Ergodic infinite group extensions of geodesic flows on translation surfaces

We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Fr\c{a}czek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces.


Introduction
In 1986 Kerchoff, Masur and Smillie showed that the the geodesic flow on a compact translation surface is ergodic in almost every direction [17]. On the other hand, the study of the dynamical properties on periodic infinite translation surfaces is in its infancy. There is a natural necessary and sufficient condition for the recurrence of the geodesic flow in almost every direction for Z or R covers [11]. Generic non-periodic translation surfaces are almost surely recurrent [29]. The almost sure diffusion rate in the periodic full occupancy wind-tree model, a certain important example of infinite translation surface, has also recently been understood [7]. We turn to the question of ergodicity. In [12], a natural example the "simple staircase" was shown to be ergodic in a.e. direction. In fact, the authors noticed that the ergodicity reduces to a classically studied question of ergodicity of cylinder flows (group valued skew products) over circle rotations initiated by Conze and studied by many authors, see [4] for a good survey. The question of ergodicity for other periodic translation surfaces reduces to the question of ergodicity of cylinder flows over interval exchange transformations.
Recently, Fraczek and Ulcigrai have shown that certain periodic infinite translation surfaces are not ergodic in almost every direction [8]. More specifically, if the compact square-tiled translation surface M is in the stratum H (2), and the infinite translation surfaceM is a Z unramified cover of M , then the infinite surface is not ergodic in a.e. direction. They showed a similar result for some Z 2 extensions such as the periodic full occupancy wind-tree model.
In this article, we study the case when the surfaceM is a ramified cover of M . This is generically the case. In the language of ergodic theory the flow onM is a Gvalued skew product over the geodesic flow on M for some Abelian locally compact group G. Our result is a sufficient condition for both recurrence and almost sure Date: December 21, 2013. The first author is supported by the Center for Advanced Studies at Ben Gurion University of the Negev as well as the Israel Council for Higher Education, and was partially supported by the Erwin Schrödinger International Institute for Mathematical Physics during preparation of this manuscript. This research is partially supported by the ANR project Perturbations. ergodicity to hold. This condition is based on ideas developed by M. Boshernitzan [2] and K. Schmidt [26].
While our theorem is somewhat technical to state, it is applicable to many concrete examples. Suppose that M is a square tiled surface with a single-cylinder direction (for example the torus), then any reasonable method of producing a Z d (or more generally an Abelian locally compact group G) infinite surfaceM which is a skew product over M is generically ergodic in almost every direction. We also produce concrete examples of staircases ergodic in almost every direction such as the staircase in Figure 1. In the appendix, we give a procedure for determining which square-tiled surfaces have a single-cylinder direction, as well as a quick method for constructing such surfaces and determining their genus as well as the number and type of singularities.  J. Chaika and P. Hubert have recently announced the following related result [3]: for a given f : [0, 1) → R with integral 0, which is the finite sum of characteristic functions of intervals, for a set of full measure of interval exchange transformations T the skew product T f (x, i) = (T x, i+f (x)) into the closed subgroup of R generated by the values of f (x) is ergodic. Their methods of proof are close to ours.

Translation surfaces
In this section we will give background material on translation surfaces as well as some definitions for later use, see [15,21,31] for details. A translation surface is a 2-dimensional manifold M and a finite set of points D = {d 1 , d 2 , . . . , d m } and an open cover of M \ D by sets {U α } together with charts φ α : U α → R 2 such that for all α, β, with U α ∩ U β = ∅, φ α • φ −1 β is a translation on its domain of definition and at each singular point d i ∈ D the surface has a 2πn cone singularity for some integer n ≥ 1.
The surface M has a flat metric obtained by pulling back the Euclidean metric on the plane via the coordinate charts. This metric is defined away from the set D. In this metric geodesics that do not go through singularities project via the charts to straight lines in the plane in a fixed direction, and such geodesics are either periodic or simple (do not intersect themselves). If the geodesic in direction θ starting at x ∈ M avoids D then we define the geodesic flow ϕ θ t (x) to the the point obtained after moving in the direction θ for a time t starting at x. Since θ is usually fixed, we will often suppress the θ dependence of the flow and simply write ϕ t (x). If the geodesic hits D, then the geodesic flow ϕ t (x) is defined up to this time. The geodesic flow preserves the surface area measure µ.
Let Γ ⊂ SL 2 (R) denote the Veech group of the surface, the group of affine transformations (seen in SL 2 (R) as acting on the charts of M ) which preserve the structure of M , and therefore permute the singularities D. If SL 2 (R)/Γ is of finite volume (i.e. Γ is a lattice), we say that M is a Veech surface.
Throughout the article we will assume that M is a compact translation surface; by [17] the geodesic flow is (uniquely) ergodic in almost every θ. We will form skew products over these flows in the following manner: let G be a locally compact Abelian group with translation-invariant metric · , whose identity element we denote 0, with Haar measure ν, and for a finite collection of geodesic path segments {γ i } (disjoint without loss of generality) whose union we denote γ, let f : γ → G be constant on each γ i and 0 elsewhere. Given a point x ∈ M and a direction θ which is not parallel to any γ i , denote Then the transformationφ t , fromM = M × G to itself, is given bỹ and it is this flow which we study;φ t preserves the measureμ = µ × ν. An essential value of a general skew product {M ,μ,φ t } is some g ∈ G such that for every ǫ > 0 and A ⊂ M with µ(A) > 0, there is some t > 0 such that We denote by E(φ t ) the union of all essential values. Then if E(φ t ) is nonempty, it is a closed subgroup of G, and {M ,φ t } is ergodic if and only if E(φ t ) = G and {M, ϕ t } is ergodic (see [1] for example). If 0 ∈ E(φ t ), then the system is called recurrent.
Definition 2.1. We will call a sequence of sets C n a quasi-rigidity sequence of sets in the direction θ if there is a sequence t n → ∞, a sequence ǫ n → 0 and a fixed ǫ > 0 such that That is, each set is of nontrivial measure, the map ϕ tn acts almost like the identity on these sets, and they are nearly invariant under the flow ϕ 1 (the use of ϕ 1 is arbitrary; any fixed positive time would do).
Lemma 2.2. Suppose that {M, ϕ t } is ergodic and that for some g ∈ G we have a quasi-rigidity sequence of sets C n such that Then g ∈ E(φ t ).
Proof. This is merely a restatement for continuous flows of [5,Cor. 2.6]. In that reference, it is assumed that G = R d , but the techniques apply to any locally compact Abelian group with Haar measure and translation-invariant metric, as we have assumed. A similar remark appears as [16,Thm. 11]; the notion of restricting a search for essential values to certain subsets whose measure is bounded away from zero is a common step in the study of periodic group extensions of finite-measure systems. The error compared to the flow in direction θ relative to the width of the cylinders is bounded for periodic approximations, bounded away from zero for fairly good periodic approximations, and converging to zero for good periodic approximations.
Definition 2.3. If the geodesic flow ϕ t in a particular direction θ decomposes the space M into a finite union of sets on which ϕ t is periodic (see Figure 2), the direction θ is said to admit a representation by periodic cylinders. Given a sequence {θ i } of directions which admit a representation by periodic cylinders, denote the cylinders by A i,j , the width of A i,j by α i,j , and the height of each cylinder (i.e. the period of the geodesic flow in this cylinder) by h i,j .
The quantity If furthermore E i → 0 we call the sequence {θ i } a good sequence of periodic approximations. If there is some δ > 0 such that for all i we have E i ∈ (δ, 1/2) we will call the sequence a fairly good sequence of periodic approximations. See again Figure 2. If inf {µ(A i,j ) = h i,j · α i,j : i = 1, 2, . . . , j = 1, 2, . . . , n i } > 0, we will say that this sequence has cylinders of comparable measures. Proof. That µ(C k ) is bounded away from zero follows from the condition that the cylinders have comparable measures. As the sequence θ i was taken to be a sequence of good periodic approximations, the relative error converges to zero. It now follows from the fact that each C k is bounded away from the boundary of A i k ,j that for each x ∈ C k , the orbit through time h i k ,j of x remains in A i k ,j , from which it follows that j (the difference is exactly those x so close to the boundary of C k that they flow out of the subcylinder in unit time), which certainly converges to zero. Definition 2.5. Suppose that the geodesic flow in direction θ admits a representation by a single periodic cylinder. Then θ will be called a single-cylinder direction.
Single-cylinder directions are of great convenience in studying translation surfaces. It was shown in [13] that if M is square-tiled with a prime number of tiles and in H(2) (genus two with a single singularity), then M has a single-cylinder direction. In [22,Cor. A.2] McMullen extended this result to all square-tiled surfaces in H(2). Kontsevich and Zorich showed that there is a dense set of square tiled surfaces in any stratum with single cylinder directions [18,Lemma 18]. Finally Lanneau and Nguyen have identified certain substrata of H(4) and H(6) which have single cylinder directions [19]. See Appendix A for a general technique for constructing square-tiled surfaces with a single-cylinder direction.
The following is a standard argument in the study of translation surfaces, but hardly elementary: Lemma 2.6. Suppose that {M, ϕ t } has at least one direction θ ′ which admits a representation by N periodic cylinders, and suppose that M is a Veech surface with Veech group Γ. Then for almost every direction θ, there is a sequence {θ i } of directions, chosen from the directions Γ(θ ′ ), which are a good sequence of periodic approximations, with N cylinders of comparable measures, to the flow in the direction θ. Furthermore, for almost every direction θ there is a sequence {θ i } taken from the directions Γ(θ ′ ) of fairly good periodic approximations with N cylinders of comparable measures.
Proof. Any element γ ∈ Γ will transform the flow in direction θ ′ to a flow in another direction which must necessarily also admit a representation by periodic cylinders. The number of cylinders and their measures are not changed by the action of γ ∈ SL(2, R), so all that must be shown is that for almost every θ, a sequence may be chosen out of this orbit which are a good or fairly good sequence of periodic approximations to the flow in the direction of θ.
The assumption that M is a Veech surface ensures that Γ, the Veech group, is Fuchsian of the first kind. The fact that almost-every direction has a sequence of good approximations coming from the orbit under Γ of the single-cylinder direction follows directly from [23, §9] (see [28] for a more general case). We state the application of this theorem to our setting in the language of this article: for every ε > 0, there exists a set of full measure Θ(ε) ⊂ S 1 such that for any θ ∈ Θ(ε), there exists a sequence of N cylinder directions with slopes p n /q n such that |θ − p n /q n | ≤ ε/q 2 n . Similarly as in [14], the conclusions for good approximations follow from Patterson's theorem. The appropriate lower bound for fairly good approximations follows from the ergodicity techniques used to prove similar statements in [16, §2.2]: the orbit of a completely periodic direction under Γ will intersect arbitrary sets of positive measure infinitely often in SL 2 (R)/Γ. This quotient is of finite volume and has finitely many cusps by assumption, so a nontrivial annular neighborhood around a generic slope will also contain infinitely many points from the orbit Γ(θ ′ ); the use of a fixed annulus as opposed to a sequence of shrinking neighborhoods distinguishes fairly good approximations from good approximations.
In particular, if M is a Veech surface, then if the flow in direction θ ′ is not periodic the surface admits a representation by periodic cylinders of comparable measures with whatever strength in approximation we desire. The conclusion of Lemma 2.6 can in fact be strengthened to state that the set of exceptional directions (those which cannot be approximated to the desired accuracy from the orbit under Γ of any given single-cylinder direction) is not just of measure zero, but indeed of Hausdorff dimension less than one. See again [16] for similar statements; we state only that almost-every direction can be so approximated in the interest of simplicity and generic appeal.

Generic Constructions
Definition 3.1. Let θ, the direction of the geodesic flow, be fixed such that {M, ϕ t } admits a sequence of periodic approximations {θ i }. LetD denote an arbitrary finite subset of M . Then a point x ∈ (D \ D) is said to avoid D ∪D in direction θ if for some fixed ǫ > 0 there is a sequence {ǫ i } so that each of the following holds: • for each i, x belongs to the interior of some A sequence of neighborhoods C i for which x satisfies this definition for the same ǫ > 0 will be called a sequence of orbit-neighborhoods for x. See Figure 4 for an illustration (the sets C i,1 and C i,2 will be defined later). If every point inD avoids D ∪D, and if D ∩D = ∅, then we will say thatD is a self-avoiding set in direction θ.
Note that we merely need some sequence of periodic approximations for each θ; in situations where a canonical method for finding approximations exists, a subsequence of this canonical set suffices.
Proposition 3.2. Suppose θ has a good sequence of periodic approximations and fix P ∈ N. Then almost every elementD ∈ M P (with respect to the product measure) is a self-avoiding set in the direction θ.
Proof. It suffices to show that the proposition is true for P = 1; larger sets may then be inductively constructed one point at a time by presuming x 1 through x P −1 are in D for the purposes of finding x P . Let θ be fixed such that {θ i } is a good sequence of periodic approximations. For a fixed ǫ > 0, let B i = B i (ǫ) be the complement of the ǫ ′ -neighborhood of the boundaries of the cylinders, where Figure 3. Now note that if x belongs to infinitely many B i (ǫ), it is self-avoiding singleton: for sufficiently large i we have the error E i < ǫ. As for this choice of ǫ the set of self-avoiding singletons in direction θ is of measure at least (1 − ǫ), and ǫ was arbitrary.
Corollary 3.3. Suppose that almost every θ admits a sequence of good approximations. Then almost every subsetD ∈ M P is self-avoiding in almost every direction.
Proof. We certainly have almost surely thatD ∩ D = ∅. For a full-measure set of θ, a full-measure set ofD is self-avoiding in direction θ. Apply Fubini's theorem to reverse the order of quantifiers. Let S = {s 1 , . . . , s n } be a finite subset of M (possibly including points in D), L i > 0 and τ i ∈ S 1 (i = 1, . . . , n) be for the moment arbitrary. Let γ r be a geodesic of length L r > 0 in direction τ r originating at s r . Note that we write "a" geodesic since if s r ∈ D there may be several possible such geodesics and if a geodesic originating at s r arrives at a point of D, there are multiple possible continuations. Although generically this will not happen, a priori we allow all such geodesic segments. Let a r,1 , . . . , a r,N (r) be a finite set of points in γ r (N (r) is arbitrary but finite); assume that a r,N (r) = ϕ Lr (s r ) (the terminus of γ r ) and set a r,0 = s r . Denote by A the (finite) collection of all a r,j together with any d ∈ D such that d ∈ γ i for some i.
Corollary 3.4. For fixed n, for almost every construction of γ r and A (with initial points s i chosen according to a probability measure on M absolutely continuous with respect to µ, lengths l r chosen according to some probability measure absolutely continuous with respect to Lebesgue on R + and directions τ r by a measure absolutely continuous with respect to Lebesgue measure on S 1 , and A then chosen according to a measure on γ which is absolutely continuous with respect to Lebesgue measure), the set A is a self-avoiding set.
Proof. We apply Proposition 3.2, noting simply that we certainly have almostsurely that no γ i crosses a singularity d ∈ D.
Let G be an Abelian locally compact group with translation invariant metric endowed with Haar measure ν, and we will use the segments γ i to define our skew product {M ,φ t } as in §2. Define a function f : M → G to be piecewise constant on each γ i with discontinuities a subset of A; denote the value of f between a i,j−1 and a i,j by f i,j . Finally, denote The function f is defined to be identity off the segments γ i ; the values σ i,j are the 'jumps' seen in f while flowing along the path γ i .
with orbit neighborhoods C k of heights h k → ∞ will be called essential in direction θ if there is some P such that along an infinite subsequence of orbit neighborhoods we have sup Showing that points are essential is in general a very demanding requirement. Single-cylinder directions, however, provide a ready approach. Proposition 3.6 (Koksma-type inequality). Suppose that M is a translation surface and the function f is defined as above. Assume further that θ ′ is a single cylinder direction, with the cylinder height given by h (and width 1/h). Suppose that the flow in the single cylinder direction θ ′ , sees an average value of the identity in G. Then, for every x not on the boundary of the cylinder we have Proof. We denote by q i,j the projected length of the geodesic segment connecting a i,j−1 to a i,j in the direction transverse to θ ′ . That the average value of the sums in the cylinder is identity is equivalent to This condition is sufficient for recurrence of the flow in the event that G = Z or R (see [25]), and is clearly necessary in all cases where G contains elements of infinite order.
Given a particular segment, the points a i,j−1 and a i,j are in the cylinder (possibly on the boundary), and the segment between them completely crosses the cylinder some number of times. Denote this number by P i,j . There are also possibly two partial crossings corresponding to the segment leaving a i,j−1 and arriving into a i,j (in P i,j = 0 then there is only one such segment); denote by p ′ i,j and p ′′ i,j respectively the lengths of these two segments when measured in the transverse direction to the flow. We have the latter of which we rewrite as As the entire space is shown in this single periodic cylinder, the entire segment is represented by these pieces, so (1) translates to The periodic orbit of any point x in the direction θ ′ (ignoring the orbits containing the points a i,j ) crosses the segment from a i,j−1 to a i,j a total of according to whether it crosses neither 'partial segment,' one of them, or both. Then we have Theorem 3.7. Let a = a j,i be an essential point with respect to a good sequence of periodic approximations. Then σ = σ i,j is an essential value of the skew product {M ,φ t } for the flow in direction θ. In particular, the skew product is recurrent.
Proof. Using a good sequence of periodic approximations allows us to consider for almost every x, for sufficiently large i the flow in direction θ and the flow in direction θ i are indistinguishable through length h i,j , at least insofar as the sums S hi,j (x) are concerned. The point a has a sequence of orbit-neighborhoods C k which remain in the same periodic cylinder A i k ,j . The only discontinuity of f in C k is the point a by definition. So we split the orbit-neighborhood into two sets of equal measure C k,1 and C k,2 so that the sum S hi k (x) is constant on each. Furthermore, the difference between the two values taken is exactly σ. See Figure 4. Without loss of generality, then, assume that for x ∈ C k,1 and y ∈ C k,2 we have As there is a uniform bound P across all values and G is locally compact, by passing to a subsequence (which we continue to denote {i k }) we may find some limit Both C k,1 and C k,2 are a quasi-rigidity sequence sets by Lemma 2.4, so by Lemma 2.2, both g and g + σ are in E(φ t ). As the set of essential values is a group, σ ∈ E(φ t ).  That is, for a generic method of making 'cuts' γ i on the surface M and using them to form a G-fold cover, as long as the values f i,j include generators for the group G, the extension is ergodic in almost every direction provided we may show the discontinuities are essential points in almost-every direction.
Example 3.9. Let M be the two-torus T = [0, 1) 2 with standard identifications (which is flat without any singularities). Almost every direction θ has a good sequence of single-cylinder periodic directions given by a subsequence of the continued fraction convergents to θ, and the flow {M, ϕ t } is ergodic in any irrational direction. Let {g i } generate the group G, and for each i construct a pair of parallel geodesics of equal length, at random starting points, in random directions, and of random lengths (with the relevant distributions are all absolutely continuous with respect to Lebesgue). Define f to be ±g i on these pairs. Then in any direction not parallel to such a segment, the projection of f to the base segment [0, 1) × {0} will be of bounded variation (Var(f ) = 4 |g i |) and have integral zero, so by a combination of Propositions 3.2 and 3.6, Theorem 3.7, and Corollary 3.8, for generic such constructions the skew product {M ,φ t } is ergodic in almost every direction.
Instead of generating 'parallel pairs' of cuts, we could generate single cuts which are subdivided appropriately. For example, suppose that that a i g i + b i g ′ i = 0, where a i and b i are in R, and the smallest closed subgroup in G containing all of the {g i , g ′ i } is G itself. Generate i cuts of random length in random directions, and subdivide each cut into two portions whose ratio is a i /b i ; let f take value g i on the segment of proportional length b i and value g ′ i on the other.
Note that any 'reasonable' method for generating cuts on M which ensures that the transfer function f has zero integral in almost-every direction will also generically ensure that the skew product {M ,φ t } is ergodic in almost every direction, provided that bounds on sums in cylinders may be established; by Proposition 3. 6 Veech surfaces which admit a single-cylinder direction will automatically obey such a bound.
Example 3.10. Let M be square-tiled : a finite cover of the torus such that M may be considered to be tiled by squares which share corners. Then M is a Veech surface; this fact was first shown by Veech using Boshernitzan's criterion in [30], but modern proofs center around the use of arithmetic subgroups of SL 2 (R) (see e.g. [13, App. C], [9,Thm. 5.5]). Assume further that M has a single-cylinder direction, so that almost every direction θ admits a good sequence of periodic singlecylinder directions. By the same argument as in the previous example, then, for a generic method of constructing f the corresponding flow is ergodic in almost every direction.
See Appendix A for material relating to determining which square-tiled surfaces have a single-cylinder direction as well as for a simple method of generating such surfaces. While convenient, the assumption that M admits a single-cylinder direction is not necessary to apply the ideas and results of this section; an example of a class of surfaces with no single-cylinder directions for which certain constructions generically produce ergodic covers is addressed in Example 4.6 and the remark following that example.
We remark that the idea of comparing ergodic sums on two different sides of a discontinuity goes back at least as far as [26] and can be found more recently in [6], and the notion of finding generic points bounded a positive proportional distance from the orbits of singularities can be found in [2]. A similar approach to studying ergodicity of real-valued skew products over rank-one interval exchange transformations can be found in [3].

Generalized Staircases
In this section we shall apply the same principles as in our generic constructions to certain specific skew products of square-tiled surfaces called generalized staircases. It is direct to see that if M consists of a single square then only trivial natural staircases are possible. If we represent the torus with two squares, however, then a natural staircase which is a ramified cover becomes possible. This cover of the torus was presented in [12], where is was shown to be equivalent to a previously studied cylinder transformation, known to be ergodic in all irrational directions.
Generalized staircases can be parameterized by the base points and lengths of their respective cuts. The results of the previous section may be readily adapted under the addition restriction that the cuts are parallel to sides to ensure that generic generalized staircases are ergodic in almost every direction. However specific directions can have quite different behavior.
Example 4.2. Let M = T be the two-torus. Suppose that θ is a direction which admits a good sequence of periodic approximations. Then there exist uncountably many choices of β, r such that with cuts as in Figure 5, the skew product model of the corresponding generalized staircase {T,φ t } is nonregular (the only essential values are {0, ∞}). This staircase is the same as the skew product considered by Conze in [4,Thm. 4.2] where exactly this conclusion is shown. Therefore the geodesic flow is recurrent (0 is an essential value), but not ergodic (not all of Z are essential values).
The situation of natural staircases is in some sense more interesting: the severe restrictions imposed by the integer endpoints of the cuts present an immediate obstacle to directly applying the techniques of the previous section without modification. If the staircase is given by an unramified cover, then all endpoints of the cuts are singularities, and there is no adapting our techniques (indeed, for M ∈ H(2) the flow will not be ergodic in almost every direction by [8]). So assume that at least one of the cuts has an endpoint (with integer coordinates, as we consider a natural staircase) which is not a singularity. In a square-tiled surface, the directions +1 r −1 β + r β θ Figure 5. A generalized staircase over the 2-torus which is nonregular for uncountably many parameters β, r.
which admit representation by periodic cylinders correspond to rational-slope flows on the plane. The endpoints of our cuts are all integer points in the plane, as are all of the singularitiesD; the endpoints of our cuts will always belong to the edge of the periodic cylinders and therefore orbit into singularities in the directions of the periodic approximations, a rather insurmountable impediment to directly apply Definition 3.1.
Thus far, however, the only identifications of the edges of the cylinders we have used is the identification of the top to the bottom, but there are also certain identifications on the sides of the cylinders. It is illustrative to make explicit which directions for square-tiled surfaces form different types of periodic approximation. Let M be square-tiled on k squares. The flow in direction θ induces a rotation by θ ′ = 1/θ mod 1 on S 1 when the bases of all squares are projected to a single circle. Let the continued fraction expansion of θ ′ be given by Then the periodic approximations corresponding to the sequence of the convergents p n /q n to θ ′ . The relative error E n of the approximation is no more than kq n q n θ ′ (the standard error of q n θ ′ multiplied by the number of squares, as a single cylinder must wind through all of them, and divided by the width 1/q n ), and it is standard that k a n+1 + 1 ≤ kq n q n θ ≤ k a n+1 .
Therefore a good sequence of periodic approximations corresponds to a subsequence of partial quotients diverging, while a fairly good sequence of periodic approximations corresponds to a subsequence of partial quotients at least as large as 2k + 1, but bounded above. Both situations are clearly satisfied for generic θ ′ . It is not immediately clear that if M admits a single-cylinder direction that both conditions are generically satisfied along the subsequence of p n /q n which correspond to these single-cylinder directions, but this fact holds by the Patterson/Sullivan theorem used in Lemma 2.6.
where the sum is with respect to the flow in the periodic direction θ i . Then e i is a finite essential value of {M ,φ t } in direction θ. If d is the endpoint of several cuts, then the essential value is given by the sum of the values of f on the cuts terminating at d. In both cases the staircase is recurrent. Figure 6. A modified orbit-neighborhood used in staircases: C i,1 and C i,2 are the two colored parallelograms. The region outside the cylinder appears somewhere inside via the appropriate identification and there are no singularities within C i,1 or C i,2 . No cuts have endpoints within these neighborhoods other than the specified cuts with values ±e i with common endpoint d. The time h ′ flow (to traverse the entire vertical distance) acts as horizontal translation by E n /q n ∈ [δ/q n , 1/2q n ] on this set. The constant δ is from the definition of fairly good approximation.
Proof. The singularities and ramification points are arranged along the edges of the periodic cylinders A i,j . The flow in the direction θ i induces the rotation by p i /q i on the projection of all horizontal bases of the k squares to a single circle. Note that the vertical distance between successive singularities and ramification points on the boundary of the cylinder (i.e. the geodesic distance in direction θ i between them) must be at least q i , and the height of any cylinder is no more than kq i , where k is the number of squares in M . Consider the flow of a vertical interval of length q i both above and below d on both sides of A i,j , and arrange the cylinder so that this interval is at the bottom of one side. We flow this interval forward until intersecting the height of d within the cylinder, and backwards to the height of the base of the cylinder; this notion is well-defined despite the flow exiting the cylinder in one of these directions as there are no singularities near d (see Figure 6). No other discontinuities of f are in these vertical segments, then, and furthermore the cuts with endpoint d cuts across the cylinder A i,j from exactly one of the two copies of d on the boundary of A i,j (because d is an endpoint of the cut). See Figure 6; the two halves of this orbit neighborhood are C i,1 and C i,2 ; because there are no singularities within height q i of d along the boundary of the cylinder there is no problem extending these neighborhoods outside the boundary. The distances are maintained as drawn, and any cut besides those with endpoint d which might intersect these neighborhoods must pass completely through them; the only endpoint of any cut seen in C i,1 and C i,2 is d. The difference in the ergodic sums is therefore exactly a sum of ±e i , the values of f on the cuts terminating at d. As the error E i is not larger than half the width of A i,j (as we have a fairly good sequence of periodic directions), these neighborhoods may flow for time to traverse the entire height of the cylinder without intersecting the sides (except for the small bit at the bottom where there are no singularities, as mentioned prior). The measure of each orbit neighborhood can be measured by the product of the vertical length of the segment between d and the nearest singularity (which is at least q i as we are square-tiled) and the horizontal drift; the error term E i , which is a positive portion of 1/q i as we have assumed only a fairly good sequence of periodic approximations; the measure of each C i,m is at least δ, where δ > 0 is the fixed constant from the definition of a fairly good sequence of periodic approximations. Therefore the sets C i,1 and C i,2 again form a quasi-rigidity sequence of sets (their near-invariance under ϕ 1 is clear from construction). As in Theorem 3.7, the the sum of the values of f on the cuts terminating at d is the difference in sums between infinitely many of the two C i,1 and C i,2 , and is therefore an essential value. Proof. As M admits a single-cylinder periodic direction, almost every θ has a fairly good sequence of single-cylinder periodic directions by Lemma 2.6. As there is only one cylinder to consider, the relevant bound on sums used in Theorem 4.3 is satisfied by Proposition 3.6. As we assume that each cut has a non-singularity endpoint not shared by any other cut, the group of essential values therefore contains each e i . Example 4.5. Consider the surface on at least five squares given by Figure 7; one row of two squares and one row of at least three. The first presentation with cuts as marked on the left and its associated staircase is not ergodic in almost every direction by [8,Thm. 1.4]; The surface is in H(2) and both cuts are nontrivial in H 1 (M, Z). The presentation on the right, however, has both cuts with an endpoint which is not a singularity; by Theorem 4.3, then, the flow on the corresponding staircase is ergodic in almost every direction. Note that the two compact surfaces are the same, and the only difference is in the choice of cuts! The condition that there is some direction θ ′ such that {M, ϕ t } decomposes into a single periodic cylinder is convenient but not necessary: Example 4.6. Let M be the square-tiled surface in Figure 8, and assume that the number of tiles in each row is odd and at least five. Then if we construct a Zstaircase using the cuts marked, this staircase is ergodic in almost every direction. This surface, however, admits no single-cylinder directions.
Proof. That each row has more than three squares ensures that the two cuts with value +1 do not join together to form a loop; each cut of value +1 therefore has both endpoints not singularities. The assumption that the number of squares in each row is odd guarantees the existence of an integer involution point; a vertex around which we may rotate the entire figure by π. Suppose that some flow in slope q/p > 1 is a single-cylinder direction. This flow induces a rotation by p/q on the bases of the squares; a periodic cylinder will be of width 1/q and may be taken to begin with an interval of length 1/q extending horizontally from this involution point. That this surface has no single-cylinder directions follows from Corollary A.5, where this particular example is referenced.
The surface clearly has a decomposition into two cylinders of equal measure and length: the horizontal flow achieves such a decomposition, and the two cylinders are exchanged by the involution. By setting the involution point to be the origin, the involution itself may be represented as −Id. Of course −Id is in the center of Γ, the Veech group. It follows that any image of these two cylinders under any element of Γ also consists of two components which are exchanged by the involution.
Note that f , the function used to define the staircase, is invariant under this involution. Combining this fact with the fact that the involution exchanges the two cylinders, we see that the ergodic sum in each periodic cylinder is equal to the other. As their sum must be the average of the function, each periodic cylinder sees a total sum of zero. We may then consider the endpoint of the cut with value +1 and directly apply the same construction in Figure 6 to find an essential value of ±1 for this staircase.
Note that any probabilistic technique of generating covers (not restricted to natural staircases, so the cuts are simple geodesic segments as in §3) on these surfaces which respects the following  H(1, 1) and has no single-cylinder directions. We form a staircase using the indicated values on the specified unit intervals.
will generically produce covers which are ergodic in almost-every direction, exactly via the techniques of §3.

Appendix A. Permutations and parallelogram-tiled surfaces
While the aforementioned results of Hubert-Lelievre, McMullen, Kontsevich-Zorich and Lanneau-Nguyen guarantee for certain square-tiled M the existence of directions θ i which decompose M into a single periodic cylinder (providing the bound on ergodic sums necessary in the definition of essential points, Definition 3.5), it is of interest to quickly determine which square-tiled surfaces admit such a direction in general.
Any square-tiled translation surface, with k total squares, may be identified with a pair of permutations in Σ k (the permutation group of k elements) (σ h , σ v ). From a combinatorial perspective it is unimportant that the tiles are squares; any parallelogram tiling would suffice (and still allows a notion of 'top/bottom' and 'left/right'). The permutation σ h is given by the identification of the right side of the tile indexed by i with the left side of the tile indexed by σ h (i), while the top of tile i is identified with the bottom of the tile of index σ v (i): see Figure 9. The observation that square-tiled surfaces may be indexed by pairs of permutations appears in [27]. It is clear that any choice of (σ h , σ v ) along with a choice of parallelogram tile defines a square-tiled translation surface M , and that any choice of charts for a surface M which gives a tiling by parallelograms can be identified with such a pair of permutations. In the search for single-cylinder directions the distinction between squares and parallelograms is unimportant (the two being distinguished only by an affine transformation), so we will use squares without loss of generality.  Proof. Suppose that there is a proper subset S ⊂ Z k which is invariant under each of σ h , σ v . Then the collection of squares indexed by S contains no identification to squares indexed by Z k \ S, thereby forming at least two connected components. On the other hand, if there are no proper invariant subsets, then given any two indices i and j, there is a sequence of σ tn which connect the square indexed by i to that indexed by j through some sequence of side identifications (where t n ∈ {h, v}).
Consider the upper-right corner of the tile indexed by i. The commutatorσ exactly encodes the trajectory of a counter-clockwise circle around this corner, and the conclusion is direct from this observation. Let V be the number of cycles in [σ v , σ h ]. Then M is given as a surface with k faces (the tiles), 2k edges (each edge is used in exactly two tiles) and V vertices. By computing the Euler characteristic, the genus g is given by assuming that M is connected (otherwise we could consider each connected component separately).
Corollary A.2. Letσ be an arbitrary element of the commutator subgroup of Σ k . Then the number of cycles inσ is either odd or even according to whether k is odd or even.
Proof. It is not difficult to show (for example, by induction on n) that the commutator subgroup of Σ k is A k , the alternating subgroup, and that A k is in fact exactly the set of commutators. The claim may be then shown directly for any element of A n . However, it may also be derived here from the Euler characteristic and Theorem A.1: the genus must be an integer. In the event thatσ is the commutator of two permutations with a shared proper invariant subset, the corresponding translation surface is not connected, but we may apply the same reasoning to each connected component. Now suppose that we flow along M in a rational direction (recall that without loss of generality we are using unit squares as tiles). Assume that the slope is q/p ≥ 1, so that in the time necessary for the flow to traverse a unit of vertical distance, it traverses a horizontal distance of p/q. Then we define a new tiling of M by kq parallelograms: the new tile of index (i − 1)q + j (for i = 1, . . . , k and j = 1, . . . , q) is defined to be the parallelogram given by flowing the interval [(j − 1)/q, j/q) in the base of the square of index i for a vertical distance of one. See Figure 10. If the flow were of slope less than one, we could perform an analogous retiling using intervals of the vertical edges of the original tiles, flowed to a horizontal distance of one.
The new tiling can also be described with a pair of permutations σ ′ h , σ ′ v . We are free to choose which direction of these new parallelograms should correspond to 'vertical'; whether the direction q/p was larger or less than one, there is a natural choice. As our figure uses a flow of slope larger than one, we let the flow correspond to the 'vertical' direction in the new tiles.
Proof. The horizontal identification is direct (refer again to Figure 10). For the vertical identifications, note that flowing to traverse one unit of vertical distance, we either flow only through the top of a square, or first across the right edge and then through the top, exactly according to whether the base of the interval is less than (1 − p/q) or not.
These permutations are of limited practical use: the number of tiles increases with the complexity of the slope. We may, however, reduce the problem to a permutation on k elements if we relax the requirements on the parallelograms. In a tiling, it is necessary for all singularities of the metric to belong to the set of corners of the tiles. Without this requirement, however, we may cover M with k parallelograms with sides horizontal and of slope q/p: the parallelogram of index i is given by flowing the interval [0, 1/q) on the bottom of the square of index i through a vertical distance of q.
It is direct to see that the k parallelograms defined in this way are disjoint: the induced rotation on the bases is of p/q for each unit of vertical distance traversed, so a vertical distance of q is traversed before the left endpoint of this interval intersects an integer point. These tiles must therefore cover M , as we construct k such tiles and each has equal measure to the original squares. It is not a proper tiling, however, as the singularities will presumably belong to the sides of these parallelograms. Now compare with the proper retiling by kq parallelograms, each of the k parallelograms in the improper tiling is tiled by q proper parallelograms, so the proper tiling is a subtiling of the improper tiling. The permutationσ ∈ Σ k σ(i) = (σ ′ v ) q ((i − 1)q + 1) is the permutation induced by this subtiling in the flow direction.
Then the following is clear by our construction: Theorem A.4. The flow in slope q/p is a single-cylinder direction if and only if σ is a single cycle on k elements.
See [20] for background in combinatorics on words; we will not present that background here. A binary word of finite length is called a Lyndon word if each left factor has an average strictly smaller than the overall average of the word, i.e. it is strictly smaller in lexicographic order than all of its rotations.
Then our Farey-series-based construction generates all (binary, finite) Sturmian Lyndon words; the correspondence is made quite explicit by considering the geometric construction of Sturmian words by counting crossings of vertical versus horizontal integer-grid segments, and the Lyndon property is quickly seen to be inherited by the given order of composition under the assumption that both ω p/q and ω p ′ /q ′ are Lyndon words. That of all primitive Sturmian words of a given length and density there is only one which is also a Lyndon word is fairly direct to show; a proof appears in [24].
Note that we definedσ to track the orbit of the interval [0, 1/q) on the base of a square. Considering a different interval or a vertical segment would amount to performing a fixed cyclic transformation on the sequence of permutations used to defineσ. That is, instead of considerinĝ σ = τ 1 • τ 2 • · · · • τ n where each τ i is either σ h or σ v , we would consider τ k • τ k+1 • · · · • τ n • τ 1 • · · · • τ k−1 .
This new permutation is naturally seen to be in the same conjugacy class asσ, and therefore has the same number and length of cycles. Similarly, as the labeling of the tiles was arbitrary, it is only the conjugacy class/cycle decomposition of the resulting permutation that is of interest. It would be of considerable interest to classify the permutation pairs (σ h , σ v ) on k elements which generate a "Farey-Sturmian" composition in the described manner which is a k-cycle. Together with the cycle decomposition of the commutator [σ v , σ h ], knowing which pairs can generate kcycles would classify the square-tiled surfaces with a single-cylinder direction! Corollary A.5. If σ h and σ v generate a subgroup of Σ k which contains no k-cycle, then M admits no single-cylinder direction.
Proof. For slopes larger than one the result is immediate. For slopes less than one, we would perform an analogous construction as indicated previously with the same result. Alternately, any cylinder decomposition for a slope less than one may be mapped via the Veech group to a slope larger than one.
Both σ h and σ v are in A 8 , while a complete 8-cycle has seven transpositions. Therefore this surface has no single-cylinder directions. This result also follows from the fact that the Veech group is all of SL 2 (Z) [10]. Therefore all periodic decompositions are of the same type: two cylinders of equal heights and measures. Proof. The statement about the possibility of ergodic d-dimensional natural staircases follows from Corollary 4.4, using the vertices which are not singularities (fixed points underσ) as endpoints of distinct cuts whose values generate Z d . If there are no cycles of length one, then all integer points are singularities. All cuts are therefore nontrivial in the relative homology group H 1 (M, D, Z). In the event that M ∈ H(2), then, the geodesic flow is not ergodic in almost every direction, again citing [8].
The above techniques may be used to quickly generate surfaces with singlecylinder directions. Assuming that a surface is tiled by parallelograms one through k such that vertical flow is a single cylinder, the permutation σ h therefore encodes the stratum of the surface. Letting the tiles be given in order, so that σ v (i) = i + 1 mod k, the singularities may be deduced from the cycle decomposition of the commutator [σ v , σ h ](i) = σ −1 h (σ h (i) + 1) − 1. For arbitrary dimension d the techniques of this appendix may be used to study dcube tiled translation manifolds through the use of d permutations (for the relevant face identifications in each dimension). A single-cylinder direction therefore gives rise to a parallelepiped tiling for which one of the permutations is a single cycle. The geometric data is still encoded by the permutations, if more involved to extract. For example, the number of connected components still corresponds directly to the number of invariant sets in the group generated by the given permutations, and the singularities of the flat metric at certain (d − 2)-dimensional simplices involves commutators of the permutations. As geodesic flows in higher dimensions are poorly understood at present, we have phrased all the above in the particular case d = 2.