Minimality of the Ehrenfest wind-tree model

We consider aperiodic wind-tree models, and show that for a generic (in the sense of Baire) configuration the wind-tree dynamics is minimal in almost all directions, and has a dense set of periodic points.


INTRODUCTION
In 1912 Paul and Tatyana Ehrenfest proposed the wind-tree model in order to interpret the ergodic hypothesis of Boltzmann [8]. In the Ehrenfest wind-tree model, a point particle (the "wind") moves freely on the plane and collides with the usual law of geometric optics with irregularly placed identical square scatterers (the "trees"). Nowadays we would say "randomly placed", but the notion of "randomness" was not made precise; in fact, it would have been impossible to do so before Kolmogorov laid the foundations of probability theory in the 1930s. The wind-tree model has been intensively studied by physicists, see for example [3], [7], [10], [13], [27], [28] and the references therein.
From the mathematically rigorous point of view, there have been many recent results about the dynamical properties of a periodic version of wind-tree models: scatterers are identical square obstacles with one obstacle centered at each lattice point. The periodic wind-tree model has been shown to be recurrent ( [12], [16], [1]), to have abnormal diffusion ( [6], [5]), and to have an absence of ergodicity in almost every direction ( [9]). Periodic wind-tree models naturally yield infinite periodic translation surfaces. Ergodicity in almost every direction for such surfaces have been obtained only in a few situations [15], [14], [22].
On the other hand for randomly placed obstacles, from the mathematically rigorous point of view, up to now it has only been shown that if at each point of the lattice Z 2 we either center a square obstacle of fixed size or omit it in a random way, then the generic in the sense of Baire wind-tree model is recurrent and has a dense set of periodic points ( [24]).
In this article we continue the study of the Baire generic properties of windtree models. We study a random version of the wind-tree model: the plane is tiled by one by one cells with corners on the lattice Z 2 , and in each cell we place a square tree of a fixed size with the center chosen randomly. Our main result is that for the generic in the sense of Baire wind-tree model, for almost all directions the wind-tree model is minimal, in stark contrast to the situation for the periodic wind-tree model, which can not have a minimal direction. 1 This result can be viewed as a topological version of the Ehrenfest's question.
The method of proof is by approximation by finite wind-tree models where the dynamics is well understood. There is a long history of proving results about billiard dynamics by approximation which began with the article of Katok and Zemlyakov [17]. This method was used in several of the results on wind-tree models mentioned above [16], [1], [24]. See [23] for a survey of some other usages in billiards. The idea of approximating infinite measure systems by compact systems was first studied in [19].
The structure of the article is as follows. In Section 2 we give formal statements of our results. In Section 3 we collect the notation necessary for our setup. Sections 4, 5 and 6 are devoted to the proof of different parts of the main theorem. Our proofs hold in a more general setting than the one described above, for example we can vary the size of the square, or use certain other polygonal trees. We discuss such extensions of our result in Section 7. Finally in Appendix A we discuss the relationship between the usual convention on the orbit of singular points for interval exchange transformations and for polygonal billiards. These conventions are not the same. Since we are studying minimality in this article, a careful comparison is made, and certain known results are reproved for a class of maps we call eligible. In particular the IETs arising from billiards with the billiard convention for orbits arriving at corners of the polygon are eligible maps.

STATEMENTS OF RESULTS
We consider the plane R 2 tiled by one by one closed square cells with corners on the lattice Z 2 . Fix r ∈ [1/4, 1/2). We consider the set of 2r by 2r squares, with vertical and horizontal sides, centered at (a, b) contained in the unit cell [0, 1] 2 . This set is naturally parametrized by with the usual topology inherited from R 2 . Our parameter space is A Z 2 with the product topology. It is a Baire space. Each parameter g = (a i , j , b i , j ) (i , j )∈Z 2 ∈ A Z 2 corresponds to a wind-tree table in the plane in the following manner: the tree inside the cell corresponding to the lattice point (i , j ) ∈ Z 2 is a 2r by 2r square with center at position (a i , j , b i , j ) + (i , j ). The wind-tree table B g is the plane R 2 with the interiors of the union of these trees removed. Note that trees can intersect only at the boundary of cells.
Fix a direction θ ∈ S 1 . The billiard flow in the direction θ is the free motion on the interior of B g with elastic collision from the boundary of B g (the boundary of the union of the trees). The billiard map T g θ in the direction θ on the table is the first return to the boundary. If the flow orbit arrives at a corner of the table, the collision is not well defined, and we choose not to define the billiard map, i.e., the orbit stops at the last collision with the boundary before reaching the corner; also backward orbits starting at a corner of a tree are not defined, but forward orbits starting at a corner are defined. Once launched in the direction θ, the billiard direction can only achieve four directions {±θ, ±(θ − π)}; thus the phase space Ω g θ of the billiard map T g θ is a subset of the cartesian product of the boundary with these four directions. It contains precisely the pairs (s, φ) such that at s the direction φ points to the interior of the table, i.e., away from the trees. The billiard map will be called minimal if the orbit of every point is dense. The set of periodic points is called locally dense if there exists a G δ -subset of the boundary which is of full measure, such that for every s in this set, there is a dense set of inner-pointing directions θ ∈ S 1 for which (s, θ) is periodic. We call a forward (resp. backward) T g θ -orbit a forward (resp. backward) escape orbit if it visits any compact set only a finite number of times.

THEOREM 2.1.
There is a dense G δ set of parameters G such that for each g ∈ G i) for a dense-G δ set of full measure of θ the billiard map T g θ is minimal and has forward and backward escape orbits, ii) the map T g has a dense set of periodic points, iii) if r is rational then the map T g has a locally dense set of periodic points, iv) no two trees intersect.
All the sets mentioned in the theorem depend on the fixed parameter r . From our definition of minimality we conclude COROLLARY 2.2. The backward orbit of any forward escape orbit is dense (and vice versa) for each g ∈ G .
COROLLARY 2.3. The billiard flow on the wind-tree table is also minimal for each g ∈ G .
We would like to point out that there is an old theorem of Gottshalk stating that (a stronger version of) minimality is impossible in non-compact, locally compact spaces [11,Theorem B]; more precisely, for a homeomorphism of a locally compact metric space X , if the forward orbit of every point y ∈ Y is dense in Y , then Y is compact. This result does not apply directly to our situation: our map is not a homeomorphism, the dynamics is not defined everywhere, and where it is not defined it is discontinuous. There is a standard way of changing the topology to make the map a homeomorphism (this construction is well described in the context of interval exchanges in [20, Section 2.1.2]). For any wind-tree table g ∈ A Z 2 , including the periodic ones, the topology obtained from Ω g θ will be locally compact but not compact. Thus Gottshalk's result apply, and the wind-tree model can never be forward minimal. In fact, in Theorem 2.1.i we construct examples of escape orbits.

NOTATIONS AND PREPARATORY REMARKS
As already mentioned in the previous section the billiard map T g θ is not defined at a point whose next collision is with a corner, and the inverse billiard JOURNAL OF MODERN DYNAMICS VOLUME 10, 2016, 209-228 map (T g θ ) −1 is not defined at a corner. In the world of billiards or (flat surfaces), a saddle connection is a flow-orbit going from a corner of a tree to some corner (maybe the same one). Because of the above convention, for the map there is a saddle connection starting at a point x if, for some k ≥ 0, T k (x) is defined but T k+1 (x) and T −1 (x) are not defined; then the saddle connection is the orbit A direction θ is called exceptional if there exists a saddle connection for T g θ . As there are countably many corners, there are at most countably many saddle connections and thus at most countably many exceptional directions. For any positive integer N , we define R N to be the closed rhombus (square) {(x, y) : |x| + |y| ≤ N + 1 2 } and we define then Ω . Let E N be the set of pairs (i , j ) so that the interior of the (i , j )-th cell is contained in R N , and let R N be the interior of the union of the closed cells indexed by E N . Let us also define Ω g θ,N to be Ω Suppose that N is an integer satisfying N ≥ 2. We will call a parameter N -tactful if for each cell inside the rhombus R N , the corresponding tree is contained in the interior of its cell. We will call an N -tactful parameter N -ringed if the boundary of R N is completely covered by trees. We call a parameter tactful if it is N -tactful for all N .
For N -ringed parameters there is a compact connected rational billiard table R N ∩ B f , called the N -ringed table, contained in the rhombus R N (see Figure 1).
The corresponding phase space is Ω  There are at most countably many exceptional directions, and for all nonexceptional directions, Ω We need to describe Ω g θ,N more concretely for any N -tactful g . Note that if the tree is contained in the interior of a cell then if s is a corner of this tree there are three directions pointing to the interior of the table, while for all other s there are only two such directions (see Figure 3). Intersecting trees can have slightly different behavior, but we do not need to describe it since they will not occur in our proof.
We think of the contribution of each tree to Ω g θ,N as the union of four closed intervals indexed by φ ∈ {±θ, ±(θ − π)}. Each of these intervals corresponds to the cartesian product of the two intersecting sides of the tree with a fixed inner pointing direction φ (see Figure 3). As before, the word inner means pointing into the table, so away from the tree. In the proof we will think of each of these intervals as I = [0, 2 2r ] since it corresponds to the diagonal (of length 2 2r ) of the tree centered at (a, b). Note that the billiard map in a fixed direction has a natural invariant measure. Let p be the arc-length parameter on I ; then I = I 1 ∪ I 2 where each I i is an interval and the invariant measure is of the form K i d p on I i with K i an explicit constant. We stick to the use of I in order to avoid manipulating the constants K i (which are direction dependent) all the time.
To make our map orientation preserving we choose the orientation of these intervals in the following way: use the clockwise orientation inherited from the tree for φ ∈ {θ, θ − π} and the counterclockwise orientation of the other two values of φ. In particular this parametrization does not depend on the angle φ (see Figures 3 and 4). Despite the fact that these "intervals" come naturally as subsets of R 2 , we will think of Ω g θ, N and Ω g θ as formal disjoint unions of onedimensional intervals.  For any N -tactful g , let J g N be the collection of all the intervals as described arising from the trees in R N . Note that the trees straddling the rhombus do not contribute to this collection.
For each tree t ∈ A let U (t , ε) be the the standard ε-neighborhood in R 2 intersected with the interior of A in R 2 . For any parameter g

PROOF OF MINIMALITY IN THEOREM 2.1
Proof. In the Appendix we study a class of maps called eligible maps. The proof of minimality in the theorem is based on Lemma A.1 of the Appendix which gives a necessary and sufficient condition for the minimality of an eligible map. We start with some remarks on the applicability of this lemma. First of all note that in the proof we will only need to apply Lemma A.1 to maps which are tactful. However these maps are not eligible. Ω Furthermore it suffices to show this for a countable basis of intervals.
By Corollary A.4 in the Appendix for any N -ringed f , any ( f , N )-non-exceptional direction θ and any interval I ⊂Ω Now consider the following perturbation of f : the new configuration g is arbitrary in the cells which do not intersect in such a way that the new parameter is still N -tactful and the trees in the cells covering the boundary of R N are replaced by close trees in such a way that the configuration is N + 1-tactful (see Figure 2). The main idea is if equation (1) holds for an open interval I inΩ f θ,N there exists an ε > 0 such that for all g ∈ U N ( f , ε), the equation (1) still holds for g and I , namely Here we can write the same interval I since there is a natural identification betweenΩ g θ,N andΩ f θ,N which will be made explicit in the proof. We recall that each tree contributes four intervals to the phase space of the wind-tree transformation in a given direction. Each of these intervals is a copy of the interval [0, 2 2r ]. Since inΩ g θ,N only the trees contained in R N are contributing, J We will say that there is a *saddle connection starting at a point . There are at most countably many *exceptional directions. Any non-*exceptional direction is non-exceptional, thus equation (1) still holds for any non-*exceptional direction θ, for f an N -ringed configuration.
To prove that the billiard mapT g θ in a given direction is minimal, it suffices to show that there exists infinitely many N such that For any N -tactful g , let C g N (K , L) be the collection of all the connected components of (T By Baire's theorem the set of configurations which are tactful is dense since for each N the set of all N -tactful configurations is an open dense set. Thus we can consider a countable dense set of parameters which are N -tactful for all N . By modifying the parameters we can assume that each one is N -ringed for a certain N still maintaining the density. Call this countable dense set { f i }, with f i being N i -ringed. We also assume N i +1 > N i . Suppose ε i are strictly positive. Let Clearly G is a G δ -dense set. We claim that there is a choice for ε i such that every parameter in G gives rise to a wind-tree model which is minimal in almost all directions. Fix f i . We already proved that equation (3) holds be the two integers given by equation (3). For the sake of simplicity, we will denote by We describe the set ∂C i (θ) exactly. Without loss of generality let us suppose for backward orbits. Second, let us consider the following sub-collection of the forward orbits of corners of trees T k θ (x) : x ∈ ∂J N i , 0 ≤ k ≤ L i , and similarly for backward iterates. Third, consider the iterates of the endpoints of I for times between K i (θ) and L i (θ). Then ∂C i (θ) is the restriction of the union of these three collections to Ω f i θ,N i . For each θ this is a finite collection of points. Note that each point in ∂C i (θ) is the endpoint of exactly one interval in C i (θ) because θ is not ( f i , N i )-*exceptional. As we vary the parameters (g , θ), clearly ∂C i (θ) will change. Moreover, if the direction θ is a non-( f i , N i )-*exceptional direction, then the points in the set change continuously in the following sense: each point in ∂C i (θ) has ( f i , θ) as a point of continuity.
Let Θ i be the set of all directions θ that are not ( f i , N i )-*exceptional. This set is of measure one since its complement is countable. For every θ ∈ Θ the points in the collection ∂C i (θ) are all distinct. Recall that Ω f i θ,N i is a union of oriented intervals, so the intersection of ∂C i (θ) with any of these intervals has a natural total order (which is a strict order). These orders induce a strict partial order in ∂C i (θ).
So, for every fixed θ ∈ Θ i it is possible to choose continuously δ i (θ) > 0 such that the strict partial order on ∂C i (θ ) is preserved for all (g , θ ) in the open set U N i ( f i , δ i (θ)) × B (θ, δ i (θ)) (where B denotes a ball in S 1 ); and thus, equation (3) holds with the same K i (θ) and L i (θ) for every such (g , θ ).
Furthermore, let us suppose now that K i (θ) and L i (θ) are optimal for equation (3) to hold. By this we mean that −K i (θ) + L i (θ) is minimal, and then if there are several choices K i (θ) is chosen maximal still satisfying K i (θ) < 0. Then

PROOF OF THE EXISTENCE OF ESCAPE ORBITS.
Proof. We will prove the existence of an escape orbit for the parameter set G and direction set Θ defined in the proof of minimality in Section 4. For any g ∈ G and θ ∈ Θ, as shown above θ will be a minimal direction of T = T g θ .
Let us consider f an N -ringed parameter, ε > 0, N > N , and f an N -ringed parameter in U N ( f , ε). Then, for any ε > 0, the set ε). Let us now fix such an ε , and let θ be a direction which is far for horizontal and vertical in the following sense: min(| tan(θ)|, | cot(θ)|) ≥ ε 2r . Then for any g ∈ U N ( f , ε ) we have that Ω g θ,N +1 Ω g θ,N is visited by any orbit starting in Ω g θ,N before reaching Ω g Ω g θ,N +1 since it makes a collision with one of the squares in the ring. In particular for any point x ∈ Ω g θ,N its orbit cannot escape to Ω g Ω g θ,N +1 without a collision in the ring of obstacles Λ g θ,N := Ω g θ,N +1 Ω g θ,N . Similarly in the opposite way, an orbit that is already outside cannot come inside without a collision on the ring of obstacles.
Let g ∈ G and let us consider an approximating sequence f i j such that g ∈ U N i j ( f i j , ε i j ) for all j . For simplicity of notation let Λ j = Λ g θ,N i j . For any nonhorizontal and non-vertical direction θ, the condition min(| tan(θ)|, | cot(θ)|) ≥ ε i j 2r discussed above is verified for j large enough since ε i is a decreasing sequence going to zero. Thus, we can choose J so large that for any j ≥ J , the ε i j is sufficiently small such that for any x ∈ Ω g θ,N i j the T orbit of x must visit the set Λ j before reaching Λ j +1 . The same is true in the opposite direction: no orbit can go from Ω to Λ j without a collision on Λ j +1 . (A similar statement holds for T −1 .) Thus the far away dynamics of T can be understood via the following transformations. For any j > J and any x ∈ Λ j , let S + (x) be T k (x), the first visit to Λ j ±1 . (Note that this is not a first return map to the union of the Λ j and it is not Where defined, these transformations satisfy Now suppose θ ∈ Θ. Note that θ is not vertical nor horizontal because these directions are ( f i , N i )-exceptional for every i . Consider the compact set Ω The set A j ,1 is non-empty since the (forward) T -orbit of any corner of a tree in Λ j is dense in Ω g θ , thus it has to get out of Ω g θ, j +1 and in doing so it is forced to have a collision in Λ j +1 . Thus the last time this orbit visits Λ j before visiting Λ j +1 will be an element of A j ,1 . Now inductively define the set A j ,n+1 := x ∈ A j ,n : ∃ k > 0, such that ∀i = 1, 2, . . . , k − 1, For each n ≥ 1 the set A j ,n is non-empty by a similar reasoning as above. Clearly A j ,n+1 ⊂ A j ,n , thus since Λ j is compact, B j := ∩ n≥1 A j ,n is non-empty. We claim that if x is in this intersection, then x is in all of the A j ,n , and thus the forward orbit of x never returns to Λ j .
Suppose not, and let m := min{n ≥ 1 : x ∈ A j ,n A j ,n }. This implies that for some k, T i (x) ∈ Λ j for i = 1, . . . , k − 1 and T k (x) is not defined; in fact, T k (x) would arrive at a corner of a tree of the obstacle ring Λ j +m . More precisely, choose a sequence (x ) ⊂ A j ,m such that x → x and T k (x) ∈ Λ j +m . Then we let y := lim →∞ T k (x ). Since our direction θ is non-exceptional, the forward orbit of y is infinite. Furthermore, we have y ∈ A j +m,n for all n, thus y is a forward orbit which never visits Λ j and which is backward singular. Since Λ j has nonempty interior (it contains intervals), this contradicts the minimality of T , and thus the forward orbit of x is not singular.
Since T g θ is minimal, every point in B j +1 must have come from Λ j , thus we have B j +1 ⊆ R + • S + B j . This implies for all j ≥ J ; here the first two equalities use the relations in equations (4) and the last equality follows from the definitions of B j and R + .
Let C j +n := S − • R − n B j +n . Iterating the above computation shows that these sets are nested, C j +n+1 ⊂ C j +n for all n ≥ 0. The set n≥0 C j +n is nonempty since it is contained in the compact set Λ j . The forward orbit of any point in this set is either singular or an escape orbit. The argument that such forward orbits are non-singular is identical to the one given above for B j .
Finally we remark that if (s, θ) is a forward escape orbit, then (s, θ + π) is a backward escape orbit, and vice versa.

PROOF OF DENSITY AND LOCAL DENSITY OF PERIODIC POINTS
Proof. The idea behind the proof is similar to what has been done for minimality in Section 4. We first apply a known result to N -ringed parameters.
In this section, the direction θ varies in the proof, so we abandon the notation T g θ and we note the billiard transformation in the wind-tree by T g (s, θ).
For each point x = (s, θ) ∈ Ω g θ and each p such that T g p (x) exists, let us consider the set of directions for which the orbit starting at s hits the same sequence of sides up to time p: θ : T g i (s, θ ) and T g i (s, θ) lie on the same side of the same tree for i = 1, . . . , p .
This set is an open interval. We will note by θ g − (x, p) and θ g + (x, p) the lower and the upper bound of this interval. We also consider the interval (t g − , t g + ) where t ± is the spatial coordinate of T g p (s, θ g ± ). Fix an N -ringed parameter f and x = (s, θ) ∈ Ω f θ,N such that (T f ) p (x) exists. Remember that the identification between the phase space is a formal identity map discussed in Section 4. Since (T g ) p is locally continuous at x, both θ g − (x, p) and θ g + (x, p), and thus t g − (x, p) and t g + (x, p), vary continuously with respect to g in a sufficiently small neighborhood of f . Let (s g * , θ) := T g p (s, θ), then, in a sufficiently small neighborhood of f , s g * varies continuously with respect to g . By definition of t g ± , for all s ∈ (t g − , t g + ), there exists an orbit starting at (s, θ ) and ending at (s , θ ) for some θ ∈ θ g − (x, p), θ g + (x, p) . Now, suppose that x = (s, θ) is T f -periodic of period p. Note that s f * = s ∈ t f − , t f + . So there exists θ g * (s) such that s, θ g * (s) is T g -periodic and its period is a divisor of p. Furthermore we can assume that this neighborhood V (x) of f is so small that (s, θ g * ) is 1 N -close to (s, θ) (with respect to a fixed usual norm). We will use the following theorem:

THEOREM ([4, Theorem 1]). In a rational polygon, periodic points of the billiard flow are dense in the phase space.
This theorem immediately implies that the same is true for the billiard map. In particular periodic points are dense in Ω Clearly G is a dense G δ . We have shown that every parameter in G gives rise to a wind-tree model with dense periodic points. Now additionally suppose r is rational. In this case, we can use a stronger property on periodic orbits. It is a part of a special case of Veech's famous theorem known as the Veech dichotomy: [21,Theorem 5.10]). If a polygon P is square tiled, then every non-singular orbit in a non-exception direction is periodic.
We will call a parameter f rationally N -ringed if f is N -ringed and all a i , j , b i , j are rational for all (i , j ) ∈ E N . The key property here is that for any rationally N -ringed parameter the N -ringed table is square-tiled and thus we can apply Veech's theorem: there exists a countable dense set {θ j } ⊂ S 1 such that every non-singular point of the form (s, θ j ) ∈ Ω f θ j ,N is periodic. We call such a direction a periodic direction.
We assume θ j are enumerated so that the maximal combinatorial length of the periodic orbits inside Ω f θ j ,N is increasing with j . Consider the smallest ( f ) such that θ 1 , . . . , θ ( f ) is 1 N -dense. Let f be a rationally N -ringed parameter. Consider a periodic direction θ and the set Ω f θ,N with saddle connections removed. We decompose this set into its periodic orbit structure; more precisely this decomposition consists of a finite collection of intervals permuted by the dynamics such that the boundary of each interval from this decomposition is in a saddle connection. We call this collection of intervals D( f , θ). For each I ∈ D( f , θ), all points in I are periodic of the same period p, and we call p−1 i =0 T f i (I ) a periodic cylinder. In the general case, we presented a construction that associates to every T f -periodic point x = (s, θ) and every g in a small enough neighborhood U 1 (s, θ) of f an angle θ g * (x) such that (s, θ g * (x)) is T g -periodic.
Because the periodic points come in cylinders, as described above for f , the angles θ g * (s, θ) and θ g * (s , θ) will coincide for s in an open interval around s if g ∈ U 1 (s, θ) ∩U 1 (s , θ).
For each interval I in D( f , θ), we can thus find an interval I ⊂ I containing at least 1 − 1 ( f )·N proportion of points of I such that the intersection θ) is open. For all g ∈ U N ( f ) and all s, s ∈ I we have θ g * (s, θ) = θ g * (s , θ). Furthermore we can assume that this neighborhood U N ( f ) of f is so small that θ g * is 1 N -close to θ (with respect to a fixed usual norm). Let { f i } ⊂ A Z 2 be countable dense and such that each f i is rationally N iringed for some N i . Let Clearly G is a dense G δ . We claim that every parameter in G gives rise to a wind-tree model with locally dense periodic orbits. For each parameter g ∈ G , there exists an infinite subsequence ( I ∈D( f k ,θ j ) I is of measure at least 1 − 1 N k m k and thus the complement of the infinite measure G δ set K k≥K is of zero measure.

GENERALIZATIONS
Our results hold in a much larger framework. In the proof of minimality we only used that N -ringed configurations are dense in the space of all configurations and that they are rational polygonal billiard tables. For the local density of periodic orbits we also used that N -ringed configurations which are Veech polygonal billiard tables are dense. Now we give some examples where these properties hold.
1) We stay in the setup discussed in the article but additionally allow the empty tree denoted by , thus the parameter space is { } ∪ {(a, b) : r ≤ a ≤ 1 − r, r ≤ b ≤ 1−r }. Ehrenfest specifically required that the average distance A between neighboring squares is large compared to 2r . For any probability distribution m on the continuous part of the space of parameters, if we add a δ function on the empty tree, then for c < 1 large enough, the distribution cδ + (1 − c)m verifies almost surely this requirement. However our result tells nothing about a full measure set of parameters for Lebesgue measure. 2) Instead of fixing r ∈ [ 1 4 , 1 2 ), we fix r between 0 and 1 2 . If r ∈ 1 2(n+1) , 1 2n , place at most n 2 copies of trees in each cell. We can then form N -ringed configuration in a more general sense where we replace the rhombus by an appropriate curve around the origin. One can do so using just n + 1 copies of the tree in each cell.
3) Instead of fixed size squares we use all vertical horizontal squares contained in the unit cell [0, 1] 2 . This set is naturally parametrized by min(a, b, 1 − a, 1 − b) where a 2r by 2r square tree is centered at the point (a, b). More generally we call a polygon a VH-tree if the sides alternate between vertical and horizontal. For example, a VH-tree with 4 sides is a rectangle, and one with 6 sides is a figure L. We can use various subsets of VH-trees, for example all VH-trees with at most 2M sides (M ≥ 4 fixed) contained in the unit cell. Or we can use the VH-trees with 12 sides and fixed side length r ∈ [1/4, 1/3) (called + signs). Many other interesting subclasses can be considered. 4) Fix a rational triangle P , and consider the set of all rescalings of P contained in the unit cell [0, 1] 2 oriented in such a way that they have either a vertical or horizontal side. 5) One can also change the cell structure to the hexagonal tiling and consider appropriate polygonal trees. For example, one can use appropriate classes of equilateral triangular trees or hexagonal trees. discrete set of points to a complement of a (possibly different) discrete set of points. We call such a map eligible.
REMARK. If a map T : X X is eligible there exists a set S ⊂ X that is discrete and such that T restricted to X S is an homeomorphism. If S 1 and S 2 are two such sets, then S 1 ∩ S 2 is also such a set. Indeed, S 1 ∩ S 2 is discrete and T is a homeomorphism on X (S 1 ∩ S 2 ) = (X S 1 )∪(X S 2 ). More generally, any intersection of such sets is also such a set. Thus, there exists a minimal discrete set (w.r.t. inclusion) such that T is a homeomorphism outside this set.

DEFINITION.
Let T : X X be an eligible map. We denote by Sing(T ) the minimal discrete subset of X such that T restricted to X Sing(T ) is a homeomorphism. We call every point in Sing(T ) a singularity of T .

REMARK. If T is eligible, then T −1 is also eligible and
Sing T −1 = X T X Sing(T ) .
Next, we redefine the notions of images and preimages of sets in a way that makes clear that we will never apply the eligible transformation on its singular points.

DEFINITION. Let T : X X be an eligible map. For any
The preimage of A by T is its image by T −1 , thus: We then define T k (A) by recurrence for any integer k, as follows: for any k ≥ 0, • In a similar way, the past orbit is • The orbit of a point is the union of its past and future orbit: We define the orbit of a set in a similar way. Let A ⊂ X .
• The future orbit of this set is the collection of all the positive iterates of T on A. It is noted by O + (x 0 ), thus • In a similar way, the past orbit is • The orbit of a point is the union of its past and future orbit: A half orbit is a future or past orbit. We say that a point orbit or half orbit is singular if it is an orbit or half orbit of a singular point (that is, a point in Sing(T ) ∪ Sing(T −1 )).
Note that the image of a non-empty set may be empty and an orbit may be finite.

DEFINITION.
Let T : X X be an eligible map. A connection is a finite nonperiodic orbit. Thus, it is both an orbit of a singularity of T and an orbit of a singularity of T −1 .
REMARK. If x 0 ∈ X is singular for both T and T −1 , then {x 0 } is a connection.

DEFINITION.
Let T : X X be an eligible map. We say that T is minimal if and only if every orbit is dense.

REMARK.
Let T : X X be an eligible map. If T is minimal and has connections, then X is finite. A.3. Keane's minimality criterion. Keane has shown that interval exchange transformations with no connections are minimal [18]. Keane proves this fact with the usual convention that the IET is defined at singular points via left continuity. This convention does not agree with our convention that billiard orbits stop when they arrive at a corner. Thus we do not define IETs at singular points, and this makes an IET an eligible map. Moreover, Keane considered IETs defined on a single interval while in our context the arising IETs are naturally defined on a finite disjoint union of intervals. More precisely, we have the following definition.
DEFINITION. An IET is an eligible map T : X X where X ⊂ R is a finite union of open, bounded intervals whose closures are disjoint, and T is a translation on each connected component of X Sing(T ). We call T reducible if a non-trivial finite union of connected components of X is invariant, and otherwise we call T irreducible.

REMARK.
The billiard map restricted to Ω g θ as described in the article is not an eligible map. The billiard map is defined on a disjoint union of closed intervals. If we restrict it to the interior of these intervals it becomes an eligible map. Furthermore if we restrict to the inside of a ringed configuration it is an IET.
Keane's result remains nonetheless true with a slight adjustment; for completeness we give a proof here. Proof. Since there are a finite number of singularities, there are a finite number of trajectories starting at points of I that hit a singularity before crossing I again. By shortening I to a subinterval I with one endpoint x we can assume that no trajectory leaving I hits a singularity before returning to I . Now consider the forward iterates T i (I ). By the definition of I , these are intervals of the same length for each i ≥ 0 until the interval returns and overlaps I . The interval I must return and overlap I in a finite time since the total length of X is finite. Let j be the minimum number of iterates needed until T i (I ) overlaps I . T j (x) = x since x is not periodic.
If T j (x) ∈ I , we are done (see Figure 6 left). Otherwise (see Figure 6 right) it is the trajectory leaving the other endpoint y of I which returns to I at time j and for some z ∈ I we have T j (z) = x. We now consider the interval I with endpoints z and x and apply the previous analysis to it. Orbiting I in the forward direction it must return to I at a certain (minimal) time k > j . We The two ways I and T j (I ) can overlap have either T k (z) ∈ I or T k (x) ∈ T j (I ), but the first can not happen since it implies that T k (x) ∈ T j (I ) or, equivalently, T k− j (x) ∈ I , which contradicts the minimality of k. Thus T k (x) ∈ I as required.
Proof of Theorem A.2. By way of contradiction suppose there is a non-periodic infinite trajectory O (x) which is not dense. Let A = X be the set of limit points of O (x). Then A is invariant under the map T . Since A = X and T is irreducible one can choose a trajectory O (y) ⊂ int(X ) ∩ A int(A) (here the closure and the interior are taken in R). Note that O (y) ⊂ A since A is closed. We will show that this implies that there is a saddle connection. Either the trajectory of y is a saddle connection (and we are done) or O (y) is infinite in at least one of the two directions. We will show that this implies that either there is an open neighborhood of y contained in A, a contradiction to y being a boundary point, or there is another saddle connection.
Let I be an open interval with y an endpoint. It is enough to show that there exists an open interval (y, z) ⊂ I which is contained in A. Doing this on both sides will yield our open neighborhood. Lemma A.3 implies that O (y) hits I again at some point z. If the interval (y, z) ⊂ A we are done. Suppose not. Then there exists w ∈ (y, z) which is not in A. Since A is closed, there is a largest open subinterval I ⊂ (y, z) containing w which is in the complement of A. Let v be the endpoint of I closest to y. Then, since A is closed, v ∈ A and the trajectory through v must be a saddle connection. For if it were infinite in either direction, it would intersect I . Now consider a ∈ X X . By the definition of IET any point b ∈ int(X ) close enough to a is such that the open interval J whose endpoints are b and a is included in X , a is in J , and there is an open interval J ⊂ X Sing(T ) such that T (J ) = J (so a is in T (J ) = J ).
By assumption k∈Z T k (I ) covers J and J , thus k∈Z T k (I ) covers a. Repeating this for all points a ∈ X X , it shows that k∈Z T k (I ) is a countable open cover of the compact set X , and thus there exists K , L such that L k=K T k (I ) is a open cover of X . This immediately implies that L k=K T k (I ) is an open cover of X .
A.4. Application to rational polygonal billiards. A polygon P is a compact, finitely connected, planar domain whose boundary ∂P consists of a finite union of segments. We play billiards in P . Take any point s ∈ ∂P any θ ∈ S 1 such that the vector (s, θ) points into the interior of P , flow (s, θ) until it hits the boundary ∂P , and then reflect the direction with the usual law of geometric optics-angle of incidence equals angle of reflection to produce the point (s , θ ) = T (s, θ). T is called the billiard map; it is not defined if s is a corner of the polygon. The inverse T −1 is defined if s is not a corner. A polygon is called rational if the angle between any pair of sides is a rational multiple of π. Suppose that the angles are π m i n i with m i and n i relatively prime; let N be the least common multiple of the n i , and D N be the dihedral group generated by reflections in the lines through the origin that meet at angle π N . Let D N (θ) denote the D N orbit of a direction θ ∈ S 1 . The following theorem is a compilation of the well known results, see for example [21, Sections 1.5 and 1.7]: THEOREM A.5. Suppose P is a rational polygon. i) For each θ ∈ S 1 any orbit starting in the direction θ only takes directions in the set D N (θ). ii) For each θ ∈ S 1 the restriction T θ of the map T to ∂P ×D N (θ) is an interval exchange transformation in the sense defined in this appendix. iii) If θ is non-exceptional, then T θ is irreducible and has no connections.
Combining this theorem with Corollary A.4 yields COROLLARY A.6. If P is a rational polygon and θ is non-exceptional, then the map T θ : X → X is minimal, and thus the orbit of every open interval I covers X .