachedA and suppose that v < v . K A 1 2 For m =1, 2 we have b( m( ,h )) = (vm, ) and B A A ∞ pr m( ,h ) = ( )◦ (vm, )= ◦ (vm, ). B Ae A B A ∪ ∞ A ∪ ∞ Moreover, CS0( m( ,h )) is the set of unit tangent vectors based on (vm, ) that B eA A ∞ point into ( )◦, and pr(CS0( m( ,h ))) = b( m( ,h )). B A e B A A B A A Corollary 7.23. Let BS. Then := cl(pr( )) is a cell in H and b( ) a side B ∈ e B eB B of . Moreover, pr( )= ◦ b( ) and pr( )◦ = ◦. B B e Be∪ B B eB e The development of a symbolic dynamics for the geodesic flow on Y via the e e e family BS of cells in SH is based on the following properties of the cells in SH: It B uses that cl(pr( )) is a convex polyhedron of which each side is a complete geodesic segmente and thatB each side is the image under some element g Γ of thee complete ∈ e SYMBOLIC DYNAMICS 2211 geodesic segment b( 0) for some cell 0 in SH. It further uses that BS is a B B fundamental set for Γ in SH and that g. cl(pr( )) g Γ, BS is a tesselation { B | ∈ B ∈ } S of H. Moreover, onee needs that b( ) ise a vertical side of pr( ) and that CSe0( ) is B B B the set of unit tangent vectors based on b( ) thate point intoe pr(e )◦. It does not B B use that cl(pr( )) BS is thee set of all cells in H nor doese one need thate for { B | B ∈ } some cells 1, 2 BS one has cl(pr( 1)) =e (pr( 2)). This meanse that one has the freedomB toB performe ∈ e (horizontal)e translationsB B ofB single cells in SH by elements in Γ . Thee followinge e definition is motivatede by thise fact. We will see that in some situations∞ the family of shifted cells in SH will induce a symbolic dynamics which has a generating function for the future part while the symbolic dynamics that is constructed from the original family of cells in SH does not have one.

Definition 7.24. Each map T: BS Γ (“translation”) is called a shift map for → ∞ BS. The family BS,T :e= T . BS e B B B ∈ is called the family of cells in SH associated to A, S and T. Each element of BS,T is called a shifted cell in SHe. e e e e For each BS,T define b( ) := pr( ) ∂ pr( ) and let CS0( ) be the sete of B ∈ B B ∩ B B unit tangent vectors in that are based on b( ) but do not point along ∂ pr( ). e e B e e B e e B Let T be a shift map for BS. e e e Remark 7.25. The results of Propositions 7.17 and 7.20-7.22 remain true for BS T e , after the obvious changes. More precisely, the union BS,T is disjoint and a fundamental set for Γ in SH, and if BS, then pr T( ). = T( ). pr( )e and B ∈ S B B B B b T( ). = T( ).b( ). Then CS0 T( ). is the set of unite tangent vectors based B B B B B B T T e e e e e e on ( ).b( ) that point into ( ).pr( )◦. eB e B e e B Be e 8. Geometrice e symbolic dynamics.e eLet Γ be a geometrically finite subgroup of PSL2(R) of which is a cuspidal point and which satisfies (A). Suppose that the set of relevant isometric∞ spheres is non-empty. Let A be a basal family of precells in H and denote the family of cells in H assigned to A by B. Suppose that S is a set of choices associated to A and let BS be the family of cells in SH associated to A and S. Fix a shift map T for BS and denote the family of cells in SH associated to A, S and T by BS,T. Recall the set CSe 0( ) for BS,T from Definition 7.24. We set e B B ∈ CS0 BS,eT := CS0 ande CS eBS,Te := π CS0 BS,T . e e B BS,T B∈[ e e c e e In Section 8.1, we will use the results from Section 7 to show that CS satisfies (C1) and hence is a cross section for the geodesic flow on Y w.r.t. certain measures µ. It will turn out that the measures µ are characterized by the conditionc that NIC (see Remark 6.5)be a µ-null set. 8.1. Geometric cross section. Recall the set BS from Section 6.

Lemma 8.1. Let BS,T. Then pr( ) is a convex polyhedron and ∂ pr( ) con- B ∈ B B sists of complete geodesic segments. Moreover, we have that pr( )◦ BS = and B ∩ ∅ ∂ pr( ) BS and pr(e e) BS = b( ) ande that b( ) is a connected componente of BS. B ⊆ B ∩ B B e e e e e 2212 ANKE D. POHL

Proposition 8.2. We have CS = CS(BS,T). Moreover, the union

CS0 BS,T = CS0 BS,T c c e B B ∈ [ is disjoint and CS0(BS,T) ise a set of representativese e foreCS.

Proof. We start by showing that CS(BS,T) CS. Let BS,T. Then there ex- e ⊆ cB ∈ ists a (unique) BS such that = T( ). . Lemma 8.1 shows that b( ) B1 ∈ B B1 B1 B1 is a connected component of BS.c Thee set CSc0( ) consistse e of unit tangent vec- B1 tors based on b(e ) whiche are not tangente toe it.e Therefore, CS0( ) CS. Nowe B1 B1 ⊆ b( ) = T( ). and CS0( ) = T( ). CS0( )e with T( ) Γ. Thus, we see B B1 B1 B B1 B1 B1 ∈ that π(b( )) eπ(BS) = BS and π(CS0( )) π(CS) = CS. Thise shows that B ⊆ B ⊆ CS(eBS,T) eCS.e e e e e ⊆ Conversely,e let v CS. Wec will show thate there is a unique c BS,T and a unique ∈ 1 B ∈ vc eCS0( ) suchc that π(v)= v. Pick any w π− (v). Remark 7.25 shows that the ∈ B ∈ set := b BcS,T is a fundamental set for Γ in SH.e Hencee there exists a P {B | B ∈ } 1 unique paire ( ,g) BS,T Γb such that v := g.w . Note that π− (CS) = CS. S B ∈ × ∈ B Thus, v CSe ande hencee pr(v) pr( ) BS. Lemma 8.1 shows that pr(v) b( ). ∈ 1 ∈ B ∩ ∈ B Therefore, v e π− (b(e )) . Since v CS, it doese not point along b(c). Hence ∈ B ∩ B ∈ B v does not point along ∂ pr( ), whiche shows that v CS0( ). This proves thate B ∈ B CS CS(BS,T). e e e ⊆ 1 To see the uniqueness of e and v suppose that w π−e(v). Let ( ,g ) B 1 ∈ B1 1 ∈ BcS,T cΓ bee the unique pair such that g1.w1 1. There exists a unique element × 1 ∈ B 1 h Γ such that h.w = w1.e Then g1hg− .v = g.w1 and v,g1hgb − .v e . Now ∈ 1 ∈ P e being a fundamental set shows that g hg− e= id, which proves that g .w = P 1 1 1 g h.w = g.w = v and = . This completes the proof. 1 B1 B Corollary 8.3. Let γ be a geodesic on Y which intersects CS in t. Then there is e e a unique geodesic γ on H which intersects CS0(BS,T) in t such that π(γ)= γ. b c Definition 8.4. Let γ be a geodesic on Y which intersects CS in γ (t ). If e 0 0 b s := min t>t0 γ0(t) CS b ∈ c b exists, we call s the first return timeof γ0(t0 ) and γ0(s) the next point of intersection b c of γ and CS. Let γ be a geodesic on H. If γ0(t) CS, then we say that γ intersects ∈ CS in t. If there is a sequence (tn)n Nb with limnb tn = and γ0(tn) CS for ∈ →∞ ∞ ∈ all n N,c then γ is said to intersect CS infinitely often in the future. Analogously, b ∈ if we find a sequence (tn)n N with limn tn = and γ0(tn) CS for all n N, then γ is said to intersect ∈CS infinitely→∞ often in the−∞ past. Suppose∈ that γ intersects∈ CS in t0. If s := min t>t γ0(t) CS 0 ∈ exists, we call s the first return timeof γ0(t0 ) and γ0(s) the next point of intersection of γ and CS. Analogously, we define the previous point of intersection of γ and CS resp. of γ and CS. c Remark 8.5. A geodesic γ on Y intersects CS if and only if some (and henceb any) 1 1 representative of γ on H intersects π− (CS). Recall that CS = π− (CS), and that CS is the set of unit tangentb vectors based onc BS but which are not tangent to BS. Since BS is a totallyb geodesic submanifoldc of H (see Proposition 6.3),c a geodesic γ SYMBOLIC DYNAMICS 2213 on H intersects CS if and only if γ intersects BS transversely. Again because BS is totally geodesic, the geodesic γ intersects BS transversely if and only if γ intersects BS and is not contained in BS. Therefore, a geodesic γ on Y intersects CS if and only if some (and thus any) representative γ of γ on H intersects BS and γ(R) BS. 6⊆ A similar argument simplifies the search for previousb and next points ofc intersection. To make this precise, suppose that γ is ab geodesic on Y which intersects CS in γ0(t ) and that γ is a representative of γ on H. Then γ0(t ) CS. There is a next 0 0 ∈ point of intersection of γ and CS if and onlyb if there is a next point of intersectionc ofb γ and CS. The hypothesis that γ0(t0)b CS implies that γ(R) is not contained in ∈ BS. Hence each intersectionb ofcγ and BS is transversal. Then there is a next point of intersection of γ and CS if and only if γ((t , )) intersects BS. Suppose that 0 ∞ there is a next point of intersection. If γ0(s) is the next point of intersection of γ and CS, then and only then γ0(s) is the next point of intersection of γ and CS. In this case, s = min t>t γ(t) BS . { 0 | ∈ } Likewise, there was a previousb point of intersection of γ and CS ifb and onlyc if there was a previous point of intersection of γ and CS. Further, there was a previous point of intersection of γ and CS if and only if γ(( ,t )) intersectsc BS. If there −∞ 0 b was a previous point of intersection, then γ0(s) is the previous point of intersection of γ and CS if and only if γ0(s) was the previous point of intersection of γ and CS. Moreover, s = max t

Deﬁnition 8.7. Let BS,T and suppose that b( ) is the complete geodesic B ∈ B segment (a, ) with a R. We assign to two intervals I( ) and J( ) which are given as follows:∞ ∈e e B e B B e e e (a, ) if pr( ) z H Re z a , I := ∞ B ⊆{ ∈ | ≥ } B (( ,a) if pr( ) z H Re z a , −∞ Be ⊆{ ∈ | ≤ } e and e ( ,a) if pr( ) z H Re z a , J := −∞ B ⊆{ ∈ | ≥ } B ((a, ) if pr( ) z H Re z a . ∞ Be ⊆{ ∈ | ≤ } e We note that the combination of Remarke7.25 with Propositions 5.10(i) and 7.20, Propositions 5.11(i) and 7.21 resp. Proposition 7.22 shows that indeed each BS,T B ∈ gets assigned a pair I( ),J( ) of intervals. B B e e e e 2214 ANKE D. POHL

Lemma 8.8. Let BS,T. For each v CS0( ) let γv denote the geodesic on H B ∈ ∈ B determined by v. If v CS0( ), then (γv( ),γv( )) I( ) J( ). Conversely, ∈ B ∞ −∞ ∈ B × B if (x, y) I( ) Je( ),e then there exists a uniquee element v in CS0( ) such that ∈ B × B B (γv( ),γv( )) = (x, y). e e e ∞ −∞ e e e Let BS,T and g Γ. Suppose that I( ) = (a, ). Then B ∈ ∈ B ∞ (g.a, g. ) if g.a < g. , e e g.I = ∞ e ∞ B ((g.a, ] ( , g. ) if g. < g.a, ∞ ∪ −∞ ∞ ∞ and e (g.a, ] ( , g. ) if g.a < g. , g.J = ∞ ∪ −∞ ∞ ∞ B ((g.a, g. ) if g. < g.a. ∞ ∞ Here, the interval (b,e ] denotes the union of the interval (b, ) with the point ∞ ∞ ∂gH. Hence, the set I := (b, ] ( ,c) is connected as a subset of ∂gH. ∞The ∈ interpretation of I is more eluminating∞ ∪ −∞ in the ball model: Via the Cayley 1 transform the set ∂gH is homeomorphic to the unit sphere S . Let b0 := (b), C 1 C c0 := (c) and I0 := (I). Then I0 is the connected component of S r b0,c0 which containsC ( ). C { } C ∞ Suppose now that I( ) = ( ,a). Then B −∞ ( , g.a) (g.( ), ] if g.a < g.( ), g.I = e −∞ ∪ −∞ ∞ −∞ B ((g.( ), g.a) if g.( ) < g.a, −∞ −∞ and e (g.( ), g.a) if g.a < g.( ), g.J = −∞ −∞ B (( , g.a) (g.( ), ] if g.( ) < g.a. −∞ ∪ −∞ ∞ −∞ eα β Note that for g = γ δ we have h i αt + β α + βs α + βs g.( ) = lim = lim = lim = g. . −∞ t γt + δ s 0 γ + δs s 0 γ + δs ∞ &−∞ % & In particular, id .( )= . −∞ ∞ Let a,b R and set (a,b) := (min(a,b), max(a,b)) and ∈ + (a,b) := (max(a,b), ] ( , min(a,b)). − ∞ ∪ −∞ Proposition 8.9. Let BS,T and suppose that S is a side of pr( ). Then there B ∈ B exist exactly two pairs ( ,g ), ( ,g ) BS,T Γ such that S = gj .b( j). Moreover, B1 1 B2 2 ∈ × B g . cl(pr( )) = cl(pr( e)) ande g . cl(pr( )) cl(pr( )) = S or vicee versa. The 1 B1 B 2 B2 ∩ B union g . CS0( ) g .eCS0( )eis disjointe and equals the set of alle unit tangent 1 B1 ∪ 2 B2 vectors ine CS that aree based on S. Let ea,b ∂gH bee the endpoints of S. Then ∈ g1.I( 1) g1.Je( 1) = (a,b)+e (a,b) and g2.I( 2) g2.J( 2) = (a,b) (a,b)+ or viceB versa.× B × − B × B − × e e e e Proof. Let D0 denote the set of unit tangent vectors in CS that are based on S. By Lemma 8.1, S is a connected component of BS. Hence D0 is the set of unit tangent vectors based on S but not tangent to S. The complete geodesic segment S divides H into two open half-spaces H1,H2 such that H is the disjoint union H S H . Moreover, pr( )◦ is contained in H or H , say pr( )◦ H . Then 1 ∪ ∪ 2 B 1 2 B ⊆ 1 e e SYMBOLIC DYNAMICS 2215

D0 decomposes into the disjoint union D0 D0 where D0 denotes the non-empty 1 ∪ 2 j set of elements in D0 that point into Hj . For j =1, 2 pick vj D0 . Since CS0(BS,T) ∈ j is a set of representatives for CS = π(CS) (see Proposition 8.2), there exists a e uniquely determined pair ( j ,gj) BS,T Γ such that vj gj. CS0( j). We will B ∈ × ∈ B show that S = gj .b( j). Assumec on the contrary that S = gj .b( j). Since S and B 6 B gj.b( j ) are complete geodesice segments,e the intersection of S and gj .b(e j) in pr(vj ) B B is transversal. Recalle that S ∂ pr( ) and b( j ) ∂ pr( j ) ande that ∂ pr( 0)◦ = ⊆ B B ⊆ B B ∂ pr(e0) for each 0 BS,T. Then there exists ε> 0 such that Bε(pr(vj ))e pr( )◦ = B B ∈ ∩ B Bε(pr(vj )) H and e e e e ∩ 1 e e e e Bε pr(vj ) gj . pr j ◦ H = . ∩ B ∩ 1 6 ∅ Hence pr( )◦ gj . pr( j )◦ = . Proposition 5.14 in combination with Remark 7.25 B ∩ B 6 ∅ e shows that cl(pr( )) = gj . cl(pr( j )). But then e B e B ∂ pr( )= gj .∂ pr( j ), e e B B which implies that S = gj .b( j). This is a contradiction, and hence S = gj .b( j). B e e B Then Lemma 8.8 implies that gj .I( j ) gj .J( j ) equals (a,b)+ (a,b) or (a,b) B × B × − − × (a,b)+. On the other hand e e e e ∂gH ∂gH = γv( ),γv( ) v D0 = γv( ),γv( ) v D0 1 × 2 ∞ −∞ ∈ 1 −∞ ∞ ∈ 2 equals (a,b)+ (a,b) or (a,b) (a,b )+. Therefore, again by Lemma 8.8, we have × − − × gj. CS0( j ) = D0 . This shows that the union g . CS0( ) g . CS0( ) is disjoint B j 1 B1 ∪ 2 B2 and equals D0. We havee cl(pr( )) H1 and g1. cl(pr( 1)) H1 withe S ∂ pr( )e g1.∂ pr( 1). Let z S. Then thereB ⊆ exists ε> 0 such thatB ⊆ ⊆ B ∩ B ∈ e e e e Bε(z) pr ◦ = Bε(z) H = Bε(z) g . pr ◦. ∩ B ∩ 1 ∩ 1 B1 Hence pr( )◦ g . pr( )◦ = . As above we ﬁnd that cl(pr( )) = g . cl(pr( )). B ∩ 1 B1 e6 ∅ Be 1 B1 Finally, g2. cl(pr( 2)) H2 with e B ⊆e e e S g . cl pr H H H = S. e ⊆ 2 B2 ∩ 1 ⊆ 2 ∩ 1 Hence cl(pr( )) g . cl(pr( )) = S. B ∩ 2 B2 e Let BS,T and suppose that S is a side of pr( ). Let ( ,g ), ( ,g ) be the B ∈ e e B B1 1 B2 2 two elements in BS,T Γ such that S = gj .b( j) and g . cl(pr( )) = cl(pr( )) and × B 1 B1 B g2. cl(pr(e 2))e cl(pr( )) = S. Then we deﬁne e e e B ∩ e B e e e p ,S := ,g and n ,S := ,g . e Be B1 1 B B2 2

Remark 8.10. Let BS,Tand suppose that S is a side of pr( ). We will show eB ∈ e e e B how one eﬀectively ﬁnds the elements p( ,S) and n( ,S). Let e e B B e , k := p ,S and , k := n ,S . B1 1 B e B2 2 e B Suppose that is the (unique) element in BS such that T( ). = and suppose 0 e e e e0 0 B B B B 1 further that j0 BS such that T( j0 ). j0 = j for j = 1, 2. Set S0 := T( 0)− .S. Be ∈ B B eB e e e B Then S0 is a side of pr( 0). For j =1, 2 we have B e e 1 e 1 e e 1 e S0 = T 0 − .S = T 0 − kj .b j = T 0 − kj T 0 .b 0 B e B B B Bj Bj e e e e e e 2216 ANKE D. POHL and

kj . cl pr j = kj T 0 . cl pr 0 . B Bj Bj e e e Moreover, cl(pr( )) = T( 0). cl(pr( 0)). Then k1. cl(pr( 1)) = cl(pr( )) is equivalent to B B B B B e e e e e 1 T 0 − k T 0 . cl pr 0 = cl pr 0 , B 1 B1 B1 B e e e e and k . cl(pr( )) cl(pr( )) = S is equivalent to 2 B2 ∩ B

e 1 e T 0 − k T 0 . cl pr 0 cl pr 0 = S0. B 2 B2 B2 ∩ B e e e 1e Therefore, ( 1, k1) = p( ,S) if and only if ( 10 , T( 0)− k1T( 10 )) = p( 0,S0), and B B 1 B B B B ( , k )= n( ,S) if and only if ( 0 , T( 0)− k T( 0 )) = n( 0,S0). B2 2 B B2 B 2 B2 B By Corollarye 7.23, thee sets 0 := cl(pr( e0)) ande 0 := cl(pr(e 0 ))e are A-cells B B Bj Bj ine H. Supposee ﬁrst that 0 arisese frome the non-cuspidale e basal precell 0 in H. Then there is a unique elementB ( ,h ) Se such that for some eh ΓA the pair A A ∈ ∈ ( 0,h) is contained in the cycle in A Γ determined by ( ,h ). Necessarily, A × A A is non-cuspidal. Let ( j ,hj ) be the cycle in A Γ determined by A A j=1,...,k × ( ,h ). Then 0 = m for some m 1,..., cyl( ) and hence 0 = ( m) A A A A ∈ { A } B B A and 0 = m( ,h ). For j = 1,...,k set g1 := id and gj+1 := hj gj . Propo- B B A A sition 5.10(iii) states that ( m) = gm. ( ) and Proposition 5.10(i) shows that B A 1 B A1 S0 ise the geodesice segment [gmgj− . ,gmgj−+1. ] for some j 1,...,k . Then 1 1 ∞ ∞ ∈ { } gjg− .S0 = [ ,h− . ]. Let n 1,..., cyl( ) such that n j mod cyl( ). m ∞ j ∞ ∈ { A } ≡ A Then hn = hj by Lemma 7.11. Proposition 7.20 shows that b( n( ,h )) = A 1 1 1 B A [ ,hn− . ]= gj gm− .S0. We claim that ( j ( ,h ),gmgj− )= p( 0,S0). For this is A ∞ ∞ 1 B A B e remains to show that gmgj− . cl(pr( j ( ,h ))) = cl(pr( 0)). Proposition 7.20 shows B A eA B e that cl(pr( j ( ,h ))) = ( n) and Lemma 7.11 implies that ( n)= ( j). Let B A A B A e e B A B A1 v be the vertex of to which is attached. Then gj .v ( j )◦ and gmgj− gj .v = e K A 1 ∈B A gm.v ( m)◦. Therefore gmgj− . ( j )◦ ( m)◦ = . From Proposition 5.14(1) ∈B A 1 B A ∩B A 6 ∅ it follows that gmgj− . ( j ) = ( m). Recall that ( m) = cl(pr( 0)). Hence 1 B A B A B A B ( j( ,h ),gmgj− )= p( 0,S0) and B A A B e

e e 1 1 − T j( ,h ) . j ,h , T 0 gmgj− T j( ,h ) = p ,S . B A A B A A B B A A B e e e e e Analogously one proceeds if 0 is cuspidal or a strip precell. A Now we show how one determines ( , k ). Suppose again that 0 arises from B2 2 B the non-cuspidal basal precell 0 in H. We use the notation from the determination A of p( 0,S0). By Corollary 4.10 there ise a unique pair ( ,s) A Z such that B s s s 1 A ∈ × b( n( ,h )) tλ. = and tλ. = n. Then tλ− gjgm− .S0 is a side of the cell ( ) A B eA ∩ A 6 ∅ A 6 A b s B1 A in H. As before, we determine ( 3, k3) BS Γ such that k3.b( 3)= tλ− gj gm− .S0 e b bB ∈ ×1 B b and k3. cl(pr( 3)) = ( ). Recall that gj gm− .S0 is a side of ( j) = ( n). We B B A e e B Ae B A e b SYMBOLIC DYNAMICS 2217 have 1 s 1 s gmg− t k . cl pr cl pr 0 = gmg− t . ( ) ( m) j λ 3 B3 ∩ B j λ B A ∩B A 1 s 1 = gmg− t . ( ) gmg− . ( j ) e e j λ B Ab ∩ j B A 1 s = gmg− . t . ( ) ( j) j λ BbA ∩B A 1 s = gmg− .t . ( ) ( n) j λ B Ab ∩B A 1 1 = gmgj− gjgm− .S0 b = S0.

1 s Thus n( 0,S0) = ( ,gmg− t k ) and B B3 j λ 3 1 s 1 n ,S = T . , T 0 gmg− t k T − . e eB B3 B3 B j λ 3 B3 H If 0 arises from a cuspidale or stripe precelle e in , then the constructione of n( ,S) is analogous.B B e Proposition 8.11. Let γ be a geodesic on Y and suppose that γ intersects CS in γ0(t ). Let γ be the unique geodesic on H such that γ0(t ) CS0(BS,T) and 0 0 ∈ π(γ0(t )) = γ0(t ). Let b BS,T be the unique shifted cell in SH forb which we havec 0 0 B ∈ γ0(tb ) CS0( ). e 0 ∈ B (i) Thereb is a next pointe ofe intersection of γ and CS if and only if γ( ) does not e ∞ belong to ∂g pr( ). B (ii) Suppose that γ( ) / ∂g pr( ). Then there is a unique side S of pr( ) in- ∞ ∈ B B tersected by γ((et , )). Suppose that ( ,g) = n( ,S). The next point of 0 ∞ B1 B intersection is on g. CS0( ).e e B1 (iii) Let ( 0,h)= n( ,b( )). Then there wase a previouse point of intersection of γ B B B and CS if and only if γ( e ) / h.∂g pr( 0). −∞ ∈ B (iv) Supposee that γ( e )e/ h.∂g pr( 0). Then there is a unique side S of h. pr( 0) −∞ ∈ B 1 1 B intersected by γ(( ,t )). Let ( ,h− ek) = p( 0,h− S). Then the previous −∞ 0 B2 B point of intersection was on k. CSe 0( 2). e e B e Proof. We start by proving (i). Recall from Remark 8.5 that there is a next point e of intersection of γ and CS if and only if γ((t , )) intersects BS. Since γ0(t ) 0 ∞ 0 ∈ CS0( ), Proposition 7.20 resp. 7.21 resp. 7.22 in combination with Remark 7.25 B shows that γ0(t ) points into pr( )◦. Lemma 8.1 states that pr( )◦ BS = and 0 B B ∩ ∅ ∂ pr(e) BS. Hence γ((t , )) does not intersect BS if and only if γ((t , )) B ⊆ 0 ∞ 0 ∞ ⊆ pr( )◦. In this case, e e B e γ( ) cl g (pr( )) ∂gH = ∂g pr( ). e ∞ ∈ H B ∩ B Conversely, if γ( ) ∂ pr( ), then γ((t , )) pr( ) or γ((t , )) ∂ pr( ). g e 0 ◦ e 0 In the latter case,∞ Lemma∈ 8.1B shows that∞γ((t⊆, ))B BS. Hence,∞ if⊆ γ( )B 0 ∞ ⊆ ∞ ∈ ∂g pr( ), then γ((t , )) pr(e )◦. e e B 0 ∞ ⊆ B Suppose now that γ( ) / ∂g pr( ). The previous argument shows that the ∞ ∈ B geodesice segment γ((t , )) intersectse ∂ pr( ), say γ(t ) ∂ pr( ) with t (t , ). 0 ∞ B 1 ∈ B 1 ∈ 0 ∞ If there was an element t (t , )r et with γ(t ) ∂ pr( ), then there is a side 2 ∈ 0 ∞ { 1} 2 ∈ B S of pr( ) such that γ(R) = S, where thee equality followse from the fact that S is a completeB geodesic segment. But then, by Lemma 8.1e , γ(R) BS, which ⊆ e 2218 ANKE D. POHL contradicts to γ0(t ) CS. Thus, γ(t ) is the only intersection point of ∂ pr( ) 0 ∈ 1 B and γ((t , )). Since γ((t ,t )) pr( )◦, γ0(t ) is the next point of intersection of 0 ∞ 0 1 ⊆ B 0 γ and CS. Moreover, γ0(t ) points out of pr( ), since otherwise γ((t , )) woulde 1 B 1 ∞ intersect ∂ pr( ) which would lead toe a contradiction as before. Proposition 8.2 B states that there is a unique pair ( ,g) BS,eT Γ such that γ0(t ) g. CS0( ). B1 ∈ × 1 ∈ B1 Then γ0(t ) pointse into g. pr( )◦. Let S be the side of pr( ) with γ(t ) S. By 1 B1 B 1 ∈ Proposition 8.9, either g. cl(pr( ))e = cl(pr(e)) or g. cl(pr( )) cl(pr( )) = S.e In B1 B B1 ∩ B the first case, γ0(t1) points intoe pr( )◦, which is a contradiction.e Therefore e B e e e g. cl(pr( )) cl(pr( )) = S, Be1 ∩ B which shows that ( ,g)= n( ,S). This completes the proof of (ii). B1 B e e Let ( 0,h)= n( ,b( )). Since γ(t ) b( ) and γ0(t ) CS0( ), Proposition 8.9 B B B 0 ∈ B 0 ∈ B implies that γ(t ) e h.b( 0) ande γ0(t ) / h. CS0( 0). Since γ(R) h.∂ pr( 0), the 0 ∈ B 0 ∈ B 6⊆ B unit tangente vectoreγ0(te) points out of pr( 0e). Because the intersectione of γ(R) and 0 B h.b( 0) is transversal ande pr( 0) is a convex polyhedrone with non-empty interior,e B B γ((t0 ε,t0)) h. pr( )◦ = for each εe > 0. As before we find that there was a previouse− point∩ of intersectionB 6 ∅e of γ and CS if and only if γ(( ,t )) intersects −∞ 0 ∂ pr( 0) and that thise is the case if and only if γ( ) / h.∂g pr( 0). B −∞ ∈ B Suppose that γ( ) / h.∂g pr( 0). As before, there is a unique t 1 ( ,t0) −∞ ∈ B − ∈ −∞ suche that γ(t 1) h.∂g pr( 0). Let S be the side of h. pr( 0) withe γ(t 1) S. − ∈ B B − ∈ Necessarily, γ((t 1,t0) h. pr( 0)◦e, which shows that γ0(t 1) points into h. pr( 0)◦ − ⊆ B − B and that γ0(t 1) is the previouse point of intersection. Let ( 2,e k) BS,T Γ be the − B ∈ × unique pair such that γ0(t 1) k.e CS0( 2) (see Proposition 8.2). By Proposition e8.9, − ∈ B we have either k. cl(pr( )) = h. cl(pr( 0)) or k. cl(pr( )) e h. cl(pr(e 0)) = S. In B2 B B2 ∩ B the latter case, γ0(t 1) points out of eh. pr( 0)◦ which is a contradiction. Hence 1 − B 1 1 h− k. cl(pr( 2) = cl(pr(e 0)), which showse that ( 2,h− ek)= p( 0,h− Se). B B e B B Corollary 8.12.e Let γ bee a geodesic on Y and supposee that γ doese not intersect CS infinitely often in the future. If γ intersects CS at all, then there exists t R such ∈ that γ0(t) CS and γb((t, )) BS = . Analogously, supposeb that η is a geodesicc ∈ ∞ ∩ ∅ on Y which does not intersect CSb infinitely oftenc in the past. If η intersects CS at all, thenb therec exists bt R such thatc η0(t) CS and η(( ,t)) BSb = . ∈ c ∈ −∞ ∩ ∅ c Example 8.13. Recall the setting of Example 7.18. We considerb the two shift c c maps T id, and b b 1 ≡ 1 1 T2 1 := − and T2 j := id for j =2,..., 6. B 0 1 B For simplicity sete 1 := T2 1 . 1 and CS0e1 := T2 1 . CS10 . Further we set B− B B − B 1 0 2 1 3 2 4 1 g := ,e g := e e , g := e , g := , 1 5 1 2 5 −2 3 5 −3 4 5 −1 − − − 4 5 1 1 1 0 g := , g := , g := . 5 5 −6 6 0 1 7 −5 1 − − Figure 19 shows the translates of the sets CSj0 which are necessary to determine the location of the next point of intersection if the shift map is T1, and Figure 20 those if T2 is the chosen shift map. SYMBOLIC DYNAMICS 2219

CS20 CS40 CS60 CS50 g4.CS40 CS30 CS10 g6.CS20

g1.CS20 g2.CS50 g3.CS30 g4.CS60 g5.CS10 0 1 2 3 4 1 5 5 5 5

Figure 19. The translates of CS0 relevant for determination of the location of the next point of intersection for the shift map T1.

g7.CS40 CS0 1CS20 CS40 CS60 CS50 CS30 g6.CS20 −

g7.CS0 1 g1.CS20 g2.CS50 g3.CS30 g4.CS60 g4.CS0 1 − − 1 0 1 2 3 4 1 − 5 5 5 5 5

Figure 20. The translates of CS0 relevant for determination of the location of the next point of intersection for the shift map T2.

Recall the set bd from Section 6. The following characterization is now obvious. Proposition 8.14. Let γ be a geodesic on Y . (i) γ intersects CS infinitely often in the future if and only if γ( ) / π(bd). ∞ ∈ (ii) γ intersects CS infinitelyb often in the past if and only if γ( ) / π(bd). −∞ ∈ b c b Recall the set NIC from Remark 6.5. b c b Theorem 8.15. Let µ be a measure on the space of geodesics on Y . Then CS is a cross section w. r. t. µ for the geodesic flow on Y if and only if µ(NIC) = 0. c Proof. Proposition 6.4 shows that CS satisfies (C2). Let γ be a geodesic on Y . Then Proposition 8.14 implies that γ intersects CS infinitely often in the past and the future if and only if γ / NIC. Thisc completes the proof.b ∈ c Let denote the set of unit tangentb vectors to the geodesics in NIC and set E b CS := CSr . st E Corollary 8.16. Let µ be a measure on the space of geodesics on Y such that c c µ(NIC) = 0. Then CSst is the maximal strong cross section w. r. t. µ contained in CS. c 8.2. Geometric coding sequences and geometric symbolic dynamics. A c label of a unit tangent vector in CS or CS is a symbol which is assigned to this vector. The labeling of CS resp. CS is the assigment of the labels to its elements. The set of labels is commonly calledc the alphabet of the arising symbolic dynamics. We establish a labelingc of CS and CS in the following way: Let v CS0(BS,T) ∈ and suppose that BS,T is the unique shifted cell in SH such that v CS0( ). Let γ be the geodesicB ∈ on H determinedc by v. ∈ e B e e e 2220 ANKE D. POHL

Suppose ﬁrst that γ( ) / ∂g pr( ). Proposition 8.11(ii) states that there is a ∞ ∈ B unique side S of pr( ) intersected by γ((0, )) and that the next point of inter- B ∞ section of γ and CS is on g. CS0( )e if ( ,g)= n( ,S). We assign to v the label B1 B1 B ( ,g). e B1 Suppose now that γ( ) ∂g pr(e ). Propositione e8.11(i) shows that there is no nexte point of intersection∞ of∈γ and CS.B Let ε be an abstract symbol which is not contained in Γ. Then we label v by εe(“end” or “empty word”). Let v CS. By Proposition 8.2 there is a unique v CS0(BS,T) such that ∈ 1 ∈ π(v)= v. We endow v and each element in π− (v) with the label of v. Theb followingc proposition is the key result for the determinatione of the set of labels. b b b

Proposition 8.17. Let BS,T and suppose that Sj , j = 1,...,k, are the sides B ∈ of pr( ). For j =1,...,k set ( j ,gj) := n( ,Sj ). Let v CS0( ) and suppose that B B B ∈ B γ is the geodesic determinede bye v. Then γ( ) gj .I( j) if and only if γ((0, )) ∞ ∈ B ∞ intersectse Sj . Moreover, if Sj e= b( ), thene gj.I( j ) I( ).e If Sj = b( ), then 6 B B ⊆ B B gj.I( j )= J( ). e B B e e e e Proof. This follows from Propositions 8.9 and 8.11 and Lemma 8.8 with similar e e arguments as in the proof of Proposition 8.9.

For BS,T let Sides( ) denote the set of sides of pr( ). B ∈ B B Corollary 8.18. Let BS,T. For each S Sides( ) set ( S,gS) := n( ,S). e e B ∈e ∈ B e B B Then I( ) is the disjoint union B e e e e e e I( )= I( ) ∂g pr( ) gS.I( S). B B ∩ B ∪ e e B S Sides([ )rb( ) e e e ∈ B B e Let Σ denote the set of labels. Corollary 8.19. The set Σ of labels is given by Σ= ε { } ∪ ( ,g) BS,T Γ 0 BS,T S Sides( 0)rb( 0): ( ,g)= n( 0,S) . ∪ B ∈ × ∃ B ∈ ∃ ∈ B B B B n o Moreover, Σ is finite. e e e e e e e e Proof. Note that for each 0 BS,T we have ∂g pr( 0) I( 0) = . Thus, ε is a B ∈ B ∩ B 6 ∅ label. Then the equality for Σ follows immediately from Corollary 8.18. Since BS,T is finite and each shifted celle in SeH has only finitely manye sides,e Σ is finite. e Example 8.20. For the Hecke triangle group Gn let A = , S = ( ,Un) and {A} { A } T id. Then BS,T = 1( ,Un) . Set := 1( ,Un). Then the set of labels is (see≡ Figure 7) {B A } B B A e Σ=e ε, ,U kS ke 1e,...,n 1 . B n ∈{ − } n o Example 8.21. Recall Example 8.13. If the shift map is T1, then the set of labels e is Σ= ε, ,g , , id , ,g , , id , ,g , , id , ,g , , id , B2 1 B4 B5 2 B6 B3 3 B5 B6 4 B3 1,g5 , 2,g6 , 4,g4 . e e e e e Be Be Be e e e SYMBOLIC DYNAMICS 2221

With the shift map T2, the set of labels is

Σ= ε, 4,g7 , 1,g7 , 2,g1 , 4, id , 5,g2 , 6, id , 3,g3 , B B− B B B B B 5, id, 6,g4, 3, id, 1,g4 , 2,g6 . e e Be Be Be Be− eB Definition and Remark 8.22. Let v CS0(BS T) and suppose that γ is the e e e, e e geodesic on H determined by v. Proposition∈ 8.11 implies that there is a unique sequence (tn)n J in R which satisfies the followinge properties: ∈ (i) J = Z (a,b) for some interval (a,b) with a,b Z and 0 (a,b), ∩ ∈ ∪ {±∞} ∈ (ii) the sequence (tn)n J is increasing, ∈ (iii) t0 = 0, (iv) for each n J we have γ0(tn) CS and ∈ ∈ γ0((tn,tn+1)) CS = and γ0((tn 1,tn)) CS = ∩ ∅ − ∩ ∅ where we set tb := if b< and ta := if a> . ∞ ∞ −∞ −∞ The sequence (tn)n J is said to be the sequence of intersection times of v (with respect to CS). ∈ 1 − Let v CS and set v := π 0 Be (v). Then the sequence of intersection ∈ |CS ( S,T) 1 times (w. r. t. CS) of v and of each w π− (v) is defined to be the sequence of intersectionb c times of v. ∈ b Now we define the geometricb coding sequences.b Definition 8.23. For each s Σ we set ∈ CSs := v CS v is labeled with s ∈ and c b c b 1 CSs := π− CSs = v CS v is labeled with s . ∈ Let v CS and let (t ) be the sequence of intersection times of v. Suppose that n n J c γ is the∈ geodesic on Y determined∈ by v. The geometric coding sequence of v is the sequencec (an)n J in Σ defined by b ∈ b b an := s if andb only if γ0(tn) CSs b ∈ for each n J. Let v ∈CS. The geometric coding sequence bof v is definedc to be the geometric coding sequence∈ of π(v).

Proposition 8.24. Let v CS0. Suppose that (tn)n J is the sequence of intersec- ∈ ∈ tion times of v and that (an)n J is the geometric coding sequence of v. Let γ be the geodesic on H determined by v∈. Suppose that J = Z (a,b) with a,b Z . ∩ ∈ ∪ {±∞} (i) If b = , then an Σr ε for each n J. ∞ ∈ { } ∈ (ii) If b< , then an Σr ε for each n (a,b 2] Z and ab 1 = ε. ∞ ∈ { } ∈ − ∩ − (iii) Suppose that an = ( n,hn) for n (a,b 1) Z and set B ∈ − ∩ g0 := h0 if b 2, e ≥ gn := gnhn for n [0,b 2) Z, +1 +1 ∈ − ∩ g 1 := id, − 1 g (n+1) := g nh−n for n [1, (a + 1)) Z. − − − ∈ − ∩ Then γ0(tn ) gn. CS0( n) for each n (a,b 1) Z. +1 ∈ B ∈ − ∩ e 2222 ANKE D. POHL

Proof. We start with some preliminary considerations which will prove (i) and (ii) and simplify the argumentation for (iii). Let n J and consider w := γ0(tn). The ∈ definition of geometric coding sequences shows that γ0(tn) CSa . Since CS is ∈ n the disjoint union k Γ k. CS0(BS,T) (see Proposition 8.2), there is a unique k Γ 1 ∈ 1 ∈ such that k− .w CS0(BS,T). The label of k− .w is an. Let η be the geodesic on H ∈1S 1 determined by k− .w. Note thateη(t) := k− .γ(t+tn) for each t R. The definition ∈ of labels shows that an e= ε if and only if there is no next point of intersection of η and CS. In this case γ0((tn, )) CS = and hence b = n + 1. This shows (i) and ∞ ∩ ∅ (ii). Suppose now that an = ( ,g). Then there is a next point of intersection of η B 1 and CS, say η0(s), and this is on g. CS0( ). Then k− .γ0(s + tn) g. CS0( ) and 1 B ∈ B k− .γ0((tn,s + tn)) CS = . Hencee tn = s + tn and γ0(tn ) kg. CS0( ). ∩ ∅ +1 +1 ∈ B Now we show (iii). Suppose that b 2.e Then v = γ0(t ) is labeled with ( e ,h ). ≥ 0 B0 0 Hence for the next point of intersection γ0(t1) of γ and CS we have e e γ0(t ) h . CS0 = g . CS0 . 1 ∈ 0 B0 0 B0 Suppose that we have already shown that e e γ0(tn ) gn. CS0 n +1 ∈ B for some n [0,b 1) Z and that b n+3. By(i) resp. (ii), γ0((tn+1, )) CS = ∈ − ∩ ≥ e ∞ ∩ 6 ∅ and hence γ0(tn+1) is labeled with ( n+1,hn+1). Our preliminary considerations show that B γ0(tn ) gnhn . CSe0 n = gn . CS0 n . +2 ∈ +1 B +1 +1 B +1 Therefore γ0(tn+1) gn. CS0( n) for each n [0,b 1) Z. ∈ B e ∈ − ∩ e Suppose that a 2. The element γ0(t 1) is labeled with ( 1,h 1). Since ≤ − − B− − γ0(t 1) k. CS0(BS,T) for somee k Γ, our preliminary considerations show that − ∈ ∈ γ0(t0) kh 1. CS0( 1). Because γ0(t0) = v CS0(BS,T), Propositione 8.2 implies ∈ −1 e B− ∈ that k = h−1 and − e 1e γ0(t0) CS0 1 = g 1. CS0 1 and γ0(t 1) h−1. CS0 BS,T = g 2. CS0 BS,T . ∈ B− − B− − ∈ − − Suppose that we have already shown that e e e e γ0(t (n 1)) g n. CS0 n and γ0(t n) g (n+1). CS0 BS,T − − ∈ − B− − ∈ − for some n [1, a) Z and suppose that a n 2. Then γ0(t n 1) exists and ∈ − ∩ e ≤− − − −e is labeled with ( n 1,h n 1). Since γ0(t n 1) h. CS0(BS,T) for some h Γ, we B− − − − − − ∈ ∈ know that γ0(t n) hh n 1. CS0( n 1). Therefore − e ∈ − − B− − e γ0(t n) hh (n+1). CS0 (n+1) g (n+1). CS0 BS,T . − ∈ − e B− ∩ − Proposition 8.2 implies that hh (n+1) = g (n+1) , γ0(t n) g (n+1). CS0( (n+1)) − e − − −e − and ∈ B 1 S T S T γ0(t (n+1)) g (n+1)h−(n+1). CS0 B , = g (n+2). CS0 B , . e − ∈ − − − Therefore γ0(tn+1) gn. CS0( n) for each n (a, 1] Z. This completes the e e proof. ∈ B ∈ − ∩ e Let Λ denote the set of geometric coding sequences and let Λσ be the subset of Λ which contains the geometric coding sequences (an)n (a,b) Z with a,b Z ∈ ∩ ∈ ∪ {±∞} for which b 2. Let Σall denote the set of all finite and one- or two-sided infinite sequences in≥ Σ. The left shift σ : Σall Σall, → σ (an)n J := ak+1 for all k J ∈ k ∈ SYMBOLIC DYNAMICS 2223 induces a partially defined map σ :Λ Λ resp. a map σ :Λσ Λ. Suppose that → → Seq: CS Λ is the map which assigns to v CS the geometric coding sequence of v. Recall→ the first return map R from Section∈ 2.3. Propositionc 8.25. The diagram b c b R CS / CS

Seq Seq c c σ Λ / Λ commutes. In particular, for v CS, the element R(v) is deﬁned if and only if ∈ Seq(v) Λσ. ∈ c Proof. This follows immediatelyb from the deﬁnition of geometricb coding sequences, Propositionsb 8.11 and 8.24.

1 Set CSst := π− (CSst) and CSst0 (BS,T) := CSst CS0(BS,T) and let Λst denote the set of two-sided inﬁnite geometric coding sequences.∩ c e e Remark 8.26. The set of geometric coding sequences of elements in CSst (or only CS0 (BS,T)) is Λ . Moreover, Λ Λσ. st st st ⊆ In the following we will show that (Λ , σ) is a symbolic dynamics for the geodesic e st ﬂow on Φ. Elementary convex geometry proves the following lemma. Lemma 8.27. Suppose that x, y ∂ Hrbd, x

γv( ),γv( ) = γw( ),γw( ) . ∞ −∞ 6 ∞ −∞ Z Assume on the contrary that ( j ,hj) j Z = ( j0 , kj ) j Z. Let (tn)n be the B ∈ B ∈ ∈ sequence of intersection times of v and (sn)n Z be that of w. Prop 8.24(iii) implies ∈ that for each n Z, the elementse pr(γ0 (tn)) and pr(e γ0 (sn)) are on the same con- ∈ v w nected component of BS. For each connected component S of BS let H1,S,H2,S denote the open convex half spaces such that H is the disjoint union

H = H ,S S H ,S. 1 ∪ ∪ 2 2224 ANKE D. POHL

Suppose first that γv( ) = γw( ). Proposition 8.14 shows that ∞ 6 ∞ γv( ),γw( ) / bd . ∞ ∞ ∈ By Lemma 8.27 we find a connected component S of BS such that γv( ) ∂gH ,Sr ∞ ∈ 1 ∂gS and γw( ) ∂gH ,S r ∂gS (or vice versa). Since BS is a manifold, each ∞ ∈ 2 connected component of BS other than S is either contained in H1,S or in H2,S. In particular, we may assume that pr(v), pr(w) H ,S. Then ∈ 1 γv([0, )) H ,S and γw((t, )) H ,S ∞ ⊆ 1 ∞ ⊆ 2 for some t > 0. Hence there is n N such that pr(γ0 (sn)) H ,S, which implies ∈ w ∈ 2 that pr(γv0 (tn)) and pr(γw0 (sn)) are not on the same connected component of BS. Suppose now that γv( ) = γw( ) and let S be a connected component of −∞ 6 −∞ BS auch that γv( ) ∂gH ,S r∂gS and γw( ) ∂gH ,S r∂gS (or vice versa). −∞ ∈ 1 −∞ ∈ 2 Again, we may assume that pr(v), pr(w) H ,S. Then ∈ 1 γv(( , 0]) H ,S and γw( ,s)) H ,S −∞ ⊆ 1 −∞ ⊆ 2 for some s < 0. Thus we find n N such that pr(γw0 (s n)) H2,S. Hence ∈ − ∈ pr(γv0 (t n)) and pr(γw0 (s n)) are not on the same connected component of BS. In both cases− we find a contradiction.− Therefore the geometric coding sequences are not equal.

Corollary 8.29. The map Seq c : CSst Λst is bijective. |CSst → Remark 8.30. If there is more than one shifted cell in SH or if there is a strip c precell in H, then the map Seq: CSrCS ΛrΛ is not injective. This is due to st → st the decision to label each v CS0( ), for each BS,T, with the same label ε if ∈ B B ∈ γv( ) ∂g pr( ) without distinguishingc c between different points in ∂g pr( ) and without∞ ∈ distinguishingB between differente shifted cellse e in SH. B e 1 e − Let Cod := Seq c :Λst CSst. |CSst → Corollary 8.31. The diagram c R CSst / CSst O O Codc c Cod σ Λst / Λst commutes and (Λst, σ) is a symbolic dynamics for the geodesic flow on Y . We end this section with the explanation of the acronyms NC and NIC (cf. Remark 6.5).

Remark 8.32. Let v be a unit tangent vector in SH based on BS and let γ be the geodesic determined by v. Then v has no geometric coding sequence if and only if v / CS. By Proposition 8.6 this is the case if and only if γ NC.c This is the reason ∈ b ∈ b why NC stands for “notb coded”. b b Supposec now that v CS. Then the geometric codingb sequence is not two-sided ∈ infinite if and only if γ does not intersect CS infinitely often in the past and the future, which by Propositionc 8.14 is equivalent to γ NIC. This explains why NIC b ∈ stands for “not infinitelyb often coded”. c b SYMBOLIC DYNAMICS 2225

9. Reduction and arithmetic symbolic dynamics. Let Γ be a geometrically finite subgroup of PSL2(R) of which is a cuspidal point and which satisfies (A). Suppose that the set of relevant isometric∞ spheres is non-empty. Fix a basal family A of precells in H and let B be the family of cells in H assigned to A. Let S be a set of choices associated to A and suppose that BS is the family of cells in SH associated to A and S. Let T be a shift map for BS and let BS,T denote the family of cells in SH associated to A, S and T. e Recall the geometric symbolic dynamics for thee geodesice flow on Y which we constructed in Section 8 with respect to A, S and T. In particular, recall the set CS0(BS,T) of representatives for the cross section CS = CS(BS,T), its subsets CS0( ) B for BS,T, and the definition of the labeling of CS. B ∈ Lete v CS0( ) for some BS,T and considerc the geodesicc e γv on H determinede ∈ B B ∈ by ve. Supposee that (an)n J is the geometric codingc sequence of v. The combi- ∈ nation of Propositionse 8.17eande8.11 allows to determine the label a0 of v from the location of γv( ), and then to iteratively reconstruct the complete future part ∞ (an)n [0, ) J of the geometric coding sequence of v. Hence, if the unit tangent ∈ ∞ ∩ vector v CS0(BS,T) is known, or more precisely, if the shifted cell in SH with ∈ B v CS0( ) is known, then one can reconstruct at least the future part of the geo- ∈ B metric coding sequencee of v. However, if γv intersects CS0(BS,T) in moree than one point, thene one cannot derive the shifted cell in SH from the endpoints of γv. B In this case, the induced discrete dynamical system on ∂gHewhich is conjugate to

(CS, R)or(CSst, R) is given by local diﬀeomorphismse which have to keep the shifted cells in their deﬁnitions. These discrete dynamical systems are used in [38, 36] to providec a dynamicalc approach to Maass cusp forms. Here, we will propose a second way to deduce discrete dynamical systems on ∂gH.

For these, we restrict in Section 9.1 our cross section CS to a subset CSred(BS,T) (resp. to CSst,red(BS,T) for the strong cross section CSst) such that for any v c c e ∈ CSred, the endpoints of γv completely determine v. We will show that CSred(BS,T) c e c and CSst,red(BS,T) are cross sections for the geodesic flow on Y w. r. t. to the same c c e measure as CS and CSst. More precisely, it will turn out that exactly those geodesics on Ycwhich intersecte CS at all also intersect CSred(BS,T) at all, and that exactly those whichc intersectc CS infinitely often in the future and the past also intersect CSred(BS,T) infinitely oftenc in the future andc the past.e Moreover, CSst,red(BS,T) is the maximal strongc cross section contained in CSred(B choices,T). In contrast to c e c e CS and CSst, the sets CSred(BS,T) and CSst,red(BS,T) do depend on the choice of c e the family BS,T. Moreover, we will deduce discrete dynamical systems (DS, F ) and c c c e c e (DSst, Fst) with DSst DS R R which are conjugate to (CSred(BS,T), R) resp. e ⊆ ⊆ × f e (CSst,red(BS,T), R). fAll constructionse f in thisf process are of geometrical naturec and effe ectively per- formablec ine a finite number of steps. Moreover, the set of labels is finite.

9.1. Reduced cross section. The set I( ) BS,T decomposes into two sequences { B | B ∈ } e e e BS,T := I , I , ... I ,k I1 B1 1 ⊇ B1 2 ⊇ ⊇ B1 1 n o e e e e 2226 ANKE D. POHL where I( ,j) = (aj , ) and a b >...>b k . B2 −∞e 1 e 2 e 2 e Set I( 1,k1+1) := and Be ∅ I ,j := I ,j rI ,j for j =1,...,k , e red B1 B1 B1 +1 1 and set I( ,k ):= and B2 2+1 e ∅ e e I ,j := I ,j rI ,j for j =1,...,k . e red B2 B2 B2 +1 2 For each BS,T set B ∈ e e e CS0 := v CS0 γv( ),γv( ) I J e rede B ∈ B ∞ −∞ ∈ red B × B n o and e e e e CSred0 BS,T := CSred0 . e e B BS,T B∈[ Deﬁne e e 1 CSred BS,T := π CSred0 BS,T and CSred BS,T := π− CSred BS,T . Further set c e e e c e CS0 := CS0 CS for each BS,T, st,red B red B ∩ st B ∈ CS0 BS,T := CS0 BS,T CS = CS0 , st,red e red e st est,rede ∩ e e B BS,T B∈[ e e e CS , BS,T := CS BS,T CS , st red red ∩ st and CSst,redBS,T := CSred BS,T CSst. e e ∩ Remark 9.1. Let BS,T. Note that the sets I ( ) and CS0 ( ) not only cB ∈ e c e cred B red B depend on but also on the choice of the complete family BS,T. The set CSred0 ( ) is identicalB to e e e e B e v CS0 γv (0, ) CS0 BS,T = e . e ∈ B ∞ ∩ ∅ In other words, if we say that v CS 0( ) has an inner intersection if and only if e ∈ B e γv((0, )) CS0(BS,T) = , then CSred0 ( ) is the subset of CS0( ) of all elements without∞ inner∩ intersection.6 ∅ eB B e e e Remark 9.2. By Proposition 8.2, the union

CS0(BS,T)= CS0( ) BS,T B B ∈ [ is disjoint and CS0(BS,T) is a set of representatives for CS. Since CS0 ( ) is a e e e e red B subset of CS0( ) for each BS,T and CSred(BS,T) = π(CSred0 (BS,T)), the set B e B ∈ c e CS0(BS,T) is a set of representatives for CSred(BS,T) and CSred0 (BS,T) is the disjoint e e e c e e union g Γ g. CSred0 (BS,T). Further, one easily sees that e ∈ c e e 1 BS T BS T S e π− CSst,red , = CSst,red , and c e e π CSst0 ,red BS,T = CSst,red BS,T .

Moreover, CSst0 ,red(BS,T) is a set of representatives for CSst,red(BS,T). e c e e c e SYMBOLIC DYNAMICS 2227

Example 9.3. Recall Example 8.13. Suppose first that the shift map is T1. Then I = ( , 1), I = (0, ), I = 1 , , I = 2 , , B1 −∞ B2 ∞ B4 5 ∞ B6 5 ∞ 3 4 I 5 = , , I 3 = , . Be 5 ∞ Be 5 ∞ e e Therefore we have e e I = ( , 1), I = 0, 1 , I = 1 , 2 , red B1 −∞ red B2 5 red B4 5 5 2 3 3 4 4 Ired 6 = , , Ired 5 = , , Ired 3 = , . Be 5 5 Be 5 5 Be 5 ∞ If the shift map isT2, then we find e e e 1 1 2 Ired 1 = ( , 0), Ired 2 = 0, , Ired 4 = , , B− −∞ B 5 B 5 5 2 3 3 4 4 Ired 6 = , , Ired 5 = , , Ired 3 = , . eB 5 5 Be 5 5 Be 5 ∞ Note that with T2, the sets Ired( ) are pairwise disjoint, whereas with T1 they are not. e · e e Lemma 9.4. Let (x, y) Ired J . ∈ e e B × B BS,T B∈[ e e Then there is a unique v CSred0 (BS,T) such that (x, y)= γv( ),γv( ) . Con- ∈ ∞ −∞ BS T versely, if v CSred0 ( , ), then ∈ e γ ( ),γ ( ) Ired J . e v v ∞ −∞ ∈ e e B × B BS,T B∈[ e e Proof. The combination of the definition of CSred0 (BS,T) and Lemma 8.8 shows that there is at least one v CS0 (BS,T) such that γv( ),γv( ) = (x, y) and that ∈ red ∞ −∞ BS T e for each , there there is at most one such v. By construction, B ∈ e I a J a I b J b = e e red B × B ∩ red B × B ∅ for a, b BS,T, a = b. Hence there is a unique such v CS0 (BS,T). The B B ∈ B 6 eB e e e ∈ red remaining assertion is clear from the definition of the sets CSred0 ( ) for BS,T. e e e e e B Be ∈ Let l (BS,T) be the number of connected components of BS of the form (a, ) 1 e e e ∞ (geodesic segment) with a a ak and let l (BS,T) be the number of connected 1 ≤ ≤ 1 2 componentse of BS of the form (a, ) with bk a b . Define ∞ 2 ≤ ≤ 1 e l BS,T := max l1 BS,T ,l2 BS,T . n o Proposition 9.5. Let ve CS0(BS,T) ande supposee that η is the geodesic on H ∈ determined by v. Let (sj )j (α,β) Z be the geometric coding sequence of v. Suppose ∈ ∩ that sj = ( j ,hj) for j =0,...,β e 2. For j =0,...,β 2 set B − − g 1 := id and gj := gj 1hj . e − − If α = 1, then let 1 be the shifted cell in SH such that v CS0( 1). Then − B− ∈ B− s := min t 0 η0(t) CSred BS,T e 0 ≥ ∈ e exists and κ e η0(s ) gl. CSred0 l 0 ∈ B l= 1 [− e 2228 ANKE D. POHL where κ := min l(BS,T) 1,β 2 . More precisely, η0(s ) gl. CSred0 ( l) for { − − } 0 ∈ B l 1,...,κ if and only if η( ) gl.Ired( l) and η( ) / gk.Ired( k) for k = ∈ {− } ∞ ∈ B ∞ ∈ B 1,...,l 1. Moreover,e if v CSst0 (BS,T), then η0(s0) CSst,red(BS,T). e − − ∈ e ∈ e Proof. Let (tn)n (α,β) Z be the sequence of intersection times of v (with respect ∈ ∩ e e to CS). Proposition 8.24(iii) respectively the choice of 1 shows that η0(tn) B− ∈ gn 1. CS0( n 1) for each n [0,β) Z. Since CSred(BS,T) CS, the minimum s0 − B − ∈ ∩ e⊆ exists if and only if η0(tm) gm 1. CSred0 ( m 1) for some m [0,β) Z. In this e ∈ − B − e ∈ ∩ case, s0 = tn and η0(s0) gn 1. CSred0 ( n 1) where ∈ − B − e n := min m [0,β) Z η0(tm) gm 1. CSred0 m 1 . ∈ ∩ e ∈ − B − Suppose that s = tn. Note that for each m [0,β) Z we have that η( ) is in 0 ∈ ∩ e −∞ J( m 1). Then the definition of CSred0 ( ) shows that B − · n = min m [0,β) Z η( ) gm 1.Ired m 1 . e ∈ ∩ ∞ ∈ − B − Hence it remains to show that the element s0 exists and that s 0 = tn for some e n 0,...,κ +1 . ∈{ } W.l.o.g. suppose that I( 1) 1(BS,T). Then I( 1) = (a, ) for some a R. B− ∈ I B− ∞ ∈ Let c1 < c2 < ... < ck be the increasing sequence in R such that c1 = a and ck = ak1 and such that thee set e e

(cj , ) j =1,...,k { ∞ | } of geodesic segments is the set of connected components of BS of the form (c, ) ∞ with a c ak1 . Then k l(BS,T). Let (cji , ) i =1,...,m be its subfamily (indexed≤ by≤ 1,...,m ) of≤ geodesic segments{ such∞ | that for each }i 1,...,m we { } ∈{ } have (cj , )= b( 0) for some e 0 BS,T such that b( 0) (BS,T) and i ∞ Bi Bi ∈ Bi ∈ I1 cj cj ... cj . e e1 ≤ e 2 ≤ ≤ m e e m The deﬁnition of Ired( ) shows that I( 1) is the disjoint union i Ired( i0). More- · B− =1 B over, J( i0) J( 1) for i =1,...,m. From Lemma 8.8 we know that B ⊇ B− e S e η( ), η( ) I( 1) J( 1). e e ∞ −∞ ∈ B− × B− Hence there is a uniquei 1,...,m such that ∈{ } e e η( ), η( ) I ( 0) J( 0). ∞ −∞ ∈ red Bi × Bi

In turn, η intersects CSred0 ( i0). Now, if η does not intersect CSred0 ( 10 )=CSred0 ( 1), B e e B B− then η([0, )) intersects (c2, ) and we have g0.b( 0) = (c2, ). If η does not in- ∞ e ∞ B ∞ e e tersect g0. CSred0 ( 0), then η([0, )) intersects (c3, ) and hence g1.b( 1) = (c3, ) and so on. This showsB that ∞ ∞e B ∞ e e CS0( i0)=CS0( ji 2)= gji 2. CS0( ji 2) B B − − B − and hence η0(tji 1) gji 2. CSred0 ( ji 2), where ji 1 k l(BS,T). If β < , − ∈ −e Be − −κ e≤ ≤ ∞ then ji 1 β 1. Thus, s0 exists and η0(s0) l= 1 gl. CSred0 ( l). Finally, if − ≤ − e ∈ − e B v CSst0 (BS,T), then η0(s0) CSst and hence η0(s0) CSst,red(BS,T). ∈ ∈ S∈ e Recalle the shift map σ :Λ Λ from Section 8.2. The followinge proposition is a consequence of Proposition 9.5→. SYMBOLIC DYNAMICS 2229

Proposition 9.6. Let v CSred0 (BS,T) and suppose that η is the geodesic on H ∈ determined by v. Let sj j (α,β) Z be the geometric coding sequence of v. Suppose ∈ ∩ e that sj = ( j ,hj) for j =0,...,β 2. B − (i) Set g0 := h0 and for j =0,...,β 3 deﬁne gj+1 := gj hj+1. If there is a next e − point of intersection of η and CSred(BS,T), then this is on κ gl. CSered0 l B l=0 [ e where κ := min l(S, T),β 2 . In this case it is on gl. CSred0 ( l) if and only if { − } B η( ) gl.Ired( l) and η( ) / gk.Ired( k) for k =0,...,l 1. If β 2, then ∞ ∈ B ∞ ∈ B − ≥ there is a next point of intersection of η and CSred(BS,T). e e e (ii) Suppose that v CSst0 ,red(BS,T). Let (tn)n Z be the sequence of intersection ∈ ∈ times of v (w.r.t. CS). Then there was a previouse point of intersection of η and CSred(BS,T) and this ise contained in

η0(t n) n =1,...,l BS,T +1 . e − Recall from Remark 6.5 that NIC denotes the set of geodesics on Y with at least e one endpoint contained in π(bd). Corollary 9.7. Let µ be a measure on the space of geodesics on Y . The sets CSred(BS,T) and CSst,red(BS,T) are cross sections w. r. t. µ if and only if NIC is a µ-null set. Moreover, CSst,red(BS,T) is the maximal strong cross section contained c e c e in CSred(BS,T). c e 9.2. Reduced coding sequences and arithmetic symbolic dynamics. Anal- c e ogous to the labeling of CS in Section 8.2 we define a labeling of CSred(BS,T). Let v CS0 (BS,T) and let γ denote the geodesic on H determined by v. Suppose ∈ red first that γ((0, )) CS (BS,T) = . Proposition 9.6 implies that theree is a next ∞ ∩ red 6 ∅ point of intersectione of γ and CS (BS,T) and that this is on g. CS0 ( ) for a red red B (uniquely determined) pair (e ,g) BS,T Γ. We endow v with the label ( ,g). B ∈ × B Suppose now that γ((0, )) CS e(BS,T) = . Then there is no nexte point of ∞ ∩ red ∅ intersection of γ and CSred(BeS,T). Wee label v by ε. e 1 e − BS T : 0 e Let v CSred( , ) and let v = π CS (BS T) (v). The label of v and of each ∈ e | red , 1 element in π− (v) is defined to be the label of v. c e Supposeb that Σred denotes the set of labels of CSredb(BS,T). b Remark 9.8. Recallb from Corollary 8.19 that Σ is finite. Then Proposition 9.6 c e implies that also Σred is finite. Moreover, Remark 8.10 shows that the elements of Σ can be effectively determined. From Proposition 9.6 then follows that also the elements of Σred can be effectively determined. The following definition is analogous to the corresponding definitions in Sec- tion 8.2.

Deﬁnition 9.9. Let v CSred0 (BS,T) and suppose that γ is the geodesic on H determined by v. Propositions∈ 9.5 and 9.6 imply that there is a unique sequence (tn)n J in R which satisﬁes the followinge properties: ∈ (i) J = Z (a,b) for some interval (a,b) with a,b Z and 0 (a,b), ∩ ∈ ∪ {±∞} ∈ 2230 ANKE D. POHL

(ii) the sequence (tn)n J is increasing, ∈ (iii) t0 = 0, (iv) for each n J we have γ0(tn) CS (BS,T) and ∈ ∈ red γ0((tn,tn+1)) CSred(BS,T)= and γ0((tn 1,tn)) CSred(BS,T)= ∩ ∅ e − ∩ ∅ where we set tb := if b< and ta := if a> . ∞ e ∞ −∞ −∞ e The sequence (tn)n J is said to be the sequence of intersection times of v with ∈ respect to CSred(BS,T). 1 − BS T : 0 e Let v CSred( , ) and set v = π CS (BS T) (v). Then the sequence of ∈ e | red , 1 intersection times w. r. t. CSred(BS,T) of v and of each w π− (v) is defined to be b c e b∈ the sequence of intersection times of v w.r.t. CSred(BS,T). For each s Σred set e b b ∈ e CS ,s BS,T := v CS BS,T v is labeled with s red ∈ red and c e b c e b 1 CS ,s BS,T := π− CS BS,T = v CS BS,T v is labeled with s . red red ∈ red Let v CS red(BS,T) and let (tn)nJ be the sequence of intersection times of v ∈ e c e ∈ e w.r.t. CSred(BS,T). Suppose that γ is the geodesic on Y determined by v. The reducedb codingc sequencee of v is the sequence (an)n J in Σred defined by b ∈ e an := s if and onlyb if γ0(tn) CSred,s BS,T b b ∈ for each n J. ∈ c e Let w CSred(BS,T). The reduced codingb sequence of w is defined to be the reduced coding∈ sequence of π(w). Let Λred denote thee set of reduced coding sequences and let Λred,σ be the subset of Λred consisting of the reduced coding sequences (an)n (a,b) Z with a,b Z ∈ ∩ ∈ ∪{±∞} for which b 2. Further, let Λst,red denote the set of two-sided infinite reduced ≥ all coding sequences. Let Σred be the set of all finite and one- or two-sided infinite sequences in Σ . Finally, let Seq : CS (BS,T) Λ be the map which assigns red red red → red to v CSred(BS,T) the reduced coding sequence of v. ∈ c e The proofs of Propositions 9.10 are analogous to those of the corresponding statementsb c ine Section 8.2. b

Proposition 9.10. (1) Let v CSred0 (BS,T). Suppose that (tn)n J is the sequence ∈ ∈ of intersection times of v and that (an)n J is the reduced coding sequence of v. ∈ Let γ be the geodesic on H determinede by v. Suppose that J = Z (a,b) with a,b Z . ∩ ∈ ∪ {±∞} (i) If b = , then an Σred r ε for each n J. ∞ ∈ { } ∈ (ii) If b< , then an Σred r ε for each n (a,b 2] Z and ab 1 = ε. ∞ ∈ { } ∈ − ∩ − (iii) Suppose that an = ( n,hn) for n (a,b 1) Z and set B ∈ − ∩ g0 := h0 if b 2, e ≥ gn := gnhn for n [0,b 2) Z, +1 +1 ∈ − ∩ g 1 := id, − 1 g (n+1) := g nh−n for n [1, (a + 1)) Z. − − − ∈ − ∩ Then γ0(tn ) gn. CSred0 ( n) for each n (a,b 1) Z. +1 ∈ B ∈ − ∩ e SYMBOLIC DYNAMICS 2231

(2) Let v, w CSst0 ,red(BS,T). If the reduced coding sequences of v and w are equal, then v =∈w. all all (3) The left shift σ : Σrede Σred induces a partially deﬁned map σ :Λred Λred → → resp. a map σ :Λred,σ Λred. Moreover, Λst,red Λred,σ and σ restricts to a → ⊆ map Λst,red Λst,red. → (4) The map Seqred c e : CSst,red(BS,T) Λst,red is bijective. CSst,red(BS,T) | 1 → − (5) Let Codred := Seqred c Be . Then the diagrams |CSst,red(cS,T) e R R CSred BS,T / CSred BS,T CSst,red BS,T / CSst,red BS,T O O red red cSeqred e c Seqe red cCod e c Code σ σ Λred / Λred Λst,red / Λst,red

commute and (Λst,red, σ) is a symbolic dynamics for the geodesic ﬂow on Y .

We will now show that the reduced coding sequence of v CSred(BS,T) can be completely constructed from the knowledge of the pair τ(v). ∈ b c e Definition 9.11. Let BS,T. Define B ∈ b Σ := s Σ v CS0 : v is labeled with s red B e ∈e red ∃ ∈ red B and for s Σ ( ) set ∈ red Be e Ds := I g.I 0 if s = 0,g e B red B ∩ red B B and e e e e Dε := I r Ds s Σ r ε . B red B B ∈ red B { } [ T Example 9.12. Recalle the Examplee 8.13.e Suppose first thate the shift map is 1. Then we have Σ = ε, ,g , ,g , ,g , ,g , ,g , red B1 B1 5 B4 4 B6 4 B5 4 B3 4 Σred 2 = ε, 2,g1, 4,g1, 6,g1, 5,g1, 3,g1 , Be Be Be Be Be Be Σred 3 = ε, 1,g5, 2,g6, 4,g6, 6,g6, 5,g6 , 3,g6 , Be Be Be Be Be Be B Σred 4 = ε, 5,g2, 3,g2 , Be Be Be e e e e Σred 5 = ε, 6,g4, 5,g4 , 3,g4 , Be Be Be B Σred 6 = ε, 3,g3 . Be Be e e Hence e e D = 4 , 1 ,D = , 3 ,D = 3 , 7 , e ,g 1 5 e ,g 1 5 e ,g 1 5 10 B1 5 B B4 4 B − ∞ B6 4 B 7 11 11 4 4 D = , ,D = , ,Dε = , e ,g e1 10 15 e ,g e1 15 5 e1 5 B5 4 B B3 4 B B and e e e

D = 0, 1 ,D = 1 , 2 ,D = 2 , 3 , e ,g 2 10 e ,g 2 10 15 e ,g 2 15 20 B2 1 B B4 1 B B6 1 B 3 4 4 1 1 D = , ,D = , ,Dε = , e ,g e2 20 25 e ,g e2 25 5 e2 5 B5 1 B B3 1 B B e e e 2232 ANKE D. POHL and

D = 4 , 1 ,D = 1, 6 ,D = 6 , 7 , e ,g 3 5 e ,g 3 5 e ,g 3 5 5 B1 5 B B2 6 B B4 6 B 7 8 8 9 9 D e 3 = , ,D e 3 = , ,D e 3 = , , ,g6 e 5 5 ,g6 e 5 5 ,g6 e 5 B6 B B5 B B3 B ∞ Dε 3 = 1 , Be { } e e and e 1 3 3 2 2 D e 4 = , ,D e 4 = , ,Dε 4 = , ,g2 5 10 ,g2 10 5 5 B5 B B3 B B and e e e

D = 3 , 7 ,D = 7 , 11 ,D = 11 , 4 , e ,g 5 5 10 e ,g 5 10 15 e ,g 5 15 5 B6 4 B B5 4 B B3 4 B 4 Dε 5 = , Be 5 e e and e 2 3 3 D = , ,Dε = . e ,g 6 5 5 6 5 B3 3 B B Suppose nowe that the shift map is Te . Then Σ , Σ , Σ and 2 red B2 red B4 red B5 T Σred 6 are as for 1. The sets D 2 ,D 4 ,D 5 and D 6 remain unchangedB as well. We have ∗ B ∗ B e∗ B e ∗ B e e e e e e Σred 1 = ε, 1,g7 , 4,g7 , 6,g7 , 5,g7 , 3,g7 B− B− B B B B and e e e e e e

Σred 3 = ε, 1,g4 , 2,g2 , 4,g6 , 6,g6 , 5,g6 , 3,g6 . B B− B B B B B Therefore e e e e e e e 1 2 D e 1 = , 0 ,D e 1 = , , −1,g7 − 5 4,g7 − 5 B B − B B − ∞ − 2 3 3 4 D e e 1 = 5 , 10 ,D e e 1 = 10 , 15 , 6,g7 − 5,g7 − B B − − B B − − 4 1 1 D e e 1 = 15 , 5 ,D ε e 1 = 5 , 3,g7 − − B B − − B − and e e

4 6 D e 3 = , 1 ,D e 3 = 1, , − ,g4 5 ,g6 5 B 1 B B2 B D = 6 , 7,D = 7 , 8 , e ,g e3 5 5 e ,g e3 5 5 B4 6 B B6 6 B D = 8 , 9 ,D = 9 , , e ,g e3 5 5 e ,g e3 5 B5 6 B B3 6 B ∞ Dε 3 = 1 . Be { } e The next corollary follows immediately from Proposition 9.6. e

Corollary 9.13. Let BS,T. Then Ired( ) decomposes into the disjoint union B ∈ B Ds( ) s Σred( ) . Let v CSred0 ( ) and suppose that γ is the geodesic on H { B | ∈ B } ∈ B determined by v. Thenev ise labeled with s if ande only if γ( ) belongs to Ds( ). S e e e ∞ B e SYMBOLIC DYNAMICS 2233

Our next goal is to ﬁnd a discrete dynamical system on the geodesic boundary of H which is conjugate to (CSred(BS,T), R). To that end we set DS := I J . c e red e e B × B BS,T B∈[ f e e For BS,T and s Σ ( ) we set B ∈ ∈ red B Ds := Ds J . e e e B B × B We deﬁne the partial map F : DS DS by e e→ e e 1 1 F e e (x, y) := (g− .x, g− .y) Ds( ) e| fB f if s = ( 0,g) Σred( ) ande BS,T. RecallB the∈ map B B ∈ e e e e τ : CS BS,T ∂gH ∂gH, v γv( ),γv( ) red → × 7→ ∞ −∞ 1 : 0 e − H where v = π CS (BS T) (v) and γv is the geodesic on determined by v. | redc , e b Proposition 9.14. The setbDS is the disjoint union

DS = Ds BS,T, s Σred . f B B ∈ ∈ B [ n o If (x, y) DS, then there is a unique element v CSred0 (BS,T) such that ∈ f e e e e ∈ e γv( ),γv( ) = (x, y). f ∞ −∞ e If and only if (x, y) Ds( ), the element v is labeled with s. Moreover, the partial ∈ B map F is well-defined and the discrete dynamical system (DS, F ) is conjugate to (CSred(BS,T), R) via τ.e e e f e Proof. The sets I ( ) J( ), BS,T, are pairwise disjoint by construction. c e red B × B B ∈ Corollary 9.13 states that for each BS,T the set I ( ) is the disjoint union B ∈ red B Ds( ) s Σred( e) . Therefore,e e thee sets Ds( ), BS,T, s Σred( ), are pairwise{ B disjoint| ∈ and B } e e B B ∈e ∈ B S e e e e e e e DS = Ds BS,T, s Σ . B B ∈ ∈ red B n o This implies that F is well-defined.[ Let (x, y) DS. By Lemma 9.4 there is f e e e e ∈ e a unique BS,T and a unique v CS0 ( ) such that γv( ),γv( ) = B ∈ ∈ red B ∞ −∞ (x, y). Corollary 9.13e shows that v is labeled with s fΣ if and only if γv( ) ∈ red ∞ ∈ Ds( ), hencee ife (x, y) Ds( ). It remains toe show that (DS, F ) is conjugate to B ∈ B (CSred(BS,T), R) by τ. Lemma 9.4 shows that τ is a bijection between CSred(BS,T) 1 e e e : 0 e − f e BS T and DS. Let v CSred and v = π CS (BS T) (v). Suppose that , is c e ∈ | red , Bc ∈ e the (unique) shifted cell in SH such that v CSred0 ( ), and let (sj )j (α,β) Z be the f c ∈ B ∈ e ∩ e reduced codingb sequence of v. Recall that s0 is the labelb of v and v. Corollary 9.13 shows that γv( ) Ds ( ). The map R is definede for v if and only if s = ε. In ∞ ∈ 0 B 0 6 precisely this case, F is defined for τ(v). b Suppose that s = ε, saye s = ( 0,g). Then the nextb intersection of γv and 0 6 0 B CS (BS,T) is on g.eCS0 ( 0), say it isb w. Then R(v)= π(w)=: w and red red B e 1 1 0 e − π CS (BS T) (w)= g− .w. e e | red , b b b 2234 ANKE D. POHL

1 Let η be the geodesic on H determined by g− .w. We have to show that F (τ(v)) = η( ), η( ) . To that end note that g.η(R)= γv(R) and hence η( ), η( ) = ∞1 −∞ 1 ∞ −∞ g− .γv( ),g− .γv( ) . Since τ(v) Ds ( ), the deﬁnition of F showse thatb ∞ −∞ ∈ 0 B 1 1 F (τ(v)) = F (γv( ),γv( )) = g− .γv( ),g− .γv( ) = η( ), η( ) . ∞ −∞ b e ∞e −∞ e ∞ −∞ Thus, (DS, F ) is conjugate to (CS (BS,T), R) by τ. e b e red Thef followinge corollary provesc thate we can reconstruct the future part of the reduced coding sequence of v CS (BS,T) from τ(v). ∈ red Corollary 9.15. Let v CSred(BS,T) and suppose that (sj )j J is the reduced coding b c e b ∈ sequence of v. Then ∈ b c e j sj = s if and only if F τ(v) Ds for some BS,T ∈ B B ∈ j for each j J N0. For j N0 rJ, the mapF is not deﬁned for τ(v). ∈ ∩ ∈ e b e e e e The next proposition shows that we can also reconstruct the past part of the e b reduced coding sequence of v CS (BS,T) from τ(v). Its proof is constructive. ∈ red Proposition 9.16. (i) The elements of b c e b 1 e S T red g− .D( ,g) 0 0 B , , ,g Σ 0 B B B ∈ B ∈ B n o are pairwise disjoint. e e e e e e (ii) Let v CSred(BS,T) and suppose that (aj )j J is the reduced coding sequence of v. Then∈ 1 J if and only if ∈ − ∈ b c e 1 e S T red τ(v) g− .D( ,g) 0 0 B , , ,g Σ 0 . b ∈ B B B ∈ B ∈ B n o In this case, [ b e e e e e e 1 e a 1 = ,g if and only if τ(v) g− .D( ,g) 0 − B ∈ B B

for some 0 BS,T and ( ,g) Σred( 0). B ∈ e B ∈ B b e e Proof. We will prove (ii), which directly implies (i). To that end set e e e e 1 − : 0 e v = π CS (BS T) (v) | red , and suppose that (tj )j J is the sequence of intersection times of v with respect to ∈ b CSred(BS,T).

Suppose ﬁrst that v CSred0 ( ) and that 1 J. Then there exists a (unique) e ∈ B −1 ∈ pair ( 0,g) BS,T Γ such that γv0 (t 1) g− . CSred0 ( 0). Since the unit tangent B ∈ × e − ∈ B vector γv0 (t0) = v is contained in CSred0 ( ), the element γv0 (t 1) is labeled with B − ( ,g).e Hence (e ,g) Σred( 0). Then e B B ∈ B e 1 1 τ(v)= γv( ),γv( ) g− .I 0 g− .J 0 I J e ∞e −∞ e∈ red B × B ∩ red B × B 1 1 = g− .I 0 I g− .J 0 J b red Be ∩ red B e × Be ∩ Be 1 = g− . I 0 gI J g.J redeB ∩ rede B × B e∩ Be 1 e g− .D( ,g) 0 . ⊆ B Be e e e e e SYMBOLIC DYNAMICS 2235

1 e S T Conversely suppose that τ(v) g− .D( ,g)( 0) for some 0 B , and some element ∈ B B B ∈ ( ,g) Σred( 0). Consider the geodesic η := g.γv. Then B ∈ B b e e e e e η( ), η( ) = g.τ(v) D( ,g) 0 . e e ∞ −∞ ∈ B B By Proposition 9.14, there is a unique u CS0 (BS,T) such that ∈ bred e e γu( ),γu( ) = η( ), η( ) . ∞ −∞ ∞ e −∞ Moreover, u is labeled with ( ,g). Let (sk)k K be the sequence of intersection B ∈ times of u w.r.t. CSred(BS,T). Then 1 K and, by Proposition 9.14, e ∈ τ π(γ0 (s )) = τ R(π(u)) = F τ(π(u)) = F γu( ),γu( ) u 1 e ∞ −∞ 1 1 = g− .γu( ),g− .γu( ) = γv( ),γv( ) = τ(v). ∞ e −∞ e ∞ −∞ This shows that γu0 (s1)= g.v = g.γv0 (t0). Then b 1 g− .γ0 (s ) γ0 ( , 0) CS BS,T . u 0 ∈ v −∞ ∩ red Hence, there was a previous point of intersection of γ and CS (BS T) and this ve red , 1 1 is g− .γu0 (s0). Recall that g− .γu0 (s0) is labeled with ( ,g). This completes the proof. B e e Let F : DS DS be the partial map defined by bk → − e e0 : Fbk g 1D e ( )(x, y) = (g.x, g.y) e f f | (B,g) B for 0 BS,T and ( ,g) eΣred( 0). B ∈ B ∈ B Corollarye e 9.17. e(i) The partiale map Fbk is well-defined. (ii) Let v CSred(BS,T) and suppose that (sj )j J is the reduced coding sequence of ∈ ∈ v. For each j J ( , 1] and eache ( ,g) Σred we have ∈ ∩ −∞ − B ∈ b c e j 1 e sj = ( ,g) if and only if Fbk τ(v) g− .D( ,g) 0 b B e ∈ B B j for some 0 BS,T. For j Z< rJ, the map Fbk is not defined for τ(v). B e∈ ∈ 0 e b e e We end this section with the statement of the discrete dynamical system which e e e b is conjugate to the strong reduced cross section CSst,red(BS,T). The set of labels of CSst,red(BS,T) is given by c e Σst,red :=Σred r ε . c e { } For each BS,T set B ∈ Σst,red :=Σred r ε . e e B B { } Recall the set bd from Section 6. For s Σst,red( ) set e ∈ e B D ,s := Ds rbd st B B e and e e

D ,s := D ,s J rbd . st B st B × B Further let e e e e DSst := Ired rbd J rbd e e B × B BS,T B∈[ f e e 2236 ANKE D. POHL and deﬁne the map F : DS DS by st st → st 1 1 F e e (x, y) := (g− .x, g− .y) st D ,s( ) e f| st Bf if s = ( 0,g) Σ , ( ) and BS,T. The map F is the “restriction” of F to the B ∈ st red B e B ∈ st strong reduced cross section CSst,red(BS,T). In particular, the following proposition is the “reduced”e analogone of Propositione e 9.14. e e c e Proposition 9.18. (i) The set DSst is the disjoint union

DSst = Dst,s BS,T, s Σst,red . Bf B ∈ ∈ B [ n o If (x, y) DSfst, then theree ise a uniquee e element v CSest0 ,red(BS,T) such that ∈ ∈ γv( ),γv( ) = (x, y). If and only if (x, y) Dst,s( ), the element v is labeled∞ withf−∞s. ∈ B e (ii) The map Fst is well-deﬁned and the discrete dynamicale systeme DSst, Fst is st red BS T conjugate to CS , ( , ), R via τ. e f e 9.3. Generating function for the future part. Suppose that the sets I ( ), c e red B BS,T, are pairwise disjoint. Set B ∈ e DS := I e e red e e B BS,T B∈[ e and consider the partial map F : DS DS given by → 1 F e x := g− .x Ds( ) | B if s = ( 0,g) Σ ( ) and BS,T. B ∈ red B B ∈ Proposition 9.19. (i) The set DS is the disjoint union e e e e

DS = Ds BS,T, s Σred . B B ∈ ∈ B [ n o If x DS, then there is (a non-unique)e e e v CSred0 (BS,eT) such that γv( )= x. ∈ ∈ ∞ Suppose that v CSred0 (BS,T) with γv( ) = x and let (an)n J be the reduced ∈ ∞ ∈ coding sequence of v. Then a0 = s if and only if x eDs( ) for some BS,T. (ii) The partial map F is well-deﬁned.e ∈ B B ∈ e e e Proof. Suppose that , BS,T and s Σ ( ), s Σ ( ) such that B1 B2 ∈ 1 ∈ red B1 2 ∈ red B2 Ds ( ) Ds ( ) = . Pick x Ds ( ) Ds ( ). Then x I ( ) I ( ), 1 B1 ∩ 2 B2 6 ∅ ∈ 1 B1 ∩ 2 B2 ∈ red B1 ∩ red B2 which implies that 1 e= e2. Nowe Corollary 9.13 yieldse that s1 = s2.e Therefore the unione in (i) is disjointe B andB hence F ise well-deﬁned.e Corollary 9.13 showse thate the union equals DS. e e Let (x, y) DS. Then (x, y) Ds( ) if and only if x Ds( ). Proposition 9.14 implies the remaining∈ statements∈ of (Bi). ∈ B f e e e Proposition 9.19 shows that

F, (DSs( )) e e e BS,T,s Σ ( ) B B∈ ∈ red B is like a generating function for the futuree part of the symbolic dynamics (Λred, σ). In comparison with a real generating function, the map i:Λred DS is missing. Indeed, if there are strip precells in H, then there is no unique choice→ for the map i. SYMBOLIC DYNAMICS 2237

To overcome this problem, we restrict ourselves to the strong reduced cross section CSst,red(BS,T).

Proposition 9.20. Let v, w CSst0 ,red(BS,T). Suppose that (an)n Z is the reduced c e ∈ ∈ coding sequence of v and (bn)n Z the one of w. If (an)n N = (bn)n N , then ∈ ∈ 0 ∈ 0 γv( )= γw( ). e ∞ ∞ Proof. The proof of Proposition 8.28 shows the corresponding statement for geometric coding sequences. The proof of the present statement is analogous. We set DSst := Ired rbd e e B BS,T B∈[ and deﬁne the map F : DS DS by e st st → st 1 F e x := g− .x st D ,s( ) | st B if s = ( 0,g) Σ , ( ) and BS,T. Further deﬁne i:Λ , DS by B ∈ st red B B ∈ st red → st i (an)n Z := γv( ), e e e e ∈ ∞ where v CS0 (BS,T) is the unit tangent vector with reduced coding sequence ∈ st,red (an)n N. Proposition 2 shows that v is unique, and Proposition 9.20 shows that i ∈ only depends on (ane)n N . Therefore ∈ 0 F , i, (Ds( )) e e e st BS,T,s Σ , ( ) B B∈ ∈ st red B is a generating function for the future part of the symbolic dynamics (Λ , σ). e st,red Example 9.21. For the Hecke triangle group Gn and its family of shifted cells BS,T = from Example 8.20 we have I( ) = Ired( ) and Σred = Σ. Obviously, the associated{B} symbolic dynamics (Λ, σ) hasB a generatingB function for the future part.e Recalle the set bd from Section 6. Heree we havee bd = Gn. . Then ∞ + + DS = R and DS = R rGn. . st ∞ Since there is only one (shifted) cell in SH, we omit from the notation in the B following. We have Dg = (g.0, g. ) and ∞ k e D ,g = (g.0, g. )rGn. for g U S k =1,...,n 1 . st ∞ ∞ ∈ n − The generating function for the future part of (Λ, σ) is F : DS DS, → 1 k F D x := g− .x for g U S k =1,...,n 1 .. | g ∈ n − For the symbolic dynamics (Λ , σ) arising from the strong cross section CS the st st generating function for the future part is F : DS DS , st st → st 1 k c F D x := g− .x for g U S k =1,...,n 1 . st| st,g ∈ n − Example 9.22. Recall Example 9.3. If the shift map is T2, then the sets Ired( ) are pairwise disjoint and hence there is a generating function for the future part· of the symbolic dynamics. In contrast, if the shift map is T1, the sets Ired( ) are not disjoint. Suppose that γ is a geodesic on H such that γ( ) = 1 . Then· γ ∞ 2 intersects CSred0 BS,T1 in, say, v. Example 9.12 shows that one cannot decide whether v CS0 6 and hence is labeled with 3,g3 , or whether v CS0 1 ∈ eB B ∈ B and thus is labeled with 4,g4 . This shows that the symbolic dynamics arising from the shift mapeT doesB not have a generatinge function for the future part. e 1 e 2238 ANKE D. POHL

10. Transfer operators. Suppose that (X,f) is a discrete dynamical system, where X is a set and f is a self-map of X. Further let ψ : X C be a function. The transfer operator of (X,f) with potential ψ is deﬁned→ by L ϕ(x)= eψ(y)ϕ(y) L y f −1(x) ∈X with some appropriate space of complex-valued functions on X as domain of definition. They originate from thermodynamic formalism. For more background information, we refer to e. g., [9, Section 14], [41], [29]. Here, we only consider potentials of the form

ψ(y)= β log f 0(y) , − | | where β C. Let Γ∈ be a geometrically ﬁnite subgroup of PSL(2, R) of which is a cuspidal point and which satisﬁes (A). Suppose that the set of relevant isometric∞ spheres is non-empty. Fix a basal family A of precells in H and let B be the family of cells in H assigned to A. Let S be a set of choices associated to A and suppose that BS is the family of cells in SH associated to A and S. Let T be a shift map for BS and let BS,T denote the family of cells in SH associated to A, S and T. e

We restrict ourselves to the strong reduced cross section CSst,red(BS,T).e Recall e the discrete dynamical system (DSst, Fst) from Section 9.2 as well as the set Σst,red, its subsets Σ , ( ) and the sets D ,s( ). The local inversesc of F aree st red B st B st 1 e e −e eFst Dst,s Dst,se F e,s := Fst e e : B → B |Dst,s (x, y) (g.x, g.y) B B 7→ e e e e e for s Σ , e ( ), e BS,T. To abbreviate, we set ∈ st red B B ∈

E e,s := Fst Dst,s e e e B B for s Σst,red( ), BS,T. e e e e For∈ two setsBM,NB ∈let Fct(M,N) denote the set of functions from M to N. Then the transfer operatore e withe parameter β C ∈ β : Fct(D , C) Fct(D , C) L st → st associated to Fst is given by e e β ( βϕ) (x, y) := det F 0e (x, y) ϕ F e (x, y) 1 e (x, y), e ,s ,s EBe,s L e e e B B BS,T s Σ , ( ) B∈X ∈ stXred B e e e e e where 1E e is the characteristic function of E ,s. Note that the set E ,s, the domain B,s B B of definition of F e,s, in general is not open. A priori, it is not even clear whether B e e E e,s is dense in itself. To avoid any problems with well-definedness, the derivative of B e F e,s shall be defined as the restriction to E e,s of the derivative of F e : F (Ds) Ds, B B B → (x, y) (g.x, g.y). Moreover, the maps F e and F are extended arbitrarily on ,s 0e,s e 7→ eB B e e e Dst rE e,s. Let βB C and consider the map e e e e ∈ τ ,β : Γ Fct(D , C) Fct(D , C), (g, ϕ) τ ,β(g)ϕ 2 × st → st 7→ 2 e e SYMBOLIC DYNAMICS 2239 where 1 β β τ ,β(g− )ϕ(x, y) := g0(x) g0(y) ϕ(g.x, g.y). 2 | | | | The map τ2,β is an action of Γ on Fct(Dst, C), and reminiscent of principal series representations. The transfer operator becomes e 1 e βϕ = 1E e e0 τ2,β(g− )ϕ. L B,(B ,g) · e e e0 e BS,T ( ,g) Σ , ( ) B∈X B ∈Xst red B

Suppose now that the sets I ( ), BS,T, are pairwise disjoint so that the map red B B ∈ Fst from Section 9.3 is a generating function for the future part of (Λst,red, σ). Its local inverses are e e e 1 − Fst Dst,s Dst,s F e,s := Fst e : B → B |Dst,s x g.x B B 7→ e e for s = ( 0,g) Σ , ( ), BS,T. If we set B ∈ st red B B ∈

E e,s := Fst Dst,s , e e e Be B then the transfer operator with parameter β associated to Fst is the map e β : Fct(DS , C) Fct(DS , C) L st → st given by 1 βϕ = 1E e e0 τβ(g− )ϕ, L B,(B ,g) e e e0 e BS,T ( ,g) Σ , ( ) B∈X B ∈Xst red B where τβ : Γ Fct(DS , C) Fct(DS , C), (g, ϕ) τβ(g)ϕ × st → st 7→ with 1 β τβ (g− )ϕ(x)= g0(x) ϕ(g.x). | | For extensive examples of such transfer operators we refer to [21, 38, 36].

Acknowledgments. The author would like to thank the referee for a thorough reading of the manuscript.

REFERENCES

[1] R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Am. Math. Soc., New Ser., 25 (1991), 229–334. [2] P. Arnoux, Le codage du flot géodésique sur la surface modulaire, (French) [Coding of the geodesic flow on the modular surface] Enseign. Math. (2), 40 (1994), 29–48. [3] E. Artin, Ein mechanisches system mit quasiergodischen bahnen, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170–175. [4] R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomol- ogy, preprint, available on www.staff.science.uu.nl/∼brugg103/algemeen/prpr.html, 2013. [5] U. Bunke and M. Olbrich, Gamma-cohomology and the selberg zeta function, J. reine angew. Math., 467 (1995), 199–219. [6] , Resolutions of distribution globalizations of Harish-Chandra modules and cohomology, J. reine angew. Math., 497 (1998), 47–81. [7] R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions, J. reine angew. Math., 492 (1997), 1–39. [8] R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Publ. Math., Inst. Hautes Etud.´ Sci., 50 (1979), 153–170. [9] P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quan- tum, Niels Bohr Institute, Copenhagen, 2008, ChaosBook.org. 2240 ANKE D. POHL

[10] C.-H. Chang and D. Mayer, The transfer operator approach to Selberg’s zeta function and modular and Maass wave forms for PSL(2, Z), Emerging applications of number theory (Min- neapolis, MN, 1996), Springer, New York, IMA Vol. Math. Appl., 109 (1999), 73–141. [11] , Eigenfunctions of the transfer operators and the period functions for modular groups, Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999), Amer. Math. Soc., Providence, RI, Contemp. Math., 290 (2001), 1–40. [12] , An Extension of the Thermodynamic Formalism Approach to Selberg’s Zeta Function for General Modular Groups, Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, pp. 523–562. [13] A. Deitmar and J. Hilgert, Cohomology of arithmetic groups with infinite dimensional coef- ficient spaces, Doc. Math., 10 (2005), 199–216 (electronic). [14] , A Lewis correspondence for submodular groups, Forum Math., 19 (2007), 1075–1099. [15] I. Efrat, Dynamics of the continued fraction map and the spectral theory of SL(2, Z), Invent. Math., 114 (1993), 207–218. [16] M. Fraczek, D. Mayer and T. Mühlenbruch, A realization of the Hecke algebra on the space of period functions for Γ0(n), J. Reine Angew. Math., 603 (2007), 133–163. [17] L. Ford, Automorphic Functions, Chelsea publishing company, New York, 1972. [18] D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487–521. [19] J. Hadamard, Les surfaces àcourbures opposées et leurs lignes géodésiques, J. Math. Pures et Appl., 4 (1898), 27–73; see also: Oeuvres Complètes de Jacques Hadamard, 2 (1698), 729–775. [20] J. Hilgert, D. Mayer and H. Movasati, Transfer operators for Γ0(n) and the Hecke operators for the period functions of PSL(2, Z), Math. Proc. Cambridge Philos. Soc., 139 (2005), 81–116. [21] J. Hilgert and A. Pohl, Symbolic Dynamics for the Geodesic Flow on Locally Symmetric Orbifolds of Rank One, Infinite dimensional harmonic analysis IV, 97C111, World Sci. Publ., Hackensack, NJ, 2009. [22] S. Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. [23] J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math., 127 (1997), 271–306. [24] J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math.(2), 153 (2001), 191–258. [25] B. Maskit, On Poincaré’s theorem for fundamental polygons, Advances in Math., 7 (1971), 219–230. [26] D. Mayer, On a ζ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195–203. [27] , On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311–333. [28] , The thermodynamic formalism approach to Selberg’s zeta function for PSL(2, Z), Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55–60. [29] D. Mayer, Transfer Operators, the Selberg Zeta Function and the Lewis-Zagier Theory of Periodic Functions, Hyperbolic geometry and applications in quantum chaos and cosmology, 146–174, London Math. Soc. Lecture Note Ser., 397, Cambridge Univ. Press, Cambridge, 2012. [30] D. Mayer, T. Mühlenbruch, and F. Strömberg, The transfer operator for the Hecke triangle groups, Discrete Contin. Dyn. Syst., 32 (2012), 2453–2484. [31] T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups, Ergodic Theory Dynam. Systems, 17 (1997), 1147–1181. [32] M. Möller and A. Pohl, Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant, Ergodic Theory Dynam. Systems, 33 (2013), 247–283. [33] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn., 2 (2008), 581–627. [34] A. Pohl, Symbolic Dynamics for the Geodesic Flow on Locally Symmetric Good Orbifolds of Rank One, 2009, dissertation, Universität Paderborn, available at http://d-nb.info/gnd/137984863. [35] , Ford fundamental domains in symmetric spaces of rank one, Geom. Dedicata, 147 (2010), 219–276. [36] , A dynamical approach to Maass cusp forms, J. Mod. Dyn., 6 (2012), 563–596. SYMBOLIC DYNAMICS 2241

[37] , Odd and Even Maass Cusp Forms for Hecke Triangle Groups, and the Billiard Flow, arXiv:1303.0528, 2013. [38] , Period Functions for Maass Cusp Forms for Γ0(p): A transfer Operator Approach, International Mathematics Research Notices, 14 (2013), 3250–3273. [39] M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. in Math., 85 (1991), 161–192. [40] J. Ratcliﬀe, Foundations of Hyperbolic Manifolds, second ed., Graduate Texts in Mathematics, 149. Springer, New York, 2006. [41] D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (2002), 887–895. [42] C. Series, Symbolic dynamics for geodesic ﬂows, Acta Math., 146 (1981), 103–128. [43] , The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69–80. [44] L. Vulakh, Farey polytopes and continued fractions associated with discrete hyperbolic groups, Trans. Amer. Math. Soc., 351 (1999), 2295–2323. Received May 2013; revised August 2013. E-mail address: [email protected]