arXiv:0806.2729v1 [math.DS] 17 Jun 2008

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW ON LOCALLY SYMMETRIC ORBIFOLDS OF RANK ONE

J. HILGERT AND A. D. POHL Abstract. We present a strategy for a geometric construction of cross sections for the geodesic ?ow on locally symmetric orbifolds of rank one. We work it out in detail for ¦£\H, where H is the upper half plane and ¦£ = P¦£0 (p), p prime. Its associated discrete dynamical system naturally induces a symbolic dynamics on R. The transfer operator produced from this symbolic dynamics has a particularly simple structure.

1. Introduction We consider the upper half plane H := {z ¡Ê C | Im z > 0} with the Riemannian metric given by the line element ds2 = y ?2 (dx2 + dy 2 ) as a model for two-dimensional real hyperbolic space. The group of orientation-preserving isometries can be identi?ed with PSL(2, R) via the action az + b g¡¤z = cz + d

b for g = a d ¡Ê ¦£ and z ¡Ê H. The geodesic compacti?cation H of H c will be identi?ed with the one-point compacti?cation of the closure of H in C, hence H = {z ¡Ê C | Im z ¡Ý 0} ¡È {¡Þ}. The action of PSL(2, R) extends continuously to ?H = R ¡È {¡Þ}. The geodesics on H are the semicircles centered on the real line and the vertical lines. All geodesics shall be oriented and parametrized by arc length. The (unit speed) geodesic ?ow on H is the dynamical system

¦µ:

R ¡Á SH (t, v)

¡ú SH ¡ä ¡ú ¦Ãv (t),

2000 Mathematics Subject Classi?cation. Primary: 37D40, Secondary: 11J70. Key words and phrases. cross section, symbolic dynamics, transfer operator.

1

2

J. HILGERT AND A. D. POHL

where SH denotes the unit tangent bundle (sphere bundle) of H and ¡ä ¦Ãv : R ¡ú H is the unique (unit speed) geodesic that satis?es ¦Ãv (0) = v. Let ¦£ be a properly discontinuous subgroup of PSL(2, R). The unit tangent bundle SY of the locally symmetric orbifold Y := ¦£\H shall be identi?ed with ¦£\SH, where we consider the induced ¦£-action on SH. Let ¦Ð : H ¡ú Y and ¦Ð : SH ¡ú SY denote the canonical projection maps. The geodesics on Y are in bijection with the ¦£-equivalence classes of geodesics on H, and the geodesic ?ow on Y is given by ¦µ := ¦Ð ? ¦µ ? (id ¡Á¦Ð ?1 ) : R ¡Á SY ¡ú SY. Here, ¦Ð ?1 shall be read as an arbitrary section of ¦Ð. The de?nition of ¦µ is independent of this choice. For the modular group PSL(2, Z), Series [6] geometrically constructed an amazingly simple cross section1 for the geodesic ?ow on the modular surface PSL(2, Z)\H. Its associated discrete dynamical system is naturally related to a symbolic dynamics on R, namely by continued fraction expansions of the limit points of the geodesics. The Gauss map is a generating function for the future part of this symbolic dynamics. In [5], Mayer investigated the transfer operator L¦Â with parameter ¦Â of the Gauss map. His work and that of Lewis and Zagier [4] have shown that there is an isomorphism between the space of Maass cusp forms for PSL(2, Z) with eigenvalue ¦Â(1 ? ¦Â) and the space of real-analytic eigenfunctions of L¦Â that have eigenvalue ¡À1 and satisfy certain growth conditions. In this article we present, for the examples ¦£ = P¦£0 (p), p prime, a geometric construction of a cross section Cred for the geodesic ?ow on ¦£\H such that, as in Series¡¯ work, its associated discrete dynamical system (Cred , R) naturally gives rise to a symbolic dynamics on R. More precisely, (Cred , R) is conjugate to a discrete dynamical system (D, F ), where D is an open subset of R ¡Á R. The domain D will be seen to naturally decompose into a ?nite disjoint union ¦Á¡ÊA D¦Á of open subsets D¦Á such that F |D¦Á : D¦Á ¡ú F (D¦Á ) is a di?eomorphism of the form e (x, y) ¡ú (h?1 x, h?1 y) for some h¦Á ¡Ê ¦£. Hence we get a canonical sym¦Á ¦Á bolic dynamics for ¦µ on the alphabet A or {h¦Á | ¦Á ¡Ê A}. If prx denotes the projection onto the ?rst component, then the sets D¦Á := prx (D¦Á ) are pairwise disjoint. This means that F is e?ectively determined by the map F : D ¡ú D, where D := prx (D) and F |D¦Á := h?1 . In addition, ¦Á F is a generating function for the future part of the symbolic dynamics.

1The concepts from symbolic dynamics are explained in Section 2.

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW

3

Because of the particular structure of F , its associated transfer operator can be described completely by a ?nite number of group elements in ¦£. For the modular surface, this construction gives a two-term transfer operator from which one can easily read o? the three-term functional equation used in the proof of the Lewis-Zagier isomorphism mentioned above and whose eigenfunctions with eigenvalue 1 that satisfy some growth conditions are in bijection with the Maass cusp forms. There are two main methods to construct cross sections and symbolic dynamics. The geometric coding consists in taking a fundamental domain for ¦£ in H with side pairing and the sequences of sides cut by a geodesic as coding sequences. The cross section is a set of unit tangent vectors based at the boundary of the fundamental domain. In general, it is very di?cult, if not impossible, to ?nd a conjugate dynamical system on the boundary. In contrast, the arithmetic coding starts with a symbolic dynamics on (parts of) the boundary which has a generating function for the future part similar to F from above and asks for a cross section that reproduces this function. Usually, writing down such a cross section is a non-trivial task. A good overview of geometric and arithmetic coding is the survey article [3]. Our method, which we call cusp expansion, is sketched in the following. The description applies to ¦£ = P¦£0 (p), ¦£ = PSL(2, Z) and some other groups as will be discussed in Section 4. We ?x a so-called Ford fundamental domain F for ¦£ in H. To each vertex v (that is, an inner vertex or a representative other than ¡Þ of a cusp) of F we assign a set A(v), which we refer to as a precell. Two precells overlap at most at boundaries, and the union of all precells coincides with F . Then we glue together some ¦£-translates of precells to obtain a cell B(v) assigned to v. The purpose of these cells is to construct a family of ?nite-sided n-gons with all vertices in ?H such that each cell has two vertical boundary components (in other words, ¡Þ is a vertex of each cell) and each non-vertical boundary component of a cell is a ¦£-translate of a vertical boundary component of some cell. Further, the family of all cells shall give a tiling of H in the sense that the ¦£-translates of all cells cover H and two ¦£-translates of cells are either disjoint or identical or coincide at exactly one boundary component. Let C denote the set of unit tangent vectors based at the boundary of some cell but that are not tangent to this boundary. We will see that C := ¦Ð(C) is a cross section. In general, (C, R) is not conjugate to a discrete dynamical system (D, F ) with D ? R ¡Á R. But C contains a subset Cred for which (Cred , R) is naturally conjugate to a dynamical system on some subset of R ¡Á R. The set Cred

4

J. HILGERT AND A. D. POHL

is itself a cross section and can be constructed e?ectively from C. For the proofs of these properties we extend the notions of precells and cells to SH. Each precell A(v) induces a precell A(v) in SH is a geometric way. From these precells we construct (in an e?ective way which, however, involves choices) a ?nite family {B(vk )}k¡ÊA of cells in SH such that each B(vk ) projects to a ¦£-translate of a cell in H. The union of all B(vk ) turns out to be a fundamental set for ¦£ in SH. This property and the interplay of the cells in SH and H imply the properties of C and Cred . The decomposition of F into precells and the construction of cells from precells is based on [7]. Our constructions di?er from Vulakh¡¯s in two aspects: one is the way in which cusps are treated, the other di?erence is that we extend the considerations to cells in the unit tangent bundle. Section 2 contains the necessary background on symbolic dynamics. In Section 3 we develop the cusp expansion for P¦£0 (p) in detail and apply it to PSL(2, Z). We end, in Section 4, with a brief discussion on potential extensions of this method to other locally symmetric orbifolds of rank one.

2. Symbolic dynamics Let C be a subset of SY . A geodesic ¦Ã on Y is said to intersect C ? in?nitely often in future if there is a sequence (tn )n¡ÊN with limn¡ú¡Þ tn = ¡Þ and ¦Ã ¡ä (tn ) ¡Ê C for all n ¡Ê N. Analogously, ¦Ã is said to intersect C ? ? in?nitely often in past if we ?nd a sequence (tn )n¡ÊN with limn¡ú¡Þ tn = ?¡Þ and ¦Ã ¡ä (tn ) ¡Ê C for all n ¡Ê N. Let ? be a measure on the space of ? geodesics on Y . A cross section C (w. r. t. ?) for the geodesic ?ow ¦µ is a subset of SY such that (C1) ?-almost every geodesic ¦Ã on Y intersects C in?nitely often in past ? and future, (C2) each intersection of ¦Ã and C is discrete in time: if ¦Ã ¡ä (t) ¡Ê C, then ? ? there is ¦Å > 0 such that ¦Ã ¡ä ((t ? ¦Å, t + ¦Å)) ¡É C = {? ¡ä (t)}. ? ¦Ã Each Borel probability measure on SY which is invariant under {¦µt }t¡ÊR induces a measure on the space of geodesics. We shall see that the cross sections constructed in Section 3 are indeed cross sections w. r. t. any choice of a measure of this type. Let pr : SY ¡ú Y denote the canonical projection on base points. If pr(C) is a totally geodesic submanifold of Y and C does not contain elements tangent to pr(C), then C automatically satis?es (C2).

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW

5

Suppose that C is a cross section for ¦µ. If C in addition satis?es the property that each geodesic intersecting C at all intersects it in?nitely often in past and future, then C will be called a strong cross section, otherwise a weak cross section. Clearly, every weak cross section contains a strong cross section. The ?rst return map of ¦µ w. r. t. a strong cross section C is the map R: C v ? ¡ú ¡ú C ¦Ãv ¡ä (t0 )

where ¦Ð(v) = v , ¦Ð(¦Ãv ) = ¦Ãv , the geodesic ¦Ãv is chosen as in the de?nition ? of ¦µ, and ¦Ãv ¡ä (t) ¡Ê C for all t ¡Ê (0, t0 ). In this case, t0 is the ?rst return / time of v resp. of ¦Ãv . For a weak cross section C, the ?rst return map ? can only be de?ned on a subset of C. In general, this subset is larger than the maximal strong cross section contained in C. Suppose that C is a strong cross section and let A be an at most countable set. Decompose C into a disjoint union ¦Á¡ÊA C¦Á . To each v ¡Ê C we assign the (two-sided in?nite) coding sequence (an )n¡ÊZ ¡Ê AZ ? de?ned by v an := ¦Á i? Rn (?) ¡Ê C¦Á . Note that R is invertible and let ¦« be the set of all sequences that arise in this way. Then ¦« is invariant under the left shift ¦Ò : AZ ¡ú AZ , (¦Ò((an )n¡ÊZ ))k := ak+1 . Suppose that the map C ¡ú ¦« is also injective, which it will be in our case. Then we have the natural map Cod : ¦« ¡ú C which maps a coding sequence to the element in C it was assigned to. Obviously, R?Cod = Cod ?¦Ò. The pair (¦«, ¦Ò) is called a symbolic dynamics for ¦µ. If C is only a weak cross section and hence R is only partially de?ned, then ¦« also contains one- or two-sided ?nite coding sequences. Let C ¡ä be a set of representatives for the cross section C, that is, C ¡ä ? SH and ¦Ð|C ¡ä is a bijection C ¡ä ¡ú C. For each v ¡Ê C ¡ä let ¦Ãv be the geodesic determined by v. De?ne ¦Ó : C ¡ú ?H ¡Á ?H by ¦Ó (?) := (¦Ãv (¡Þ), ¦Ãv (?¡Þ)) v v where v = (¦Ð|C ¡ä )?1 (?). For some cross sections C it is possible to choose C ¡ä in a such way that ¦Ó is a bijection between C and some subset D of R ¡Á R. In this case the dynamical system (C, R) is conjugate to (D, F ), where F := ¦Ó ? R ? ¦Ó ?1 is the induced selfmap on D (partially de?ned if C is only a weak cross section). Moreover, to construct a symbolic dynamics for ¦µ, one can start with a decomposition of D into pairwise disjoint subsets D¦Á , ¦Á ¡Ê A.

6

J. HILGERT AND A. D. POHL

Finally, let (¦«, ¦Ò) be a symbolic dynamics with alphabet A. Suppose that we have a map i : ¦« ¡ú D for some D ? R such that i((an )n¡ÊZ ) depends only on (an )n¡ÊN0 , a (partial) selfmap F : D ¡ú D, and a decomposition of D into a disjoint union ¦Á¡ÊA D¦Á such that F (i((an )n¡ÊZ )) ¡Ê D¦Á ? a1 = ¦Á for all (an )n¡ÊZ ¡Ê ¦«. Then F , more precisely the triple (F, i, (D¦Á )¦Á¡ÊA ), is called a generating function for the future part of (¦«, ¦Ò). If such a generating function exists, then the future part of a coding sequence is independent of the past part. 3. Cusp expansions We consider the groups ¦£ := P¦£0 (p) :=

a b c d

¡Ê PSL(2, Z) c ¡Ô 0 mod p

for p prime. A subset F of H is a fundamental region for ¦£ in H if F is open, gF ¡É F = ? for all g ¡Ê ¦£ \ {id} and H = g¡Ê¦£ gF. If, in addition, F is connected, it is a fundamental domain. A fundamental set for ¦£ in H is a subset of H which contains precisely one representative of each ¦£-orbit. Clearly, each fundamental region is contained in a fundamental set. Changing H into SH in these de?nitions, one gets the notions of fundamental region, domain and set for ¦£ in SH. The isometric sphere b of an element g = a d ¡Ê ¦£ is the set I(g) := {z ¡Ê H | |cz + d| = 1}, c its exterior is de?ned by ext I(g) := {z ¡Ê H | |cz + d| > 1}. Note that I(g) is a (complete) geodesic arc if g does not ?x ¡Þ. The stabilizer of b ¡Þ in ¦£ is ¦£¡Þ = {( 1 1 ) | b ¡Ê Z }. Hence F¡Þ := {z ¡Ê H | 0 < Re z < 1} 0 is a fundamental domain for ¦£¡Þ in H. Following through the proofs of Thm. 22 in [1], we ?nd that F := F¡Þ ¡É ext I(g)

g¡Ê¦£\¦£¡Þ

is a fundamental domain for ¦£ in H. To construct F if su?ces to consider the isometric spheres Iq := {z ¡Ê H | |pz ? q| = 1} for q = 1, . . . , p ? 1. p?1 Then F = F¡Þ ¡É q=1 ext Iq . From F we can read o? that ¦£ has two cusps,

3 represented by ¡Þ and v0 := 0, see Fig. 1. The points vk := 2k+1 + i 2p 2p for k = 1, . . . , p ? 2 will be called the inner vertices of F . Note that vk is the intersection point of Ik and Ik+1 . Let vp?1 := 1 denote the other representative of the cusp 0 that is seen by F . Further let mk denote the i element of Ik with maximal imaginary part, that is, mk := k + p . p ¡Ì

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW

7

For a, b ¡Ê H let [a, b] denote the geodesic arc in H from a to b containing the points a and b, let [a, b) be the geodesic arc from a to b containing a but not b, and de?ne (a, b] and (a, b) analogously. For a, b ¡Ê R ¡È {¡Þ} the context will always clarify whether (a, b) refers to a geodesic arc or a real interval. Then the boundary components of F are the geodesic arcs (v0 , ¡Þ), (vp?1 , ¡Þ) and [vk , vk+1 ] for k = 0, . . . , p ? 2. For each vertex vk (k = 0, . . . , p ? 1) the set A(vk ) := z ¡Ê F

k p

¡Ü Re z ¡Ü

k+1 p

is the precell attached to vk . In other words, A(v0 ) and A(vp?1 ) are the A(v0 ) m1 A(v1 ) v1 m2 A(v2 ) v2 m3 A(v3 ) v3 m4 A(v4 )

v0 = 0

v4 = 1

Figure 1. Decomposition of F into precells for p = 5. hyperbolic triangles with vertices v0 , m1 , ¡Þ resp. vp?1 , mp?1 , ¡Þ. The precell A(vk ) for an inner vertex vk is the hyperbolic quadrangle with vertices mk , vk , mk+1 , ¡Þ. Obviously, the precells overlap only at boundaries and

p?1

F=

A(vk ).

k=0

Let D be a subset of H and x ¡Ê D. A unit tangent vector v at x is said to point into D if the geodesic ¦Ã, determined by ¦Ã(0) = x and ¦Ã ¡ä (0) = v runs into D, i. e., if there exists ¦Å > 0 such that ¦Ã((0, ¦Å)) ? D. The unit tangent vector v is said to point along the boundary of D if there is ¦Å > 0 such that ¦Ã((0, ¦Å)) ? ?D. It is said to point out of D if it points into H \ D. For each A(vk ) let A(vk ) be the set of unit tangent vectors that point into its interior A(vk )? . The boundary of A(vk ) then consists of the unit tangent vectors that are tangent to ?A(vk ). Lemma 3.1. The disjoint union

p?1

V :=

k=0

A(vk )

8

J. HILGERT AND A. D. POHL p?1 k=0

is a fundamental region for ¦£ in SH. Its boundary is

? A(vk ).

One easily checks that the determining element g ¡Ê ¦£ of the isometric sphere I(g) is unique up to left multiplication by ¦£¡Þ and that gI(g) = I(g ?1 ). This implies that for each k ¡Ê {1, . . . , p ? 1} there is a unique l ¡Ê {1, . . . , p ? 1} such that Ik is determined by gkl := E.g., for k = 1 we have l = p ? 1 and g1,p?1 = p?1 ?1 . We now de?ne p ?1 for each vk , k = 0, . . . , p ? 1, a set B(vk ), which we call a cell, as follows. For 1 ¡Ü k ¡Ü p ? 2 we set B(vk ) := {gA(vl ) | l = 1, . . . , p ? 1, g ¡Ê ¦£, gvl = vk }, and further B(v0 ) := A(v0 ) ¡È gp?1,1 A(vp?1 ) resp. B(vp?1 ) := A(vp?1 ) ¡È g1,p?1 A(v0 ). The ¦£-translates of the family {B(vk )}0¡Ük¡Üp?1 cover H, since the ¦£-translates of {A(vk )}0¡Ük¡Üp?1 do so. Lemma 3.2 shows that the family of cells meets the other demands from Sec. 1 as well. Lemma 3.2. For k = 0, . . . , p?1, the cell B(vk ) is the hyperbolic triangle with vertices k , k+1 and ¡Þ. If k ¡Ê {0, p ? 1}, then the non-vertical p p boundary component of B(vk ) is

1 gp?1,1 ¡¤ (vp?1 , ¡Þ) = v0 , p l ? 1+kl p p ?k

¡Ê ¦£.

resp.

g1,p?1 ¡¤ (¡Þ, v0 ) =

p?1 p , v1

.

If 1 ¡Ü k ¡Ü p ? 2, then there are unique a, b ¡Ê {1, . . . , p ? 2} such that The non-vertical boundary component of B(vk ) is ga,k+1 ¡¤

a+1 p ,¡Þ

B(vk ) = A(vk ) ¡È ga,k+1 A(va ) ¡È gb+1,k A(vb ). =

k k+1 p, p

b = gb+1,k ¡¤ ¡Þ, p .

Moreover, if g?B(vk )¡ÉB(vl ) = ? for some g ¡Ê ¦£ and k, l ¡Ê {0, . . . , p?1}, then g?B(vk ) ¡É B(vl ) ? ?B(vl ).

¡ä For k = 0, . . . , p ? 1 let Ck denote the set of unit tangent vectors based k on p + iR+ that point into B(vk )? , hence ¡ä Ck = X ¡Ê SH ? ? X = a ?x | k +iy + b ?y | k +iy , a > 0, b ¡Ê R, y > 0 .

p p

Further set which is the set of unit tangent vectors based on 1 + iR+ that point into B(vp?1 )? . Let pr : SH ¡ú H denote the canonical projection onto base points.

¡ä Cp := X ¡Ê SH ? ? X = a ?x |1+iy + b ?y |1+iy , a < 0, b ¡Ê R, y > 0 ,

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW ¡ä C0 ¡ä C1 ¡ä C2 ¡ä C3 ¡ä C4 ¡ä C5

9

B(v0 )

B(v1 )

B(v2 )

B(v3 )

B(v4 )

0

1 5

2 5

3 5

4 5

1

¡ä Figure 2. The sets Ck for p = 5.

Proposition 3.3. There are pairwise disjoint subsets B(vk ) of SH, k = 0, . . . , p, such that

¡ä (i) Ck ? B(vk ), (ii) the disjoint union

p k=0

B(vk ) is a fundamental set for ¦£ in SH,

(iii) pr(B(vp )) = B(vp?1 ) and pr(B(vk )) = B(vk ) for k = 0, . . . , p ? 1. Proof. We pick an extension of the fundamental region V = A(vk ) from Lemma 3.1 to a fundamental set E for ¦£ in SH by adding a suitable subset of ? A(vk ) to V . For z ¡Ê F let Ez denote the set of unit tangent vectors in E that are based at z. For each k = 0, . . . , p ? 1 and each z ¡Ê A(vk )? 1 2 pick a partition of Ez into three non-empty disjoint subsets Wk,z , Wk,z , 3 Wk,z . For z in a non-vertical boundary component of A(vk ), k = p?1 and 1 2 3 Re(z) ¡Ê { k , k+1 } let Wk,z := Ez and Wk,z = Wk,z := ?. For z contained p p in a non-vertical boundary component of A(vp?1 ) and Re(z) ¡Ê { p?1 , 1} / p 1 3 divide Ez into two non-empty disjoint subsets Wp?1,z , Wp?1,z and let 2 Wp?1,z := ?. For z with Re(z) = k/p for some k ¡Ê {0, . . . , p ? 1} let 1 Wk,z be the subset of Ez of all unit tangent vectors that point into A(vk ) 3 1 2 and let Wk,z := Ez \ Wk,z and Wk,z := ?. For z with Re(z) = 1 let 2 3 1 Wp?1,z := Ez and Wp?1,z = Wp?1,z := ?. Now we de?ne B(vk ) as follows. For 1 ¡Ü k ¡Ü p ? 2 let a, b be as in Lemma 3.2. Then B(vk ) :=

z¡ÊA(vk ) 1 Wk,z ¡È ga,k+1 ¡¤ 2 Wa,z ¡È gb+1,k ¡¤ 3 Wb,z .

z¡ÊA(va )

z¡ÊA(vb )

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J. HILGERT AND A. D. POHL

For k ¡Ê {0, p ? 1, p} we set B(v0 ) :=

z¡ÊA(v0 ) 1 W0,z

¡È

gp?1,1 ¡¤ ¡È ¡È

2 Wp?1,z z¡ÊA(vp?1 ) 2 W0,z z¡ÊA(v0 ) 3 W0,z . z¡ÊA(v0 )

B(vp?1 ) :=

z¡ÊA(vp?1 )

1 Wp?1,z

g1,p?1 ¡¤ g1,p?1 ¡¤

B(vp ) :=

z¡ÊA(vp?1 )

3 Wp?1,z

By Lemma 3.2 one sees that the B(vk ) satisfy all the claims of the proposition. For reasons that will become clear later (see Prop. 3.6) we shift B(vp ) ¡ä and Cp one unit to the left, hence let B(v?1 ) := Further we set C ¡ä :=

p 1 ?1 ¡¤ B(vp ) 0 1 p?1 ¡ä k=?1 Cp and ¡ä and C?1 := 1 ?1 0 1 ¡ä ¡¤ Cp .

C :=

k=0 y>0

S¦Ð( k +iy) Y

p

? ¦Ð ¡Ày 2 ?y | k +iy

p

.

The set C is the union of all unit tangent vectors in SY which are based at the boundary of some ¦Ð(B(vk )), k = 0, . . . , p ? 1, but are not tangent to it. We have the following lemma which shows that C ¡ä is a set of representatives for C. Lemma 3.4. ¦Ð|C ¡ä : C ¡ä ¡ú C is a bijection. The next major goal is the following theorem.

Let C := ¦£ ¡¤ C ¡ä = ¦Ð ?1 (C) and B := ¦£ ¡¤ p?1 ?B(vk ) = pr(C). A k=0 geodesic ¦Ã on Y intersects C i? some (and hence any) representative of ? ¦Ã intersects C. Since C is the set of all unit tangent vectors based in B ? that are not tangent to B, and since B is totally geodesic, a geodesic ¦Ã on H intersects C if and only if it intersects B transversely. Hence C satis?es (C2). Suppose that the geodesic ¦Ã intersects C in ¦Ã ¡ä (t0 ). By Lemma 3.4 there ? ? is a unique geodesic ¦Ã on H such that ¦Ã ¡ä (t0 ) ¡Ê C ¡ä and ¦Ð(¦Ã ¡ä (t0 )) = ¦Ã ¡ä (t0 ). ? If s := min{t > t0 | ¦Ã ¡ä (t) ¡Ê C} = min{t > t0 | ¦Ã(t) ¡Ê B}

Theorem 3.5. C is a cross section w. r. t. any measure induced by a Borel probability measure on SY invariant under {¦µt }t¡ÊR .

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW

11

exists, then s is the ?rst return time of ¦Ã ¡ä (t0 ) and R(? ¡ä (t0 )) = ¦Ð(¦Ã ¡ä (s)). ? ¦Ã We call ¦Ã ¡ä (s), resp. ¦Ã ¡ä (s), the next point of intersection of ¦Ã with C, resp. ? of ¦Ã with C. It might happen that ¦Ã ¡ä (s) ¡Ê C ¡ä . Then we say that ¦Ã ¡ä (t0 ) ? and ¦Ã ¡ä (t0 ) are elements with interior intersection in future. The next ? point of exterior intersection of ¦Ã and C shall be ¦Ã ¡ä (r), where r := min{t > t0 | ? g ¡Ê ¦£ \ {id} : ¦Ã ¡ä (t) ¡Ê gC ¡ä } if r exists. Analogously, we de?ne previous point of intersection, previous point of exterior intersection and elements with interior intersection in past. Prop. 3.6 below discusses which geodesics intersect C at all, which ones intersect it in?nitely often, and, moreover, if there is a next point, resp. was a previous point, of exterior intersection of the corresponding geodesic on H and on which copy of C ¡ä this point lies. Figs. 3 and 4 show the relevant ¦£-translates of C ¡ä . To simplify the statement let NC(¦£) denote the (?nite) set of geodesics on Y which have a representative on H which is tangent to B, and let h1,?¡Þ := h?1,?1 := h0,0 :=

?1 p ?1 p 1 0 p1 0 ?1 0 ?1

hc,k := gc?1,k+1 h?1,p?1 := g1,p?1 h0,p := ( 1 1 ) . 01

for 1 ¡Ü k ¡Ü p ? 2

Finally, let A := {?¡Þ, ?1, . . . , p}. Proposition 3.6. Let ¦Ã be a geodesic in Y . ? (i) (ii) (iii) (iv) ¦Ã intersects C i? ¦Ã ¡Ê NC(¦£). ? ? ¦Ã intersects C in?nitely often in future i? ¦Ã (¡Þ) ¡Ê ¦£{0, ¡Þ}. ? ? / ¦Ã intersects C in?nitely often in past i? ¦Ã (?¡Þ) ¡Ê ¦£{0, ¡Þ}. ? ? / Suppose that ¦Ã intersects C in ¦Ã ¡ä (t0 ). Let ¦Ã be the unique lifted ? ? geodesic on H such that ¦Ã ¡ä (t0 ) ¡Ê C ¡ä and ¦Ð(¦Ã ¡ä (t0 )) = ¦Ã ¡ä (t0 ). ? (a) There is a next point of exterior intersection of ¦Ã and C i? ¦Ã(¡Þ) ¡Ê k k = ?1, . . . , p . / p (b) If k < ¦Ã(¡Þ) < hc,k ¡Þ, k ¡Ê A \ {?1, p ? 1}, resp. h?1,?1 ¡Þ < p ¦Ã(¡Þ) < 0 or h?1,p?1 ¡Þ < ¦Ã(¡Þ) < 1, then the next point of ¡ä exterior intersection is on hc,k ¡¤ Cc . (c) There was a previous point of exterior intersection of ¦Ã and C 1 / i? ¦Ã(¡Þ) < 0 and ¦Ã(?¡Þ) = p , or ¦Ã(¡Þ) > 0 and ¦Ã(?¡Þ) ¡Ê k k = ?1, . . . , p ? 2 . p

12

J. HILGERT AND A. D. POHL

(d) We have the following cases: ¦Ã(?¡Þ) ¡Ê ¦Ã(¡Þ) ¡Ê

1 ? ¡Þ, ? p 1 ? p, 0 1 0, p 1 0, p 1 p, ¡Þ

prev pt of ext intsec on

¡ä h?1 ¡¤ C0 0,0 ?1 ¡ä h?1,?1 ¡¤ C?1 ?1 ¡ä h?1,?1 ¡¤ C?1 ¡ä h?1 ?1,p?1 ¡¤ Cp?1 ?1 ¡ä hk+1,b ¡¤ Cb ¡ä h?1 ¡¤ Cp?1 0,p

k k+1 p, p

, 1¡Ük ¡Üp?2

¡ä C1

(0, ¡Þ) 1 p, ¡Þ (?¡Þ, 0) (?¡Þ, 0) k+1 p ,¡Þ

¡ä C2 ¡ä C3

(0, ¡Þ)

¡ä h1,?¡Þ C1

¡ä ¡ä C?1 C0

¡ä C4

¡ä h0,5 C0

¡ä h?1,?1 C?1

¡ä h0,0 C0

¡ä h3,1 C3

¡ä h4,2 C4

¡ä h2,3 C2

¡ä h?1,4 C?1

1 ?5

0

1 5

2 5

3 5

4 5

1

Figure 3. The light gray part is the set C ¡ä , the dark ¡ä gray parts are ¦£-translates of the Ck as indicated. This ?gure shows the translates important for determining the next point of exterior intersection.

¡ä h?1 C4 0,5 ¡ä h?1 C0 0,0

¡ä ¡ä C?1 C0

¡ä h?1 C?1 ?1,?1

¡ä ¡ä h?1 C4 C1 ?1,4

¡ä C2 ¡ä h?1 C3 2,3 ¡ä h?1 C1 3,1

¡ä C3 ¡ä h?1 C2 4,2

¡ä C4

?1 5

0

1 5

2 5

3 5

4 5

Figure 4. Here one can read o? the previous point of exterior intersection. Prop. 3.6 shows that exactly the geodesics on Y with one or two limit points in the cusps do not intersect C in?nitely often. Let E denote the set of unit tangent vectors to these geodesics. Adapting the proof in [2], p. 59, to our situation, we get that E is a null set w. r. t. any Borel probability measure on SY which is invariant under {¦µt }t¡ÊR . Therefore C also satis?es (C1) and hence is a cross section. The maximal strong cross section contained in C is Cs := C \ E.

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW

13

Recall the map ¦Ó from Section 2. If v ¡Ê C is an element with interior ? intersection in future, then ¦Ó (?) = ¦Ó (R(?)) since the geodesics on H v v corresponding to v resp. to R(?) only di?er by their parametrization. ? v Likewise, if v ¡Ê C is an element with interior intersection in past, then ? ¦Ó (?) = ¦Ó (R?1 (?)). Figs. 3 and 4 clearly show that C contains both, v v elements with interior intersection in future and in past. Hence it is impossible to ?nd a dynamical system on ?H that is conjugate to (C, R). To overcome this problem we reduce C to another cross section. Let P denote the set of elements in C that have interior intersection in future or past and set Cred := C \ P resp. Cred,s := Cs \ P.

Prop. 3.6 implies that each geodesic on Y that intersects C in?nitely often, also intersects Cred,s in?nitely often. Therefore both Cred and Cred,s are cross sections and Cred,s is a strong cross section. For 0 ¡Ü k ¡Ü p ? 1 let Dk := Further set

1 D?¡Þ := ? ¡Þ, ? p ¡Á (0, ¡Þ), k k+1 p, p

¡Á ? ¡Þ, k . p

D?1 := ? 1 , 0 ¡Á (0, ¡Þ), p Dk,s := Dk \ ¦£{0, ¡Þ} ¡Á ¦£{0, ¡Þ}

k¡ÊA

Dp := (1, ¡Þ) ¡Á (?¡Þ, 1),

and hk := hc,k for k ¡Ê A. De?ne the selfmap F on D := Fs on Ds := k¡ÊA Dk,s by F |Dk := h?1 k resp. Fs |Dk,s := h?1 . k

Dk resp.

Then Prop. 3.6 implies that the two diagrams Cred

¦Ó R

/C red

¦Ó

Cred,s

¦Ó

R

/ Cred,s

¦Ó

D

e F

/D

Ds

e Fs

/D s

commute. The sets Dk resp. Dk,s satisfy all the properties announced in Sec. 1. Therefore we naturally get a symbolic dynamics with alphabet A and a natural generating function for its future part.

14

J. HILGERT AND A. D. POHL

3.1. Associated transfer operators. We restrict ourselves to the strong cross section Cred,s . Let D := prx (Ds ) = R \ ¦£{0, ¡Þ} and Dk := prx (Dk,s ) for k ¡Ê A. The discrete dynamical system (Ds , Fs ) induces a symbolic dynamics with generating function F : D ¡ú D given by F |Dk := h?1 for k ¡Ê A. Its local inverses are k

?1 Fk := (F |Dk ) : F (Dk ) ¡ú Dk , x ¡ú hk x.

Then the transfer operator with parameter ¦Â L¦Â : {densities on D} ¡ú {densities on D} associated to F is given by (L¦Â ?)(x) =

k¡ÊA ¡ä Fk (x)¦Â ?(Fk (x))¦ÖF (Dk ) (x),

where ¦ÖF (Dk ) is the characteristic function of F (Dk ), and the maps Fk ¡ä and Fk are extended arbitrarily on D \ F (Dk ). 3.2. The case of the modular surface. We apply the cusp expansion to the modular group ¦£ = PSL(2, Z). All proofs are omitted as they are analogous to those for P¦£0 (p). The set F := {z ¡Ê H | 0 < Re z < 1, |z| > 1, |z ? 1| > 1} is a fundamental domain for ¦£ in H. Note that F = F¡Þ ¡É ext I(g)

g¡Ê¦£\¦£¡Þ

¡Ì with F¡Þ := {z ¡Ê H | 0 < Re z < 1}. The point v := 1 (1 + i 3) is the 2 only (inner) vertex of F . Its associated precell A(v) coincides with F . The cell B(v) is the hyperbolic triangle with vertices 0, 1, ¡Þ. As before, set Y := ¦£\H and let ¦Ð : SH ¡ú SY denote the canonical projection. Further, let C be the set of unit tangent vectors in SY which are based on ¦Ð(B(v)), but that are not tangent to it. Then C is a weak cross section with set of representatives C ¡ä := X ¡Ê SH

? ? X = a ?x |iy + b ?y |iy , y > 0, a > 0, b ¡Ê R .

The elements of C ¡ä are the unit tangent vectors based on iR+ that point into B(v)? . One easily sees that C does not contain elements with interior

SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW

15

intersection. Let Cs denote the maximal strong cross section contained in C. Then (Cs , R) is conjugate to (Ds , Fs ), where Ds := D0,s ¡È D1,s and D0,s := (0, 1) \ Q ¡Á (?¡Þ, 0) \ Q , D1,s := (1, ¡Þ) \ Q ¡Á (?¡Þ, 0) \ Q , Fs |D0,s := e Fs |D1,s := e

1 0 ?1 1 1 ?1 0 1

, .

Note that Q = ¦£¡Þ. Set D := R+ \ Q, D0 := (0, 1) \ Q and D1 := (1, ¡Þ) \ 1 0 Q. Then the function F : D ¡ú D, given by F |D0 := ?1 1 , F |D1 := 1 ?1 0 1 , is the ?rst component of Fs restricted to the ?rst variable. Its associated transfer operator with parameter ¦Â is (L¦Â ?) (x) = ?(x + 1) + (x + 1)?2¦Â ?

x x+1

,

x ¡Ê D.

If we extend L¦Â to act on densities on R+ by the obvious formula, then the eigenfunctions of L¦Â with eigenvalue 1 are precisely the solutions of ?(x) = ?(x + 1) + (x + 1)?2¦Â ?

x x+1

,

x ¡Ê R+ .

This is exactly the functional equation used in [4]. We do not think that it is a coincidence that eigenfunctions of Mayer¡¯s transfer operator are related to those of L¦Â , since F is closely related to the Farey process or slow continued fractions, whereas the Gauss map is the generating function for (ordinary) continued fractions. 4. Outlook The construction of a symbolic dynamics for the geodesic ?ow on ¦£\H via the method of cusp expansion seems to work for a wide class of subgroups ¦£ of PSL(2, R). Some necessary properties of such a group ¦£ are that ¡Þ is a cusp and that the boundary of ext I(g)

g¡Ê¦£\¦£¡Þ

contains the maxima of the relevant isometric spheres. For example, cusp expansion is applicable to Hecke triangle groups, thus also to certain nonarithmetic groups. The precells in H can be de?ned independently of the choice of a fundamental domain for ¦£ in H. But if there is no close relation between the union of precells and a fundamental domain for ¦£, we do not expect to be able to construct symbolic dynamics with this method. On the positive side, we do expect it to work for certain groups acting on higher-dimensional real hyperbolic spaces and, more generally, on rank one symmetric spaces of non-compact type.

16

J. HILGERT AND A. D. POHL

References

1. L. Ford, Automorphic functions, Chelsea publishing company, New York, 1972. 2. S. Katok and I. Ugarcovici, Arithmetic coding of geodesics on the modular surface via continued fractions, European women in mathematics¡ªMarseille 2003, CWI Tract, vol. 135, Centrum Wisk. Inform., Amsterdam, 2005, pp. 59¨C77. 3. , Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 87¨C132 (electronic). 4. J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. (2) 153 (2001), no. 1, 191¨C258. 5. D. Mayer, The thermodynamic formalism approach to Selberg¡¯s zeta function for PSL(2, Z), Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 1, 55¨C60. 6. C. Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), no. 1, 69¨C80. 7. L. Vulakh, Farey polytopes and continued fractions associated with discrete hyperbolic groups, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2295¨C2323. ¡§ ¡§ Institut fur Mathematik, Universitat Paderborn, 33095 Paderborn, Germany E-mail address: {hilgert,pohl}@math.upb.de