Excursion and return times of a geodesics to a subset of a hyperbolic Riemann surface

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to a subsurface with geodesic boundary. As a consequence we get the average time a generic geodesic spends in such a subsurface. Related results are obtained for excursions into a collar neighborhood of a simple closed geodesic and the associated distribution of excursion depths.


Introduction
The geodesic flow on the unit tangent bundle of a finite area hyperbolic Riemann surface is ergodic [13]. One important, well known consequence is that on average, the time spent by a generic geodesic in a subset of the surface is equal to the relative area of the set. At first glance it does not appear that this approach tells us anything directly about other specific aspects of the behavior of the geodesic relative to the set; for example, about the average rate at which the geodesic returns to the set, the average length of each visit to the set -called the excursion time-or the average time between visits: all related quantities. The reason for this is that these values are determined by certain aspects of the geometry of the sets beyond just the relative area. Nevertheless, for a reasonably large and interesting selection of sets, once the geometry is properly accounted for, it is the classical Ergodic Theorem for flows from which one can infer the existence of a limit and compute its value.
In the cases considered here, where the subset is a subsurface with geodesic boundary or a collar neighborhood of a simple closed geodesic, it is possible to provide precise descriptions of these types of behavior for a generic geodesics relative to the set. In the first instance the return time depends only on the length of the boundary and the area of the surface, whereas excursion times are determined by the relative area along with the length of the boundary. In the second case the return time depends on the geodesics length, the width of its collar and the area of the surface, while excursion times only depend on the width of the collar. This last fact is at first surprising, yet it mirrors the result from [10] where it was shown that 2000 Mathematics Subject Classification. 30F35(primary), 32Q45, 37E35, 53D25 (secondary) . the average excursion time into an embedded cusp neighborhood is π, independent of the area of the neighborhood and furthermore, independent of the surface. In the latter case it is also possible to describe the distributions of the maximal depths of geodesic excursions-again something that depends only on the geometry of the collar. This is reminiscent of results from [5] and [9].
The return time to a cusp relative to the depth of the excursion was studied by Sullivan in [15], leading in various directions, to generalizations and tangents. One of these tangents was followed by Nakada in [12], where he found the average return time to a cusp neighborhood of a finite volume 3-manifold, as one step in his study of approximation properties of rationals in an imaginary quadratic number field. His approach was taken up by Stratmann [14], who used it to get estimates for the average return time of a generic geodesic in a hyperbolic manifold-where generic means with respect to the Liouville-Patterson measure on the unit tangent bundle. We employed their methods in [10] to determine average return and excursion times to a cusp neighborhood on a finite area hyperbolic surface and, like Nakada and Stratmann, we used these values to establish metrical results for approximation by the cusps of a Fuchsian group.

Background and main results
2.1. Geodesic boundary. A finite area hyperbolic 2-orbifold S is the quotient of the Poincaré upper-half plane H by a discrete group G ⊂ PSL 2 (R).
Let γ 1 , . . . , γ n be a collection of disjoint simple closed geodesics on S, which together bound a subsurface M of S. We write Γ = ∪γ i for the boundary of M . The length of a closed geodesic β is written l(β). The total length of the boundary of M is then l(Γ) = l(γ i ).
Each vector v in the unit tangent bundle T 1 S uniquely determines a geodesic ray α v : [0, ∞) → S which is the projection to S of the forward orbit {G t (v) | t ∈ [0, ∞)}, of the geodesic flow on the unit tangent bundle. If α v := α v ([0, ∞)) intersects the boundary Γ of M infinitely often, then there is a sequence of pairs of parameter values In other words, α v meets the set M in precisely the arcs We shall refer to these arcs as the excursions of α v into M and to the sequence of parameters as the excursion parameters of α v . Let #X denote the cardinality of the set X. area(T ) shall denote the hyperbolic area of T ⊂ S. Area is measured in the hyperbolic metric.
Then the asymptotic average rate at which a geodesic returns to the set M is given as follows.
Since geodesics are parameterized by arc length, the length of the arc α[(t i , s i )] is s i − t i . One consequence of the theorem is the value for the asymptotic average length of an excursion.
Remark 1. The average distance between the starting points of the first n + 1 excursions is computed by the sum Since n is essentially #{i | t i ≤ t n }, by taking the limit as n → ∞, we see that the average distance between consecutive excursions is generically equal to the inverse of the value for the limit in (2.2), π area(S) l(Γ) .

Collar neighborhoods.
Let γ be a simple closed geodesic on S. Let C r (γ) = C r be the collar neighborhood of γ. This is the set of points within a fixed distance r of γ. If r is not too large (r < R l(γ) = log coth l(γ) 4 , see Section 2.5), the r-collar about γ is divided by γ into disjoint open cylinders denoted A r and B r , which we will call half-collars. If a subsurface M is specified, then we shall suppose that A r ⊂ M. Let λ Ar = ∂C r ∩ ∂A r and λ Br = ∂C r ∩ ∂B r , where ∂D denotes the boundary of D in S. The values r, l(γ) and area(C r ) are not independent but rather, the width of the collar and the length of γ determine one another and the two of them determine the area of the collar. This will be made precise in Section 4.1, Proposition 2.
We shall suppose the geodesic ray α v , determined by the direction v in In other words, these parameters determine the segments of α v in C r with both endpoints on its boundary. We will focus on those that enter C r at λ Ar . P r is called the set of excursion parameters.
In this case the analogue of Corollary 1, the asymptotic average length of an excursion into a collar neighborhood, is given by the following.
2.3. The distribution of collar excursions. In [9] the number theoretic distributions from the Doeblin-Lenstra Conjecture [5], [7] and the related Theorem of Bosma [4] were reinterpreted and expanded in a geometric setting. The analogous notions can be defined in terms of the distribution of the depths of maximal penetration of geodesic excursions into a collar neighborhood of a simple closed geodesic. Given a geodesic α v with excur- is the distance between the point c and the geodesic γ. This is the depth of the i th excursion. Using Theorem 2 we show, in the next corollary, that the depths are nicely distributed, in a fashion that is independent of l(γ). Choose R 0 < R l(γ) .
Corollary 3. For r ≤ R 0 and almost all v ∈ T 1 S the limit is defined and takes the value The unit tangent bundle of the hyperbolic plane H can be given as a cartesian product T 1 H = H × S 1 . In these coordinates the natural invariant measure for the geodesic flowG has the form m = dAdφ. There is another set of very useful coordinates in which T 1 H is described, up to measure zero, as the the set of triples (x, y, t) ∈ R 3 with x = y. Then (x, y, t) corresponds to the vector v =α(t) ∈ T 1 H, where α is the geodesic in H oriented from the endpoint α − = x to α + = y and parameterized so that α(0) is the Euclidean midpoint of the semicircle α(R). The geodesic flow on T 1 H satisfiesG s (x, y, t) = (x, y, t + s). Furthermore, the geodesic flow on T 1 S has the invariant probability measure µ, whose lift to T 1 H is equal to [13] As a consequence of the Ergodic Theorem and the Poincaré Recurrence Theorem, [8], E has full measure in T 1 S.

2.5.
The normalization of G. Given a simple closed geodesic γ on S we may suppose that the Fuchsian group G has been normalized so that the imaginary axis I in H covers γ. Then the stabilizer of I in G is generated by the transformation g(z) = ζz with log ζ = l(γ). The arcγ = {it | 1 ≤ t < ζ} ⊂ I is mapped injectively onto γ by the covering projection.
C r (γ) is the image of the r-neighborhood C r (I) of I under the covering projection. It follows from the Collar Lemma (see [1] and [6] for the many references) that there is a value R l(γ) = log coth l(γ) 4 so that for r < R l(γ) , the collar C r (I) is mapped disjointly from itself by any h ∈ G which is not a power of g. Consequently, C r (γ) is a cylinder embedded in S, as we have been assuming. Henceforth, we shall suppose that r < R l(γ) .
The curves λ Ar and λ Br lift to straight linesλ Ar andλ Br bounding C r (I). They eminate from the origin and make an angle φ with I. We shall further stipulate that the lift of the half-collar A r in C r (I) lies in the right halfplane.

Proofs for regions with geodesic boundary
3.1. The single geodesic. Let L * (γ) = L * (γ, A r ) be the subset of the unit tangent bundle over γ where v ∈ L * (γ) if there exists τ 0 so that for 0 < t < τ 0 the points of α v (t) lie in A r . This is the subset of vectors based on γ that point into A r . Note that the set does not depend on the choice of r < R l(γ) . Then L(γ) = L * (γ) ∩ E is a cross-section for the geodesic flow on T 1 S, [3]. In other words, for almost all v ∈ T 1 S there exists an increasing sequence of values τ i so that G τ i (v) ∈ L(γ). Given Analysis of the thickened section is the main tool in the proof of Theorem 1.
For a given x ∈ R + and t > 0, the point y at which the geodesic xy with endpoints x and y meets the point it ∈ I, is the solution to the equation This is y = − t 2 x . Therefore for a given x > 0, is the interval of values y ∈ R for which xy intersects the intervalγ.
For x ∈ R and y ∈ I x , let t xy denote the parameter value for which the unit tangent vector (x, y, t xy ) ∈ L(γ). Then Consequently, 3.2. Many geodesics. We shall prove a theorem that is slightly more general than Theorem 1. Let {γ i } n i=1 be a finite sequence of mutually disjoint simple closed geodesics, except that we allow geodesics to appear twice in the list. To each γ i in the sequence, associate a distinct halfcollars A i . With Γ as before and A = ∪A i , define L(Γ, A) = ∪L(γ i , A i ) and L ε (Γ, A) = ∪L ε (γ i , A i ). In this case there is not necessarily a subsurface M with which the half-collars are associated.
Suppose v ∈ E. Then G t (v) will meet L(Γ, A) in a sequence of points G τ j (v). In other words, α v (τ j ) will lie on one of the γ i with a tangent pointing into A i . Let N v (t) denote the number of times the orbit lies in L(Γ, A) or the number of times α v crosses one of the geodesics in the direction stipulated by a half-collar.

Theorem 3. For almost all
Proof. The characteristic function of a set Y is written χ Y . For ε small we have the inequalities (3.6) Divide through by tε and let t → ∞. By the Ergodic Theorem and Proposition 1, for almost all v ∈ T 1 S, the left and right hand limits converge to Proof of Theorem 1. The theorem follows from Theorem 3 by choosing Γ to be the boundary of the M and taking the half-collar A i associated to γ i to be the one inside M .

Excursion times.
Proof of Corollary 1. With the possible exception of the very first, the sum of the first n excursion lengths is The average can then be written in the form The value n on the right is #{i | t i < s n }, from Theorem 1. By that theorem, the limit as n → ∞ of s n /n exists and is the inverse of the right-hand side of equation (2.2). By the Ergodic Theorem the limit of the first factor also exists and is the µ-measure of T 1 M ; this is, the area of M divided by the area of S. We should note that in both instances convergence is almost everywhere. Taken together this completes the proof.

Preliminaries.
Recall that φ is the angle made by the rayλ A and the imaginary axis. Write sin φ+ i cos φ = a+ ib = p φ . The relationship between area(C r ), r and φ is given in the following proposition.
Proposition 2. area(C r (γ)) = 2l(γ) tan φ = 2l(γ) sinh r. In particular, Proof. In order to see how r and φ are related, we look at the geodesic segment that lies on the unit circle centered at 0 running between the imaginary axis and the lineλ A . Using polar coordinates with the hyperbolic metric in H we see that Therefore, e r = sec φ + tan φ = 1 + tan 2 φ + tan φ. Solving for tan φ gives sinh r.
The area of C r (γ) is the area of the region in C r (γ) between the two circles centered at the origin with radii 1 and ζ respectively. Again, computing with the hyperbolic metric in polar coordinates we have The following simple, geometric lemma will be very useful.

Lemma 1.
(1) Given x > 0, if the geodesic xy in H passes through x and the point tp φ , t > 0, then (2) The geodesic xy in H, x > y, tangent toλ A at the point tp φ has endpoints Consequently, given x, xy it tangent toλ A when Proof. As in formula (3.2) the geodesic xy contains tp φ if With p φ = a+ib, we square both sides and simplify to get xy−tax−tay+t 2 = 0. Solving for y gives (4.1). Suppose that xy is tangent toλ A at tp φ . Let w be the midpoint on the real axis between y and x. Then the line from w to tp φ is orthogonal tõ λ A . Equating slopes we have tb ta−w = − a b or w = t a . Then the radius of the semi-circle xy is |tp φ − t a | = b a t. With center and radius in hand it is easy to write down x and y which gives (4.2). Formula (4.3) follows.

4.2.
The thickened section. In order to prove Theorem 2 we shall recycle the approach taken in the proof of Theorem 1, only this time the thickened section will be defined with respect to the boundary λ Ar of the collar neighborhood of γ. The situation is somewhat more involved.
Let s denote the segment ofλ A between the points p φ and ζp φ . It is a preimage of λ Ar under the covering projection. Henceforth x will alway be a positive number.
We define a notion of intersection between the directed geodesic β = xy and the arc s that only counts the first point at which the geodesic intersects λ A . More precisely, if β ∩λ A = ∅ then set xy∩ s = ∅. And if β intersectsλ A in the points β(t 1 ) and β(t 2 ), t 1 ≤ t 2 , then We shall define several sections of the unit tangent bundle over λ Ar by specifying sets of triples (x, y, t xy ) in the unit tangent bundle over s. For i = 0, 1, 2, 3 let J i * (s) = {(x, y, t xy )|R i and xy∩ s = ∅} where R 0 , . . . , R 3 denote respectively the conditions, R 0 : x > 0, y ∈ R, R 1 : 0 < y < x, R 2 : 0 < x < y, R 3 : y < 0 < x. Given x, y, if xy∩ s = ∅ then there is a unique t xy so that the point (x, y, t xy ) lies in the intersection.
As in section 3.1, define J i (s) = E ∩ J i * (s) and the thickened sections J i ε (s) = {(x, y, t)|(x, y, t xy ) ∈ J i (s) and t xy ≤ t ≤ t xy + ε}. J i ε (s) projects to the thickened section J i ε (λ Ar ) in the unit tangent bundle of S over the boundary of the collar. Note that in formula 3.3 we subtracted ǫ because A r was to the right ofγ. Now it is to the left ofλ Ar .
In order to see what these sets represent, note that up to measure zero J 0 (λ Ar ) is equal to the set of all vectors over λ Ar pointing into the collar. J 3 (λ Ar ) is the subset determining geodesics that cross γ. By elementary hyperbolic geometry such geodesics will then exit the collar at λ B . Removing J 3 (λ Ar ) from J 0 (λ Ar ) results in two disjoint sets, which are J 1 (λ Ar ) and J 2 (λ Ar ). Geodesics determined by vectors from these subsets enter λ Ar from different sides of J 3 (λ Ar ) and exit the collar at λ Ar . Together these are all the geodesics that enter and exit via λ Ar .
Proof. The computation is done in (x, y, t) coordinates for the sets J i ε (s). We shall describe the limits of integration by specifying potential x values and then giving the corresponding set of y values using Lemma 1. Three cases are distinguished for the domain of x. In all we have y < x. For now we make the additional assumption that ζa < (1 + b)/a. If x is larger than ζ(1 + b)/a then for all y ∈ R, xy∩ s = ∅. Thus, the first case to consider is when (1 + b)/a < x < ζ(1 + b)/a. It follows from (4.2) that for such an x there is a point y so that xy is tangent toλ A and that this point of tangency lies in s. It also follows from (4.3) that given x, the interval of corresponding y values will vary between ( 1−b 1+b )x, the point at which xy is tangent to s and axζ−ζ 2 x−aζ where xy meets ζp θ , the upper endpoint of s. Thus the measure of the set of vectors in L ε (s) determined by these ( The second case is when ζa < x < (1 + b)/a. According to the lemma, as y varies between ax−1 x−a and axζ−ζ 2 x−aζ , xy∩ s varies between p φ and ζp φ , taking on all values in s. Writing out the integral as in the previous case, we see that the measure of the corresponding subset of L ε (s) is equal to M 2 where π area(S) The final case is where a < x < ζa. Given x in this interval, xy will meet s at p φ when y = ax−1 x−a and as y goes to −∞, xy limits at the vertical line intersecting s in the point xp φ . In this case the measure of the associated subset of L ε (s) is M 3 with Using Proposition 2 and the facts a = sin φ, b = cos φ and log ζ = l(γ), we get which is the proposition under the assumption that ζa < (1 + b)/a. If we reverse the inequality a similar computation yields the same result.

4.3.
Returns to a collar. Proof of Theorem 2. To begin we show that µ(J 3 ε (λ Ar )) = µ(L ε (γ)) = εl(γ) π area(S) . (4.4) It was observed in Section 3.1 that given v ∈ E, G t (v) will meet L(Γ, A) in a sequence of points G τ j (v), j ∈ Z + . Each time α v meets the geodesic γ from the A side, it must have either originated in A or else crossed λ Ar first, before proceeding to γ. Conversely, each time α v crosses λ Ar so that its tangent lies in J 3 ε (λ Ar ), it must go on to cross γ. Thus, for each j ≥ 2 there will be a corresponding value η j so that G η j (v) ∈ J 3 ε (λ Ar ) and if G η (v) ∈ J 3 ε (λ Ar ), then η = η j for some j. In other words, the section J 3 ε (λ Ar ) counts crossing of γ from the A side exactly as does L ε (γ).
Let P = {(t i , s i )} and P ′ = {(t ′ i , s ′ i )} be the excursion parameters of α v . For r > R l(γ) and v ∈ E define the counting functions N 0 v (r)(t) = #{i | t i < t}, and N 1 v (r)(t) = #{i | t ′ i < t}. Taking the role of L(γ) from the proof of Theorem 1 the sections J 0 (λ Ar ) andĴ (λ Ar ) = J 1 (λ Ar ) ∪ J 2 (λ Ar ) count crossings by α v of λ Ar that go into A and crossing of λ Ar by α v into A that exit A through λ Ar , respectively. Thus, rewriting formula (3.6) in the first instance gives, for ε > 0 sufficiently small, t 0 χ J 0 ε (λ Ar ) (G τ (v))dτ − 2ε ≤ εN 0 v (r)(t) ≤ t 0 χ J 0 ε (λ Ar ) (G τ (v))dτ + 2ε. Again, dividing by tε, letting t go to infinity and applying the Ergodic Theorem proves the first part of Theorem 2. The proof of the second part of the theorem follows if J 0 and N 0 are replaced byĴ and N 1 . By the Ergodic Theorem and Theorem 2, this equals area(C r ) area(S) × π area (S) l(γ) cosh r = 2π tanh r.
Here, the later equality follows from Proposition 2.
Proof of Corollary 3. We argue as in [10]. Write N v (r)(t) = #{j | t j < t} for the function that counts returns to the radius r collar. Using Theorem 2, the distribution δ(r) can be written as This completes the proof of Corollary 2.