An infinite surface with the lattice property II: Dynamics of pseudo-Anosovs

We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfies a mixing-type formula with polynomial decay. Then we consider the extent to which the action of these hyperbolic affine automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves and on homology classes. These objects are exponentially attracted by the expanding and contracting foliations but exhibit polynomial decay. We are able to work out exact asymptotics of these limiting quantities because of special integral formula for algebraic intersection number which is attuned to the geometry of the surface and its deformations.


W. PATRICK HOOPER
A . We study the behavior of hyperbolic a ne automorphisms of a translation surface which is in nite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We nd that hyperbolic a ne automorphisms are not recurrent and yet their action restricted to cylinders satis es a mixing-type formula with polynomial decay. en we consider the extent to which the action of these hyperbolic a ne automorphisms satisfy urston's de nition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves and on homology classes. ese objects are exponentially a racted by the expanding and contracting foliations but exhibit polynomial decay. We are able to work out exact asymptotics of these limiting quantities because of special integral formula for algebraic intersection number which is a uned to the geometry of the surface and its deformations. I Translation surfaces built from two copies of a regular polygon as depicted below were studied by Veech and proven to have beautiful properties [Vee89]. Perhaps most surprisingly, these surfaces admit a ne symmetries distinct from the obvious Euclidean symmetries. An understanding of these symmetries allowed Veech to prove his famed dichotomy theorem: In all but countably many directions every trajectory equidistributes, and the countably many exceptional directions are completely periodic. Veech also used this symmetry group to answer natural counting problems on these surfaces. Let X n denote the double regular n-gon surface. In [Hoo14], we showed that by choosing a ne maps A n of the plane which send three consecutive vertices of the regular n-gon to the points F 1. Veech , P 1 and P5 4 are shown.
(−1, 1), (0, 0) and (1, 1), the sequence of surfaces P cos π n = A n (X n ) converges to the in nite area surface P 1 built from two polygonal parabolas, the convex hulls of the sets {(n, n 2 ) : n ∈ Z} and {(n, −n 2 ) : n ∈ Z}. e limiting surface P 1 is depicted in the center of Figure 2.
An a ne automorphism of a translation surface S is a homeomorphism φ : S → S which is a real a ne map in local coordinates. Using the natural identi cation between tangent spaces to non-singular points of S with the plane, we see that the derivative Dφ of such a map must be constant and so we interpret Dφ as an element of GL(2, R). e subgroup of GL(2, R) consisting of derivatives of a ne homeomorphisms of S is called the surface's Veech group. (We allow orientation reversing elements in the Veech group, which di ers from conventions in some other articles.) For the surfaces we will consider, the Veech groups are contained in SL ± (2, R), the group of 2 × 2 matrices with real entries and determinant ±1.
In [Hoo14], it was shown that the symmetries of P cos π n persist to the limiting surface P 1 and beyond: ere are surfaces P c de ned for c ≥ 1 and all these surfaces have topologically conjugate a ne automorphism group actions [Hoo14,eorem 7]. e Veech groups of P c vary continuously in c and lie in the image of the representation ρ c : G → SL ± (2, R) where G = (C 2 * C 2 * C 2 ) × C 2 with C 2 denoting the cyclic group of order two and We call a matrix in SL ± (2, R) hyperbolic if it has two eigenvalues with distinct absolute values and call an a ne automorphism hyperbolic if its derivative is hyperbolic. A hyperbolic matrix in SL ± (2, R) has two real eigenvalues λ u and λ s with |λ u | > 1, |λ s | < 1 and λ u λ s = ±1. e Veech group ρ 1 (G) of P 1 has numerous hyperbolic elements.
A signi cant goal of this paper is to address the extent to which hyperbolic a ne automorphisms of in nite surfaces satisfy the de ning properties of pseudo-Anosov homeomorphisms of closed surfaces. e example we study is very special which enables us to say more about these questions than we'd expect to be able to for a general surface, so in particular we expect this paper sets some limits on what we could hope to be true in general.
For closed surfaces, the dynamics of hyperbolic a ne automorphisms are well known to be mixing since they admit Markov partitions [FLP12,§10.5]. On the surface P 1 we observe that cylinders satisfy a mixing-type formula but with polynomial decay. A cylinder on a translation surface is a subset isometric to R/cZ × [0, h] for some circumference c > 0 and height h > 0. We say two sequences a n and b n are asymptotic and write a n ∼ b n if lim n→∞ an bn = 1. eorem 1. Let A and B be cylinders in P 1 and φ : P 1 → P 1 be a hyperbolic a ne automorphism with derivative Dφ = ρ 1 (g). en is a positive constant which can be computed using the formula above where we use λ u c to denotes the eigenvalue of ρ c (g) with greatest absolute value.
Note that λ u c is real and varies analytically in a neighborhood of c = 1 because the entries of the matrix ρ c (g) are real polynomials in c and ρ 1 (g) is hyperbolic (and hyperbolicity is stable under perturbation of the matrix). is ensures that the quantity [ d dc λ u c ] c=1 is a well-de ned real number. It will follow from later work (Lemma 5) that the quantity β is positive.
As a consequence of this theorem, we note that no hyperbolic φ is recurrent because of the n −3 2 decay rate seen above: Corollary 2. If φ : P 1 → P 1 is hyperbolic then its action on P 1 is totally dissipative: there is a countable collection W of Lebesgue-measurable subsets of P 1 so that (1) the Lebesgue measure of the complement P 1 W ∈W W is zero, and (2) each W ∈ W is wandering in the sense that the collection {φ −n (W ) : n ≥ 0} is pairwise disjoint.
Because of non-recurrence, we tend to think points as the wrong objects to act to study hyperbolic a ne automorphisms of P 1 . Fortunately, some famous observations of urston suggest that acting on simple closed curves might be more natural.
We brie y recall urston's de nition of a pseudo-Anosov homeomorphism of a closed surface M following [ u86]. Let S = S(M ) be the collection of homotopically non-trivial simple closed curves up to isotopy on M . Le ing i : S × S → R denote geometric intersection number (i.e., the minimum number of transverse intersections among curves from the isotopy classes), we have an induced map i * : S → R S de ned by i * (α)(β) = i(α, β) so that the image is contained in the non-negative cone of R S . We let PR S denote the projectivization R S /(R {0}) and P : R S → PR S denote the projectivization map. We endow R S with the product topology and PR S with the quotient topology.
urston observed that for a compact surface M the projectivized image P • i * S(M ) has compact closure which we will denote by PS(M ). As long as the surface M has negative Euler characteristic, the closure PS(M ) is a sphere of dimension one less than the dimension of the Teichmüller space T (M ) (and forms its urston boundary). e space PS(M ) was identi ed with the space of projective measured foliations on M which urston also introduced and gives a geometric meaning to PS. Homeomorphisms of M naturally act on PS(M ) and isotopic homeomorphisms act in the same way. A pseudo-Anosov homeomorphism of M is a homeomorphism φ : M → M for which there are non-zero µ u and µ s in R S such that their projections lie in PS(M ) and so that there is a λ > 0 so that µ u • φ −1 = λµ u and µ s • φ = λ −1 µ s . e action of a pseudo-Anosov homeomorphism φ on PS(M ) is analogous the action of a hyperbolic isometry on the boundary of hyperbolic space: For any α ∈ S(M ), see [FLP12,Corollary 12.3]. urston showed that more generally for any P (ν) ∈ PS(M ) is paper investigates the extent to which the above results hold for the surface P 1 . We begin with trying to emulate the above de nitions for P 1 .
Perhaps we have been slightly abusing terminology to call P 1 a "surface". It has two in nite cone singularities coming from the vertices of the polygonal parabola. ese singularities do not have neighborhoods locally homeomorphic to an open subset of the plane, so we de ne P • 1 to be P 1 with these singularities removed. e space P • 1 is an in nite genus topological surface. We de ne S = S(P • 1 ) to be the collection of simple closed curves in P • 1 up to isotopy and de ne i * : S → R S , PR S and P : R S → PR S as above. We de ne only this time we note that PS is not compact. (If α n is a sequence of simple closed curves exiting every compact subset of P • 1 then lim i * (α n ) = 0 ∈ R S and no subsequence of P • i * (α n ) converges in PR S .) We show: eorem 3. Fix a hyperbolic a ne automorphism φ : P 1 → P 1 and let λ u 1 ∈ R denote the expanding eigenvalue of Dφ. Let µ u and µ s be the elements of R S(P • 1 ) corresponding to the transverse measures on P 1 to foliations parallel to the expanding and contracting eigenspaces of Dφ respectively. en: where β is given as in eorem 1 and u u and u s denote expanding and contracting unit eigenvectors of Dφ. In particular, we have P . On the other hand, it is not true (2) holds for every P (ν) ∈ PS(P • 1 ) {P (µ u ), P (µ s )}. For every direction θ of irrational slope and every c > 1, there is a direction θ and a homeomorphism h : P 1 → P c (in the isotopy class of a standard identi cation between these surfaces) which carries the foliation on P 1 in direction θ to the foliation in direction θ on P c . Furthermore, if θ was stabilized by a hyperbolic a ne automorphism φ : P 1 → P 1 then θ is stabilized by an a ne automorphism φ : P c → P c which is the same up to the canonical identi cation and isotopy. We can pull back the transverse measure on P c in direction θ to obtain another measured on the foliation of P 1 in direction θ which is stabilized by φ. is was proved in a more general context in [Hoo15,eorem 4.4]. is new measure corresponds to a distinct φ-invariant element of R S(P • 1 ) . In fact we can see this element lives in PS(P • 1 ) in many cases because the straight-line ow in many eigendirections of pseudo-Anosov homeomorphisms is ergodic [Hoo15,eorem 4.5], so that we can obtain projective approximations by simple closed curves by owing point forward until it returns close and closing it up. As we increase the ow time, convergence to P (µ u ) is guaranteed for almost every starting point by the ratio ergodic theorem.
We have shown that elements in P (S(P • 1 )) are a racted under φ and φ −1 -orbits respectively by P (µ u ) and P (µ s ), but noted that this does not hold on the closure PS(P • 1 ), so it is natural to wonder how a ractive P (µ u ) and P (µ s ) are in other contexts.
Since P 1 is a translation surface, it is natural to orient our foliations in each direction and to consider our transverse measures to be signed measures. Informally, let H 1 (P • 1 ; R) be real weighted nite sums of homology classes of closed curves in P • 1 , and let H 1 (P 1 , Σ; R) be real weighted nite sums of homology classes of closed curves and curves joining the singularities of P 1 . Algebraic intersection number gives a weakly non-degenerate bilinear map , R be the collection of linear maps from H 1 (P 1 , Σ; R) to R and let PH = H/(R {0}) be the projectivization of this space. In a parallel construction to geometric intersection number, we get a map induced by algebraic intersection and a projectivization map: Given an oriented arc on P 1 we can li it to the universal cover and project it to the plane under the developing map. e holonomy vector of the arc is the di erence of the developed end point and the starting point. Holonomy gives rise to linear maps hol 1 : H 1 (P • 1 ; R) → R 2 and hol 1 : H 1 (P 1 , Σ; R) → R 2 , where we write hol 1 to indicate we are computing holonomy on P 1 . Given a direction described by a unit vector u ∈ R 2 we get an element of H using the linear map If φ : P 1 → P 1 is an a ne homeomorphism with hyperbolic derivative, then the choice of unit unstable and stable eigenvectors of Dφ give rise via (3) to respective elements µ u and µ s of H satisfying µ u • φ −1 = λ u 1 µ u and µ s • φ −1 = λ s 1 µ s where λ u 1 and λ s 1 are the expanding and contracting eigenvalues of Dφ. We show that the projectivized classes P (µ u ) and P (µ s ) respectively a ract and repel every element of P •∩ * H 1 (P • 1 ; R) , but unlike in prior results the rate of polynomial decay depends on the chosen homology class.
Moreover, there is a descending sequence of subspaces indexed by N . . so that: (1) Each S j+1 is codimension one in S j (i.e., S j+1 is the kernel of a surjective linear map S j → R).
We remark that statement (3) also holds in the opposite direction, i.e., n j+ 3 2 (λ s 1 ) −n ∩ * φ −n (γ) converges to a multiple of µ s , but the rate given by j is determined by a di erent sequence of nested subspaces. is theorem is a consequence of our eorem 19 which is stronger in that it gives a formula for the constant appearing in the limit in statement (3). e proof of eorem 4 contains a concrete description of the subspaces S j and gives a slightly di erent formula for the limiting constant; see (34).
Again we could consider trying to extend this type of convergence to a larger space, and PH itself would be a natural candidate but this space also contains xed points corresponding to measured foliations on P c . It is not clear to the author if there is a natural intermediate space between P • ∩ * H 1 (P • 1 ; R) and PH in which we see P (µ u ) and P (µ s ) as the global a ractor and repeller.
We will now brie y discuss the method we use to prove these results. e main idea is to completely understand algebraic intersection numbers, and we provide a formula for computing the algebraic intersection numbers between curves on P 1 . It rst needs to be observed that the surfaces P c for c ≥ 1 are canonically homeomorphic, with the homeomorphisms coming from viewing P c for c ≥ 1 as a parameterized deformation of translation surfaces. Fixing a curve γ ∈ P 1 representing a homology class we can use these canonical homeomorphisms to obtain corresponding curves in P c . We observe that the holonomies of these curves measured on P c denoted hol c γ ∈ R 2 depend polynomially in c. us this quantity makes sense for all c ∈ R. In Lemma 12, we show that for any two curves γ and σ representing homology classes on P 1 , their algebraic intersection number is given by is result gives a mechanism to reduce the eorems above to questions involving the asymptotics of certain trigonometric integrals.
Organization of article.
• In §1 we provide a condensed description of main ideas we use from the theory of translation surfaces. We give more formal descriptions of the homological spaces mentioned above. • In §2 we investigate the continuous family of representations that arises out of considering the Veech groups of the surfaces P c . is family of representations is studied in the abstract and we prove results about eigenvalues that are crucial for our later arguments. • Section 3 addresses the geometry and dynamics of P 1 .
-In §3.1 we review the construction of the surfaces P c .
-In §3.2 we compute generating sets for the homological spaces we work with.
-In §3.3 we explain how holonomies of homology classes deform as we vary c and formalize our intersection number formula described in (4). -In §3.4 we describe the a ne automorphism groups of the surfaces P c for c ≥ 1.
-In §3.5 we prove eorem 19, which is the most important result in the paper: it gives an asymptotic formula for algebraic intersections of the form φ n (γ) ∩ σ. We use it to prove eorem 4. -In §3.6 we prove our asymptotic formula for intersections of cylinder described by eorem 1. e key is the observation that the area of intersection of two cylinders is largely governed by algebraic intersection numbers between the core curves.
-In §3.7 we consider geometric intersection numbers and prove eorem 3.

B
A translation surface is a topological surface with an atlas of charts to the plane so that the transition functions are translations. Equivalently, an translation surface S is a surface whose universal cover is equipped with a local homeomorphism called the developing map dev from the universal coverS to R 2 so that for any deck transformation ∆ :S →S, there is a translation T : R 2 → R 2 so that dev • ∆ = T • dev . Such a surface should be considered equivalent to the surface obtained by post-composing the developing map with a translation. Note that this de nition does not allow for cone singularities on the surface but they may be treated as punctures.
In this paper we will be considering surfaces of in nite genus, but our surfaces will be decomposable into countably many triangles. Each translation surface we consider S will be a countable disjoint union of triangles with edges glued together pairwise by translations in such a way so that each point on the interior of an edge has a neighborhood isometric to an open subset of the plane. (Figure 6 depicts such a decomposition for P 1 .) We will use Σ to denote the singularities which are the equivalence classes of the vertices of the triangles in S. e surface S Σ is a translation surface, while S itself is not a surface if countably many triangles meet at a singularity.
Let R be a ring containing Z such as Z or R. For us, rst relative homology over R is H 1 (S, Σ; R) may be viewed as the R-module generated by oriented edges of the triangulation of S and subject to the conditions that the sum of two copies of the same edge with opposite orientations is zero as is the sum of edges oriented as the boundary of a triangle. Less formally, H 1 (S, Σ; R) may be viewed as homology classes consisting of nite weighted sums of curves joining points in Σ and closed curves subject to the condition that they pass through only nitely many triangles. Any such curve is homotopic to a union of edges and the resulting class in H 1 (S, Σ; R) is independent of the choice of such a homotopy.
Let S • denote S Σ. We de ne H 1 (S • ; Z) to be the abelianization of π 1 (S • ). Note that by compactness a closed curve in S • only intersects nitely many triangles de ning S. We de ne Again, we may consider H 1 (S • ; R) to represent all nite sums of closed curves in S • weighted by elements of R.
We use ∩ to denote the algebraic intersection number which is bilinear. We follow the convention that α ∩ β is positive if α is moving rightward and β is moving upward. With R as above, algebraic intersection extends to a bilinear map Let γ : [0, 1] → S be a curve and letγ : [0, 1] →S be a li to the universal cover. e holonomy vector of γ is Observe that this quantity is independent of the choice of li . Holonomy yields linear maps which send a weighted sum of curves to the corresponding weighted sum of holonomy vectors. We use hol for both maps because the following map diagram commutes Let S be a translation surface built as above by gluing together triangles in a countable set T . Let f : S → S be a homeomorphism preserving the singular set Σ and satisfying the condition that for any T ∈ T , the image f (T Σ) intersects only nitely many triangles in T . en f acts naturally on the homological spaces de ned above. We denote this action by f * . Letf :S →S be a li to the universal cover. We say that f is an a ne automorphism of S if there is an a ne map A : is notion does not depend on the li f . e derivative of an a ne automorphism f : S → S is the element Df ∈ GL(2, R) which is given by A post-composed by a translation (so that the origin is xed). We say f is hyperbolic if Df is a hyperbolic matrix, i.e., Df has distinct eigenvalues and their absolute values di er. Observe that if f is an a ne automorphism then for any homology class σ we have e collection of all a ne automorphisms of S forms a group which we denote by Aff (S). In this paper, the Veech group of S is D Aff (S) ⊂ GL(2, R). (Some papers consider the Veech group to be D Aff (S) ∩ SL(2, R).)

A
For each real c ∈ R let Γ c denote the subgroup of SL ± (2, R) generated by −I together with the involutions We will later see that Γ c realizes the Veech group DAff (P c ) for c ≥ 1, but in this section we will be interested in proving some results about how the deformation Γ c e ects hyperbolic elements of Γ 1 . e group P GL(2, R) is naturally identi ed with the isometry group of the hyperbolic plane, and each of these three matrices act by re ections in a line in the hyperbolic plane (called the axis of the re ection). Figure 4 depicts the axes of re ections corresponding to the matrices above. Observe that the relationship between the lines A and C changes as c varies. We will be concerned with the groups Γ c when c ≥ −1. Observe that C −1 = −B −1 . When −1 < c < 1, the axes of re ections A c and C c intersect at an angle of θ where cos θ = c. When θ = 2π n , the group Γ c is discrete with the triangle formed by the re ection axes forming a fundamental domain for the group action. When c = 1, these re ection axes become tangent at in nity, and Γ c becomes the congruence two subgroup of SL ± (2, Z) whose fundamental domain is the ideal triangle enclosed by the three re ection axes. When c ≥ 1, the group stays discrete but is no longer a la ice since the fundamental domain which consists of the enclosed ultra-ideal triangle has in nite area.
We will think of these groups as a continuous family of representations of the abstract group where C 2 = Z/2Z is the cyclic group of order two. We de ne the representation ) sending the generators of G to A c , B c , C c and −I respectively.
For any z ∈ C, we de ne i.e., we evaluate ρ(g) with c = z. Below and throughout this paper, we abuse notation by identifying the free variable c temporarily with a constant, so we will simply write ρ c rather than ρ z . e following is the main result we will need about this family of representations.
Lemma 5. Let g ∈ G be such that ρ 1 (g) has a real eigenvalue λ u 1 with |λ u 1 | > 1. For each c ∈ R, let λ u c denote the choice of an eigenvalue realizing the maximum value of the absolute value of an eigenvalue of ρ c (g). en: ( is positive. We remark that the constant β appeared in eorems 1 and 3. e remainder of the section is devoted to a proof of the Lemma and the following consequence which involves the operator norm · on real 2 × 2 matrices de ned using the Euclidean metric on R 2 . e group PGL(2, R) is the isometry group of the hyperbolic plane. For a hyperbolic G ∈ PGL(2, R) the eigenvalue λ u of largest absolute value has the geometric signi cance: (9) inf x∈H 2 dist(x, Gx) = 2 log |λ u |. Moreover, this in mum is achieved. e collection of points where this in mum is achieved is a geodesic in H 2 called the axis of the hyperbolic isometry G. e axis has a canonical orientation determined by the direction G translates the geodesic. If G belongs to a discrete group Γ, then this axis projects to a curve of length 2 log |λ u | in the quotient H 2 /Γ.
We will need to extend these ideas to triangular billiard tables which are not quotients of H 2 by a discrete group. Let ∆ be any triangle in H 2 (with the case of interest being the triangles shown in Figure 4). We label the edges by the {A, B, C} and we will use a , b , and c to denote the bi-in nite geodesics in H 2 that contain the marked edges. Let G ∼ = C 2 * C 2 * C 2 represent G modulo the central C 2 in G. Given ∆ we get a representation ρ ∆ : G → PGL(2, R) where we send the generators g A , g B , g C ∈ G to the re ections in a , b , and c respectively.
Each conjugacy class [g] in G has a representative of the form where L i = L i+1 for any i ∈ {0, . . . , n − 1} and addition taken modulo n. Moreover this representative is unique up to cyclic permutations. We de ne the orbit-class Ω([g]) to be the collection of all closed recti able curves in H 2 that visit the lines L 0 , L 1 , . . . , L n−1 in that order. We de ne the orbit-length of [g] to be L( We have the following lemma. Lemma 7. Fix ∆ and notation as above. Let g ∈ G and let [g] be its conjugacy class. Let λ u denote the eigenvalue of ρ ∆ (g) with largest absolute value. en 2 log |λ u | ≤ L([g]).
e proof is essentially the "unfolding" construction from polygonal billiards: Proof. e statement is certainly true unless |λ u | > 1. Since eigenvalues are a conjugacy invariant, we may assume that g is given as in (10). If the conclusion is false, there is a γ ∈ Ω(G) with length less than 2 log |λ u |. We will draw a contradiction by construct a new path γ in H 2 with length equal to that of γ such that ρ ∆ (g) translates the starting point of γ to the ending point of γ violating (9). We may assume γ begins on the geodesic L 0 and then travels to L 1 and so on. For i ∈ {0, . . . , n − 1}, let γ i be the portion of γ which travels from L i−1 to L i with subscripts taken modulo n so that is makes h n = g as given in (10).
Statement (2) holds by construction since γ 0 = γ 0 (as h 0 is the identity). To see statement (1) observe that γ i terminates at L i at the same point at which γ i+1 starts. us, From (1) we see that a point on γ 0 is joined to the corresponding point on γ n by a path of length equal to the length of γ, and from (2) we see that these points di er by an application of the hyperbolic isometry ρ ∆ (h n ). Recalling that h n = g, we see this violates (9).
We will need to introduce one more idea. e Klein model for the hyperbolic plane consists of Geodesics in the Klein model are Euclidean line segments. Distance between points in the Klein model may be computed in two ways. Let P 1 and P 2 be two points in KH 2 , and let P 1 P 2 be the Euclidean line through them. Let Q 1 and Q 2 be the two points of ∂KH 2 ∩ P 1 P 2 , chosen so that P 1 is closest in the Euclidean metric to Q 1 . en the distance between P 1 and P 2 is given by the logarithm of the cross ratio, Alternately, we can compute distance by using the metric tensor ds.
e distance between two points can be computed by integrating the metric tensor over the geodesic path between them. See [CFKP97] or any hyperbolic geometry text for more details.
As discussed above the projectivization of ρ c (G) to a subgroup of PGL(2, R) is generated by the re ections in the sides of a triangle ∆ c in H 2 . See Figure 4. When c = cos θ ≤ 1, the triangle has two ideal vertices and one vertex with angle θ. For our purposes, we will think of ∆ c ⊂ KH 2 . We de ne where P 1 = (−1, 0), P 2 = (1, 0), and P 3 = (0, . It is an elementary check that this triangle is isometric to the triangle used to de ne the re ection group ρ c (G). e re ection lines of the elements A c , B c , and C c are given by A = P 3 P 1 , B = P 1 P 2 , and C = P 2 P 3 respectively. Fixing c observe that ρ c : G → SL ± (2, R) projectivizes to a representation ρ c : G → PGL(2, R). e generators g A , g B , g C ∈ G are mapped under ρ c respectively to A c , B c and C c of (5) viewed as elements of PGL(2, R). en A , B , and C are the xed point sets of these hyperbolic re ections.
Proof of Lemma 5, statement (1). Choose g ∈ G so that ρ 1 (g) has a real eigenvalue λ u 1 with |λ u 1 | > 1. Let G 1 = ρ 1 (g) which we think of as a hyperbolic transformation of H 2 . In the case of c = 1, the group ρ 1 (G ) is discrete and ∆ 1 is a fundamental domain for the action. us, the projection of the axis of G 1 to H 2 /ρ 1 (G) minimizes length in the orbit-class Ω(g). In particular, this billiard path γ 1 realizes the in mum discussed in Lemma 7.
Let |λ u c | denote the greatest absolute value of an eigenvalue of G c = ρ c (g). We will use Lemma 7 to show that |λ u c | < |λ u 1 | for all −1 ≤ c < 1. In coordinates on the closure of KH 2 consider the linear map Observe from our formula for the vertices that M c (∆ 1 ) = ∆ c . We claim that M c shortens every line segment in ∆ 1 except line segments contained in the side P 1 P 2 . Consider a segment XY with nite length in ∆ 1 . Let P Q be the geodesic containing XY , so that P, Q ∈ ∂KH 2 with P the closest to X as in the le side of Figure 5. en by (11), And, let P , Q ∈ ∂KH 2 be the points where the geodesic M c (P )M c (Q) intersects the boundary. en, It is a standard computation that dist KH 2 (X, Y ) > dist KH 2 (M c (X), M c (Y )) so long as the line segment P Q strictly contains the line segment M c (P )M c (Q), i.e., as long as M c (P ) = P or M c (Q) = Q . In particular, the only time this inequality could be false is when P Q ⊂ {(x, y)|y = 0}. Our claim is proved.
To nish the proof, we apply the claim to the billiard path γ 1 constructed in the rst paragraph. No nite length billiard path can have a segment contained in the line y = 0, therefore length M c (γ 1 ) < length γ 1 .
whenever −1 ≤ c < 1. en by Lemma 7, for all such c, 2 log |λ u c | ≤ length M c (γ 1 ) < length γ 1 = 2 log |λ u 1 |. us |λ u c | < |λ u 1 |. Proof of Lemma 5, statement (2). Let g ∈ G be chosen so that ρ 1 (g) has an eigenvalue λ u 1 so that |λ u 1 | > 1. By possibly replacing ρ 1 (g) with −Iρ 1 (g) we may assume λ u 1 > 1. Let λ u c be the largest eigenvalue of G c = ρ c (g), which by continuity of c → G c satis es λ u c > 1 in a neighborhood of 1. We must show e quantity d dc λ u c exists at c = 1, since entries of G c vary polynomially in c. us we can a ord to just look for a one-sided derivative and we will restrict a ention to the case of c < 1 so that we can make use of the map M c : ∆ 1 → ∆ c of (14).
Let m = √ 1+c √ 2 . We have d dc m = 1 2 √ 2+2c , and d dc m| c=1 = 1 4 . By (12), the KH 2 length of the tangent vector i = (1, 0) at the point (x, y) ∈ KH is given by We compute d dc which is positive on all of KH 2 other than those points where y = 0. Note, we are perturbing c in the negative direction. is says that o the line y = 0, M c is compressing every horizontal vector enough to be detected by the rst derivative. Let J 1 be the KH 2 length of the tangent vector j = (0, 1) at the point (x, y) ∈ KH and J 2 be the KH 2 length of the vector M c (j) = (0, m) at the point M c (x, y) = (x, my). We have In this case, M c is compressing every vertical vector enough to be detected by the rst derivative. e argument concludes in the same manner as the previous proof. Let γ 1 be the billiard path on ∆ 1 corresponding to G 1 . e argument above tells us that d dc length(M c (γ 1 )) = k > 0. But for c < 1, 2 log λ u c ≤ length(M c (γ 1 )) = length(γ 1 ) − k(1 − c) + higher order terms.
By taking a straight forward derivative, we get Proof of Proposition 6. Fix a c 0 < 1. e function c → |λ u c | is continuous so a ains its maximum in [−1, c 0 ]. By (1) of Lemma 5 we know this maximum is less than λ u 1 , so we can select ξ so that |λ u c | < ξ < λ u 1 for all c ∈ [−1, c 0 ]. We will show that ξ −n ρ c (g) n tends to the zero matrix uniformly in the operator norm.
First observe that it su ces to nd an N ∈ N so that n ≥ N implies ρ c (g) n < ξ n for c ∈ [−1, c 0 ]. Suppose this statement is true and x an > 0. We will nd an N so that n > N implies that the operator norm ξ −n ρ c (g) n < . By continuity of c → ρ c (g) N we can set en choose K su ciently large so that Any n > (K + 1)N can be wri en in the form n = kN + n for some k > K and some n satisfying 0 ≤ n < N so that by submultiplicativity of the operator norm we have e main idea is to continuously diagonalize the matrix action of ρ c (g) n . However there is a minor di culty that for some values of c the matrix might not be diagonalizable. e matrices ρ c (g) have constant determinant ±1 and so the eigenvalues are distinct except in the case that the determinant is one and the trace is ±2. ere are only nitely many c at which the trace of ρ c (g) is ±2 since the trace is polynomial in c and is not constant (by statement (2) of Lemma 5).
Let c 1 , . . . , c k denote the nitely many values of c in [−1, c 0 ] for which ρ c (g) has trace ±2. Observe that if P ∈ SL(2, R) is a any matrix with determinant one and trace ±2, then the operator norm P n grows asymptotically linearly. We conclude that for each i ∈ {1, . . . , k} there is a N i so that n > N i implies that ρ c i (g) n < ξ n . en by continuity and submultiplicativity of the operator norm (similar to the argument above), we can nd an r i > 0 and a N i > N i so that (Note that at this point u and s are just being used to distinguish eigenvectors and do not necessarily correspond to expanding and contracting directions.) Let P u c be the projection matrix so that v u is an eigenvector with eigenvalue 1 and so that v s is an eigenvector with eigenvalue zero. Let P s c be the matrix with the v u and v s playing opposite roles. en P s c and P u c vary continuously for c ∈ J and we have (16) ρ c (g) n = (λ u c ) n P u c + (λ s c ) n P s c for all n ∈ N and c ∈ J.
Le ing M denote the supremum of the operator norms of all matrices of the form P u c and P s c with c ∈ [−1, c 0 ] we see that ρ c (g) n ≤ M (|λ u c | n + |λ s c | n ) ≤ 2M (ξ ) n where ξ is the supremum of |λ u c | and |λ s c | over c ∈ J. en by de nition of ξ we have ξ < ξ and thus 2M ( ξ ξ ) n tends exponentially to zero. In particular, there is a N J so that n > N implies that ρ c (g) n < ξ n for all c ∈ J.
Se ing N = max{N 1 , . . . , N k , N J } we see that for any n > N we have ρ c (g) n < ξ n . From our rst observation this proves the proposition.
3. T 3.1. Construction. We follow the construction of the parabola surface as an a ne limit of Veech's surfaces built from 2 regular n-gons as described in [Hoo14]. Consider the regular ngon in R 2 to be the convex hull of the orbit of (1, 0) under the rotation matrix ree points on this orbit are given by R −1 t (1, 0) = (cos t, − sin t), (1, 0) and R −1 t (1, 0) = (cos t, sin t). ere is an a ne transformation C t : R 2 → R 2 of the plane which carries these three points to (−1, 1), (0, 0) and (1, 1), respectively and a calculation shows e image of the regular polygon under C t is the polygon Q + c whose vertices lie in the orbit of (0, 0) under T c = C t • R t • C −1 t which we compute to be the a ne map Observe that because T c is an a ne transformation de ned with coe cients in Z[c], for any k ∈ Z, the k-th vertex T k c (0, 0) of Q + c has coordinates which are polynomial in c. From the above it can be observed that when c = cos 2π n that Q + c is an n-gon. When c = 1, the T c -orbit is given by T n c (0, 0) = (n, n 2 ), and we interpret Q + c as a polygonal parabola. Similarly, when c > 1, Q + c should be interpreted as a polygonal hyperbola. We let Q − c be the image of Q + c under rotation by π about the origin. To form a surface P c for some c = cos 2π n or c ≥ 1, we identify each edge e of Q + c by translation with the image of the edge of Q − c obtained by applying this rotation. ese surfaces are depicted in Figure 2. 3.2. Homological generators. We use Σ to denote the collection of two singularities of P 1 .
Proposition 8. e saddle connections in the common boundaries of the parabolas Q + 1 and Q − 1 generate H 1 (P 1 , Σ; Z).
We denote these saddle connections by σ i for i ∈ Z. See Figure 6.
Proof. e surface P 1 is triangulated by the saddle connections in the set {σ i } together with horizontal and slope one saddle connections. See Figure 6. Observe that by using the relation that the sum of edges around a triangle is zero, we can inductively write each horizontal and slope one saddle connection as a sum of the σ i .
For each integer j = 0 we de ne γ j to be the closed geodesic which travels within Q + 1 from the midpoint of σ 0 to the midpoint of σ j and then travel back within Q − 1 . See the right side of Figure 6. Let P • 1 = P 1 Σ. Proposition 9. e curves γ j generate H 1 (P • 1 ; Z). Proof. Consider the polygonal region R in R 2 formed by gluing together Q + 1 and Q − 1 (not including vertices) along the interior of the edge σ 0 . e region R is simply connected. To form P • 1 we glue along the interiors of edges σ j for j = 0. us the fundamental group of P • 1 is generated by curves within R which cross over exactly one of these edges. ese are our γ j curves, and this means that they also generate the abelianization of the fundamental group H 1 (P • 1 ; Z).

Deformation and holonomy.
e surfaces P c for c ≥ 1 are all homeomorphic by homeomorphisms P c → P c respecting the decompositions P c = Q + c ∪ Q − c and P c = Q + c ∪ Q − c and sending the orbit n → T n c (0, 0) of vertices of Q + c to the orbit n → T n c (0, 0) of vertices of Q + c . is uniquely characterizes the homeomorphism up to isotopy. e paper [Hoo14] noted that the surfaces P c for c ≥ 1 are all naturally homeomorphic, and proved that the surfaces have the same geodesics in a coding sense and have a ne automorphism groups which act in the same way (up to the natural homeomorphism on the surfaces).
For each c ≥ 1 we have notion of holonomy on P c . Using the canonical homeomorphism P 1 → P c we can evaluate this holonomy on classes in P 1 giving us a family of holonomy maps hol c : H 1 (P 1 , Σ, Z) → R 2 de ned for c ≥ 1.
Observe that T c has determinant one and entries which are polynomial in c. us all vertices of Q + c have coordinates which are integer polynomials in c. It follows that for any σ ∈ H 1 (P 1 , Σ; Z) the map c → hol c σ lies in Z[c] 2 . We de ne the deformation holonomy map of the family of surfaces P c to be the map hol : so that for c ≥ 1 we have hol (σ)(c) = hol c (σ).
By tensoring with R we extend hol to We have the following: Proposition 10. e map hol of (20) is an isomorphism of R-modules.
Proof. For d ≥ 0 let P d denote the collection of polynomials of degree at most d with coe cients in R. Because we have normalized three vertices joined by σ −1 and σ 0 in our construction of Q + c we can see that (21) hol σ 0 = (1, 1) and hol σ −1 = (1, −1).
Both these vectors lie in P 2 0 . Recall that T c carries each vertex of Q + c to the subsequent vertex. From the de nition of T c in (19) we see that for all j ∈ Z, Just considering the cs appearing in these matrices we can prove inductively that for k > 0, hol σ k ∈ (2c) k , (2c) k + P 2 k−1 and hol σ −k−1 ∈ (2c) k , −(2c) k + P 2 k−1 . To prove the proposition, it su ces to show that for every n ≥ 0, the restriction of hol to span R {σ −n−1 , σ −n , . . . , σ n−1 , σ n } is an isomorphism to P 2 n . Noting that there are 2n + 2 vectors in the list which matches the dimension of the space P 2 n , we see it su ces to prove that hol span R {σ −n−1 , σ −n , . . . , σ n−1 , σ n } ⊃ P 2 n . We prove this by induction. From 21 we see that the statement is true when n = 0. Now assuming the statement holds for n = k − 1, i.e., hol span R {σ −k−2 , . . . , σ k−1 } ⊃ P 2 k−1 , we see from (22) that by adding σ k and σ −k−1 to the list, the image of the new span contains P 2 k . Now consider the related map To understand it, consider the exact sequence of groups is is the standard relative homology long exact sequence in our se ing, where we have used the observation that H * (P 1 ; R) ∼ = H * (P • 1 ; R) since the two singularities are isolated and of innite cone type guaranteeing that P 1 is homotopy equivalent to P • 1 . e holonomy map factors through the inclusion ι * , i.e., hol • ι * = hol and the above exact sequence says that ι * is an inclusion so we see that the holonomy map (23) is an isomorphism to a subspace of R[c] 2 . To gure out what that subspace is observe that ι * H 1 (P • 1 ; R) = ker ∂ * . We can enumerate the two singularities as s 0 representing the identi ed vertices {(n, n 2 )} of the polygonal parabola with even coordinates and s 1 representing the identi ed vertices with odd coordinates. e image of δ * is the collection elements of H 0 (Σ; R) of the form x[s 1 ] − x[s 0 ] for some x ∈ R. We can recover this value of x as the image of H 1 (P 1 , Σ; R) using the map which allows us to de ne the short exact sequence Proposition 11. For σ ∈ hol (P 1 , Σ; R) and (x, y) = hol (σ) the quantity y(−1) gives the value of (σ). us we have the isomorphism Since the map sending σ to y(−1) is linear, it su ces to check that (σ) agrees with y(−1) on our basis σ j for hol (P 1 , Σ; R). Observe that (σ j ) = (−1) j . Using formulas from the proof of Proposition 10 we can compute that hol −1 (σ j ) = (−1) j (2j + 1, 1), and note that the y-coordinate matches (σ j ).
We will now prove the integral formula mentioned in the introduction.
Lemma 12 (Intersection as integration). e algebraic intersection number of any γ ∈ H 1 (P • 1 ; R) and any σ ∈ H 1 (P 1 , Σ; R) is given by 1 2π where the wedge product in the integral yields an element of Z[c] interpreted as a function of t by se ing c = cos t.
Proof. Observe that the integral expression is bilinear in γ and σ. Algebraic intersection number is bilinear as well, so it su ces to check the formula on our generating sets for H 1 (P • 1 ; Z) and H 1 (P 1 , Σ; Z). Observe that γ j (oriented to move from σ 1 to σ j in Q + 1 ) intersects σ j with positive sign and σ 0 with negative sign. us γ j ∩ σ k = δ j,k − δ 0,k where δ a,b equals 1 if a = b is zero otherwise. We will show that the integral evaluates to the same expression.
First we need to compute hol γ j and hol σ k . Let v n = T n c (0, 0) ∈ Z[c] 2 be the n-th vertex of Q + c viewed as a polynomial in c. Observe Using the fact that T c = C t •R t •C −1 t and the de nition of C t in (18), we see v n = C t (cos nt, sin nt). erefore (24) hol γ j = C t cos (j + 1)t + cos(jt) − cos(t) − 1, sin (j + 1)t + sin(jt) − sin(t) , hol σ k = C t cos (k + 1)t − cos(kt), sin (k + 1)t − sin(kt) . e transformation C t is a ne and scales signed area by a multiplicative constant. Namely, Le ing v = C −1 t ( hol γ j ) and w = C −1 t ( hol σ k ) be the quantities in parenthesis enclosed in (24), we see that the quantity being integrated is v ∧ w sin t = 2 cos (k − j)t − 2 cos(kt), where we have done signi cant simplifying using trigonometric identities. It follows that as desired.
For our later discussion of geometric intersection numbers, it will be useful for us to extend our notion of algebraic intersection number to a bilinear map is extension has geometric meaning for saddle connections.
Lemma 13. Let σ 1 and σ 2 be saddle connections and let i(σ 1 , σ 2 ) denote the number of (unsigned) intersections of σ 1 and σ 2 not counting those that occur at the singularities. en Proof. ere two cases to consider. First suppose one of the curves joins a singularity to itself, say σ 1 has two endpoints at the singularity s * . en σ 1 is really a loop and we can apply a homotopy only deforming σ 1 in a small neighborhood of the s * which makes σ 1 into a new closed loopσ 1 . We will detail a way to obtainσ 1 so that it is fairly easy to see what is happening. For r ∈ (0, √ 2) observe that the ball of radius r about s * contains no complete saddle connections. By choosing r ∈ (0, √ 2) su ciently small we can arrange that the ball B r (s * ) only intersects σ 1 and σ 2 in segments of length r at the start and end of the saddle connections. ( e ball will only intersect σ 2 if it starts or ends at the same singularity.) Chop o the segments of σ 1 that are within the ball. Since this singularity is an in nite cone singularity, the boundary ∂B r (s * ) is homeomorphic to the real line, so we can join the points of where σ 1 hits the ball by a unique arc ∂B r (s * ). Call the resulting loopσ 1 . e added arc may cross σ 2 if the saddle connection σ 2 starts or ends at s * . If σ 2 has one endpoint at s * , we have introduced at most one new crossing so that the result holds in this case. If σ 2 both starts and ends at s * , then the added arc may cross twice, but if it does then the signs ascribed to the intersections are opposite. Again we have shown that |σ 1 ∩ σ 2 | is within one of i(σ 1 , σ 2 ).
If we can not arrange to be in the rst case, then both σ 1 and σ 2 join distinct singularities. Orientation is irrelevant for the statement we are trying to prove, so we can assume σ 1 moves from singularity s 0 to singularity s 1 and σ 2 moves from s 1 to s 0 . Let γ 1 be the curve formed by concatenating σ 1 and σ 2 . Because our extended de nition of ∩ is bilinear and alternating, we have σ 1 ∩ σ 2 = γ 1 ∩ σ 2 . We can again make γ 1 into a closed curveγ 1 following the method of the previous paragraph.
is time we choose r small enough so that the two balls B r (s 0 ) and B r (s 1 ) do not intersect and so that the balls only intersect the saddle connections in initial and terminal segments. Soγ 1 follows σ 1 outside of the two balls then wraps around the boundary of B r (s 1 ), then follows a σ 2 until it hits B r (s 1 ) and closes up following the boundary of B r (s 0 ). Actually, it is be er not to follow σ 2 exactly, instead we follow a parallel arc which stays on one side of σ 2 . Again the arcs in the boundary of the balls may have introduced one or two new crossings, but if we introduce two then they occur with opposite signs. Again we have shown that |σ 1 ∩ σ 2 | is within one of i(σ 1 , σ 2 ).
3.4. A ne automorphisms. e a ne automorphism group was investigated carefully in [Hoo14].
eorem 14 ( eorem 3 [Hoo14]). For c ≥ 1, the group D Aff (P c ) is generated by A c , B c , C c and −I as de ned in (5).
As a consequence we see that for each c ≥ 1, there is a canonical isomorphism where G and ρ c are de ned as in (6) and (8) respectively. e actions are essentially the same in a topological sense: eorem 16 ( eorem 7 [Hoo14]). For c, c ≥ 1 and any g ∈ G, the automorphisms Φ c (g) : P c → P c and Φ c (g) : P c → P c are the same up to conjugation by the canonical homomorphism P c → P c and isotopy.
is has the following consequence for our deformation holonomy map.
Proof. Fixing a c ≥ 1, we see that hol c • Φ c (g) = ρ c (g) · hol c (γ) since D • Φ c (g) = ρ c (g). is expression represents the equation in the proposition evaluated at a speci c c ≥ 1, but we have veri ed it for uncountably many values (all c ≥ 1). Fixing any γ, the expression claims equality of two elements of Z[c] 2 . e entries are polynomial and we have veri ed the equation on in nitely many values, so the equation holds for all c.
3.5. Asymptotics of algebraic intersections. In this subsection we state our main result involving asymptotic algebraic intersection numbers of homology classes. e main ideas involve using hol to convert the homology classes to elements of R[c] 2 , use Proposition 17 to convert an a ne automorphism's action to the action of an element ρ(g) ∈ SL ± (2, Z[c]), and to use our integral formula to evaluate the intersection numbers.
It is useful to notice that since ρ 1 (g) is hyperbolic, the matrices ρ c (g) are diagonalizable in a neighborhood of c = 1. We can express this diagonalization in the form (26) ρ c (g) = λ u c P u c + λ s c P s c where λ u c and λ s c are analytic real valued functions of c corresponding to the expanding and contracting eigenvalues of ρ c (g) and P u c denote the projection matrices (matrices with one eigenvalue equal one and one equal zero) with the same eigenvectors as ρ c (g). Again we just interpret these quantities as de ned and analytic in a neighborhood of c = 1.
Let v, w ∈ R[c] 2 . We de ne (27) v w = (P u c v) ∧ w which is a real valued analytic function de ned in a neighborhood of c = 1. is quantity relates to constants in our asymptotics and it will be important to know that this function can not be identically zero.
Lemma 18. Let ρ 1 (g) be hyperbolic and de ne P u c and as above. en so long as v, w ∈ R[c] 2 are both non-zero, the real analytic function v w is not identically zero.
Proof. First observe that P u c + P s c = I so that v w = (P u c v) ∧ (P u c w + P s c w) = (P u c v) ∧ (P s c w). Since the functions are analytic and the eigendirections are transverse, the only way v w could be identically zero is if either P u c v was identically zero or P s c w was identically zero. We will approach this by contradiction and without loss of generality we may assume P u c v ≡ 0. is says that v(c) lies in the stable eigenspace of ρ c (g) for all c su ciently close to one. at is, us it follows that λ s c is a rational function, i.e., there are polynomials p, q ∈ R[c] so that λ s c = p q in a neighborhood of c = 1 wherever q(c) = 0 (which holds an open set since v has non-zero polynomial entries). Furthermore, we can assume that p q is reduced in the sense that p and q share no roots in C. Since det ρ c (g) ≡ ±1, we know that λ u c = ± q p . e sum of the eigenvalues is the trace of ρ c (g) which we will denote by t ∈ R[c]. us we have the identity (28) p 2 ± q 2 = tpq.
By Lemma 5, we know d dc λ u c | c=1 = 0, so that at least one of p and q is non-constant. is means that one of those, say p, two has a root z ∈ C. en z is also a root of tpq. us it follows from (28) that z must be a root of q 2 and therefore also of q, but this contradicts our assumption that p and q do not share a common root. e following is our main technical result. It describes the asymptotics of homology classes under a hyperbolic a ne automorphism. eorem 19. Suppose φ : P 1 → P 1 is hyperbolic a ne automorphism with derivative Dφ = ρ 1 (g). Let γ ∈ H 1 (P • 1 ; R) and σ ∈ H 1 (P 1 , Σ; R) be non-zero classes. De ne quantities as above in (26) and (27). Let k ≥ 0 be an integer and κ ∈ R be nonzero so that the Taylor expansion of hol γ hol σ about c = 1 is of the form (Note that this quantity is not identically zero by Lemma 18.) en the sequence of algebraic intersection numbers φ n * (γ) ∩ σ is asymptotic to a constant times n −k− 3 2 (λ u 1 ) n , and in fact: where Γ denotes the gamma function and β = 1 as in eorem 1 of the introduction. is result also holds for γ, σ ∈ H 1 (P 1 , Σ; R) with algebraic intersection numbers computed as in (25).
Proof. Fix γ and σ. We will compute intersections using the integral in Lemma 5 with this being the de nition in the case γ, σ ∈ H 1 (P 1 , Σ; R). Let c = cos t throughout this proof. Let v c = hol γ and w c = hol σ which are both elements of R[c] 2 . Determine k and κ as stated in the theorem. By our integral formula for intersection numbers and Proposition 17, Here, the quantity (ρ c (g) n v c ) ∧ w c lies in R[c] and we integrate with respect to t while taking c = cos t.
We will be demonstrating that is asymptotic to the quantity on the right side of the equation in the theorem. Let λ u c be an eigenvalue for ρ c (g) realizing the maximum absolute value of an eigenvalue. From Lemma 5 we know that d dc |λ u c | c=1 > 0 and thus we can nd an interval [0, ] on which t → |λ u cos t | is decreasing and takes values larger than one. We can split the integral of (29) into two pieces at . e contribution of the interval [ , π] to (29) can be wri en as where ξ is some quantity so that 1 < ξ < |λ u 1 | so that ξ −n ρ c (g) n tends to zero uniformly for t ∈ [ , π] as obtainable from Proposition 6. Observe that the fraction in front of (30) decays to zero because of the exponential growth of λ u 1 ξ n . e integral in (30) also tends to zero because v c , w c and 1−cos t are continuous in t and therefore bounded in absolute value by a constant. On the other hand, from our use of Proposition 6, the quantity ξ −n ρ c (g) n decays to zero uniformly. Now consider the interval [0, ]. As in (26), on this interval we can write ρ c (g) n = (λ u c ) n P u c + (λ s c ) n P s c for n ≥ 0.
us we can split the contribution of [0, ] to (29) into unstable and stable parts. e stable part has the form n k+ 3 which tends to zero exponentially since |λ s c | = |λ u c | −1 < 1 so that the integral is uniformly bounded while the fraction in front decays exponentially. e unstable part is more interesting and using the fact that λ u c has constant sign for t ∈ [0, ] we can write it as where is used as in (27). We will use a theorem of Erdélyi following [Olv14,Ch. 3 §8] to study the asymptotics of the sequence of integrals (temporarily ignoring the fraction in front). e integral can be wri en in the form 0 e −np(t) q(t) dt where p(t) = − ln |λ u c | and q(t) = (v c w c )(1 − cos t).
To apply Erdélyi's theorem we need to know the rst few terms of a series expansion for p(t) and q(t) about t = 0. By the chain rule, in a neighborhood of t = 0 we have us we have p(0) = − ln |λ u 1 |, p (0) = 0, and p (0) = 1 which is positive by statement (2) of Lemma 5. In addition, we know that t = 0 is the location of the minimum of p(t) on [0, ]. Recalling that a Taylor series expansion for v c w c = hol γ hol σ in terms of c was given as a hypothesis of the eorem, we have . By [Olv14,eorem 8.1], the sequence of values of integrals has the asymptotic form Plugging these values in for the integral in (31), we obtain the limit stated in the theorem.
Proof of eorem 4. It su ces to prove the second statement, since the statement on convergence of projective classes to P (µ u ) and P (µ s ) follows from statement (3) in the theorem. Fix φ and notation as above. Given a non-zero γ ∈ H 1 (S, Σ; R) observe that P u c hol γ has coordinates which are real analytic functions of c. As a consequence of Lemma 18, P u c hol γ is not identically zero. en by analyticity, the following constant is de ned e value of k can be interpreted as a function k : H 1 (S, Σ; R) → Z ∪ {+∞} where we assign k(γ) as in (32) except if γ = 0 in which case we assign +∞. en we can de ne the subsets in the theorem to be S j = {γ : k(γ) ≥ j}. Fixing k we observe that the map D k : γ → d k dc k P u c hol γ is linear and consequently the S j are subspaces and each S j+1 is codimension at most one in S j . Recalling Proposition 11, we see that we can construct pairs of polynomials (p, q) in the image of hol H 1 (S, Σ; R) so that the minimal k ≥ 0 such that [ d k dc k P u c (p, q)] c=1 = 0 is arbitrary: Such p(c), q(c) need to approximate the stable direction of ρ c (g) at c = 1 to order k − 1 but not order k. It follows that S j+1 is always codimension one in S j . Finally because of the previous paragraph we know that if γ = 0 then k(γ) = 0 so that j≥0 S j = {0}. We have proved statements (1) and (2) of the theorem.
To see (3) x a non-zero γ and let k be as in (32). We need to show that there is a constant L so that for any σ ∈ H 1 (S, Σ; R), lim n→+∞ n k+ 3 2 (λ u 1 ) n φ n (γ) ∩ σ = Lµ u (σ), and by de nition µ u (σ) = u u 1 ∧ hol 1 (σ) where u u denotes a choice of a unit unstable eigenvector of ρ 1 (g). It su ces to prove this for elements of the form σ l which generate H 1 (S; Σ; R) by Proposition 9. eorem 19 tells us that c denote the continuous choice of unstable unit eigenvectors of ρ c (g) de ned in a neighborhood of c = 1, which extends the choice of u u 1 used in the de nition of µ u . en we can de ne the real-analytic function f (c) so that Here f (c) = ± P u c hol γ and may be changing signs as c passes through 1. To compute κ we just need to look at the lowest order terms in this expression computed at the point c = 1. Observe that hol 1 σ l ∈ Z 2 while u u 1 has quadratic irrational slope so we know that u u 1 ∧ hol 1 σ l is a nonzero constant. On the other hand using our de nition of k we know that hol γ hol σ l vanishes to order k − 1 and so we must have f = α(c − 1) k + O((c − 1) k+1 ) and with this de nition of α we have κ = α(u u 1 ∧ hol 1 σ l ) = αµ u (σ l ) so we see by plugging in to (33) that e constant in front of µ u (σ l ) is now independent of l so we see that 3.6. Asymptotics of cylinder intersections. e goal of this section is to prove eorem 1 and Corollary 2 of the introduction.
We rst need to make a general remark about cylinders intersection on a translation surface. Let C be a cylinder on a translation surface S. A core curve γ C of C is a closed geodesic in C. Such closed geodesic must wind once around the circumference of C.
Proposition 20. Let A and B be cylinders on a translation surface S with core curves γ A and γ B . Assuming A and B are not parallel (i.e., hol γ A ∧ hol γ B = 0), we have where γ A ∩ γ B denotes algebraic intersection number.
Proof. Cylinders on a translation surface intersect in a union of parallelograms, which are isometric and di er only by parallel translation. e number of these parallelograms is the absolute value of the algebraic intersection number between the core curves. us, we need to show that the area of one such parallelogram is given by us, the quotient given in (35) is |a∧b|, the area of the parallelogram formed by the intersection.
Proof of eorem 1. Let A and B be cylinders on P 1 and let γ A and γ B be their core curves and let v, w ∈ Z 2 be their holonomies. Let φ be a hyperbolic automorphism of P 1 . en Dφ = ρ 1 (g) is a hyperbolic element of SL ± (2, Z). As a consequence the eigenvalues of Dφ are quadratic irrationals and so (P u 1 v) ∧ w is non-zero. In the context of eorem 19 this implies that k = 0 and κ = (P u 1 v) ∧ w and so the theorem gives that For su ciently large values of n, the cylinders φ n (A) and B are not parallel, so we may apply Proposition 20 to obtain that e quantity |(λ u 1 ) −n ρ 1 (g) n v∧w| converges to |κ|, so this gives us the expression in the statement of the theorem.
Proof of Corollary 2. Fix φ : P 1 → P 1 hyperbolic. We will explain how to produce the wandering sets W i,k so that P 1 i,k W i,k has zero measure. Let A i be the i-th horizontal cylinder as depicted in Figure 3. By eorem 1 applied to φ −1 we have is means that the set of points in A i which return in nitely o en to A i has measure zero. For k ≥ 0, let W i,k to be the set of points in A i which return exactly k times to A i under the action of φ −1 . From the remarks above we see that A i k≥0 W i,k is measure zero. Since P 1 = i A i we see that W = {W i,k } satis es the statements in Corollary 2.
3.7. Geometric intersection numbers. e space P 1 is non-positively curved in the sense that its universal cover is CAT (0). is guarantees that all homotopically non-trivial closed curves have geodesic representatives in the metric sense. In particular every such curve has a realization as a sequence of saddle connections σ 1 , . . . , σ k so that the endpoint singularity of σ i coincides with the start singularity of σ i+1 (mod k) and the angle made between the two saddle connections at this singularity is at least π. A closed metric geodesic may also be a closed non-singular straightline trajectory, but by moving to the boundary of the corresponding cylinder we can nd a metric geodesic representative of this homotopy class consisting of a sequence of saddle connections.
We will now describe a way to compute the geometric intersection numbers between two nontrivial homotopy classes of closed curves in the punctured surface P • 1 . First we may nd metric geodesic representatives α = α 1 ∪ . . . ∪ α k and γ = γ 1 ∪ . . . ∪ γ l for the curves in P 1 where the α i and γ j are saddle connections. ere are two types of intersections between α and γ: those that occur at singularities and those that do not. e unit tangent bundle space T 1 s * at an in nite cone singularity s * is naturally homeomorphic to a line and we can make it a metric line using angle coordinates, identifying it with the universal cover of R/2πZ. Two saddle connections meeting at a singularity s * thus determine an interval I in T 1 s * . Our metric geodesics α and γ thus determine two sequences of intervals I 1 , . . . , I k and J 1 , . . . J l in the pair of lines T 1 s 0 ∪ T 1 s 1 . We say two intervals I and J are linked if I contains an endpoint of J and J contains an endpoint of I. Proof. We will show that we can nd representatives of the classes α and γ which realize the claimed intersection number. en we will argue that the representatives realize the intersection number.
ere is an r > 0 so that the open balls B r (s 0 ) and B r (s 1 ) do not intersect and only intersect the saddle connections in the set {α i }∪{γ j } in segments of length r where the saddle connections enter and exit the singularities. Let H be the hyperbolic upper half plane. Because the singularities are in nite cone singularities, the balls are homeomorphic to H ∪ {∞} via a sending radial lines in the ball to vertical lines in H and sending the singularity to ∞. Consecutive arcs of saddle connections of α and γ intersecting the balls B r (s * ) are then sent to a vertical geodesic headed to up in nity followed by a vertical geodesic back down to the real line in ∂H. We can straighten such an arcs to a geodesic in H joining the places where the two arcs pass through the real axis. Do this for all the visits of the saddle connections to the singularities. Observe that two geodesics with distinct endpoints in H joining points in R intersect if and only if the corresponding intervals in R link. Transversality guarantees this distinct endpoint condition for the geodesics constructed as above. us performing this action results in a pair of curvesα andγ that intersect in precisely the number of times stated in the theorem.
We must argue that the number of intersections betweenα andγ is minimal among curves in their homotopy classes. For this it su ces to show that we can not nd a simple closed curve formed from an arc ofα and an arc ofγ which is homotopically trivial; see [FLP12,Proposition 3.10]. Such arcs must join together at a pair of intersections forα andγ. Existence of such a curve is ruled out by the Gauss-Bonnet theorem which promises that for a closed polygon in the plane, the total exterior angle is 2π. Indeed suppose we had two such arcs ofα andγ which formed a simple closed homotopically trivial loopη. enη bounds a topological disk. By deforming back to the at geodesics α and γ we get a curve η that bounds a at polygonal disk. By transversality, the interior angles at the intersections are positive, so the exterior angles at these intersection points are each strictly less than π.
e other vertices of η must come from visits of α or γ to the singularities, and since they are metric geodesics the interior angles are at least π and so the exterior angles are negative. We conclude that the total exterior angle of η is less than 2π, which contradicts the existence of η.
Lemma 13 tells us how to estimate (within 1) the number of interior intersections of saddle connections. Combining this lemma with the above result yields: Proposition 22. Let α = α 1 ∪ . . . ∪ α k and γ = γ 1 ∪ . . . ∪ γ l be transverse closed metric geodesics in P 1 . en the geometric intersection number i(α, γ) is within 2kl of Note that the identity in the equation above holds by de nition see (25).
Proof. We get an error of up to kl from possible linked intervals in Proposition 21 and another error of up to one from comparing each integral to the number of interior intersections of the corresponding saddle connections (see Lemma 13). Now consider an a ne automorphism φ : P 1 → P 1 . Let α = α 1 ∪ . . . ∪ α k be a closed metric geodesic on P 1 . Observe that φ(α) = φ(α 1 ) ∪ . . . ∪ φ(α k ) is also a closed metric geodesic because this image satis es the angle condition on consecutive saddle connections. As a consequence we see that when considering the geometric intersection number i φ n (α), γ we can use the integral formula above and the only e ect is that we introduce a uniformly bounded error.
Fix φ : P 1 → P 1 with hyperbolic derivative Dφ. Let u u and u s be unstable and stable unit eigenvectors for Dφ.
en the stable and unstable elements µ u , µ s ∈ R S(P • 1 ) corresponding to the transverse measures on P 1 to foliations parallel to the expanding and contracting directions are de ned by evaluating them on a simple closed curve with metric geodesic representative α = α 1 ∪ . . . ∪ α k by Proof of eorem 3. Statement (1) is standard: In the unstable direction we have . A similar argument works in the stable direction. Now x a homotopically non-trivial simple closed curve and let α = α 1 ∪ . . . ∪ α k be a metric geodesic representative. We will prove that n 3 2 |λ u 1 | n i * • φ n (α) converges to a constant times µ s with the constant as given in the theorem. To prove this let γ be another homotopically non-trivial simple closed curve and let γ = γ 1 ∪. . .∪γ k be a metric geodesic representative. Fixing an i and a j, and observe that hol 1 α i and hol 1 γ j are non-zero vectors in Z 2 so that (P 1 u hol 1 α i ) ∧ hol 1 γ k = 0. is tells us that k = 0 in eorem 19 and |κ| = |(P 1 u hol 1 α i ) ∧ hol 1 γ k | = u s ∧ hol 1 α i u s ∧ u u u u ∧ hol 1 γ k = µ s (α i )µ u (γ j ) |u u ∧ u s | , where we are slightly abusing notation by applying (36) to saddle connections (but this is justi ed if we think of these functions as determining measured foliations.) en from eorem 19 we get is is equivalent to the rst limiting statement in statement (2) of the theorem. e second limit can be obtained by switching φ for φ −1 and stable for unstable.

S
So ware was used in several ways in this paper. SageMath [ e18] and the FlatSurf SageMath Module [DH] were used to experimentally check the results in this paper. FlatSurf was also used to generate the gures of translation surfaces.
A eorem 1 was rst proved while the author was a postdoc at Northwestern and the author would like to thank John Franks and Amie Wilkinson for helpful conversations at the time. e author would also like to thank Barak Weiss for some more recent conversations, and the anonymous referee for suggesting a number of improvements.
is article is based upon work supported by the National Science Foundation under Grant Number DMS-1500965 as well as a PSC-CUNY Award (funded by e Professional Sta Congress and e City University of New York).