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Transactions of the American Mathematical Society Dimension of escaping geodesics
Dimension of escaping geodesics
Gönye, Zsuzsannaयह पुस्तक आपको कितनी अच्छी लगी?
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360
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english
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Transactions of the American Mathematical Society
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10.1090/s0002994708045133
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May, 2008
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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 10, October 2008, Pages 5589–5602 S 00029947(08)045133 Article electronically published on May 22, 2008 DIMENSION OF ESCAPING GEODESICS ZSUZSANNA GÖNYE Abstract. Suppose M = B/G is a hyperbolic manifold. Consider the set of escaping geodesic rays γ(t) originating at a ﬁxed point p of the manifold M , i.e. dist(γ(t), p) → ∞. We investigate those escaping geodesics which escape at the fastest possible rate, and ﬁnd the Hausdorﬀ dimension of the corresponding terminal points on the boundary of B. In dimension 2, for a geometrically inﬁnite Fuchsian group, if the injectivity radius of M = B/G is bounded above and away from zero, then these points have full dimension. In dimension 3, when G is a geometrically inﬁnite and topologically tame Kleinian group, if the injectivity radius of M = B/G is bounded away from zero, the dimension of these points is 2, which is again maximal. 1. Introduction Consider G, a discrete, torsion free group of isometries of the hyperbolic metric on the hyperbolic three ball B; i.e. a Kleinian group. Passing to the quotient B/G by identiﬁcation of the Gequivalent points we obtain the quotient space M , which is a manifold. Suppose this group G is nonelementary, and denote its limit set by Λ. A point x on the boundary of the ball is a nonconical point if there is a geodesic ray ending at x so that the projection of this geodesic will eventually leave any compact set and tend to the ideal boundary. Among these points there is a subset that escapes to the ideal boundary at the fastest possible rate; these are called deep points. The original deﬁnition of deep points is due to McMullen ([14]). A point is a deep point if there is a geodesic ray γ : [0, ∞) → C(Λ) in the convex hull parameterized by arclength and terminating at x, so that for some δ > 0 dist γ(t), ∂C(Λ) ≥δ t for all t, i.e. the depth of γ inside the convex hull of the limit set Λ increases linearly with the hyperbolic length. (All the necessar; y deﬁnitions will be given later in the text.) We can generalize this notation by taking any Lipschitz function φ(t) : [1, ∞) → [1, ∞) with the property of limt→∞ φ(t) = ∞. We ﬁx a point z0 ∈ M = B/G, and Received by the editors November 29, 2005 and, in revised form, March 9, 2007. 2000 Mathematics Subject Classiﬁcation. Primary 30F40, 28A78; Secondary 30F35. Key words and phrases. Fuchsian groups, Kleinian groups, escaping geodesics, deep points, Hausdorﬀ dimension. c 2008 American Mathematical Society Reverts to public domain 28 years from publication 5589 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5590 ZSUZSANNA GÖNYE consider the set of geodesic rays starting at z0 parameterized by the hyperbolic arclength. Deﬁne the set of geodesics in the convex core which escape at a rate φ as dist γ(t), z0 1 C ≤ ≤C . Γφ = γ : C φ(t) C C Let Λφ denote the terminating points of the geodesics in ΓC φ , and let Λφ = C Λφ . The main theorem of this paper is the following: Theorem 1.1. Suppose G is a geometrically inﬁnite, topologically tame Kleinian group, M = B/G has injectivity radius bounded away from zero and there is a Green’s function on M . Let φ(t) : [1, ∞) → [1, ∞) be a Lipschitz function satisfying limt→∞ φ(t) = ∞; then the Hausdorﬀ dimension of Λφ is 2. The relevant deﬁnitions will be given later in Sections 2, 3 and 4. The idea of the proof is as follows: we can ﬁnd a positive harmonic function u on the manifold M which tends to 0 in the geometrically ﬁnite ends of M (Lemma 4.1). Lifting u up to the covering space B, we get a hyperbolic harmonic function U on the ball B. This hyperbolic harmonic function is a Poisson integral of some positive measure µ, which is supported on the limit set. Using this measure µ, we construct a Bloch martingale {fn } on the dyadic squares Q of length 2−n by deﬁning fn as µ Q(x) . fn (x) = m Q(x) With the help of a technical lemma (Lemma 5.2) we can ﬁnd a Cantor set, which has Hausdorﬀ dimension two (Lemma 3.3), on which the martingale grows approximately at the same rate as the given Lipschitz function φ, i.e. fn (Q) 1 ≤ C. ≤ C φ(n) The martingale fn (x) for x ∈ Q has bounded distance from the harmonic function U on the top of the Carleson square drawn over Q (Lemma 4.2); therefore U (zQ ) 1 ≤ C. ≤ C φ(n) Finally, U (z) gives the distance approximately from γ(t) to the base point, which gives an estimate for dist(γ(t), ∂C(Λ)) on manifolds speciﬁed in the main theorem. An analogous theorem can also be given for Fuchsian groups: Theorem 1.2. Suppose G is a geometrically inﬁnite Fuchsian group, M = B/G has injectivity radius bounded and bounded away from zero, and there is a Green’s function on M . Let φ(t) : [1, ∞) → [1, ∞) be a Lipschitz function satisfying limt→∞ φ(t) = ∞; then the Hausdorﬀ dimension of Λφ is 1. Fernández and Melián in [9] extensively studied the size of the set of escaping geodesics starting at a point of the hyperbolic A geodesic ray γ(t) origi surface. nating at p is called escaping if limt→∞ dist γ(t), p = +∞. These rays may leave the convex core. It is known that if there is a Green’s function on the surface, then the set of escaping geodesics has full measure. If there is no Green’s function, then the Hausdorﬀ dimension of the terminal points is still 1. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse DIMENSION OF ESCAPING GEODESICS 5591 Taking φ(t) = t in Theorems 1.1 and 1.2 we get the dimension of deep points in sets described in the theorem above. Corollary 1.3. If G is a geometrically inﬁnite, topologically tame Kleinian group and M = B/G has injectivity radius bounded away from zero and has a Green’s function, then the deep points have dimension 2. 2. Definitions and notation Let M (R̄d ) denote the orientation preserving Möbius transformations in the ddimensional extended space R̄d = Rd ∪ {∞}. The subgroup of M (R̄d ) which preserves the upper halfplane H = {x ∈ R̄d : xd > 0} or the unit ball B = {x ∈ R̄d : x < 1} will be denoted by M (H) and M (B), respectively. A discrete group G of M (B) in dimension 3 is called a Kleinian group. A Fuchsian group is a Kleinian group that stabilizes a round disc on ∂B, the sphere at inﬁnity. In this paper we consider only nonelementary groups, that is, G has no ﬁnite orbit in H3 = {x ∈ R̄3 : x3 > 0}. If G is a discrete subgroup of M (B), the orbit G(a) of any point a ∈ B can accumulate only on the boundary of B. So we call a point x ∈ S = ∂B a limit point, if there is an orbit G(a) accumulating at x. The limit set is the set of limit points and is denoted by Λ(G) or simply by Λ. The complementary set S\Λ of Λ is called the ordinary set, and is denoted by Ω ([3]). Let G be a Kleinian group. Then the quotient space Ω/G, which is obtained from the ordinary set of G by identifying equivalent points under the mappings of G, is a marked (possibly disconnected) Riemann surface ([13]). If Ω/G is a ﬁnite marked Riemann surface (i.e. a ﬁnite union of compact surfaces, each with at most a ﬁnite number of punctures), then G is called analytically ﬁnite. The Ahlfors ﬁniteness theorem shows that G is analytically ﬁnite if it is ﬁnitely generated. A Möbius group G is called geometrically ﬁnite if some convex fundamental polyhedron has ﬁnitely many faces. In dimensions 2 and 3 the standard deﬁnition of geometric ﬁniteness is that the Dirichlet region must have ﬁnitely many faces. It is known that this criterion implies that every Dirichlet region and every convex fundamental polyhedron has ﬁnitely many faces ([3], [15]). Moreover, geometric ﬁniteness implies that the group is ﬁnitely generated, and therefore analytically ﬁnite. The convex hull of Λ ⊂ S = ∂B, denoted by C(Λ), is the smallest convex subset of B containing all geodesics with both endpoints in Λ. The convex core of a hyperbolic manifold M = B/G is given by the quotient C(Λ)/G and denoted by C(M ). For x ∈ M the injectivity radius, inj(x), is half the distance between the two closest distinct lifts of x to B. In the theorem we assume that the injectivity radius is bounded away from zero uniformly on M , which in dimensions 2 and 3 implies that G has no parabolic elements. A Kleinian group is called topologically tame, if the corresponding quotient manifold M = B/G is homeomorphic to the interior of a compact 3manifold with boundary. This implies that the convex core C(M ) consists of a compact piece and a ﬁnite number of ends Ej , which are topologically equivalent to S × R+ for some compact surface S. D. Calegari and D. Gabai in [6] and I. Agol in [1] proved the Marden conjecture, where all complete hyperbolic 3manifolds with ﬁnitely generated fundamental group are topologically tame. We note here that in [7] Canary License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5592 ZSUZSANNA GÖNYE showed that topological tameness is equivalent to analytical tameness in dimension 3. Moreover, if G is topologically tame, then there is an upper bound for the injectivity radius inside the convex core. In the Introduction we gave the deﬁnition of a deep point deﬁned by a geodesic ray in the convex hull of Λ. An equivalent deﬁnition can also be given on the quotient manifold, as in [4]. A point x ∈ Λ is deep if the geodesic ray γ ending at x satisﬁes dist γ̃(t), M \C(M ) ≥δ>0 t for all t ≥ t0 , where γ̃ denotes the curve on the quotient space which corresponds to γ. 3. Dyadic martingale and Hausdorff dimension An nth generation dyadic cube in Rd is Qn = x = (x1 , x2 , . . . , xd ) : ai ≤ xi < ai + 2−n , 1 ≤ i ≤ d where a = (a1 , a2 , . . . , ad ), the corner of the cube, has coordinates in the form i ai = m 2n with an integer mi . The collection of these dyadic cubes is denoted by Dn . For any given point x ∈ Rd let Qn (x) denote that unique nth generation dyadic cube from Dn which contains the point x, and let Qn  denote the sidelength of Qn . The mth generation descendants of Qn are the dyadic subcubes of Qn with sidelength of 2−m Qn . There are 2md of them. Suppose Q0 is a unit cube in Rd . Then a sequence of functions {fn }∞ n=0 is said to be a dyadic martingale on Q0 if (1) fn is measurable on each Qn ∈ D, (2) Q1n  Qn fn < ∞, (3) Q1m  Qm fn = fm for all m < n. In addition to this usual deﬁnition, we also require that fn must be constant on the nth generation dyadic cubes. Since in this paper we use only dyadic martingales, so we often omit the “dyadic” attribute. There is a standard way to construct a martingale from a ﬁnite measure. If a ﬁnite measure µ is given on Q0 , then the functions µ Qn (x) (3.1) fn (x) = Qn (x)d deﬁne a dyadic martingale, where Qn (x)d (or just Qn (x) if the notation is clear from the text) denotes the ddimensional Lebesgue measure of Qn (x). We deﬁne the martingale diﬀerences as ∆fn (x) = fn+1 (x) − fn (x) and the martingale square function as ∞ Sf (x) = 1/2 χQn (x) ∆fn 2∞ . n=1 A martingale is called Bloch if supn ∆fn ∞ < ∞. If {fn } is an L1 bounded martingale, then fn converges a.e. to a function f with f 1 < ∞. For more results on the convergence of martingales, you may see [11]. We will need the following two estimates for dyadic martingales. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse DIMENSION OF ESCAPING GEODESICS 5593 Lemma 3.1. Let fn be a dyadic martingale on Q0 ⊂ Rd with limit function f . Suppose Sf ∞ < ∞. Then for λ ≥ 0, λ2 . x ∈ Q0 : f (x) − f0 (x) ≥ λ ≤ exp − 2Sf 2∞ The proof of this lemma is due to Herman Rubin and can be found in the paper of ChangWilsonWolﬀ [8, Theorem 3.1]. Lemma 3.2. Suppose µ is a probability measure on X. Suppose F is a measurable, real valued function on X so that X F dµ = 0 and F 4 ≤ BF 2 . Then 1 1 µ x : F (x) ≤ − √ F 2 . ≥ 2 64B 12 8B The proof is given in [4]. Suppose φ is an increasing, continuous function from [0, ∞) to itself such that φ(0) = 0. For a given set E we deﬁne the Hausdorﬀ content as φ H∞ (E) = inf φ(rj ) : E ⊂ ∪j D(xj , rj ) , where D(xj , rj ) denotes a ball of radius rj centered at xj . Especially, if φ(t) = tα φ α we denote H∞ by H∞ . The Hausdorﬀ dimension of a set E is α (E) = 0 . dimH (E) = inf α : H∞ For more details and examples on Hausdorﬀ dimension you may see [5]. In the proof of our theorem we will also need the following lemma. Lemma 3.3. Suppose En is a union of closed dyadic cubes of generation kn so that E0 ⊃ E1 ⊃ E2 ⊃ ... and there are constants N and with (1) kn − kn+1  = N for all n. (2) If Q ∈ En is generation kn , then En+1 ∩ Qd ≥ Qd . If E = n En , then dimH (E) ≥ d − C(N, ), where C(N, ) → 0 whenever > 0 is ﬁxed and N → ∞. The proof of this lemma can be found in [10], [12] and [16]. 4. The hyperbolic space and harmonic functions The unit ball B in Rn is the disc model for the ndimensional hyperbolic space equipped with the hyperbolic metric (4.1) dρ = 2dx . 1 − x2 An alternative model of the hyperbolic nspace is the upper halfplane H = x = (x1 , x2 , . . . , xn ) : xn > 0 ⊂ Rn equipped with the metric (4.2) dρ = dx . xn Using the hyperbolic metric deﬁned in B = x ∈ Rn : x < 1 by (4.1) we may construct the hyperbolic volume and area element, the normal derivative and License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5594 ZSUZSANNA GÖNYE gradient in the hyperbolic ball as 2n dx1 dx2 · · · dxn , (1 − x2 )n 2n−1 dσ , dσH = (1 − x2 )n−1 ∂v 1 − x2 ∂v , = ∂nH 2 ∂n 1 − x2 ∇H u = ∇u. 2 On the upper halfplane H = x = (x1 , . . . , xn ) ∈ Rn , xn > 0 these are dVH = dx1 · · · dxn , xnn dσ dσH = n−1 , xn ∂v ∂v = xn , ∂nH ∂n ∇H u = xn ∇u, dVH = respectively. A more detailed description can be found in [2] and [15]. The hyperbolic LaplaceBeltrami operator for the unit ball B ⊂ Rn is given by 2(n − 2)r ∂ (1 − r 2 )2 ∆+ , ∆H = 4 1 − r 2 ∂r where r = x. On the upper halfplane this is n−2 ∂ ∆H = x2n ∆ − . xn ∂xn A function f is called hyperbolically harmonic if it satisﬁes the hyperbolic Laplace equation, ∆H f = 0. We deﬁne the Green’s function on a quotient manifold M as follows. F is a Green’s function on M with a pole at the projection of a point a, if there exists a function f : B\{G(a)} → R such that the projection of f is F and the following are true for f : • f is a hyperbolic harmonic function on B\{G(a)}, • f ◦ g =f for all g ∈ G, 1 1 exists, i.e. f has singularity z−a at the point a, • limz→a f (z) − z−a • f is the smallest positive function with these properties. The hyperbolic version of Green’s formula is ∂v ∂u u dσH . (u∆H v − v∆H u)dVH = −v ∂nH ∂nH D ∂D We will need the following existence theorem, which was proven in [4]. Lemma 4.1. Suppose G is topologically tame, geometrically inﬁnite, M = B/G has an injectivity radius bounded below by > 0 and Green’s function G(w, z) exists on M . Then there exists a positive harmonic function U on M such that supz∈M ∇U (z) ≤ 1 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse DIMENSION OF ESCAPING GEODESICS 5595 and U tends to zero in the geometrically ﬁnite ends of M . If, in addition, G is topologically tame, then for any a0 > 0 there are constants a1 and a2 so that ∇U 2 dV ≥ a2 , B(z,a1 ) for every z such that dist z, C(M ) ≤ a0 . Moreover, U (z) tends to +∞ in the geometrically inﬁnite ends as dist z, ∂C(M ) → ∞. If Q is a cube in Rn , then Q̂ = Q × 0, (Q) is called the Carleson cube in Rn+1 + with base Q, and let zQ denote the center of Q̂. is the hyperbolic Poisson integral of thepositive Lemma 4.2. Suppose U on Rn+1 + U (z) ≤ 1. For a square Q ∈ Rn , let Qt = (x, t) : measure µ and satisﬁes ∇ H x ∈ Q . Then there is an A < ∞ so that ≤ A, U (zQ ) − 1 U (x, t)dx Q Q for any 0 < t ≤ (Q), where (Q) denotes the sidelength of Q, and U (zQ ) − 1 ≤ A. dµ Q Q The proof of this lemma in dimension n = 2 was given in [4], and it can be proven in higher dimension on a similar way as in [10]. 5. Two lemmas on martingales Deﬁne a martingale on the dyadic cubes using the positive measure µ described in Lemma 4.1 deﬁned by equation (3.1). According to Lemma 4.2, there exists a constant A so that U (zQn (x) ) − fn (x) ≤ A. To prove our main theorem we will need the following two lemmas for dyadic martingales. Lemma 5.1. Suppose µ is a positive measure on the cube [0, 1]d , d ≥ 1, so that n (x)) the corresponding dyadic martingale deﬁned by fn (x) = µ(Q Qn (x) is Bloch and 1 2 Qn  ∆fn 2 ≥ δ > 0 whenever fn (x) ≥ 1 on Qn (x). We claim that there is an > 0 and M < ∞ so that for any suﬃciently large n, there is a constant C for which the following holds. Let Q be any dyadic cube, and let fQ denote the function in the martingale deﬁned by Q, i.e. fQ = µ(Q) Q on Q. Suppose fQ ≥ C. Then among dn the 2 nth generation descendants of Q, at least 2dn satisfy M n ≥ fQ − fQ ≥ 1, and at least 2dn satisfy −M n ≤ fQ − fQ ≤ −1. Proof. Suppose supn ∆fn ∞ = L < ∞ and Q1n  ∆fn 22 ≥ δ > 0 whenever 6 fn (x) ≥ 1, and ﬁx an with 0 < ≤ min 216δL12 , 1 . By an appropriate scaling we may assume that Q = 1. Then the martingale square function for the sequence {f0 , f1 , ..., fn } is n−1 (5.1) χQj (x) ∆fj 2∞ Sf (x) = 1/2 ≤ √ j=0 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse nL. 5596 ZSUZSANNA GÖNYE 4 Let F = ∆f0 +...+∆fn−1 , and suppose that n > 64L δ 3 and that fQ ≥ 1+nL = C. Then fj ≥ 1 for all 0 ≤ j ≤ n − 1, so ∆fj 22 ≥ δ for all 0 ≤ j ≤ n − 1. The system n−1 ∆fj j=0 is orthogonal, therefore n−1 nL2 ≥ F 22 = (5.2) ∆fj 22 ≥ nδ. j=0 Let λ(t) = {x : F (x) > t} deﬁne the distribution function of F . Then F  = p p (5.3) ∞ tp−1 λ(t)dt, 0 2 2 F ∞ − 2St and by Lemma 3.1, λ(t) ≤ e t2 ≤ e− 2L2 n . Therefore F 44 (5.4) ∞ F  = 4 t3 λ(t)dt 0 ∞ 2 3 − 2Lt 2 n ≤4 t e dt 0 ∞ = 8L4 n2 ye−y dy = 4 0 4 2 = 8L n . √ √ √ Hence F 4 ≤ 4 8L n = Bδ n ≤ BF 2 with the constant B = can apply Lemma 3.2 so (5.5) µ x ∈ Q : F (x) ≤ − √ Using the fact that F 2 ≥ get that (5.6) 1 F 2 8B 2 ≥ Now we 1 . 64B 12 √ √ nδ and the assumptions that n > 8L2 √ 3 δ = √ √8 B 2 δ 1 √ nδ µ {x ∈ Q : F (x) ≤ −1} ≥ µ x ∈ Q : F (x) ≤ − √ 8B 2 1 ≥ µ x ∈ Q : F (x) ≤ − √ F 2 8B 2 1 ≥ 64B 12 δ6 = 15 12 2 L ≥ 2. Switching F with −F , with the same assumptions, we get (5.7) √ 4 √8L . δ µ {x ∈ Q : F (x) ≥ 1} ≥ 2. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse we DIMENSION OF ESCAPING GEODESICS 5597 Next, consider the √ following subsequence {f0 , ..., fn }. By Lemma 3.1, for a positive constant M ≥ L −2 ln , 2 2 x ∈ Q : fn (x) − f0 (x) ≥ nM ≤ exp −n M 2 2Sf ∞ −nM 2 ≤ exp (5.8) 2L2 n M2 ≤ exp − 2 2L ≤ for every n ≥ 1 and < 1. Repeating this argument for {−f0 , ..., −fn } we get that x ∈ Q : fn (x) − f0 (x) ≤ −nM ≤ . (5.9) Therefore, for every suﬃciently large n there is a constant C = 1 + nL so that if fQ ≥ C, then µ {x ∈ Q : 1 ≤ F (x) = fn (x) − f0 (x) ≤ nM } ≥ and µ {x ∈ Q : −1 ≥ F (x) = fn (x) − f0 (x) ≥ −nM } ≥ . In other words, among all of the 2dn nth generation descendants of Q, at least 2dn satisfy the inequality M n ≥ fQ − fQ ≥ 1, and at least 2dn of them satisfy −M n ≤ fQ − fQ ≤ −1. Lemma 5.2. Suppose G is a topologically tame and geometrically inﬁnite Kleinian group, so that M = B/G has injectivity radius bounded below by some positive epsilon and there exists a Green’s function on M. Let U be a positive harmonic function on M for which supz∈M ∇H U (z) ≤ 1, and let {fm } denote the corresponding martingale as deﬁned by (3.1). Then this martingale {fm } has bounded diﬀerences away from zero in the L2 norm, whenever its value is larger than some ﬁxed constant C. Proof. Suppose fm > C on the dyadic cube Qm . From Lemma 4.2 we know that U (zQ ) − fm  ≤ A, where zQ denotes the center of the Carleson square in Rn+1 + with base Qm . Since G is topologically tame and inj(z) ≥ > 0, the convex core C(M ) can be written as a compact part and a ﬁnite number of ends Ej , each of which is topologically equivalent to S × R+ for some compact surface S ([7]). We may suppose that we are already in such an end. First, we will show that for the given constant A, there exists a constant L, so that for all v ∈ C(M ) with U (v) ≥ C we can ﬁnd another point w with ρ(v, w) ≤ L and U (v) − U (w) ≥ 6A. Lemma 4.1 says that there exist r and a so that (5.10) ∇U 2 dV ≥ a B(z,r) for every z ∈ C(M ). Consider a geodesic ray on M originating at the point v and going to inﬁnity in the convex core. We may put disjoint balls of radius r along this geodesic, say N balls, and denote w the endpoint, so ρ(v, w) = 2rN . Cut Ej at v and at w, and call these surfaces Σ1 and Σ2 , respectively, and let T denote License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5598 ZSUZSANNA GÖNYE the part of Ej between these cuts. We may also assume that U (v) = 0. Green’s Theorem says that ∂f ∂g − g dσ. f ∆g − g∆f dVH = f (5.11) ∂n ∂n T Σ1 ∪Σ2 Let f = 1 and g = U 2 ; then 2 (5.12) ∆(U )dV = Σ1 ∪Σ2 T By elementary calculations we get that 2 ∇U  dV = (5.13) Σ1 ∪Σ2 T ∂(U 2 ) dσ. ∂n U ∂U dσ. ∂n Using Lemma 4.1, we can estimate the lefthand side of (5.13) by (5.14) ∇U 2 ≥ N a. T For the estimation of the righthand side of (5.13) we can use that ∇U  ≤ 1, so ∂U dσ ≤ diam(Σ1 )area(Σ1 ), (5.15a) U ∂n Σ1 ∂U dσ ≤ U (w) + diam(Σ2 ) area(Σ2 ). (5.15b) U ∂n Σ2 Since E = S × R+ , diam(Σi ) ≤ D and area(Σi ) ≤ S along the entire end Ej . Using the estimations (5.14) and (5.15) in (5.13), we get that (5.16) N a ≤ DS + U (w) + D S, and so Na − 2D ≤ U (w). S Therefore, we can choose a uniform N large enough so that U (v) − U (w) ≥ 6A and let L = 2rN . Next, we show that there is a point w such that U (zQ ) − U (w) ≥ 3A while ρ(zQ , w) ≤ 3L. We start at the point zQ on the Carleson square and go straight down toward the boundary by hyperbolic distance 2L; we call this point v. As we have just shown above, there exists a w such that ρ(v, w) ≤ L while U (v)−U (w) ≥ 6A, which means that either U (zQ )−U (v) ≥ 3A or U (zQ )−U (w) ≥ 3A. Assume the latter is true. Finally, we show that there is a subfamily {fmi } in the original martingale sequence with bounded diﬀerences away from zero in L2 norm while mi+1 − mi ≤ 3L for all i. From Lemma 4.2, U (zQ ) − fm  ≤ A and we can ﬁnd a w such that U (zQ ) − U (w) ≥ 3A, but ρ(zQ , w) ≤ 3L. We may also assume that w is in the middle of a Carleson square, since ∇U  ≤ 1. This Carleson square is diﬀerent from the original Qm , call it Qm , and let fm be the martingale function determined by the size of this square. Then from Lemma 4.2 we have that U (w) − fm  ≤ A and U (zQ ) − fm  ≤ A, while U (zQ ) − U (w) ≥ 3A. So fm − fm  ≥ A on Qm , while m − m  ≤ 3L. Therefore Qm  1 1 A2 3L = δ > 0. fm − fm 2 dx ≥ (5.18) Qm  Qm Qm  2 (5.17) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse DIMENSION OF ESCAPING GEODESICS 5599 6. The proof of the theorem Suppose G is a topologically tame, geometrically inﬁnite Kleinian group and the quotient manifold M = B/G has injectivity radius bounded away from zero. This implies that G has no parabolic elements. Suppose φ(t) : [1, ∞) → [1, ∞) is a Lipschitz function, i.e. φ(s) − φ(t) ≤ Bs − t for some B < ∞, and satisﬁes limt→∞ φ(t) = ∞. Fix a point z0 ∈ M and consider the set of geodesic rays starting at z0 parameterized by hyperbolic arclength. Deﬁne the set of geodesics in the convex core which escape at rate φ as dist γ(t), z0 −1 ≤C . = γ : C ≤ ΓC φ φ(t) C Let ΛC φ denote the terminal points of these geodesics, and let Λφ = C Λφ . Theorem 6.1. Suppose G is a geometrically inﬁnite, topologically tame Kleinian group and M = B/G has injectivity radius bounded away from zero and there is a Green’s function on M . Let φ(t) : [1, ∞) → [1, ∞) be a Lipschitz function satisfying limt→∞ φ(t) = ∞; then dimH (Λφ ) = 2. The analogous theorem for Fuchsian groups: Theorem 6.2. Suppose G is a geometrically inﬁnite Fuchsian group, M = B/G has bounded injectivity radius which is also bounded away from zero and there is a Green’s function on M . Let φ(t) : [1, ∞) → [1, ∞) be a Lipschitz function satisfying limt→∞ φ(t) = ∞; then dimH (Λφ ) = 1. Proof of Theorem 6.1. By Lemma 4.1 there exists a positive harmonic function U on M with supz∈M ∇U (z) ≤ 1 and U tends to zero in the geometrically ﬁnite ends of M . This U lifts to a hyperbolic harmonic function (which we will also call U ) on B, and this function is a Poisson integral of some positive measure µ supported on thelimit set. Consider the corresponding dyadic Bloch martingale µ(Q(x)) . fQ (x) = Q(x) Q∈D Using Lemma 5.2 we may pass to a subsequence of {fQ } for which the martingale diﬀerences are bounded away from zero whenever the value of the martingale is not less than a constant C. Notice that even if we work with this subsequence we can still use Lemma 5.1 because there are upper and lower bounds for the number of generations we skip. To simplify our indexes we will suppose that {fQ } itself is a Bloch martingale with diﬀerences bounded away from zero whenever the value is not less than C. Using Lemma 5.1 we can create a Cantor set {El } of nested dyadic cubes, where the dyadic martingale is comparable to the function φ. As in Lemma 5.1 ﬁnd appropriate > 0, M < ∞, and ﬁx a suﬃciently large N and the corresponding constant C. Since U tends to inﬁnity on the geometrically inﬁnite ends we may suppose that fQ ≥ C, except maybe for ﬁnitely many generations of cubes. First, notice that we may suppose that the function φ is Lipschitz with a Lipschitz constant N1 , i.e. φ(x) − φ(y) ≤ N1 x − y. In case φ(x) − φ(y) ≤ Bx − y with a 1 larger constant B than N1 , we can rescale our function by choosing Φ(x) = BN φ(x). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5600 ZSUZSANNA GÖNYE Deﬁne E0 as the collection of those largest cubes Q where fQ ≥ C and let k0 be the number which denotes the generation of these cubes. Then there is a positive constant D so that fk0 (Q) − φ(k0 ) ≤ D on all Q ∈ E0 . We may also assume that D ≥ M N . We deﬁne the sets {En } inductively as follows. Suppose we already have the set En deﬁned, and the quotient (or the diﬀerence) of fkl and φ(kl ) is bounded 1 on all the previous sets, say D ≤ fkl − φ(kl ) ≤ D for all l ≤ n. Let kn+1 = kn + N . Since φ is a Lipschitz function with constant N1 , so φ(kn )−1 ≤ φ(kn+1 ) ≤ φ(kn )+1. To choose the next generation of cubes, for each Q ∈ En we compare fkn (Q) to φ(kn+1 ): If fkn (Q) < φ(kn+1 ), then choose those N th generation descendants Q of Q for which M N ≥ fkn+1 − fkn ≥ 1. According to Lemma 5.1 there are at least 2dN of them, and then fkn+1 (Q ) = fkn + a, where a ∈ [1, M N ]. Therefore, fkn+1 (Q ) − φ(kn+1 ) ≤ D because (6.1) fkn+1 − φ(kn+1 ) = fkn + a − φ(kn+1 ) < a ≤ M N ≤ D and φ(kn+1 ) − fkn+1 = φ(kn+1 ) − fkn − a ≤ φ(kn ) − fkn + 1 − a ≤D+1−a ≤ D. (6.2) If fkn (Q) ≥ φ(kn+1 ), then choose those N th generation descendants Q of Q for which −M N ≤ fkn+1 − fkn ≤ −1. From Lemma 5.1 we know that there are at least 2dN of them, and then fkn (Q ) = fkn − a, where a ∈ [1, M N ]. Therefore fkn+1 (Q ) − φ(kn+1 ) ≤ D, because (6.3) fkn+1 − φ(kn+1 ) = fkn − a − φ(kn+1 ) > −a ≥ −M N ≥ −D and φ(kn+1 ) − fkn+1 = φ(kn+1 ) − fkn + a ≥ φ(kn ) − fkn − 1 + a ≥ −D + a − 1 ≥ −D. (6.4) So we deﬁne the next imbedded set En+1 by {all the chosen descendants of Q}. (6.5) En+1 = Q∈En Then En+1 ⊂ En , and for all Q ∈ En+1 we have fkn+1 (Q ) − φ(kn+1 ) ≤ D; moreover, Qd = Qd 2dN for all Q ∈ En . Since lim t→∞ φ(t) = ∞, the inequality fn (Q) − φ(n) ≤ D implies n (Q) that the quotient fφ(n) is also bounded above. Moreover, it is also bounded away from zero for suﬃciently large values of n. Therefore, for all Q ⊂ En , fkn (Q) 1 ≤ D. (6.7) ≤ D φ(kn ) (6.6) En+1 ∩ Qd ≥ 2dN License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse DIMENSION OF ESCAPING GEODESICS 5601 The nested sets {En } deﬁned this way satisfy the requirements of Lemma 3.3, so the Hausdorﬀ dimension of the set E = n En is (6.8) dimH (E) ≥ d − C(N, ) with limN →∞ C(N, ) = 0. According to Theorem 3 in Sullivan’s paper [17] U (z) 1 ≤ C, ≤ (6.9) C dist(z, z0 ) and Lemma 4.2 states that U (z) − fQz  ≤ A, which we can combine with the fn (Q) 1 D ≤ φ(n) ≤ D, to get distγ(n), z 1 0 (6.10) ≤ ≤ C . C φ(n) inequality (6.7), This shows that dimH (Λφ ) = d. Choosing φ(t) = t the set Λφ determines the deep points, and using Theorem 6.1 we get the following corollary: Corollary 6.3. If G is a noncompact, geometrically inﬁnite and topologically tame Kleinian group, M = B/G has injectivity radius bounded away from zero, and there exists a Green’s function on M , then the deep points have dimension 2. Similarly in the Fuchsian case: Corollary 6.4. If G is a geometrically inﬁnite Fuchsian group so that there exists a Green’s function on M = B/G and the injectivity radius is bounded away from zero as well as bounded from above, then the deep points have dimension 1. The dimension of escaping points has been studied, and recently J. L. Fernández and M. V. Melián (in [9]) showed that the escaping geodesics on a ﬁnitely generated divergence type Riemann surface have dimension 1. Deep points form a smaller set, but perhaps one could derive a similar theorem for an intermediate set, for Λφ as deﬁned in Section 1 with some function φ. Acknowledgement The author would like to give special thanks to her advisor, Christopher J. Bishop, for the many helpful consultations and for his useful comments and suggestions on the previous drafts of this work. References [1] I. Agol, Tameness of Hyperbolic 3manifolds, preprint, available at arXiv:math.GT/0405568, 2004. [2] L. V. 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MR1041575 (91i:58104) [16] S. Rohde, The boundary behavior of Bloch functions, J. London Math. Soc. (2) 48 (1993), no. 3, 488–499. MR1241783 (94k:30083) [17] D. Sullivan, Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and Hausdorﬀ dimension two, Geometry Symposium 504 (1980), 127–144. MR655423 (83h:53054) Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794 Current address: Department of Mathematics, University of West Hungary, Szombathely, H9700, Hungary Email address: zgonye@ttmk.nyme.hu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse