Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 359–372

A NOTE ON COUNTING CUSPIDAL EXCURSIONS

Bernd Stratmann

Universität Göttingen, Mathematisches Institut, SFB 170 Bunsenstr. 3–5, D-37073 Göttingen, Germany; stratman@cfgauss.uni-math.gwdg.de

Abstract. For geometrically finite Kleinian groups with parabolic elements we study that part of the Lagrange spectrum which does not lie in the Markov spectrum. Using the ergodicity of the associated geodesic flow with respect to the Liouville–Patterson measure, we obtain an estimate for the asymptotic frequency with which recurrent geodesics enter certain cusp regions. In particular, this allows a quantitative description of the logarithmic affinity of geodesic excursions for the cusps.

1. Introduction

We consider geometrically finite (N + 1)-manifolds M of constant negative curvature with cusps. These are manifolds which admit a representation by a finite sided, convex, fundamental polyhedron Δ, a subset of the universal covering space. The model for the universal covering space in use is the Poincaré ball model (D^{N+1}, d), where d denotes the hyperbolic metric on the (N + 1)-dimensional unit ball. M can be thought of as being a compact manifold M_{o} to which a finite number of ‘cusps’ (‘imploding contact with infinity’) and possibly a finite number of ‘funnels’ (‘exploding contact with infinity’) are attached. The fundamental group G of M is a discrete subgroup of Con(N), the group of all orientation preserving diffeomorphisms of D^{N+1} , and is usually referred to as a geometrically finite Kleinian group (we exclude the possibility of G being elementary). For the investigation of Kleinian groups a certain parameter δ(G) has proved to be of central interest. For any positive s form the Dirichlet series

∑_{g∈G} e^{−sd(0,g0)},

then δ = δ(G) is defined to be the exponent of convergence of this series and is called the exponent of convergence of G.

In this paper we study a certain quantitative aspect of the geodesic dynamic on M. Let ℛ(Δ) denote the recurrent part of the geodesic flow on M. ℛ(Δ) is a subset of the unit tangent bundle 𝒮 (Δ). Each element of ℛ(Δ) gives rise to a recurrent geodesic on M, that is a geodesic which returns infinitely often into the compact part M_{o} of M, and neither starts nor ends in a funnel. For obvious geometric reasons the dynamic inside the funnels, that is, geodesics which either start or end in a funnel, is of minor interest here. A consequence of the existence of funnels is that the relevant ‘canonical measure’ is no longer comparable to the Lebesque measure. In the geometrically finite case the relevant measure turns out to be the Patterson measure, which we denote in the following by μ (for a construction of μ we refer to [11] and [10]). It was shown by Sullivan ([16], [10, Theorem 4.4.4]), that μ is an ergodic probability measure (with respect to the action of G on L(G)). The support of μ is the limit set L(G), the set of accumulation points of the G-orbit of a point x in D^{N+1} . The Patterson measure can be viewed as a measure on the fibre ℛ_{x}(Δ) and gives rise to a geodesic flow invariant, ergodic measure ν̃ on ℛ(Δ). The measure ν̃ will be called the Liouville– Patterson measure.

1991 Mathematics Subject Classification: Primary 30F40.

Research supported by the Sonderforschungsbereich 170, University of Göttingen.

In [15, Proposition 4.9] we worked out measure theoretical results concerning the ‘speed’ with which geodesic rays gradually occupy the cusp regions of M. In particular we gave an elementary proof of a generalized version of Sullivan’s logarithmic law for geodesics (LLG). This law considers the ‘distance function’

ℛ_{o}(Δ) × R^{+} ∋ (\underline{v}, t) ⟼ N_{t}(\underline{v}) ∈ R^{+},

where N_{t}(\underline{v}) measures the hyperbolic distance between a suitable chosen compact region M_{o} in M and the point one reaches after ‘travelling’ the hyperbolic distance t along the geodesic ray corresponding to \underline{v}. The ‘logarithmic law for geodesics’ then states that, for μ-almost all \underline{v} in the fibre ℛ_{o}(Δ),

(LLG)  lim sup_{t→∞} \frac{N_{t}(\underline{v})}{log t} = \frac{1}{2δ − k_{max}} ,

where k_{max} denotes the maximal occurring rank for a cusp in M. This result expresses the affinity of the geodesics for the cusp regions; but it does not give any information concerning the ‘rate’ with which a geodesic eventually makes its way deeper and deeper into the cusps.

In this paper we add to the qualitative results of [15] a quantitative aspect concerning the ‘frequency’ with which certain cusp regions are visited by ν̃-almost all recurrent geodesics. In order to illustrate our results, we assume that the compact part M_{o} is chosen to be sufficiently large. By C we denote a connected component of M \ M_{o} which represents a cusp of M; and for a positive number λ we let C(λ) denote that subregion of C which lies at a hyperbolic distance λ from M_{o}. We then consider the ‘counting function’

ℛ(Δ) × R^{+} × (0, 1] ∋ (\underline{v}, t, ε) ⟼ α_{t}^{p}(\underline{v}, ε) ∈ N,

where α_{t}^{p}(\underline{v}, ε) counts the number of visits to C(log(1/ε)) made after starting at the base point of \underline{v} in D^{N+1} and then ‘travelling’ a hyperbolic distance t along the geodesic in M which is determined by \underline{v}. Generalizing results obtained for cofinite Kleinian groups by Moeckel and Nakada ([8], [9]), we derive a law for the asymptotic frequency of cusp excursions (LAF) for the general case of geometrically finite Kleinian groups with parabolic elements. This law states the existence of positive constants c and c′, depending on the group G, such that, for all sufficiently small, positive ε and for ν̃-almost all \underline{v} in ℛ(Δ),

(LAF)  c ε^{2δ−k(p)} ≤ lim_{t→∞} \frac{α_{t}^{p}(\underline{v}, ε)}{t} ≤ c′ ε^{2δ−k(p)};

where k(p) denotes the rank of the cusp p under consideration.

An immediate consequence of this result is an estimate for the ‘asymptotic expectation’ of the number of cusp visits to a particular parabolic cusp with respect to the number of cusp visits to some other parabolic cusp. To be precise, we deduce the existence of positive constants c_{o} and c′_{o} such that, for all sufficiently small, positive ε, for all pairs (p, q) of different cusps and for ν̃-almost all \underline{v} and \underline{w} in ℛ(Δ),

c_{o} ε^{k(q)−k(p)} ≤ lim_{t→∞} \frac{α_{t}^{p}(\underline{v}, ε)}{α_{t}^{q}(\underline{w}, ε)} ≤ c′_{o} ε^{k(q)−k(p)}.

Finally, we convert the statement of the law (LAF) into the language of hyperbolic rays which start at the origin in D^{N+1} and terminate in the limit set of the Kleinian group G. We derive for μ-almost all limit points an asymptotic estimate on the number of ‘ε-squeezed standard horoballs’ which are intersected after ‘travelling’ a hyperbolic distance t along those rays.

Also, we would like to point out that the results of this paper have a natural interpretation in terms of elementary number theory. For this we recall some well known facts from the theory of metric Diophantine approximation. For a positive number c, an irrational number θ is called c-approximable if the inequality

|θ − \frac{p}{q}| < \frac{c}{q^{2}}

can be fulfilled for infinitely many reduced fractions p/q, with q positive. Perron ([12]) first noticed that one can associate to each irrational number θ a positive number

ϱ(θ) := inf_{c>0} {c : θ is c-approximable}.

For c greater than or equal to the Hurwitz number 1/\sqrt{5}, each irrational number is c-approximable and thus the Lagrange spectrum

ℒ := {ϱ(θ) : θ irrational}

is a subset of the interval (0, 1/\sqrt{5} ]. Markov showed that the intersection of ℒ with the interval (1/3, 1/\sqrt{5} ] comprises a set ℳ of countably many numbers which accumulate only at the value 1/3 ([7]). The set ℳ is called the Markov spectrum. In [3] it was shown in detail how to derive the continued fraction expansion of an irrational number from a coding for geodesics on the modular surface P SL_{2}(R)/P SL_{2}(Z) (see also [13], [14]). In particular it turns out that the set ℳ corresponds to the set of simple (not self-intersecting) geodesic loops on that surface. The results in this paper thus allow statements concerning the complement of ℳ in ℒ .

If λ denotes the 1-dimensional Lebesgue measure and if we use the usual notation for the regular continued fraction expansion of an element θ in the unit interval, namely

θ = \frac{1}{θ_{1} + \frac{1}{θ_{2} + ⋯}} = [θ_{1}, θ_{2}, …],

then the results in this paper give rise to the following number theoretical fact.

For all N in N and for λ-almost all θ = [θ_{1}, θ_{2}, …] in the unit interval we have

lim_{n→∞} \frac{card\{m : 1 ≤ m ≤ n and θ_{m} ≥ N\}}{n + ∑_{i=1}^{n} log θ_{i}} = k_{o} · log(1 + \frac{1}{N} );

where

k_{o} := (log 2 + ∑_{n=1}^{∞} log n · log(1 + \frac{1}{n(n + 2)}))^{−1} = (log 5.2 …)^{−1}.

We remark that this result may be derived also from an application of the ergodic theorem to the continued fraction map (see also [6]).

The author would like to thank the Sonderforschungsbereich 170 at the University of Göttingen for its support and hospitality.

2. Preliminaries

In this section we recall some concepts of the theory of Kleinian groups and their associated geodesic dynamic. These concepts should provide the reader with the necessary background for an understanding of the following section. As already mentioned in the introduction, G < Con(N) always denotes a non-elementary, geometrically finite Kleinian group which contains parabolic elements; by Δ we denote a convex, finite sided fundamental polyhedron for the discontinuous action of G on D^{N+1} . If H(L(G)) denotes the convex hull of the limit set L(G), we may then assume without loss of generality, that the origin 0 in D^{N+1} is an element of Δ ∩ H(L(G)). Our list of required concepts begins with a brief resumé on the subject of parabolic fixed points.

Parabolic cusps. Let P denote a complete set of inequivalent parabolic fixed points of G. It is well known that P is a finite set of points in S^{N}. We assume that each element of P lies at the boundary of Δ. If P is chosen in this fashion, then the elements of P are called basic parabolic fixed points. To each basic parabolic fixed point p we associate a horoball H_{p} of radius r(p), that is a Euclidian N -ball in D^{N+1} of radius r(p) which ‘touches’ S^{N} at the point p. For p in P we let G_{p} denote the stabilizer of p in G. Then it is a well known fact that the radii r_{p} of the ‘basic horoballs’ H_{p} can be chosen such that the set ⋃_{p∈P} {g(H_{p}) : g ∈ G/G_{p}} comprises a set of pairwise disjoint horoballs ([2, p. 248]). If in particular the radii r(p) are chosen to be maximal with respect to this disjointness property (which we shall assume from now on), then the G-orbit of these horoballs represents the so called standard set of horoballs and this set will be denoted by {H_{g(p)}(r_{g}) : g ∈ 𝒯_{p}, p ∈ P }, where 𝒯_{p}, the so called top representation of G_{p} in G, denotes a geometrically chosen set of coset representatives of G/G_{p} (for more details on this the reader is referred to [15, Chapter 2]). For a positive ε less than or equal to 1 and for p in P, we define H_{ε}^{p} ⊆ H_{p} to be the horoball at p of radius ε r(p).

A consequence of the Bieberbach theorem is that, for each p in P, the stabilizer G_{p} contains a subgroup G_{p}^{∗} of finite index such that G_{p}^{∗} is isomorphic to Z^{k(p)} for some k(p) in N. The number k(p) is called the rank of p. It then follows that there exists a ‘fundamental domain’ Q_{∞}^{p} for the action of G_{p}^{∗} on S^{N} \ {p} (Q_{∞}^{p} is the image under the Cayley transformation of the ‘fundamental parallelopiped’ introduced in [15]). By this we mean an N -dimensional, closed subset of S^{N}, whose spherical diameter q_{∞}(p) is chosen to be minimal with respect to the following properties:

1. L(G) \ {p} ⊂ G_{p}^{∗}(Q_{∞}^{p});

2. for all distinct g and h in G_{p}^{∗},

g(int (Q_{∞}^{p})) ∩ (int (Q_{∞}^{p})) = ∅;

3. Q_{∞}^{p} lies ‘opposite’ p, i.e., every geodesic with initial point in Q_{∞}^{p} and end point p has non-trivial intersection with Δ ∩ ∂H_{p}. We let Q_{∞}^{p} × {p} denote the set of geodesics which have their initial point in

Q_{∞}^{p} and which end in p. Further, for positive ε, we define the subset Q_{ε}^{p} of ∂H_{ε}^{p} by

Q_{ε}^{p} := (Q_{∞}^{p} × {p}) ∩ ∂H_{ε}^{p}.

The hyperbolic diameter of Q_{ε}^{p} will be denoted by q_{ε}(p).

Geodesic flows. If 𝒮 (D^{N+1}) denotes the unit tangent bundle over D^{N+1} , then 𝒮 (D^{N+1}) is the union of fibres 𝒮_{z}(D^{N+1}), where the union is taken with respect to all base points z in D^{N+1} . The canonical projection pr : 𝒮 (D^{N+1}) → D^{N+1} maps an element of 𝒮 (D^{N+1}) to the base point of the fibre of which it is an element.

On 𝒮 (D^{N+1}) we introduce the metric d^{∗}, which is induced by an additive combination of the hyperbolic metric d and the metric θ, where θ is defined as follows. Consider in 𝒮 (D^{N+1}) two elements \underline{v} and \underline{w} such that pr(\underline{v}) and pr(\underline{w}) are close. Move \underline{v} from pr(\underline{v}) to pr(\underline{w}) by parallel displacement, which leads to an element \underline{v}′. Then, θ(\underline{v}, \underline{w}) is defined to be the angle (at most π) between \underline{v}′ and \underline{w}.

As usual, an element \underline{v} in 𝒮 (D^{N+1}) is parameterized by a triple (v^{−}, v^{+}, s) in ((S^{N} × S^{N}) \ {diag .}) × R, where v^{−}, respectively v^{+} , denotes the initial respectively end point of the unique geodesic γ(\underline{v}) determined by \underline{v}, and s is the ‘signed’ hyperbolic distance between the summit of the geodesic γ(\underline{v}) (i.e., that point on γ(\underline{v}) which lies at maximal Euclidian distance from the boundary S^{N}) and the base point pr(\underline{v}); ‘signed’ means that s is positive if and only if pr(\underline{v}) is an element of that half of γ(\underline{v}) which lies between v^{+} and the summit.

Let Φ_{t} denote the geodesic flow on 𝒮 (D^{N+1}). We recall that Φ_{t} is defined for t in R and \underline{v} = (v^{−}, v^{+}, s) in 𝒮 (D^{N+1}), by

Φ_{t}(v^{−}, v^{+}, s) = (v^{−}, v^{+}, s + t).

On 𝒮 (D^{N+1}) we introduce an equivalence relation ‘∼’ as follows. Two elements \underline{v} and \underline{w} of 𝒮 (D^{N+1}) are equivalent if and only if there exists an element g in G such that γ(\underline{v}) = g(γ(\underline{w})) and pr(\underline{v}) = g(pr(\underline{w})) are satisfied. The unit tangent bundle over Δ is then defined by

𝒮 (Δ) := 𝒮 (D^{N+1})/ ∼ .

Factoring out this equivalence relation, we see that the flow Φ_{t} on 𝒮 (D^{N+1}) induces a flow ϕ_{t} on 𝒮 (Δ), and ϕ_{t} is called the geodesic flow on 𝒮 (Δ).

In this paper we are mainly interested in a certain subset ℛ(Δ) of 𝒮 (Δ). Roughly speaking, the elements of ℛ(Δ) give rise to geodesics on the manifold D^{N+1}/G which are recurrent in either their future or their past, but neither start nor end in a funnel. To be precise, ℛ(Δ) is defined as follows:

ℛ(Δ) := \{\underline{v} ∈ 𝒮 (Δ) : v^{−}, v^{+} ∈ L(G) and pr(\underline{v}) ∈ Δ\}.

If ℒ (G) denotes the set of loxodromic fixed points of G, then it is known ([1, Theorem 5.3.8]) that (ℒ (G)×ℒ (G))\{diag .} is dense in (L(G)×L(G))\{diag .}. Since each element of (ℒ (G) × ℒ (G)) \ {diag .} gives rise to a closed geodesic on the manifold D^{N+1}/G, it follows that the set ℛ(Δ) admits an interpretation as the closure of the set of all closed geodesics on D^{N+1}/G.

Local cross sections. In order to define special subsets of ℛ(Δ), we recall the notion of a local cross section with respect to some fixed time. For this let ϱ denote some positive number. A closed subset 𝒞 of 𝒮 (Δ) which lies transversal to the flow ϕ_{t}, is called a local cross section for ϕ_{t} with respect to the time ϱ if one has for each element \underline{v} in 𝒞 ,

⋃_{|t|<ϱ} ϕ_{t}(\underline{v}) ∩ 𝒞 = \underline{v}.

In the following section a central part is played by certain subsets of ℛ(Δ). The elements of these sets are based on the boundary of ‘ε-squeezed’ standard horoballs and point inward into these horoballs. To be precise, for positive ε less than or equal to 1 and for p in P, we define the following cross section which is based on Δ ∩ ∂H_{ε}^{p},

𝒞^{p}(ε) := \{\underline{v} = (v^{−}, v^{+}, s) ∈ ℛ(Δ) : v^{−} ∈ Q_{∞}^{p}, pr(\underline{v}) ∈ Q_{ε}^{p} and s < 0\}.

From the construction it is evident that there exists a positive constant τ = τ(G), such that, for each p in P and for each positive ε less than or equal to 1, the section 𝒞^{p}(ε) is a local cross section for ϕ_{t} with respect to τ.

Canonical measures. As we have already said in the introduction, we denote by μ the Patterson measure. For geometrically finite Kleinian groups it is well known that μ is an ergodic and ‘δ -conformal’ probability measure which is supported on the limit set L(G) and which has no atomic part ([10], [16]). By δ -conformal we mean that for each Borel subset E of S^{N} and for each g in G,

μ(g(E)) = ∫_{E} P (g^{−1}(0), ξ)^{δ(G)} dμ(ξ);

where the Poisson kernel P is defined as usual for z in D^{N+1} and ξ in S^{N} by P (z, ξ) := (1 − |z|^{2}) · |z − ξ|^{−2} .

For g in Con(N) and x, y in D^{N+1} ∪ S^{N}, an elementary calculation shows that

|g(x) − g(y)|^{2} = |g′(x)| · |g′(y)| · |x − y|^{2}.

Using this estimate we obtain a (G × G)-invariant measure ν on (L(G) × L(G)) \ {diag .} by

dν(ξ, η) := \frac{dμ(ξ) dμ(η)}{|ξ − η|^{2δ}}.

Using the above mentioned parametrization of 𝒮 (Δ), it is now possible to define on 𝒮 (Δ) a measure ν̃ by

dν̃(ξ, η, s) := dν(ξ, η) · ds.

The measure ν̃ is called the Liouville–Patterson measure. For the geometrically finite case it was shown by Sullivan ([16]) that ν̃ is a finite and ϕ_{t} -invariant measure, and further that the geodesic flow is ergodic with respect to ν̃. Because of its finiteness, we can assume in the following without loss of generality that ν̃ is actually a probability measure on ℛ(Δ).

Geometry of horoballs. For ξ in S^{N} let s_{ξ} denote the hyperbolic ray in D^{N+1} with initial point the origin and with end point ξ. For positive t, we denote by ξ_{t} the point on s_{ξ} which lies at a hyperbolic distance t from the origin.

Further, we let s_{ξ_{t}} be the geodesic segment on s_{ξ} which lies between ξ_{t} and the origin. The ‘shadow projection’ Π is a map from D^{N+1} onto S^{N} which is defined for subsets A of D^{N+1} by

Π(A) := {ξ ∈ S^{N} : s_{ξ} ∩ A ≠ ∅}.

For the moment let us fix a horoball H(1) in D^{N+1} with diameter 1. Thus in particular 0 is an element of H(1). For positive ε less than or equal to 1 let H(ε) ⊆ H(1) denote the horoball of radius ε and with the same point of tangency at S^{N} as H(1). Let η(ε) in S^{N} be chosen such that s_{η(ε)} is tangent to H(ε). If further z(ε) denotes the unique point in D^{N+1} at which s_{η(ε)} intersects ∂H(1) \ {0}, then we derive the following ‘horoball excursion formula’.

Lemma 1. There exist positive constants c_{1} and c_{2} such that, for each positive ε less than or equal to 1, we have

(HEF1)  c_{1} ε^{2} ≤ 1 − |z(ε)| ≤ c_{2} ε^{2}.

Proof. Let z^{∗}(ε) be the point of tangency of s_{η(ε)} at H(ε). From the theorem of Pythagoras we deduce that

(1 − ε)^{2} = |z^{∗}(ε)|^{2} + ε^{2},

and thus

2ε = 1 − |z^{∗}(ε)|^{2}.

On the other hand, it is well known ([1]) that there exist universal, positive constants c and c′ such that

c (1 − |z^{∗}(ε)|) ≤ e^{−d(0,z^{∗}(ε))} ≤ c′ (1 − |z^{∗}(ε)|),

and thus

\frac{1}{2}c (1 − |z^{∗}(ε)|^{2}) ≤ e^{−d(0,z^{∗}(ε))} ≤ c′ (1 − |z^{∗}(ε)|^{2}).

Combining these two estimates, it follows that

c ε ≤ e^{−d(0,z^{∗}(ε))} ≤ 2c′ ε.

Since d(0, z(ε)) = 2d(0, z^{∗}(ε)), we have that

c^{2} ε^{2} ≤ e^{−d(0,z(ε))} ≤ 4c^{′^{2}} ε^{2},

and hence

\frac{c^{2}}{c′} ε^{2} ≤ 1 − |z(ε)| ≤ \frac{4c^{′^{2}}}{c} ε^{2}.

For a fixed, positive number q we now partition ∂H(1) into spherical annuli of hyperbolic width q; i.e., we define annuli inductively as follows:

A_{1} := {z ∈ ∂H(1) : d(0, z) ≤ q};

and for n in N greater than 1, let

A_{n} := {z ∈ ∂H(1) \ ⋃_{k=1}^{n−1} A_{k} : 0 < d(z, A_{n−1}) ≤ q}.

The following ‘horoball excursion formula’ is easily obtained from Lemma 1 and we shall therefore omit the proof.

Lemma 2. There exist positive constants c_{3} and c_{4} , depending on q, such that, for each n in N and z in A_{n}, we have

(HEF2)  \frac{c_{3}}{n^{2}} ≤ 1 − |z| ≤ \frac{c_{4}}{n^{2}} .

3. Asymptotic frequencies

In this section we shall prove our main result. We begin by giving an estimate for the Liouville–Patterson measure of the local cross sections with respect to the time τ which were introduced in the preceding section. The estimate is required in the proof of our theorem.

Proposition 1. There exist positive constants k_{1}, k_{2} and ε_{o} such that, for each p in P and for each positive ε less than ε_{o}, we have

k_{1} ε^{2δ−k(p)} ≤ ν(𝒞^{p}(ε)) ≤ k_{2} ε^{2δ−k(p)}.

Proof. Let p in P be given. For convenience, we assume that the origin 0 in D^{N+1} is an element of H_{p} ∩ H(L(G)) (for the kind of arguments we will give in the following, this simplifying assumption changes only the constants which occur by a factor which depends on the hyperbolic distance between 0 and ∂H_{p}). It is known that (∂H_{p} ∩ H(L(G)))/G_{p}^{∗} is a compact subset of D^{N+1} ([17], [15, Chapter 2]). Using this fact it follows that we can choose a ‘horospherical fundamental domain’ Q_{1}^{p} on ∂H_{p}, that is a fundamental domain for the action of G_{p}^{∗} on H_{p} whose hyperbolic diameter q_{1}(p) is minimal with respect to this property. We can further assume that Q_{1}^{p} is chosen such that it contains the origin. Now, we define horospherical annuli on ∂H_{p} of constant hyperbolic width q_{1}(p) as follows.

Let

D_{1}^{p} := {z ∈ ∂H_{p} : d(0, z) ≤ q_{1}(p)};

and for m in N greater than 1, let

D_{m}^{p} := {z ∈ ∂H_{p} \ ⋃_{k=1}^{m−1} D_{k}^{p} : 0 < d(z, D_{m−1}^{p}) ≤ q_{1}(p)}.

An elementary Euclidean volume argument then implies the existence of positive constants c_{1} and c_{2} such that

(1)  c_{1} n^{k(p)−1} ≤ card{g ∈ G_{p}^{∗} : g(0) ∈ D_{n}^{p}} ≤ c_{2} n^{k(p)−1}.

Let g be in G_{p}^{∗} such that g(0) is contained in D_{n}^{p} for some n in N. Using the formula (HEF 2), derived in the preceding section, one obtains the existence of positive constants c_{3} and c_{4} , depending on q_{1}(p), such that

c_{3} n^{−2} ≤ 1 − |g(0)| ≤ c_{4} n^{−2}.

Combining this estimate with the fact that |ξ − g^{−1}(0)| is bounded from above and below for all ξ in Q_{∞}^{p}, it follows that

(2) μ(g(Q_{∞}^{p})) = ∫_{Q_{∞}^{p}} P (g^{−1}(0), ξ)^{δ} dμ(ξ) ≍ μ(Q_{∞}^{p}) · (1 − |g(0)|^{2})^{δ} ≍ n^{−2δ};

Here and in the following, the sign ‘≍’ is used to denote that the quotient of the two related quantities is bounded from below and above.

Combining (1) and (2), we obtain the existence of positive constants c_{5} and c_{6} such that

(3)  c_{5} n^{k(p)−1−2δ} ≤ μ( ⋃_{{g∈G^{∗}_{p} \atop g(0)∈D^{p}_{n}}} g(Q_{∞}^{p})) ≤ c_{6} n^{k(p)−1−2δ}.

Let ε be positive and less than or equal to 1. As mentioned in the preceding section, a cross section of the geodesic flow ϕ_{t} admits a representation as a subset of S^{N} × S^{N}. In particular, there exists a uniquely determined subset R_{ε}^{p} of S^{N} such that 𝒞^{p}(ε) is represented in this fashion by Q_{∞}^{p} ×R_{ε}^{p}. From the construction it then follows that there exists a positive constant ϱ_{o}, depending on q_{1}(p), such that

(4)  Π(H_{p}(ϱ_{o}ε)) ⊂ R_{ε}^{p} ⊂ Π(H_{p}(ϱ_{o}^{−1}ε)).

In order to compute the μ measure of R_{ε}^{p}, let elements ξ^{+} and ξ^{−} of S^{N} be chosen such that s_{ξ^{±}} is tangential to H_{p}(ϱ_{o}^{±1}ε). An application of the formula (HEF 1) gives that the hyperbolic length of the segment s_{ξ^{±}} \ {0} ∩ H_{p} is equal to log(ϱ_{o}^{±1}ε)^{−2} (apart from an additive constant). This implies that there exist positive numbers κ^{+}(ε) and κ^{−}(ε) such that

(5)  κ^{±}(ε) ≍ ε^{−1}

and

(6)  s_{ξ^{±}} \ {0} ∩ ∂H_{p} ∈ D_{κ^{±}(ε)}^{p}.

Using (3), (5) and (6), we deduce

\begin{aligned}μ(Π(H_{p}(ϱ_{o}^{±1}ε))) & ≍ ∑_{m=κ^{±}(ε)}^{∞} ∑_{{g∈G^{∗}_{p} \atop g(0)∈D^{p}_{m}}} g(Q_{∞}^{p}) \\ & ≍ ∑_{m=κ^{±}(ε)}^{∞} m^{k(p)−1−2δ} ≍ (κ^{±}(ε))^{k(p)−2δ} ≍ ε^{2δ−k(p)}. \end{aligned}

Using (4), this implies the existence of positive constants c_{7} and c_{8} such that

(7)  c_{7} ε^{2δ−k(p)} ≤ μ(R_{ε}^{p}) ≤ c_{8} ε^{2δ−k(p)}.

Let ε chosen to be sufficiently small; i.e., let ε be positive and less than ε_{o}(p), where the positive constant ε_{o}(p), depending on q_{1}(p), is chosen such that

R_{ε_{o}(p)}^{p} ∩ Q_{∞}^{p} = ∅.

Now, there exists a positive constant c_{9} , depending on ε_{o}(p) and thus on q_{1}(p), such that, for all (ξ, η) in Q_{∞}^{p} × R_{ε}^{p},

(8)  c_{9} < |ξ − η| ≤ π.

Using (7) and (8), we deduce, for ε less than ε_{o}(p),

ν(𝒞^{p}(ε)) = ∫_{Q_{∞}^{p}} ∫_{R_{ε}^{p}} \frac{dμ(ξ) dμ(η)}{|ξ − η|^{2δ}} ≍ μ(Q_{∞}^{p}) · μ(R_{ε}^{p}) ≍ ε^{2δ−k(p)}.

If we define ε_{o} := min{ε_{o}(p) : p ∈ P }, and adjust the constants in order to get rid of the simplifying assumption that 0 is an element of ∂H_{p}, the proposition follows.

We may now turn to the main result of this paper. For this, we recall the definition of the ‘counting function’ under consideration. For \underline{v} in ℛ(Δ), p in P, positive ε less than ε_{o} and for positive t, we define

α_{t}^{p}(\underline{v}, ε) := card\{s ∈ [0, t] : ϕ_{s}(\underline{v}) ∈ 𝒞^{p}(ε)\}.

This function counts, while one is ‘travelling’ the hyperbolic distance t, the number of visits of the geodesic ray determined by \underline{v} into the particular cusp region which is given by H_{ε}^{p}. For this function we have the following result.

Theorem. For all p in P, for all positive ε less than ε_{o} and for ν̃-almost all \underline{v} in ℛ(Δ), we have

(LAF)  k_{1} ε^{2δ−k(p)} ≤ lim_{t→∞} \frac{α_{t}^{p}(\underline{v}, ε)}{t} ≤ k_{2} ε^{2δ−k(p)}

(k_{1}, k_{2} and ε_{o} are the constants of Proposition 1).

Proof. Let p be in P and let ε be positive and less than ε_{o}. We have seen in the preceding section that 𝒞^{p}(ε) is a local cross section for the flow ϕ_{t} with respect to the time τ. Here, τ is some fixed, positive number which depends on G. We may define the ‘τ -flow box’ 𝒞̃^{p} (ε) by

𝒞̃^{p} (ε) := ⋃_{0<u<τ} g_{−u}(𝒞^{p}(ε)).

Let χ_{(·)} denote the characteristic function. For \underline{v} in ℛ(Δ) and positive t, we have

(9)  τ · (α_{t}^{p}(\underline{v}, ε) − 1) ≤ ∫_{0}^{t} χ_{𝒞̃^{p} (ε)}(ϕ_{s}(\underline{v})) ds ≤ τ · (α_{t}^{p}(\underline{v}, ε) + 1).

Using the ergodicity of the measure ν̃ and applying Hopf’s generalization of the Birkhoff ergodic theorem ([4], [5]), it follows, for ν̃-almost all \underline{v} in ℛ(Δ), that

(10)  lim_{t→∞} \frac{1}{t} ∫_{0}^{t} χ_{𝒞̃^{p} (ε)}(ϕ_{s}(\underline{v})) ds = ν̃(𝒞̃^{p} (ε)).

The fact that 𝒞^{p}(ε) is a local cross section with respect to τ implies that

(11)  ν̃(𝒞̃^{p} (ε)) = τ · ν(𝒞^{p}(ε)).

Combining (9), (10) and (11), we derive, for ν̃-almost all \underline{v} in ℛ(Δ),

lim_{t→∞} \frac{α_{t}^{p}(\underline{v}, ε)}{t} = ν(𝒞^{p}(ε)).

Using Proposition 1, it now follows that

k_{1} ε^{2δ−k(p)} ≤ lim_{t→∞} \frac{α_{t}^{p}(\underline{v}, ε)}{t} ≤ k_{2} ε^{2δ−k(p)}.

An immediate consequence of the preceding theorem is a statement concerning the relative asymptotic frequency of cusps visits. For this we define, for p and q in P, \underline{v} and \underline{w} in ℛ(Δ) and positive ε less than ε_{o}, the relative asymptotic frequency R_{ε}^{p,q}(\underline{v}, \underline{w}) of the p-cusp visits of the geodesic corresponding to \underline{v} with respect to the q -cusp visits of the geodesic corresponding to \underline{w}, by

R_{ε}^{p,q}(\underline{v}, \underline{w}) := lim_{t→∞} \frac{α_{t}^{p}(\underline{v}, ε)}{α_{t}^{q}(\underline{w}, ε)}.

Corollary. There exist positive constants k_{3} and k_{4} such that, for each p and q in P, for each positive ε less than ε_{o} and for ν̃-almost all \underline{v} and \underline{w} in ℛ(Δ), we have

k_{3} ε^{k(q)−k(p)} ≤ R_{ε}^{p,q}(\underline{v}, \underline{w}) ≤ k_{4} ε^{k(q)−k(p)}.

This corollary reflects the way in which the geodesic dynamic is effected by the interplay of cusps of different ranks. For example, let us consider a geometrically finite 3-manifold with two cusps p and q such that k(p) = 2 and k(q) = 1. It is intuitively clear that the proportion of the limit set in the shadow of a standard horoball associated to p is in a certain sense greater than the proportion of the limit set in the shadow of an equally sized standard horoball associated to q. In this particular case, the corollary states that for a recurrent geodesic the ‘asymptotic expectation’ of visiting the cusp at p is roughly ε^{−1} times higher than the ‘asymptotic expectation’ of visiting the cusp at q.

Finally, we convert the statement of the Theorem into a statement concerning the asymptotic frequencies with which geodesic rays emanating from the origin in D^{N+1} intersect ‘ε-squeezed’ standard horoballs. For this we consider the ‘counting function’ β_{t}^{p}(ξ, ε), an analogue of α_{t}^{p}(\underline{v}, ε). The function β_{t}^{p}(ξ, ε) is defined for ξ in L(G), p in P, positive ε less than ε_{o} and positive t, by

β_{t}^{p}(ξ, ε) := card{g ∈ 𝒯_{p} : s_{ξ_{u}} ∩ H_{g(p)}(εr_{g}) ≠ ∅ for some u ∈ (0, t]}.

We obtain the following result.

Proposition 2. For each p in P, for all positive ε less than ε_{o} and for μ-almost all ξ in L(G), we have

k_{1} ε^{2δ−k(p)} ≤ lim_{t→∞} \frac{β_{t}^{p}(\underline{v}, ε)}{t} ≤ k_{2} ε^{2δ−k(p)}

(k_{1}, k_{2} and ε_{o} are the constants of Proposition 1).

Proof. Consider the set of leaves of the strong stable foliation on 𝒮 (Δ) whose images under the projection pr contain the origin in Δ; i.e., consider the set

W_{o}^{ss} := ⋃_{\underline{v}∈𝒮_{o}(Δ)} W^{ss}(\underline{v}),

where W^{ss}(\underline{v}) is defined for \underline{v} in 𝒮 (Δ) by

W^{ss}(\underline{v}) := \{\underline{w} ∈ 𝒮 (Δ) : lim_{t→∞} d^{∗}(ϕ_{t}(\underline{v}), ϕ_{t}(\underline{w})) = 0\}.

It is clear that if (LAF) is satisfied for some \underline{v} in 𝒮_{o}(Δ), then (LAF) is also satisfied for each \underline{w} in W^{ss}(\underline{v}). On the other hand, if some \underline{w} in 𝒮 (Δ) satisfies (LAF), then (LAF) also holds for ϕ_{t}(\underline{w}) for all t in R. Thus we have

– if (LAF) holds for \underline{v} in 𝒮_{o}(Δ), (LAF) holds for each \underline{w} in ⋃_{t∈R} ϕ_{t}(W^{ss}(\underline{v})). The proposition now follows from our theorem when we recall that 0 is assumed to be an element of H(L(G)) and that the following inclusion is satisfied:

ℛ(Δ) ⊆ ⋃_{t∈R} ϕ_{t}(W_{o}^{ss}).

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Received 15 March 1994