aa r X i v : . [ m a t h . D S ] O c t Convergence of Siegel-Veech constants
Benjamin Dozier ∗ Abstract
We show that for any weakly convergent sequence of ergodic SL ( R )-invariant probabilitymeasures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech con-stants converge to the Siegel-Veech constant of the limit measure. Together with a measureequidistribution result due to Eskin-Mirzakhani-Mohammadi, this yields the (previously con-jectured) convergence of sequences of Siegel-Veech constants associated to Teichm¨uller curvesin genus two.The proof uses a recurrence result closely related to techniques developed by Eskin-Masur.We also use this recurrence result to get an asymptotic quadratic upper bound, with a uniformconstant depending only on the stratum, for the number of saddle connections of length at most R on a unit-area translation surface. Basic Definitions. A translation surface is a pair X = ( M, ω ), where M is a Riemann surface,and ω is a holomorphic 1-form. Away from its zeroes, ω defines a flat (Euclidean) metric. Themetric has a conical singularity of cone angle 2( n + 1) π at each zero of order n .A saddle connection is a geodesic segment that starts and ends at zeroes (we allow the endpointsto coincide), with no zeroes on the interior of the segment. We can also consider closed loops nothitting zeroes that are geodesic with respect to the flat metric. Whenever there is one of these, therewill always be a continuous family of parallel closed geodesic loops with the same length. We referto a maximal such family as a cylinder . Every cylinder is bounded by a union of saddle connectionsparallel to the cylinder.The bundle Ω M g of holomorphic 1-forms over M g (the moduli space of genus g Riemann sur-faces), with zero section removed, can be thought of as the moduli space of translation surfaces.This bundle breaks up into strata of translation surfaces that have the same multiplicities of thezeroes of ω . We denote by H ( m , . . . , m k ) the stratum of unit-area surfaces with k zeroes of order m , . . . , m k .There is an action of SL ( R ) on each stratum H which will play a central role in our discussion.To see the action, we first observe that by cutting along saddle connections, we can represent everytranslation surface as a set of polygons in the plane, such that every side is paired up with a parallelside of equal length. Since SL ( R ) acts on polygons in the plane, preserving the property of a pairof sides being parallel and equal length, the group acts on H . We will work mostly with elements ofthe following form: g t = (cid:18) e − t e t (cid:19) , r θ = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) . ∗ Department of Mathematics, Stanford University, [email protected] . Supported in part by NSF grant DGE-114747. nvariant measures. On each H there is a canonical probability measure µ MV in the Lebesguemeasure class, called the Masur-Veech measure , which is SL ( R )-invariant and in fact ergodic([Mas82], [Vee82]). This measure is defined in terms of the periods of the 1-form ω .There is a rich interplay between dynamics on an individual translation surface X (e.g. propertiesof saddle connections or cylinders) and the dynamics of the SL ( R ) action on strata. In particular,by the seminal work of Eskin-Mirzakhani [EM13] and Eskin-Mirzakhani-Mohammadi [EMM15], theorbit closure SL ( R ) X supports a canonical SL ( R )-invariant ergodic probability measure, andproperties of this measure are closely connected to dynamics on individual translation surfaces. Siegel-Veech constants.
Let N ( X, R ) be the number of cylinders on X of length at most R .The study of asymptotics of this function as R → ∞ is of central importance and has inspired muchof the work on spaces of translation surfaces. By work of Masur ([Mas88] and [Mas90]), for a fixed X , there are quadratic upper and lower bounds for the growth of N ( X, R ) in terms of R . If µ isan ergodic SL ( R )-invariant probability measure on H , then by Eskin-Masur ([EM01]) there existsa constant c ( µ ), the cylinder Siegel-Veech constant associated to µ , characterized by the propertythat for µ -a.e. X in H , N ( X, R ) ∼ c ( µ ) · R as R → ∞ . It is an important open question whether for every translation surface X , there existssome c such that N ( X, R ) ∼ cR , as R → ∞ . This question is closely connected to the classificationof measures on H that are invariant under the unipotent subgroup (cid:26)(cid:18) ∗ (cid:19)(cid:27) ⊂ SL ( R ).For more information on translation surfaces, the reader can consult one of many surveys availableon the topic, for instance [Zor06] or [Wri15]. The first new result states that if measures converge, then the corresponding Siegel-Veech constantsdo as well.
Theorem 1.1.
Suppose µ , µ , . . . are ergodic SL ( R ) -invariant probability measures on H , and that µ n → η , in the weak-* topology, where η is another ergodic SL ( R ) -invariant probability measure.Then the Siegel-Veech constants satisfy c ( µ n ) → c ( η ) . In Section 2, we use Theorem 1.1 to prove convergence, for the stratum H (2), of Siegel-Veechconstants for non-arithmetic Teichm¨uller curves (numerical evidence of this was found by Bainbridge[Bai07]), and for arithmetic Teichm¨uller curves (conjectured by Leli`evre, based on numerical evidenceand proof in a restricted case [Lel06]). This involves a new type of application of the measureequidistribution result of Eskin-Mirzakhani-Mohammadi in [EMM15]. Remark 1.1.
We will work with the cylinder Siegel-Veech constant for concreteness, but the resultand proof work for other Siegel-Veech constants as well, in particular for the saddle connectionSiegel-Veech constant (which counts saddle connections rather than cylinders) and the area Siegel-Veech constant (which is formed from counts of cylinders weighted by the area of the cylinder). Theresult and proof also works with a pair (
M, q ), where q is a holomorphic quadratic differential (alsoknown as a half-translation surface ). Remark 1.2.
Note that if we define the Siegel-Veech constant of X to be the Siegel-Veech constantof the canonical measure whose support is the orbit closure SL ( R ) X , then this does not define acontinuous function on H . This is because special surfaces with small orbit closure, for instance Veechsurfaces, will often have Siegel-Veech constants different from c ( µ MV ), the Siegel-Veech constant forthe Masur-Veech measure on H (see Section 1.5 for references), while a dense subset of surfaces willhave Siegel-Veech constant equal to c ( µ MV ). 2 .3 Uniform asymptotic quadratic upper bound The second new result gives a uniform quadratic upper bound on the number of cylinders, whichholds, asymptotically, for all surfaces in a stratum.
Theorem 1.2.
Given H a unit-area stratum, there exists a constant c max such that for any surface X ∈ H , N ( X, R ) ≤ c max R for all R ≥ R ( X ) , where R : H → R is an explicit function of the length of the shortest saddleconnection on X (and of the genus of the stratum). Note that the function R will not in general be bounded for a fixed stratum. This is because asurface in a fixed stratum can have arbitrarily many short saddle connections (for instance, by takinga surface with a short slit, and then gluing in a cylinder with small height and circumference). But,according to Theorem 1.2, as we increase R , the effect of these short saddle connections eventuallydiminishes. Remark 1.3.
As in the case of Theorem 1.1, the result and proof work if we replace the count ofcylinders with the count of saddle connections, or the count of cylinders weighted by the area of thecylinder. The result and proof also work with a pair (
M, q ), where q is a holomorphic quadraticdifferential. The main tool needed in the proofs of both Theorem 1.1 and Theorem 1.2 is the following proposition,which may be of independent interest. It is a recurrence-type result which controls the length of theshortest saddle connection, on average, over translation surfaces on a large “circle” centered at any X . Here we deduce the proposition directly from a more general result proved in [Doz17]; in thatpaper the more general result is used to study the distribution of angles of saddle connections.Let ℓ ( X ) denote the length of the shortest saddle connection on X , and let g t = (cid:18) e − t e t (cid:19) , r θ = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) . Proposition 1.1.
For any stratum H and < δ < / , there exists a function α : H → R ≥ andconstants c , b such that for any X ∈ H , Z π ℓ ( g T r θ X ) δ dθ ≤ c e − (1 − δ ) T α ( X ) + b, for all T > . The function α ( X ) is bounded above by an explicit function of ℓ ( X ) (and the genusof the stratum). When we use Proposition 1.1, it will be crucial that the constant b does not depend on the surface X . Some related results appear in [EM01] (also see [Ath06]), but the above formulation with anadditive constant which does not depend on the surface is new. Proof of Proposition 1.1.
This is a special case of Proposition 2.1 in [Doz17]. That Propositioninvolves integrating over any subinterval I ⊂ [0 , π ]; the above is simply the case when I equals[0 , π ]. The proof in that paper follows the strategy used by Eskin-Masur ([EM01]), but keeps trackof the constant b above. The approach is to use the “system of integral inequalities”, which wasfirst developed for the proof of the quantitative Oppenheim conjecture by Eskin-Margulis-Mozes([EMM98]), who were working in the context of lattices. A key technical aspect unique to thetranslation surfaces context involves combining “complexes” of saddle connections. (cid:4) .5 Previous work A good deal of progress has been made in understanding the Siegel-Veech constants of Veech surfaces,which often lead to explicit expressions for the quadratic growth rates for billiards on polygons. Inhis foundational paper [Vee89], Veech used Eisenstein series to show that all Veech surfaces satisfyan exact quadratic asymptotic for the growth rate of cylinders, and he gave a way of computing theconstants. He computes the constants for translation surfaces arising from unfolding certain isoscelestriangles. Gutkin-Judge [GJ00] give a different formula for computing the Siegel-Veech constant, theproof of which uses softer ergodic-theoretic results related to counting horocycles in the hyperbolicplane. Vorobets [Vor96] discovered similar results independently.Schmoll [Sch02] studied the problem of counting cylinders and saddle connections on tori withadditional marked points. Along similar lines, Eskin-Masur-Schmoll [EMS03] study translationsurfaces that are branched covers of tori. Using Ratner theory, they get exact quadratic asymptoticsfor all these surfaces, and they explicitly compute the constants for certain surfaces arising frombilliards in rectangles with barriers. In complementary work, Eskin-Marklof-Morris [EMWM06]study the case of branched covers of Veech surfaces that are not tori. They also get exact quadraticasymptotics for these surfaces, and they explicitly compute the constants for certain (non-Veech)surfaces that arise from unfolding triangles. Their proof modifies the techniques of Ratner to workin their setting, where the relevant moduli space is not homogeneous, but shares some importantproperties with homogeneous spaces. Bainbridge-Smillie-Weiss ([BSW16]) show that in the eigenfornloci in H (1 , all surfaces satisfy exact quadratic growth asymptotics.The results discussed above apply only to special translation surfaces. In the opposite direction,one can ask about Siegel-Veech constants for the Masur-Veech measure on a whole stratum. Eskin-Masur-Zorich [EMZ03] give a general method for computing these in terms of the volumes of strataand neighborhoods of certain parts of the boundary of strata. Results of Eskin-Okounkov [EO01]allow one to compute these volumes.The convergence of the area Siegel-Veech constants was proven by Matheus-M¨oller-Yoccoz ([MMY15],Remark 1.10). That proof relies on the close connection between area Siegel-Veech constants andsums of Lyapunov exponents and does not work for other Siegel-Veech constants. • Section 2 gives an application to Siegel-Veech constants associated to Veech surfaces in genus2, an application showing that Siegel-Veech constants are bounded in a fixed stratum, andsome results on the set of Siegel-Veech constants associated to all the measures on a stratum. • Section 3 contains the proof of Theorem 1.1 (Convergence of Siegel-Veech constants), assumingthe recurrence-type result Proposition 1.1. • Section 4 contains the proof of Theorem 1.2 (Uniform asymptotic quadratic upper bound),assuming Proposition 1.1. • Section 5 poses open questions about the size of the extremal Siegel-Veech constants in eachstratum, and discusses some results in this direction.
I would like to thank Maryam Mirzakhani, my thesis advisor, for guiding me with numerous stim-ulating conversations and suggestions. I am also very grateful to Alex Wright, for many helpfuldiscussions and detailed feedback. 4
Applications H (2) We give an application of Theorem 1.1 in genus two, which we then use to give proofs of theconvergence of certain sequences of Siegel-Veech constants. The application uses in a crucial waythe equidistribution result of Eskin-Mirzakhani-Mohammadi [EMM15].Let µ MV be the Masur-Veech measure on the stratum H (2). Theorem 2.1.
Let { C n } be a sequence of distinct closed SL ( R ) -orbits in H (2) , and let µ n be theergodic SL ( R ) -invariant probability measure whose support is C n . Then lim n →∞ c ( µ n ) = c ( µ MV ) = 10 π . Recall that for surfaces generating a closed SL ( R ) orbit (known as Veech surfaces), the genericquadratic growth constant for the whole orbit actually equals the constant for every surface in theorbit ([Vee89], Proposition 3.10). Thus the above result implies that the quadratic growth constantsfor a sequence of distinct Veech surfaces in H (2) tend to the constant for the whole stratum. Proof.
We claim that lim n →∞ µ n = µ MV . By the equidistribution result in [EMM15] (Corollary2.5), if this were not the case, there would exist a subsequence k n → ∞ and N an affine invariantsubmanifold of H (2), with each C k n contained in N . (An affine invariant manifold is the image ofa proper immersion from a connected manifold to a stratum that is cut out locally by homogeneousreal linear equations in period coordinates). Now N cannot be a single closed SL ( R )-orbit, since itcontains infinitely many distinct closed SL ( R )-orbits. Then by McMullen’s classification ([McM07],Theorem 1.2), N must be the whole stratum H (2), contradiction, establishing the claim.Now Theorem 1.1 gives lim n →∞ c ( µ n ) = c ( µ MV ). By [EMZ03] (Example 14.7, second case), c ( µ MV ) = π (the normalization for the Siegel-Veech constant used in that paper differs from oursby a factor of π ). (cid:4) The two corollaries below, which apply to non-arithmetic and arithmetic Veech surfaces, respec-tively, follow immediately from Theorem 2.1.
Corollary 2.1 (Convergence for non-arithmetic Veech surfaces) . Let D be a positive integer that isnot a perfect square, with D ≡ , , and let E D be the SL ( R ) -orbit in H (2) of a pair ( M, ω ) for which the Jacobian Jac( M ) admits real multiplication by O D , the ring of integers in Q [ √ D ] ,with ω an eigenform (these orbits are known to be closed). Let µ D be the ergodic SL ( R ) -invariantprobability measure whose support is E D . Then lim D →∞ c ( µ D ) = c ( µ MV ) = 10 π . Bainbridge found a formula for c ( µ D ) and numerical evidence suggesting that the above conver-gence holds ([Bai07], discussion after Theorem 14.1). Corollary 2.2 (Convergence for arithmetic Veech surfaces) . Let { S n } be a sequence of square-tiledsurfaces in H (2) , where S n is tiled by exactly k n squares, with k n → ∞ . Let µ n be the ergodic SL ( R ) -invariant probability measure whose support is the (closed) orbit SL ( R ) S n . Then lim n →∞ c ( µ n ) = c ( µ MV ) = 10 π . k n , the number of squares, is a prime, and found numerical evidence for the general case ([Lel06]). Remark 2.1.
One can also use the strategy above to prove that the Siegel-Veech constants corre-sponding to the eigenform loci in H (1 ,
1) (which are no longer just closed orbits) converge to theSiegel-Veech constant for H (1 , D = 5 locus ([Bai10], Theorem 1.5). For the arithmetic eigenform loci, convergence was proven byEskin-Masur-Schmoll ([EMS03], Theorem 1.3); here the sequence of Siegel-Veech constants is noteventually constant. Theorem 2.2.
Fix a stratum H . There exists a bound B (depending on the stratum) such that forany ergodic SL ( R ) -invariant probability measure µ on H , c ( µ ) ≤ B. We give two different proofs.
Proof via Theorem 1.1.
Suppose, for the sake of contradiction, that µ n is a sequence of ergodic SL ( R )-invariant probability measures on H , with c ( µ n ) → ∞ . By passing to a subsequence, andapplying the equidistribution theorem [EMM15] (Corollary 2.5), we can assume that µ n → η , where η is another ergodic SL ( R )-invariant probability measure. Then Theorem 1.1 gives thatlim n →∞ c ( µ n ) = c ( η ) < ∞ , contradicting our assumption. (cid:4) Proof via Theorem 1.2.
We claim that for any such µ , c ( µ ) ≤ c max , where c max is the constant inTheorem 1.2. By [EM01], there exists some X ∈ H (in fact the following will hold for µ -a.e. X )such that N ( X, R ) ∼ c ( µ ) R . Hence by Theorem 1.2, c ( µ ) ≤ c max . (cid:4) Theorem 2.3.
Fix a stratum H . There exists a bound β > (depending on the stratum) such thatfor any ergodic SL ( R ) -invariant probability measure µ on H , c ( µ ) ≥ β. Proof.
The proof strategy is the same as that of the proof of Theorem 2.2 via Theorem 1.1 above.Suppose, for the sake of contradiction, that µ n is a sequence of ergodic SL ( R )-invariant prob-ability measures on H , with c ( µ n ) →
0. By passing to a subsequence, and applying the equidistri-bution theorem [EMM15] (Corollary 2.5), we can assume that µ n → η , where η is another ergodic SL ( R )-invariant probability measure. Then Theorem 1.1 gives thatlim n →∞ c ( µ n ) = c ( η ) , which we claim is positive. Indeed, by [EM01], the Siegel-Veech constant gives the quadratic growthrate of cylinders for an η typical surface, and by [Mas88], the constant must be positive. Since westarted by assuming the limit is zero, we have a contradiction. (cid:4) .3 The set of Siegel-Veech constants Using Theorem 1.1, we can easily prove several results showing that the set of Siegel-Veech constantsof all the measures for a fixed stratum is not too complicated.
Theorem 2.4.
Fix a stratum H . Let S ( H ) = { c ( µ ) : µ an ergodic SL ( R ) -invariant probability measure on H} . Then S ( H ) is closed as a subset of R . This will follow as the n = 0, N = H case of Proposition 2.1 below.By Eskin-Mirzakhani-Mohammadi [EMM15], the ergodic SL ( R )-invariant probability measureson H are in bijection with the set of affine invariant submanifolds M of H . We define c ( M ) to equal c ( µ ), where µ is the measure corresponding to M . Proposition 2.1.
Fix an affine invariant submanifold N of H . Let S n ( N ) = { c ( M ) : M ⊂ N , dim C M ≥ n } . Then S n ( N ) is closed as a subset of R .Proof. Suppose x ∈ R and x = lim k →∞ c ( M k ) for some sequence M k of affine invariant submani-folds, corresponding to measures µ k . Consider the set of all affine invariant submanifolds that containinfinitely many of the M k , and pick an element M that is minimal (with respect to inclusion) in thisset. The set is non-empty (since N is in it), and a minimal element exists because the longest chain(with respect to inclusion) has cardinality at most dim C ( N ) < ∞ . Note that dim C M ≥ n , hence c ( M ) ∈ S n ( N ). Now by equidistribution ([EMM15], Corollary 2.5), we can find a subsequence j k such that µ j k converges to µ , where µ is the measure corresponding to M .By Theorem 1.1, c ( M ) = c ( µ ) = lim k →∞ c ( µ j k ) = lim k →∞ c ( M j k ) = x . Hence x ∈ S n ( N ), andwe are done. (cid:4) Given a closed X ⊂ R , we define the derived set X ∗ to be the set obtained from X by removingall the isolated points. We let X ∗ n = ( · · · ( X ∗ ) ∗ · · · ) ∗ , where there are n occurrences of ∗ . We definethe rank , rank( X ), to be the smallest n for which X ∗ n = {} ; if no such n exists, we declare the rankto be infinity. (Note that this is a slight variation of the usual notion of Cantor-Bendixson rank .) Theorem 2.5.
Let
N ⊂ H be an affine invariant submanifold with dim C N = d . Then rank S n ( N ) ≤ d − n + 1 . In particular, rank S ( H ) ≤ dim C H + 1 .Proof. We argue by downwards induction on n .The base case is n = d . Here the only affine invariant submanifold of N with dimension at least d is N itself. Thus S d ( N ) = { c ( N ) } , which has rank 1, so this gives the base case.For the inductive step assume the result for n .We claim that S n − ( N ) ∗ ⊂ S n ( N ). To see this, let x ∈ S n − ( N ) ∗ , which means we can finda sequence of distinct M i of dimension at least n − i c ( M i ) = x . As in the proofof Proposition 2.1, using [EMM15] (Corollary 2.5) we can find N containing all M j i for somesubsequence j i , and c ( N ) = lim i c ( M j i ) = x . Since N contains distinct manifolds of dimension atleast n −
1, it must have dimension at least n (this uses the fact that affine invariant submanifoldscome from proper immersions). So x = c ( N ) ∈ S n ( N ). This completes the proof of the claim.Now for any closed sets A ⊂ B ⊂ R , it follows immediately from our definition that rank( A ) ≤ rank( B ). Then, from the definition of rank, this fact, the claim above, and the inductive assumption,rank( S n − ( N )) ≤ rank( S n − ( N ) ∗ ) + 1 ≤ rank( S n ( N )) + 1 ≤ ( d − n + 1) + 1 = d − ( n −
1) + 1 , which completes the induction. (cid:4) Proof of Theorem 1.1 (Convergence of Siegel-Veech con-stants)
Proof of Theorem 1.1.
The proof would be immediate if strata were compact, but they are not. Therecurrence-type result Proposition 1.1 allows us to get around this.The Siegel-Veech constant can be defined by c ( µ ) = R H ˆ f dµ R R f dλ , for any f : R → R continuous and compactly supported. Here ˆ f : H → R is the Siegel-Veechtransform of f given by ˆ f ( X ) := X c ∈ Λ( X ) f ( c ) , where Λ( X ) ⊂ R is the multi-set of holonomies of cylinders. The holonomy of a saddle connection s is the element of C (which we identify with R ) given by integrating the 1-form ω along any of theperiodic geodesics defining the cylinder.Hence to prove Theorem 1.1, it suffices to show thatlim n →∞ Z H ˆ f dµ n = Z H ˆ f dη for all such f . Note that if ˆ f were compactly supported, this would follow immediately from thedefinition of weak-* convergence. The idea is to approximate ˆ f by compactly supported functions,and bound the integral of the error term using integrability results from [EM01]. The key point isthat we need a bound for the error term that is independent of the particular measure µ .Let C K = { X ∈ H : ℓ ( X ) ≤ K } . These sets are compact. Now let χ K be a continuous function H → [0 ,
1] whose value is 1 on C K and 0 on H\ C K +1 . Define ˆ f K = ˆ f · χ K . Note that ˆ f K is compactlysupported, hence lim n →∞ Z H ˆ f K dµ n = Z H ˆ f K dη. It remains to show that by choosing K large, we can make (cid:12)(cid:12)(cid:12)R H ˆ f dµ − R H ˆ f K dµ (cid:12)(cid:12)(cid:12) uniformly smallfor any choice of ergodic SL ( R )-invariant probability measure µ . We can assume f , and hence ˆ f ,are non-negative, since we only need to show the equality for some f for which R H ˆ f dη is non-zero.Since f is compactly supported, it is dominated by some multiple of an indicator function of a largeball. By [EM01] Theorem 5.1(a), it follows that ˆ f < C/ℓ δ for some C and 0 < δ < / (cid:12)(cid:12)(cid:12)(cid:12)Z H ˆ f dµ − Z H ˆ f K dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z H\ C K ˆ f dµ ≤ C Z H\ C K ℓ δ dµ (1) ≤ C K δ ′ Z H\ C K ℓ δ + δ ′ dµ, (2)where we choose δ ′ > δ + δ ′ < /
2. We introduce this extra δ ′ to get the K δ ′ term inthe above, which will allow us to get decay as K → ∞ . By [EM01] Lemma 5.5, the last integralabove is finite; we need to show the somewhat stronger statement that it is bounded from aboveindependent of the choice of µ . 8he idea is to replace the integral of ℓ δ + δ ′ over the whole stratum by the integral over largecircles centered at a µ -generic point, using Nevo’s theorem, and then use Proposition 1.1 to boundthe integrals over circles. To apply Nevo’s theorem, we need to choose a smoothing function φ : R → R that is non-negative, smooth, and compactly supported (having the smoothing is fine forour purposes; Nevo’s result should also be true without having to smooth).Now by Nevo’s theorem (see [EM01] Theorem 1.5) and Proposition 1.1, for µ a.e. X ∈ H , Z H ℓ δ + δ ′ dµ · Z ∞−∞ φ ( t ) dt = lim τ →∞ Z ∞−∞ φ ( τ − t ) (cid:18) π Z π ℓ ( g t r θ X ) δ + δ ′ dθ (cid:19) dt ≤ b Z ∞−∞ φ ( t ) dt. Hence R H ℓ δ + δ ′ dµ ≤ b , for any µ . Plugging into (2) gives, for all µ , (cid:12)(cid:12)(cid:12)(cid:12)Z H ˆ f dµ − Z H ˆ f K dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C · bK δ ′ . (3)Now we put everything together. Fix ǫ >
0. Choose K large so that C · b/K δ ′ < ǫ/
3. Nowchoose N such that (cid:12)(cid:12)(cid:12)(cid:12)Z H ˆ f K dµ n − Z H ˆ f K dη (cid:12)(cid:12)(cid:12)(cid:12) < ǫ/ n ≥ N .Then by the triangle inequality, and (3), (4), for n ≥ N , (cid:12)(cid:12)(cid:12)(cid:12)Z H ˆ f dµ n − Z H ˆ f dη (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z H ˆ f dµ n − Z H ˆ f K dµ n (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z H ˆ f K dµ n − Z H ˆ f K dη (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z H ˆ f K dη − Z H ˆ f dη (cid:12)(cid:12)(cid:12)(cid:12) ≤ C · b/K δ ′ + ǫ/ C · b/K δ ′ < ǫ/
3) = ǫ, and we are done. (cid:4) Remark 3.1.
Some similar ideas appear in the proof of Theorem 2.4 in [EMS03].
Proof of Theorem 1.2.
The proof is a modification of the Eskin-Masur proof of Masur’s original(non-uniform) quadratic upper bound (Theorem 5.4 in [EM01]; the result was originally proved in[Mas90]).By [EM01] Proposition 3.5, there is an absolute constant c such that N ( X, R ) − N ( X, R ) ≤ cR Z π N ( g log R r θ X, dθ. The proof of this involves taking the indicator function of the trapezoid with vertices ( ± , , ( ± , g t to the trapezoid makes it long and thin,and then rotating it around using r θ allows one to count saddle connections whose holonomy hasabsolute value between R and 2 R . The right-hand side involves a radius of 4 since the originaltrapezoid is contained in a ball of radius 4. 9ow we apply Theorem 5.1(a) in [EM01], and then Proposition 1.1 to get N ( X, R ) − N ( X, R ) ≤ cR Z π N ( g log R r θ X, dθ ≤ cR Z π c ℓ ( g log R r θ X ) δ dθ ≤ cc R (cid:16) c e − (log R )(1 − δ ) α ( X ) + b (cid:17) . The constants c, c , c , b do not depend on X or R . For R ≥ R ( X ) large, we can make c e − (log R )(1 − δ ) α ( X ) < b ,and so we get N ( X, R ) − N ( X, R ) ≤ bcc R . A straight-forward geometric series argument then gives the desired inequality.We also see that the function R ( X ) can be chosen to depend only on α ( X ) (in an explicitway). The function α ( X ) is itself bounded by an explicit function of ℓ ( X ) (and the genus of thestratum). (cid:4) Question 5.1.
Given H , what is sup c ( µ ), where the sup ranges over all ergodic SL ( R )-invariantprobability measures on H ? In particular, what are the asymptotics of this quantity as the genus ofthe stratum tends to infinity?We can show that the asymptotic growth rate of sup c ( µ ) is somewhere between quadratic andexponential, as a function of genus (at least along some sequence of strata with genus tending toinfinity). We now explain how to get these upper and lower bounds.By taking branched covers of a fixed translation surface in which the preimages of all the singularpoints are branch points, we can exhibit a family of surfaces with genus g → ∞ for which the numberof cylinders of length at most R is at least kg R for some fixed constant k >
0. This shows thatsup c ( µ ) for these stratum is at least kg .On the other hand, by Theorem 2.2, the supremum is finite, and in fact any constant c max forwhich Theorem 1.2 holds gives an upper bound for the supremum. If we keep track of the c max coming from the proof of Theorem 1.2, we find it grows at most exponentially in the genus, but itseems hard to do better than exponential using our method. The exponential nature of the methodarises from an induction on the complexity of certain “complexes” of saddle connections in the proofof the generalization of Proposition 1.1 given in [Doz17]. Remark 5.1.
If we work with quadratic differentials and allow simple poles (corresponding to pointswith cone angle π ), then it is not clear that there is any bound for the analogue of c max that dependsonly on the genus, since there are infinitely many strata in a given genus.The corresponding question about the smallest Siegel-Veech constant in each stratum is alsointeresting. Question 5.2.
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