On ergodicity of foliations on $\mathbb{Z}^d$-covers of half-translation surfaces and some applications to periodic systems of Eaton lenses

We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for $\mathbb{Z}^d$-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents.


Introduction
1.1. Periodic Eaton lens distributions in the plane. An Eaton lens is a circular lens on the plane R 2 which acts as a perfect retroreflector, i.e. so that each ray of light after passing through the Eaton lens is directed back toward its source, see Figure 1. More precisely, if an Eaton lens is of radius R > 0 then the refractive Figure 1. Light rays passing through an Eaton lens and its flat counterpart index (RI for short) inside lens depends only on the distance from the center r and is given by the formula n(x, y) = n(r) = 2R/r − 1. The refractive index n(x, y) is constant and equals 1 outside the lens.
In this paper we consider dynamics of light rays in periodic Eaton lens distributions in the plane R 2 ∼ = C. As a simple example take a lattice Λ ⊂ R 2 and consider an Eaton lens of radius R > 0 centered at each lattice point of Λ. This configuration of lenses will be denoted by L(Λ, R) 2000 Mathematics Subject Classification. 37A40, 37F40, 37D40. The first author was partially supported by the Narodowe Centrum Nauki Grant 2014/13/B/ST1/03153. The second author was partially supported by Simons Collaboration Grant 318898.
Let us call an Eaton lens distribution, say L, in R 2 admissible, if no pair of lenses intersects. For every admissible Eaton lens configuration L the dynamics of the light rays can be considered as a geodesic flow (g L t ) t∈R on the unit tangent bundle of R 2 with lens centers removed, see Section A for details. The Riemannian metric inducing the flow is given by g (x,y) = n(x, y) · (dx ⊗ dx + dy ⊗ dy), where n(x, y) is the refractive index at point (x, y).
Since each Eaton lens in L acts as a perfect retroreflector, for any given slope θ ∈ R/πZ there is an invariant set P L,θ in the unit tangent bundle, such that all trajectories on P L,θ have direction θ or θ + π outside the lenses. The restriction of the geodesic flow (g L t ) t∈R to P L,θ will be denoted by (g L,θ t ) t∈R . Moreover, (g L,θ t ) t∈R possesses a natural invariant infinite measure µ L,θ equivalent to the Lebesgue measure on P L,θ , see Section A for details. With respect to this setting we consider measure-theoretic questions.
In [17] for example the authors have shown, that simple periodic Eaton lens configurations, for example L(Λ, R), have the opposite behavior of ergodicity. More precisely, a light ray in an Eaton lens configuration is called trapped, if the ray never leaves a strip parallel to a line in R 2 . The trapping phenomenon observed in [17] was extended in [16] to the following result: Theorem 1.1. If L(Λ, R) is an admissible configuration then for a.e. direction θ ∈ R/πZ there exist constants C = C(Λ, R, θ) > 0 and v = v(Λ, R, θ) ∈ R/πZ, such that every orbit in P L(Λ,R),θ is trapped in an infinite band of width C > 0 in direction v.
Knieper and Glasmachers [18,19] have trapping results for geodesic flows on Riemannian planes. Among other things Theorem 2.4 in [19] says, that for all Riemann metrics on the plane that are pull backs of Riemann metrics on a torus with vanishing topological entropy, the geodesics are trapped. Nevertheless the trapping phenomena obtained in [18,19] and [17,16] have different flavors. The former is transient whereas the latter is recurrent.
Let us further mention that Artigiani describes a set of exceptional triples (Λ, R, θ) for which the flow (g L(Λ,R),θ t ) t∈R is ergodic in [2].
In this paper we investigate ergodicity and trapping for more complicated periodic Eaton lens distributions. In fact, given a lattice Λ ⊂ C let us denote a Λ-periodic distribution of k Eaton lenses with center c i ∈ C and radius r i ≥ 0 for i = 1, . . . , k by L(Λ, c 1 , . . . , c k , r 1 , . . . , r k ). Of course, we will only consider admissible configurations. If the list of Eaton lenses has centrally symmetric pairs, we write ±c i for their centers and list their common radius only once. We adopt the convention that if the radius of a lens is zero then this lens disappears.
For a random choice of admissible parameters in this family of configurations in Section5 we prove trapping. Theorem 1.2. For every lattice Λ ⊂ C, every vector of centers c ∈ C k and almost every r ∈ R k >0 such that L(Λ, c, r) is admissible the geodesic flow on P L(Λ,c,r),θ is trapped for a.e. θ ∈ R/πZ.
An admissible ergodic Eaton lens configuration in the plane. As a consequence we have that the set of parameters (Λ, c, r, θ) for which (g L(Λ,c,r),θ t ) t∈R is ergodic is very rare. Despite this, in this paper, we find exceptional one-dimensional ergodic sets (piecewise smooth curves) of parameters such that a random choice inside such a curve provides an ergodic behavior of light rays. In fact the configurations we found are curves θ −→ L(Λ θ , c 1 (θ), . . . , c k (θ), r 1 (θ), . . . , r k (θ)) parameterized with the angle θ ∈ R/πZ. We should stress that results of [16] essentially show, that ergodic curves do not exists when k = 1.
We devote part of the paper showing several curves of ergodic Eaton lens configurations in the plane, see Figures 29,30 and 28. For some of those curves we describe admissible Eaton lens configurations only for an interval of slopes in R/πZ. Reduction to quadratic differentials and cyclic pillow case covers. The dynamical results for periodic Eaton lens distributions in the plane rely on the equivalence of the Eaton dynamics in a fixed direction, say θ, to the (dynamics on a) direction foliation F θ (q) of a quadratic differential q on the plane. Starting from a (slit-fold) quadratic differential, the connection is made by replacing a slitfold, as shown in Figure 4, by an Eaton lens. For a given direction the dynamical equivalence of a slit-fold and an Eaton lens is motivated by Figure 1. This equivalence is described in detail in Section A. We distinguish two objects, a flat lens is a two-dimensional replacement of an Eaton lens perpendicular to the light direction, that does not change the future and the past of the light in the complement of the Eaton lens that is replaced, see Figure 1. A slit-fold on the other hand is a flat lens in the language of quadratic differentials. In fact a slit-fold is constructed by removing a line segment, say [a, b] with a, b ∈ C, from the plane (or any flat surface), then a closure is taken so that the removed segment is replaced by two parallel and disjoint segments. Then for each segment one identifies those pairs of points, that have equal distance from the segments center point. Once this is done we obtain a slit-fold that we denote by a, b on the given surface, see Figure 4. The  single slit-fold a, b defines a quadratic differential on the plane with two singular points located on the (doubled) centers of the segment and a zero at its (identified) endpoints. Alternatively that quadratic differential on the plane is obtained as quotient of the abelian differential defined by gluing two copies of the slit plane For fixed k ∈ N the set S k of quadratic differentials made of k disjoint slit folds is a subset of Q((−1) 2k , 2 k ), the vector space of genus one quadratic differentials that have 2k singular points and k cone points of order 2. Disjoint means, the cone points of different slit-folds do not fall together. We will use the superset S k ⊃ S k of the quadratic differentials that are made of exactly k slit-folds, including the ones with merged cone points. Let us consider three special quadratic surfaces X 2 ∈ S 2 , Figure 5. Torus quadratic differentials X 2 , X 3 and X 4 . X 3 ∈ S 3 and X 4 ∈ S 4 drawn on Figure 5. Theorem 1.4. Let X = X k for k = 2, 3, 4 and denote by X its universal cover (quadratic differential on the plane). Then for almost every θ ∈ R/πZ the foliation in the direction θ on X is ergodic.
Those ergodic foliations on the plane can be converted into ergodic curves of admissible Eaton lens distributions.
The ergodicity of universal covers of quadratic surfaces in S k on the other hand is rather exceptional. If X ∈ S k satisfies a separation condition on slit folds (which is an open condition) then the foliation in the direction θ on X is trapped for a.e. θ ∈ R/πZ, see Corollary 5.6 for details.
The following more general ergodicity result supplies the key to the proof of Theorem 1.4 and Theorem 1.3. Theorem 1.5. Let (X, q) be a quadratic differential on a compact, connected surface such that all Lyapunov exponents of the Kontsevich-Zorich cocycle of (X, q) are zero. Then for every connected Z d -cover ( X, q), with d ≤ 2g, almost every directional foliation on ( X, q) is ergodic. This result is in fact a consequence of the more general Theorem 4.6 that provides a criterion on ergodicity for translation flows on Z d -covers of compact translation surfaces. We would like to mention that a similar result was obtained independently by Avila, Delecroix, Hubert and Matheus but it was never published (communicated by Pascal Hubert). Some related research was also recently done by Hooper who studied ergodicity of directional flows on translation surfaces with infinite area, see e.g. [22].

2.
Ergodic slit-fold configurations on planes by cyclic pillowcase covers.
In this section we outline the strategy to construct the ergodic quadratic differentials on the plane assuming the validity of Theorem 1.5. Theorem 1.5 reduces the problem of ergodicity from cyclic quadratic differentials in the plane to quadratic differentials (T , q) on the torus T with zero Lyapunov exponents. A recent criterion of Grivaux and Hubert [20] implies that a cyclic cover of the pillowcase has zero Lyapunov exponents, if it is branched at (exactly) three points. Now it turns out that there is a only a short list of those branched cyclic covers T → P. Recall, Figure 6. The pillowcase quadratic differential in polygonal representation the pillowcase P is a quadratic differential q P on the sphere S 2 . To characterize it, consider the quadratic differential dz 2 on R 2 ∼ = C. It is invariant under translations and the central reflection −id : R 2 → R 2 . Thus it descends to the torus T 2 := R 2 /Z 2 defining a quadratic differential invariant under the hyperelliptic involution ϕ : T 2 → T 2 induced by the central reflection of R 2 . So it further descends to a quadratic differential q P on the quotient sphere S 2 = T 2 /ϕ. The pillowcase the pair P = (S 2 , q P ), see Figure 6. Putting the result from [20] on cyclic pillowcase covers and Theorem 1.5 together one has: Corollary 2.1. Let π : X → P be a finite cyclic cover branched over three of the singular points of P and let q = π * q P be the pull back quadratic differential to X. If ( X, q) → (X, q) is a connected Z d -cover with d ≤ 2g, then almost every directional foliation on ( X, q) is ergodic.
We further present a list of relevant pillow-case covers: Up to the action of SL 2 (Z) on covers and up to isomorphy, there are three cyclic covers (T, q) → P that are branched over exactly three cone points of q P . The degree of each such cover is 3, 4 or 6. Figure 7 shows polygonal one strip representations of one cyclic pillowcase cover in each degree. We note that the quadratic differential on the degree 3 cover has the Ornithorynque (see [11] for the description of the surface) as its orientation cover and the quadratic differential on the degree 4 cover has the Eierlegende Wollmilchsau (see also [11]) as its orientation cover. There are particular questions regarding the conversion of a quadratic differential to an admissible Eaton lens distribution in the plane. In order to convert the torus differentials from Proposition 2.2 to Eaton lens distributions one needs a cover that is a slit-fold differential in the plane. We Eaton differentials and skeletons. For a fixed direction the (long term) Eaton lens dynamics on the plane or a torus is equivalent to the dynamics on a particular slit-fold, so we call a quadratic differential that is given by a union of slit-folds a pre-Eaton differential. The radius of an Eaton lens replacing a slit-fold depends on the angle between the light ray and the slit-fold, a light direction needs to be specified for such a replacement. Recall that a configuration of Eaton lenses is admissible, if no pair of Eaton lenses intersects. A pre-Eaton differential q is called an Eaton differential, if there is a nonempty open interval I ⊂ R such that for every (light) direction θ ∈ I mod π the direction foliation F θ (q) is measure equivalent to the geodesic flow of an admissible Eaton lens configuration, whose lens centers and radii depend continuously on θ ∈ I. We further call an Eaton differential maximal, if I → R/πZ, x → x mod π is onto. Finally let us call a (pre-)Eaton differential ergodic, if its direction foliations are ergodic in almost every direction. Note, that a pre-Eaton differential must be located on a torus, or a plane, since it has no singular points besides the ones of its slit-folds. So it is enough to present a pre-Eaton differential by a union of slit-folds, that we will call skeleton. Below we introduce and use geometric as well as algebraic presentations of skeletons.
Proof of Theorem 1.4. Pre-Eaton differentials are obtained from all three torus differentials in Figure 7, by first cutting vertically through their center and then rotating one of the halfes underneath the other as in Figure 8. Up to rescaling the resulting pre-Eaton differentials are X 2 (from the degree 3 cover), X 3 (from the degree 4 cover) and X 4 (from the degree 6 cover) as shown in Figure 5. It follows, that X 2 , X 3 and X 4 are cyclic covers of the pillowcase and branched over exactly three singularities of P. Passing to their universal covers we obtain three pre-Eaton differentials X 2 , X 3 , X 4 on the plane. In view of Corollary 2.1 almost every directional foliation for every such differential is ergodic.
Below we call the quadratic differential X 3 on the complex plane obtained from the degree 4 pillowcase cover X 3 the Wollmilchsau differential, see Figure 9.  . The quadratic surface X 3 (rotated by π/2) and its universal cover X 3 (Wollmilchsau differential) Theorem 2.3. The Wollmilchsau differential is an ergodic, maximal Eaton differential.
Ergodicity follows because Theorem 1.5 applies. To show the other statements of the Theorem we need to describe an Eaton lens configuration depending continuously on θ ∈ R/πZ and show that it is admissible. This is done in Proposition 2.4, see the comment after that.
Eaton lenses may overlap when placed at slit-fold centers. To resolve this problem we deform the measured foliation tangential to its direction θ ∈ R/πZ to a measure equivalent foliation by moving slit-folds parallel to θ. More precisely take a direction foliation F θ (q) of a quadratic differential q that contains a slit-fold. Then changing the location of the slit-fold while keeping its endpoints (and therefore its center points) on the same leaves of F θ (q) is called a railed motion. Changing a slit-fold skeleton using railed motions is called a railed deformation. In terms of Teichmüller Theory railed deformations are isotopies, or Whitehead moves that preserve the transverse measure of a measured foliation. In particular, two measured foliations that differ by railed deformations are Whitehead equivalent. A Whitehead move is a deformation of a foliated surface that collapses a leaf connecting two singular points, or it is the inverse of such a deformation, see [27, page 116]. Figure 11 shows railed deformations deforming skeletons into disjoint slit-folds. Each of those consists of several Whitehead moves. Some railed motions are shown in Figure 10 to the left. After performing a railed deformation, appropriately sized Eaton lenses are placed at the slit-fold centers.  ) t∈R is equivalent to the ergodicity of the directional foliation generated by the Wollmilchsau differential X 3 in the direction θ.
Proof. For this proof we will use complex coordinates on the plane. Let us consider the situation for light directions θ ∈ [0, π/4] first. For those angles the Eaton lens configurations are periodic with respect to the lattice Λ := Z4i ⊕ Z(4 + 2i). Therefore it is enough to show that Eaton lenses centered inside the strip S = {z ∈ C; | z| ≤ 2} are pairwise disjoint and do not leave the strip, i.e. do not cross the boundary of the strip.
Modulo the action of Λ there are three Eaton lenses on γ W (θ). The first one L 0 (θ) has radius r 0 (θ) = 2 sin θ and is centered at the origin. Then there is a pair of lenses denoted by L ± (θ) centered at c ± (θ) = ±(1 + i(1 + tan θ)), both of radius r ± (θ) = cos θ, see Figure 2. Since the radius of the Eaton lenses L ± (θ) is less then 1 and the radius of L 0 (θ) is bounded by 2, the lenses in the Λ orbit of any one of those three Eaton lenses are pairwise disjoint. For the same reason the Z4i orbit of all three Eaton lenses lies in the strip S.
The line in direction θ through the point i contains the center of L + (θ) since its slope is tan θ. The distance of that line to its parallel through the origin, denoted by t ± (θ), is cos θ, equaling the radius of L + (θ). So the lines t ± (θ) and t ± (θ) + 2i are tangent to L + (θ). Then by central symmetry the lines t ± (θ) and t ± (θ) − 2i are tangents to L − (θ). It follows that L + (θ) + 4ni lies between the lines t ± (θ) + 4ni and t ± (θ) + (4n + 2)i and L − (θ) + 4ni lies between the lines t ± (θ) + (4n − 2)i and t ± (θ) + 4ni for every n ∈ Z. Therefore, no pair of Eaton lenses in the Z4i orbits of L ± (θ) intersect. Since the Z(4 + 2i) translates of S cover the whole plane, intersecting only in their boundary lines, we conclude that no pair of Eaton lenses in the Λ orbits of L ± (θ) intersect.
Since L 0 (θ), the lens in the origin, has radius 2 sin θ the line in direction θ through −2, denoted by t 0 (θ), is tangent to it. By reflection symmetry with respect to the vertical axis, the line through 2 in direction π − θ is also a tangent to L 0 (θ). Let us denote this (tangent-)line by t 0 (θ), we shall see it is also tangent to L + (θ). Indeed, the reflection of t 0 (θ) with respect to the vertical through the center of L + (θ) is the tangent t ± (θ). Since the centers of L + (θ) and L 0 (θ) lie on different sides of their common tangent t 0 (θ) these lenses do not intersect. By central symmetry the same is true for L − (θ) and L 0 (θ). Since all three lenses L ± (θ) and L 0 (θ) in the parallelogram in S bounded by t ± (θ)±2i are disjoint and these parallelograms have a (modulo boundary) disjoint Λ orbit, we conclude that the lens distribution given by γ W (θ) is disjoint for all θ ∈ [0, π/4]. ) t∈R is measure equivalent to the direction θ dynamics defined by the surface X 3 . First the results of Appendix A imply, that for given θ ∈ [0, π/4] the ergodicity of the geodesic flow is equivalent to the ergodicity of the measured foliation defined by the slit-fold distribution obtained from the flat lens representation of Eaton lenses. That is, for given θ ∈ [0, π/4] we replace every Eaton lens by a slit-fold centered at the lens' center, perpendicular to θ and with length equal to the diameter of the lens. In fact modulo Λ we obtain the slit-folds S ± (θ) := ± 1 + cos θ sin θ + i(1 + tan θ − cos 2 θ), 1 − cos θ sin θ + i(1 + tan θ + cos 2 θ) through the centers of L ± (θ) and S 0 (θ) := − 2 sin 2 θ + 2i sin θ cos θ, 2 sin 2 θ − 2i sin θ cos θ through the origin, see Figure 11. The endpoints of the slit-fold S + (θ) lie on the lines (and direction θ foliation leaves) t ± (θ) and t ± (θ) + 2i. That means we can perform a railed deformation of S + (θ) along those leaves terminating in the slit-fold 0, 2i . By central symmetry there is a railed deformation of S − (θ) to the slit-fold 0, −2i . The end points of the slit-fold S 0 (θ) are located on the direction θ foliation leaves through the point ±2, so S 0 (θ) has a railed deformation to the slit-fold − 2, 2 . But that means the skeleton S + ∪ S − ∪ S 0 is Whitehead equivalent to the skeleton − 2, 2 ∪ 0, 2i ∪ 0, −2i . The Λ orbit of the latter is the Wollmilchsau skeleton in the plane, showing the claim on equivalence of ergodicity for angles θ ∈ [0, π/4].
The strategy we have just used to replace an Eaton lens with a slit-fold is the same for every angle. Let us describe this process for the slit-folds in the Wollmilchsau skeleton: For a fixed direction θ ∈ R/πZ a slit-fold, say S, replaces an Eaton lens, say L, if the two lines in direction θ through the endpoints of S are tangent to L.
Step by step, the flat lens equivalent to L is in the quadratic differential interpretation the slit-fold S L perpendicular to the direction θ with diameter and center matching those of L. In that case, the endpoints of S L lie on the two said tangents to L and therefore there is a railed deformation of S L to S. If, as in our case, more than one slit-fold is involved it must be checked that the tangent segments between S and S L do not cross another slit-fold. This is illustrated in Figure 11 for an angle θ ∈ [0, π/4] (left) and for an angle θ ∈ [π/4, π/2] (right). This same strategy is applied for the angles θ ∈ [π/4, π/2] below. The tangent lines necessary to show equivalence to the Wollmilchsau skeleton are also needed to show admissibility.
For the angles θ ∈ [π/4, π/2] the lattice of translation depends on the angle. In fact Λ θ := Z4i + Z(2l(θ) + 2i), where l(θ) = 2 − cot θ(1 − cot θ). While L 0 (θ) is still centered at the origin, now with radius r 0 (θ) = l(θ) sin θ = 1 sin θ + sin θ − cos θ the other two lenses L ± (θ) as before of radius cos θ are now centered at c ± (θ) = ±(cot θ + 2i), see Figure 3. In particular the radii of the lenses L ± (θ) are bounded by 1 < l(θ) ≤ 2 and the radius of the lens L 0 (θ) is bounded by l(θ) ≤ 2. Because the generators of the lattice Λ θ move each lens by at least twice their diameter there are no pairwise intersections possible among the lenses in one Λ θ orbit. Moreover the Z4i orbit of L − (θ) lies on the left of the vertical through the origin while the Z4i orbit of L + (θ) lies on the right of that line. As 0 ≤ cot θ ≤ 1 we have Moreover, r 0 (θ) = l(θ) sin θ ≤ l(θ). It follows that the Z4i orbits of L ± (θ) and L 0 (θ) are contained in the strip S = {z ∈ C; | z| ≤ l(θ)}. Since the Z(4 + 2i) translates of S cover the whole plane, intersecting only in their boundary lines, we conclude that no pair of Eaton lenses in the Λ θ orbits of L ± (θ) intersect.
Restricted to the Z4i orbit the lens configuration have for all θ ∈ [π/4, π/2] reflection symmetries around the coordinate axes. More precisely the Z4i orbit of each lens is invariant under the reflection at the horizontal while the Z4i orbits of L ± (θ) are interchanged by reflection at the vertical. Given these symmetries, all that remains to be seen is that L 0 (θ) does not intersect with L + (θ). To do this we find a common tangent to L 0 (θ) and L + (θ) that separates them. Let us consider the tangent line s(θ) to L 0 (θ) at the intersection point of its boundary with the half-line t 0 (θ) in direction θ through the origin. The the direction of s(θ) is π/2 − θ. The half-line t 0 (θ) + i in direction θ through the point i intersects perpendicularly s(θ) and goes through the center of L + (θ). By elementary geometry, see also Figure 3, the distance from i to the center of L + (θ) is (sin θ) −1 . The leg of the right triangle with hypothenuse the segment from 0 to i lying on t 0 (θ) has length sin θ. So the intersection point of s(θ) with t 0 (θ)+i must be at distance r 0 (θ)−sin θ = 1 sin θ −cos θ from i. But then it has distance cos θ from the center of L + (θ) and so the tangent s(θ) to L 0 (θ) is also tangent to L + (θ).
To show admissibility for one of the remaining angles, say θ ∈ [π/2, π], notice that L ± (θ) are the lenses L ± (π − θ) reflected at the vertical through the origin. We also have L 0 (θ) = L 0 (π − θ) and the lattice of translations has the same symmetry Λ θ = Λ π−θ . So the Λ θ orbits of these (reflected) lenses match the distribution given in the introduction. Since for θ = π/2 the lenses L ± (θ) are located on the vertical coordinate axis, this continuation of γ W is continuous at π/2. Moreover globally the lens distribution for θ ∈ [π/2, π] equals the one for π − θ reflected at the vertical coordinate axis. Since a reflection is an isometry, it preserves admissibility of lens distributions. Finally the Eaton lens configuration at θ = π matches that at θ = 0, since γ W (π) + 2i = γ W (0).
In particular the proof of Proposition 2.4 shows that the Wollmilchsau differential is a maximal Eaton differential. This, together with the fact that the Wollmilchsau differential appears as a cyclic pillow case cover branched over exactly three points, shows Theorem 2.3.

Quadratic differentials on tori in the determinant locus
3.1. Quadratic and Abelian differentials. In this article quadratic differentials are the fundamental objects. They appear in various presentations, analytical, polygonal and geometrical. All of those play important roles in different parts of our text.
Consider a Riemann surface X, i.e. a one dimensional complex manifold, not necessarily compact, and a quadratic differential q on X with poles of order at most one. A quadratic differential is a tensor that can locally be written as f (z) dz 2 = f (z) dz ⊗ dz, where f is a meromorphic function with poles of order at most one. Away from the poles and zeros of f one may use q to define natural coordinates on X If ζ 1 and ζ 2 are local coordinates, then dζ 1 = f (z) dz = ±dζ 2 in the intersection of the coordinate patches, so ζ 1 = ±ζ 2 + c for some c ∈ C. That way the pair (X, q) defines a maximal atlas made of natural coordinates and is therefore called half-translation surface. The maximal atlas is also called half-translation structure. The coordinate changes for any two charts from a half-translation structure are translations combined with half-turns (180 degree rotations) and this motivates the name half-translation surface. Similarly to a quadratic differential it is possible to consider an Abelian differential (holomorphic 1-form) ω on X. If Σ ⊂ X denotes the set of zeros of ω, as for quadratic differentials, away from Σ Abelian differential defines natural coordinates on X ζ = z z0 ω.
If ζ 1 and ζ 2 are local coordinates and their coordinate patches intersect then ζ 1 = ζ 2 + c for some c ∈ C. So the pair (X, ω) defines a maximal atlas made of natural coordinates and is called translation surface. Here the maximal atlas is called translation structure.
Objects on the plane that are invariant under translations pull back via natural charts to X and glue together to give global objects on the translation surface (X, ω). Among those objects are the euclidean metric, the differential dz, and constant vector fields in any given direction. In fact, the pull back of the differential dz recovers ω on the translation surface (X, ω). Similarly objects on the plane that are invariant under translations and half-turns define global objects on the half-translation surface (X, q). Here objects of interest are again the euclidean metric, the quadratic differential dz 2 (recovering q), and any direction foliation by (non-oriented) parallel lines. Since there is one line foliation on C for each angle θ ∈ R/πZ that is tangent to ± exp iθ, we denote its pullback to X by F θ (q), or F θ if there is no confusion about the quadratic differential. For a translation surface, say (X, ω), the constant unit vector field on C in direction θ ∈ R/2πZ defines a directional unit vector field V θ = V ω θ on X \ Σ. Then the corresponding directional flow (ϕ θ t ) t∈R = (ϕ ω,θ t ) t∈R (also known as translation flow ) on X \ Σ preserves the area measure µ ω given by µ ω (A) = | A i 2 ω ∧ ω|. If the surface X is compact then the measure µ ω is finite. We will use the notation (ϕ v t ) t∈R for the vertical flow (corresponding to θ = π 2 ) and (ϕ h t ) t∈R for the horizontal flow respectively (θ = 0). For every half-translation surface (X, q) there exists a unique double cover π o : ( X, q) → (X, q), the orientation cover, characterized by the property that it is branched precisely over all singular points with odd order. The pull-back q = π * o q is the square q = ω 2 of an abelian differential ω ∈ Ω(X). If M = X then the translation surface (M, ω) is called also the orientation cover of the half-translation surface (X, q). The pull-back F θ of any direction foliation F θ is orientable. This foliation coincides with the foliations determined by the directional flows (ϕ θ t ) t∈R and (ϕ θ+π t ) t∈R on (M, ω). Moreover, the ergodicity of the foliation F θ is equivalent to the ergodicity of the translation flow (ϕ θ t ) t∈R .
Particular representations of half-translation structures. The quadratic differential (dz) 2 on C is invariant under translations and rotations of 180 degrees, that group generated by those isometries are in the group of half-translations. Invariance of (dz) 2 under that group results in a variety of possible constructions of quadratic differentials, or equivalently half-translation surfaces.
Most notably a (compact) polygon in C all of whose edges appear in parallel pairs, together with an prescribed identification of edge pairs by half-translations. It is known, that any quadratic differential on a compact surface can be represented by such a polygon. A second way is to take suitable quotients of C under certain discrete groups of half-translations. Here any torus C/Λ with a lattice Λ of translations is an example. Our way to built quadratic differentials in the plane C ∼ = R 2 and on a torus is by successively adding (non-intersecting) slit-folds. Since the identifications of the edges of a slit-fold are half-translations the given quadratic differential defines a canonical new one on the surface with slit-fold. One important properties of slit-folds is that they do not change the genus of the half-translation surface to which they are added. Not only slit-folds have this property of defining quadratic differentials without changing the genus. In fact more general types of "folds" are shown in Appendix B. They are helpful in the construction of other ergodic curves.

Cyclic covers of pillowcases.
In this section we classify those quadratic differentials on tori that arise as pullbacks of the pillowcase along a covering map (cyclic covers) which is unbranched over one point. Two of those examples are quotients of the well known Ornithorynque and Eierlegende Wollmilchsau under an involution.
Given a Riemann surface X and a finite subset Σ * ⊂ X it is well known that the elements of ξ ∈ H 1 (X, Σ * ; G), G an abelian group, define a regular cover π : X ξ → X over X\Σ * branched over Σ * ⊂ X with deck transformation group G. To describe this cover formally first denote by · , · : , where · denotes the deck group action of G on X ξ . Here we consider the case when the homology group is a direct sum of cyclic groups of the kind H 1 (X, Σ * ; Z/mZ).
Let us look at the pillowcase P with underlying space X = CP 1 and take Σ ⊂ CP 1 to be the pillowcases four singular points. We are looking for pillowcase covers with at most three branch points. That means such a cover is unbranched over at least one singular point of the pillowcase. Then the result of Hubert and Griveaux [20] implies that the cover is in the determinant locus. We now construct those covers.
3.3. Differentials in the determinant locus. Take the pillowcase X = P with named singular points p 1 , p 2 , p 3 , p 4 = p F ∈ P put in clockwise order starting from the upper left. We assume the point p F is fixed under all automorphisms (and affine maps) of P. We further assume all branching of covers is restricted to the set Σ * = {p 1 , p 2 , p 3 }.
Let γ 12 , γ 23 be generators in H 1 (P, {p 1 , p 2 , p 3 }; Z/dZ) so that γ 12 is the class of the oriented horizontal path joining p 1 and p 2 and γ 23 is the class of the oriented vertical path joining p 2 and p 3 . Let γ h , γ v be generators in H 1 (P \{p 1 , p 2 , p 3 }; Z/dZ) such that γ h is the class the horizontal (right oriented) simple loop and γ v is the class of the simple loop around p 1 with counterclockwise orientation. Then Let us consider any cyclic degree d cover P ξ of P branched over Σ * which is defined by a homology class ξ = w h γ 12 + w v γ 23 ∈ H 1 (X, Σ * ; Z/dZ). Here are called weights of the cover P ξ → P. Therefore the cover is determined by the The cover defined by those data has a straightforward geometric realization. Namely, cut the pillowcase along the three line segments joining: p 1 with p 2 , p 2 with p 3 and p 3 with p F . The resulting surface is isometric to a rectangle of width 2 and height 1 in the complex plane. Let us denote this polygonal presentation of X with cuts by X c and take d labeled copies X c × {1, . . . , d} = X c 1 · · · X c d . Now identify the vertical right edge of X c i with the vertical left edge X c i+wv mod d by a translation. Then identify the right half of the upper horizontal edge of X c i with the left half of the upper horizontal edge of X c i+w h mod d using a half turn and identify the right half of the lower horizontal edge of X c i with the left half of the lower horizontal edge of X c i using a half turn. This determines X d (w h , w v ) because of the covers cyclic nature. By eventually renaming the decks we may assume that . See [5] and [12] for a more background and applications of cyclic covers.
We now determine those cyclic covers that are torus differentials, i.e. have genus 1. To calculate the genus of X d (w h , w v ) we note, that the covering has gcd(w h , d) preimages over p 1 , gcd(|w h − w v |, d) preimages over p 2 and w v = gcd(w v , d) preimages over p 3 because it is cyclic. It follows that the respective branching orders are That means we have an angle excess of (o i − 2)π around any preimage of p i for i = 1, 2, 3.
Proof. Write down the standard formula expressing the Euler characteristic of quadratic differentials in terms of total angle deficit for singular points and total angle excess for cone points: By definition the degree of the pillowcase cover Proof. A torus has vanishing Euler characteristic, thus from Proposition 3.1 we directly derive the condition Dividing by d, we see that a torus presents a positive integer solution of the problem Without restriction of generality we may assume that any solution fulfills If a = 2 then 1/b + 1/c = 1/2 which gives b ≤ 4. Therefore we obtain two possibilities (b, c) = (3,6) or (4,4). 3). It follows that we get only (3, 3, 3), (2,4,4), (2, 3, 6) as solutions. Since we obtain lcm(a, b, c) = d. It follows that d = 3, 4, 6 respectively.

3.4.
Branched pillow case covers that are torus differentials. In spite of Proposition 3.2 all we need to do to exhaust the list of possible of torus covers is to go through a short list of possible cases. Because p F is assumed to be fixed the pillowcase has no automorphisms. For d = 3, 4 and 6 we need to find the weights The weights cannot be 0 or d, because the cover must be branched over all three points p 1 , p 2 and p 3 to give a surface of genus larger than zero, the genus of the pillowcase. Thus without loss of generality we can pick the weights w h , w v from {1, . . . , d − 1}. For d = 6 we obtain the following weight pairs fulfilling the conditions: The weights tell us the number of deck changes that occur when we go over either homology class. By renaming the decks so that deck k becomes deck d − k we obtain the cover Thus those are isomorphic, in particular for d = 6 we have X 6 (1, 3) ∼ = X 6 (5, 3) and X 6 (2, 3) ∼ = X 6 (4, 3). For d = 3 and d = 4 the same line of arguments applies and leads to the following list of covers: Torus differentials of degree d = 3, 4 and 6 The group SL 2 (R) acts real linearly on the plane and defines a map on halftranslation surfaces by post composition with local coordinates. Alternatively one may take a polygon representation of the surface and apply a matrix A ∈ SL 2 (R), viewed as linear map of R 2 , to it. The edges of the polygon are then identified exactly as before the deformation. That defines an action of SL 2 (R) on surfaces with quadratic differential. We denote by A · X the deformation of X by A ∈ SL 2 (R).
Let X ξ → X be a branched G-cover over Σ * ⊂ X and determined by ξ ∈ H 1 (X, Σ * ; G). Then the deformation A · X ξ is a branched cover determined by The pillowcase is stabilized by all elements of SL 2 (Z), as one can easily check on the two (parabolic) generators P h : . Stabilized means the original pillowcase can be obtained from the deformed pillowcase by successively cutting off polygons, translating and if needed rotating them to another boundary in tune with the edge identification rules of the pillowcase.
Let us consider any cover X d (w h , w v ) = P ξ (with ξ = w h γ 12 + w v γ 23 ) and A ∈ SL 2 (Z). Since A · P = P, we have Moreover for the parabolic generators P h and P v we have This yields the action of parabolic matrices on degree d pillowcase covers: Since the group of maps generated by two involutions (x, y) → (x, y − x) and (x, y) → (y − x, y) has exactly 6 elements, so we obtain the following: The SL 2 (Z) orbit of a pillowcase cover is given by Note, that for low degree this orbit is even smaller: The orbits of degree three and four covers contain less than six tori. As can be easily seen from the proposition, compare the table of surfaces, that the relevant torus differentials of fixed degree lie on one SL 2 (Z) orbit.
Orientation covers of some pillow case covers. We consider the orientation covers of X d (2, 1), for d = 3, 4 and X d (3, 1) for d = 6 drawn on Figure 7. Recall that the orientation cover ( X, ω 2 ) → (X, q) of a quadratic differential (X, q) is uniquely characterized as the degree two cover, branched precisely over the cone points having an odd total angle (in multiples of π). There is a sheet exchanging involution ρ on X that has the preimages of the odd cone points as fixed-points. The involution is locally a rotation by π, eventually followed by a translation. Using this one may construct orientation covers given a polygonal representation. One considers two copies of the polygon and whenever two edges were identified by a rotation on the original polygon, one identifies any of those two edges as before but now to the corresponding edge of the other copy. Turning any one copy by 180 degrees the new identifications become translations and we have a translation surface. For the surfaces at hand this procedure is reflected in the Figure 14. The orientation cover of X 6 (3, 1) following Figures 12, 13 and 14. The first two are splendid specimens in the zoo of square tiled surfaces. If the name did not immidiately give it away, a look at the figures should explain the idea of a square tiled surface. In fact, X 3 (2, 1) ∼ = X 2 is the Ornithorynque and X 4 (2, 1) ∼ = X 3 is known as the Eierlegende Wollmilchsau. Both names reflect the surfaces multiple rather exceptional properties, each of them has vanishing Lyapunov exponents. To our best knowledge the orientation cover of X 6 (3, 1) is not such a well studied square tiled surface and we are not able to provide a direct reason to motivate such research.

Ergodicity of translation flows and measured foliations on infinite covers
In this section we prove a useful criterion on ergodicity for translation flows on Z d -covers (see Theorem 4.6). The key Theorem 1.5 follows directly from this criterion.
4.1. Z d covers. Let X be a Z d -cover of a compact connected surface X and let p : X → X be the covering map, i.e. there exists a properly discontinuous Z daction on X such that X/Z d is homeomorphic to X. Then p : X → X is the composition of the natural projection X → X/Z d and the homeomorphism. Denote by · , · : H 1 (X, Z) × H 1 (X, Z) → Z the algebraic intersection form. Then any and · denotes the action of Z d on X. The Z d -cover corresponding to γ will be denoted by X γ .
Remark 4.1. Note that the surface X γ is connected if and only if the group homomorphism H 1 (X, Z) ξ → γ, ξ ∈ Z d is surjective.
If q is a quadratic differential on X then the pull-back p * (q) of q by p is also a quadratic differential on X γ and will be denoted by q γ . For any θ ∈ R/πZ we denote by F θ = F γ θ the corresponding measurable foliation on ( X γ , q γ ). If (M, ω) is a compact translation surface and γ ∈ H 1 (M, Z) d is a d-tuple then the translation flow on the Z d -cover ( M γ , ω γ ) in the direction θ is denoted by ( ϕ θ t ) t∈R . Let (X, q) be a connected half-translation surface and denote by (M, ω) its orientation cover which is a translation surface. Then there exist a branched covering map π : M → X such that π * (q) = ω 2 and an idempotent σ : X → X such that π • σ = π and σ * (ω) = −ω.
The space H 1 (M, R) has an orthogonal (symplectic) splitting into spaces H + 1 (M, R) and H − 1 (M, R) of σ * -invariant and σ * -anti-invariant homology classes, respectively. Moreover, the subspace H + 1 (M, R) is canonically isomorphic to H 1 (X, R) via the map π * : H + 1 (M, R) → H 1 (X, R), so we identify both spaces. Recall that the measured foliation F θ of X is ergodic for some θ ∈ R/2πZ if and only if the translation flow (ϕ θ t ) t∈R on M is ergodic with respect to the measure µ ω (possibly infinite).
Remark 4.2. Let γ ∈ (H 1 (X, Z)) d be a d-tuple such that the Z d -cover X γ is connected. Since H + 1 (M, Z) and H 1 (X, Z) are identified, we can treat γ as a d-tuple in (H + 1 (M, Z)) d . Let us consider the corresponding Z d -cover M γ . Then the maps π : M → X and σ : M → M can be lifted to a branched covering map π : M γ → X γ and an involution σ : M γ → M γ so that π • σ = π. Then π establishes an orientation cover ( M γ , ω γ ) of the half-translation surface ( X γ , q γ ). Therefore, for every θ ∈ R/2πZ the ergodicity of the measured foliation F θ of ( X γ , q γ ) is equivalent to the ergodicity of the translation flow ( ϕ θ t ) t∈R on ( M γ , ω γ ). Note, that the measure µ ωγ is an infinite Radon measure. Denote by Q(X) the moduli space of half-translation surfaces which is also naturally stratified by the number and the types of singularities. We denote by Q(κ 1 , . . . , κ s ) the stratum of quadratic differentials (X, q) which are not the squares of Abelian differentials, and which have s singularities and their orders are (κ 1 , . . . , κ s ), where κ i ≥ −1. Then κ 1 + . . . + κ s = 4g X − 4, where g X is the genus of X.
The group SL(2, R) acts naturally on T 1 (M ) and M 1 (M ) as follows. Given a translation structure ω, consider the charts given by local primitives of the holomorphic 1-form. The new charts defined by postcomposition of these charts with an element of SL(2, R) yield a new complex structure and a new differential that is Abelian with respect to this new complex structure, thus a new translation structure. We denote by g · ω the translation structure on M obtained acting by g ∈ SL(2, R) on a translation structure ω on M .
The Teichmüller flow (g t ) t∈R is the restriction of this action to the diagonal subgroup (diag(e t , e −t )) t∈R of SL(2, R) on T 1 (M ) and M 1 (M ). We will deal also with the rotations (r θ ) θ∈R/2πZ that acts on T 1 (M ) and M 1 (M ) by r θ ω = e iθ ω. Theorem 4.3 (see [25]). For every Abelian differential ω on a compact connected surface M for almost all directions θ ∈ R/2πZ the vertical and horizontal flows on (M, r θ ω) are uniquely ergodic.
Every θ ∈ R/2πZ for which the assertion of the theorem holds is called Masur generic. The following recent result is more refined and yields Birkhoff generic elements among r θ ω for θ ∈ R/2πZ. Theorem 4.4 (see [3]). For almost all θ ∈ R/2πZ we have All directions θ ∈ R/2πZ for which the assertion of the theorem holds are called Birkhoff generic.
Let V → M be an SL(2, R)-invariant subbundle of H 1 (M, R) which is defined and continuous over M. For every ω ∈ M we denote by V ω its fiber over ω.
Let us consider the KZ-cocycle (G V t ) t∈R restricted to V. By Oseledets' theorem, there exists Lyapunov exponents of (G V t ) t∈R with respect to the measure ν M . If additionally, the subbundle V is symplectic, its Lyapunov exponents with respect to the measure ν M are: Theorem 4.5 (see [3]). Let λ V 1 = λ 1 > λ 2 > . . . > λ s−1 > λ s = −λ V 1 be distinct Lyapunov exponents of (G V t ) t∈R with respect to ν M . Then for a.e. θ ∈ R/2πZ there exists a direct splitting of the fibre V r θ ω = s i=1 U i r θ ω such that for every ξ ∈ U i r θ ω we have Each θ ∈ R/2πZ for which the assertion of the theorem holds is called Oseledets generic. Then V r θ ω has a direct splitting into unstable, central and stable subspaces The dimensions of E + r θ ω and E − r θ ω are equal to the number of positive Lyapunov exponents of (G V t ) t∈R . One of the main objectives of this paper is to prove (in Section 4.5) the following criterion on ergodicity for translation flows on Z d -covers. By Theorem 3 in [13] we have the following result that will be applied in the proof of Theorem 4.6. Suppose that (M, ω) is an orientation cover of a compact half-translation surface (X, q). Then the SL(2, R)-invariant symplectic subspace H + 1 (M, R) determines an SL(2, R)-invariant symplectic subbundle of H + 1 which is defined and continuous over M. The fibers of this bundle can be identified with the space H + 1 (M, R) = H 1 (X, R) so the dimension of each fiber is 2g X , where g X is the genus of X. The Lyapunov exponents of the bundle H + 1 are called the Lyapunov exponents of the half-translation surface (X, q). We denote by λ top (q) the largest exponent. For every α ∈ A we will denote by ξ α = ξ α (ω, I) ∈ H 1 (M, Z) the homology class of any loop formed by the segment of orbit for (ϕ θ t ) t∈R starting at any x ∈ Int I α and ending at T x together with the segment of I that joins T x and x, that we will denote by [T x, x]. Proposition 4.8 (see [15] for d = 1). For every γ ∈ H 1 (M, Z) d the directional flow ( ϕ θ t ) t∈R on the Z d -cover ( M γ , ω γ ) has a special representation over the skew product T ψγ : I × Z d → I × Z d of the form T ψγ (x, n) = (T x, n + ψ γ (x)), where ψ γ : I → Z d is a piecewise constant function given by In particular, the ergodicity of the flow ( ϕ θ t ) t∈R on ( M γ , ω γ ) is equivalent to the ergodicity of the skew product T ψγ : Since the ergodicity of the flow ( ϕ θ t ) t∈R is equivalent to the ergodicity of T ψγ , this will allow us to apply the theory of essential values of cocycles to prove Theorem 4.6 in Section 4.5.

4.4.
Ergodicity of skew products. In this subsection we recall some general facts about cocycles. For relevant background material concerning skew products and infinite measure-preserving dynamical systems, we refer the reader to [28] and [1].
Let G be a locally compact abelian second countable group. We denote by 0 its identity element, by B G its σ-algebra of Borel sets and by m G its Haar measure. Recall that, for each ergodic automorphism T : (X, B, µ) → (X, B, µ) of a standard Borel probability space, each measurable function ψ : X → G defines a skew product automorphism T ψ which preserves the σ-finite measure µ × m G : Here we use G = Z d . The function ψ : X → G determines also a cocycle ψ ( · ) : Z × X → G for the automorphism T by the formula Then T n ψ (x, g) = (T n x, g + ψ (n) (x)) for every n ∈ Z. An element g ∈ G is said to be an essential value of ψ, if for every open neighbourhood V g of g in G and any set B ∈ B, µ(B) > 0, there exists n ∈ Z such that The set of essential values of ψ is denoted by E(ψ). Proposition 4.9 (see [28]). The set of essential values E(ψ) is a closed subgroup of G and the skew product T ψ is ergodic if and only if E(ψ) = G. Proposition 4.10 (see [4]). Let (X, d) be a compact metric space, B the σalgebra of Borel sets and µ be a probability Borel measure on X. Suppose that T : (X, B, µ) → (X, B, µ) is an ergodic measure-preserving automorphism and there exists an increasing sequence of natural numbers (h n ) n≥1 and a sequence of Borel sets (C n ) n≥1 such that µ(C n ) → α > 0, µ(C n T −1 C n ) → 0 and sup x∈Cn d(x, T hn x) → 0.
If ψ : X → G is a measurable cocycle such that ψ (hn) (x) = g for all x ∈ C n , then g ∈ E(ψ).

4.5.
Prof of Theorem 4.6. In this section we prove the following result. In view of Theorems 4.3, 4.4 and 4.7, it proves Theorem 4.6.
Suppose that the directional flow (ϕ θ t ) t∈R on (M, ω) in a direction θ ∈ R/2πZ is ergodic and minimal. Let I ⊂ M \ Σ (Σ is the set of zeros of ω) be an interval transversal to the direction θ with no self-intersections. The Poincaré return map T : I → I is a minimal ergodic IET, denote by I α , α ∈ A the intervals exchanged by T . Let λ α (ω, I) stands for the length of the interval I α .
Denote by τ : I → R + the map of the first return time to I for the flow (ϕ θ t ) t∈R . Then τ is constant on each I α and denote by τ α = τ α (ω, I) > 0 its value on I α for all α ∈ A. Let us denote by δ(ω, I) > 0 the maximal number ∆ > 0 for which the set Suppose that J ⊂ I is a subinterval. Denote by S : J → J the Poincaré return map to J for the flow (ϕ θ t ) t∈R . Then S is also an IET and suppose it exchanges intervals (J α ) α∈A . The IET S is the induced transformation for T on J. Moreover, all elements of J α have the same first return time to J for the transformation T . Let us denote this return time by h α ≥ 0 for all α ∈ A. Then I is the union of disjoint towers {T j J α : 0 ≤ j < h α }, α ∈ A. Lemma 4.12. Suppose that 0 ≤ h ≤ min{h α : α ∈ A} is a number such that each T j J for 0 ≤ j < h is a subinterval of some interval I β , β ∈ A. Then for every α ∈ A we have Proof. Let ψ γ,J : J → Z d be the cocycle associated to the interval J. Then On the other hand, ψ γ,J (x) = γ, ξ α (J) for x ∈ J α , so ψ (hα) = γ, ξ α (J) on J α .
Since x 0 and T hα x 0 = Sx 0 belong to J, by assumption, for all 0 ≤ i < h the points T i T hα x 0 and T i x 0 belong the interval T i J ⊂ I β for some β ∈ A. Therefore, for every 0 ≤ i < j. It follows that ψ  Proof. Suppose, contrary to our claim, that T j J contains an end x of some interval I β . Then x = ϕ θ τ (j) (x0) (x 0 ) for some x 0 ∈ J and there is 0 ≤ s < τ (x) such that ϕ θ s x is a singular point. Therefore, ϕ θ τ (j) (x0)+s x 0 is a singular point and τ (j) (x 0 ) + s < (j + 1)|τ | ≤ h|τ | ≤ ∆, contrary to the assumption.
The following result follows directly from Lemmas A.3 and A.4 in [14].
Lemma 4.14. For every (M, ω) there exist positive constants A, C, c > 0 such that if 0 ∈ R/2πZ is Birkhoff and Masur generic then there exists a a sequence of nested horizontal intervals (I k ) k≥0 in (M, ω) and an increasing divergent sequence of real numbers (t k ) k≥0 such that t 0 = 0 and for every k ≥ 0 we have Proof of Theorem 4.11. Assume that the total area of (M, ω) is 1. Taking ω 0 = r π/2−θ ω we have 0 ∈ R/2πZ is Birkhoff and Masur generic for ω 0 . Since the flow ( ϕ θ t ) t∈R on ( M γ , ω γ ) coincides with the vertical flow on ( M γ , (ω 0 ) γ ), we need to prove the ergodcity of the latter flow.
By Lemma 4.14, there exists a sequence of nested horizontal intervals (I k ) k≥0 in (M, ω 0 ) and an increasing divergent sequence of real numbers (t k ) k≥0 such that (4.4) and (4.5) hold for k ≥ 0 and t 0 = 0.
Let I := I 0 and for the flow ( ϕ v t ) t∈R on ( M γ , (ω 0 ) γ ) denote by T : I → I and ψ : I → Z d the corresponding IET and cocycle respectively. For every k ≥ 1 the first Poincaré return map T k : I k → I k to I k for the vertical flow (ϕ v t ) t∈R on (M, ω 0 ) is an IET exchanging intervals (I k ) α , α ∈ A whose length in (M, ω 0 ) are equal to e −t k λ α (g t k ω 0 , I k ), α ∈ A, resp. In view of (4.5), the length of I k in (M, ω 0 ) is Moreover, by the definition of δ, the set x ∈ I k does not contain any singular point.
Denote by h k α ≥ 0 the first return time of the interval (I k ) α to I k for the IET T . Let h k := e t k δ(g t k ω 0 , I k )/|τ (ω 0 , I)| and C k α := Now Lemmas 4.12 and 4.13 applied to J = I k and ∆ = e t k δ(g t k ω 0 ) give for every k ≥ 1 and α ∈ A. Moreover, by (4.5), By assumption, in view of (4.4), we have Therefore for every α ∈ A the sequence { γ, ξ α (g t k ω 0 , I k ) } k≥1 in Z d is bounded.
Passing to a subsequence, if necessary, we can assume the above sequences are constant. In view of (4.6) and (4.7), Proposition 4.10 gives γ, ξ α (g t k ω 0 , I k ) ∈ E(ψ) for every α ∈ A and k ≥ 1. Recall that for every k ≥ 1 the homology classes ξ α (g t k ω 0 , I k ), α ∈ A generate H 1 (M, Z). As M γ is connected, the homomorphism is a group and contains all these vectors, we obtain E(ψ) = Z d , so the skew product T ψ is ergodic. In view of such that each subbundle V i is strongly irreducible. Denote by λ V i top the maximal Lyapunov exponent of the reduced Kontsevich-Zorich cocycle (G V i t ) t∈R and with respect to the measure ν M . As a step of the proof of Theorem 1.4 in [3] the authors showed also the following result: Theorem 4.15. If ξ ∈ V i ω is non-zero then for a.e. θ ∈ R/2πZ we have A consequence of this result is the following: Therefore, by Theorem 4.15, for a.e. θ we have which completes the proof.
The following result is a direct consequence of Theorem 4.6 and yields some relationship between the value of the Lyapunov exponent λ(ω, γ) for γ ∈ H 1 (M, Z) and the ergodic properties of translation flows on the Z d -cover ( M γ , ω γ ).
Theorem 4.17. Let (M, ω) be a compact translation surface and let γ ∈ H 1 (M, Z) d be such that M γ is connected and λ(ω, γ i ) = 0 for 1 ≤ i ≤ d. Then ( ϕ θ t ) t∈R is ergodic for almost every θ ∈ R/2πZ. Proof. We present the arguments of the proof only for d = 1. In the higher dimensional case, the proof runs along similar lines.
Let us consider the SL 2 (R)-invariant splitting (4.8) into strongly irreducible subbundles and let γ = m i=1 γ i be such that γ i ∈ V i ω . Since λ(ω, γ) = 0, by Theorem 4.15, Then V γ is a non-zero SL 2 (R)-invariant subbundle so that γ ∈ V γ ω and all Lyapunov exponents of the restricted KZ-cocycle (G V γ t ) t∈R with respect to the measure ν M vanish. Then Theorem 4.6 provides the final argument.
Finally, we can formulate a conjecture which was stated so far informally in the translation surface community. It expresses completely the relationship between the value of the Lyapunov exponent and the ergodic properties of translation flows on the Z-covers on compact surfaces.
Conjecture. Let (M, ω) be a compact translation surface and let ( M γ , ω γ ) be its connected Z-cover given by γ ∈ H 1 (M, Z). Then The claim (i) is confirmed by Theorem 4.17. The truth of the claim (ii) is suggested only by a much weaker result proved in [15].

Non-ergodicity and trapping for typical choice of periodic system of Eaton lenses
In this section we present the proof of Theorem 1.2.
Let Λ ⊂ C be a lattice. For any quadratic differential q on the torus X := C/Λ we denote by q the pullback of q by the projection map p : C → C/Λ. Denote by F θ and F θ the measured foliations in a direction θ ∈ R/πZ derived from (X, q) and (C, q) respectively. Recall that a foliation F θ trapped, if there exists a vector v ∈ S 1 ⊂ C and a constant C such that every leaf of F θ is trapped in an infinite band of width C parallel to v. Of course, every trapped foliation is highly non-ergodic.
Let (M, ω) be the orientation cover of the half-translation torus (X, q) and let π : M → X be the corresponding branched covering map. Then the space H + 1 (M, R) H 1 (X, R) of vectors invariant under the deck exchange map on homology is a two dimensional real space. Denote by γ 1 , γ 2 ∈ H 1 (X, Z) H + 1 (M, Z) two homology elements determining the Z 2 -covering p : C → X. Since γ 1 , γ 2 are linearly independent, they span the space H 1 (X, R) H + 1 (M, R). Let ( M , ω) be the Z 2 -cover of (M, ω) given by the pair (γ 1 , γ 2 ) ∈ H + 1 (M, Z) 2 . For every θ ∈ R/2πZ let M + θ be the set of points x ∈ M such that the positive semi-orbit (ϕ θ t (x)) t≥0 on (M, ω) is well defined.
Let D ⊂ M be a bounded fundamental domain of the Z 2 -cover such that the interior of D is path-connected and the boundary of D is a finite union of intervals. For every x ∈ M + θ and t > 0 define the element σ θ t (x) ∈ H 1 (M, Z) as the homology class of the loop formed by the segment of the orbit of x from x to ϕ θ t (x) closed up by the shortest curve joining ϕ θ t (x) with x that does not cross p −1 (∂D). The following result is a more general version of Theorem 3.2 in [17]. Since its proof runs essentially as in [17], we omit it. Proposition 5.1. Assume that for a direction θ ∈ R/2πZ there is a non-zero homology class ξ ∈ H + 1 (M, R) = Rγ 1 + Rγ 2 and C > 0 such that | σ θ t (x), ξ | ≤ C for every x ∈ M + θ and t > 0. If the foliation F θ has no vertical saddle connection the lifted foliation F θ is trapped.
Let M be the closure of the SL(2, R)-orbit of (M, ω) and denote by ν M the affine probability measure on M. Let us consider the restriction of the Konsevich-Zorich Recall that a.e. θ ∈ R/2πZ is Oseledets generic for the subbundle. This implies the existence of the stable subspace E − r θ ω ⊂ H + 1 (M, R) whose dimension is equal to the number of positive Lyapunov exponents of (G H + 1 t ) t∈R . Moreover, by Theorem 4.4 in [14] we have. Proposition 5.2. Suppose that π/2 − θ ∈ R/2πZ is a Birkhoff, Oseledets and Masur (BOM) generic direction for (M, ω). Then for every ξ ∈ E − r π/2−θ ω there exists C > 0 such that | σ θ t (x), ξ | ≤ C for all x ∈ M + θ and t > 0.
Since almost every direction is BOM generic, the previous two results yield the following criterion.
Proposition 5.3. Suppose that the Lyapunov exponent λ top (q) of (C/Λ, q) is positive. Then for a.e. θ ∈ R/πZ the measured foliation F θ on (C, q) is trapped.
To show the positivity of the Lyapunov exponents we will use Forni's criterion: Let Λ ⊂ C be a lattice and w ∈ Λ a non-zero vector. Let us fix a unit vector v ∈ S 1 ⊂ C linearly independent from w, a k-tuple c = (c 1 , . . . , c k ) of different points on the torus C/Λ and a k-tuple r = (r 1 , . . . , r k ) of positive numbers. Denote by q v,c,r the quadratic differential on the torus C/Λ arising from the k slit-folds parallel to v, centered at points c 1 , . . . , c k ∈ C/Λ and with radii r 1 , . . . , r k respectively. If all slit-folds are pairwise disjoint then q v,c,r ∈ Q((−1) 2k , 2 k ).
For every 1 ≤ j ≤ k denote by S j (w) ⊂ C/Λ the shadow of the j-th slit in the direction w, i.e. S j (w) = {c j + sv + tw : s ∈ [−r j , r j ], t ∈ [0, 1]}. A quadratic differential q v,c,r is called separated by the vector w ∈ Λ, if each shadow S j (w) is a proper cylinder (not the whole torus) and any two different shadows S j (w), S j (w) are either pairwise disjoint or the centers c j , c j lie on the same linear loop parallel to the vector w ∈ Λ.
Lemma 5.5. If q v,c,r is a quadratic differential on C/Λ which is separated by a non-zero vector w ∈ Λ then the Lyapunov exponent λ top (q v,c,r ) is positive.
Proof. Without loss of generality we may assume Λ = Z 2 , so w = (0, 1) and v = (1, 0). This assumption simplifies the argument. Let us divide the slit centers into N cliques (1 ≤ N ≤ k). Centers that lie on the same vertical linear loop are in a clique. Denote by x 1 , . . . , x N ∈ R/Z the horizontal coordinates of the cliques so that x 1 < x 2 < . . . < x N < x 1 + 1. We will also need cliques of the corresponding slit-folds; two slit-folds are in the same clique, if and only if their shadows in the vertical direction intersect, see Figure 15.
Suppose that the j-th clique contains m j ≥ 1 slit-folds centered at c j,l := (x j , y j,l ) ∈ C/Λ for 1 ≤ l ≤ m j so that y j,1 < y j,2 < . . . < y j,mj < y j,1 + 1. Then N j=1 m j = k. Since the quadratic differential is separated by the vertical direction, there are exactly N vertical linear loops that separate the cliques of slit-folds. For 1 ≤ j ≤ N , denote by s j a vertical upward-oriented linear loop separating the j-th and (j+1)-th cliques of slit-folds, see Figure 15. We adopt throughout the periodicity convention that the (N + 1)-th clique is the first one, i.e. x N +1 = x 1 .
Let (M v,c,r , ω v,c,r ) be the orientation cover of (C/Λ, q v,c,r ). Using Forni's criterion we will show that all Lyapunov exponents of ω v,c,r are positive. This implies the positivity of λ top (q v,c,r ). Let π : (M v,c,r , ω v,c,r ) → (C/Λ, q v,c,r ) be the natural projection. Then the holomorphic one form ω v,c,r lies in M(1 2k ) and the genus of M v,c,r is k + 1. More geometrically, M v,c,r is the translation surface made of two copies of a slitted torus C/Λ (denoted by T + -left; and T − -right), where the slits replace the slit-folds on (C/Λ, q v,c,r ), see Figure 15. Let σ : M v,c,r → M v,c,r be the involution that exchanges the slitted tori T + and T − by translation. Finally, each side of any slit on T + and T − is glued to its σ-image by a 180 degree rotation. Denote by π −1 ± : C/Λ → T ± the two branches of the inverse of π. Figure 15. The half-translation surface (C/Λ, q v,c,r ) and its orientation cover (M v,c,r , ω v,c,r ).
Note that all regular vertical orbits on (M v,c,r , ω v,c,r ) are periodic. We distinguish k + 2N such orbits: • for every 1 ≤ j ≤ N let O ± j = π −1 ± (s j ); • for every 1 ≤ j ≤ N and 1 ≤ l ≤ m j the orbit O j,l is made of two vertical segments: the first one joins π −1 + (c j,l ) and π −1 + (c j,l+1 ) inside T + and the second one joins π −1 − (c j,l+1 ) and π −1 − (c j,l ) inside T − (we adopt the convention that c j,mj +1 = c j,1 ).
Since π −1 + (c j,l ) = π −1 − (c j,l ) in M v,c,r , the above two segments together yield a periodic orbit O j,l .
From these k + 2N periodic orbits we choose k + 1, so that the surface obtained after removing the distinguished k + 1 orbits from M v,c,r is homeomorphic to the 2(k + 1)-punctured sphere. The choice of the periodic orbits depends on the parity of N . At first let us look at the surface For every 1 ≤ j ≤ N let R ± j be the region of T ± that is bounded by the orbit O ± j and the union mj+1 l=1 O j+1,l , see Figure 15. Similarly, L ± j is the region of T ± bounded by the orbit O ± j and the union mj l=1 O j,l . Then M is the union of 2N connected components and each such component A ± j is the union of L ± j ∪ R ∓ j−1 for 1 ≤ j ≤ N ; where we adopt the convention that R ± 0 = R ± N . The component A ± j is homeomorphic to the m j -punctured annulus ((m j + 2)-punctured sphere) and its boundary consists of orbits O ± j , O ∓ j−1 and O j,l for 1 ≤ l ≤ m j , see Figure 16. Odd case. If N is odd then we take: O + 1 and O j,l for 1 ≤ j ≤ N and 1 ≤ l ≤ m j . Since N j=1 m j = k, this yields a family of k + 1 vertical periodic orbits. Then the surface Since N is odd, all such junctures taken together are arranged in the following pattern:

Since each annulus A ±
j has m j punctures and appears in the above sequence exactly once, it follows that M 1 is an annulus with 2 N j=1 m j = 2k punctures. Therefore, M 1 is homeomorphic to the 2(k + 1)-punctured sphere.
Even case. If N is even then we take k + 1 vertical periodic orbits: O + 1 , O − 1 , O 1,l for 2 ≤ l ≤ m j and O j,l for 2 ≤ j ≤ N and 1 ≤ l ≤ m j . Then the surface is made of the punctured annuli A + j , A − j , 1 ≤ j ≤ N glued along the loops O + j , O − j for 2 ≤ j ≤ N and O 1,1 . Each such gluing yields a pattern Since N is even, all such junctures together are arranged in the following pattern: Since each annulus A ± j has m j punctures and appears in the above sequence exactly once, it follows that M 2 is an annulus with 2 N j=1 m j = 2k punctures. Therefore, M 1 is homeomorphic to the 2(k + 1)-punctured sphere.
Applying Proposition 5.4 to the translation surface (M v,c,r , ω v,c,r ) then yields the positivity of all Lyapunov exponents of ω v,c,r , and finally the positivity of λ top (q v,c,r ).
Lemma 5.5 combined with Proposition 5.3 leads to a trapping criterion for slitfolds systems q v,c,r . Recall that q v,c,r is the half-translation structure on C given by the system of slit-folds parallel to the vector v, centered at {c 1 , . . . , c k } + Λ and whose radii are r 1 , . . . , r k respectively. Corollary 5.6. If q v,c,r is a quadratic differential on C/Λ which is separated by a non-zero vector w ∈ Λ then the measured foliation F θ of (C, q v,c,r ) is trapped for almost every θ ∈ R/πZ. Let S be an infinite system of Eaton lenses on C and let θ ∈ R/πZ. Then P S,θ is an invariant set for the geodesic flow consisting of four copies of each lens and two copies of the complement of the lenses with planar geometry. This gives a natural projection π S,θ : P S,θ → C associating the footpoint (in C) to any unit tangent vector in P S,θ . We call the geodesic flow on P S,θ trapped if ∃ C>0 ∃ u∈C,|u|=1 ∀ t∈R ∀ x∈P S,θ | π S,θ (G S,θ t x) − π S,θ (x), u | ≤ C. Remark 5.7. Note that the geodesic flow on P S,θ is trapped, if and only if Moreover, the geodesic flow on P S,θ is trapped, if and only if the direction θ foliation on the corresponding slit-fold plane is trapped.
Let Λ be a lattice on C and let c = (c 1 , c 2 , . . . , c k ) ∈ C k be a vector such that the points c j + w are pairwise distinct for 1 ≤ j ≤ k and w ∈ Λ. Each such vector is called proper. A vector of radii r = (r 1 , r 2 , . . . , r k ) ∈ R k >0 is called (Λ, c)-admissible if dist(c i + Λ, c j + Λ) > r i + r j for i = j. Admissibility guarantees that Eaton lenses of radius r j centered at c j + Λ for 1 ≤ j ≤ k do not intersect. Recall, that such a Λ-periodic system of Eaton lenses is denoted by L(Λ, c, r). Of course, the set of (Λ, c)-admissible vectors is open in R k .
Let A = {A 1 , . . . , A m } be a partition of {1, . . . , k}. Then for every r ∈ R k and x ∈ R m denote by r(x) the vector in R k defined by r(x) j = x l r j whenever j ∈ A l . In particular, taking x = 1 = (1, . . . , 1) ∈ R k gives r(1) = r.
Denote by Adm Λ,c,A ⊂ R m >0 the set of all x ∈ R m >0 such that the vector 1(x) is (Λ, c)-admissible. This is a non-empty open subset.
Proof. First we pass to the flat version of any admissible system L(Λ, c, r) and its geodesic flow in direction θ ∈ R/πZ. The resulting object is the quadratic differential q e i(θ+π/2) ,c,r on C and its foliation F θ . The geodesic flow on P L(Λ,c,r),θ and the foliation F θ are orbit equivalent.
As a corollary we obtain the following more general version of Theorem 1.2.
Corollary 5.9. For every lattice Λ ⊂ C, every proper vector of centers c ∈ C k and every partition A of {1, . . . , k} the geodesic flow on P L(Λ,c,1(r)),θ is trapped for a.e. (r, θ) ∈ Adm Λ,c,A × R/πZ. Let us now consider the partitions {{1}, {2}, {3}} and {{1, 2, 3}}. By applying Theorem 5.8 to those, we obtain another type of results saying, that almost every small perturbation of (L θ0 , θ 0 ) leads to a trapped geodesic flow. In the first case all radii are perturbed independently whereas in the second case all radii are perturbed simultaneously.
In summary, the curves of ergodic lens distributions described in the paper are very exceptional. They are surrounded by highly non-ergodic systems. We have shown this phenomenon only for a particular "ergodic" curve, but for the other "ergodic" curves it can be shown along the same lines.
Moreover, we conjecture that the trapping property is measurably typical along many curves transversal to the ergodic curves described in the paper. An interesting and highly involved result of that type was proved in [16], where the authors consider curves arising from fixed systems of lenses for which the direction θ varies.

Appendix A. Eaton lens dynamics
To precisely describe the dynamics of light rays passing through an Eaton lens, we denote the lens of radius R > 0 and centered at (0, 0) by B R . The refractive index (RI for short) in B R depends only on the distance from the center r := x 2 + y 2 ∈ (0, R] and is given by the formula n(x, y) = n(r) = 2R/r − 1; at the center we put n(0, 0) = +∞. Suppose, for simplicity, that the refractive index n(x, y) is constant and equals 1 outside B R . Recall that the dynamics of light rays can be described as the geodesic flow on R 2 \ (0, 0) equipped with the Riemannian metric g = n·(d x⊗d x+d y⊗d y). Of course, the geodesics are straight lines or semilines outside B R . The dynamics of the geodesic flow inside B R was described for example in [21]. After passing to polar coordinates (r, θ) we use the Euler-Lagrange equation to see that any geodesic inside B R satisfies where (r 0 , θ 0 ) is a point of the geodesic minimizing the distance to the center. It follows that for any point (r, θ) of the geodesic in B R we have In particular inside of B R the geodesic is an arc of an ellipse. Let s := R 2 − (R − r 0 ) 2 and rotate the geodesic by −θ 0 . Then the equation of the ellipse becomes Since the ellipse is centered at (− √ R 2 − s 2 , 0) and (− √ R 2 − s 2 , ±s) are its in- Figure 18. Flow directions inside and outside of an Eaton lens tersection points with the boundary of B R , the geodesic has horizontal tangents at these intersecting points. Rotating everything back to the original position we see, that the direction of any geodesic is reversed after passing through B R . The only exception is the trajectory that hits the center of the lens. For this trajectory we adopt the convention, that at the center it turns and continues its motion backwards.
Now for every θ ∈ R/πZ consider the restriction of the geodesic flow (g θ t ) t∈R to its invariant subset of the unit tangent bundle of R 2 consisting of all trajectories assuming direction θ or π+θ outside B R . Denote by P θ the phase space of that flow. Since all flows (g θ t ) t∈R are isomorphic by rotations, we restrict our considerations to the horizontal flow (g t ) t∈R = (g 0 t ) t∈R . Denote by B R the interior of B R . Through every point of B R pass exactly four trajectories of (g t ) t∈R , while through every point of A R := R 2 \ B R pass exactly two, in direction 0 and π, see Figure 18. It follows, that P θ consists of four copies of B R (B ±± R ) and two copies of A R (A ± R ). Let us take a closer look at the dynamics (g t ) t∈R on the four copies of B ±± R . Since they are related by a reflective symmetry or the reversal of time, we can restrict Therefore, Let t s be the exit time of (− √ R 2 − s 2 , s) from B ++ R . Since r(t s , s) minimizes the distance to the origin, we have s 2 = R 2 − (R − r(t s , s)) 2 = r(t s , s)(2R − r(t s , s)). It follows that t s = R arcsin 1 = 1 2 πR.
Introduce new coordinates on B ++ R given by (t, s). Then the set E R = B R ∪ ([0, πR/2) × (−R, R)) is the domain of these coordinates and they coincide with the cartesian coordinates on C R . Moreover, by definition, the geodesic flow (g t ) t∈R in the new coordinates is the unit horizontal translation in positive direction.
One can define the same type of coordinates on the other copies B +− R , B −+ R and B −− R . Let us consider a measure µ on P 0 that coincides with the Lebesgue measure on A ± R and the Lebesgue measure in the new coordinates on each B ±± R . This is a (g t ) t∈R -invariant measure and we will calculate its density in the next paragraph. .
differentiating it in the direction s we obtain .
Differentiating the first equality of (A.3) in the direction s we obtain In view of (A.7), it follows that ∂θ ∂s = 1 r(2R − r) − s 2 r(2R − r) .
By (A.3), Hence, Therefore, the density of the invariant measure µ restricted to B ++ R in the cartesian coordinates is On the other copies B ±± R the measure µ is given by ξ R (±x, y) d x d y.
For every θ ∈ R/πZ the flows (g θ t ) t∈R phase space P θ is given by the rotation of P 0 by θ and the invariant measure µ θ is the rotation of µ by the same angle.
Generally instead of one Eaton lens on the plane we deal with a pattern L of infinitely many pairwise disjoint Eaton lenses on R 2 . We are interested in the dynamics of the light rays provided by the geodesic flow (g L t ) t∈R on R 2 without the centers of lenses; the Riemann metric is given by g (x,y) = n(x, y)·(dx⊗dx+dy⊗dy). The local behavior of the flow around any lens was described in detail previously. For every θ ∈ R/πZ there exists an invariant set P L,θ in the unit tangent bundle, such that all trajectories on P L,θ are tangent to ±e iθ outside the lenses. The restriction of (g L t ) t∈R to P L,θ is denoted by (g L,θ t ) t∈R . Moreover, (g L,θ t ) t∈R possesses a natural invariant measure µ L,θ equivalent to the Lebesgue measure on P L,θ . The density of µ L,θ is equal to one outside lenses and inside every lens of radius R centered at (c 1 , c 2 ) is determined by ξ R (±(x − c 1 ), y − c 2 ) depending on its copy in the phase space. Moreover, the density is continuous on P L,θ and piecewise C ∞ .
A.1. From the geodesic flow to translation surfaces and measured foliations. For simplicity we return to a single lens and the horizontal flow (g θ t ) t∈R on P 0 . Representing P 0 in (t, s) coordinates, we can treat it as the union on A ± R and D ±± R , see Figure 20. Moreover, the new coordinates give rise to a translation structure on the surface P 0 . Since the horizontal sides of D ±± R do not belong to P 0 , the surface is not closed. However, we can complete the surface by adding the horizontal sides as in Figure 21. Let us denote the completed surface by P 0 . It has two singular points with the cone angle 6π which are connected by two horizontal saddle connections labeled by A and B in Figure 21. Moreover, the flow (g t ) t∈R is measure-theoretically isomorphic to the horizontal translation flow on the translation surface P 0 .
Let us consider an involution σ : P 0 → P 0 given by the translation between upper and lower parts of P 0 in Figure 20. Then the quotient surface Q 0 = P 0 / < σ > is a half-translation surface. It has two singular points (marked by circles) having cone angle 3π connected by a horizontal saddle connection labeled by A (and then continued as A ) and two poles (marked by squares), see Figure 22.   If we consider an infinite pattern L of Eaton lenses on R 2 , then for every θ ∈ R/πZ we can similarly represent the space P L,θ as a translation surface which after a completion is a closed translation surface P L,θ . The translation flow (ϕ L,θ t ) t∈R on P L,θ in the direction θ is measure-theoretically isomorphic to the flow (g L,θ t ) t∈R . Moreover, the surface P L,θ has an natural involution σ which maps a unit vector to the vector at the same foot-point but oppositely directed. The quotient surface Q L,θ = P L,θ / < σ > is a half-translation surface that is the euclidian plane with a system of pockets each attached at the place of the corresponding lens. Each pocket is a rotated (by θ) version of the pocket in Figure 22. Its length is equal to the diameter of the corresponding lens and is perpendicular to θ. Most relevant for us, the ergodicity of measured foliation F L θ in the direction θ on Q L,θ is equivalent to the ergodicity of (ϕ L,θ t ) t∈ , and hence to the ergodicity of the flow (g L,θ t ) t∈R . The measured foliation F L θ is Whitehead equivalent to the foliation FL L θ where each attached briefcase is replaced by the slit-fold stemming from the "flat lens" representation of the same Eaton lens in direction θ, as in Figure 23. In summary, Figure 23. The half-translation equivalent to an Eaton lens instead of studying the ergodic properties of the geodesic flow (g L,θ t ) t∈R on the plane with a system of Eaton lenses it suffices to pass to the measured foliation FL L θ where each Eaton lens is replaced by the corresponding flat lens of the same center and diameter as the lens attached perpendicular to θ.

Appendix B. Folds and Skeletons
In this section we describe examples of ergodic curves obtained from other torus differentials. Starting with some of the quadratic differentials in our table one obtains quadratic differentials on the plane that are not pre-Eaton differentials. This section shows ways how to convert those into pre-Eaton differentials. In particular, the quadratic differentials on the plane we deal with, have holes that need to be removed. We model the holes by pillow-folds and then convert them to an appropriate union of slit-folds.  X d (a, b). First we convert the standard polygonal representation of a pillowcase cover X d (a, b) into a pre-Eaton differential. Recall from Section 3 that for X 3 (2, 1), X 4 (2, 1) and X 6 (3, 1) this can be done by a central cut followed by turning one half underneath the other. After the half-turn the absolute homology generators are arranged as shown in Figure  24 for X 6 (3, 1). The arrangement for the absolute homology of X 3 (2, 1) looks similar after the half-turn: The two homology generators overlap in the middle third of the rectangle representing the surface. Then consider the universal cover X 6 (3, 1) → X 6 (3, 1) determined by the pair of homology generators. Let us label the deck shifts as in Figure 24, and the decks by Z 2 . Then, starting at deck (0, 0) we reach deck (1, 0), once crossing the left third of the rectangles upper edge and we Figure 25. The universal homology cover X 6 (3, 1) reach deck (0, 1) when crossing the right third. We enter deck (1, 1) when crossing the middle third of the upper edge and so forth. The labeled tiles of Figure 25 show the cover. It has rectangular holes causing jumps of the directional dynamics in the plane. In particular the skeleton describing the quadratic differential contains boundaries of the spared rectangles besides the slit-folds, see Figure 26. Let us now Skeleton modulo Z 2 symmetry: Figure 26. Skeleton representation of C 6 (3, 1) forget the covering and just consider the skeleton on the plane. While only the dynamics outside the spared rectangles was previously defined we now extend the definition to the inside as follows: The folded parts of any rectangle are genuine slit-folds and the translation identified edges are translation identified from the inside, too. That way we obtain a quadratic differential on the whole plane that we denote by C 6 (3, 1). The notation, a combination of the standard complex plane notation together with the weight notation of the pillow case cover, will be used for other surfaces below. The "inside" of each rectangle is a pillow-case carrying invariant foliations. The natural extension promotes an easy geometric definition of the foliation: Given a direction θ ∈ R/πZ consider the unoriented lines parallel to ±e iθ in C. Then put a skeleton in the plane and identify the intersection points of the leaves with the skeleton according to the respective rules, i.e. translation or central rotation.
The extension of the quadratic differential, i.e. X 6 (3, 1), to the whole complex plane, i.e. C 6 (3, 1), is a step towards realizing the skeleton by admissible Eaton lens configurations on the plane. In fact this first step allows us to converts the "outside" quadratic differential into a pre-Eaton differential, as shown in Figure 27. Seen from the view point of Eaton lens dynamics the conversion removes the jump of leaves over the rectangular gaps and replaces it by an equivalent jumpfree dynamics. The process shown in Figure 27 performed backwards is a railed deformation moving slit-folds through other slit-folds that changes their character: A pair of slit-folds becomes a pair of translation identified lines. At the end we merge singular points which is not a railed deformation in the strict sense of the definition. Pillow-folds and chip-folds. All that can be done for rectangular folds can be done for folds built from a parallelogram. So our definitions include parallelograms. Let this is the fold configuration with n+1 slit folds parallel to [a, b]. We call this object n-chip-fold. In particular, a 1-chip-fold is a chip-fold. Analogously a, b n · |c, d| denotes the n-pillow-fold obtained by replacing all slit-folds of an n-chip-fold that are parallel to [c, d] by line segments. These line segments are identified by a translation in the direction of the vector − → ab.
Proposition B.1. Take a plane equipped with a single pillow-fold and consider a fixed direction foliation on the outside of a pillow-fold. Then there is an n-chipfold or a pair of parallel slit-folds which has an outer measured foliation Whitehead equivalent (up to a finite number of leaves) to the given measured foliation.
Proof. Since the problem is invariant under affine transformations we can consider a pillow-fold 0, ib |0, a| in the complex plane where a, b ∈ R + , so the segment [0, a] is horizontal, and [0, ib] is vertical. For fixed θ ∈ R/πZ consider the outer foliation for 0, ib |0, a|. If θ = π/2, we translate the two horizontal sides together at the center of the rectangle. The resulting skeleton consists of two vertical slit-folds. That is a Whitehead move and so the outer foliations are equivalent.
Suppose | tan θ| ≤ b a . That is, the slope of the foliation is bounded by the slope of the diagonal [0, a+ib] of the rectangle [0, ib]×[0, a]. In this case translate the two horizontal edges of the rectangle parallel to the foliation towards its inside through the vertical slit-folds, so that every point on the edges remains on the same line (including the slit-fold identification) of slope tan θ, as shown in Figure 27. Note, that the two horizontal edges form a loop. Two slit-folds appear unless | tan θ| = b a . In that case, both slit-folds fall together and we regard it as a single slit-fold centered at the center of the rectangle. By construction both outer measured foliations differ by a Whitehead move that breaks up the singular point at the vertex of the pillowfold, so they are equivalent.
For larger angles we need to use an intermediate step. In fact, if θ is not covered by the previous case(s), then there is a minimal n ≥ 2 such that | tan θ| ≤ n b a . Then change the given pillow-fold 0, ib |0, a| into an n-pillow-fold 0, ib n·|0, a/n| by putting n−1 successive a/n translates of the left vertical slit-fold into the rectangle. Then for each of the n (translation equivalent) pillow-folds the previous conversion into a union of slit-folds applies. Here we may change finitely many leaves, the ones hitting the endpoints of the n − 1 new slit-folds that are put into the pillow case. Again we find a measurably equivalent outer foliation. Note, that the inner foliation is changed by this procedure, but this is irrelevant for our claim.
Let us call a skeleton in the plane standard skeleton, if it is a countable union of pillow-folds and slit-folds, so that no pillow-fold contains other folds. For those we can use Proposition B.1 inductively to obtain: Corollary B.2. For any quadratic differential defined by a standard skeleton and any direction θ ∈ R/πZ the outer measured foliation tangential to θ in the plane is up to countably many leaves Whitehead equivalent to the direction foliation of a pre-Eaton-differential in the plane.
The skeletons we consider are special, they have exactly one unbounded component. With the boundary identifications given by the skeleton the unbounded component is homeomorphic to a plane. B.2. Other ergodic Eaton curves. Using the X 3 (2, 1), X 6 (3, 1) and X 6 (3, 2) torus differentials, we present more examples of admissible ergodic Eaton lens curves. Skeletons of the torus differentials allow us to write down differentials on the plane and represent them geometrically by arrow diagrams as in Figures 29,30 and 28. In those particular cases all folds will be horizontal and vertical in cartesian coordinates. Because the skeleton depends on the angle, see Proposition B.1, we only present the ergodic curve for small angles. We do not give a formal proof of admissibility for those Eaton lens distributions, it would go along the same lines as done in Proposition 2.4 for the Wollmilchsau differential. The figures give some clues how to work out the details, such as dividing tangent lines between some lenses.