aa r X i v : . [ m a t h . D S ] F e b SLOPE GAP DISTRIBUTIONS OF VEECH SURFACES
LUIS KUMANDURI AND JANE WANG
Abstract.
We show that the limiting gap distribution of slopes of saddle connections on a Veech translationsurface is piecewise real-analytic with finitely many points of non-analyticity. We do so by taking an explicitparameterization of a Poincar´e section to the horocycle flow on SL(2 , R ) / SL(
X, ω ) and establishing a keyfiniteness result for the first return map under this flow. Introduction
In this paper, we will study the slope gap distributions of Veech surfaces, a special type of translationsurface.
Translation surfaces can be defined geometrically as finite collections of polygons with sidesidentified in parallel opposite pairs. If we place these polygons in the complex plane C , the surface inheritsa Riemann surface structure from C , and the one form dz gives rise to a well-defined holomorphic one-formon the surface. This leads to a second equivalent definition of a translation surfaces as a pair ( X, ω ) where X is a Riemann surface and ω is a holomorphic one-form on the surface. Every translation surface locallyhas the structure of ( C , dz ), except for at finitely many points that have total angle around them 2 πk forsome integer k ≥
2. These points are called cone points and correspond to the zeros of the one-form ω . Azero of order n gives rise to a cone point of angle 2 π ( n + 1).A translation surface inherits a flat metric from C . Saddle connections are then straight line geodesicsconnecting two cone points. The holonomy vector of a saddle connection γ is then the vector describinghow far and in what direction the saddle connection travels, v γ = Z γ ω. We will be interested in the distribution of directions of these vectors for various translation surfaces.There is a natural SL(2 , R ) action on translation surfaces coming from the linear action of matrices on R , as can be seen in Figure 1. (cid:20) (cid:21) Figure 1.
A matrix in SL(2 , R ) acting on a translation surface.Sometimes, a matrix takes a surface ( X, ω ) back to itself. That is, after acting on the surface by thematrix, it is possible to cut and past the new surface so that it looks like the original surface again. Thestabilizer of a surface (
X, ω ) under the SL(2 , R ) action is called the Veech group of the surface. It will bedenoted SL(
X, ω ) or just Γ and is a subgroup of SL(2 , R ). When the Veech group SL( X, ω ) of a translationsurface has finite covolume in SL(2 , R ), the surface ( X, ω ) is called a
Veech surface . Sometimes suchsurfaces are also called lattice surfaces since SL(
X, ω ) is a lattice in SL( X, R ). Veech surfaces have manynice properties, such as satisfying the Veech dichotomy : in any direction, every infinite trajectory on thesurface is periodic or every infinite trajectory is equidistributed. For more information about translationand Veech surfaces, we refer the reader to [HS06] and [Zor06].
From work of Vorobets ([Vor05]), it is known that for almost every translation surface (
X, ω ) with respectto the Masur-Veech volume on any strata of translation surfaces (for details about Masur-Veech volume andstrata, please see [Zor06]), the angles of the saddle connections equidistribute in S . That is, if we letΛ( X, ω ) := { holonomy vectors of saddle connections of ( X, ω ) } , and we normalize the circle to have total length 1, then for any interval I ⊂ S , as we let R → ∞ , theproportion of vectors in Λ( X, ω ) of length ≤ R that have direction in the interval I converges to the lengthof I .A finer measure of the randomness of the saddle connection directions of a surface is its gap distribution ,which we will now define. The idea of the gap distribution is it records the limiting distribution of the spacingsbetween the set of angles (or in our case, slopes) of the saddle connection directions of length up to a certainlength R . We will be working with slope gap distributions rather than angle gap distributions because theslope gap distribution has deep ties to the horocycle flow on strata of translation surfaces. Thus, dynamicaltools relating to the horocycle flow can be more easily applied to analyze the slope gap distribution.Let us restrict our attention to the first quadrant and to slopes of at most 1 and define S ( X, ω ) := { slope( v ) : v ∈ Λ( X, ω ) and 0 < Re( v ) , ≤ Im( v ) ≤ Re( v ) } . We also allow ourselves to restrict to slopes of saddle connections of at most some length R in the ℓ ∞ metric, and define S R ( X, ω ) := { slope( v ) : v ∈ Λ( X, ω ) and 0 < Re( v ) , ≤ Im( v ) ≤ Re( v ) ≤ R } . We let N ( R ) denote the number of unique slopes N ( R ) := | S R ( X, ω ) | . By results of Masur ([Mas88],[Mas90]), the growth of the number of saddle connections of length at most R in any translation surface isquadratic in R . We can order the slopes: S R ( X, ω ) = { ≤ s R < s R < · · · < s RN ( R ) − } . Since N ( R ) grows quadratically in R , we now define the renormalized slope gaps of ( X, ω ) to be G R ( X, ω ) := { R ( s Ri − s Ri − ) : 1 ≤ i ≤ N ( R ) − s i ∈ S R ( X, ω ) } . If there exists a limiting probability distribution function f : [0 , ∞ ) → [0 , ∞ ) for the renormalized slopegaps lim R →∞ | G R ( X, ω ) ∩ ( a, b ) | N ( R ) = Z ba f ( x ) dx, then f is called the slope gap distribution of the translation surface ( X, ω ). If the sequence of slopes ofholonomy vectors of increasing length of a translation surface were independent and identically distributeduniform [0 ,
1] random variables, then a probability computation shows that the gap distribution would be aPoisson process of intensity 1. In all computed examples of slope gap distributions, however, this is not thecase.In [AC12], Athreya and Chaika analyzed the gap distributions for typical surfaces and showed that foralmost every translation surface (with respect to the Masur-Veech volume), the gap distribution exists.They also showed that a translation surface is a Veech surface if and only if it has no small gaps , that is, iflim inf R →∞ (min( G R ( X, ω )) >
0. In a later work [ACL15], Athreya, Chaika, and Leli´evre explicitly computedthe gap distribution of the golden L and in [Ath16], Athreya gives an overview of results and techniquesabout gap distributions. Another relevant work is a paper by Taha ([Tah19]) studying cross sections to thehorocycle and geodesic flows on quotients of SL(2 , R ) by Hecke triangle groups. The computation of slopegap distributions involved understanding the first return map of the horocycle flow to a particular transversalof a quotient of SL(2 , R ).In [UW16], Uyanik and Work computed the gap distribution of the octagon, and also showed that thegap distribution of any Veech surface exists and is piecewise real analytic. Sanchez then went on to studythe gap distributions of certain non-Veech surfaces coming from slit tori ([San20]). Up until then, all knownslope gap distributions were for Veech surfaces.While Uyanik and Work gave an algorithm to compute the gap distribution of any Veech surface andshowed that the gap distribution was piecewise analytic, their work does not say anything about the numberof points of non-analyticity of the gap distribution. We fill in that gap and show that every Veech surfacehas a gap distribution with finitely many points of non-analyticity. LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 3
Theorem 1.
Let ( X, ω ) be a translation surface with lattice Veech group. Then its slope gap distributionhas finitely many points of non-analyticity. As we will see in Section 2.3, applying Uyanik and Work’s algorithm to certain Veech surfaces results incomputations that make it seem like the gap distribution can have infinitely many points of non-analyticity.We will now give a brief outline of the paper. In Section 2.1 we will go over background information onslope gap distributions, including how to relate the gap distribution to return times to a Poincar´e sectionof the horocycle flow. In Section 2.2, we will outline the algorithm of Uyanik and Work, and observe somepossible modifications. In Section 2.3, we will see how a couple steps of Uyanik and Work’s algorithmapply to a specific Veech surface. A priori, the first return map to the Poincar´e section breaks the sectioninto infinitely many pieces, but after making some modifications to the parameterization, we will see thatthere are in fact finitely many pieces. In Section 3 we will give a proof of Theorem 1. The strategy of theproof is to apply a compactness argument to show finiteness under our modified parameterization of thePoincar´e section. We will show that on a compact set that includes the Poincar´e section, every point hasa neighborhood that can contribute at most finitely many points of non-analyticity to the gap distribution.This will give us that the slope gap distribution has finitely many points of non-analyticity overall. In Section4, we discuss a few further questions regarding slope gap distributions of translation surfaces.1.1.
Acknowledgements.
The authors would like to thank Moon Duchin for organizing the PolygonalBilliards Research Cluster held at Tufts University in 2017, where this work began, as well as all of theparticipants of the cluster. We would also like to thank Jayadev Athreya, Aaron Calderon, Jon Chaika,Samuel Leli´evre, Anthony Sanchez, Caglar Uyanik, and Grace Work for helpful conversations about limit-ing gap distributions. This work was supported by the National Science Foundation under grant numberDMSCAREER-1255442, and by the National Science Foundation Graduate Research Fellowship under grantnumbers 1122374 and 1745302. 2.
Background
A Poincar´e section for the horocycle flow.
In this section, we review a general strategy forcomputing the gap distribution of a translation surface by relating slope gap distributions to the horocycleflow. For more background and proofs of the statements given here, see [AC14] or [ACL15].Suppose that we wish to compute the slope gap distribution of a translation surface (
X, ω ). We letΛ(
X, ω ), sometimes shortened to just Λ, be the set of holonomy vectors of the surface. We may start byconsidering all of the holonomy vectors of (
X, ω ) in the first quadrant, with ℓ ∞ norm ≤ R . If we act on( X, ω ) by the matrix g − R ) = (cid:20) /R R (cid:21) , the slopes of the holonomy vectors of g − R ) ( X, ω ) in [0 , × [0 , R ] is the same as R times the slopes ofthe holonomy vectors of ( X, ω ) in [0 , R ] × [0 , R ], as we can see in Figure 2.Another important observation is that the horocycle flow h s = (cid:20) − s (cid:21) , changes slopes of holonomy vectors by s . That is,slope( h s ( z )) = slope( z ) − s for z ∈ R . As a result, slope differences are preserved by the flow h s .Now we let the Veech group of the surface be SL( X, ω ). We can then define a transversal or
Poincar´esection for the horocycle flow on SL(2 , R ) / SL(
X, ω ) to be the surfaces in the orbit of (
X, ω ) with a shorthorizontal saddle connection of length ≤
1. That is,Ω(
X, ω ) = { g SL(
X, ω ) : g Λ ∩ ((0 , × { } ) = ∅} . Then, the slope gaps of (
X, ω ) for holonomy vectors of ℓ ∞ length ≤ R are exactly 1 /R times the setof N ( R ) − X, ω ) of the surface g − R ) ( X, ω ) under the horocycle flow h s for s ∈ [0 , R ]. Here, we are thinking of return times as the amount of time between each two successive times LUIS KUMANDURI AND JANE WANG g − R ) RR R Λ( X, ω ) Λ( g − R ) ( X, ω )) Figure 2.
Upon renormalizing a surface (
X, ω ) by applying g − R ) , the slopes of thesaddle connections of ( X, ω ) scale by R .that the horocycle flow returns to the Poincar´e section. In this way, the slope gaps of ( X, ω ) are related tothe return times of the horocycle flow to the Poincar´e section. Summarizing, since G R ( X, ω ) is the set ofslope gaps renormalized by R , we have that G R ( X, ω ) = { first N ( R ) − g − R ) ( X, ω ) to Ω(
X, ω ) under h s } . For a point z in the Poincar´e section Ω( X, ω ), we denote by R h ( z ) the return time of z to Ω( X, ω ) underthe horocycle flow. Then, as one lets R → ∞ , this renormalization procedure gives us thatlim R →∞ | G R ( X, ω ) ∩ ( a, b ) | N ( R ) = µ { z ∈ Ω( X, ω ) : R h ( z ) ∈ ( a, b ) } , where µ is the unique ergodic probability measure on Ω( X, ω ) for which the first return map under h s isnot supported on a periodic orbit. Computing the slope gap distribution then reduces to finding a Poincar´esection for the horocycle flow on SL(2 , R ) / SL(
X, ω ), a suitable measure on this Poincar´e section, and thedistribution function for the first return map on the Poincar´e section.We note that this last point also makes it clear that every surface in the SL(2 , R ) orbit of a Veech surfacehas the same slope gap distribution. We also note that scaling the surface by c scales the gap distributionfrom f ( x ) to c f (cid:0) xc (cid:1) (see [UW16] for a proof of this latter fact).2.2. Computing Gap Distributions for Veech surfaces.
In [UW16], Uyanik and Work developed ageneral algorithm for computing the slope gap distribution for Veech surfaces. In particular, their algorithmfinds a parameterization for the Poincar´e section of any Veech surface and calculates the gap distribution byexamining the first return time of the horocycle flow to this Poincar´e section. In this section, we’ll go overthe basics of this algorithm. For more details about this algorithm as well as a proof of why it works, pleasesee Uyanik and Work’s original paper.We start by supposing that (
X, ω ) is a Veech surface with n < ∞ cusps. Then, we let Γ , . . . , Γ n berepresentatives of the conjugacy classes of maximal parabolic subgroups of SL( X, ω ). We are going to finda piece of the Poincar´e section for each parabolic subgroup Γ i . The idea here is that the set of shortestholonomy vectors of ( X, ω ) in each direction breaks up into S ni =1 SL(
X, ω ) v i where the v i vectors are in theeigendirections of the generators of each Γ i .The Poincar´e section is given by those elements g ∈ SL( X, R ) / SL(
X, ω ) such that g ( X, ω ) has a short(length ≤
1) horizontal holonomy vector:Ω(
X, ω ) = { g SL(
X, ω ) : g Λ ∩ ((0 , × { } ) = ∅} , where Λ is the set of holonomy vectors of ( X, ω ). Up to the action of SL(
X, ω ), these short horizontalholonomy vectors are then just g v i for a unique v i . LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 5
So Ω(
X, ω ) then breaks up into a piece for each Γ i , which we can parametrize as follows depending onwhether − I ∈ SL(
X, ω ). Case 1: − I ∈ SL ( X, ω ) . In this case, Γ i ∼ = Z ⊕ Z / Z and we can choose a generator P i for the infinitecyclic factor of Γ i that has eigenvalue 1. Up to possibly replacing P i with its inverse, we have that thereexists a C i ∈ SL(2 , R ) such that S i = C i P i C − i = (cid:20) α i (cid:21) for some α i > C i ( X, ω ) has a shortest horizontal holonomy vector of (1 , i is then parametrized by matrices M a,b = (cid:20) a b /a (cid:21) with − ≤ a < < a ≤ M a,b C i ( X, ω ) has a short horizontal holonomy vector ( | a | , S i and − I are in the Veech group of C i ( X, ω ), we need to quotient out the full setof M a,b matrices by the subgroup generated by S i and − I . The result is that the Poincar´e section pieceassociated to Γ i can be parametrized byΩ i = { ( a, b ) ∈ R : 0 < a ≤ − ( α i ) a < b ≤ } , where each ( a, b ) ∈ Ω i corresponds to g SL(
X, ω ) for g = M a,b C i . Remark 1.
We make a remark here that while Ω i is defined in this specific way in Uyanik and Work’spaper, there is actually a lot more freedom in defining Ω i . We just need to choose a fundamental domain forthe M a,b matrices under the action of h S i , − I i . For any m, c ∈ R , another such fundamental domain is Ω i = { ( a, b ) ∈ R : 0 < a ≤ and ma + c − ( α i ) a < b ≤ ma + c } . That is, instead of choosing Ω i to be a triangle whose top line is the line b = 1 for < a ≤ , we choose Ω i to be a triangle whose top line is the line b = ma + c for some slope m and b -intercept c . We see thedistinction between these two Poincar´e section pieces in Figure 3. ab b = 1 b = 1 − ( α i ) a abc b = ma + cb = ( m − α i ) a + c Figure 3.
Two possible Poincar´e section pieces Ω i . We further note that we can make similar modifications to Ω i in Case 2 below as well. In this case, therewill be another triangle with a < , and we have the freedom to choose the top line of the triangles with a > and a < independently. These modifications will be integral in our finiteness proofs. LUIS KUMANDURI AND JANE WANG
Case 2: − I SL ( X, ω ) . This case breaks up into two subcases depending on whether the generator P i of Γ i ∼ = Z has eigenvalue 1 or − P i has eigenvalue 1, then we again can find S i = C i P i C − i = (cid:20) α i (cid:21) for some α i > C i ( X, ω ) has a shortest horizontal holonomy vector of (1 , M a,b parameterize the Poincar´e section piece, but now we only can quotient out by the subgroupgenerated by S i . The result is that the Poincar´e section piece associated to Γ i can be parametrized byΩ i = { ( a, b ) ∈ R : 0 < a ≤ − ( α i ) a < b ≤ } [ { ( a, b ) ∈ R : − ≤ a < α i ) a < b ≤ } , where each ( a, b ) ∈ Ω i corresponds to g SL(
X, ω ) for g = M a,b C i .When P i has eigenvalue −
1, we can only find C i ∈ SL(2 , R ) such that S i = C i P i C − i = (cid:20) − α i − (cid:21) , where α i > C i ( X, ω ) has a shortest horizontal holonomy vector of (1 , M a,b matrices by the subgroup generated by S i . The resulting Poincar´e section piece associated to Γ i can be parameterized by Ω i = { ( a, b ) ∈ R : 0 < a ≤ − (2 α i ) a < b ≤ } , where each ( a, b ) ∈ Ω i corresponds to g SL(
X, ω ) for g = M a,b C i .Having established what each piece of the Poincar´e section associated to each Γ i looks like, we also need tofind the measure on the whole Poincar´e section. The measure on the Poincar´e section is the unique ergodicmeasure µ on Ω( X, ω ), which is a scaled copy of the Lebesgue measure on each of these pieces Ω i of R .Upon finding the Poincar´e section pieces, the return time function function of the horocycle flow at a point M a,b C i ( X, ω ) is the smallest positive slope of a holonomy vector of M a,b C i ( X, ω ) which short horizontalcomponent. That is, if v = ( x, y ) is the holonomy vector of C i ( X, ω ) such that M a,b ( x, y ) is the holonomyvector on M a,b ( x, y ) with the smallest positive slope among all holonomy vectors with a horizontal componentof length ≤
1, then the return time function at that point ( a, b ) ∈ Ω i in the Poincar´e section is given by theslope of M a,b ( x, y ), which is ya ( ax + by ) . We call such a vector v = ( x, y ) a winner or winning saddle connection . We note that while technically v is the holonomy vector of a saddle connection, we will often use the terms holonomy vector and saddleconnection interchangeably. Our proof that the slope gap distribution of a Veech surface has finitely manypoints of non-analyticity will rely on us showing that each piece Ω i of the Poincar´e section has finitely manywinners.Each such v would then be a winner on a convex polygonal piece of Ω i . Furthermore, the cumulativedistribution function of the slope gap distribution would then be given by areas between the hyperbolicreturn time function level curves and the sides of these polygons, and would therefore be piecewise realanalytic with finitely many points of non-analyticity.2.3. Examples and Difficulties.
In this section, we will give an example of difficulties that arise from thechoice of parameterization of the Poincar´e section. In particular, it is possible for there to be infinitely manywinning saddle connections under certain parameterizations, but only finitely many different winners undera different parameterization. For full computations of a gap distribution we refer to [ACL15] and [UW16].We will take the surface L in Figure 4 and analyze the winning saddle connection on the componentΩ of the Poincar´e section corresponding to the parabolic subgroup of SL( L ) generated by (cid:20) (cid:21) . L is a7-square square-tiled surface with a single cone point. LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 7
Figure 4.
The surface L with cone point in redSince (cid:20) (cid:21) is in the Veech group, and L has a length 1 horizontal saddle connection, the correspondingpiece of the Poincar´e section Ω can be parametrized by matrices M a,b = (cid:20) a b a − (cid:21) with 0 < a ≤ − a < b ≤
1. Notice that L has all saddle connections with coordinates ( n,
2) and ( n,
3) for n ∈ Z , and nosaddle connection with y -coordinate 1. Proposition 1.
In a neighborhood of the point (0 , on Ω , the winning saddle connection always has y -coordinate .Proof. Take a saddle connection v = ( n, k ) with k > M a,b v with horizontal component ≤
1. Wewill show that if k > a < there is a saddle connection w = ( m,
2) so that the slope of M a,b w isless than the slope of M a,b v , and so that M a,b w has short horizontal component. Since there are no saddleconnections with k = 1, this implies that the winning saddle connection must have y -coordinate 2.The x -coordinate of M a,b w is ma + 2 b . Since L has all ( m,
2) saddle connections, we may choose an m so that 1 − a < ma + 2 b ≤
1. The condition that slope( M a,b w ) < slope( M a,b v ), rearranges to( na + kb ) < k ma + 2 b )If k >
2, then since L is square-tiled we have that k ≥
3, and when a < / ma + 2 b > − a ≥ , thus k ( ma + 2 b ) >
1. Since M a,b v has a short horizontal component, we know that na + kb ≤ (cid:3) Let A m be the region where the saddle connection ( − m,
2) is the winning saddle connection. By Proposi-tion 1, in the top left corner of Ω , A m is the region where M a,b ( − m,
2) = (2 b − ma, a − ) has smallest slopeamong all saddle connections with y -coordinate 2 and short horizontal component. The slope is a − b − ma , sominimizing the slope is equivalent to maximizing 2 b − ma with the constraint that 2 b − ma ≤
1, or in otherwords, − m = ⌊ − ba ⌋ . But as a → b →
1, so − m ∼ − a → −∞ . This implies that there infinitely manysaddle connections that occur as winners in the top left corner of the Poincar´e section.By Remark 1 in Section 2.2, we notice that we can change the parameterization of the Poincar´e section.One problem in our previous parameterization was that there were infinitely many winners in the upper lefthand corner (0 ,
1) of our Poincar´e section. To fix this, we will change our parameterization so that the upperleft corner is at (0 , /
2) and the slope of the top line of our Poincar´e section triangle is nicely compatiblewith the (1 ,
2) holonomy vector. This will ensure that there are finitely many winners in the top left cornerand will result in finitely many winners across the entire Poincar´e section. We will prove that we can alwaysdo this for arbitrary Veech surfaces in Section 3.
LUIS KUMANDURI AND JANE WANG ab A m A m − Figure 5.
Regions A m where ( − m,
2) is a winner ab − ,
2) wins(2 ,
3) wins(2 ,
2) wins
Figure 6.
The new Poincar´e section breaks up into 3 pieces, with saddle connections (1,2)winning in the blue region, (2,3) in the yellow region, and (2,2) in the red regionWe will use the parameterization 0 < a ≤ − a < b ≤ − a . This parameterization is chosen toensure that the saddle connection (1 ,
2) of L wins in a neighborhood of the top line segment, which preventsthe problem that arises in the previous parameterization.In this case the only winners are the (1 , ,
3) and (2 ,
2) saddle connections on L .(1) (1 ,
2) wins in the region (cid:26) ( a, b ) (cid:12)(cid:12)(cid:12)(cid:12) < a ≤ , − a < b ≤ − a and 13 − a < b (cid:27) . (2) (2 ,
3) wins in the region (cid:26) ( a, b ) (cid:12)(cid:12)(cid:12)(cid:12) < a ≤ , − a < b ≤ − b (cid:27) . (3) (2 ,
2) wins in the region (cid:26) ( a, b ) (cid:12)(cid:12)(cid:12)(cid:12) < a ≤ , − a < b ≤ − a (cid:27) . LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 9
To see this, notice that the saddle connection ( x, y ) is the winner at ( a, b ) if M a,b ( x, y ) has smallest positiveslope amongst all saddle connections with short horizontal component. M a,b ( x, y ) has short horizontal inthe region 0 < a ≤ − xy a < b ≤ y − xy a . Minimizing the slope at ( a, b ) is equivalent to maximizing xy overall saddle connections with a short horizontal component.Working out the exact winners then comes down to casework. In this case, M ( a,b ) ( m,
2) never has a shorthorizontal component for m > a, b ) in the Poincar´e section, and simple casework shows where (1 , ,
2) are the winners. For saddle connections with y -coordinate greater than 2, we need to understandthose with x/y > which can potentially win against (1 ,
2) or (2 , ,
3) wins in the yellow region as (2 , a, b ) in that region. All other saddle connection with y = 3and x ≥ y ≥
4, a similar analysisshows that none of the saddle connections can appear as winners, giving the result.3.
Main Theorem
In Section 2.3, we examined the 7-square tiled surface L and saw that in one parameterization, it lookedlike the Poincar´e section would admit infinitely many winning saddle connections and therefore give thepossibility of infinitely many points of non-analyticity in the slope gap distribution. However, when westrategically chose a different parameterization of this piece of the Poincar´e section, there were now onlyfinitely many winners. Thus, this piece of the Poincar´e section could only contribute finitely many points ofnon-analyticity to the slope gap distribution. We could then have repeated this process for the other piecesof the Poincar´e section.This is one of the key ideas of the main theorem of this paper: Theorem 1.
Let ( X, ω ) be a translation surface with lattice Veech group. Then its slope gap distributionhas finitely many points of non-analyticity. This section is devoted to the proof of this theorem. We will begin by giving an outline of the proof andthen will dive into the details of each step.3.1.
Outline.
Let us begin with an outline of the proof of the main theorem, Theorem 1. The idea is thatafter choosing strategic parameterizations of each piece of the Poincar´e section of a translation surface (
X, ω )with lattice Veech group, we will use compactness arguments to show that there are finitely many winnerson each piece.(1) We begin with a translation surface (
X, ω ) with lattice Veech group and focus on a piece of itsPoincar´e section corresponding to one maximal parabolic subgroup in SL(
X, ω ). Up to multiplicationby an element of GL(2 , R ), we will assume that the generator of the parabolic subgroup has ahorizontal eigenvector and ( X, ω ) has a horizontal saddle connection of length 1. Based on propertiesof the saddle connection set of (
X, ω ), we strategically choose a parameterization T X of this Poincar´esection piece. T X will be some triangle in the plane.(2) For any saddle connection v of ( X, ω ), we will define a strip S Ω ( v ) that gives a set of points( a, b ) ∈ R > × R where v is a potential winning saddle connection on the surface M a,b ( X, ω ) ∈ T X . We willstart by showing various properties of these strips that we will make use of later on in the proof.(3) We will then show that there for every point ( a, b ) ∈ T X with b >
0, ( a, b ) has an open neighborhoodwith finitely many winning saddle connections.(4) We then move on to show that for points ( a, b ) ∈ T X with b ≤ a = 1 of T X , ( a, b ) has an open neighborhood with finitely many winning saddle connections.(5) Next, we show that on the right boundary a = 1 of T X , there are finitely many winning saddleconnections.(6) Using the finiteness on the right boundary, we show that for any point ( a, b ) ∈ T X with a = 1, ( a, b )has an open neighborhood with finitely many winning saddle connections.(7) By compactness of T X , there is a finite cover of T X with the open neighborhoods of points ( a, b ) ∈ T X that we found in our previous steps. Since each of the these open neighborhoods had finitely manywinners, we find that there are finitely many winning saddle connections across all of T X .(8) Finally, we show that finitely many winners on each piece of the Poincar´e section implies finitelymany points of non-analyticity of the slope gap distribution. Proof.
Using the method of [UW16] outlined in Section 2.2, it will suffice to show that every piece ofthe Poincar´e section can be chosen so that there are only finitely many winning saddle connections. Formost of the arguments in this section, we will fix a piece of the Poincar´e section and will work exclusivelywith it.We recall that there is a piece of the Poincar´e section for each conjugacy class of maximal parabolic sub-group in SL(
X, ω ). We will now fix such a maximal parabolic subgroup Γ i and work with the correspondingcomponent of the Poincar´e section. Without loss of generality we may assume that ( X, ω ) has a horizontalsaddle connection with x -component 1 and that Γ i is generated by P i = (cid:20) α i (cid:21) Using the notation of Section 2.2, this is essentially replacing (
X, ω ) with C i ( X, ω )Since (
X, ω ) is a Veech surface with a horizontal saddle connection, it has a horizontal cylinder decom-position ([HS06]), and therefore, for all a ∈ R there are only finitely many heights 0 ≤ h ≤ a so that ( X, ω )has a saddle connection with y -component h . Let y > X, ω ), and let x > y . Our first step is to use this saddle connection to give a parameterization of the Poincare section that isadapted to the geometry of ( X, ω ).By Remark 1, we can choose the following parameterization of this piece of the Poincar´e section, aspictured in Figure 7: T X = (cid:26) ( a, b ) (cid:12)(cid:12)(cid:12)(cid:12) < a ≤ , − x ay − na ≤ b ≤ − x ay (cid:27) . Here, n is either α i or 2 α i depending on which one is needed to fully parameterize this piece of thePoincar´e section, as described in Section 2.2. In the case where − I SL(
X, ω ) and P had eigenvalue 1, thePoincar´e section has an additional triangle with a <
0. In particular, we can choose this triangle so that itconsists of points ( − a, − b ) for ( a, b ) ∈ T X . But we note that if v were the winning saddle connection for M a,b ( X, ω ), then − v is the winning saddle connection for M − a, − b ( X, ω ), and hence if proving that there areonly finitely many winners, it suffices to consider only the portion with a > ab y b = − x y a + y b = ( − x y − n ) a + y Figure 7.
A Poincar´e section piece for (
X, ω ) with y > x > X, ω ).Our goal now is to prove that the return time function is piecewise real analytic with finitely many pieces.We will do so by proving that there are finitely many winning saddle connections v , . . . , v n ∈ Λ( X, ω ) such
LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 11 that each point ( a, b ) ∈ T X has a winner M a,b v i for some 1 ≤ i = n . We will repeat this for every T X corresponding to each maximal parabolic subgroup.To achieve this goal, we will first define an auxiliary set that will help us understand for what points( a, b ) ∈ T X a particular v ∈ Λ( X, ω ) is a candidate winner. By a candidate winner, we mean that M a,b v has a positive x coordinate at most 1 and a positive y coordinate. If v = ( x, y ), the x -coordinate conditionis the condition that 0 < ax + by ≤
1. We also note that for M a,b ( x, y ) to be be a winner, we need that a − y >
0. Since a > T X , the latter condition reduces to saying that y > Definition 1.
Given a saddle connection v = ( x, y ) with y > , we define S Ω ( v ) as the strip of points ( a, b ) ∈ R > × R such that < ax + by ≤ . This corresponds to the set of surfaces M a,b ( X, ω ) for which M a,b v is a potential winning saddle connection. ab y x Figure 8.
A strip S Ω ( v ) for v = ( x, y ). Here y >
0. The slope of the upper and lowerlines of the strip is − xy .Let us note some properties of these strips S Ω ( v ) that we will be using repeatedly in our proofs. Werecall that we are assuming without loss of generality that ( X, ω ) has a short horizontal saddle connectionof length 1. Considering the particular piece T X of the Poincar´e section, we recall that T X is parametrizedby matrices M a,b = (cid:20) a b a − (cid:21) so that M a,b ( X, ω ) has a short horizontal saddle connection of length ≤ X, ω ) is a Veech surface, it breaks up into horizontal cylinders and therefore there existsan y > y and furthermore that every saddle connectionwith positive height has height ≥ y .With these assumptions in in place, we note the following useful properties of the strips S Ω ( v ):(1) The strip S Ω ( v ) for v = ( x, y ) is sandwiched between a solid line that intersects the b -axis at y anda dotted line that intersects the b -axis at 0. Both lines have slope − xy . We also know that y ≥ y ,so y ≤ y .(2) Fixing any c >
0, there are only finitely many y coordinates of saddle connections v such that S Ω ( v )intersects the y -axis at any point ≥ c .This is because ( X, ω ) being a Veech surface and having a horizontal saddle connection impliesthat the surface breaks up into finitely many horizontal cylinders of heights h , . . . , h n and everysaddle connection with positive y -component must have a y -component that is a nonnegative linearcombination of these h i s. Since there are finitely many such y values ≤ /c , there are finitely manystrips that intersect the y axis at points ≥ c .(3) At a particular point ( a, b ) ∈ T X , the winner is the saddle connection v = ( x, y ) ∈ Λ( X, ω ) such that M a,b v = ( ax + by, a − y ) has the least slope among those saddle connections satisfying 0 < ax + by ≤ a − y >
0. Since a > Mi , this corresponds to the saddle connection with thegreatest xy with y > In terms of our strips, we’re fixing the point ( a, b ) and looking for the strip S Ω ( v ) that contains( a, b ) and has the least slope, since each strip has slope − xy .(4) For any given y >
0, there are only finitely many saddle connection vectors v = ( x, y ) of ( X, ω ) with x ≥ S Ω ( v ) intersects T X .This is because S Ω ( v ) does not intersect T X for xy larger than some constant C that depends on T X and y . Specifically, we can let C = x y + n , the negative of the slope of the bottom line thatdefines the triangle T X . Since the saddle connection set is discrete, there are finitely many x ≥ y such that xy ≤ C .With these facts established, let us first show a lemma that will imply that the return time of the horocycleflow to any point ( a, b ) ∈ T X is finite and that will be useful in proving Lemma 6 later. Lemma 1.
Let ( a, b ) ∈ T X so that ( a, is a short horizontal saddle connection of M a,b ( X, ω ) . Then M a,b ( X, ω ) has saddle connections v = ( x , y ) and v = ( x , y ) so that y , y > and < x ≤ a and ≤ x < a . (These saddle connections might be the same). This implies that every point in T X has awinning saddle connection, or equivalently, that every point in T X is in some strip S Ω ( v ) for some v = ( x, y ) with y > .Proof. Let us take a horizontal saddle connection on our surface M a,b ( X, ω ) with holonomy vector ( a, p and q . Then, we will consider developing a width a verticalstrip on our surface extending upward with the open horizontal segment from p to q as its base. Since oursurface is of finite area, this vertical strip must eventually hit a cone point r or come back to overlap ouroriginal open segment from p to q . Now we’re going to define our vectors v and v in each case. p qr v = v p r = q ... ... v = v p q ...... v v Figure 9.
The vectors v and v in the three different cases of vertical strip.In the former case when the top edge of our vertical strip hits a cone point r in the interior of the edge,the straight segment from p to r cutting through our vertical strip gives us both our saddle connections v = v .The latter case when the top edge of our vertical strip comes back to overlap our original open segmentbreaks up into two cases. If we have a complete overlap, then the saddle connection from p on the bottomedge to q on the top edge gives us our vector v and the saddle connection from p on the bottom edge to p on the top edge gives us our vector v . If we have an incomplete overlap, then the top edge contains thecone point r = p or r = q , and the saddle connection from p on the bottom edge to r on the top edge givesus both of our saddle connections v = v .In any of these cases letting v = M − a,b ( v ) gives us that S Ω ( v ) contains our initial point ( a, b ) and v is apossible winning saddle connection. (cid:3) The following lemma will help us show that there are finitely many winning saddle connections on certainsets in T X . Lemma 2.
Let S be a closed set that is a subset of S Ω ( v ) for v = ( x, y ) with y > . Then, there are finitelymany winning saddle connections on S .Proof. Let S be a closed set contained in S Ω ( v ) for a saddle connection v = ( x, y ) of ( X, ω ) with y > v is a potential winning saddle connection on all of S . That is, for any point( a, b ) ∈ S , M a,b v has positive y component and positive and short ( ≤ x component.We recall that for a point ( a ′ , b ′ ) ⊂ S to have winner v ′ = ( x ′ , y ′ ) = v = ( x, y ), we need that v ′ is a saddleconnection of ( X, ω ), x ′ y ′ > xy , and that ( a ′ , b ′ ) ⊂ S Ω ( v ′ ). LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 13
This corresponds to the strip S Ω ( v ′ ) having a smaller slope than S Ω ( v ) and still intersecting S . Giventhat S is closed and the bottom boundary of S Ω ( v ) is open, there exists an h > S iscompletely on or above the line b = − xy a + h . Furthermore, since the left boundary of S Ω ( v ) is open, S is apositive distance away from the y axis. ab y h S b = − xy a + h Figure 10.
A choice of 1 /h for a particular S .Then, for S Ω ( v ′ ) to intersect S and for x ′ y ′ > xy , we need that y ′ < h since otherwise the strip S Ω ( v ′ ) wouldhave y -intercepts 1 /y ′ ≤ /h and 0 and would have smaller slope than that of S Ω ( v ) and would thereforenot intersect S .But since ( X, ω ) is a Veech surface with a horizontal saddle connection, it decomposes into finitely manyhorizontal cylinders. Therefore, the set of possible vertical components y ′ of saddle connections are a discretesubset of R and thus, there are finitely many vertical components of saddle connections that satisfy y ′ < h .In addition, for each y ′ , since the set of corresponding x -components of saddle connections ( x ′ , y ′ ) is adiscrete set of R , there are finitely many corresponding x ′ such that x ′ y ′ > xy and S Ω ( v ′ ) intersects S . This isbecause for sufficiently large x ′ , the slope of S Ω ( v ′ ) will be too steep to meet S , which is a positive distancefrom the y -axis. Therefore, there are finitely many possible winning saddle connections on S . (cid:3) We recall that our goal is to show that every point ( a, b ) ∈ T X has a neighborhood on which there arefinitely many winners. This will allow us to use a compactness argument to prove that there are finitelymany winners on all of T X . Building off of the previous lemma, we show in the next lemma that certainpoints ( a, b ) ∈ T X have an open neighborhood on which there are finitely many winners. Lemma 3.
Let ( a, b ) be in the interior of some strip S Ω ( v ) . Then, there exists a neighborhood of ( a, b ) withfinitely many winning saddle connections.Proof. Let ( a, b ) be in the interior of the strip S Ω ( v ) for v = ( x, y ) with y > x ≥
0. Then, we canfind an ǫ > ǫ around ( a, b ) remains in the interior of the strip. That is, wechoose an ǫ > B ǫ (( a, b )) ⊂ S Ω ( v ) . We can then use Lemma 2 to conclude that there are finitely many winning saddle connections on B ǫ (( a, b ))and therefore on B ǫ (( a, b )). (cid:3) We now look at points ( a, b ) ∈ T X with b > Lemma 4.
Every point ( a, b ) ∈ T X such that b > has a neighborhood B ǫ (( a, b )) such that there are finitelymany winning saddle connections on B ǫ (( a, b )) ∩ T X . The same is also true for the point (0 , /y ) . Proof.
We recall that T X is a triangle bounded by the lines b = − x y a + y on top, the line a = 1 on theright and the line b = ( − x y − n ) a + y on the bottom.We break up the proof of this lemma into cases, depending on the location of ( a, b ) ∈ T X ∪ { (0 , /y ) } :(1) 0 < b < − x y a + y : These points are in the interior of T X . We also notice that they must be in theinterior of the strip S Ω ( v ) for v = ( x , y ). Therefore, by Lemma 3, such a point ( a, b ) must have aneighborhood with finitely many winning saddle connections.(2) b = − x y a + y : These points are on the top line of T X but have a >
0. We recall that y was chosento be the least y > X has a saddle connection ( x, y ). Then, x was the least x > x, y ) was a saddle connection of X .Let ( a, b ) be any point on the top line of T X with a > v = ( x , y ). Then ( a, b ) is on thetop line of the strip S Ω (( x , y )). We can find an ǫ > B ǫ (( a, b )) ∩ S Ω (( x , y )) is a closed subsetof S Ω (( x , y )). By Lemma 2, there are then finitely many winners on B ǫ (( a, b )) ∩ S Ω (( x , y )).(3) ( a, b ) = (0 , /y ): This point is not in T X but is the top left corner of the triangle that makes up T X .We can find a y > y such that every saddle connection ( x, y ) of X with y > y must satisfythat y ≥ y . Thus, we can then choose an ǫ > B ǫ ((0 , /y )) ∩ T X ⊂ S Ω (( x , y )) and no strip S Ω (( x, y )) for a saddle connection with y > y and x ≥ B ǫ ((0 , /y )). This would implythat the only possible winning saddle connections on B ǫ ((0 , /y )) are of the form ( x, y ) for x ≥ x .But if we fix y = y , since the set of saddle connections ( x, y ) is discrete and T X is bounded belowby the line b = ( − x y − n ) a + y , there are only finitely many saddle connections v = ( x, y ) of ( X, ω )whose strip S Ω ( v ) intersects B ǫ ((0 , /y )) (exactly those x such that x ≤ x ≤ x + ny ). We haveshown then that only finitely many strips S Ω ( v ) for holonomy vectors v that could win over ( x , y )intersect B ǫ ((0 , /y )) and therefore there are only finitely many winners on this neighborhood. (cid:3) Having established that points ( a, b ) ∈ T X with b > a, b ) ∈ T X with b ≤
0. We first show that such points that are not on the right boundaryof T X have finitely many winners. Lemma 5.
Let ( a, b ) ∈ T X satisfy that b ≤ and a < . Then, there exists a neighborhood B ǫ (( a, b )) suchthat there are finitely many winning saddle connections on B ǫ (( a, b )) ∩ T X .Proof. By Lemma 3, it suffices to show that ( a, b ) lies on the interior of a strip S Ω ( v ) for some saddleconnection v .Consider the points p ǫ = ((1 + ǫ ) a, (1 + ǫ ) b ), with winner w ǫ . Since ( a, b ) lies in the interior of T X , for ǫ > p ǫ also lies in T X . Moreover notice that p ǫ and ( a, b ) lie on the same line throughthe origin. This immediately implies that ( a, b ) lies in the interior of S Ω ( w ǫ ), as seen in Figure 3.2.( a, b ) p ǫ Figure 11.
The strip S Ω ( w ǫ ). LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 15
Indeed, by the definition of S Ω ( v ) for any holonomy vector v as a half-open strip with the open bottomboundary passing through the origin, for all points p ∈ S Ω ( v ) the points tp for 0 < t < (cid:3) The combination of our previous lemmas shows that for all ( a, b ) ∈ T X away from the right verticalboundary, there are only finitely many winners in a neighborhood of ( a, b ). We also want to show that foreach (1 , b ) on the right vertical boundary, there are only finitely many winners in a neighborhood. We willdo this in two steps. First we will show that there are finitely many winning saddle connections along theright boundary of T X . We will then use this result to prove that every point (1 , b ) on the right boundary of T X has a neighborhood with finitely many winning saddle connections.For our first result, we will need the following definition: Definition 2.
Given ( a, b ) ∈ R , define the set S Λ ( a, b ) as the strip of vectors v = ( x, y ) ∈ R such that < ax + by ≤ and y > . This corresponds to the set of vectors that are potential winners on the surface M a,b ( X, ω ) . We think of this definition as a sort of dual to Definition 1, where instead of thinking of the surfacescorresponding to a particular winning saddle connection, we think about the the set of possible coordinatesof winning saddle connections for a particular surface.
Lemma 6.
There are only finitely many winning saddle connections along the right vertical boundary a = 1 of T X .Proof. By Lemma 1, we know that every point (1 , b ) on the right boundary of T X has a winning saddleconnection. The set of b ∈ R such that (1 , b ) ∈ T X is some interval [ c, d ]. We note that since (cid:20) α (cid:21) is in the Veech group of our surface for some α >
0, it suffices to show that there are finitely manywinners for b ∈ [ c + nα, d + nα ] for any n ∈ Z . This is because ( x, y ) is the winner for b ′ if and only if( x − nαy, y ) is the winner for b ′ + nα . For convenience, we will prove that there are finitely many winnersfor b ∈ [ M, N ] = [ c + nα, d + nα ] for an n such that M, N > b , we let v b be its corresponding winning saddle connection. We wish to show that the setof vectors v b is finite. We suppose that { v b } is infinite. Then, since the set of b ∈ [ M, N ], we must be able tofind a convergent subsequence of b i ∈ R such that b i → b ′ and b ′ , b i ∈ [ M, N ] for all i . In particular, b ′ > S Λ (1 , b ′ ) cannot have a winning saddle connection, which would contradict Lemma 1.This corresponds to a saddle connection ( x, y ) in the strip S Λ (1 , b ′ ) that maximizes xy . The strip S Λ (1 , b ′ )satisfies that y > < x + b ′ y ≤
1, or alternatively that − b ′ x < y ≤ − b ′ x + b ′ . We recall that b ′ > b ′ Figure 12.
The strip S Λ (1 , b ′ ).We suppose that the winning saddle connection ( x ′ , y ′ ) for b ′ lies in the interior of S Λ (1 , b ′ ). If x ′ y ′ > x i y i and ( x ′ , y ′ ) ∈ S Λ (1 , b i ), then ( x i , y i ) could not be the winner for (1 , b i ) because ( x ′ , y ′ ) beats it and is still inthe strip S Λ (1 , b i ). We let C b ′ be the cone given by the intersection of y < y ′ x ′ x and y > y ′ x ′ − x − y ′ x ′ − . We notice that if( x i , y i ) ∈ C b ′ , then it follows that ( x ′ , y ′ ) ∈ S Λ (1 , b i ). One can see this algebraically or visually by notingthat if ( x i , y i ) is in the cone C b ′ as depicted in Figure 13, then S Λ (1 , b i ) contains ( x i , y i ) and is boundedby two lines with x -intercepts 0 and 1 and therefore must contain the point ( x ′ , y ′ ). Furthermore, the firstinequality defining the cone gives us that x ′ y ′ > x ′ y ′ .Therefore, if ( x i , y i ) is a winning saddle connection for some (1 , b i ) it cannot be in the open cone C b ′ asdefined above. Since b i → b ′ , this implies that for any ǫ > n large enough such that thestrips S Λ (1 , b i ) all lie in a region S n that is region where ( − b ′ + ǫ ) x ≤ y ≤ ( − b ′ − ǫ ) x + ( b ′ + ǫ ) and y > ǫ such that the slopes of the two bounding lines of S n are wedged between theslopes of the bounding lines of C b ′ . That is, we will choose ǫ > − b ′ − ǫ ) > y ′ x ′ and ( − b ′ + ǫ ) < y ′ x ′ − .We call this latter region S n . Figure 13 illustrates these regions. 1( x ′ , y ′ ) S n C b ′ S Λ (1 , b ′ ) Figure 13.
The strip S Λ (1 , b ′ ) with its winner ( x ′ , y ′ ) and cone C b ′ , along with the region S n containing the winners ( x i , y i ) for i ≥ n .Given these conditions, we notice that S n \C b ′ is a compact set. With the possible exception of one pointthat equals ( x ′ , y ′ ), the winning saddle connections ( x i , y i ) for i ≥ n must all be in this region. But the setof holonomy vectors of saddle connections of ( X, ω ), of which { ( x i , y i ) } is a discrete subset of R with noaccumulation points, and so there are only finitely many ( x i , y i ) ∈ S n \C b ′ . This contradicts the infinitenessof the set { ( x i , y i ) } . Hence, if S Λ ((1 , b ′ )) contained a point ( x ′ , y ′ ), it could not be in the interior of the strip.We also consider the case when ( x ′ , y ′ ) is in on the boundary of S Λ (1 , b ′ ). That is, we suppose that ( x ′ , y ′ )is on the line y = − b ′ x + b . If there exists a saddle connection in the interior of S Λ (1 , b ′ ), we can appealto the reasoning in the previous case to find a contradiction. Else, Lemma 1 guarantees that there is also aholonomy vector ( x ′′ , y ′′ ) on the open boundary y = − b ′ x of S Λ (1 , b ′ ).We now consider the cone C ′ b ′ given by the intersection of the regions y < y ′ x ′ x and y > y ′′ x ′′ − x − y ′′ x ′′ − .Similar to the previous case, we can find n large enough such that the strips S Λ (1 , b i ) all lie in a region S n that is defined by ( − b ′ + ǫ ) x ≤ y ≤ ( − b ′ − ǫ ) x + ( b ′ + ǫ ) and y >
0. Here, we again choose ǫ > S n are wedged between the slopes of the bounding lines of C ′ b ′ . That is,we will choose ǫ > − b ′ − ǫ ) > y ′ x ′ and ( − b ′ + ǫ ) < y ′′ x ′′ − . We call this latter region S n . Figure 14illustrates these regions. LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 17 x ′ , y ′ )( x ′′ , y ′′ ) S n C b ′ S Λ (1 , b ′ ) Figure 14.
The strip S Λ (1 , b ′ ) with its winner ( x ′ , y ′ ), the vector ( x ′′ , y ′′ ) on its openboundary, its cone C ′ b ′ , along with the region S n containing the winners ( x i , y i ) for i ≥ n .Since the set { ( x i , y i ) } has no accumulation points and S n \C ′ b ′ is compact, all but finitely many of the { ( x i , y i ) } for i ≥ n must lie in the cone C ′ b ′ and not be equal to ( x ′ , y ′ ) or ( x ′′ , y ′′ ). Let us consider one ofthese ( x i , y i ). The corresponding strip S Λ (1 , b i ) is the region between two parallel lines that intersect the x -axis at 1 and 0, including the line through 1 but not including the line through 0. Therefore, S Λ (1 , b i )must either contain ( x ′ , y ′ ) or ( x ′′ , y ′′ ), depending on if b i ≤ b ′ or b i > b ′ respectively. If it contains ( x ′ , y ′ ),then by similar reasoning as in the previous case ( x ′ , y ′ ) beats ( x i , y i ) and so ( x i , y i ) could not have beenthe winner for (1 , b i ). If it contains ( x ′′ , y ′′ ), then either ( x ′′ , y ′′ ) beats ( x i , y i ), which means that ( x i , y i ) wasnot the winner, or ( x i , y i ) was in the interior or S Λ ((1 , b ′ )), which contradicts that the interior of S Λ (1 , b ′ )did not contain any saddle connections. In either case, we have a contradiction.Since we found a contradiction in both the cases when there was saddle connection in the interior and onthe boundary of S Λ (1 , b ′ ), we see that there must have been only finitely many winners on the right verticalboundary of T X . (cid:3) We can now use the previous lemma to show that points on the right boundary of T X have a neighborhoodwith finitely many winners. Lemma 7.
Given any point ( a, b ) ∈ T X with a = 1 , there exists a neighborhood B ǫ (( a, b )) such that thereare finitely many winning saddle connections on B ǫ (( a, b )) ∩ T X .Proof. We suppose that we have a point ( a, b ) ∈ T X with a = 1, b = b ′ . Then, Lemma 1 guarantees that(1 , b ′ ) is in some strip S Ω ( v ). If (1 , b ′ ) is in the interior of S Ω ( v ), then Lemma 3 shows that there is aneighborhood of (1 , b ′ ) in T X with finitely many potential winners.We now consider the case where (1 , b ′ ) is not in the interior of any strip. This must then mean that (1 , b ′ )is on the top boundary of some strip S Ω ( v ). We will first deal with the case where (1 , b ′ ) is not on the topboundary of T X . Then every point (1 , b ′ + c ) for c > T X where a = 1, this then implies that (1 , b ′ ) is on the bottom boundary of some other strip S Ω ( w ), where w is the winning saddle connection for all (1 , b ′ + c ) for c > a = 1 line of T X by Lemma 6, we cannow choose an ǫ > w is the winning saddle connection for (1 , b ′ + c ) and v is thewinning saddle connection for (1 , b ′ − c ) for any 0 < c ≤ ǫ . ab T X B ǫ ((1 , b ′ )) S Ω ( v ) S Ω ( w ) Figure 15.
The winning strips S Ω ( v ) and S Ω ( w ) near (1 , b ′ ) on the right boundary of T X .We claim now that there are finitely many winning saddle connections on B ǫ ((1 , b ′ )). We recall that for apoint ( a, b ) ∈ B ǫ ((1 , b ′ )) ∩ T X to have a winning saddle connection other than v or w , there must be a strip S Ω ( u ) for a saddle connection u that is steeper (has more negative slope) than S Ω ( v ) or S Ω ( w ) (whicheveris the winner at ( a, b )) that contains ( a, b ).Shrinking ǫ if necessary, B ǫ ((1 , b ′ )) lies above the line b = − xy a + h for some h > x, y ) = v . Then,as in the proof of Lemma 2, we can show that there are finitely many strips saddle connections u of ( X, ω )with strips S Ω ( u ) intersect B ǫ ((1 , b ′ )) and that are at least as steep as S Ω ( v ).If S Ω ( u ) is at most as as steep as S Ω ( w ), then it cannot win for any point in B ǫ ((1 , b ′ )) ∩ T X since w or v would win instead.If S Ω ( u ) has steepness strictly between that of w and v , then for u to be a winner for some point( a, b ) ∈ B ǫ ((1 , b ′ )) ∩ T X , we must have that ( a, b ) ∈ S Ω ( u ) ∩ ( S Ω ( w ) \ S Ω ( v )). But then, by slope considerations, S Ω ( u ) must also intersect the a = 1 boundary of T X in B ǫ ((1 , b ′ )) above the point (1 , b ′ ). But this contradictsthat w and v were the only winners on the right boundary of T X in B ǫ ((1 , b ′ )).Hence, we have seen that only the finitely many saddle connections u with strips that intersect B ǫ ((1 , b ′ ))and have slope steeper than that of S Ω ( v ) can be winners on B ǫ ((1 , b ′ )) ∩ T X .If (1 , b ′ ) were on the top boundary of T X , then (1 , b ′ ) is on the top boundary of S Ω ( v ) where v = ( x , y ).We can again reason as in the proof of Lemma 2 that we can find an ǫ > u with S Ω ( u ) intersecting B ǫ ((1 , b ′ )) ∩ T X and with slope steeper than that of v .This shows that there are finitely many winners on B ǫ ((1 , b ′ )) ∩ T X . (cid:3) Combining these lemmas, this shows that for all points in T X there are finitely many winners in aneighborhood, and hence by compactness there are finitely many winners on T X . Proof of Theorem 1.
We will consider T X = T X ∪ { (0 , /y ) } . This is a compact set. We showed in Lemmas4, 5, and 7 that for any point ( a, b ) ∈ T X , we can find a neighborhood B ǫ (( a, b )) such that there are finitelymany possible winning saddle connections on B ǫ (( a, b )) ∩ T X . Since T X is compact, it is covered by finitelymany of these neighborhoods. Since a finite union of finite sets is finite, the set of possible winners on T X isfinite.Each winning saddle connection v i would then be a winner on a convex polygonal piece of T X . Thecumulative distribution function of the slope gap distribution would then be given by the sums of areasbetween the level curves of the hyperbolic return time functions ya ( ax + by ) as described in Section 2.2 and thesides of these polygons. Since there are finitely many polygonal pieces, the cumulative distribution functionand therefore also the slope gap distribution would be piecewise real analytic with finitely many points ofnon-analyticity. (cid:3) LOPE GAP DISTRIBUTIONS OF VEECH SURFACES 19 Further Questions
We end with a few questions for further exploration.(1) Are there bounds on the number of points of non-analyticity of a Veech surface?In [BMMM + n was found for the translation surface givenby gluing opposite sides of the 2 n -gon. While a global bound on the points of non-analyticity forthe slope gap distributions of all Veech surfaces is unlikely, we can ask if there are bounds based onthe number of cusps or other properties of the Veech group or of the surface.(2) Can anything be said about the decay of the tails of slope gap distributions of Veech surfaces?In [San20], it was shown that a certain family of non-Veech surfaces had slope gap distributionswith quadratic tail decay. That is, if f ( t ) is the slope gap distribution, then R ∞ t f ( t ) dt ∼ t − . Cana similar result be attained for Veech surfaces?(3) What can be said about the gap distributions of non-Veech surfaces?In [AC12], it was shown that the limiting slope gap distribution exists for almost every translationsurface, and in [San20], the slope gap distributions for a special family of non-Veech surfaces wereshown to be piecewise real-analytic. We can ask if the limiting slope gap distributions are alwayspiecewise real-analytic, and if so, are there always finitely many points of non-analyticity.(4) Where do the points of non-analyticity lie?Beyond just understanding the number of points of non-analyticity, we can ask about number-theoretic properties of the points themselves. In every example known to the authors of a limitingslope gap distribution, after rescaling, the points of non-analyticity lie in the trace field of the Veechgroup. Given that the gap distribution is computed by integrating areas between hyperbolas inregions related to the geometry of the surface, it is natural to conjecture that points of non-analyticitylie in quadratic extensions of the trace field. References [AC12] J. S. Athreya and J. Chaika. The distribution of gaps for saddle connection directions.
Geom. Funct. Anal. ,22(6):1491–1516, 2012.[AC14] Jayadev S. Athreya and Yitwah Cheung. A Poincar´e section for the horocycle flow on the space of lattices.
Int.Math. Res. Not. IMRN , (10):2643–2690, 2014.[ACL15] Jayadev S. Athreya, Jon Chaika, and Samuel Leli`evre. The gap distribution of slopes on the golden L. In
Recenttrends in ergodic theory and dynamical systems , volume 631 of
Contemp. Math. , pages 47–62. Amer. Math. Soc.,Providence, RI, 2015.[Ath16] Jayadev S. Athreya. Gap distributions and homogeneous dynamics. In
Geometry, topology, and dynamics innegative curvature , volume 425 of
London Math. Soc. Lecture Note Ser. , pages 1–31. Cambridge Univ. Press,Cambridge, 2016.[BMMM +
21] J. Berman, T. Mcadam, A. Miller-Murthy, C. Uyanik, and H. Wan. Slope gap distribution of saddle connectionson the 2n-gon. In preparation, 2021.[HS06] Pascal Hubert and Thomas A. Schmidt. An introduction to Veech surfaces. In
Handbook of dynamical systems.Vol. 1B , pages 501–526. Elsevier B. V., Amsterdam, 2006.[Mas88] Howard Masur. Lower bounds for the number of saddle connections and closed trajectories of a quadratic differ-ential. In
Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) , volume 10 of
Math. Sci. Res. Inst.Publ. , pages 215–228. Springer, New York, 1988.[Mas90] Howard Masur. The growth rate of trajectories of a quadratic differential.
Ergodic Theory Dynam. Systems ,10(1):151–176, 1990.[San20] Anthony Sanchez. Gaps of saddle connection directions for some branched covers of tori, 2020.[Tah19] Diaaeldin Taha. On cross sections to the geodesic and horocycle flows on quotients of SL (2 , R ) by Hecke trianglegroups G q . arXiv preprint arXiv:1906.07250 , 2019.[UW16] Caglar Uyanik and Grace Work. The distribution of gaps for saddle connections on the octagon. Int. Math. Res.Not. IMRN , (18):5569–5602, 2016.[Vor05] Yaroslav Vorobets. Periodic geodesics on generic translation surfaces. In
Algebraic and topological dynamics ,volume 385 of
Contemp. Math. , pages 205–258. Amer. Math. Soc., Providence, RI, 2005.[Zor06] Anton Zorich. Flat surfaces. In