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

We construct cross sections for the geodesic flow on the orbifolds $\Gamma\backslash H$ 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 $\Gamma$ 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.


Introduction
The construction and use of symbolic dynamics in various setups has a long history. It goes back to work of Hadamard [Had98] 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 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 PSL 2 (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 [Ser85], Series, based on work by Artin [Art24], provided a symbolic dynamics for the geodesic flow on the modular surface PSL 2 (Z)\H which relates this flow to the Gauss map x in a way that periodic geodesics on PSL 2 (Z)\H correspond to finite orbits of F . Here, H denotes the hyperbolic plane. Mayer [May76,May90,May91] investigated the transfer operator (weighted evolution operator) with parameter β ∈ C associated to F , which here takes the form He found a Banach space B such that for Re β > 1 2 , the transfer operator L F,β acts on B, is nuclear of order 0, and has a meromorphic extension L F,β to the whole complex β-plane with values in nuclear operators of order 0 on B. He showed that the Selberg zeta function Z is represented by the product of the Fredholm determinants of ± L F,β : (1) Z(β) = det(1 − L F,β ) · det(1 + L F,β ).
On the other side, Lewis and Zagier [LZ01] (see Lewis [Lew97] for even Maass cusp forms) proved that Maass cusp forms for PSL 2 (Z) with eigenvalue β(1 − β) are linearly isomorphic to real-analytic functions f on R >0 which satisfy the functional equation (2) f (x) = f (x + 1) + (x + 1) −2β f x x + 1 and are of strong decay at 0 and ∞. Functions of this kind are called period functions for PSL 2 (Z). Using this characterization of Maass cusp forms, Chang and Mayer [CM99] as well as Lewis and Zagier [LZ01] deduced that even resp. odd Maass cusp forms for PSL 2 (Z) are linearly isomorphic to the ±1-eigenspaces of Mayer's transfer operator. Efrat [Efr93] proved this earlier on a spectral level, that is, the parameters β for which there exist a ±1-eigenfunction of L F,β are just the spectral parameters of even resp. odd Maass cusp forms. In addition, Bruggeman [Bru97] provided a hyperfunction approach to the period functions for PSL 2 (Z).
By using representations and a hyperfunction approach, Deitmar and Hilgert [DH07] could induce the period functions for PSL 2 (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 [CM01a,CM01b] 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 [HMM05] and [FMM07] 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 [Pol91] considered the transfer operator family associated to the Series symbolic dynamics for cocompact lattices. 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 where h is a function compensating for multiple codings. • Fried [Fri96] developed a symbolic dynamics for the billiard flow on Λ\H for triangle groups Λ ⊂ PGL 2 (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 [Mor97] used a modified Bowen-Series symbolic dynamics for a wide class of non-cocompact lattices and investigated the associated transfer operator families. Here the problem with multiple codings arises again. • Mayer et. al. [MS08,MMS12] 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. However, 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 [BLZ13] (see [BO95,BO98,DH05] 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 fundamental 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 [MP13], 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 PSL 2 (Z). For the Hecke congruence subgroups Γ 0 (p), p prime, this transfer operator approach to Maass cusp forms is performed in [Poh12b]. Finally, in [Poh12a] it is shown for all Fuchsian lattices which are admissible here, including a 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 [MP13], 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 Fredholm determinant as in (1). For the modular group, these results reproduce Mayer's transfer operator.
In [Poh13] 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 [Efr93] 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 below 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 below we present a sketch the construction of the symbolic dynamics, discrete dynamical system and transfer operator families. Then Sections 4-9 below contain a detailed proof of the construction, and in Section 10 below we briefly discuss the structure of the arising transfer operators.

Preliminaries
2.1. Two-dimensional hyperbolic good orbifolds. We use the upper half plane H := {z ∈ C | Im z > 0} with the Riemannian metric given by the line element ds 2 = y −2 (dx 2 + dy 2 ) as model for the two-dimensional real hyperbolic space. The associated Riemannian metric will be denoted by d H . We identify the group of orientation-preserving Riemannian isometries with PSL 2 (R) via the well-known left action Let Γ be a Fuchsian group, that is, a discrete subgroup of PSL 2 (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 class of the corresponding geometric entity 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 γ v denote the (unit speed) geodesic on H determined by γ ′ v (0) = v. The (unit speed) geodesic flow on H will be denoted by Φ, hence 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 Φ := π • Φ • (id ×π −1 ) : R × SY → SY.
Here, π −1 is an arbitrary section of π. One easily sees that Φ does not depend on the choice of π −1 . Throughout, we use the convention that if e denotes an element belonging in some sense to H (like geodesics on H, subsets of H), then e denotes the corresponding element belonging to Y . The one-point compactification of the closure of H in C will be denoted by H g , It is homeomorphic to the geodesic compactification of H. The action of PSL 2 (R) extends continuously to the boundary ∂ g H = R ∪ {∞} of H in H g .
We let R := R ∪ {±∞} denote the two-point compactification of R and extend the ordering of R to R by the definition −∞ < a < ∞ for each a ∈ R.
Let I be an interval in R. A curve α : I → H is called a geodesic arc if α can be extended to a geodesic. The image of a geodesic arc is called a geodesic segment. If α is a geodesic, then α(R) is called a complete geodesic segment. A geodesic segment is called non-trivial if it contains more than one element. The geodesic segments of the geodesics on H are the semicircles centered on the real line and the vertical lines.
If α : I → H is a geodesic arc and a < b are the boundary points of I in R, then the points If α(a), α(b) ∈ ∂ g H, it will always be made clear whether we refer to a geodesic segment or an interval in R. Let U be a subset of H. The closure of U in H is denoted by U or cl(U ), its boundary is denoted by ∂U , and its interior is denoted by U • . To increase clarity, we denote the closure of a subset V of H g in H g by V g or cl H g V . Moreover, we set For a subset W ⊆ R let int R (W ) denote the interior of W in R and ∂ R W the boundary of W in R. If X is a subset of ∂ g H, then int g (X) denotes the interior of X in ∂ g H. If X ⊆ R, then int g (X) = int R (X If, in addition, F is connected, then it is a fundamental domain for Γ in H. The Fuchsian group 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 which implies that c = 0. Let | · | denote the Euclidean norm on C. The isometric sphere of g is defined as The exterior of I(g) is the set is its interior. If the representative of g is chosen such that c > 0, then the isometric sphere I(g) is the complete geodesic segment with endpoints − d c − 1 c and − d c + 1 c . If z 0 = x 0 + iy 0 is an element of I(g), then the geodesic segment (z 0 , ∞) is contained in ext I(g), and the geodesic segment (x 0 , z 0 ) belongs to int I(g). Moreover, is a partition of H into convex subsets such that ∂ ext I(g) = I(g) = ∂ int I(g). Let denote the common part of the exteriors of all isometric spheres of Γ. We call a point z ∈ ∂ g H 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 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., [For72,Kat92,Rat06,Poh10,Poh09].
Proposition 2.1. Let Γ be a geometrically finite Fuchsian group that has ∞ as cuspidal point. Then, for any r ∈ R, the set is a convex fundamental domain for Γ in H with finitely many sides.
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 γ ′ (t) ∈ CS, then we say that γ intersects CS in t. Further, γ is said to intersect CS infinitely often in the future if we find a sequence (t n ) n∈N with t n → ∞ as n → ∞ and γ ′ (t n ) ∈ CS for all n ∈ N. Analogously, γ is said to intersect CS infinitely often in the past if we find a sequence (t n ) n∈N with t n → −∞ as n → ∞ and γ ′ (t n ) ∈ CS for all n ∈ N. Let µ be a measure on the space of geodesics on Y . The set CS is called a cross section (w.r.t. µ) for the geodesic flow Φ if (C1) µ-almost every geodesic γ on Y intersects CS infinitely often in the past and in the future, (C2) each intersection of γ and CS is discrete in time: if γ ′ (t) ∈ CS, then there is We call a subset U of Y a totally geodesic suborbifold of Y if π −1 ( U ) is a totally geodesic submanifold of H. Let pr : SY → Y denote the canonical projection on base points. If pr( CS) is a totally geodesic suborbifold of Y and CS does not contain elements tangent to pr( CS), then CS automatically satisfies (C2).
Suppose that CS is a cross section for Φ. If, in addition, CS satisfies the property that each geodesic intersecting CS at all intersects it infinitely often in the past and in the future, then CS will be called a strong cross section, otherwise a weak cross section. Clearly, every weak cross section contains a strong cross section.
The first return map of Φ w. r. t. the strong cross section CS is the map where v ∈ SH with π(v) = v, π(γ v ) = γ v , and is the first return time ofv or γ v . This definition requires that t 0 = t 0 ( v) exists for each v ∈ CS, which will indeed be the case in our situation. For a weak cross section CS, the first return map can only be defined on a subset of CS. In general, this subset is larger than the maximal strong cross section contained in CS. Suppose that CS is a strong cross section and let Σ be an at most countable set. Decompose CS into a disjoint union α∈Σ CS α . To each v ∈ CS we assign the (two-sided infinite) coding sequence (a n ) n∈Z ∈ Σ Z defined by 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 σ (a n ) n∈Z k := a k+1 .
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. Obviously, the diagram 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. Let CS ′ be a set of representatives for the cross section CS, that is, CS ′ is a subset of SH such that π| CS ′ is a bijection CS ′ → CS. Relative to CS ′ , we define the map τ : where v := (π| CS ′ ) −1 (v). For some cross sections CS it is possible to choose CS ′ in such a way that τ is a bijection between CS and some subset DS of R × R. In this case the dynamical system ( CS, R) is conjugate to ( DS, F ) by τ , where F := τ • R • τ −1 is the induced selfmap on DS (partially defined if CS is only a weak cross section). Moreover, to construct a symbolic dynamics for Φ, one can start with a decomposition of DS into pairwise disjoint subsets D α , α ∈ Σ.
Finally, let (Λ, σ) be a symbolic dynamics with alphabet Σ. Suppose that we have a map i : Λ → DS for some DS ⊆ R such that i (a n ) n∈Z depends only on (a n ) n∈N0 , a (partial) selfmap F : DS → DS, and a decomposition of DS into a disjoint union α∈Σ D α such that F i((a n ) n∈Z ) ∈ D α ⇔ a 1 = α for all (a n ) n∈Z ∈ Λ. Then F , more precisely the triple F, i, (D α ) 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.

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 [HP08,MP13,Poh12b,Poh12a]. As we will see already in this sketch, the cuspidal point ∞ has 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.
c d ∈ Γ Γ ∞ with c chosen positive, and consider its isometric sphere Its point of maximal height 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 K, that is, if I ∩ ∂K contains a submanifold of H of codimension 1. If the isometric sphere I is relevant, then I ∩ ∂K is called its relevant part. An endpoint of the relevant part of a relevant isometric sphere is called a vertex of K.
From now on we impose the following condition on Γ: For each relevant isometric sphere, its summit is contained in ∂K but not a vertex of K.
The relevant isometric spheres are I(S) and its T k n -translates, k ∈ Z. (ii) Let The isometric spheres are the sets where c ∈ N and d ∈ Z. The set K is given by (see Figure 2) (iii) Let S := 0 −1 1 0 and T := [ 1 4 0 1 ], and denote by Γ the subgroup of PSL 2 (R) which is generated by S and T . The set K is indicated in Figure 3.  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 K. For that we note that there are two kinds of vertices of K. Let v be a vertex of K. If v ∈ H, then we call v an inner vertex, otherwise v is called an infinite vertex. Now we construct a family G of geodesic segments in the following way: Whenever v is an infinite vertex of K, then the (complete) geodesic segment (v, ∞) belongs to this family. Moreover, whenever 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. 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. Then is a weak cross section for the geodesic flow. By eliminating a certain set of vectors from it, we will also construct a strong cross section.
Construction of a set of representatives. The family G of geodesic segments induces a tesselation of K into hyperbolic triangles, quadrilateral and strips. We call these tesselation elements 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 CS ′ pre 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 CS ′ pre 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.
Isometric fundamental regions allow to define cycles of their sides as in the Poincaré Fundamental 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 ∂ g H. Even more, all vertices are in Γ.∞. At this point, Condition (A) is essential. We call these polyhedron cells in H. All the arising cells in H have two vertical sides, and all the non-vertical sides are Γ-translates of some vertical sides of some cells in H. The translation elements can be determined by an algorithms from the generators of the relevant isometric spheres.
Lifting this construction to the unit tangent bundle SH and using this to redistribute CS ′ pre allows us to find a set of representatives CS ′ such that CS ′ is the disjoint union CS ′ = α∈A CS ′ α for some finite index set A such that for each α ∈ A there exists a vertical side (a complete geodesic segment) of some cell in H such that CS ′ α is the set of unit tangent vectors based on this side and pointing into the cell.
The definition of precells in H and the construction of cells in H from precells is based on ideas in [Vul99]. Our construction differs from Vulakh's in three important aspects: We define three kinds of precells in H of which only one are precells in sense of Vulakh. Finally, contrary to Vulakh, we extend the considerations to precells and cells in unit tangent bundle.
Example 3.2. The set F n := z ∈ H |z| > 1, | Re z| < λn 2 is an isometric fundamental domain for the Hecke triangle group G n (see Figure 4). Hecke triangle groups have only one precell in H, up to equivalence under It is indicated in Figure 5.   . Lifting to SH and defining CS ′ for G 5 .
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 CS ′ . Whenever L is the side of some Γ-translate of some cell in H and L ′ is the set of unit tangent vectors based on that side such that all vectors are pointing into this Γ-translate or all vectors pointing out of the Γ-translate, then there is a unique pair (α, g) ∈ A × Γ such that L ′ = g. CS ′ α . Both, α and g can be determined algorithmically. Further, Γ. CS ′ is just the union of all such L ′ .
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 where α 1 , α 2 ∈ A, I α1 , I α2 , J α1 , J α2 are intervals, and g ∈ Γ depends on α 1 , α 2 .
The associated transfer operator L F,β with parameter β ∈ C is then  the forward-direction of the geodesic flow, then the discrete dynamical system for G n is F : given by the bijections (g k .0, g k .∞) G n .∞ → R >0 G n .∞, x → g −1 k .x for k = 1, . . . , n − 1 with g k := U k n S. The associated transfer operator L F,β with parameter β ∈ C is then defined on functions f : R >0 G n .∞ → C. For the modular group PSL 2 (Z) = G 3 the transfer operator becomes The 1-eigenfunctions of this transfer operator are characterized by the functional equation (2). In [MP13], 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.
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 1 } and {α 2 } from the domain and range of F , we will shrink the cross section CS and the set of representatives CS ′ in an algorithmic way to deduce a new cross section CS red with set of representatives CS ′ red . In terms of notions introduced only 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 from [HP08,Poh12b,Poh12a] 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).
Proof. Proposition 3.4 states that F is a fundamental domain for Γ in H. Its shape and side-pairings yield that it is an isometric fundamental domain. Therefore, the isometric sphere I(g 1 ) is relevant and s 1 is its relevant part. The summit of I(g 1 ) is s := 3+i 11 . One easily calculates that s ∈ int I(g 2 ). Thus, Γ does not satisfy (A). In [Vul99], Vulakh states that each geometrically finite subgroup of PSL 2 (R) for which ∞ is a cuspidal point satisfies (A). The previous example shows that this statement is not right. This property is crucial for the results in [Vul99]. Thus, Vulakh's constructions do not apply to such a huge class of groups as he claims.

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.
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 K := g∈Γ Γ∞ ext I(g).
4.1. The structure of K. We start with a short consideration of the vertex structure of K.
The set of all isometric spheres of Γ need not be locally finite. For example, in the case of the modular group PSL 2 (Z), each neighborhood of 0 in H g 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 K has no accumulation points. Moreover, if v is an inner vertex of K, then there are exactly two distinct relevant isometric spheres such that v is a common 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 K. For this let Proposition 4.1.
Let v be a one-sided infinite vertex of K. Then there exists a unique one-sided infinite vertex w of K such that the strip pr −1 ∞ ( v, w ) ∩ H is contained in K. In particular, pr −1 ∞ ( v, w ) does not intersect any isometric sphere in H, and, of all vertices of K, pr −1 ∞ ( v, w ) contains only v and w. For any one-sided infinite vertex v of K, we call the interval v, w from Proposition 4.1 a boundary interval, and we call w the one-sided infinite vertex adjacent to v. Boundary intervals will be needed for the definition and investigation of strip precells defined below.  Let v be a vertex of K. Suppose first that v is an inner vertex or a two-sided infinite vertex. Then there are (exactly) two relevant isometric spheres I 1 , I 2 with relevant parts [a 1 , v] resp. [v, b 2 ]. Let s 1 resp. s 2 be the summit of I 1 resp. I 2 . By Condition (A), the summits s 1 and s 2 do not coincide with v.
If v is a two-sided infinite vertex, then define A 1 to be the hyperbolic triangle 1 with vertices v, s 1 and ∞, and define A 2 to be the hyperbolic triangle with vertices v, s 2 and ∞. The sets A 1 and A 2 are the precells in H attached to v. Precells arising in this way are called cuspidal.
If v is an inner vertex, then let A be the hyperbolic quadrilateral with vertices s 1 , v, s 2 and ∞. The set A is the precell in H attached to v. Precells that are constructed in this way are 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 vertex w other than v such that pr −1 ∞ ( v, w ) does not contain vertices other than v and w (see Proposition 4.1). Let s be the summit of I.
Define A 1 to be the hyperbolic triangle with vertices v, s and ∞, and define A 2 to be the vertical strip pr −1 ∞ ( v, w ) ∩ H. The sets A 1 and A 2 are the precells in H attached to v. The precell A 1 is called cuspidal, and A 2 is called a strip precell.
Example 4.5. The precells in H of the group Γ from Example 3.1(iii) are up to Γ ∞ -equivalence one strip precell A 1 and two cuspidal precells A 2 , A 3 as indicated in Figure 10. Figure 10. Precells in H of Γ.
Condition (A) yields that the relation between different precells and between the precells and K are well-structured.
Proof. We use the notation from Definition 4.3 to discuss the relation between precells and K.
Suppose first that A is a non-cuspidal precell in H attached to the inner vertex v. Condition (A) implies that the summits s 1 and s 2 are contained in the relevant parts of the relevant isometric spheres I 1 and I 2 , respectively. Therefore they are contained in ∂K.
By Proposition 4.1, A is contained in K. From these observations, the statements follow easily.
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.  Let Λ be a Fuchsian group. A subset F of H is called a closed fundamental region for Λ in H if F is closed and F • is a fundamental region for Λ in H. If, in addition, F • is connected, then F is said to be a closed fundamental domain for Λ in H. Note that if F is a non-connected fundamental region for Λ in H, then F can happen to be a closed fundamental domain.
Let {A j | j ∈ J} be a family of real submanifolds (possibly with boundary) of H or H g , and let n := max{dim A j | j ∈ J}. We call the union j∈J A j essentially disjoint if for each i, j ∈ J, i = j, the intersection A i ∩ A j is contained in a real submanifold (possibly with boundary) of dimension n − 1.
Theorem 4.8. There exists a set {A j | j ∈ J}, indexed by J, of precells in H such that the (essentially disjoint) union j∈J A j is a closed fundamental region for Γ in H. The set J is finite and its cardinality does not depend on the choice of the specific set of precells. The set {A j | j ∈ J} can be chosen such that j∈J A j is a closed fundamental domain for Γ in H. In each case, the (disjoint) union j∈J A • j is a fundamental region for Γ in H.
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 infinite vertex v of K. Let λ > 0 be the unique positive number such that From Lemma 4.7 and Proposition 4.6 one easily deduces that Then Proposition 4.6(ii) implies that h k .A k = A j k , and in turn h k and j k are unique. We will show that the map ϕ : , we find that h k g = id and j = j k . Hence, ϕ is surjective. It follows that #K = #J.
It remains to show that the disjoint union P := k∈K A • k is a fundamental region for Γ in H. Obviously, P is open and contained in F • . This shows that P satisfies (F1) and (F2). Since (A • ) = A for each precell in H and K is finite, it follows that Hence, P satisfies (F3) as well, and thus it is a fundamental region for Γ in H.
Each set A := {A j | j ∈ J}, indexed by J, of precells in H with the property that F := j∈J A j is a closed fundamental region is called a basal family of precells in H or a family of basal precells in H. If, in addition, F is connected, then A is called a connected basal family of precells in H or a connected family of basal precells in H.
The proof of Theorem 4.8 shows the following statements. Let Corollary 4.10. Let A be a basal family of precells in H.
4.3. The tesselation of H by basal families of precells. The following proposition is crucial for the construction of cells in H from precells in H. Note that the element g ∈ Γ Γ ∞ in this proposition depends not only on A and b but also on the choice of the basal family A of precells in H. In this section 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 ∈ A be a basal precell that is not a strip precell, and suppose that b is a non-vertical side of A. Then there is a unique element g ∈ Γ Γ ∞ such that b ⊆ I(g) and g.b is the non-vertical side of some basal precell 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 g.b ⊆ g.I(g) = I(g −1 ), which implies that g.b is a non-vertical side.
Let h ∈ Γ Γ ∞ be any generator of I, let s be the summit of I and v the vertex of K that A is attached to. Then b = [v, s]. Further, b is contained in the relevant part of I = I(h). By Lemma 4.7, the set h.b = [h.v, h.s] is contained in the relevant part of the relevant isometric sphere I(h −1 ), the point h.v is a vertex of K and h.s is the summit of I(h −1 ). Thus, there is a unique precell A h with non-vertical side h.b. By Corollary 4.10, there is a unique basal precell A ′ and a unique m ∈ Z such that . Clearly, also g := t −m λ 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 = t n λ h.
This shows the uniqueness.
The basal precell A ′ cannot be a strip precell, since it has a non-vertical side. Finally, A is cuspidal if and only if v is an infinite vertex. This is the case if and only if gv is an infinite vertex, which is equivalent to A ′ being cuspidal. This completes the proof.
Lemma 4.12. Let A be a precell in H. Suppose that S is a vertical side of A. Then there exists a precell A ′ in H such that S is a side of A ′ and A ′ = A. In this case, S is a vertical side of A ′ .
Proposition 4.13. Let A 1 , A 2 be two precells in H and let g 1 , g 2 ∈ Γ. Suppose that g 1 .A 1 ∩ g 2 .A 2 = ∅. Then we have either g 1 .A 1 = g 2 .A 2 and g 1 g −1 2 ∈ Γ ∞ , or g 1 .A 1 ∩ g 2 .A 2 is a common side of g 1 .A 1 and g 2 .A 2 , or g 1 .A 1 ∩ g 2 .A 2 is a point which is the endpoint of some side of g 1 .A 1 and some side of g 2 .A 2 . If S is a common side of g 1 .A 1 and g 2 .A 2 , then g −1 1 .S is a vertical side of A 1 if and only if g −1 2 .S is a vertical side of A 2 . Proof. W.l.o.g. g 1 = id. Let A be a basal family of precells in H. Corollary 4.10 shows that we may assume that A 1 ∈ A. Let S 1 = [a 1 , ∞] and S 2 = [a 2 , ∞] be the vertical sides of A 1 . If A 1 has non-vertical sides, let these be A is a fundamental region for Γ in H (see Theorem 4.8). Therefore, g 2 h = id and, by Proposition 4.6(ii), and the argument from above shows that A ′ 1 = g 2 .A 2 and g 2 ∈ Γ ∞ . From this it follows that S 1 is a vertical side of A 2 . If It remains the case that g 2 .A 2 intersects A 1 is an endpoint v of some side of A 1 . By symmetry of arguments, v is an endpoint of some side of A 2 . This completes the proof.
Corollary 4.14. Let A be a basal family of precells in H. Then is a tesselation of H which satisfies in addition the property that if g 1 .A 1 = g 2 .A 2 , then g 1 = g 2 and A 1 = A 2 .
Proof. Let F := {A | A ∈ A}. Theorem 4.8 states that F is a closed fundamental region for Γ in H, hence g∈Γ g.F = H. This proves (T1). Property (T2) follows directly from Proposition 4.13. Now let (g 1 ,

Cells in H
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. 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 Poincaré Fundamental Polyhedron Theorem are defined (see e.g. [Mas71]). These are certain finite sequences of pairs (A, h) ∈ A × Γ Γ ∞ such that each cycle is determined up to cyclic permutation by any pair which belongs to it. Moreover, if (A, h A ) is an element of some cycle, then One of the crucial properties 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.

Cycles in
The definition of precells shows that A is attached to a unique (inner) vertex v of K, and A is the unique precell attached to v. Therefore we set A(v) := A. Further, A has two non-vertical sides b 1 and b 2 . Let {k 1 (A), k 2 (A)} be the two elements in Γ Γ ∞ given by Proposition 4.11 such that b j ∈ I(k j (A)) and k j (A).b j is a non-vertical side of some basal precell. Necessarily, the isometric spheres I(k 1 (A)) and I(k 2 (A)) are different, therefore k 1 (A) = k 2 (A). The set {k 1 (A), k 2 (A)} is uniquely determined by Proposition 4.11, the assignment A → k 1 (A) clearly depends on the enumeration of the non-vertical sides of A. Now w := k j (A).v is an inner vertex. Let A(w) be the (unique non-cuspidal) basal precell attached to w. Since one non-vertical side of A(w) is k j (A).b j , which is contained in the relevant isometric sphere I(k j (A) −1 ), and k j (A) −1 k j (A).b j = b j is a non-vertical side of some basal precell, namely of A, one of the elements in Γ Γ ∞ assigned to A(w) by Proposition 4.11 is k j (A) −1 .
Construction 5.1. Let A ∈ A be a non-cuspidal precell and suppose that A = A(v) is attached to the vertex v of K. We assign to A two sequences (h j ) of elements in Γ Γ ∞ using the following algorithm: (step 1) Let v 1 := v and let h 1 be either k 1 (A) or k 2 (A). Set g 1 := id, g 2 := h 1 and carry out (step 2).
Example 5.2. Recall the Hecke triangle group G n and its basal family A = {A} of precells in H from Example 3.2. The two sequences assigned to A are (U n ) n j=1 and U −1 n n j=1 .
The statements of Propositions 5.3-5.7 follow as in the Poincaré Fundamental Polyhedron Theorem, using Lemma 4.7 and elementary convex geometry. For this reason we omit the detailed proofs.
(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 (a j ) j=1,...,k and (b j ) j=1,...,k be the two sequences assigned to A. Then they are inverse to each other in the following sense: For each j = 1, . . . , k we have Further, both unions are essentially disjoint, and B is the polyhedron with the (pairwise distinct) vertices (in this order) Let h A be one of the elements in Γ Γ ∞ assigned to A by Proposition 4.11. Let (h j ) j=1,...,k be the sequence assigned to A by Construction 5.1 with h 1 = h A . For j = 1, . . . , k set g 1 := id and g j+1 := h j g j . Then the Let A ∈ A be a cuspidal precell. Suppose that b is the non-vertical side of A and let h A be the element in Γ Γ ∞ assigned to A by Proposition 4.11. Let A ′ be the (cuspidal) basal precell with non-vertical side h A .b. Then the (finite) sequence (iii) Recall the group Γ and the basal family Proposition 5.6. Let A ∈ A be a non-cuspidal precell in H and suppose that h A is one of the elements in Γ Γ ∞ assigned to A by Proposition 4.11. Let . . , k} and define the sequence (A ′ l , a l ) l=1,...,k by Then a 1 = h j is one of the elements in Γ Γ ∞ assigned to A j by Proposition 4.11 Let A be a cuspidal basal precell in H. Suppose that g is the element in Γ Γ ∞ assigned to A by Proposition 4.11 and let (A, g), (A ′ , g −1 ) be the cycle in A × Γ determined by (A, g). Define The set B(A) is well-defined because g is uniquely determined.
Let A be a non-cuspidal basal precell in H and fix an element h A in Γ Γ ∞ assigned to A by Proposition 4.11. Let (A j , h j ) j=1,...,k be the cycle in A × Γ determined by (A, h A ). For j = 1, . . . , k set g 1 := id and g j+1 := h j g j . Set 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 A ′ of precells in H, we will call the first ones A-cells and the latter ones A ′ -cells.
Example 5.9. Recall the Example 5.5. For the Hecke triangle group G 5 , Figure 11 shows the cell assigned to the family A = {A} from Example 3.2. For the group PΓ 0 (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. . Figure 13. The family of cells in H assigned to A for Γ.
In the series of the following 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 A be a non-cuspidal basal precell in H. Suppose that h A is an element in Γ Γ ∞ assigned to A by Proposition 4.11 and let (A j , h j ) j=1,...,k be the cycle in A × Γ determined by (A, h A ). For j = 1, . . . , k set g 1 := id and g j+1 := h j g j . Then the following assertions hold true.
Proof. By Proposition 5.3, B(A) is the polyhedron with vertices (in this order) Since each of its sides is a complete geodesic segment, B(A) is convex. This shows (i). The statement (ii) follows from the proof of Proposition 5.3.
To prove (iii), fix j ∈ {1, . . . , k} and recall from Proposition 5.6 the cycle . . , k set c 1 := id and c l+1 := a l c l . Then This immediately implies that the side Proposition 4.13 implies that A = A j and h = g −1 j . If b is a point, then b = z must be the endpoint of some side of g −1 l .A l . From z ∈ B(A) • it follows that z = v. Now the previous argument applies.
To show the uniqueness of j ∈ {1, . . . , k} with A = A j and h = g −1 j , suppose that there is p ∈ {1, . . . , k} with A = A p and h = g −1 p . Then g j = g p . By Proposition 5.3(iv) j = p. The remaining parts of (iv) follow from (iii).
Proposition 5.11. Let A be a cuspidal basal precell in H which is attached to the vertex v of K. Suppose that g is the element in Γ Γ ∞ assigned to A by Proposition 4.11. Let (A, g), (A ′ , g −1 ) be the cycle in A × Γ determined by (A, g). Then we have the following properties.
Then either h = id and A = A or h = g −1 and A = A ′ . In particular, A is cuspidal and h.B( A) = B(A).
The following proposition is implied by following through the glueing procedure from precells to cells. The details are easily seen by straightforward deductions.
(1) Let A be a non-cuspidal basal precell in H. Suppose that (h j ) j=1,...,k is a sequence in Γ Γ ∞ assigned to A by Construction 5.1. For j = 1, . . . , k set g 1 := id and g j+1 := h j g j . Let A ′ be a basal precell in H and g ∈ Γ such that B(A) ∩ g.B(A ′ ) = ∅. Then we have the following properties.
Suppose that h ∈ Γ Γ ∞ is the element assigned to A by Proposition 4.11. Let A ′ be a basal precell in H and g ∈ Γ such that we have Then the following assertions hold true.
(3) Let A be a basal strip precell in H. Let A ′ be a basal precell in H and g ∈ Γ such that B(A) ∩ g.B(A ′ ) = ∅. Then the following statements hold.
Corollary 5.15. The family of Γ-translates of all cells provides a tesselation of H. In particular, if B is a cell in H and S a side of B, then there exists a pair

The base manifold of the cross sections
Let Γ be a geometrically finite subgroup of PSL 2 (R) of which ∞ is a cuspidal point and which satisfies (A), and suppose that there are relevant isometric spheres. In this section we define a set CS which will turn out, in Section 8.1 below, 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 will already see that CS satisfies (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 on pr( CS) but not tangent to it. To achieve this, we start at the other end. We fix a basal family A of precells in H and consider the family B of cells in H assigned to A. 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 (future) 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 A. For future purposes we already define the sets NC(B) and bd(B) and show that also these are independent of the choice of A. Throughout let 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).
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).
Proof. Since BS is Γ-invariant by definition, we see that BS = π −1 ( BS). Therefore, BS is a totally geodesic suborbifold of Y of codimension one. Moreover, CS = π −1 ( CS) and hence CS is indeed the set of unit tangent vectors based on BS but not tangent to BS. Finally, pr( CS) = BS. Therefore, CS satisfies (C2).
Remark 6.5. Let NIC be the set of geodesics on Y of which at least one endpoint is contained in π(bd). Here, π : H g → Γ\H g denotes the extension of the canonical projection H → Y to H g . 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.
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 is for "boundary" in sense of geodesic boundary, and bd(B) is the geodesic boundary of the cell B. Moreover, which will become more sense in Section 8.2 (see Remark 8.32), NC stands for "not coded" and NC(B) for "not coded due to the cell B". Finally, NIC is for "not infinitely often coded".

Precells and cells in SH
Let Γ be a geometrically finite subgroup of PSL 2 (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. Recall the projection pr : SH → H on base points.
Remark 7.3. Let A be a precell in H and A the corresponding precell in SH. Since A is a convex polyhedron with non-empty interior, pr( A) is the precell in H to which A corresponds.
Lemma 7.4. Let A 1 , A 2 be two different precells in H. Then the precells A 1 and A 2 in SH are disjoint.
Proof. This is an immediate consequence of Proposition 4.6(ii) and Definition 7.2.  We remark that the visual boundary and the visual closure of a precell A in SH is a proper subset of the topological boundary resp. closure of A in SH. Proposition 7.7. Let {A j | j ∈ J} be a basal family of precells in H and let { A j | j ∈ J} be the set of corresponding precells in SH. Then there is a fundamental set F for Γ in SH such that indexed by J, of precells in SH such that (4) holds for some fundamental set F for Γ in SH, then the family {pr( A j ) | j ∈ J} is a basal family of precells in H.
Proof. This is a slight extension of a similar statement in [HP08].
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 { A k | k ∈ K} is a set of precells in SH, indexed by K, such that (4) holds for some fundamental set F for Γ in SH, then #K = m.  Proof. This is an obvious consequence of the definition of cycles.
Definition 7.12. Let A be a non-cuspidal basal precell in H and let h A be an element in Γ Γ ∞ assigned to A by Proposition 4.11. Suppose that (A j , h j ) j=1,...,k is the cycle in A × Γ determined by (A, h A ). We set Example 7.13. Recall Example 5.5. For the Hecke triangle group G n and its basal precell A in H we have A = A 2 and hence cyl(A) = 1. In contrast, the basal precell A(v 1 ) in H of the congruence group PΓ 0 (5) appears only once in the cycle in A × PΓ 0 (5) and therefore cyl A(v 1 ) = 3.
Construction and Definition 7.14.
which is possible by Proposition 7.7. For each basal precell A ∈ A and each z ∈ F let E z (A) denote the set of unit tangent vectors in F ∩ vc( A) based at z. Fix any enumeration of the index set J of A, say J = {j 1 , . . . , j k }. For z ∈ F and l ∈ {1, . . . , k} set for A ∈ A. Recall from Proposition 7.7 that pr( F ) = F . Thus, Suppose that A is a non-cuspidal precell in H and let v be the vertex of K to which A is attached. Let (A j , h j ) j=1,...,k be the cycle in A × Γ determined by (A, h A ). For j = 1, . . . , k set g 1 := id and g j+1 := h j g j , and let s j be the summit of I(h j ). Further, for convenience, set h 0 := h k and s 0 := s k . In the following we partition certain F (A j ) into k subsets. More precisely, we partition each element of the set { F(A j ) | j = 1, . . . , k} into k subsets. Let j ∈ {1, . . . , cyl(A)}.
j,z := F z (A j ) and W where the first part (l) of the subscript of A l,l−j+1 is calculated modulo cyl(A) and the second part (l − j + 1) is calculated modulo k.
Suppose that A is a cuspidal precell in H. Let (A 1 , h 1 ), (A 2 , h 2 ) be the cycle in A × Γ determined by (A, h A ). Set g := h 1 = h A , g 1 := id and g 2 := h 1 = g. Suppose that v is the vertex of K to which A is attached, and let s be the summit of I(g). Let j ∈ {1, 2}. We partition F (A j ) into three subsets as follows.
For Then we define Suppose that A is a strip precell in H. Let v 1 , v 2 be the two (infinite) vertices of K to which A is attached and suppose that v 1 < v 2 . We partition F (A) into two subsets as follows.
For z ∈ A • we pick any partition of F z (A) into two non-empty disjoint subsets W tangent vectors in black belong to A 1,1 , those in very light grey to A 1,2 , those in light grey to A 1,3 , those in middle grey to A 1,4 , and those in dark grey to A 1,4 . The second figure in Figure 15 shows the cell B 1 (A,  Proof. Construction 7.14 picks a fundamental set F for Γ in SH and chooses a family P := { F z (A) | z ∈ F , A ∈ A} of subsets of it. Since the union in (5) is disjoint, P is a partition of F . Recall the notation from Construction 7.14. One considers the family The elements of P 1 are pairwise disjoint and each element of P is contained in P 1 . Hence, P 1 is a partition of F . The next step is to partition each element of P 1 into a finite number of subsets. Thus, F is partitioned into some family P 2 of subsets of F . Then each element W of P 2 is translated by some element g(W ) in Γ to get the family P 3 := {g(W ).W | W ∈ P 2 }. Since F is a fundamental set for Γ in SH, the elements of P 3 are pairwise disjoint and P 3 is a fundamental set for Γ in SH. Now P 3 is partitioned into certain subsets, say into the subsets Q l , l ∈ L. Each cell B in SH is the union of the elements in some Q l( B) such that l( B 1 ) = l( B 2 ) if B 1 = B 2 . Therefore, the union B B ∈ B S is disjoint and a fundamental set for Γ in SH. Example 7.18. Recall the congruence subgroup PΓ 0 (5) and its cycles in A×PΓ 0 (5) from Example 5.5. We choose S := A(v 4 ), h −1 , A(v 1 ), h 1 as set of choices associated to A and set as well as CS ′ j := CS ′ B j for j = 1, . . . , 6. Figure 18 shows the sets CS ′ j . Lemma 7.19. Let A 1 , A 2 be two basal precells in H and let g ∈ Γ such that g. vc( A 1 ) ∩ vc( A 2 ) = ∅. Suppose that A 1 = A 2 or g = id. Then Moreover, suppose that A 1 is cuspidal or non-cuspidal and that there is a unit tangent vector w ∈ vc( A 1 ) pointing into a non-vertical side S 1 of A 1 such that gw ∈ vc( A 2 ). Then g.w points into a non-vertical side S 2 of A 2 and g.S 1 = S 2 .
Let A 1 and w be as in the claim. Further let γ be the geodesic determined by w. Then g.γ is the geodesic determined by g.w. By definition there exists ε > 0 such that γ((0, ε)) ⊆ S 1 and g.γ((0, ε)) ⊆ A 2 . Then Since the sets A 1 and g −1 .A 2 intersect in more than one point and A 1 = g −1 .A 2 , Proposition 4.13 states that A 1 ∩ g −1 .A 2 is a common side of A 1 and g −1 .A 2 . Necessarily, this side is S 1 . Proposition 4.13 shows further that g.S 1 is a nonvertical side of A 2 . Thus, g.w points along the non-vertical side g.S 1 of A 2 .  (A, h A )).
Proof. We use the notation from Construction 7.14. Let j ∈ {1, . . . , cyl(A)} and z ∈ A j . At first we show that F z (A j ) = ∅. For each choice of F we have A j ⊆ F ∩ vc( A j ). Remark 7.3 states that pr( A j ) = A j . Hence E z (A j ) ∩ A j = ∅. More precisely, if A j z denotes the set of unit tangent vectors based on z that point into The set A j z is non-empty, since A j is convex with non-empty interior. Let k ∈ J such that A k = A j . Then where the last inclusion follows from Lemma 7.19. Since A j ∩ vb( A j ) = ∅ by Lemma 7.6, it follows that Let j ∈ {1, . . . , cyl(A)} set B j := B j (A, h A ) and Note that necessarily k ≥ 3. Then Since l − j + 1 ≡ 1, 2 mod k for l ∈ {1, . . . , j − 1} ∪ {j + 2, . . . , k}, it follows that For the last equality we use that pr ∞ (s j ) = h −1 j .∞ by Lemma 4.7. Hence s j is contained in the geodesic segment pr −1 ∞ (h −1 j .∞) ∩ H = (h −1 j .∞, ∞), which shows that the union of the two geodesic segments [s j , ∞) and [s j , h −1 j .∞) is indeed (h −1 j .∞, ∞). Proposition 5.10 implies that This shows that b( B j ) = (h −1 j .∞, ∞). The set of unit tangent vectors in B j that are based on b( B j ) is the disjoint union Hence, (6) shows that D ′ j contains all unit tangent vectors of the first two kinds mentioned above. Let w be the unit tangent vector with pr(w) = s j which points into [s j , g j .v].
Suppose first that w ∈ F . Then w ∈ vc( A j ) ∩ F and therefore w ∈ E sj (A j ). Let k ∈ J with A k = A j . Assume for contradiction that w ∈ vc( A k ). Lemma 7.19 implies that [s j , g j .v] is a non-vertical side of A k , which is a contradiction. Hence w / ∈ E sj (A k ). Therefore, w ∈ F sj (A j ) and hence w ∈ D ′ j . Suppose now that w / ∈ F . Then there exists a unique g ∈ Γ {id} such that gw ∈ F . Let A be a basal precell in H such that g.w ∈ vc( A) ∩ F . Lemma 7.19 shows that g.[s j , g j .v] is a non-vertical side S of A. Thus, g −1 .A ∩ B(A j ) • = ∅. By Proposition 5.10(iv) there is a unique l ∈ {1, . . . , k} such that g = g l g −1 j and A = A l . Now [s j , g j .v] is mapped by h j to the non-vertical side [h j .s j , g j+1 .v] of A j+1 . Thus, g = h j and A = A j+1 . Then h j .w ∈ vc( A j+1 ). As before we see that w ∈ D ′ j . Moreover, pr(CS ′ ( B j )) = b( B j ).
Analogously to Proposition 7.20 one proves the following two propositions.
Proposition 7.21. Let (A, h A ) ∈ S and suppose that A is a cuspidal precell in H.
Let v be the vertex of K to which A is attached and let (A, g), The development of a symbolic dynamics for the geodesic flow on Y via the family B S of cells in SH is based on the following properties of the cells B in SH: It uses that cl(pr( B)) is a convex polyhedron of which each side is a complete geodesic segment and that each side is the image under some element g ∈ Γ of the complete geodesic segment b( B ′ ) for some cell B ′ in SH. It further uses that B S is a fundamental set for Γ in SH and that {g. cl(pr( B)) | g ∈ Γ, B ∈ B S } is a tesselation of H. Moreover, one needs that b( B) is a vertical side of pr( B) and that CS ′ ( B) is the set of unit tangent vectors based on b( B) that point into pr( B) • . It does not use that {cl(pr( B)) | B ∈ B S } is the set of all cells in H nor does one need that for some cells B 1 , B 2 ∈ B S one has cl( pr( B 1 )) = B(pr( B 2 )). This means that one has the freedom to perform (horizontal) translations of single cells in SH by elements in Γ ∞ . The following definition is motivated by this 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 has not.

Geometric symbolic dynamics
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. 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 B S be the family of cells in SH associated to A and S. 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 condition that NIC (see Remark 6.5) be a µ-null set.
8.1. Geometric cross section. Recall the set BS from Section 6. Conversely, let v ∈ CS. We will show that there is a unique B ∈ B S,T and a unique v ∈ CS ′ ( B) such that π(v) = v. Pick any w ∈ π −1 (v). Remark 7.25 shows that the set P := { B | B ∈ B S,T } is a fundamental set for Γ in SH. Hence there exists a unique pair ( B, g) ∈ B S,T × Γ such that v := g.w ∈ B. Note that π −1 ( CS) = CS. Thus, v ∈ CS and hence pr(v) ∈ pr( B) ∩ BS. Lemma 8.1 shows that pr(v) ∈ b( B). Therefore, v ∈ π −1 (b( B)) ∩ B. Since v ∈ CS, it does not point along b( B). Hence v does not point along ∂ pr( B), which shows that v ∈ CS ′ ( B). This proves that CS ⊆ CS( B S,T ).
To see the uniqueness of B and v suppose that w 1 ∈ π −1 ( v). Let ( B 1 , g 1 ) ∈ B S,T × Γ be the unique pair such that g 1 .w 1 ∈ B 1 . There exists a unique element h ∈ Γ such that h.w = w 1 . Then g 1 hg −1 .v = g.w 1 and v, g 1 hg −1 .v ∈ P. Now P being a fundamental set shows that g 1 hg −1 = id, which proves that g 1 .w 1 = g 1 h.w = g.w = v and B 1 = B. This completes the proof.
Corollary 8.3. Let γ be a geodesic on Y which intersects CS in t. Then there is a unique geodesic γ on H which intersects CS ′ ( B S,T ) in t such that π(γ) = γ.
Definition 8.4. Let γ be a geodesic on Y which intersects CS in γ ′ (t 0 ). If s := min t > t 0 γ ′ (t) ∈ CS exists, we call s the first return time of γ ′ (t 0 ) and γ ′ (s) the next point of intersection of γ and CS. Let γ be a geodesic on H. If γ ′ (t) ∈ CS, then we say that γ intersects CS in t. If there is a sequence (t n ) n∈N with lim n→∞ t n = ∞ and γ ′ (t n ) ∈ CS for all n ∈ N, then γ is said to intersect CS infinitely often in future. Analogously, if we find a sequence (t n ) n∈N with lim n→∞ t n = −∞ and γ ′ (t n ) ∈ CS for all n ∈ N, then γ is said to intersect CS infinitely often in past. Suppose that γ intersects CS in t 0 . If s := min t > t 0 γ ′ (t) ∈ CS exists, we call s the first return time of γ ′ (t 0 ) and γ ′ (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.
Remark 8.5. A geodesic γ on Y intersects CS if and only if some (and hence any) representative of γ on H intersects π −1 ( CS). Recall that CS = π −1 ( CS), and that CS is the set of unit tangent vectors based on BS but which are not tangent to BS. Since BS is a totally geodesic submanifold of H (see Proposition 6.3), a geodesic γ 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.
A similar argument simplifies the search for previous and next points of intersection. To make this precise, suppose that γ is a geodesic on Y which intersects CS in γ ′ (t 0 ) and that γ is a representative of γ on H. Suppose that we are given a geodesic γ on Y which intersects CS in γ ′ (t 0 ) and suppose that γ is the unique geodesic on H which intersects CS ′ ( B S,T ) in γ ′ (t 0 ) and which satisfies π(γ) = γ. We now characterize when there is a next point of intersection of γ and CS resp. of γ and CS, and, if there is one, where this point is located. Further we will do analogous investigations on the existence and location of previous points of intersections. To this end we need the following preparations.
Definition 8.7. Let B ∈ B S,T and suppose that b( B) is the complete geodesic segment (a, ∞) with a ∈ R. We assign to B two intervals I( B) and J( B) which are given as follows: We note that the combination of Remark 7.25 with Propositions 5.10(i) and 7.20 resp. with Propositions 5.11(i) and 7.21 resp. Proposition 7.22 shows that indeed each B ∈ B S,T gets assigned a pair I( B), J( B) of intervals. Note that for g = α β γ δ we have In Proposition 8.9. Let B ∈ B S,T and suppose that S is a side of pr( B). Then there exist exactly two pairs ( B 1 , g 1 ), ( B 2 , g 2 ) ∈ B S,T × Γ such that S = g j .b( B j ). Moreover, g 1 . cl(pr( B 1 )) = cl(pr( B)) and g 2 . cl(pr( B 2 ))∩cl(pr( B)) = S or vice versa. The union g 1 . CS ′ ( B 1 ) ∪ g 2 . CS ′ ( B 2 ) is disjoint and equals the set of all unit tangent vectors in CS that are based on S. Let a, b ∈ ∂ g H be the endpoints of S. Then Proof. Let D ′ 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 D ′ 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 H 1 , H 2 such that H is the disjoint union H 1 ∪ S ∪ H 2 . Moreover, pr( B) • is contained in H 1 or H 2 , say pr( B) • ⊆ H 1 . Then D ′ decomposes into the disjoint union D ′ 1 ∪ D ′ 2 where D ′ j denotes the non-empty set of elements in D ′ that point into H j . For j = 1, 2 pick v j ∈ D ′ j . Since CS ′ ( B S,T ) is a set of representatives for CS = π(CS) (see Proposition 8.2), there exists a uniquely determined pair ( B j , g j ) ∈ B S,T × Γ such that v j ∈ g j . CS ′ ( B j ). We will show that S = g j .b( B j ). Assume for contradiction that S = g j .b( B j ). Since S and g j .b( ch j ) are complete geodesic segments, the intersection of S and Hence pr( B) • ∩ g j . pr( B j ) • = ∅. Proposition 5.14 in combination with Remark 7.25 shows that cl(pr( B)) = g j . cl(pr( B j )). But then which implies that S = g j .b( B j ). This is a contradiction, and hence S = g j .b( B j ). Then Lemma 8.8 implies that g j .I( B j )×g j .J( B j ) equals (a, b) + ×(a, b) − or (a, b) − × (a, b) + . On the other hand Therefore, again by Lemma 8.8, we have g j . CS ′ ( B j ) = D ′ j . This shows that the union g 1 .
Let B ∈ B S,T and suppose that S is a side of pr( B). Let ( B 1 , g 1 ), ( B 2 , g 2 ) be the two elements in B S,T × Γ such that S = g j .b( B j ) and g 1 . cl(pr( B 1 )) = cl(pr( B)) and g 2 . cl(pr( B 2 )) ∩ cl(pr( B)) = S. Then we define p B, S := B 1 , g 1 and n B, S := B 2 , g 2 .
Remark 8.10. Let B ∈ B S,T and suppose that S is a side of pr( B). We will show how one effectively finds the elements p( B, S) and n( B, S). Let Moreover, cl(pr( B)) = T( B ′ ). cl(pr( B ′ )). Then k 1 . cl(pr( B 1 )) = cl(pr( B)) is equivalent to and k 2 . cl(pr( B 2 )) ∩ cl(pr( B)) = S is equivalent to  A, h A ). For j = 1, . . . , k set g 1 := id and g j+1 := h j g j . Proposition 5.10(iii) states that B(A m ) = g m .B(A) and Proposition 5.10(i) shows that S ′ is the geodesic segment [g m g −1 j .∞, g m g −1 j+1 .∞] for some j ∈ {1, . . . , k}. Then Let n ∈ {1, . . . , cyl(A)} such that n ≡ j mod cyl(A). Then h n = h j by Lemma 7.11. Proposition 7.20 shows that b ( B n (A, h A ) For this is remains to show that g m g −1 j . cl(pr ( B j (A, h A ))) = cl(pr( B ′ )). Proposition 7.20 shows that cl(pr ( B j (A, h A ))) = B(A n ) and Lemma 7.11 implies that B(A n ) = B(A j ). Let v be the vertex of K to which A is attached.
Analogously one proceeds if A ′ is cuspidal or a strip precell. Now we show how one determines ( B 2 , k 2 ). Suppose again that B ′ arises from the non-cuspidal basal precell A ′ in H. We use the notation from the determination of p( B ′ , S ′ ). By Corollary 4.10 there is a unique pair ( Thus n( B ′ , S ′ ) = ( B 3 , g m g −1 j t s λ k 3 ) and If B ′ arises from a cuspidal or strip precell in H, then the construction of n( B, S) is analogous.
(i) There is a next point of intersection of γ and CS if and only if γ(∞) does not belong to ∂ g pr( B).
Corollary 8.12. Let γ be a geodesic on Y and suppose that γ does not intersect CS infinitely often in future. If γ intersects CS at all, then there exists t ∈ R such that γ ′ (t) ∈ CS and γ((t, ∞)) ∩ BS = ∅. Analogously, suppose that η is a geodesic on Y which does not intersect CS infinitely often in past. If η intersects CS at all, then there exists t ∈ R such that η ′ (t) ∈ CS and η((−∞, t)) ∩ BS = ∅.
Example 8.13. Recall the setting of Example 7.18. We consider the two shift maps T 1 ≡ id, and  Recall the set bd from Section 6. The following characterization is now obvious.
Proposition 8.14. Let γ be a geodesic on Y . Recall the set NIC from Remark 6.5. 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 past and future if and only if γ / ∈ NIC. This completes the proof.
Let E denote the set of unit tangent vectors to the geodesics in NIC and set CS st := CS E.
Corollary 8.16. Let µ be a measure on the space of geodesics on Y such that µ(NIC) = 0. Then CS st is the maximal strong cross section w. r. t. µ contained in CS.
8.2. Geometric coding sequences and geometric symbolic dynamics. A 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 called the alphabet of the arising symbolic dynamics.
We establish a labeling of CS and CS in the following way: Let v ∈ CS ′ ( B S,T ) and suppose that B ∈ B S,T is the unique shifted cell in SH such that v ∈ CS ′ ( B). Let γ be the geodesic on H determined by v.
Suppose first that γ(∞) / ∈ ∂ g pr( B). Proposition 8.11(ii) states that there is a unique side S of pr( B) intersected by γ((0, ∞)) and that the next point of intersection of γ and CS is on g. CS ′ ( B 1 ) if ( B 1 , g) = n( B, S). We assign to v the label ( B 1 , g).
Suppose now that γ(∞) ∈ ∂ g pr( B). Proposition 8.11(i) shows that there is no next point of intersection of γ and CS. Let ε be an abstract symbol which is not contained in Γ. Then we label v by ε ("end" or "empty word").
Let v ∈ CS. By Proposition 8.2 there is a unique v ∈ CS ′ ( B S,T ) such that π(v) = v. We endow v and each element in π −1 ( v) with the labels of v.
The following proposition is the key result for the determination of the set of labels. Let Σ denote the set of labels.
Corollary 8.19. The set Σ of labels is given by Moreover, Σ is finite.
Proof. Note that for each B ′ ∈ B S,T we have ∂ g pr( B ′ ) ∩ I( B ′ ) = ∅. Thus, ε is a label. Then the claimed equality follows immediately from Corollary 8.18. Since B S,T is finite and each shifted cell in SH has only finitely many sides, Σ is finite. Example 8.21. Recall Example 8.13. If the shift map is T 1 , then the set of labels is Σ = ε, B 2 , g 1 , B 4 , id , B 5 , g 2 , B 6 , id , B 3 , g 3 , B 5 , id , B 6 , g 4 , B 3 , id , B 1 , g 5 , B 2 , g 6 , B 4 , g 4 .
The sequence (t n ) n∈J is said to be the sequence of intersection times of v (with respect to CS).
Let v ∈ CS and set v := π| CS ′ ( B S,T ) −1 ( v). Then the sequence of intersection times (w. r. t. CS) of v and of each w ∈ π −1 ( v) is defined to be the sequence of intersection times of v.
Now we define the geometric coding sequences.
Definition 8.23. For each s ∈ Σ we set Let v ∈ CS and let (t n ) n∈J be the sequence of intersection times of v. Suppose that γ is the geodesic on Y determined by v. The geometric coding sequence of v is the sequence (a n ) n∈J in Σ defined by a n := s if and only if γ ′ (t n ) ∈ CS s for each n ∈ J.
Let v ∈ CS. The geometric coding sequence of v is defined to be the geometric coding sequence of π(v).
Proposition 8.24. Let v ∈ CS ′ . Suppose that (t n ) n∈J is the sequence of intersection times of v and that (a n ) 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 ∪ {±∞}.
(ii) If b < ∞, then a n ∈ Σ {ε} for each n ∈ (a, b − 2] ∩ Z and a b−1 = ε. (iii) Suppose that a n = ( B n , h n ) for n ∈ (a, b − 1) ∩ Z and set 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 := γ ′ (t n ). The definition of geometric coding sequences shows that γ ′ (t n ) ∈ CS an . Since CS is the disjoint union k∈Γ k. CS ′ ( B S,T ) (see Proposition 8.2), there is a unique k ∈ Γ such that k −1 .w ∈ CS ′ ( B S,T ). The label of k −1 .w is a n . Let η be the geodesic on H determined by k −1 .w. Note that η(t) := k −1 .γ(t+t n ) for each t ∈ R. The definition of labels shows that a n = ε if and only if there is no next point of intersection of η and CS. In this case γ ′ ((t n , ∞)) ∩ CS = ∅ and hence b = n + 1. This shows (i) and (ii). Suppose now that a n = ( B, g). Then there is a next point of intersection of η and CS, say η ′ (s), and this is on g. CS ′ ( B). Then k −1 .γ ′ (s + t n ) ∈ g. CS ′ ( B) and k −1 .γ ′ ((t n , s + t n )) ∩ CS = ∅. Hence t n+1 = s + t n and γ ′ (t n+1 ) ∈ kg. CS ′ ( B).
Now we show (iii). Suppose that b ≥ 2. Then v = γ ′ (t 0 ) is labeled with ( B 0 , h 0 ). Hence for the next point of intersection γ ′ (t 1 ) of γ and CS we have Suppose that we have already shown that for some n ∈ [1, −a) ∩ Z and suppose that a ≤ −n − 2. Then γ ′ (t −n−1 ) exists and is labeled with ( B −n−1 , h −n−1 ). Since γ ′ (t −n−1 ) ∈ h. CS ′ ( B S,T ) for some h ∈ Γ, we know that γ ′ (t −n ) ∈ hh −n−1 . CS ′ ( B −n−1 ). Therefore . CS ′ B S,T . Therefore γ ′ (t n+1 ) ∈ g n . CS ′ ( B n ) for each n ∈ (a, −1] ∩ Z. This completes the proof. Let Λ denote the set of geometric coding sequences and let Λ σ be the subset of Λ which contains the geometric coding sequences (a n ) n∈(a,b)∩Z with a, b ∈ Z ∪ {±∞} for which b ≥ 2. Let Σ all denote the set of all finite and one-or two-sided infinite sequences in Σ. The left shift σ : Σ all → Σ all , σ (a n ) n∈J k := a k+1 for all k ∈ J induces a partially defined map σ : Λ → Λ resp. a map σ : Λ σ → 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.
In the following we will show that (Λ st , σ) is a symbolic dynamics for the geodesic flow on Φ. Elementary convex geometry proves the following lemma.
Lemma 8.27. Suppose that x, y ∈ ∂ g H bd, x < y. Then there exists a connected component S = [a, b] of BS with a, b ∈ R, a < b, such that x < a < y < b.
For the proof of the following proposition we recall that each connected component of BS is a complete geodesic segment and that it is of the form pr(g. CS ′ ( B)) for some pair ( B, g) ∈ B S,T × Γ. Conversely, for each pair ( B, g) ∈ B S,T , the set pr(g. CS ′ ( B)) is a connected component of BS. Proof. Let ( B j , h j ) j∈Z be the geometric coding sequence of v and assume that .
Proposition 8.24 shows that v ∈ CS ′ ( B −1 ) and w ∈ CS ′ ( B ′ −1 ). Lemma 8.8 implies that B −1 = B ′ −1 , which shows that the geometric coding sequences of v and w are different.
Suppose now that Assume for contradiction that ( B j , h j ) j∈Z = ( B ′ j , k j ) j∈Z . Let (t n ) n∈Z be the sequence of intersection times of v and (s n ) n∈Z be that of w. Prop 8.24(iii) implies that for each n ∈ Z, the elements pr(γ ′ v (t n )) and pr(γ ′ w (s n )) are on the same connected component of BS. For each connected component S of BS let H 1,S , H 2,S denote the open convex half spaces such that H is the disjoint union Suppose first that γ v (∞) = γ w (∞). Proposition 8.14 shows that By Lemma 8.27 we find a connected component S of BS such that γ v (∞) ∈ ∂ g H 1,S ∂ g S and γ w (∞) ∈ ∂ g H 2,S ∂ g S (or vice versa). Since BS is a manifold, each connected component of BS other than S is either contained in H 1,S or in H 2,S . In particular, we may assume that pr(v), pr(w) ∈ H 1,S . Then γ v ([0, ∞)) ⊆ H 1,S and γ w ((t, ∞)) ⊆ H 2,S for some t > 0. Hence there is n ∈ N such that pr(γ ′ w (s n )) ∈ H 2,S , which implies that pr(γ ′ v (t n )) and pr(γ ′ w (s n )) are not on the same connected component of BS. Suppose now that γ v (−∞) = γ w (−∞) and let S be a connected component of BS auch that γ v (−∞) ∈ ∂ g H 1,S ∂ g S and γ w (−∞) ∈ ∂ g H 2,S ∂ g S (or vice versa). Again, we may assume that pr(v), pr(w) ∈ H 1,S . Then γ v ((−∞, 0]) ⊆ H 1,S and γ w (−∞, s)) ⊆ H 2,S for some s < 0. Thus we find n ∈ N such that pr(γ ′ w (s −n )) ∈ H 2,S . Hence pr(γ ′ v (t −n )) and pr(γ ′ w (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. 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. This is the reason why NC stands for "not coded".
Suppose now that v ∈ CS. Then the geometric coding sequence is not two-sided infinite if and only if γ does not intersect CS infinitely often in past and future, which by Proposition 8.14 is equivalent to γ ∈ NIC. This explains why NIC is for "not infinitely often coded".

Reduction and arithmetic symbolic dynamics
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 B S is the family of cells in SH associated to A and S. Let T be a shift map for B S and let B S,T denote the family of cells in SH associated to A, S and T.
Recall the geometric symbolic dynamics for the geodesic flow on Y which we constructed in Section 8 with respect to A, S and T. In particular, recall the set CS ′ ( B S,T ) of representatives for the cross section CS = CS( B S,T ), its subsets CS ′ ( B) for B ∈ B S,T , and the definition of the labeling of CS.
Let v ∈ CS ′ ( B) for some B ∈ B S,T and consider the geodesic γ v on H determined by v. Suppose that (a n ) n∈J is the geometric coding sequence of v. The combination of Propositions 8.17 and 8.11 allows to determine the label a 0 of v from the location of γ v (∞), and then to iteratively reconstruct the complete future part (a n ) n∈[0,∞)∩J of the geometric coding sequence of v. Hence, if the unit tangent vector v ∈ CS ′ ( B S,T ) is known, or more precisely, if the shifted cell B in SH with v ∈ CS ′ ( B) is known, then one can reconstruct at least the future part of the geometric coding sequence of v. However, if γ v intersects CS ′ ( B S,T ) in more than one point, then one cannot derive the shifted cell B in SH from the endpoints of γ v . In this case, the induced discrete dynamical system on ∂ g H which is conjugate to ( CS, R) or ( CS st , R) is given by local diffeomorphisms which have to keep the shifted cells in their definitions. These discrete dynamical systems are used in [Poh12b,Poh12a] to provide a dynamical approach to Maass cusp forms. Here, we will propose a second way to deduce discrete dynamical systems on ∂ g H.
For these, we restrict, in Section 9.1, our cross section CS to a subset CS red ( B S,T ) (resp. to CS st,red ( B S,T ) for the strong cross section CS st ) such that for any v ∈ CS red , the endpoints of γ v completely determine v. We will show that CS red ( B S,T ) and CS st,red ( B S,T ) are cross sections for the geodesic flow on Y w. r. t. to the same measure as CS and CS st . More precisely, it will turn out that exactly those geodesics on Y which intersect CS at all also intersect CS red ( B S,T ) at all, and that exactly those which intersect CS infinitely often in future and past also intersect CS red (  where I( B 1,j ) = (a j , ∞) and a 1 < a 2 < . . . < a k1 , and For each B ∈ B S,T set Therefore we have I red B 1 = (−∞, 1), I red B 2 = 0, 1 5 , I red B 4 = 1 5 , 2 5 , I red B 6 = 2 5 , 3 5 , I red B 5 = 3 5 , 4 5 , I red B 3 = 4 5 , ∞ . If the shift map is T 2 , then we find I red B 2 = 0, 1 5 , I red B 4 = 1 5 , 2 5 , I red B 6 = 2 5 , 3 5 , I red B 5 = 3 5 , 4 5 , I red B 3 = 4 5 , ∞ . Note that with T 2 , the sets I red (·) are pairwise disjoint, whereas with T 1 they are not.
Proof. The combination of the definition of CS ′ red ( B S,T ) and Lemma 8.8 shows that there is at least one v ∈ CS ′ red ( B S,T ) such that γ v (∞), γ v (−∞) = (x, y) and that for each B ∈ B S,T there there is at most one such v. By construction, Proposition 9.5. Let v ∈ CS ′ ( B S,T ) and suppose that η is the geodesic on H determined by v. Let (s j ) j∈(α,β)∩Z be the geometric coding sequence of v. Suppose that s j = ( B j , h j ) for j = 0, . . . , β − 2. For j = 0, . . . , β − 2 set g −1 := id and g j := g j−1 h j .
Hence it remains to show that the element s 0 exists and that s 0 = t n for some n ∈ {0, . . . , κ + 1}.
Proposition 9.6. Let v ∈ CS ′ red ( B S,T ) and suppose that η is the geodesic on H determined by v. Let s j j∈(α,β)∩Z be the geometric coding sequence of v. Suppose that s j = ( B j , h j ) for j = 0, . . . , β − 2. Recall from Remark 6.5 that NIC denotes the set of geodesics on Y with at least one endpoint contained in π(bd). Let v ∈ CS ′ red ( B S,T ) and let γ denote the geodesic on H determined by v. Suppose first that γ((0, ∞)) ∩ CS red ( B S,T ) = ∅. Proposition 9.6 implies that there is a next point of intersection of γ and CS red ( B S,T ) and that this is on g. CS ′ red ( B) for a (uniquely determined) pair ( B, g) ∈ B S,T × Γ. We endow v with the label ( B, g).
Suppose now that γ((0, ∞)) ∩ CS red ( B S,T ) = ∅. Then there is no next point of intersection of γ and CS red ( B S,T ). We label v by ε.
Let v ∈ CS red ( B S,T ) and let v := π| CS ′ red ( B S,T ) −1 ( v). The label of v and of each element in π −1 ( v) is defined to be the label of v.
Suppose that Σ red denotes the set of labels of CS red ( B S,T ).
Remark 9.8. Recall from Corollary 8.19 that Σ is finite. Then Proposition 9.6 implies that also Σ red 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 Σ red can be effectively determined.
The following definition is analogous to the corresponding definitions in Section 8.2.
Definition 9.9. Let v ∈ CS ′ red ( B S,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 (t n ) n∈J in R which satisfies the following properties: (i) J = Z ∩ (a, b) for some interval (a, b) with a, b ∈ Z ∪ {±∞} and 0 ∈ (a, b), (ii) the sequence (t n ) n∈J is increasing, (iii) t 0 = 0, (iv) for each n ∈ J we have γ ′ (t n ) ∈ CS red ( B S,T ) and The sequence (t n ) n∈J is said to be the sequence of intersection times of v with respect to CS red ( B S,T ).
Let v ∈ CS red ( B S,T ) and set v := π| CS ′ Let v ∈ CS red ( B S,T ) and let (t n ) n∈J be the sequence of intersection times of v w. r. t. CS red ( B S,T ). Suppose that γ is the geodesic on Y determined by v. The reduced coding sequence of v is the sequence (a n ) n∈J in Σ red defined by a n := s if and only if γ ′ (t n ) ∈ CS red,s B S,T for each n ∈ J. Let w ∈ CS red ( B S,T ). The reduced coding sequence of w is defined to be the reduced coding sequence of π(w).
Let Λ red denote the set of reduced coding sequences and let Λ red,σ be the subset of Λ red consisting of the reduced coding sequences (a n ) n∈(a,b)∩Z with a, b ∈ Z ∪ {±∞} for which b ≥ 2. Further, let Λ st,red denote the set of two-sided infinite reduced coding sequences. Let Σ all red be the set of all finite and one-or two-sided infinite sequences in Σ red . Finally, let Seq red : CS red ( B S,T ) → Λ red be the map which assigns to v ∈ CS red ( B S,T ) the reduced coding sequence of v.
The proofs of Propositions 9.10 are analogous to those of the corresponding statements in Section 8.2.
Proposition 9.10. (1) Let v ∈ CS ′ red ( B S,T ). Suppose that (t n ) n∈J is the sequence of intersection times of v and that (a n ) n∈J is the reduced coding sequence of v. Let γ be the geodesic on H determined by v. Suppose that J = Z ∩ (a, b) with a, b ∈ Z ∪ {±∞}.
(2) Let v, w ∈ CS ′ st,red ( B S,T ). If the reduced coding sequences of v and w are equal, then v = w.
commute and (Λ st,red , σ) is a symbolic dynamics for the geodesic flow on Y .
We will now show that the reduced coding sequence of v ∈ CS red ( B S,T ) can be completely constructed from the knowledge of the pair τ ( v).
Definition 9.11. Let B ∈ B S,T . Define Example 9.12. Recall the Example 8.13. Suppose first that the shift map is T 1 . Then we have Σ red B 1 = ε, B 1 , g 5 , B 4 , g 4 , B 6 , g 4 , B 5 , g 4 , B 3 , g 4 , Σ red B 2 = ε, B 2 , g 1 , B 4 , g 1 , B 6 , g 1 , B 5 , g 1 , B 3 , g 1 , Σ red B 3 = ε, B 1 , g 5 , B 2 , g 6 , B 4 , g 6 , B 6 , g 6 , B 5 , g 6 , B 3 , g 6 , Σ red B 4 = ε, B 5 , g 2 , B 3 , g 2 , The next corollary follows immediately from Proposition 9.6. Our next goal is to find a discrete dynamical system on the geodesic boundary of H which is conjugate to ( CS red ( B S,T ), R). To that end we set For B ∈ B S,T and s ∈ Σ red ( B) we set We define the partial map F : DS → DS by Recall the map −1 ( v) and γ v is the geodesic on H determined by v.
Proposition 9.14. The set DS is the disjoint union This implies that F is well-defined. Let (x, y) ∈ DS. By Lemma 9.4 there is a unique B ∈ B S,T and a unique v ∈ CS ′ red ( B) such that γ v (∞), γ v (−∞) = (x, y). Corollary 9.13 shows that v is labeled with s ∈ Σ red if and only if γ v (∞) ∈ D s ( B), hence if (x, y) ∈ D s ( B). It remains to show that ( DS, F ) is conjugate to ( CS red ( B S,T ), R) by τ . Lemma 9.4 shows that τ is a bijection between CS red ( B S,T ) and DS. Let v ∈ CS red and v := π| CS ′ red ( B S,T ) −1 ( v). Suppose that B ∈ B S,T is the (unique) shifted cell in SH such that v ∈ CS ′ red ( B), and let (s j ) j∈(α,β)∩Z be the reduced coding sequence of v. Recall that s 0 is the label of v and v. Corollary 9.13 shows that γ v (∞) ∈ D s0 ( B). The map R is defined for v if and only if s 0 = ε. In precisely this case, F is defined for τ ( v).
The following corollary proves that we can reconstruct the future part of the reduced coding sequence of v ∈ CS red ( B S,T ) from τ ( v). for each j ∈ J ∩ N 0 . For j ∈ N 0 J, the map F j is not defined for τ ( v).
The next proposition shows that we can also reconstruct the past part of the reduced coding sequence of v ∈ CS red ( B S,T ) from τ ( v). Its proof is constructive. Proof. We will prove (ii), which directly implies (i). To that end set and suppose that (t j ) j∈J is the sequence of intersection times of v with respect to CS red ( B S,T ). Suppose first that v ∈ CS ′ red ( B) and that −1 ∈ J. Then there exists a (unique) pair ( B ′ , g) ∈ B S,T × Γ such that γ ′ v (t −1 ) ∈ g −1 . CS ′ red ( B ′ ). Since the unit tangent vector γ ′ v (t 0 ) = v is contained in CS ′ red ( B), the element γ ′ v (t −1 ) is labeled with ( B, g). Hence ( B, g) ∈ Σ red ( B ′ ). Then Conversely suppose that τ ( v) ∈ g −1 . D ( B,g) ( B ′ ) for some B ′ ∈ B S,T and some element ( B, g) ∈ Σ red ( B ′ ). Consider the geodesic η := g.γ v . Then η(∞), η(−∞) = g.τ ( v) ∈ D ( B,g) B ′ .
Corollary 9.17. (i) The partial map F bk is well-defined.
(ii) Let v ∈ CS red ( B S,T ) and suppose that (s j ) j∈J is the reduced coding sequence of v. For each j ∈ J ∩ (−∞, −1] and each ( B, g) ∈ Σ red we have s j = ( B, g) if and only if F j bk τ ( v) ∈ g −1 . D ( B,g) B ′ for some B ′ ∈ B S,T . For j ∈ Z <0 J, the map F j bk is not defined for τ ( v). We end this section with the statement of the discrete dynamical system which is conjugate to the strong reduced cross section CS st,red ( B S,T ).
The set of labels of CS st,red ( B S,T ) is given by if s = ( B ′ , g) ∈ Σ st,red ( B) and B ∈ B S,T . The map F st is the "restriction" of F to the strong reduced cross section CS st,red ( B S,T ). In particular, the following proposition is the "reduced" analogon of Proposition 9.14. If x ∈ DS, then there is (a non-unique) v ∈ CS ′ red ( B S,T ) such that γ v (∞) = x. Suppose that v ∈ CS ′ red ( B S,T ) with γ v (∞) = x and let (a n ) n∈J be the reduced coding sequence of v. Then a 0 = s if and only if x ∈ D s ( B) for some B ∈ B S,T . (ii) The partial map F is well-defined.
Let (x, y) ∈ DS. Then (x, y) ∈ D s ( B) if and only if x ∈ D s ( B). Proposition 9.14 implies the remaining statements of (i).
Proposition 9.19 shows that is like a generating function for the future part of the symbolic dynamics (Λ red , σ). In comparison with a real generating function, the map i : Λ red → DS is missing. Indeed, if there are strip precells in H, then there is no unique choice for the map i.
To overcome this problem, we restrict ourselves to the strong reduced cross section CS st,red ( B S,T ).
Proposition 9.20. Let v, w ∈ CS ′ st,red ( B S,T ). Suppose that (a n ) n∈Z is the reduced coding sequence of v and (b n ) n∈Z that of w. If (a n ) n∈N0 = (b n ) n∈N0 , then γ v (∞) = γ w (∞).
Proof. The proof of Proposition 8.28 shows the corresponding statement for geometric coding sequences. The proof of the present statement is analogous. where v ∈ CS ′ st,red ( B S,T ) is the unit tangent vector with reduced coding sequence (a n ) n∈N . Proposition 2 shows that v is unique, and Proposition 9.20 shows that i only depends on (a n ) n∈N0 . Therefore is a generating function for the future part of the symbolic dynamics (Λ st,red , σ).
Example 9.21. For the Hecke triangle group G n and its family of shifted cells B S,T = { B} from Example 8.20 we have I( B) = I red ( B) and Σ red = Σ. Obviously, the associated symbolic dynamics (Λ, σ) has a generating function for the future part. Recall the set bd from Section 6. Here we have bd = G n .∞. Then DS = R + and DS st = R + G n .∞.
Since there is only one (shifted) cell in SH, we omit B from the notation in the following. We have D g = (g.0, g.∞) and D st,g = (g.0, g.∞) G n .∞ for g ∈ U k n S k = 1, . . . , n − 1 . The generating function for the future part of (Λ, σ) is F : DS → DS, F | Dg x := g −1 .x for g ∈ U k n S k = 1, . . . , n − 1 .. For the symbolic dynamics (Λ st , σ) arising from the strong cross section CS st the generating function for the future part is F st : DS st → DS st , F st | Dst,g x := g −1 .x for g ∈ U k n S k = 1, . . . , n − 1 . Example 9.22. Recall Example 9.3. If the shift map is T 2 , then the sets I red (·) 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 T 1 , the sets I red (·) are not disjoint. Suppose that γ is a geodesic on H such that γ(∞) = 1 2 . Then γ intersects CS ′ red B S,T1 in, say, v. Example 9.12 shows that one cannot decide whether v ∈ CS ′ B 6 and hence is labeled with B 3 , g 3 , or whether v ∈ CS ′ B 1 and thus is labeled with B 4 , g 4 . This shows that the symbolic dynamics arising from the shift map T 1 does not have a generating function for the future part.

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 L of (X, f ) with potential ψ is defined by 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., [CAM + 08, Section 14], [Rue02], [May12].
Here, we only consider potentials of the form ψ(y) = −β log |f ′ (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 B S is the family of cells in SH associated to A and S. Let T be a shift map for B S and let B S,T denote the family of cells in SH associated to A, S and T.
Suppose now that the sets I red ( B), B ∈ B S,T , are pairwise disjoint so that the map F st from Section 9.3 is a generating function for the future part of (Λ st,red , σ). Its local inverses are For extensive examples of arising transfer operators we refer to [HP08,Poh12b,Poh12a].