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arXiv:1905.04435v2 [math.GT] 5 Jul 2021

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS ON

HYPERBOLIC SURFACES

ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADI

Abstract.We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at mostLon a compact surface equipped with a Riemannian metric of negative curvature. The proof relieson the exponential mixing rate for the Teichm¨uller geodesic flow.

1.Introduction

Letg≥2, and letSbe a compact Riemann surface of genusg. LetT(S) be the Teichm¨uller space of complete hyperbolic metrics onS, and let

M(S) =T(S)/Modg

be the corresponding moduli space, where Mod gis the mapping class group ofS. LetM? M(S). Problems related to the asymptotic growth rate of the number of closed geodesics onMhave been long studied. In particular, thanks to works of Delsart, Huber, and Selberg we have the following: There exists someδ=δ(M)>0 so that the number of closed geodesics of length at mostLonMequals (1) Li(eL) +OM(eL-δ), where Li(x) =?x 2dt logt; see [Bus] and references there. More generally, the growth rate of the number of closed geodesics on a negatively curved compact manifold was studied by Margulis, [Mar]. His proof,which is different from the above mentioned works, is based on the mixing property of theMargulis measure for the geodesic flow. In the constant negative curvature case, Margulis" method combined with an exponential mixing rate for the geodesic flow, also providesan estimate like (1) - albeit with a weaker power savingδ, see e.g. [MMO].

1.1.Simple closed geodesics.The aforementioned fundamental results do not provide

any estimates for the number of simple closed geodesics onM. Indeed, very few closed geodesics onMare simple, [BS2], and it is hard to discern them inπ1(M), [BS1]. More explicitly, it was shown in [Ri] that the number ofsimpleclosed geodesics of length at most

LonMis bounded above and below byOM(L6g-6).

A.E. acknowledges support by the NSF and the Simons Foundation. A.M. acknowledges support by the NSF and Alfred P. Sloan Research Fellowship. 1

2 ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADIIn her PhD thesis, [Mir1] and [Mir2], Mirzakhani proved an asymptotic growth rate for the

number of simple closed geodesics of a given topological type on a hyperbolic surfaceM- recall that two simple closed geodesicsγandγ?onMare of the same topological type if there exists someg?Modgso thatγ?=gγ. LetXbe a compact surface equipped with a Riemannian metric of negative curvature. We emphasize that the curvature is not assumed to be constant; indeed, elements inM(S) will be denoted byMto minimize the confusion. By a multi-geodesicγonXwe mean γ=?di=1aiγiwhereγi"s are disjoint, essential, simple closed geodesics, andai>0 for all onX. The multi-geodesicγwill be called integral (resp. rational) ifai?Z(resp.ai?Q).

Given a rational multi-geodesicγ0onX, define

s Mirzakhani, [Mir2, Thm. 1.1], proved the following estimate whenMis a hyperbolic surface: (2)sM(γ0,L)≂nγ0(M)L6g-6, wherenγ0:M(S)→R+(theMirzakhanifunction) is a continuous proper function; geo- metric informations carried bynγ0are also studied in [Mir2]. In this paper we obtain a quatitative version of (2); moreover, our approach allows us to prove such a result in the more general setting ofvariablenegative curvature. Theorem 1.1.There exists someκ=κ(g)>0so that the following holds. LetXbe a compact surface of genusgequipped with a Riemannian metric of negative curvature. Let

0be a rational multi-geodesic onX. Then

s

X(γ0,L) =nγ0(X)L6g-6+Oγ0,X(L6g-6-κ)

wherenγ0(X)is a positive constant which depends onγ0andX. The proof of Theorem 1.1 is based on the study of a related counting problem in the space of geodesic measured laminations onS, `a la Mirzakhani. The space of measured laminations onS, which we denote byML(S), is a piecewise linear integral manifold homeomorphic toR6g-6; but it does not have a natural differentiable structure, [Th1]. Train tracks were introduced by Thurston as a powerful technical device for understanding measured laminations. Roughly speaking, train tracks are induced by squeezing almost parallel strands of a very long simple closed geodesic to simple arcs on a surface; they provide linear charts forML(S).

The mapping class group Mod

gofSacts naturally onML(S). Moreover, there is a natural Mod g-invariant locally finite measure onML(S), the Thurston measureμTh, given by the piecewise linear integral structure onML(S), [Th1]. For any open subsetU? ML(S) and anyt >0, we have

Th(tU) =t6g-6μTh(U).

On the other hand, any metric of negative curvatureXonSinduces the length function λ?→?X(λ) onML(S), which satisfies?X(tλ) =t?X(λ) for allt >0. It is proved in [Mir1,

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS 3

App. A] that?Mis a convex function onML(S) whenMis a hyperbolic surface. This fact remains valid in the more general setting of variable negative curvature, see§5.5. The source of the polynomially effective error term in Theorem1.1 is the exponential mixing property of the Teichm¨uller geodesic flow proved by Avila, Gou¨ezel, and Yoccoz, [AGY, AR, AG]. We combine this estimate with ideas developed by Margulis in his PhD thesis, [Mar], to prove the following theorem which is of independent interest - see Theorem 7.1 for a more general statement. Letτbe a train track and letU(τ) be the corresponding train track chart. For every λ?U(τ) we let?λ?τdenote the sum of the weights ofλinU(τ), see§5. Theorem 1.2.There exists someκ1=κ1(g)>0so the following holds. Letτbe a maximal train track. LetL≥1and letγ0be a simple close curve onM. There exists a constant c

γ0>0so that

It is worth noting that in view of Theorem 1.2, the asymptoticbehavior of the number of points in one Mod points in this cone agree up to multiplicative constant. Theorem 1.2, in the more general form Theorem 7.1, plays a crucial role in our analysis. Indeed, using the aforementioned convexity of the length function, we will prove Theorem 1.1 using Theorem 7.1 in§8. It is an intriguing problem to investigate the asymptotic behavior of functions similar to and different fromsX(γ0,L) or the complexity considered in Theorem 1.2. For instance, for a suitable formulation of a combinatorial length - usingintersection numbers - the count is exactly a polynomial, see [FLP]. We also refer the reader to [CMP] where a related problem is studied for thrice punctured sphere.

1.2.Outline of the paper.In§2 we collect some preliminary results. In§3 we prove an

equidistribution result with an error term, Proposition 3.2, which may be of independent interest; see, e.g. [KM, LMir]. The proof of this proposition is based on the exponential mixing rate for the Teichm¨uller geodesic flow, [AGY], and the so calledthickeningtech- nique, see [Mar, EMc]. In§4 we prove Proposition 4.1; this proposition is one of the main ingredients in the proof, and could be compared to argumentsin [Mar, Chap. 6]. We will recall some basic facts aboutML(S), and study the relation between the linear structures onML(S) and the space of quadratic differentials in§5 and§6. The orbital counting in sec- tors ofML(S) is studied in§7; the main result here is Theorem 7.1. We prove Theorem 1.1 in§8.

1.3.Acknowledgement.This project originated in fall of 2015 when the authors were

members of the Institute for Advanced Study (IAS), we thank the IAS for its hospitality. We thank C. McMullen, K. Rafi, and A. Zorich for helpful discussions. We also thank F. Arana-Herrera, H. Oh, and A. Wright for their comments on an earlier version of this

4 ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADIpaper. We are in debt to G. Margulis and F. Arana-Herrera for drawing our attention to the

case of variable negative curvature, and to K. Rafi for providing the proof of Theorem 5.1. Last, but not least, we thank the anonymous referee for theircareful reading and several helpful comments.

2.Preliminaries and notation

LetQ(S) denote the moduli space of quadratic differentials onS, and letQ1(S) be the

moduli space of quadratic differentials with area one onS. For anyα= (α1,...,αk,ς) with?αi= 4g-4 andς? {±1}, defineQ1(α) to be (a connected component) of the stratum

of quadratic differentials consisting of pairs (M,q) whereM? M(S) andqis a unit area quadratic differential onMwhose zeros have multiplicitiesα1,...,αkandς= 1 ifqis the quare of an abelian differential and-1 otherwise. Then Q

1(S) =?

αQ

1(α).

PutQ(α) :={tq:t?R,q? Q1(α)}. Let Σ?Sbe a set ofkdistinct marked points. LetQ1T(α) denote the space of quadratic differentials (M,q) equipped with an equivalence class of homeomorphismsf:S→Mthat send the marked points to the zeros ofq. The equivalence relation is isotopy rel marked points. Letπ:Q1T(α)→ Q1(α) be the forgetful map which forgets the markingf; this is an infinite degree branched covering. Similarly, let Ω(S) denote the moduli space of Abelian differentials onS, and let Ω1(S) be the moduli space of area one Abelian differentials. For anyα= (α1,...,αk), we letH(α) denote the corresponding stratum, and letH1(α) denote the area one abelian differentials.

Note that passing to a branched double cover

ˆMofM, we may realizeQ1(α) as anaffine

invariant submanifoldinH1(ˆα) corresponding to odd cohomology classes onˆM, see§2.1. However, even ifqbelongs to a compact subset ofQ1(S), the complex structure onˆM may have very short closed curves in the hyperbolic metric, e.g. a short saddle connection between two distinct zeros on (M,q) could lift to a short loop inˆM. Note however that if ˆM,ω) is the aforementioned double cover of (M,q), then the length of the shortest saddle connection inωis bounded by the length of the shortest saddle connection inq, i.e. compact subsets ofQ1(α) lift to compact subsets ofH1(ˆα).

2.1.Period coordinates.Letx= (M,ω)? H(α), and let Σ?Mbe the set of zeros of

ω. Passing to a finite cover, which we continue to denote byH(α), we assume there are no orbifold points inH(α). Define the period map

Φ :H(α)→H1(M,Σ,C).

Let us recall that Φ can be defined as follows. Let #Σ =k. Fix a triangulationTof the surface by saddle connections ofx, that is: 2g+k-1 directed edgesδ1,...,δ2g+k-1which form a basis forH1(M,Σ,Z). Define

Φ(x) =??

iω?

2g+k-1

i=1.

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS 5

Note that this map depends on the triangulationT. IfT?is any other triangulation, and ?is the corresponding period map, then Φ?◦Φ-1is linear. For anyx? H(α), there is a neighborhoodB(x) ofxso that the restriction of Φ toB(x) is a homeomorphism onto Φ(B(x)), see§2.9. We always chooseB(x) small enough so that, using the Gauss-Manin connection, the triangulation aty?B(x) can be identified with the triangulation atx. We define the period coordinates atx= (M,q)? Q(α) as follows. Ifς= 1, thenqis a square of an abelian differential, and we may define period coordinates as above. Ifς=-1, we use the orienting double coverH(ˆα) to define the period coordinates: in this case there is a canonical injection fromQ(α) intoH(ˆα). Any Riemann surface in the image of this map is equipped with an involution. This way we get the period map fromQ(α) toH1odd(M,Σ,C) - the anti-invariant subspace of the cohomology for the involution. Puth:= 2g+k-2 ifς= 1 andh:= 2g+k-3 ifς=-1; the numberhis the topological entropy of the Teichm¨uller geodesic flow onQ1(α).

2.2. SL(2,R)-action onH1(α).Letx? H1(α), we write Φ(x) as a 2×nmatrix. The action

ofg=?a b c d? ?SL(2,R) in these coordinates is linear. We choose a fundamental domain for the action of the mapping class group and think of the dynamics on the fundamental domain. Then, the SL(2,R)-action becomes?x1... xn y

1... yn?

?→?a b c d?? x1... xn y

1... yn?

A(g,x),

whereA(g,x)?Sp(2g,Z)? Zk-1is theKontsevich-Zorich cocycle. That is:A(g,x) is the change of basis one needs to perform to return the pointgxto the fundamental domain. It can be interpreted as the monodromy of the Gauss-Manin connection restricted to the orbit of SL(2,R).

In the sequel, we letat=?et0

0e-t? ,ut=?1t 0 1? , and ¯ut=?1 0 t1?

We have the following.

Theorem 2.1(Veech-Masur).The spaceH1(α)carries a natural measureμin the Lebesgue measure class such that (1)H1(α)has finite measure, (2)μisSL(2,R)-invariant and ergodic. More generally, for any affine invariant manifold,M ? H1(α), we letμdenote the SL(2,R)- invariant affine measure onM. In particular, all the strata inQ1(S) are equipped with such invariant measures.

2.3.Mapping class group action.We denote elements in Modgusing bold letters, e.g.,

gdenotes an element in Modg. The action of ModgonQ1T(α) commutes with the action of SL(2,R), we will however denote both these actions as left action and write, e.g.g·˜xor simply denoted byg˜x.

6 ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADI2.4.The constants.In the sequel we will useκ•andN•,•= 1,2,...to denote various

constants. Unless it is explicitly mentioned otherwise, these constants are allowed only to depend on the genus. The constantsκ•are meant to indicate small positive numbers while N •are used for constants which are expected to be>1. We will also use the notationA?B. This expression means: there exists a constantc >0 otherwise noted) not on anything else. We writeA?BifA?B?A. If a constant (implicit or explicit) depends on another parameter othersthan the genus, we will make this clear by writing, e.g.??,C(x), etc. We also adopt the following?-notation. We writeB=A±?ifB=A±cwherec >0 depends only on the genus. Similarly, one definesB?A?,B?A?. Finally, we also writeA?B? ifA??B?A?(possibly with different exponents).

2.5.Modified Hodge norm.LetMbe a Riemann surface. By definition,Mhas a

complex structure. LetHMdenote the set of holomorphic 1-forms onM. One can define theHodge inner productonHMby ?ω,η?=i 2? M We have a natural mapr:H1(M,R)→ HMwhich sends a cohomology classc?H1(M,R) to the holomorphic 1-formr(c)? HMsuch that the real part ofr(c) (which is a harmonic

1-form) representsc. We can thus define the Hodge inner product onH1(M,R) by?c1,c2?=

?r(c1),r(c2)?. Then ?c1,c2?=? M c

1? ?c2,

where?denotes the Hodge star operator and we choose harmonic representatives ofc1and ?c2to evaluate the integral. We denote the associated norm by? · ?M. This is theHodge norm, see [FK]. Ifx= (M,ω)? H1(α), we will often write? · ?H,xto denote the Hodge norm? · ?Mon H

1(M,R). Since?·?H,xdepends only onM, we have?c?H,kx=?c?H,xfor allc?H1(M,R)

and allk?SO(2). LetE(x) = span{[Re(ω)],[Im(ω)]}- the spaceE(x) is often referred to as thestandard space. We let (3)p:H1(M,Σ,R)→H1(M,R) denote the natural projection;pdefines an isomorphism betweenE(x) andp(E(x))? H

1(M,R).

For our applications in the sequel (and in order to account for thelossof hyperbolicity in the thin part of the moduli space) we need to consider a modification of the Hodge norm.

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS 7

The classescαand?cα.Letαbe a homology class inH1(M,R). We let?cα?H1(M,R) be the cohomology class so that? M

ω? ?cα

for allω?H1(M,R). Then,? M ?cα? ?cβ=i(α,β), wherei(·,·) denotes the algebraic intersection number. Let?denote the Hodge star opera- tor, and let c

α=?-1(?cα).

Then, for anyω?H1(M,R) we have

?ω,cα?=? M

ω? ?cα=?

where?·,·?is the Hodge inner product. We note that?cαis a purely topological construction which depends only onα, butcαdepends also on the complex structure ofM. Fix??>0 (theMargulis constant) so that any two geodesics of hyperbolic length less than ?must be disjoint. Letσdenote the hyperbolic metric in the conformal class ofM. For any closed curveαon M, let?M(α) denote the length of the geodesic representative ofαin the metricσ.

We recall the following.

Theorem 2.2.[ABEM, Thm. 3.1]For any constantL >1there exists a constantc >1, such that for any simple closed curveαwith?M(α)< L, we have (4) 1 Furthermore, if?M(α)< ??andβis the shortest simple closed curve crossingα, then 1 Short bases.Suppose (M,ω)? H1(α). Fix?1< ??and letα1,...,αkbe the curves with in the flat metric defined byωwithi(αi,βi) = 1. We can pick simple closed curvesγr, depending only on the genus, and so that theαj,βjandγjform a symplectic basisSfor H

1(M,R). We will call such a basisshort.A short basis is not unique, and in the following

we fix some measurable choice of a short basis at each point ofH1(α). We recall the definition of a modified Hodge norm from [EMM]; this is similar (but not the same) to the one defined in [ABEM]. The modified norm is defined on the tangent space to the space of pairs (M,ω) whereMis a Riemann surface andωis a holomorphic

1-form onM. Unlike the Hodge norm, the modified Hodge norm will depend not only on

the complex structure onMbut also on the choice of a holomorphic 1-formωonM. Let

8 ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADIWe can write anyθ?H1(M,R) as

(5)θ=k? i=1a i(?cαi) +k? i=1b i?αi(σ)1/2(?cβi) +2g-2k? r=1u i(?cγr),

We then define

(6)?θ???x=?θ?H,x+? k? i=1|ai|+k? i=1|bi|+2g-2k? r=1|ur|? Note that? · ???depends on the choice of a short basis; however, switching toa different short basis can change? · ???by at most a fixed multiplicative constant depending only on the genus. (7)? ?cαi???x?1, see§2.4 for the notation?. Similarly, we have (8)? ?cβi???x? ? ?cβi?H,x?1 ?M(αi)1/2. In addition, in view of Theorem 2.2, ifγis any other moderate length curve onM,??cγ???x? ? ?cγ?H,x=O(1). Thus, ifBis a short basis atx= (M,ω), then for anyγ? B, (9) Ext

By Ext

γ(x) we mean the extremal length ofγinM, wherex= (M,ω). Remark.From the construction, we see that the modified Hodge norm is greater than the Hodge norm. Also, if the flat length of shortest curve in the flat metric defined byω is greater than?1, then for any cohomology classc, for someNdepending on?1and the genus, i.e., the modified Hodge norm is within a multiplicative constant of the Hodge norm. Note however that for a fixed absolute cohomology classc,?c???xis not a continuous function ofx, asxvaries in a Teichm¨uller disk; this is due to the dependence on the choice of a short basis. To remedy this, we pick a positive, continuous,SO(2)-bi-invariant functionφ on SL(2,R) which is supported on a neighborhood of the identity with?

SL(2,R)φ(g)dg= 1,

and define ?c??x=?c?H,x+?

SL(2,R)?c???gxφ(g)dg.

It follows from [EMM, Lemma 7.4] that for a fixedc, log?c??xis uniformly continuous asx varies in a Teichm¨uller disk. In fact, there is a constantm0such that for allx? H1(α), all c?H1(M,R) and allt >0, Remark 2.3.Even though?·??xis uniformly continuous as long asxvaries in a Teichm¨uller disk, it may be only measurable in general (because of the choice of short basis).

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS 9

2.6.Relative cohomology.Forc?H1(M,Σ,R) andx= (M,ω)? H1(α), letpx(c)

denote the harmonic representative ofp(c), wherep:H1(M,Σ,R)→H1(M,R) is the natural map. We viewpx(c) as an element ofH1(M,Σ,R). Then, (similarly to [EMM,§7], see also [ABEM] and [EMR]) we define the modified Hodge norm? ??onH1(M,Σ,R) as follows. ?c??x=?p(c)??x+? (z,z?)?Σ×Σ????? z,z?(c-px(c))????? whereγz,z?is any path connecting the zeroeszandz?ofω. Sincec-px(c) represents the zero class in absolute cohomology, the integral does not depend on the choice ofγz,z?. Note that the? · ??norm onH1(M,Σ,R) is invariant under the action of SO(2). As above, we pick a positive continuous SO(2)-bi-invariantfunctionφon SL(2,R) supported on a neighborhood of the identity such that?

SL(2,R)φ(g)dg= 1, and define

(12)?c?x=?

SL(2,R)?c??gxφ(g)dg.

Then, the? · ?xnorm onH1(M,Σ,R) is also invariant under the action of SO(2).

By [EMM, Lemma 7.5] there exists someN1so that

2.7.The AGY-norm.Let?·?AGY,xdenote the norm defined in [AGY,§2.2.2]. We recall

the definition: letx= (M,ω)? H1(α). For anyc?H1(M,Σ,C), define (14)?c?AGY,x= sup

γ|c(γ)|

|Φ(x)(γ)| where the supremum is taken over all saddle connections ofω. This defines a norm and the corresponding Finsler metric is complete, see [AGY]. We note that? ?xand? ?AGY,xare commensurable to each other on compact subsets of H

1(α).

For everyx= (M,q)? Q1(α), we define the norms? ?xand? ?AGY,xusing the branched double cover

ˆM.

Lemma 2.4.Letc?H1(M,Σ,C),t≥0ands?[0,1]. Then Proof.This is proved in [AG, Lemma 5.2], see also [AGY, eq. (2.13)],we recall the argument. Writec=a?+ib?and Φ(x) =a+ib. Then the definition (14), implies that for allt≥0 and (16) ?(atus)?c?AGY,atusx= sup |et(a(γ)+sb(γ))+ie-tb(γ)|

γ|a?(γ)+sb?(γ)+ib?(γ)|

|a(γ)+sb(γ)+ib(γ)|. sincez=u-susz, we also get that|usz| ≥ |z|/2.

This observation and (16) imply that

γ|a?(γ)+ib?(γ)|

|a(γ)+ib(γ)|.

The lower bound follows similarly.?

2.8.Non-divergence results.Recall thatQ1(α) is realized as an affine invariant sub-

manifold inH1(ˆα), moreover, compact subsets ofQ1(α) lift to compact subsets ofH1(ˆα).

Letu:H1(ˆα)→[2,∞] be the function constructed in [EMas] and [Ath]. Theorem 2.5.There exists a compact subsetK?α? Q1(α)and someN2>0with the following property. For everyt0and everyx? Q1(α), there exists such thatx?=atusx?K?α. Proof.The stratumQ1(α) is an affine invariant submanifold inH1(ˆα). The claim thus follows from [Ath, Thm. 2.2] and [AG, Lemma 6.3] applied withδ= 1/2.?

2.9.Period box.Let ˜x= (M,q)? Q1T(α). For everyr >0 define

R Let nowr >0 be so that Φ-1is a homeomorphism onRr(˜x)∩Φ(Q1T(α)). Put B r(˜x) = Φ-1? R r(˜x)? The open subsetBr(˜x) will be called aperiod boxof radiusrcentered at ˜x. Thanks to [AG, following. the following hold. Let˜x? Q1T(α)be a lift ofx. Then (1)The restriction of the covering mapπtoBr(˜x)is injective. (2)For all˜x1,˜x2?Br(˜x), the Teichm¨uller distance between˜x1and˜x2is at most1. Proof.The argument is similar to the one used in the proof of [EMM, Lemma 8.2]. t≥0 ands?[-1,1] wheredTdenotes the Tichm¨uller distance. Moreover, there exist a constantC≥1 so that C whereK?α? Q1(α) is the compact set introduced in Theorem 2.5.

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS 11

so thatBr(x)(x) is embedded in the sense that the projection from the Teichm¨uller space Q

1T(α) to the Moduli spaceQ1(α) restricted toBr(x)(˜x) is injective. Letr0= infx?K?αr(x).

By compactness ofK?α,r0>0. Decreasingr0if necessary, we assume that for allx?K?αand all ˜x1,˜x2?Br0(˜x), the Teichm¨uller distance between ˜x1and ˜x2is at most 1.

LetN≥1 be so that

whereN2is as in Theorem 2.5. We will show thatN3=Nsatisfies the claims in the lemma. First note that in view of [AG,

Prop. 5.3],B(˜x) :=Bu(x)-N(˜x) is well defined for allx? Q1(α) and all the lifts ˜x? Q1T(α).

Suppose now that there existsx? Q1(α) and ˜x1,˜x2?B(˜x) such that ˜x2=g˜x1for someg

in the mapping class group. Write

Letx?i=aτusxi, and put ˜x?i=aτus˜xi; also put ˜x?=aτus˜x. Then, in view of (15) we have

where for the last estimate we used (17) and the fact thatu(x)≥2. However, ˜x?2=g˜x?1, thus, bothx?1andx?2belong to the projection ofBr0(˜x?); this contradicts the fact that B r0(x?) is embedded. This contradiction shows thatBu(x)-N(x) is embedded, establishing part (1). We now turn to part (2). We use the above notation. Let ˜x1,˜x2?Bu(x)-N(x), and define x i=aτusxi?K?αand ˜x?i=aτus˜xias above. Then (18) implies that d d where we used (17) andu(x)≥2 in the last inequality. The proof is complete.?

For everyx? Q1(α) we put

(19)r(x) =u(x)-N3; for every compact subsetK? Q1(α), letr(K) = inf{r(x) :x?K}. lift ofx. We refer toBr(x) as the ball of radiusrcentered atx.

12 ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADI2.10.Horospherical foliation.Given a pointx= (M,q)? Q1(α), the tangent space

T xQ1(α) decomposes as T xQ1(α) =Rv(x)?Eu(x)?Es(x) wherev(x) with?v(x)?AGY,x= 1 determines the direction of the Teichm¨uller geodesic flow, E u(x) = TxQ1(α)∩DΦ-1x?H1(†,‡,R)?,and E s(x) = TxQ1(α)∩DΦ-1x?iH1(†,‡,R)?.

where (†,‡) = (M,Σ) ifς= 1 and (†,‡) = (ˆM,ˆΣ) ifς=-1 - recall thatˆMis the orienting

double cover ofMand we use Φ to locally identifyRQ1(α) withH1(M,Σ,C) ifς= 1 and with theH1odd(ˆM,ˆΣ,C) ifς=-1.

If Φ(x) =a+ibfor somex? Q1(α), then

(20)Eu(x) ={a??H1(M,Σ,R) :i(a?,b) = 0}, andEs(x) ={ib??iH1(M,Σ,R) :i(a,b?) = 0}whenς= 1. Similarly, one can defineEu,s in the caseς=-1. The subspacesEu,s(x) depend smoothly onx, moreover, they are integrable. We denote the corresponding leaves byWu(x) andWs(x), respectively. Also put W cu(x) :={atWu(x) :t?R} andWcs(x) :={atWs(x) :t?R}. Letμuxandμsxdenote the leafwise measures of the natural measureμalongWu(x) and W s(x), respectively. Theny?→μu,syis constant alongWu,s(x), respectively, and we have (21) (at)?μux=e-htμuatxand(at)?μsx=ehtμsatx; see also [AG,§4] where these measures are defined using volume forms. IfBr(x) is a period box centered atx, thenμ|Br(x)has a product structure as dLeb×dμs× dμu, see e.g. [AG, Prop. 4.1]. B u,sr(x) = the connected component ofxinBr(x)∩Wu,s(x).

DefineB•r(x) for•= cu,cs similarly.

We also denote functions which are supported on the leavesWu,Wcu, etc. using the same superscript, e.g.,φudenotes a function which is supported on a leafWu(x). We use the norm? · ?AGY,xto induce a metricdWu,s(x)onBu,sr(x) for 0< r < r(x). Hence notions such as diam etc. refer to this metric. Let˜W•(˜x) denote the foliation•inQ1T(α), and defineB•(˜x) accordingly.

Letwu,s?Eu,s(x). Then

see [AG, Lemma 5.2]. Moreover, we have the following uniformhyperbolicity estimate.

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS 13

Proposition 2.7.LetK? Q1(α)be a compact subset. There exist someκ2(K)and some t

0=t0(K)with the following property. Lett≥t0; suppose thatx,atx?K, moreover,

assume that |{τ?[0,t] :aτx?K}| ≥t/3. Then for allw?Es(x)and allt≥t0. Proof.Let? ?ABEM,xdenote the modified Hodge norm defined in [ABEM,§3]. LetCbe a constant so that for ally?K. In view of [ABEM, Thm. 3.15], there exists someκ3(K) so that under our assumptions in this proposition we have

We now compute

The claim thus holds withκ2=κ3/2 andt0=4logC

κ3.?

Lemma 2.8.LetK?αbe as in Theorem 2.5. There is a positive constantN4and for every

0< θ <1there existsκ4(θ), and a compact subsetKα(θ)?K?αwith the following properties.

H u t(x,θ) :=?y?Bur(x) :|{τ?[0,t] :aτy?Kα(θ)}| ≥θt?.

Then for everyt≥N4logu(x), we have

u

Proof.See [AG, Prop. 6.1].?

We apply the above withθ= 0.5, and put

(25)Kα=Kα(0.5),κ4:=κ4(0.5),andHut(x) :=Hut(x,0.5) for the rest of the paper.

We have the following corollary

Corollary 2.9.Letx? Q1(α), and lett≥N4logu(x). For everyy?Hut(x)and everyquotesdbs_dbs33.pdfusesText_39

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