Horospherical dynamics in invariant subvarieties

We consider the horospherical foliation on any invariant subvariety in the moduli space of translation surfaces. This foliation can be described dynamically as the strong unstable foliation for the geodesic flow on the invariant subvariety, and geometrically, it is induced by the canonical splitting of $\mathbb{C}$-valued cohomology into its real and imaginary parts. We define a natural volume form on the leaves of this foliation, and define horospherical measures as those measures whose conditional measures on leaves are given by the volume form. We show that the natural measures on invariant subvarieties, and in particular, the Masur-Veech measures on strata, are horospherical. We show that these measures are the unique horospherical measures giving zero mass to the set of surfaces with no horizontal saddle connections, extending work of Lindenstrauss-Mirzakhani and Hamenstaedt for principal strata. We describe all the leaf closures for the horospherical foliation.


Introduction
It is an interesting fact that geometric questions about rational polygonal billiards can be addressed by studying the dynamics on moduli spaces of translation surfaces.This is one of many reasons to study the dynamics on moduli spaces of translation surfaces -see the surveys [MT02,Zor06,FM14,Wri15b] for other motivation and a survey of results.We remind the reader that this moduli space is partitioned into strata, which correspond to translation surfaces of a fixed topological type.The group G def = SL 2 (R) acts on each stratum.The horocycle flow is given by The analogy between dynamics on strata and homogeneous dynamics has been fruitful.In the setting of homogeneous dynamics U -actions and G-actions were analyzed in work of Ratner which showed that orbit closures and ergodic invariant probability measures are surprisingly well-behaved.The dynamics of G-actions (and moreover the dynamics of its subgroup P of upper triangular matrices) on strata were analyzed in two papers [EM18,EMM15] where it was shown that orbit closures and ergodic invariant measures have nice descriptions (see Section 2.2 for a precise statement).The situation for the U -action on the strata of the moduli spaces is now known to be more complicated due to the work of Chaika-Smillie-Weiss [CSW20].
The G-orbit closures are endowed with a wealth of geometrical structures, among which is the horospherical foliation which plays the role of the strong unstable manifold foliation for the one parameter diagonal subgroup which is called the geodesic flow (see §3.2).In §3 we will define horospherical measures.Loosely speaking, the horospherical leaves are endowed with affine structures and the horospherical measures are those for which the conditional measures on theses leaves are translation invariant with respect to these affine structures.In the setting of homogeneous dynamics, there is a corresponding notion of horospherical dynamics.It has been established by Dani in [Dan78] and [Dan81] about a decade prior to the work of Ratner that these dynamical systems are also well-behaved.This paper is concerned with showing that horospherical measures and horospherical leaves in strata are also well-behaved.
1.1.Statement of results.All measures considered in this paper are Borel regular Radon measures on strata of translation surfaces.Any G-orbit closure M (1) ⊂ H (1) supports a unique ergodic G-invariant finite smooth measure; we will refer to this measure as the special flat measure on M (1) .The following are the main results of this paper.
Theorem 1.1.The special flat measure on any G-orbit closure is horospherical.
We will say that a measure µ is saddle connection free if µ-a.e.surface has no horizontal saddle connections.
Theorem 1.2.Up to scaling, the only saddle connection free horospherical measure on a G-orbit closure is the special flat measure.
We emphasize that horospherical measures are a a priori not assumed to be finite.It is thus a consequence of Theorem 1.2 that horospherical measures are finite under the saddle connection free assumption; it seems likely, but we were not able to prove, that all horospherical ergodic measures are finite.Theorem 1.2 was announced without proof in [BSW22, Claim 1, §9].The saddle connection free assumption cannot be removed; for example, the length measure on a periodic horocycle trajectory in a closed G-orbit is horospherical.In §5 we will give more interesting examples of invariant subvarieties and horospherical measures on them, which are not the special flat measure.We will also classify (see §5.1) all the horospherical measures on the simplest nontrivial invariant subvarieties, namely the eigenform loci in H(1, 1).
If a surface has a horizontal cylinder then so does any surface on its horospherical leaf.We will say that a leaf of the horospherical foliation is cylinder-free if all surfaces on the leaf have no horizontal cylinders.We say that a measure µ on M is cylinder-free if µ-a.e.surface has no horizontal cylinders.In §5 we give examples of horospherical measures which are not special flat and for which almost every point has a horizontal saddle connection.For these measures it is also the case that almost every point has a cylinder.It seems likely that this is always the case; or in other words, that in Theorem 1.2 the condition 'saddle connection free' can be weakened to 'cylinder-free'.The analogous assertion about orbit closures is true: Theorem 1.3.Any cylinder-free leaf for the horospherical foliation of a G-orbit closure is dense in that G-orbit closure.
The proof of Theorem 1.3 uses a statement of independent interest (Theorem 7.2), about extending horizontal saddle connections while staying inside invariant suborbifolds.This result was explained to us by Paul Apisa and Alex Wright, and its proof is given in Appendix A.
The geodesic flow is the restriction of the G-action to the subgroup (1) A def = {g t : t ∈ R} ⊂ G, where g t def = e t 0 0 e −t .
Answering a question of Forni, we prove: Theorem 1.4.For any finite horospherical measure µ on M, the pushforward measures, g t * µ, converge to the special flat measure on M, with respect to the weak- * topology, as t → +∞.
Related results are proved in [For21]; we stress however that the notion of 'horospherical measure' used in [For21] is different from the one we use here.From a dynamical perspective, the horospherical foliation is the strong unstable foliation for the geodesic flow.Our arguments yield a simpler proof of the following theorem.
Theorem 1.5 ( [EM18,EMM15]).The special flat measure is the unique A-invariant horospherical measure on any G-orbit closure.Any leaf for the weak-unstable foliation on any G-orbit closure is dense.
Remark 1.6.Note that we do not assume that the measure is finite in Theorem 1.5.If we assumed finiteness, then the first statement would follow immediately from Theorem 1.4.Also note that Theorem 1.4 is false for infinite measures, as the following example shows.Take M = GL + 2 (R)q ≃ GL + 2 (R)/Γ a closed orbit of a Veech surface with Veech group Γ; then a horospherical measure in this case is just a U -invariant measure.For s > 0, let ν s be a normalized length measure on a periodic U -orbit of length s, and let µ = 1.2.Further motivation, prior work, and some ideas from the proofs.The work of Eskin, Mirzakhani and Mohammadi gives a very detailed understanding of invariant measures and sets for the G-action and the P -action on strata of translation surfaces.A central remaining open problem is to understand horocycle invariant ergodic measures.Such an understanding would have an application to the fundamental problem of asymptotic growth of saddle connections on translation surfaces or rational billiards (see [EM01]).As we will see in §3, horospherical measures are horocycle-invariant; thus understanding horospherical measures can be seen as a contribution to the problem of understanding general horocycle-invariant measures.
A previous measure rigidity result for horospherical measures was obtained in 2008, independently by Lindenstrauss and Mirzakhani [LM08] and by Hamenstädt [Ham09].They were interested in understanding mapping class group invariant measures on the space of measured laminations.By a 'duality principle' (see [LM08,§5]) this question is very closely related to the problem of classifying horospherical measures on the principal stratum.
Our argument for Theorem 1.2 follows [LM08], which in turn is inspired by ideas of Dani [Dan78] and Margulis [Mar04] The main ingredients are the mixing of the A-action, the use of dynamical boxes and how they transform under the A-action, and nondivergence results for the U -action (which in the present context were obtained in [MW02]).After the requisite preparations, this argument is given in §4.In order to carry out the details of this argument, we give a precise description of horospherical measures and special flat measures, and their decomposition into conditional measures in flow boxes in §3.Theorem 1.3 is proved in §7.Theorems 1.4 and 1.5 are proved in §6.
1.3.Acknowledgements.We are grateful to Paul Apisa and Alex Wright for providing the proof of Theorem 7.2.The proof is given in Appendix A. We are also grateful to Giovanni Forni for useful comments.We acknowledge support from grants BSF 2016256, ISF 2019/19 and ISF-NSFC 3739/21.

Preliminaries
In this section we introduce our objects of study and set up our notation.There are many approaches to these definitions.In our approach, the linear orbifold structure (or affine orbifold structure) given by period coordinates will be important and we will stress this point of view in what follows.A suitable reference for the theory utilizing this point of view is [BSW22, §2], and unless stated otherwise, our notation, terminology and assumptions are as in [BSW22].See also [MT02,Zor06,FM14,Wri15b].See [Gol] for a general discussion of affine manifolds.
2.1.Strata and period coordinates.Let S be a connected, compact orientable surface of genus g, Σ = {ξ 1 , . . ., ξ k } ⊂ S a finite set, a 1 , . . ., a k non-negative integers with a i = 2g−2, and H = H(a 1 , . . ., a k ) the corresponding stratum of translation surfaces.We let H m = H m (a 1 , . . ., a k ) denote the stratum of marked translation surfaces and π : H m → H the forgetful mapping.It will be useful to assume that singular points are labeled, or equivalently, H = H m /Mod(S, Σ), where Mod(S, Σ) is the group of isotopy classes of orientation-preserving homeomorphisms of S fixing Σ, up to an isotopy fixing Σ.We will typically denote elements of H by the letter q when we want to consider them as points of H, and by the letter M or M q when we want to consider their underlying topological or geometrical properties as spaces in their own right.Points in H m will be typically denoted by boldface letters such as q.
We recall the definition of the map dev : H m → H 1 S, Σ; R 2 .For an oriented path γ in M q which is either closed or has endpoints at singularities, let hol(M q , γ) def = γ dx q , γ dy q , where dx q and dy q are the 1-forms on M q inherited from the the forms dx and dy on the plane.Given q ∈ H m represented by f : S → M q , where M q is a translation surface, we define dev(q The map dev is also known in the literature as the period map.There is an open cover {U τ } of H m , indexed by triangulations τ of S with triangles whose vertices are in Σ, such that the restricted maps are homeomorphisms onto their image.The charts ϕ τ give an atlas with affine overlap maps and endow H m with a structure of affine manifold.This atlas of charts {(U τ , ϕ τ )} is known as the period coordinate atlas.
The Mod(S, Σ)-action on H m is properly discontinuous and affine, and hence H inherits the structure of affine orbifold, and the map π : H m → H is an orbifold covering map.We can associate to any affine manifold a holonomy cover and a developing map.In this case H m is a cover with trivial holonomy and dev plays the role of a developing map of H (see [Gol]).
The group GL + 2 (R) acts on translation surfaces in H and H m by modifying planar charts.It acts on H 1 S, Σ; R 2 via its action on the coefficients R 2 .The GL + 2 (R)action commutes with the Mod(S, Σ)-action, and thus the map π is GL + 2 (R)equivariant for these actions.The GL + 2 (R)-action on H m is free, since dev(gq) = dev(q) for any nontrivial g ∈ GL + 2 (R).We have a coordinate splitting of R 2 and we write R 2 = R x ⊕R y to distinguish the two summands in this splitting.There is a corresponding splitting of cohomology We refer to the summands in this splitting as the horizontal space and vertical space respectively.It can also be useful to identify the coefficients with C and consider H 1 (S, Σ; C).This is the most natural choice when we are considering Abelian differentials.An Rstructure on a complex vector space V is given by a choice of a real subspace W ⊂ V so that V = W ⊕ iW .If V is equipped with an R-structure we say that a complex subspace More generally, if V is a complex vector space with an R-structure, then GL + 2 (R) acts on V , with the matrix a b c d Lemma 2.1.Let V be a complex vector space with an R-structure, and V ′ be a real subspace.The following are equivalent: (1) V ′ is invariant under the action of GL + 2 (R).
Proof.The implication (2) =⇒ (1) is clear from the definitions.We prove (1) implies (2).If V ′ is invariant under GL + 2 (R), then since it is a closed subset of V , it is mapped into itself by any 2-by-2 matrix, invertible or not.Let From the definition of the GL + 2 (R) action, one sees that multiplication by a corresponds to multiplication by i, and multiplication by b and c correspond to projections onto the two summands in (2).Invariance by a implies that V ′ is a complex subspace, and from the relations bV ′ ⊂ V ′ and cV ′ ⊂ V ′ and b + c = Id, we see that V ′ is defined over R.
Remark 2.2.If V has an R-structure, then so does its dual space, so it makes sense to say that a linear function on V is real.A complex subspace of V is defined over R if and only it cut out by real linear functions.We will not use this description in this paper.
We have a restriction map Res : H 1 (S, Σ; R 2 ) → H 1 (S; R 2 ) (given by restricting a cochain to closed paths).Since Res is topologically defined, its kernel ker(Res) is Mod(S, Σ)-invariant.Moreover our convention that singular points are marked implies that the Mod(S, Σ)-action on ker(Res) is trivial.
Define the real REL space For any v ∈ Z the constant vector field on H 1 (S, Σ; R 2 ) in direction v pulls back to a well-defined vector field on H m via the local diffeomorphism dev.Since monodromy acts trivially on Z, this descends to a vector field on H. Integrating this vector field gives a locally defined real REL flow (corresponding to v) (t, q) → Rel tv (q).For every q ∈ H a trajectory is defined for t ∈ I q , where the domain of definition is an open interval of R which contains 0. This interval is all of R if the underlying surface M q has no horizontal saddle connections.If q ∈ H, s ∈ R and t ∈ I q then t ∈ I usq , and Rel tv (u s q) = u s Rel tv (q).The set (4) as well as the sets I q (v), are explicitly described in [BSW22, Thm.6.1].

Invariant subvarieties.
In this subsection, we introduce our notion of invariant subvarieties and irreducible invariant subvarieties.It will be shown in [SY], using the work of Eskin-Mirzakhani [EM18] and Eskin-Mirzakhani-Mohammadi [EMM15], that an irreducible invariant subvariety is exactly a GL + 2 (R)-orbit closure while an invariant subvariety is a finite union of such GL + 2 (R)-orbit closures.Definition 2.3.A d-dimensional linear manifold is a submanifold L of H m which is a connected component of dev −1 (V ) where V is a d-dimensional complex subspace of H 1 S, Σ; R 2 defined over R.
Since the developing map is equivariant and Mod(S, Σ) acts linearly on the space H 1 S, Σ; R 2 , it follows that Mod(S, Σ) takes a d-dimensional linear manifold to a d-dimensional linear manifold.If L is a linear manifold corresponding to V L ⊂ H 1 S, Σ; R 2 , we denote by Γ L be the subgroup of Mod(S, Σ) that preserves L. Since the developing map dev is Mod(S, Σ)-equivariant, we get an induced action of Γ L on V L .We say that L is an equilinear manifold if furthermore we have det (γ| VL ) = ±1 for every γ ∈ Γ L .
Definition 2.4.A d-dimensional invariant subvariety is a subset M ⊂ H such that π −1 (M) is a locally finite union of d-dimensional equilinear manifolds.
We will write d = dim(M); in some texts this is referred to as the complex dimension of M. The term "invariant" in the definition of invariant subvariety is justified by the following: Proposition 2.5.An invariant subvariety is closed and The set π −1 (M) is closed because it is a locally finite union of closed sets, and this implies that M is closed.
Since π is GL + 2 (R)-equivariant, it is enough to prove that π −1 (M) is GL + 2 (R)invariant.Let L be a linear submanifold contained in π −1 (M) which maps to V L under dev.By definition, V L is defined over R and by Lemma 2.1 it is invariant under the action of GL + 2 (R) on H 1 S, Σ; R 2 .Since dev is GL + 2 (R)-equivariant the action of GL + 2 (R) on H m preserves dev −1 (V L ).Since GL + 2 (R) is connected, the action of GL + 2 (R) on H m preserves L. Since π −1 (M) is a union of linear submanifolds it follows that it is invariant under GL + 2 (R).Definition 2.6.A d-dimensional invariant subvariety is said to be irreducible if it cannot be written as a union of two proper distinct d-dimensional invariant subvarieties.
We have the following equivalent characterization: Proposition 2.7.Let M be a d-dimensional invariant subvariety.Then M is irreducible if and only if for any d-dimensional equilinear manifold L ⊂ π −1 (M), we have For the proof of Proposition 2.7 we will need the following: Lemma 2.8.If L and L ′ are distinct d-dimensional linear submanifolds, then π(L) ∩ π(L ′ ) is a meager subset of π(L) and of π(L ′ ).
Proof.We first show that π −1 (π(L) ∩ π(L ′ )) is a countable union of sets of dimension less than d.We have: . This is a complex subspace of positive codimension so its inverse image is a nowhere dense subset of the d- Proof of Proposition 2.7.Say that M is irreducible and let L be a d-dimensional equilinear manifold in π −1 (M).If (5) does not hold, we can write π −1 (M) as a countable union of orbits of distinct linear submanifolds L 1 , L 2 , . .., as where L = L 1 and the list {L i } contains more than one element.We have We define Since M is irreducible, and since we have assumed that (5) fails, we have M = B.This implies A ⊂ B, and hence π(L 1 ) = ℓ π(L 1 )∩π(L ℓ ).According to Lemma 2.8 π(L 1 ) ∩ π(L ℓ ) is a meager subset of π(L 1 ) so our decomposition of π(L 1 ) expresses π(L 1 ) as a meager set and violates the Baire category theorem.We conclude that M = A which is what we wanted to show.Now assume that for any d-dimensional equilinear manifold L ⊂ π −1 (M) we have (5).Suppose we have a decomposition M = A ∪ B where where both collections {L i }, {L ′ k } are Mod(S, Σ)-invariant and comprised of ddimensional equilinear manifolds.By (5), π −1 (A) and π −1 (B) are either empty or equal to π −1 (M).Thus A and B are not proper subsets of M.
It follows from Proposition 2.7 that if M is a d-dimensional irreducible invariant subvariety and L is a d-dimensional equilinear manifold contained in π −1 (M), then π(L) = M.This motivates the following definition that will be used throughout the text Definition 2.9.
The following result establishes the link between GL + 2 (R)-orbit closures and invariant subvarieties.In the forthcoming [SY], it will be deduced from the results of [EMM15,EM18].
Convention 2.11.From now on we will make the standing assumption that all the invariant subvarieties we will consider are irreducible.
Let M be a d-dimensional invariant subvariety.We conclude this section by constructing a Radon measure supported on M which will be defined up to a multiplicative constant.This will require some constructions which are summarized in Appendix B. Let L be a lift of M, let V L = dev(L) and let Γ L be the stabilizer in Mod(S, Σ) of L. Let α be a volume form on L that is obtained as the pullback by dev of an element of the top degree exterior power of V L .The group GL + 2 (R) acts smoothly on H m .Denoting by g * the pull-back operator on differential forms corresponding to the action of g ∈ GL + 2 (R) on H m , we have The volume form α defines a measure on L that we denote by µ L .Since L is an equilinear manifold, the measure µ L is Γ L -invariant.Furthermore, since Mod(S, Σ) acts transitively on the set of irreducible components, it can be arranged that for any γ ∈ Mod(S, Σ), γ * m L = m L•γ .This means that the sum μM = where the sum ranges over the lifts of M, is a Mod(S, Σ)-invariant measure on H m .The measure μM is a Radon measure, which follows from the fact that the collection of irreducible components is locally finite.Using Proposition B.3, there is a unique Radon measure µ M on H m such that for any f ∈ C c (H m ), we have where (as in equation ( 44)).The measure µ M is supported on M. It follows from Lemma B.2 that it is SL 2 (R)-invariant.We call it the linear measure on M.
Notice that this is a slight abuse of language as µ M is only determined up to a multiplicative constant.
2.3.Area one locus, cone construction, and special linear measures.Let q ∈ H m , let q = π(q), and let M = M q be the underlying translation surface.
The area of M can be expressed using period coordinates as follows.We define a Hermitian form on H 1 (S, Σ; C) by (See [BSW22, §2.5] for a topological interpretation of equation ( 7).)The area of M is then given by (dev(q), dev(q)).This is thus a quadratic formula in period coordinates.
For the purposes of this paper we will use a related real valued bracket α, β involving the pairing of horizontal and vertical classes.Say that on a marked surface M q we have a 1-form α corresponding to an element of H 1 (S, Σ; R x ) (a horizontal form) and a 1-form β corresponding to an element of H 1 (S, Σ; R y ) (a vertical form).Then α, β = S α ∧ β and this gives (8) area(M q ) = dx q , dy q = S dx q ∧ dy q .
We denote the subset of surfaces in H m and H of area one by H m and H (1) .More generally, when M is an invariant subvariety and L is a lift of M, we also denote by M (1) and L (1) their intersection with the area-one locus.The latter are G-invariant and invariant under real REL flows (where defined).
We recall that there is a rescaling action of R * + on H that corresponds to the action of the subgroup of GL + 2 (R) of scalar matrices with positive coefficients.We consider the cone measure m M on M (1) defined for any Borel subset A ⊂ M (1) by When M is the whole stratum H, the measure m H is called the Masur-Veech measure.More generally, we shall call the measure m M the special flat measure on M. If L is a lift of M, we can perform the same cone construction with the measure µ L and we denote by m L the corresponding measure.Let mM be the pre-image of m M under π, that is the unique measure on H m such that for any f ∈ C c (H m ), 2.4.The sup-norm Finsler metric.We now recall the sup-norm Finsler metric on H m .This structure was studied by Avila, Gouëzel and Yoccoz, for proofs and more details see [AGY06] and [AG10].Let • denote the Euclidean norm on R 2 .For a translation surface q, denote by Λ q the collection of saddle connections on M q and let ℓ q (σ) = hol q (σ) be the length of σ ∈ Λ q .For β ∈ H1 (M q , Σ q ; R 2 ) we set (12) We now define a Finsler metric for H m .Let f : S → M q be a marking map representing a marked surface q ∈ H m .Using period coordinates we can identify the tangent space to H m at q with H 1 (S, Σ; R 2 ).Then (13) is a norm on H 1 (S, Σ; R 2 ).It satisfies the equivariance property where q • h is represented by the marking map f • h.The map is continuous.The Finsler metric defines a distance function 1 on H m which we call the sup-norm distance and define as follows: (15) dist(q 0 , q 1 ) where γ ranges over smooth paths γ : [0, 1] → H with γ(0) = q 0 and γ(1) = q 1 .The topology induced by the sup norm distance on H m is the one induced by period coordinates, and the resulting metric space is proper and complete.We can use the distance function on H m to define a distance function on H by dist(q 0 , q 1 ) = inf{dist(q 0 , q 1 ) :

Horospherical measures
Let M be an invariant subvariety of dimension n.The goal of this section is to define the horospherical foliation on M and the related horospherical measures, which are our object of study in this paper.These objects will be defined via their counterparts for the irreducible components of π −1 (M).
3.1.Boxes.We now define a notion of boxes.They will be used throughout the text and will play two roles: boxes give local coordinates on invariant subvarieties (more precisely, on the irreducible components of their pre-image by π) that are convenient for the study of horospherical measures; additionally, they will be used in a mixing argument in the proof of Theorem 1.2.
From now on, we identify H 1 (S, Σ, C) with H 1 (S, Σ, R 2 ) as in §2.1.Let V ⊂ H 1 (S, Σ, C) be a complex linear subspace defined over R. We have and denote by ( 17) the projections corresponding to the direct sum decomposition (16), and by π ′ x the projection from π −1 x (V x {0}) to the projective space P(V x ).Finally let (18) Ψ : Proof.Say that (x 0 , y 0 ) ∈ V (1) is mapped by Ψ to (x 0 , y 0 ) in P(V x ) × V y .We will construct a local inverse.Since x 0 , y 0 = 1 we can find neighborhoods U x of x 0 in V x and U y of y 0 in V y so that x, y > 0 for x ∈ U x and y ∈ U y .We define maps , y and (20) . The map ψ is smooth and descends in a well-defined way to define ψ.We see that Ψ • ψ is the identity map, i.e., ψ is a local inverse of Ψ. Definition 3.2 (Boxes).Let L be a lift of M and let V = dev(L).A box in L is a relatively compact subset B ⊂ L (1) together with a diffeomorphism ϕ : U ′ x ×U y → B such that, in the notations above, • U ′ x and U y are open sets in P(V x ) and V y respectively.
x × {y}).The composition in the second item in Definition 3.2 makes sense since dev(L (1) ) ⊂ V (1) , in light of equation (8).It should be understood as a choice of a suitable parameterization for boxes.Note that the data ϕ, U ′ x × U y are implicit in the notion of a box, but in order to avoid excessive notation we simply write B.
More generally, a box in π −1 (M) is a box in one of the irreducible components of π −1 (M).Such a box B will be called regular if for any γ ∈ Mod(S, Σ) either B • γ ∩ B = ∅ or γ ∈ Γ, where Γ is the stabilizer in Mod(S, Σ) of B (i.e. the set of γ ∈ Mod(S, Σ) such that B • γ = B).When B is regular, the map π induces a homeomorphism B/Γ → π(B).In particular the image of a regular box by π is an open subset of M. Since Mod(S, Σ) acts diagonally on P(H 1 (S, Σ, R x )) × H 1 (S, Σ, R y ), the set of boxes is preserved by the action of Mod(S, Σ).Furthermore, a finite intersection of boxes is a box.Thus, by Lemma 3.1, for every q ∈ π −1 (M), there is a regular box in π −1 (M) containing q.
Remark 3.3.There is an asymmetry in the definition of a box; we could equally well define a box using V x and P(V y ), but we will make no use of that kind of box.

Definition of the horospherical foliation.
Recall that a smooth map of manifolds is a submersion if its derivative is of full rank at every point.The implicit function theorem implies that the connected components of the fibers of a submersion are the leaves of a foliation.Definition 3.4.Let L be a lift of M and let V be the linear space on which L is modeled.The foliations on L (1) induced by the submersions π ′ x • dev : L (1) → P(V x ) and π y • dev : L (1) → V y , are called the weak stable and strong unstable foliations.They are denoted respectively by W s L and W uu L .The leaf of q ∈ L (1) for the weak stable foliation is denoted by W s L (q) and the leaf of q for the strong unstable foliation is denoted by W uu L (q).It follows from Lemma 3.1 that these foliations are well-defined, and the leaves of these foliations are everywhere transverse.
Lemma 3.5.The action of Mod(S, Σ) permutes the leaves of W uu L .For any leaf F , the restriction dev| F is a local homeomorphism to an affine subspace of V and with respect to this affine structure, the subgroup Γ L def = {γ ∈ Mod(S, Σ) : L • γ = L} acts on the leaves of W uu L by affine maps.Proof.The monodromy preserves the product splitting V = V x ⊕ V y and acts linearly on each factor.Thus the monodromy acts projectively on P(V x ).Since dev is monodromy equivariant, the leaves of the foliations W s L and W uu L are permuted by the action of Mod(S, Σ).
For the second assertion, it is clear from the definitions that dev maps the leaf F to a set of the form {(x, y 0 ) ∈ V : x ∈ V x and x, y 0 = 1} for some fixed y 0 ∈ V y , and by Lemma 3.1, the map dev| F is a local diffeomorphism.The last assertion follows from the Mod(S, Σ)-equivariance of dev and the fact that Mod(S, Σ) preserves the bracket •, • .
Remark 3.6.Lemma 3.5 equips the leaves of the foliation W uu with an affine manifold structure.This structure need not be geodesically complete.Using real Rel deformations, one easily constructs affine geodesics in a leaf W uu L (q) which contain a surface with a horizontal saddle connection whose length goes to zero as one moves along the leaf.There are additional sources of non-completeness involving surfaces whose horizontal foliation is minimal but not uniquely ergodic, see [MW14].Furthermore, using [MW14, Thm.1.2], one can show that each leaf W uu L (q) is mapped by the developing map homeomorphically to an explicitly described convex domain in H 1 (S, Σ; R x ), defined by finitely many linear inequalities and equalities.
It follows from Lemma 3.5 that the partition of L (1) given by the leaves W uu L induces a partition of M (1) .We denote it by W uu and if q ∈ M, we denote by W uu (q) the element of the partition that contains q.We emphasize that W uu does not depend on the choice of a particular irreducible component used to define it.This is a consequence of the fact that Mod(S, Σ) acts on H 1 (S, Σ, C) by real endomorphisms and thus preserves the splitting into real and imaginary parts of cohomology classes.Definition 3.7.A horosphere is an element of the partition of W uu .Remark 3.8.Occasionally, we may call the partition W uu the horospherical foliation of M, even though M is generally not a manifold.Even if this will play no role in the rest of the paper, we justify this choice of terminology for the sake of completeness: the invariant subvariety M can be seen to have the structure of a properly immersed manifold M, i.e., is the image of a manifold N under a proper orbifold immersion f : N → H and there is a foliation on N whose leaves are sent to horospheres by f .We can choose N to be the quotient of L by a finite-index torsion-free normal subgroup Γ 0 of Mod(S, Σ) and f : L/Γ 0 → H, qΓ 0 → π(q).By Lemma 3.5, the horospherical foliation on L descends to a foliation on the manifold L/Γ 0 .The leaves of this foliation are indeed mapped to horospheres and f is an orbifold immersion.The fact that it is proper follows from the fact that the collection of irreducible components of π −1 (M) is locally finite.
Reversing the roles of π x and π y , and defining π ′ y in an analogous fashion, we also define the strong stable and unstable foliations W ss L and W u L as those induced by the submersion π x • dev, π ′ y • dev respectively.Lemma 3.5 holds for these foliations as well, with obvious modifications.Summarizing: for every q ∈ L we have , the leaves W ss L (q) and W uu L (q) have a natural affine structure and, for n = dim(M), we have As we saw in §2.4, the sup-norm Finsler metric induces a distance function on H m as a path metric.We will induce distance functions on leaves of the stable and strong stable foliations using the same approach.For q 0 , q 1 ∈ H m belonging to the same stable (respectively, strong stable) leaf, we define dist (s) (q 0 , q 1 ) (respectively, dist (ss) (q 0 , q 1 )) by the formula in equation ( 15), but making the additional requirement that the entire path γ is contained in the stable (respectively strong stable) leaf of the q i .We similarly define dist (s) (q 0 , q 1 ) and dist (ss) (q 0 , q 1 ) for q 0 , q 1 ∈ H belonging to the same stable (respectively, strong stable) leaf.We will call the distance functions dist (s) , dist (ss) the stable (resp.strong stable) sup-norm distance function.
These distance functions have the following properties: Proposition 3.9.Let L a lift of M and let q 0 , q 1 ∈ L.
(2) If q 0 , q 1 are in the same strong stable leaf then for all t ≥ 0, dist (ss) (g t q 0 , g t q 1 ) ≤ dist (ss) (q 0 , q 1 ).And the same holds for the strong unstable leaf.
(4) Statements (1), (2) and (3) also hold in H, for q 0 , q 1 in place of q 0 , q 1 .Proof.Assertion (1) is obvious from definitions, and assertions (2) and (3) are proved in [AG10,§5] (where what we call the strong stable foliation is referred to as the stable foliation).The assertions for H follows from the corresponding ones for H m .
Remark 3.10.Almost everywhere, the horospheres W uu (q) and W ss (q) are actually the unstable and stable manifolds of the geodesic flow.That is, for any q, and almost every (with respect to the measure class induced by the affine structure on leaves) q 1 ∈ W uu (q), q 2 ∈ W ss (q), we have dist(g t q, g t q 1 ) −→ t→−∞ 0 and dist(g t q, g t q 2 ) −→ t→∞ 0. This is proved in [Vee86] (see also [FM14]) for M = H (1) .The same result for general invariant subvarieties can be proved by adapting the arguments used in [FM14].
3.3.Definition of horospherical measures.Let L be a lift of M as in Subsection 2.2 and let V ⊂ H 1 (S, Σ, C) be the subspace on which L is modeled.We write V = V x ⊕ V y as in equation (2).Let η x and η y be the translation invariant volume forms on V x and V y determined by a choice of an element of the top degree wedge power of V x and V y .Define We recall that the measure µ L on L was defined in Section 2.2 as the integral of a volume form α. From now on, this form will be chosen so that α = α x ∧ α y .We define the Euler vector field E on H m such that for any q ∈ H m , This vector field can be thought of as the tangent vector to the rescaling action, which justifies our choice of terminology.Notice furthermore that the image of E by dev is the usual Euler vector field e(v) = v on H 1 (S, Σ; C).This is due to the fact that dev is GL + 2 (R)-equivariant.Since L is a linear manifold, the vector field E is tangent to it.We use this to define the form i.e., the contraction of α x by the Euler field E. The restriction of β x to the leaves of W uu L induces a volume form.We denote by ν βx the induced measures.We emphasize that this defines a system of measures, one on each leaf W uu L (q), so one should write ν L βx,q instead of ν βx ; we omit this in our notation.We say that a measure m on L is horospherical if it is supported on L (1) and its conditional measures on the leaves of W uu L are given by the measures ν βx .More precisely, this means that for any box B in L, there is a measure λ on U y such that for any compactly supported continuous function f : Remark 3.11.The measure λ is a so-called 'transverse measure' for the horospherical foliation.This means it is a system of measures on sets tranverse to the foliation which is invariant under holonomy along leaves, see [CC03, Vol. 1, 10.1.13& 11.5.2].According to the theory of transverse measures equation (23) yields as a bijection between horospherical measures and transverse measures.We will not be using this point of view in this paper.
Remark 3.12.Let φ t be a smooth flow acting on L. A measure ν is said to be invariant if for any t ∈ R we have (φ t ) * ν = ν.This definition is equivalent to requiring that the conditional measures of ν on the orbits of φ t be multiples of the Lebesgue measure dt, i.e., invariant under the maps φ s x → φ t+s x for any fixed t.
The equivalence can be shown by disintegrating ν on flow boxes, i.e., boxes whose horizontal plaques are pieces of φ t -orbits.By Lemma 3.5, leaves of W uu L are modeled on linear subspaces, and thus one could try and define horospherical measures as those that are invariant under translation along the leaves.However, these translations are not part of a globally defined group action; for instance trajectories might escape to infinity in finite time.Our definition of horospherical measures is inspired by the second characterization of invariant measures, where the foliation by orbits of φ t is replaced by the strong unstable foliation and the translation invariant measure dt is replaced by ν βx .
In order to define a notion of horospherical measures on M, we first need some terminology: let ν be a Radon measure on M and let ν be its pre-image by π as in equation (10) (see also Appendix B).By construction, the measure ν is supported on π −1 (M).If L is a lift of M, then the restriction of the measure ν to L is called the lift of ν corresponding to L.More generally, a lift of ν is a measure of the form ν| L where L is any lift of M. For instance, the measures m L in equation ( 11) are the lifts of m M .Definition 3.13 (Horospherical measure).A Radon measure ν on M is horospherical if its lifts are horospherical.
By Proposition 2.7, it is enough that one of the lifts is horospherical, as the action of Mod(S, Σ) preserves the set of horospherical measures on H m .By definition, a horospherical measure on M is supported on M (1) .We have the following useful local disintegration formula: Proposition 3.14.Let ν be a horospherical measure on M. For any regular box ϕ : , there is a measure λ on U y such that for any compactly supported continuous function f : M → R, denoting B = π(B) we have Proof.Let L be a lift of M in which B is contained.We denote by Γ the stabilizer in Mod(S, Σ) of B, and by ν the pre-image of ν under π.By definition, the measure ν| L is horospherical, and we let λ 0 be a measure on U y as in equation ( 23).We set |Γ| λ 0 , and claim that λ satisfies equation (24).Indeed, let f ∈ C c (H) and assume for now that the support of f is contained in B. Let h be the function that is equal to f • π on B and 0 elsewhere.This function is continuous and its support is contained in B by construction.Using that the stabilizer of B in Mod(S, Σ) is also Γ, we calculate that for any q ∈ H, we have Hm h dθ q = |Γ|f (q).We have which is what we wanted.In case the support of f is arbitrary, we pick a sequence ψ n of uniformly bounded smooth functions with support contained in B and that converge pointwise to 1 B , the indicator function of B, and we apply the previous computation to ψ n f in place of f .We have Passing to the limit using Lebesgue's dominated convergence, we obtain equation (24).
3.4.The special flat measures are horospherical.In this subsection we prove Theorem 1.1, which gives us our first examples of horospherical measures.Namely we will show that the Masur-Veech measures on strata, and more generally, the special flat measures defined in equation ( 9), are horospherical.
Let M be an invariant subvariety and let L be a lift of M. In order to establish Theorem 1.1, we shall first establish that the measure m L as in equation ( 11) is horospherical.This will be achieved in Proposition 3.17.We need some preparatory results.We recall that the measure m L is obtained by the cone construction applied to µ L , i.e., for any Borel set A ⊂ L (1) , m L (A) = µ L (cone(A)), and the measure µ L is itself obtained by integration of α = α x ∧ α y , where α x and α y are as in equation (21).Let β def = ι E α.By construction, β induces a volume form on L (1) and we denote by µ β the measure obtained by integration of β.The following relates the measure µ β and the cone measure m L .Lemma 3.15.We have Proof.The proof is an application of Stokes' theorem.It follows from equations ( 6) and ( 22) that the Lie derivative of α with respect to the Euler vector field satisfies contained in one chart for the manifold structure on L. Notice that the only part of the boundary ∂cone(U ) of cone(U ) to which E is not tangent is U itself.In particular, the only part of ∂cone(U ) on which ι E α does not vanish is U .We have from Stokes' formula (for manifolds with corners, see e.g.[Lee13]) that It follows from the Cartan formula that L E (α) = dι E α + ι E dα.Since α is closed, we deduce that dι E (α) = L E (α).Gathering everything, we obtain This is true for all U as above, and these open sets generate the Borel σ-algebra on L (1) .
We introduce two new vector fields E x and E y on H m : By definition we have E = E x + E y , and for any q ∈ H m , we have For the proof of Theorem 1.1, we will also need the following calculation: Lemma 3.16.The restrictions of the forms α x and α y to L (1) satisfy Proof.Let n = dim(M).We begin by observing that on restriction to L (1) , we have Indeed, we deduce from equation ( 26) that E x −E y is tangent to L (1) .In particular, since L (1) has dimension 2n− 1 (as a real vector space), any family of 2n− 1 linearly independent vector fields that are tangent to L (1) contain E x − E x in their span.This implies equation ( 27).Now we calculate: The last equality follows from the Leibniz formula for contractions Now, notice that E x is tangent to the fibers of π y • dev.Since α y = (π y • dev) * η y , we deduce that ι Ex α y = 0. Similarly, we prove that ι Ey α x = 0 and thus Proposition 3.17.The measure m L is horospherical.
Proof.It follows from Lemma 3.15 that m L is given, up to a multiplicative constant, by integration of the differential form β = ι E (α x ∧ α y ).Lemma 3.16 implies that β = 2β x ∧α y .Notice that both the forms β x and α y are basic, i.e., they are obtained by pullback of forms on P(V x ) and V y by the projections dev • π ′ x and dev • π y .Indeed, we have α y = (dev•π y ) * η y and using Lemma 3.1, we can build a differential form β ′ x on P(V x ) such that (dev x .We have: If we let λ be the measure on U y given by integration of the form Proof of Theorem 1.1.By definition, in order to prove that m M is horospherical, we need to show that the lifts of m M are horospherical.We recall from equation ( 11) that the lifts of m M are given by the m L .Theorem 1.1 is then a consequence of Proposition 3.17.
3.5.The horocycle flow, real REL, and horospherical measures.In this subsection we will show that the horocycle flow and the real Rel deformations move points in their horospherical leaf, and preserve horospherical measures.
This implies that the horocycle flow maps W uu L (q) to itself and since π is GL + 2 (R)equivariant, we deduce that horospheres are preserved by the horocycle flow.
We also deduce from equation (28) that the horocycle flow preserves the form α x .Since the horocycle flow commutes with the rescaling action, it also preserves the Euler vector field, from which we deduce that for any s ∈ R we have u s * β x = β x .In particular, the horocycle flow preserves the measures ν βx .
Let ν be a horospherical measure and let s 0 > 0. We claim that for any q ∈ M such that the orbit segment {u s q : s ∈ R, |s| ≤ s 0 } is embedded, i.e., U q is not a periodic horocycle orbit with period of length smaller than 2s 0 , there is an open set U ⊂ M containing q such that for any compactly supported continuous function f with support contained in U and any s ∈ R with |s| ≤ s 0 , we have: To see this, let q ∈ H m be such that π(q) = q and define Let Γ be the stabilizer in Mod(S, Σ) of q.Since the GL + 2 (R)-action on H m commutes with Mod(S, Σ), the group Γ acts trivially on σ and since U q is not a periodic orbit with period smaller than 2s 0 , we have that that σ • γ ∩ σ = ∅ for any γ ∈ Mod(S, Σ), unless γ ∈ Γ.By thickening σ, we can find a box B ⊂ π −1 (M) containing σ and up to replacing B with ∩ γ∈Γ B • γ, we can assume that B is regular.By construction, for any s ∈ R with |s| ≤ s 0 , the surface u s q belongs to B = π(B) and lies on the plaque of q.Since the horocycle flow acts continuously, there is a neighborhood U ⊂ B around q such that for any q ′ ∈ π −1 (U) ∩ B and |s| ≤ s 0 , we have u s q ′ ∈ B. Let λ be a transverse measure on U y , i.e., a measure as in equation ( 24), and let f be a continuous function with support contained in The second equality follows from the fact that π is GL + 2 (R)-equivariant and the fact that for any y ∈ U y , the action of u s maps supp(f • π) ∩ L y inside L y , which in turns is implied by our choice of U and the fact that the horocycle flow maps the leaves of W uu L into themselves.For any f ∈ C c (H), define It is easy to see that s f is always positive, and using the first part of the proof together with a partition of unity argument, we can show that equation (29) holds for f and s ∈ R with |s| ≤ s f .Furthermore, notice that for any s ∈ R, we have This proves that ν is horocycle flow-invariant.
For any irreducible invariant subvariety M, we let (30) where V is the model space of some lift of π −1 (M) and Z is the real REL space.
Notice that the space Z M actually does not depend on the choice of particular lift.This is a consequence of Proposition 2.7 together with with the fact that Mod(S, Σ) acts trivially on ker(Res).
Proposition 3.19.Let v ∈ Z M , q ∈ M, and suppose the Rel flow Rel v (q) is defined.Then Rel v (q) ∈ W uu (q).If ν is a horospherical measure and Rel v (q) is defined for ν-a.e.q, then ν is invariant under the (almost everywhere defined) map q → Rel v (q).
Proof.Let Z (q) be as in equation (4).Since Z M ⊂ V , where V is the subspace that M is modeled on, we have Rel v (q) ∈ M if q ∈ M and v ∈ Z M ∩ Z (q) .The only properties of the horocycle flow which were used in the proof of Proposition 3.18 are that u s0 preserves the horospheres, and acts on them by translations.The same properties are valid for the action of Rel v for v ∈ Z M .Indeed, Rel v sends surfaces of area one to surfaces of area one, and if dev(q) = (x, y) then dev(Rel v q) = (x + v, y).
If a measure µ on M is saddle-connection free, then for µ-a.e.q ∈ M, Rel v (q) is defined for every v ∈ Z M , and satisfies the 'group law' property Following [Wri15a], we say that an irreducible invariant subvariety M is of rank one if dim(Res(V )) = 2, where V is the model space of any lift of π −1 (M).In the rank one case we have the following converse to Propositions 3.18 and 3.19: Proposition 3.20.If M is an invariant subvariety of rank one and µ is a saddle connection-free measure, then µ is horospherical if and only if it is invariant under the horocycle and the real Rel flows.
Proof.By a dimension count, we see that when M has rank one, the dimension of horospheres is the same as dim(Z M ) + 1.This means that the horosphere that is the group action generated by the horocycle flow and real Rel acts transitively on the horospheres.As we saw in the proofs of Propositions 3.18 and 3.19, this action is by translations, with respect to the affine structure on W uu (q) afforded by Lemma 3.5.Since the measures ν βx are the unique (up to scaling) translation-invariant measures on the affine manifolds L (1) , the invariance of µ under the horocycle and real Rel flows implies that the conditional measures on the plaques in a box are given by ν βx .
3.6.Further properties.Let X be a manifold with a foliation, and a Borel measure µ.We say that µ is ergodic for the foliation if any Borel subset B which is a union of leaves satisfies either µ(B) = 0 or µ(X B) = 0.For instance we have: Proposition 3.21.The special flat measure on an invariant subvariety is ergodic for the horospherical foliation.
Proof.This follows from Proposition 3.18, the ergodicity of the special flat measure with respect to the A-action (see e.g.[FM14, Section 4]), and the Mautner phenomenon (see [EW11]).
Denote by P (uu) (M) the collection of horospherical measures on M with total mass at most one.The following standard results in ergodic theory are valid in the context of horospherical measures: Proposition 3.22.For the horospherical foliation on any invariant subvariety M (1) , we have: (1) The space P (uu) (M), with the weak- * topology, is a compact convex set.
(2) A horospherical probability measure is ergodic if and only if it is an extreme point of P (uu) (M).
We say that two surfaces q, q ′ ∈ H are horizontally equivalent if there is a homeomorphism M q → M q ′ of the underlying surfaces that preserves the labels of singularities and maps the union of the horizontal saddle connections of M q bijectively to the union of those of M q ′ .Note that a horizontal equivalence only preserves certain horizontal structure.It preserves saddle connections but need not preserve the horizontal foliation.
Proposition 3.23.Any two surfaces in the same horospherical leaf are horizontally equivalent.
Proof.It suffices to show this upstairs; that is, we let q, q ′ ∈ H m with q ′ ∈ W uu L (q), let f : S → M q and f ′ : S → M q ′ be marking maps representing q, q ′ and show that, if f and f ′ are carefully chosen, f ′ • f −1 gives a bijection of horizontal saddle connections.We first discuss q ′ ∈ W uu L (q) which are sufficiently close to q.Let f : S → M q be a marking map representing q and let σ 1 , . . ., σ r be the horizontal saddle connections on M q .Let τ be a triangulation of S such that the segments f −1 (σ i ) are edges of triangles.Let f ′ be constructed from τ , so that the map f ′ • f −1 is affine on each triangle of τ (see the discussion of comparison maps in [BSW22, §2.4]).Let U = U τ be the corresponding neighborhood of q.Then for any surface q ′ ∈ U, represented by f ′ : S → M q ′ , the paths f ′ • f −1 (σ i ) are represented by saddle connections on M q ′ .Furthermore, if q ′ ∈ U ∩ W uu L (q) then these paths are horizontal saddle connections, so that f ′ • f −1 : M q → M q ′ is a homeomorphism mapping the horizontal saddle connections of M q injectively to horizontal saddle connections on M q ′ .Now choose q max ∈ W uu L (q) so that it has the maximal number of horizontal saddle connections.We will show that the set V of surfaces in W uu L (q) which are horizontally equivalent to q max is open and closed, and this will conclude the proof.By the preceding discussion, V is open in W uu L (q).Furthermore, if q n → q ∞ is a convergent sequence of surfaces in W uu L (q), with q n ∈ V, then the horizontal saddle connections on the surfaces M qn have length bounded uniformly from above and below, and so converge to paths on M q∞ which are represented by horizontal saddle connections or by finite concatenations of horizontal saddle connections.Sequences of paths which are distinct on the surfaces M qn cannot converge to the same paths on M q∞ because they issue from different singularities, or from different prongs at the same singularity.Thus q ∞ has at least the same number of saddle connections as q max , and so, by maximality, q ∞ ∈ V.This completes the proof.
From Proposition 3.23 we deduce: Corollary 3.24.If ν is an ergodic horospherical measure then there is a subset M ′ ⊂ M of full ν-measure such any two surfaces in M ′ are horizontally equivalent.
Remark 3.25.In [BSW22, Def.5.1], using boundary marked surfaces, topological horizontal equivalence is introduced.In this definition the homeomorphism M q → M q ′ is required to preserve additional structure, e.g. the angular differences between saddle connections at each singular point.Proposition 3.23 and Corollary 3.24 hold for this finer notion of equivalence as well.

Saddle connection free horospherical measures
In this section we will prove Theorem 1.2.We first state and prove some auxiliary statements.
4.1.The Jacobian distortion in a box.The different plaques in a box can be compared to each other using the structure of a box.Namely, let ϕ : U ′ x × U y → B ⊂ L (1) be a box.For any point y ∈ U y we define where L y is the plaque of y in B (see Definition 3.2).
For any two points y 0 and y 1 in U y , the map ϕ y0,y1 def = ϕ y1 • ϕ −1 y0 is a diffeomorphism between the plaques L y0 and L y1 in B, identifying points parameterized by the same point in U ′ x .Define where x q = π x • dev(q).The diffeomorphism ϕ y0,y1 is not measure preserving.Instead, we have the following: Proposition 4.1.(Jacobian calculation) For any two points y 0 , y 1 ∈ U y we have Proof.For any y ∈ U y , write where π x is the projection in equation ( 17).Then Ly is an open subset of the affine hyperplane By Definition 3.2 the map is a diffeomorphism with inverse π x • dev.We denote by e x the Euler vector field on V x .Notice that F * β x is the restriction to Ly of ι ex η x .Indeed, we calculate The map F gives a chart of L y in which β x is ι ex η x .We shall perform our calculation in these charts and verify equation (31) in Ly instead of L y .Let y 0 , y 1 ∈ U y , and set h : The map ϕ y0,y1 : L y0 → L y1 is expressed in charts simply as the map φy0,y1 : This implies by the product rule that Hence, denoting d = dim(M), for v 1 , . . ., v d−1 in the tangent space to Ly0 at x we have: This is Formula (31).
Notice that for any y 0 , y 1 ∈ U y and [x] ∈ U ′ x , we have .
This leads us to define the distortion of B as follows: Remark 4.2.The quantity x,y0 x,y1 has the following geometric interpretation.The points ϕ([x], y 0 ) and ϕ([x], y 1 ) are in the same weak stable leaf, and this leaf is further foliated by strong stable leaves.The geodesic flow maps a given weak stable leaf to itself, permuting the strong stable leaves inside it.The choice t = log x,y0 x,y1 is the value of t ∈ R for which g t maps ϕ([x], y 0 ) to the strong stable leaf of ϕ([x], y 1 ).
The distortion can be used to bound the variation of the mass of the horospherical plaques of B with respect to the measures ν βx .Indeed, by an easy change of variables, using ϕ y0,y1 we have From this it follows that (33) The distortion of a box is well-behaved with respect to the geodesic flow.For t ∈ R and any B in π −1 (M), we write x ×U y → L (1) be the parametrization of B, where L is an irreducible component of π −1 (M) and let Ūy be the image of U y under multiplication by e −t , and let φ([x], y) def = g t • ϕ([x], e t y).Using the fact that the geodesic flow preserves the splitting into stable and horospherical foliation, and acts on V y by multiplication by e −t , we see that φ : 1 a parameterization of B t as in Definition 3.2.Also, for i = 0, 1, if ([x], ȳi ) ∈ U ′ x × Ūy , where ȳi = e −t y i , then x, ȳ0 x, ȳ1 = x, y 0 x, y 1 .
This implies that the distortion of B is the same as the distortion of B t .The last statement follows from the fact that the actions of GL + 2 (R) and Mod(S, Σ) commute.

Thickness of a box. We now introduce the notion of the thickness of a box.
To define this quantity we use the sup-norm Finsler metric of §2.4 to induce a distance function on leaves of the stable foliation.We rely on work of Avila and Gouezel [AG10, §5], who defined a similar distance function on the leaves of the strong stable foliation.
For a subset of a stable leaf, we denote by diam (s) its diameter with respect to the distance function dist (s) .We define the thickness of the box B as that is, the maximal diameter of a plaque for the stable foliation.
We will need boxes whose thickness is also well-behaved under the geodesic flow.Similarly to Proposition 4.3, we have: Proposition 4.4.For any ε > 0 and any q ∈ π −1 (M (1) ), there is a regular box B in π −1 (M) containing q such that for any t ≥ 0, τ B t ≤ ε.
Proof.Let L be a lift of M that contains q and let Γ be the stabilizer in Mod(S, Σ) of q.Since Mod(S, Σ) acts properly discontinuously on H m , there is a neighborhood V containing q such that for any γ ∈ Mod(S, Σ), either V • Γ ∩ V = ∅ or γ ∈ Γ.By Lemma 3.1, let B ⊂ V be a box containing q and let φ : Ū ′ x × Ūy → B be the parametrization of B. Let dev(q) = (x 0 , y 0 ), let Û ′ x be a neighborhood of [x 0 ] whose closure is contained in Ū ′ x , and let That is, C is a bounded subset of a horospherical leaf, contained in a plaque of B, and with closure in the interior of B. Let ε 1 ∈ 0, ε 4 be small enough so that and let ε 2 ∈ 0, ε 4 such that q 2 ∈ W ss L (q 1 ) : dist (ss) (q 1 , q 2 ) < ε 2 is contained in B. Such numbers ε 1 , ε 2 exist because C is bounded, and C 2 contains a neighborhood of q.We can therefore let U ′ x ⊂ Û ′ x and U y ⊂ Ūy be small enough open sets so that B = φ(U ′ x × U y ) contains q and is contained in C 2 .Since B is contained in V, we may replace B by ∩ γ∈Γ B • γ and we can assume that B is regular, with stabilizer Γ.
For q ∈ B, let L s (q) be the plaque through q for the weak stable foliation, that is, the connected component of q in B ∩ W s L (q).For each q 2 ∈ B there is a point q 0 , which is the unique point in the intersection C ∩ L s (q 2 ), and a path from q 0 to q 2 which is a concatenation of two paths γ 1 and γ 2 .The path γ 1 = {g t q 0 : t ∈ I} from q 0 to q 1 goes along a geodesic arc, where I is an interval of length at most ε 1 .The path γ 2 from q 1 to q 2 has sup-norm length at most ε 2 and is contained in W ss L (q 2 ).Since ε 1 , ε 2 < ε 4 , each point in any stable plaque in B is within distance at most ε 2 from the unique point at the intersection of this plaque with C, where the distance is measured using the distance function dist (s) .Concatenating such paths we see that the diameter of any stable plaque in B is at most ε, and this implies the same bound for stable plaques in B. That is, the thickness of B is less than ε.By Proposition 3.9, the lengths of geodesic paths and of paths in strong stable leaves, do not increase when pushed by g t for t ≥ 0. Thus the same argument (using the pushes of γ 1 and γ 2 by g t ) give the required upper bound on the thickness of B t .
For a compactly supported continuous function f on M, we denote by ω f its continuity modulus with respect to the sup-norm distance function.In particular, ω f (t) → 0 as t → 0+ and |f (q 1 ) − f (q 2 )| ≤ ω f (dist(q 1 , q 2 )) for any q 1 , q 2 ∈ M.
The following key lemma says that for any horospherical measure ν, any regular box B and any test function f , the integral of f with respect to ν| B can be approximated by the integral of f • π with respect to ν βx on any one horospherical plaque of B, provided that B has small distortion and small thickness.We recall that B ⊂ M is defined as the image of B by π.
Lemma 4.5.Let ν be a horospherical measure, let f ∈ C c M (1) and let B be a regular box such that ν(B) > 0. Then for any y ∈ U y , Proof.For y, y ′ ∈ U y , let ϕ y,y ′ : L y → L y ′ be as in §4.1.On the one hand, for any y ′ ∈ U y we have The second inequality follows from the fact that, by definition of the thickness, for any [x] ∈ U ′ x , the distance between the points ϕ([x]) and ϕ(ϕ y ′ ,y ([x])), with respect to the distance function dist (s) , is at most τ B and thus also with respect to the distance function dist, together with the fact that π is a contraction.
On the other hand, by the definition of δ B , we have: The last inequality follows from Proposition 4.1 and the definition of δ B .Using equation (33) we deduce that for any y, y ′ ∈ V y , Let y 0 ∈ U y and let λ be a measure on U y as in equation ( 24).Notice that ν(B) = Uy ν βx (L y ) dλ(y).Therefore 4.3.Mixing of geodesics, nondivergence of horocycles.We recall the following useful results: Lemma 4.6 (Nondivergence of the horocycle flow [MW02]).For any ε > 0 and c > 0 there is a compact K ⊂ M (1) such that for any q ∈ M (1) , one of the following holds: • The surface q has a horizontal saddle connection of length smaller than c.
For any 0 < c ≤ ∞, let M <c be the subset the subset of M (1) consisting of surfaces which have a horizontal saddle connection of length smaller than c, and let M ≥c = M (1) − M <c .We deduce the following corollary.
Lemma 4.7.For any 0 < ε < 1 and 0 < c ≤ ∞, there is a compact K ⊂ M (1)  such that for any U -invariant measure µ on M (1) , Proof.Given ε and c, let K be a compact set given by Lemma 4.6 (if c = ∞, we can apply Lemma 4.6 to any finite c).An application of a generalisation of the Birkhoff ergodic theorem for locally finite measures (see [Kre85, Thm.2.3] for a general formulation) to the invariant measure µ and the function 1 K shows that there is a non-negative function By Lemma 4.6, we have that for almost every q ∈ M ≥c , f (q) > 1 − ε.As a consequence, Lemma 4.7 will be used at several places in this text.The first fact we deduce from it is the following: Lemma 4.8.Let ν be a saddle connection free horospherical measure and let δ > 0. Then there is a regular box B ⊂ π −1 (M), a constant c > 0 and an unbounded increasing sequence of times t i such that: (a) For all i ≥ 0, ν(B ti ) > cν(M), where B ti = π(B ti ).(b) Both the thickness and distortion of each B ti are smaller than δ.
In particular it follows from (a) that ν is finite.
Proof.Let K be a compact subset as in Lemma 4.7 for ε = 1 2 , c = ∞, and denote ν t def = (g −t ) * ν.By Proposition 3.18, ν is U -invariant, and since g t normalizes U , the same holds for ν t .Since ν is saddle-connection free, so is ν t .So, applying Lemma 4.7 to ν t , For every δ > 0, using Proposition 4.4, K ∩ M (1) can be covered by the image by π of regular boxes B 1 , . . ., B N whose distortion is smaller than δ, and for which the thickness of g t (B j ) is smaller than δ, for each j and each t ≥ 0. By Lemma 4.3, the distortion of g t (B j ) is also less than δ for each j and each t ≥ 0. Let c def = 1 2N .For each t, there is j = j(t) ∈ {1, . . ., N } such that Let t i → ∞ be a sequence along which j = j(t i ) is constant.Then (a) and (b) hold for B = B j .
Lemma 4.9 (Mixing of the geodesic flow).For any invariant subvariety M, the geodesic flow is mixing with respect to the special flat measure on M (1) .
For a proof and detailed discussion of this result and its quantitative strengthenings, see [FM14,Chap. 4] or [EMM22].

4.4.
Putting it all together.We have gathered all the ingredients needed to give the proof of one of our main results.
Proof of Theorem 1.2.Let ν be a saddle connection free horospherical measure.We assume first that ν is ergodic for the horospherical foliation.We will show that the special flat measure m M is absolutely continuous with respect to ν.To see this, let A be a Borel set of positive measure for m M .Since m M is a Radon measure, in particular inner regular, there is a compact K contained in A such that m M (K) > 0. Let U be an open set that contains A and let f : M (1) → [0, 1] be a continuous function whose support is contained in U and that evaluates to 1 on K.Such a function exists by Urysohn's Lemma.Let ε > 0, and choose δ > 0 so that By Lemma 4.8, there is c > 0, a regular box B and t i → ∞ such that for each i, τ B t i < δ and δ B < δ, and ν(B ti ) ≥ cν(M).Applying Lemma 4.5 to both ν and m M we obtain By mixing of the geodesic flow with respect to m M , there is i > Since ε was chosen arbitrarily, we have proven ν(U ) ≥ c ν(M) m M (K).Since this holds for an arbitrary open U containing A, and ν(M) is finite, we deduce by outer regularity of the measure ν that ν(A) is positive.This completes the proof that m M ≪ ν.It follows from Proposition 3.22 that m M = cν for some c ≥ 0, and since m M is nonzero, c > 0 and ν = 1 c m M .For general ν, we obtain from the case just discussed that all the ergodic components of the measure ν are proportional to the special flat measure and thus ν itself is proportional to the special flat measure.

Examples of horospherical measures
The simplest example of a horospherical measure which is not the special flat measure occurs when M is a closed GL + 2 (R)-orbit.In this case the leaves of the horospherical foliation are the U -orbits, and the length measure on a closed periodic U -orbit is a horospherical measure; indeed, in this case, the transverse measure λ in equation ( 24) is atomic.
In order to obtain more complicated examples, we use the following: Proposition 5.1.Let W uu (q) be a closed horosphere in M. Then W uu (q) is the support of a horospherical measure ν whose lifts are the measures ν L βx,q where L is a lift of M and π(q) = q.
Proof.The horosphere W uu (q) is closed if and only the collection W uu L (q) is locally finite, where q ranges over π −1 (q) and L ranges over the lifts of M that contain q.Each of the W uu L (q) carries the Radon measure ν L βx,q and the measure is a Mod(S, Σ)-invariant Radon measure on π −1 (M).Let ν be the Radon measure on M whose lift is ν (see Proposition B.3).The measure ν is horospherical by construction.
To construct an example of a closed horosphere, we use horizontally periodic surfaces, i.e., surfaces which can be represented as a finite union of horizontal cylinders.Let M = H(1, 1).This stratum is an invariant subvariety of dimension 5 (see Definition 2.4), and thus its horospherical leaves have real dimension 4. Let a, b be real numbers with a, b ∈ (0, 1) and 0 < b < min(a, 1 − a), let τ 1 , τ 2 ∈ S 1 def = R/Z, and set τ1 def = aτ 1 and τ2 def = (1 − a)τ 2 , so that τ1 , τ2 take values in circles of circumference a, 1 − a respectively.Define the surface q = q a,b,τ1,τ2 ∈ M by the polygonal representation shown in Figure 1.In the horizontal direction it is comprised of two cylinders, each of height 0.5, and of areas 0.5a and 0.5(1 − a).The parameters τ 1 , τ 2 are called twist parameters.Changing them by adding an integer amounts to performing the corresponding number of Dehn twists in the two cylinders, and thus does not change the surface q.
It is clear that varying the parameters a, b, τ 1 , τ 2 results in surfaces that belong to the horospherical leaf of q, and thus, by a dimension count, they locally parameterize the leaf of q.In either of the cases b → 0 or b → min(a, 1 − a), the surfaces q a,b,τ1,τ2 have shorter and shorter horizontal saddle connections on the boundaries of the cylinders, and thus exit compact subsets of M (1) .This means that the horosphere W uu (q) is closed and that the map is a proper embedding whose image is W uu (q).
It can be checked that in this case the map (a, b, τ1 , τ2 ) → dev(q a,b,τ1,τ2 ) is affine in charts.Thus the horospherical measure can be written explicitly (up to scaling) as dν(q a,b,τ1,τ2 ) = da db dτ 1 dτ 2 .
Remark 5.2.For the horospherical measure constructed in the preceding example, the space Z M (defined in equation (30)) is one dimensional, and for every surface q in the support of this measure, Z (q) (defined in equation (4)) is a bounded interval.Moreover, for any v ∈ Z M there is a positive measure set of surfaces q (with small values of a) for which Rel v (q) is not defined.This shows that the hypothesis in Proposition 3.19, that Rel v (q) is defined, is not always satisfied.(More explicitly, a leaf of the real REL foliation is given by varying b.) It is no coincidence that the closed horospheres in the two preceding examples consist of horizontally periodic surfaces.
Proposition 5.3.For any M and any q ∈ M, the surface M q is horizontally periodic if and only if W uu (q) is closed.In this case every surface in W uu (q) is horizontally periodic, and the horospherical measure on W uu (q) constructed in Proposition 5.1 is finite.
Proof.Suppose first that M q is horizontally periodic, and let f : S → M q be a marking map representing q ∈ π −1 (q).Let C 1 , . . ., C s be the horizontal cylinders on M q , and let c j , h j denote respectively the circumference and height of C j .Since the area of M q is one, (34) Let α 1 , . . ., α r , α r+1 , . . ., α r+s be a collection of oriented paths in S with endpoints in Σ which satisfy the following: taken to lie in the bounded set [0, c j ).Also, as the parameters t i leave compact subsets of the bounded domain described above, at least one of the horizontal saddle connections on the corresponding surface has length going to zero.This implies that the bounded set of surfaces we have just described by varying the parameters t i , τ j projects to the entire leaf W uu (q), that this leaf is properly embedded, and that all surfaces in this leaf are horizontally periodic.
Furthermore, we can use equation (34) to express c 1 as a function of c 2 , . . ., c s (a constant function when s = 1), and using the linear equations defining L, we can write some of the variables c j , τ j , t i as linear combinations of a linearly independent set of variables.We can then write the horospherical measure up to scaling as dν(q) = j=J1 dc j j∈J2 dτ j i∈J3 dt i , for some subsets of indices, and thus the preceding discussion shows that the total measure of the leaf is bounded.Now suppose that M q is not horizontally periodic.According to [SW04], the horocycle orbit U q consists of surfaces that are not horizontally periodic, but there is q ′ ∈ U q such that M q ′ is horizontally periodic.By Proposition 3.18, U q ⊂ W uu (q), and thus q ′ ∈ W uu (q).Since M q is not horizontally periodic, according to the first part of the proof, q ′ / ∈ W uu (q).This shows that the leaf W uu (q) has an accumulation point that is not contained in the leaf, which is to say that W uu (q) is not closed.5.1.Classification of horospherical measures in the eigenform loci in H(1, 1).The stratum H(1, contains a countable collection of complex 3-dimensional invariant subvarieties known as eigenform loci.This terminology is due to Mc-Mullen, who gave a complete classification of these invariant subvarieties in a sequence of papers (see [McM07] and references therein), following the first such examples discovered by Calta [Cal04].The horocycle invariant measures and orbitclosures for the U -action on an eigenform locus, were classified in [BSW22] (these classification results require Theorem 1.2 of the present work).We can classify the horospherical measures inside eigenform loci as follows: Theorem 5.4.Let M be an eigenform locus in H(1, 1), and let ν be an ergodic horospherical measure on M. Then either ν is the special flat measure m M or ν is the measure given by Proposition 5.1 on a closed horosphere W uu (q) of a horizontally periodic surface q ∈ M.
Proof.Suppose that ν is neither the special flat measure nor the measure supported on a closed horosphere of a horizontally periodic surface.This immediately rules out cases (1), (2), and (7) of the classification of U -ergodic measures in [BSW22, Thm.9.1].By Theorem 1.2, ν cannot be saddle-connection free, which rules out case (5).In each of the three remaining cases (3), (4), and (6), ν-a.e.surface has exactly one horizontal saddle connection or exactly two homologous horizontal saddle connections forming a horizontal slit.We conclude that in each of these three cases, ν(M ≥∞ ) = 0, and moreover that for ν-a.e.surface we can lengthen or shorten all horizontal saddle connections by moving in the real REL leaf.It follows that Rel s (M ≥c ) and M ≥c+s differ on a set of ν measure zero.Since ν is RELinvariant, ν(M ≥c ) = ν(Rel s (M ≥c )), and we conclude that the quantity ν(M ≥c ) does not depend on c for any finite c.
By Lemma 4.7 applied to any positive c, this quantity is bounded by ν(K) for some compact set K, and is therefore finite.So, taking the limit as c → 0, we see that ν(M) = ν(M ≥c ) for any c, and then taking the limit as c → ∞, we conclude that ν(M) = ν(M ≥∞ ), which is equal to 0 from above.This absurdity rules out the remaining cases (3), (4), and (6).5.2.An example of horospherical measure in H(2).Since there is currently no classification of horospherical measures in H(2), it is of interest to give examples.In this subsection we construct an ergodic horospherical measure which is not the special flat measure and is not supported on one properly embedded horospherical leaf.Its support is contained in the four-dimensional invariant subvariety M = H(2), the genus two stratum consisting of surfaces with one singular point of order two.
Recall from Corollary 3.24 that for a given ergodic horospherical measure, almost all surfaces are horizontally equivalent.In Figure 2 we show a typical surface q for our horospherical measure, and a typical topological picture of its horizontal saddle connections.These saddle connections will be denoted by δ and δ ′ .They disconnect the surface into a horizontal cylinder C, shaded gray in Figure 2, and a torus T .
A surface in H(2) with two horizontal saddle connections, bounding a horizontal cylinder.On the right, the corresponding horizontal saddle connection diagram.
Let x be the length of δ and δ ′ , let η be a saddle connection passing from top to bottom of the cylinder C, and let its holonomy be (a, τ ).Fix q ∈ π −1 (q).The height of C is constant and equal to a in a neighborhood of q in W uu L (q).The area of C is ax, and hence Moreover, changing τ by an integer multiple of x amounts to performing a Dehn twist in C so does not change the surface M q .Thus we may take When varying surfaces within their horospherical leaves, we change horizontal components of all saddle connections, and thus changing τ and x we stay in the horospherical leaf.Similarly, by Proposition 3.18, u s ′ q ∈ W uu (q) for every s ′ .Moreover, if M q = C ∪ T as above, the surface u (T ) s M q obtained by performing the horocycle flow on T and leaving C unchanged is also in W uu (q).It is easy to check that changing the three parameters x, τ, s gives a linear mapping in period coordinates, and that the three corresponding tangent directions in directions in T q (M) are linearly independent.Since dim(M) = 4, the dimension of the horospherical leaves in M is three, so the variables x, τ, s give an affine parameterization of a neighborhood of q in W uu (q).Since the height a of C remains constant in W uu (q), by equations ( 38) and (39), the variables x, τ take values in the bounded domain We construct a bundle B with base ∆, and a homogeneous space fiber, as follows.
Let Tor def = G/SL 2 (Z), the space of tori of some fixed area.This area is usually taken to be one, but by rescaling, can be taken to be any fixed number.For each x ∈ 0, 1 a , let Tor(x) denote the space of tori of area 1 − ax and with an embedded horizontal segment of length x.This is the complement in Tor of a closed set with empty interior (consisting of periodic horocycles of period at most x).Define B to be the bundle with base ∆ and such that the fiber over (x, τ ) ∈ ∆ is Tor(x).
Let µ be the G-invariant probability measure on Tor.Since the set of surfaces which do not admit an embedded horizontal segment of some length is of µ-measure zero, we can also think of µ as a probability measure µ x on Tor(x).For (x, τ ) ∈ ∆ let C = C(x, τ ) be a cylinder of height a, circumference x and twist τ .We have a map Ψ : defined by gluing the torus T from Tor(x), with a slit of length x, to the cylinder C(x, τ ).Let The image Ψ(B) is a five-dimensional properly embedded submanifold of M, consisting of all surfaces that can be presented as in Figure 2 for some fixed choice of a > 0. Along any sequence of elements (x, τ ) ∈ ∆ leaving compact subsets, we have either x → 0 or the area 1 − ax of T goes to zero, and in both cases the surfaces in the image of Ψ have short saddle connections.This shows that Ψ(B) is properly embedded.Since ν is invariant under translations using the affine coordinates x, τ, s, it is a finite horospherical measure supported on Ψ(B).

The geodesic flow and weak unstable foliation
Proof of Theorem 1.4.Let µ be a finite horospherical measure, and let µ t def = g t * µ.Our goal is to show that µ t → t→∞ m M .In order to prove that µ t → m M , it is enough to show that in any subsequence t n → ∞ one can find a further subsequence t ′ n so that µ t ′ n → m M .This will be accomplished in two steps.In the first step we will pass to a subsequence along which µ t ′ n → µ ∞ , and show that µ ∞ is also a probability measure.In the second step we show that µ ∞ is saddle connection free.Since µ ∞ is also horospherical by item (1) of Proposition 3.22, an application of Theorem 1.2 then completes the proof.
Since µ(M) is finite, we can renormalize so that µ(M) = 1.For the first step, we need to show that the sequence of measures {µ tn } is tight, i.e., for any ε > 0 there is a compact K ⊂ M such that for all large enough n, µ tn (K) ≥ 1 − ε.For this we will use Lemma 4.7.
Since µ(M) = 1, there is a c small enough that µ(M ≥c ) > 1 − ε 2 .By Lemma 4.7 there is a compact K ⊂ M such that for ever U -invariant measure ν.Applying this to ν = µ t for any t ≥ 0 gives where the penultimate inequality uses M ≥c ⊂ M ≥e −t c .By tightness, there exists a subsequential limit that is a probability measure.Now, letting µ ∞ be any limit along a subsequence t n , it remains to show that µ ∞ is saddle-connection free.We will show that for any ε > 0 and any Next, choose n large enough so that e −tn C < c and where the last inequality uses M <e −tn C ⊂ M <c .Since ε and C were arbitrary, we conclude that µ ∞ is saddle-connection free, and this concludes the proof.
Proof of Theorem 1.5.Let ν be a horospherical measure that is invariant by the geodesic flow.We will show that ν(M <∞ ) = 0. Since and so in particular it is finite.Therefore, in the limit as c → ∞, we see that ν(M [c,∞) ) = 0, and then again taking the limit as c → 0 we conclude ν(M <∞ ) = 0. Finally, by Theorem 1.2 we conclude that ν is the special flat measure.We now show that any leaf for the weak-unstable foliation is dense.Let q ∈ M (1) 1 , let U be an open set contained in M (1) and let f be a nonzero non negative compactly supported function whose support is contained in U .In order to show U ∩ W u (q) = ∅ we will show that there is p ∈ W u (q) such that f (p) > 0. Let ε def = M f dµ M > 0, let ω f denote the continuity modulus of f with respect to the sup-norm distance function, and let q ∈ π −1 (q).Using Propositions 4.3 and 4.4, let B be a regular box containing q such that for any t ≥ 0, the box . Let m M be the special flat measure on M (1) .By mixing of the geodesic flow (Proposition 4.9), there is T > 0 such that for any t > T , we have not intersect, except perhaps at singular points.Our convention is that cylinders are closed, and thus a cylinder and a saddle connection on one of its boundary components are not considered to be disjoint.We recall the notion of M-equivalence of cylinders, introduced in [Wri15a].Let M be an invariant subvariety, let q ∈ M and let C 1 , C 2 be two parallel cylinders in M q .The cylinders are called M-parallel if there is a neighborhood U of q in M, such that C 1 , C 2 remain parallel for all q ′ ∈ U.More precisely: • there is a lift L of M and open V ⊂ L and U ⊂ M such that q ∈ U, π| V : V → U is a homeomorphism and dev is injective on V; • for q ∈ V with q = π(q), represented by a marking map f : S → M q , and for any q ′ ∈ V, represented by f ′ : S → M q ′ , the sets Being M-parallel is clearly an equivalence relation.For a cylinder C on a translation surface M , we denote by G C the subgroup of GL + 2 (R) fixing the holonomy of the core curve of C.
. ., C r are parallel on M and g ∈ G C1 then the cylinder surgery corresponding to g, C 1 , . . ., C r is a modification of the surface M obtained by applying g to the C i and leaving the complement M r i=1 C i untouched.For example if C is horizontal then the elements of G C are of the form 1 s 0 t , with t > 0. The cylinder surgery of such a matrix with t = 1 consists of cylinder shears with shear parameter s.The cylinder surgery with s = 0 consists of cylinder stretches with stretch parameter t.By an appropriate conjugation, the definition of cylinder shears and stretches is extended to non-horizontal cylinders.We have: ]).For M, any q ∈ M, and an M-parallel equivalence class of cylinders C 1 , . . ., C r on M q , if g ∈ G Ci then the surface obtained from M q by cylinder surgery corresponding to g, C 1 , . . ., C r is also in M.
Suppose q ∈ M and C 1 , . . ., C r are M-parallel cylinders on M q , which are not necessarily a full equivalence class of M-parallel cylinders.Let L be a lift of M, let q ∈ L ∩ π −1 (q) and let V ⊂ H 1 (S, Σ; C) such that dev(L) = V.Varying g ∈ G C1 gives rise to a two dimensional collection (in the previous example, corresponding to possible choices of the parameters s, t) of surfaces, obtained from M q by cylinder surgery corresponding to g, C 1 , . . ., C r .This collection corresponds to a complex affine line in period coordinates.A generator for this complex line is where h i is the height of C i , γ i is the core curve of C i and γ * i is the dual class to γ i in H 1 (S, Σ).Moving along the line tangent to σ {Ci,hi} in M amounts to performing cylinder shears in each of the C i , and moving along the line tangent to i • σ {Ci,hi} in M amounts to performing cylinder stretches.
Below we will be interested in such one-parameter families of deformations that preserve M and are tangent to σ {Ci,hi} as in equation ( 41), in which the {C i } might not be a full equivalence class of M-parallel cylinders, and the h i might not be their heights.Note that for cylinder shears, such surgeries are well-defined for any value of the shear parameter s, and for cylinder stretches, they are well-defined as long as t > 0.
When σ {Ci,hi} ∈ V for q, L, V as above we simply say that σ {Ci,hi} is contained in the tangent space to M at q. Proposition A.2.If M is an invariant subvariety and q ∈ M is not horizontally periodic, then there is a nonempty collection of M-parallel cylinders C 1 , . . ., C r on M q which consists of cylinders disjoint from all horizontal saddle connections on M q , and positive h 1 , . . ., h r such that the class σ {Ci,hi} as in equation (41) is contained in the tangent space to M at q. Furthermore, there is constant A 0 > 0, depending only on M, so that if q ∈ M has no horizontal cylinders then one can choose a collection of cylinders with these properties so that, in addition, the sum of the areas of the cylinders is at least A 0 .We note that the h i in Proposition A.2 might not be the heights of the C i .The vector σ {Ci,hi} is tangent to the line in M obtained by applying different cylinders shears to each C i .In order to formalize this, for g i ∈ G Ci for each i, we define the cylinder surgery corresponding to {C i , g i } to be the modification of M q obtained by applying g i to each C i , leaving the complement M r i=1 C i untouched.With this terminology, the line tangent to σ {Ci,hi} is the collection of surfaces obtained by cylinder surgery corresponding to {C i , g i }, where s ∈ R and g i performs a cylinder shear with parameter sh i in C i .Similarly, the line tangent to i • σ {Ci,hi} is the collection of surfaces obtained by cylinder surgery corresponding to {C i , g i }, where s ∈ R and g i performs a cylinder stretch with parameter sh i in C i .
Proof of Proposition A.2.The validity of both statements is unchanged if we replace the surface q with some surface q ′ in its horocycle orbit U q.This follows from the facts that the U -action is linear in charts, preserves horizontal saddle connections and maps M-equivalent cylinders to M-equivalent cylinders preserving their area.According to [SW04], there is q ∞ in the closure of U q which is horizontally periodic, and we will see that the required properties hold for all q ′ sufficiently close to q ∞ .Let M ∞ be the underlying surface of q ∞ , let C 1 , . . ., C r be the horizontal cylinders on M ∞ , and for each i, A i , c i , h i , γ i denote respectively the area, circumference, height, and core curve of C i .Here we consider γ i as an element of H 1 (S, Σ) by using a marking f : S → M ∞ corresponding to q ∞ ∈ π −1 (q ∞ ).By Proposition A.1, σ {Ci,hi} belongs to the tangent space of M at q.
For any θ 0 > 0 there is a neighborhood U = U(θ 0 ) of q ∞ in M such that if q ′ ∈ U then the underlying surface has r cylinders C ′ 1 , . . ., C ′ r of circumferences c ′ A i , and with directions of core curves in (−θ 0 , θ 0 ).If C is a cylinder and σ is a saddle connection on a translation surface, recall that we say that σ crosses C if it intersects all the core curves of C. Since a cylinder contains no singularities in its interior, if a saddle connection intersects the interior of a cylinder, then it must cross it.Let s be the maximal length of a horizontal saddle connection on M q and let θ 0 be small enough so that a horizontal segment of length s cannot cross a cylinder of direction θ satisfying 0 < |θ| < θ 0 , with circumference at most c and area at least A.
By making U smaller, so that it is an evenly covered neighborhood of q ∞ , we can ensure that σ {C ′ i ,hi} belongs to the tangent space of M at q ′ .Indeed, if V is a connected component of π −1 (U) and q ′ , q ∞ ∈ V are preimages of q ′ , q ∞ respectively, then the core curves of the cylinders C ′ i , C i map to the same elements γ i ∈ H 1 (S, Σ) under the corresponding marking maps, and thus σ {C ′ i ,hi} = σ {Ci,hi} .Now suppose that q ′ ∈ U q ∩ U, and let M ′ be the underlying surface.Since q is not horizontally periodic, neither is q ′ .Therefore there is an equivalence class C 1 , . . ., C r of M-parallel cylinders on M ∞ , so that the corresponding cylinders C ′ 1 , . . ., C ′ r are not horizontal cylinders on M ′ , and satisfy the bounds in equation (42).Furthermore the maximal length of a horizontal saddle connection on M ′ is s, since the horocycle flow maps horizontal saddle connections to horizontal saddle connections of the same length.By choice of θ 0 , the cylinders in this equivalence class are all disjoint from horizontal saddle connections on M ′ .This proves the first assertion.
Let t be an upper bound on the number of horizontal cylinders for a surface in M and let A 0 def = 1 2t .The argument above works for any collection of M-parallel cylinders C 1 , . . ., C r which are horizontal on q ∞ and are not horizontal on q ′ .If q has no horizontal cylinders then neither does q ′ , and we can apply the argument with any equivalence class of M-parallel horizontal cylinders C 1 , . . ., C r on M ∞ .One of these classes must have total area at least 1 t , and thus for U sufficiently small, the sums of the areas of the corresponding cylinders C ′ 1 , . . ., C ′ r is at least A 0 .
Proof of Theorem 7.2.We first prove the first assertion.Let C 1 , . . ., C r be the Mparallel cylinders on M q , and h 1 , . . ., h r the positive numbers provided by Proposition A.2.We define q ′ as g • ϕ(q), where g ∈ GL + 2 (R) and ϕ is a cylinder surgery corresponding to {C i , g i }, and g i is the cylinder stretch with parameter sh i for some s < 0. Although the maps g, ϕ do not preserve the area of the surface, we will choose parameters so that their composition does.Moreover neither of these maps changes the vertical component of the holonomy of any curve.The GL + 2 (R)action preserves M, and by Proposition A.2, so does ϕ.The map g will increase the length of all horizontal saddle connections on M q , and the cylinder surgery ϕ will not affect their length, since the cylinders C i are disjoint from the horizontal saddle connections on q.
The area of ϕ(q) is smaller than the area of q since all of the cylinders C i are stretched by a negative parameter.Let A be the sum of the areas of the cylinders C 1 , . . ., C r .Then by choosing the parameter s appropriately, we can arrange so that the area of ϕ(q) is 1 − A 2 .We now set (43) and g def = t 0 0 1 .
Then g increases the lengths of horizontal saddle connections by a factor t > 1, and multiplies the area of ϕ(q) by t.This completes the proof of the first assertion.
For the second assertion, we use the second assertion in Proposition A.2 to choose the cylinders so the sum of their areas satisfies A ≥ A 0 .This ensures that the horizontal saddle connections on q ′ def = g • ϕ(q) are longer than the horizontal saddle connections on q by a factor of at least t, where t > 1 is as in equation (43) and t − 1 is bounded away from 0. In light of Proposition 3.23, q ′ ∈ W uu (q) will not have horizontal cylinders either.So we can apply the above argument iteratively, at each stage obtaining surfaces in W uu (q) with longer and longer horizontal saddle connections.Since the lengths of these horizontal saddle connections grows by a definite amount in each step, after finitely many steps they will all be longer than T .
Appendix B. Measures on H and Mod(S, Σ)-invariant measures on H m The goal of this section is to prove a result on the correspondence between Radon measures on H and Mod(S, Σ)-invariant Radon measures on H m .This result is part of the folklore but we were not able to find a reference; see [Fur73,Prop. 1.3] for an analogous result in a restricted setting.
We state the result in a general setting.Let X be a paracompact manifold and Γ a discrete group acting properly discontinuously on X.We will write the Γ-action as an action on the right.Let X = X/Γ be the quotient space and π : X → X the quotient map.If Γ acts freely then X is a manifold and π is a covering map.If Γ does not act freely then we can view X as an orbifold and π as a regular orbifold covering map (although no knowledge of orbifolds is assumed in this section).We do not assume that the action on Γ is faithful but, since the action of Γ is proper, the subgroup of Γ that acts trivially on X must be finite.
For q ∈ X let Γ(q) be the stabilizer of q in Γ.For any q ∈ X, we define a measure on X by (44) θ q def = q∈π −1 (q) |Γ(q)| • δ q , where δ q is the Dirac mass at q.The measure θ q is supported on π −1 (q).For any f ∈ C c ( X) and q ∈ X we have (45) X f dθ π(q) = γ∈Γ f (q • γ).
It follows from the fact that Γ acts properly discontinuously on X that the sum on the right-hand side is finite.
Definition B.1.Given a Radon measure ν on X we define a Radon measure ν on X, called the pre-image of ν, by the formula X f dν = X X f dθ q dν(q) for any f ∈ C c ( X). ( 46) Equation ( 46) defines a unique Radon measure ν on X in light of the Riesz Representation Theorem.To see that (46) converges, note that the integrand q → F (q) def = X f dθ q is a Borel function, which is supported on the compact set π(supp f ), and is bounded by D f ∞ , where D def = #{γ ∈ Γ : (supp f ) • γ ∩ supp f = ∅} is finite since the Γ-action is properly discontinuous.
By equation (45) the measures θ q are all Γ-invariant, and since ν is an average of the measures θ q , we have: Lemma B.2.The measure ν is Γ-invariant.
The following converse can be understood as a disintegration theorem for Γinvariant Radon measures on X.
In line (51) we used Claim B.5.In line (52) we used the definition of the pre-image applied to the function of compact support f • g.