Counting closed geodesics in Moduli space

We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.

Theorem 1.4 (Rafi, Hammenstadt) For any compact K, and sufficiently large R, Previous Results. The first results on this problem are due to Veech [Ve]. He proved that there exists a constant c 2 such that and conjectured that c 2 = h. In a remark in a paper by Ursula Hamenstadt [H1] (see also [H2]), in which the main focus is different, she proves that c 2 ≤ (6g − 6 + 2n)(6g − 5 + 2n). Sasha Bufetov [Bu] proved the formula whereÑ (R) is the number of periodic orbits of the Rauzy-Veech induction such that the log of the norm of the renormalization matrix is at most R. (This is a closely related problem; essentiallỹ N (R) counts closed geodesics on a certain finite cover of M g ). However the equation (1) does not easily imply Very recently, Kasra Rafi [Ra1] proved Corollary 1.3 (which implies (2)) for the case of the fivepunctured sphere.
Remarks. We note that (2) is an immediate consequence of Theorem 1.1, which is a bit more precise. In order to prove Theorem 1.1 one needs Corollary 1.3 and certain recurrence results for geodesics, which are based on [Ath].

A system of inequalities.
Suppose 0 < s < 1 (in fact we will be using s = 1/2 only). Let τ ≫ 1 be a parameter to be chosen later. (In particular we will assume e −(1−s)τ < 1/2.) Let A τ be the operator of averaging over a ball of radius τ in Teichmüller space. So if f is a real-valued function on Teichmüller space, then (A τ f )(X) = 1 m(B(X, τ )) B (X,τ ) f (y) dm(y).
Remark. In [EM], [Ath] and [EMM] the average is over spheres. In this context, we use balls, since Minsky's product region theorem gives us much more precise information about balls then about spheres.
Let m be the maximal number of disjoint curves on a closed surface of genus g. Choose K > e 2mτ , and pick constants ǫ 1 < ǫ 2 < · · · < ǫ m < 1/K 3 such that for all 1 ≤ i ≤ m − 1, Note that K and ǫ 1 , . . . , ǫ m are constants which depend only on τ and the genus g.
For 1 ≤ i ≤ m and X ∈ T g let ℓ i (X) denote the extremal length of the i'th shortest curve on X. Let f 0 = 1 and for 1 ≤ j ≤ m let Note that f j is invariant under the action of the mapping class group, and thus descends to a function on M g . Let u(X) = m k=1 f j (X).
Let ǫ ′ j = ǫ j /(mK 2 ). Let W j = {X ∈ T g : ℓ j+1 (X) > ǫ ′ j }. Note that W 0 is compact, and on W j there are at most j short curves. If X ∈ W j−1 then X has at least j short curves, and thus if X ∈ W j \ W j−1 then X has exactly j short curves.
In this subsection, we prove the following: Proposition 2.1 Set s = 1/2. Then we may write where b(X) is a bounded function which vanishes outside the compact set W 0 , and for all j and for all X ∈ W j−1 , c(X) ≤ C ′ j τ j e −jτ , where C ′ j depends only on the genus.
We now begin the proof of Proposition 2.1. We recall the following: Lemma 2.2 There exists L 0 > 0 (depending only on g) and for every L > L 0 there exist constants 0 < c 1 < c 2 such that for all X ∈ T g , c 1 ≤ m(B(X, L)) ≤ c 2 The constants c 1 and c 2 depend on L and g (but not on X).
Proof. Suppose X ∈ T g . Let α 1 , . . . , α k be the curves on X with hyperbolic length less then ǫ (where ǫ is the Bers constant). Let a i denote the hyperbolic length of a i . Then the extremal length Ext X (α i ) ≈ a i (by e.g. [Mi]). It follows from the Kerchhoff formula for the Teichmüller distance CITE... that for all Y ∈ B(X, L), where C depends only on the genus. Then, in view of the definition of extremal length, for any area 1 holomorphic quadratic differential q on Y ∈ B(X, L), where ℓ q (·) denotes length in the flat metric defined by q, and C depends only on the genus. Thus any flat metric in the conformal class of a surface in B(X, L) has curves of flat length at most L √ Ca i . Let F be a fundamental domain for the action of Γ on T g . Then, in view of (5), and the definition of the measure m(·), for any γ ∈ Γ, Let I X,L denote the set of elements γ ∈ Γ such that B(X, L) ∩ γF is non-empty. We will now estimate the size of I X,L . Note that up to uniformly bounded index (the bound depending on g and L) I X,L consists only of twists around the α i . By [Mi], the number of twists around α i which one can take and still stay in B(X, L) is O(L 2 /a i ). Thus, Now the upper bound of Lemma 2.2 follows from (6). We now briefly outline the proof of the lower bound. Let R ⊂ Q g /Γ be the set of flat structures such that each S ∈ R has flat cylinders C i with width (i.e. core curve) between √ a i and √ a i /2 and height between 1/(g √ a i ) and 1/(2g √ a i ), and the rest of the arcs in a triangulation of S ∈ R have length comparable to 1. It is easy to verify using the definition of the measures µ and m that where c depends only on the genus. Note that by [Ra2] for any Y ∈ π(R), the only short curves on Y (in the hyperbolic or extremal metric) are the core curves of the cylinders C i , and the extremal length of the core curve of C i is within a constant multiple of a i . Thus, there exists a constant L ′ depending only on the genus such that π(R) ⊂ B(X, L ′ ).
Note that the above equation takes place in T g /Γ. We may think of it as taking place in T g if we identify π(R) with a subset of the fundamental domain F . Then, for any γ ∈ I X,L ′ , Thus, in view of (7) and (8), where c depends only on the genus.
Sketch of proof. The product region theorem [Mi] states that P (α 1 , . . . , α j ) can be identified with a subset of (H 2 ) j × T ′ (where T ′ is the quotient Teichmüller space obtained by collapsing all the α i ), and the Teichmüller metric on P (α 1 , . . . , α j ) is within an additive constant of the supremum metric on (H 2 ) j × T ′ . Let m ′ denote the product measure on (H 2 ) j × T ′ , and let A ′ τ denote the averaging operator with respect to the product measure m ′ , i.e. for a real-valued function f , We first establish the lemma with A τ replaced by A ′ τ . We may write X = (X 1 , . . . , X j , X ′ ) where X k is in the k'th copy of the hyperbolic plane and X ′ ∈ T ′ . Because of the product region theorem B(X, τ ) is essentially B(X 1 , τ )×. . .×B(X j , τ )×B ′ (X ′ , τ ) where for 1 ≤ k ≤ j, B(X k , τ ) is a ball of radius τ in the hyperbolic plane and B ′ (X, τ ) a ball of radius τ in T ′ . Also, by assumption, for any Y ∈ B(X, τ ) the set of k shortest curves on Y is where for Y j ∈ H 2 , ℓ(Y j ) is the flat length of the shortest curve in the torus parametrized by Y j . (The exponent is 2s instead of s since on the torus extremal length is the square of flat length). Hence, where Vol is the standard volume form on H 2 , and the notation A ≈ B means that the ratio A/B is bounded from above and below in terms of the genus. Now the integral in the parenthesis, i.e. an average of ℓ −s over a ball in a hyperbolic plane is essentially done in [EM,Lemma 7.4] (except that there the average is over spheres, but to get the average over balls one just makes an extra integral over the radius). One gets for 1/2 < s < 1, and for s = 1/2, Substituting these expressions into (10) completes the proof of Lemma 2.3 with A ′ τ instead of A τ . Recall that a set N is an (c, 2c) separated net on a metric space X if N ⊂ X , every point of X is within 2c of a net point, and the minimal distance between net points is at least c. In view of Lemma 2.2, for any (2, 2c) separated net N in any Teichmüller space (including H 2 ), for τ ≫ 1 and any X, where c 1 , c 2 depend only on c and the genus. Similarly, in view of the form of the function f k , Let d(·, ·) denote the Teichmüller metric and let d ′ (·, ·) denote the supremum metric on (H 2 ) j ×T ′ . Minsky's product region theorem states that there exists a constant β > 0 depending only on the genus such that for all X, Y ∈ P (α 1 , . . . , α j ), |d(X, Y ) − d ′ (X, Y )| < β. Choose L ≫ β, and choose a (L, 2L)-separated net N k in each factor. Let N be the product of the N k . Then N is an (L − β, 2L + β)-separated net in P (α 1 , . . . , α j ). Now in view of (11), (In the above equation, A ≈ B means that the ratio A/B is bounded by two constants depending only on β, L and g, and thus ultimately only on g). Similarly, using (12), we can show that Remark. The proof works even if at some point Y ∈ B(X, τ ) there are short curves other then {α 1 , . . . , α j } (but these other curves are longer then the maximum of the lengths of the α j at Y ). This is used in the next lemma.
Lemma 2.4 For 1 ≤ j ≤ m, let u j (X) = m k=j f j (X). Suppose ℓ j (X) < ǫ j . Then (assuming τ is large enough), where c j are as in (9). In particular, letting j = 1, and noting that the set {X ∈ M g : ℓ 1 (X) > ǫ 1 } is compact, we have for all X ∈ T g , Proof. Note that for any 1 ≤ i ≤ m, and any X ∈ T g , (this is because in B(X, τ ) the extremal length of any curve cannot change by more then e 2τ ). We divide the set {j, j +1, . . . m} into two disjoint subsets: Let I 1 be the set of k ∈ {j, j +1, . . . m} such that and let I 2 be the set of k ∈ {j, j + 1, . . . m} such that the opposite inequality to (15) holds. Suppose k ∈ I 1 . Then, by (14), Now suppose k ∈ I 2 . We claim that Indeed, if k = j then (17) is true by assumption. If k > j then where we have used the inequality opposite to (15) in the last estimate. Thus (17) follows. We now claim that under the assumption that k ∈ I 2 we have If k = m this is clear from (17) (since in the case where ℓ m (X) is small, there are no other short curves on X). Now if k < m, then where again we used the inequality opposite to (15) in the last estimate. Thus, Now (18) follows from (19), (17) and (3). Now in view of (17) and (18), Lemma 2.3 can be applied to f k . Thus, for k ∈ I 2 , where for the last inequality we assumed that τ was large enough so that c k < c j for k > j. Now (13) follows from (16) and (20).

A uniform estimate for the measure of a ball.
Proposition 2.5 There exists a constant C 2 such that for any X, any δ ′′ > 0 and any sufficiently large τ , the volume of any B(X, τ ) is bounded by C 2 e (h+δ ′′ )τ .
Proof. See Appendix.

Proof of Theorem 1.2
We discretize Teichmüller space by fixing a (1, 2) separated net N ⊂ T g ; this means that the distance between any two net points is at least 1, and any point in Teichmüller space is within distance 2 of a net point. Let We note that there exist constants κ 1 and κ 2 such that for all X ∈ T g , where the κ i depend on τ and the ǫ i (and thus ultimately only on τ and the genus).
Trajectories of the random walk. Suppose R ≫ τ and let n be the integer part of R/τ . By a trajectory of the random walk we mean a map λ : {0, n − 1} → T g such that for all 0 < k ≤ n − 1 we have d(λ k , λ k−1 ) ≤ τ and also λ k belongs to the net N we are using in T g . Let P(X, R) denote the set of all trajectories for which d(λ 0 , X) = O(1). It is a corollary of Proposition 2.5 that where | · | denotes the cardinality of a set. We say that a trajectory is almost closed in the quotient if the distance in M g between the projection to M g of λ(0) and the projection to M g of λ n−1 is O(1).
Let δ > 0 be a constant to be chosen later. (We will have δ < ǫ ′ j for 1 ≤ j ≤ m, where the ǫ ′ j are as in §2.1). For j ∈ N, let P j (X, δ, R) denote the set of all trajectories starting within O(1) of X for which at any point, there are at least j curves of length at most δ. LetP j (X, δ, R) denote the subset of these trajectories which is almost closed in the quotient.
Lemma 2.6 For any j ∈ N, and any ǫ ′ > 0 there exists C > 0 such that for τ large enough (depending on ǫ ′ and the genus g), and δ > 0 small enough (depending on τ , ǫ ′ and g), Here C depends on τ , δ, ǫ ′ and g.
Proof. Let R = nτ , and let where where the elements of P j (X, δ, R, r) are the trajectories λ belonging to P j (X, δ, R) but truncated after k = r/τ steps. Then where in the next to last line we estimated a sum over N ∩ B(λ k−1 , τ ) by a constant C times an integral over B(λ k−1 , τ ). Note that for λ ∈ P j (X, δ, R), the number of curves shorter then δ on λ k−1 is at least j. Thus, if δ is small enough, (depending on the the ǫ ′ j and thus ultimately only on τ and the genus), λ k−1 ∈ W j−1 . Then, from Proposition 2.1, and assuming τ is large enough so that Proposition 2.5 holds with δ ′′ < ǫ ′ /2, we get Now iterating (25) n = R/τ times we get We now choose τ so that log(CC ′ j τ ) τ < ǫ ′ /2. Now the lemma follows from (21), (24), and (26).
Let N j (X, δ, R) be the number of conjugacy classes of closed geodesics of length at most R which pass within O(1) of the point X and always have at least j curves of length at most δ.
Lemma 2.7 For any ǫ ′ > 0 we may choose τ large enough (depending only on ǫ ′ ) so that for any X ∈ T g , any δ < 1/2 and any sufficiently large R (depending only on ǫ ′ , τ ) we have Proof. Let I X denote the subset of the mapping class group which moves X by at most O(1). Then up to uniformly bounded index, I X consists only of Dehn twists around curves which are short on X. Now consider a closed geodesic λ in M g which passes within O(1) of p(X) (recall that p denotes the natural map from T g → M g ). Let [λ] denote the corresponding conjugacy class in Γ. Then there are approximately |I X | lifts of [λ] to T g which start within O(1) of X. Each lift γ is a geodesic segment of length equal to the length of λ.
We can mark points distance τ apart on γ, and replace these points by the nearest net points. (This replacement is the cause of the ǫ ′ ). This gives a map Ψ from lifts of geodesics to trajectories. If the original geodesic λ has length at most (1 − ǫ ′ )R and always has j curves shorter then δ, then the resulting trajectory belongs toP j (X, 2δ, R).
If two geodesic segments map to the same trajectory, then the segments fellow travel within O(1) of each other. In particular if g 1 and g 2 are the pseudo-anosov elements corresponding to the two geodesics, then d(g −1 2 g 1 X, X) = O(1), thus g −1 2 g 1 ∈ I X . We now consider all possible geodesics contributing to N j (X, δ, (1 − ǫ ′ )R); for each of these we consider all the possible lifts which pass near X, and then for each lift consider the associated random walk trajectory. We get: (the factor of |I X | on the left hand side is due to the fact that we are considering all possible lifts which pass near X, and the factor of |I X | on the right is the maximum possible number of times a given random walk trajectory can occur as a result of this process). Thus, the factor of |I X | cancels, and the lemma follows.
The following lemma is due to Veech [Ve].
Lemma 2.8 Suppose λ ∈ M g is a closed geodesic of length at most R. Then for any X ∈ λ, Proof. We reprodce the proof for completeness. LetX be some point in T g with p(X) = X. Suppose the esitmate is false, and let α be a curve onX with hyperbolic length less then ǫ ′ 0 e −(6g−4)R . Let γ be the element of the mapping class group associated to the lift of λ passing throughX.
Proof of Theorem 1.2. Let ǫ ′ = ǫ/8. By Lemma 2.7 and Lemma 2.6 we can choose τ and δ so that (27) holds and also (23) holds with δ replaced by 2δ. We get, for sufficiently large R, Finally N j (δ, R) is at most X N j (X, δ, R), where we have to let X vary over a net points within distance 1 of a fundamental domain for the action of the mapping class group. In view of Lemma 2.8, the number of relevant points in the net is at most polynomial in R. Thus Theorem 1.2 follows.
For a compact subset K of M g and a number θ > 0 let N K (θ, R) denote the number of closed geodesics γ of length at most R such that γ spends at least θ-fraction of the time outside K.
Theorem 3.1 Suppose θ > 0. Then there exists a compact subset K of M g and δ > 0 such that for sufficiently large R, Proof. In view of Corollary A Teichmüller geodesic γ is in fact a path in the unit tangent bundle of M g , i.e. the space Q g /Γ, where Q g is the space of unit area holomophic quadratic differentials on surfaces of genus g. Let P(1, . . . , 1) ⊂ Q g denote the principal stratum, i.e. the set of pairs (M, q) where q is a holomoprhic quadratic differential on M with simple zeroes. As above, for a compact subset K of P(1, . . . , 1) and θ > 0 we denote by N K (θ, R) denote the number of closed geodesics γ of length at most R such that γ spends at least θ-fraction of the time outside K.
Theorem 3.2 Suppose θ > 0. Then there exists a compact subset K ⊂ P(1, . . . , 1) and δ > 0 such that for sufficiently large R, The rest of this subsection will consist of the proof of Theorem 3.2. Choosing hyperbolic neighborhoods of points. If S ∈ Q g is a pair (M, q) where M is a genus g surface and q is a quadratic differential on M , then we let ℓ(S) denote the length of the shortest saddle connection on S (in the flat metric defined by q). Let d T (·, ·) denote the Tecihmller metric. Suppose K 1 ⊂ M g is a compact set. For simplicity, we denote the preimage of K 1 in T g by the same letter.
As in [ABEM,§2], we denote the strong unstable, ubstable, stable and strong stable foliations of the geodesic flow by F uu , F u , F s and F ss respectively; for a given quadratic differential q, We consider the distance function defined by the modified Hodge norm d E on each horosphere F ss and let d E (·, ·) denote the Euclidean metric as defined in [ABEM,§8.4].
• P3: There exists C 1 > 0 such that if d H (q 1 , q 2 ) < 1, q 1 ∈ F ss (q 2 ) and s ≥ 0 • P4: Moreover, given ǫ, β > 0 , there exists C 0 , α > 0 such that for any q 1 ∈ F ss (q 2 ) with d H (q 1 , q 2 ) < 1, and s ≥ 0 if |{t| t ∈ [0, s], ℓ(g t q 1 ) > ǫ}| > βs, and g s q 1 ∈ K 1 , then Note that in this case, by (31), there exists L 0 ( depending only on K 1 , β and ǫ) such that for s > L 0 , (30) implies that : Proof of Lemma 3.3. Let B E (q, r) denote the radius r ball with center q in the Euclidean metric. Since K 1 is compact, there is a number ρ 1 > 0 depending only on K 1 and ǫ such that B E (q, ρ 1 ) is contained in one fundamental domain for the action of the mapping class group Γ. Now let S 0 = S(x) denote the sphere at x, i.e. the set of unit area holomorhic quadratic differentials on the surface x. Let γ(t) = g t (q x,p0 ). For q ∈ S 0 near γ(0) = q x,p0 , let in other words, we can choose t(q) ∈ R be such that f (q) ∈ F uu (q x,p0 ) and g t(q) q and f (q) are on the same leaf of F ss . Then clearly f (q x,p0 ) = q x,p0 , and there exists a number ρ 3 > 0 depending only on K 1 such that the restriction of f to B E (q x,p0 , ρ 3 ) ∩ S 0 is a homeomorphism onto a neighborhood of q x,p0 in F uu (q x,p0 ). In particular, for any q ∈ B E (q x,p0 , ρ 3 )∩S 0 , we know that F uu (q x,p0 )∩F s (q) = ∅.
This is straightforward (in view of the non-uniform hyperboliclity as in (29) and (31)) but somewhat tedious argument. Let Then V is relatively open as a subset of F u (q x,p0 ). Let ∂V denote the boundary of V viewed as a subset of F u (q x,p0 ). Therefore we can choose ρ 2 > 0 depending only on K 1 such that for all q ′ ∈ ∂V , By (29) and the fact that ∂V ⊂ F u (q x,p0 ), this implies that for some constant ρ ′ 2 , (depending only on K 1 ), all q ′ ∈ ∂V and all L > 0, Note that U ⊂ t∈R g t S 0 ∼ = R × S 0 . Let ∂U denote the boundary of U viewed as a subset of R × S 0 . Suppose q 1 ∈ ∂U . We may write q 1 = g t q for some q ∈ S 0 . Then the fact that q 1 ∈ ∂U implies that either d E (q, q x,p0 ) = ρ 3 /2 or |t| = ρ 3 . In either case, let q 2 = g t+t(q) f (q). Then q 2 ∈ ∂V , q 1 and q 2 are on the same leaf of F ss , and where C depends only on K 1 . Hence, by (29), we have In order to prove the claim, we show that there exists L 0 (depending only on K 1 and ǫ) such that for L > L 0 , Suppose that (36) fails. Then: • by (35), d E (g L q 2 , γ L ) ≤ C 2 , where C 2 only depends on K 1 . On the other hand, we can choose |t 0 | ≤ ρ 3 such that g t0 q 2 ∈ F ss (q x,p0 ). Using (31) for g L+t0 q 2 ∈ F ss (γ L ), we get that where ǫ ′ , ǫ 0 only depend on K 1 , and ǫ.
Let q 1 = q x,p1 , and (as in the previous lemma) let q = f (q x,p2 ) be the unique point in F uu (q 1 ) ∩ F s (q x,p2 ). Now let q 2 be a quadratic differential of area 1 on the geodesic joining x to p 2 such that q 2 ∈ F ss (q); in particular, we have q ∈ F uu (q 1 ), q ∈ F ss (q 2 ).
As a result, from (29) we get d E (g r q, g r q 2 ) < ǫ 0 /4 and d E (g r q 1 , g r q) < C 1 , where C 1 only depends on K 1 , and ǫ. Consider the map between the points on the geodesic [xp 1 ] to the points on [xp 2 ] as follows: We can choose, 0 ≤ s 0 ≤ r such that |{t |s < t < r , ℓ(g t q 1 ) ≥ ǫ}| = r/6, and let A = {t |0 < t < s 0 , ℓ(g s q 1 ) ≥ ǫ}.
It is easy to check that |A| > r/3. We claim that for s ∈ A, we have This is because: • For any s ∈ A, (30) for g r q and g r q 1 and the interval (0, r − s) holds. Hence, by (31), d E (g s q 1 , g s q) = d E (g s−r (g r q 1 ), g s−r (g r q)) < ǫ 0 /4; • by (29), d E (g s q, g s q 2 ) < ǫ 0 /4; • Finally, since q 1 ∈ F uu (q), d E (g s q 1 , g s q 2 ) ≤ ǫ 0 .
Proof of Theorem 3.2. Choose θ 1 > 0 . Let K 1 ⊂ M g be such that Theorem 3.1 holds for K = K 1 , and θ = θ 1 . Let K 2 be a compact subset of P(1, . . . , 1)/Γ such that K 2 ⊂ p −1 (K 1 ), and let K 3 be a subset of the interior of K 2 . We may choose these sets so that µ(K 3 ) > (1/2), where µ is the Lebesque measure on Q g /Γ normalized so that µ(Q g /Γ) = 1. We also choose K 2 and K 3 to be symmetric, i.e if q ∈ K 2 then −q ∈ K 2 (and same for K 3 ). Then there exists ǫ > 0 such that for X ∈ K 3 , ℓ(X) > ǫ. Let c 0 be as in Lemma 3.4. We now choose a (c 1 , c 2 ) separated net N on T g , which c 1 < c 0 , c 2 < c 0 . We may assume that N ∩ p −1 (K 1 ) is invariant under the action of the mapping class group. Suppose X ∈ T g , and let S(X) denote the unit sphere at X, i.e. the set of area 1 holomoprhic quadratic differentials on X. Let so that V (X, T ) is the subset of B(X, T ) consisting of points Y ∈ B(X, T ) such that the geodesic from X to Y spends more then half the time outside K 2 . By [ABEM,Theorem 6.2], for any θ 1 > 0, there exists T > 0 such that for any τ > T and any where Nbhd a (A) denotes the set of points within Teichmller distance a of the set A. Then, since K 1 is compact and θ 1 is arbitrary, this implies that for any θ 2 > 0 there exists T > 0 such that for any τ > T and any X ∈ N ∩ p −1 (K 1 ), By the compactness of K 1 and [ABEM, Theorem 1.2 and Theorem 10.1] there exists C 1 > 1 such that for τ sufficiently large and any X ∈ N ∩ p −1 (K 1 ), Thus, for any θ 3 > 0 there exists T > 0 such that for τ > T , From now on we assume that τ is sufficiently large so that (40) holds. Let K ′ 1 = Nbhd c2 (K 1 ), and let G(R) denote the set of closed geodesics in M g of length at most R, and let G K ′ 1 (θ 3 , R) ⊂ G(R) denote the subset which contributes to N K ′ 1 (θ 3 , R). In view of Theorem 3.1, it is enough to show that there exists δ 0 > 0 such that for R sufficiently large, As in §2, we associate a random walk trajectory Φ(γ) to each closed geodesic γ ∈ G(R). Let P 1 (R) = Φ(G(R) \ G K ′ 1 (θ 3 , R)) denote the set of resulting trajectories. Note that by construction, every trajectory in P 1 spends at most θ 3 fraction of the time outside K 1 .

(b)
The length of γ ′ is within ǫ of L.
Outline of Proof. This is very similar to the proof of Lemma 3.4. In view of the hyperbolicity statement (31), there is a neighborhood of γ(0) (of size at most c 0 where c 0 is as in Lemma 3.4) such that the time L geodesic flow restricted to the neighborhood expands along the leaves of F uu and contracts along the leaves of F ss , in the metric d H . Then the contraction mapping principle (applied first to the map on F ss and then to the inverse of the map on F uu ) allows us to find a fixed point for the geodesic flow near γ(0).
In view of Theorem 3.2, the proof essentially reduces to the now standard hyperbolic dynamics argument of Margulis [Mar], see also [KH]. The argument below is not rigorous: its aim is to recall some of the key ideas with emphasis on what is different in this setting. We refer the reader to e.g. [KH,§20.6] for the missing details of the argument. Choose any θ > 0. Fix a compact set K ⊂ P(1, . . . , 1) such that Theorem 3.2 holds, and µ(K) > (1 − θ/2). Let ǫ ′ > 0 be such that ℓ(q) > ǫ ′ for all q ∈ K. Let K 1 = π −1 (π(K)). Let U ⊂ K 1 be a small box in the tangent space Q g . We assume that the Teichmüller diameter of U is at most c 0 , where c 0 is as in Lemma 3.4 (with ǫ ′ instead of ǫ).
Recall that the geodesic flow on Q g is mixing, i.e. for A, B ⊂ (Q g ), µ(g t (A) ∩ B) ≈ µ(A)µ(B) for large values of t. We now want to consider what happens to U under the action of the flow. We can think of the action of the geodesic flow as stretching U along F uu and contracting U along F ss . Because of the mixing property of the flow, we see the image of U under the flow intersecting U many times.
Let R be large enough that g R (U ) ∩ U has more than one component. We say that a connected component is "regular" if for some q ∈ U , (so that the geodesic segment spends at least half the time in K). Note that in view of Lemma 3.4, if (43) holds for some q ∈ U then for all q ∈ U , where K ′ ⊃ K is also a compact subset of P(1, . . . , 1). It follows from the closing lemma (Lemma 4.1) that each regular component connected component of the inersection g R (U ) ∩ U contains exactly one closed geodesic of length between R and R + ǫ, where ǫ is related to the diameter of U . Conversely, each closed geodesic of length between R and R+ǫ which spends at least half the time in K belongs to a regular connected component of g R (U )∩U . Thus, where C(U, R) is the number of regular connected components of the intersection g R (U ) ∩ U , N 1 (U, R, R + ǫ) is the number of closed geodesics of length between R and R + ǫ which intersect U and spend at least (1 − θ)-fraction of the time in K, and N 2 (U, R, R + ǫ) is the number of closed geodesics of length between R and R + ǫ which intersect U and spend at least (1 − 2θ)-fraction of the time in K ′ . Note that the irregular connected components are all contained in the set By the ergodic theorem, µ(E R ) → 0 as R → ∞. Thus, for R sufficiently large, most of the measure of g R U ∩ U is contained in regular connected components. Now we can estimate, where A is a single connected component. The flow contracts the measure along F ss by e −hR , so the area of each component of the intersection has area about µ(U )e −hR . By the mixing property, we know that the numerator in the preceding equation is about µ(U ) 2 , so we get that (In the above we made the simplifying assumption that all regular intersections have the same area. This can be removed with a bit of care: see [KH,§20.6] for the details). Combining (46) with (45) and Theorem 3.2, we get Let N 1 (R, R + ǫ) denote the number of closed geodesics of length between R and R + ǫ which spend at least θ-fraction of the time in K. Of course not all geodesics contributing to N 1 (R, R + ǫ) will pass through U . To catch all of them, we create a tiling of K 1 by boxes U i . Naively, the number of such geodesics of length R passing through U i is about µ(U i )e hR , so the total number of geodesics is µ(U i )e hR = e hR µ(U i ) = e hR µ(K 1 ) ≈ e hR . This is wrong, of course, because a geodesic will pass through many boxes. If the width of the boxes is ǫ, and the length of the geodesic is about R, and it spends at least 1 − θ fraction of the time in K 1 , we get that each of our paths passed through about R/ǫ boxes. This shows that our counting was off by a factor of (1 + O(θ))R/ǫ, so we get Thus the total number N 1 (R) of geodesics of length less then R which stay at least half the time in K can be estimated as follows: The sum on the right telescopes, and we get Since ǫ > 0 and θ > 0 are arbitrary, the theorem follows.
A Appendix: Proof of Proposition 2.5 Let ∆ be a net in M g , and let ∆ ⊂ π −1 ( ∆) be a net in T g ; in other words, there are constants c 1 , c 2 > 0 such that 1) : given X ∈ T g there exists Z ∈ ∆ such that d T (X, Z) ≤ c 1 , and 2) : for any Z 1 = Z 2 ∈ ∆, we have d T (Z 1 , Z 2 ) ≥ c 2 . In this case π(∆) = ∆. Here π : T g → M g is the natural projection to the moduli space. We recall that there exists a constant C 2 > 0 such that for any X ∈ T g |π(B(X, τ )) ∩ ∆| ≤ C 2 τ 6g−6 .

Now this implies Proposition 2.5 in view of Lemma 2.2.
Proof of Proposition A.1. Given X ∈ T g , let ℓ γ (X) = Ext γ (X). Fix a very small ǫ 0 . Here we say γ is short on X if ℓ γ (X) ≤ ǫ 0 . Let A X denote the set of all short simple closed curves on X. Given X, Y ∈ T g , and B ⊂ A X let F τ (X, Y, B) = {gY |g ∈ Γ, d T (X, gY ) ≤ τ, A X ∩ A gY = B} ⊂ F τ (X, Y ).
On the other hand, since we can approximate a geodesic by points in the net ∆, we have |F τ (X, Y )| ≤ |P|, also by the definition, P ⊂ W ∈Z P(W ).
We are using (51) to obtain the last inequality. Now we have, where c g = O(g 2 ).