Triangle tiling billiards draw fractals only if aimed at the circumcenter

Transcription

1 Triangle tiling billiards draw fractals only if aimed at the circumcenter Pascal Hubert, Olga Paris-Romaskevich arxiv: v [math.ds] 31 May 018 Abstract. Consider a periodic tiling of a plane by congruent triangles which is obtained from an equilateral tiling by a linear transformation. We study a billiard in this tiling defined in the following way. A ball follows straight line segments and bounces from the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to 1. We show that almost all the trajectories of such a tiling exhibit integrable behavior : they are either closed or linearly escaping. An exceptional family of trajectories has a very different, non-integrable behavior. These trajectories approach Rauzy-like fractals. We give a more precise description of these exceptional trajectories. Our proofs use the connection of this system with the dynamics of interval exchange transformations with flips. This connection was first noticed in the work of P. Baird-Smith, D. Davis, E. Fromm and S. Iyer. 1 Introduction. 1.1 Main character: tiling billiards. This article is devoted to the study of a new object which is born from a union of two classical objects, one coming from the world of geometry and one from the world of dynamics : tilings on one side and billiards on the other side. The idea is very simple: a ball follows along straight line segments and bounces from the boundaries of the tiles in a fixed tiling on the plane. Once the ball reaches a boundary of some tile, it bounces of in a neighbouring tile following a reflection rule defined by a coefficient of refraction k. The refraction coefficient can change from tile to tile and a tiling can be arbitrary, see for example Figure 1 for a tiling trajectory in a tiling coming from Matisse s collage. The first interesting case is a tiling billiard in a very simple periodic tiling with changing (from tile to tile) refraction coefficients. For example, in [14] P. Glendinning studies the dynamics of tiling billiards on a standard infinite chess board tiling with different refraction indices for black and white tiles. The second article studying tiling billiards (and introducing them by that name) is [10]. In it, D. Davis, K. DiPietro, J. Rustad and A. St Laurent define negative tiling billiards for which the refraction coefficient between tiles is equal to 1 and they study the basic properties of such billiards. In this setting, [9] deals with the trihexagonal tiling and its ergodic properties as well as the work [8] deals with negative triangle tiling billiards. By a triangle tiling one understands a periodic tiling by congruent triangles in which two of each neighboring triangles have different orientations and form a parallelogramm. These four articles are the only published ones we know of concerning tiling billiards. Diana Davis pointed to us that A. Mascarenhas and B. Fluegel wrote a preprint [0] in which they Aix Marseille Université, CNRS, Centrale Marseille, IM - UMR 7373, F Marseille, France. Univ Rennes, CNRS, IRMAR - UMR 665, F Rennes, France 1

2 defined and studied negative refraction and it s basic properties from the physical point of view, and gave several examples of simple tilings and billiard trajectories in them. Unfortunately, this paper has never been published and is even not on-line. In [8] the authors show that triangle tiling billiards are very special. This dynamical system appears to have a first integral : the distance between a segment of a trajectory in a triangle and its circumcenter. This fact is crucial in order to prove that all bounded trajectories in this system are closed and not self-intersecting. The work [8] starts the exploration of fruitful connections between such negative triangle tiling billiards and interval exchange transformations with flips. Using this connection, the authors manage to find a singular trajectory (it has a branching point - a vertex in the tiling) in a triangle tiling billiard that exhibits fractal behavior. Our work is largely inspired by the connections pointed out in [8]. We continue the study of negative triangle tiling billiards. Our main result is that the fractal-like behavior discovered in [8] can be exhibited only by a trajectory that passes through the centers of circumcircles of all the triangles that it hits. Our proof uses in a crucial way the modified Rauzy induction for interval exchange transformations with flips that was introduced by Nogueira [3]. This induction stops for almost any IET with at least one flip. The proof of our main result is based on two ideas. First, there exists an invariant hyperspace for the Rauzy induction procedure that corresponds exactly to the space of trajectories hitting the circumcenters. For this first part we need some computer assisted calculations (not very heavy but still...). The second ingredient in the proof is a more precise understanding of the structure of permutations corresponding to the stopping points of the induction. The goal of this presentation is to make the previous paragraph clear to anybody who had read the rest of the article. 1. Plan of the paper. In Section and in particular its Subsection.1, we introduce triangle tiling billiards as well as three other systems that seem unrelated at first sight but are actually the same up to the change of coordinates, see Subsection.. In Subsection.3 we remind the reader the results from [8], especially since this work is not yet published. We describe the heuristics of the behavior of triangle tiling billiards. We also precise the connection of triangle tiling billiards with interval exchange transformations on the circle with flips. Sections 3, 4 and 5 are devoted to these IETs. In Section 3, we introduce our main tool - modified Rauzy induction. In Section 4, we prove Theorem which is our first ingredient. It gives a necessary condition for the minimality of a fully flipped rotational 3-IET on the circle S 1 of length 1: rotation has to be equal to 1. In Section 5, we define a notion of integrability for interval exchange transformations with flips and we prove Theorem 4 which is our second ingredient. It shows that almost all fully flipped rotational 3-IET on the circle are integrable. Heuristically, integrability corresponds to the behavior of an IET which is reduced to rotations and flips on subcylinders of an associated translation surface. In Section 6 we go back to tiling billiards. In Theorem 6 (which is essentialy the consequence of Theorems and 4) we describe qualitatively the behavior of almost all trajectories of triangle tiling billiards: those which do not pass through the circumcenter. In Section 7, we discuss the special case of trajectories passing through a circumcenter and an associated possible fractal behavior as well as the relations of this behavior with previously studied objects such as SAF-invariant, Arnoux-Rauzy family of interval exchange transformations, Rauzy gasket and the Hooper-Weiss curve. In this Section we also give a geometric interpretations of arguments of Sections 3. Section 8 is devoted to the generalizations of triangle tiling billiards to the case of quadrilateral tilings.

3 Figure 1: A part of a tiling billiard trajectory in the tiling defined by one of Matisse s collages. Initital painting (except for a black tiling billiard trajectory): H. Matisse, L Escargot (Snail), 1953, Gouache on paper, cut and pasted on paper mounted on canvas, Tate Museum, Succession Henri Matisse/DACS 018 Four dynamical systems. We will consider four dynamical systems that seem quite different from the first glance..1 Definitions. Take some triangle. Suppose that this triangle has its angles equal to α 1, α and α 3. From now on till the end of the article the vertices corresponding to these angles are denoted A, B and C and the sides facing these vertices are denoted a, b and c. (I) Triangle tiling billiards. Definition 1. Consider a periodic tiling of the plane by triangular tiles which are all congruent to. This tiling is obtained by cutting the plane by three families of equidistant parallel lines, see Figure. Define a billiard in this tiling in a following way. A particle on the plane follows a straight line till it hits the side of some tile. Then the trajectory continues in the neighbouring tile, following the rule of negative refraction with coefficient 1. In other words, the oriented angle that the trajectory makes with the side of the tile, changes its sign but keeps the same absolute value. We call the dynamical system defined in this way a triangle tiling billiard. We are interested in the dynamics of such a billiard, for different parameters of the system (α 1, α, α 3 ) and different initial conditions of the trajectory. Note that the size of doesn t matter since the dynamical behavior of a tiling billiard is invariant under homothety. 3

4 Figure : The first six segments of a piecewise linear trajectory in a negative triangle tiling billiard. The refraction coefficient k is equal to 1. (II) Reflection in a circumcircle. Definition. Consider a circle on the complex plane of radius 1 centered at the origin. Fix for any τ (0, 1) an oriented chord l in this circle that connects the point e πiτ with the point 1 and is headed to 1. Then we inscribe a triangle with angles α 1, α, α 3 in this circle. We suppose that the orientation of the triangle is chosen in a way that the vertices A, B and C are placed in a counter-clockwise manner. There is one free parameter X [0, 1) in this system which correponds to the position of the vertex A = e πix. The dynamical system that interests us is the following. Suppose that the chord l intersects some side of, for example the side a. We consider the triangle which is congruent to, shares the side a with it and is inscribed in the same circle but is different from and doesn t have the same orientation as. The line l intersects into two points : one on the side a and one on one of two other sides of. Now let us take one more reflection of with respect to the diameter of the circle perpendicular to the line l. One obtains a new triangle whose orientation is the same as the orientation of the initial triangle, see Figure 3. The map X X is a dynamical system which we will call reflection in a circumcircle. This system may seem quite tricky, and one could ask why we just don t reflect with respect to the sides crossed by the line l each time and consider X X as a dynamical system: the passage to the triangle is needed in order to make the direction of the line l an invariant of the system - for this one has to change the coordinate on the circle. Indeed, for the trajectory in the triangle the direction of the line l is changed to the opposite. This change is just a technical detail since the squares of these two maps coincide: X = X, see Figure 3. Note that the reflection in a circumcircle is defined on some subset of the circle of initial conditions X [0, 1) since for some values of parameter X the line l won t intersect the triangle. (III) Reflections in a triangle with respect to a circle of tangency. This system was suggested to us by Shigeki Akiyama. Definition 3. Fix some triangle inscribed in a unit circle as previously and a parameter τ (0, 1). Consider a circle T confocal to the circumcircle of of a smaller raidus, equal to r = cos πτ. Consider a dynamical system defined on the (one-dimensional) space of oriented segments connecting two sides of the triangle and tangent to T, in a following way. For some oriented segment x consider its positive (with respect to segment orientation) intersection with one of the sides of the triangle. Let 4

5 τ X X l X 0 Figure 3: Reflection in a circumcircle. X T X Figure 4: Triangletangent system. it be some point X. Consider then a point X on the same side of the triangle but symmetrical with respect to the middle of the side. Construct a new oriented segment x passing by X and tangent to the circle T. By symmetry, x exists. The dynamical system x x is of interest for us. We call this system a reflection in a triangle with respect to a circle of tangency, see Figure 4. (IV) Fully flipped 3-interval exchange transformations on S 1. Definition 4. Let τ [0, 1) and l j := α j, j = 1,, 3. Let a circle of length 1 be subdivided into three π intervals of lengths l 1, l and l 3 correspondingly : [0, 1) = [0, l 1 ) [l 1, l 1 + l ) [l 1 + l, 1). Consider a following interval exchange transformation with flips: each one of the intervals is flipped and shifted by τ [0, 1). This is a continuous outside a three-point set {l 1, l 1 + l, 1} map F : S 1 S 1, see Figure 5. We call such a map a fully flipped 3-interval exchange transformation on a circle. Note that T = F is an interval exchange transformation (without flips). An explicit formula for F is the following: 5

6 0 l 1 l 1 + l 1 0 l 1 + τ mod 1 l 1 τ 1 Figure 5: Fully flipped 3-IET. Along this paper we draw the shapes (triangles and quadrilaterals) over the intervals of continuity of interval exchange transformations to make it visually more clear for the reader when the interval is flipped and when the interval is not flipped by an IET. The orientation of the form drawn over an interval changes if this interval is flipped and is not changed otherwise. We learned the idea of this graphic representation from Diana Davis. x + l 1 + τ mod 1 if x I a := [0, l 1 ) F (x) = x + l + τ mod 1 if x I b := [l 1, l 1 + l ). x + l 3 + τ mod 1 x I c := [l 1 + l, 1) We denote the set of all the transformations of this kind by CET 3 τ. The space of these transfomations is parametrized by a three-dimensional space C S 1 which is a direct product of a simplex C of lengths l j, j = 1,, 3 and a circle, τ S 1. Here C = {(l 1, l, l 3 ) R 3 l j 0, l 1 + l + l 3 = 1}.. Connections. Suppose that a trajectory of a tiling billiard passes through an edge e common to two triangles, and. When one folds these two adjacent triangles and across their common edge e one onto each other, the two parts of the trajectory of a tiling billiard that belong to and fold one onto each other and belong to the same straight line after such a folding. This simple but crucial folding remark gives a Proposition 1 ([8]). For any (negative) triangle tiling billiard the following holds. 1. Every trajectory crosses each triangle in the tiling at most once.. Along the trajectory, the distance between a segment of a trajectory in some triangle that it crosses and its circumcenter is invariant from one triangle to the other. Moreover, the circumcenter of each triangle stays at the same side from the (oriented) segment of the trajectory. 3. All bounded trajectories are closed. This Proposition shows that the systems (I) and (II) depict in fact the same system but in different coordinates. Instead of unfolding a billiard trajectory as in (I), one can fold it on the fixed circumcircle of the starting triangle. The direction τ of a bouncing particle with respect to the circumcenter is fixed along the dynamics. We will use the system (II) a lot as a convenient way to look at the system (I) in the following. The system (III) gives another view on the parameter τ. One can keep the triangle fixed and change the direction of a bouncing line. The line will bounce from the same side after one iteration. The reflection rule holds: the angle of incidence is equal to the angle of reflection. The difference is 6

7 that the point in which the trajectory rebounces, changes. But all the segments of trajectories are tangent to some circle with the center in the circumcenter of. This circle is a first integral for the dynamics of (III) as much as the direction of the line l is a first integral for the dynamics of (II). One can also see that the system (II) is an interval exchange transformation with flips with respect to the coordinate X on the circle (at least on the subset of the circle where this system is defined). Hence the system (II) is a restriction of the system (IV ) on some subset of the circle (which depends on the parameters α j, j = 1,, 3 and τ), see [8] for more details. We are interested in the understanding of the behavior of trajectories in triangle tiling billiards. What does a generic trajectory do? What are the possible behaviors of generic trajectories? What are the exceptional behaviors and for what values of parameters do they occur? In this article we will mostly work with the systems of type (IV ), i.e. with fully flipped interval exchange transformations. Then we will draw some conclusions for systems of type (II) by using the connections between (II) and (IV ) discovered in [8]. In the end, we give an answer (to some extent) to all of these three questions..3 Qualitative behavior of orbits in triangle tiling billiards. Proposition 1 gives a following classification (that we will use as well as definition) of different orbit behavior in a (negative) triangle tiling billiard. Definition 5. Any trajectory of a triangle tiling billiard is one of these three types. Closed (periodic) orbits. plane. A trajectory is closed if it is a closed piecewise linear curve in the Drift-periodic orbits. A trajectory is drift-periodic if it is invariant under a translation of the plane. In other words, it corresponds to a periodic orbit of the system (II) (see Subsection.1) and its trajectory in the tiling is unbounded. Escaping orbits. A trajectory is escaping if it is not periodic (neither closed nor drift-periodic). In [8] it is proven that drift-periodic orbits occur only in the tilings corresponding to the triangles with rationally dependent angles (the ratios of the angles being rational numbers). Even more, if the angles are rationally dependent, any orbit is periodic (closed or drift-periodic). An important remark is that closed orbits come by opem families. Indeed, an orbit close to a periodic one, by continuity, will continue the path in the same sequence of triangles and, coming back to the same triangle, by Proposition 1, it closes up. Escaping orbits happen to have two types of qualitatively different behaviors. We will distinguish between linearly escaping orbits and non-linearly escaping orbits. Definition 6. A trajectory of a tiling billiard is linearly escaping if it stays at a bounded distance from some fixed line in the plane and is not drift-periodic. An escaping trajectory which is not linearly escaping, is non-linearly escaping. See Figure 6 for some examples of trajectories. Our main goal in the following is to show that the only non-linearly escaping trajectories are all passing through the circumcenters of the corresponding triangles in the tiling. Otherwise, if the angles of the triangle are rationally independent, the behavior is either periodic or linearly escaping. These properties of triangle tiling billiards come as a consequence of the behavior of interval exchange transformations with flips to which we devote the next three Sections. 7

8 Figure 6: Examples of different types of trajectories in triangle tiling billiards. First row : 1. and. are periodic trajectories of periods 10 and 114 respectively; 3. is a drift-periodic trajecotory in the triangle with angles π, π3, π6 of a drift period 6. Second row: 1. and. are linearly escaping trajectories : they do not have any drift period but they stay in a bounded distance from some fixed line in R with an irrational slope; 3. represents the first steps of an infinite non-linearly escaping trajectory which spirals out to infinity. The pictures are drawn by the program [15] authored by P. Hooper and A. St Laurent. 8

9 3 Modified Rauzy induction for interval exchange transformations with flips. 3.1 Definitions. Here we will be working with interval exchange transformations on the interval and on the circle. Definition 7. An interval exchange transformation (IET) is a bijection T : [0, 1) [0, 1) with the property that there exist the points 0 = a 0 < a 1 <... < a n = 1 and the numbers t 1,... t n R such that T [ai 1,a i ) (x) = x + t i, 1 i n. The set of all such transformations is denoted by IET n. An interval exchange transformation with flips (IET with flips) is a bijection F : [0, 1) [0, 1) with the property that there exist the points 0 = a 0 < a 1 <... < a n = 1, the numbers t 1,... t n R as well as the vector k = (k 1,..., k n ) (1, 1, 1,..., 1) with k i 1, 1 such that F [ai 1,a i ) (x) = k i x + t i, 1 i n. If k i = 1 then the interval [a i 1, a i ) is flipped, otherwise if k i = 1 it is shifted, and not flipped. The set of all such transformations is denoted by IETF n. A fully flipped interval exchange transformation (fully flipped IET) is an interval exchange transformation with flips for which k = ( 1, 1,..., 1). We denote the set of all such transformations by FET n. Obviously, FET n IETF n. Definition 7 can be generalized in a natural way to the n-interval exchanges on the circle. For our purposes, we will define a subclass CET n τ of fully flipped interval exchange transformations on the circle that is related to the problem of reflections in a circumcircle. Definition 8. Fix non-zero numbers l 1, l,..., l n R + such that l l n = 1. Consider the points a 0 = 0, a 1 = l 1, a = l 1 + l,..., a i = l l i,..., a n = 1 on the circle of unit length. A fully flipped interval exchange transformation on the circle belongs to CET n τ if it has a following form: F (x) = x + l i + τ mod 1 x [a i 1, a i ) i = 1,... n. Here τ S 1 is a parameter. We will denote this set as CET n τ. This is the set of fully flipped τ-shifted n-interval exchange transformations on the circle. Note that this last definition is the generalization of the set CET 3 τ, defined as (IV ) in Section. Remark. The set CET n τ can be viewed as a subset of FET n+1 IETF n+1 by cutting one of the intervals in two halves (the interval whose image contains 0 as an interior point). Suppose T is an IET, T IET n. By assigning to each interval [a i 1, a i ) its number i [[1, n]] in the partition of the interval [0, 1), one associates in a natural way a permutation σ S n corresponding to an interval exchange transformation T. Indeed, [0, 1) = T [0, 1) is a disjoint union of intervals T [a i 1, a i ) in a new order (from left to right) which gives a permutation σ. A trivial permutation corresponds to the identity map. In this article we will usually denote the intervals not by their numbers in the sequence but by their labels (usually majuscule letters of latin alphabet A, B, C,...). One can associate a permutation to a map F IETF n in the same way as we just did it for T IET n. Now we will also add an additional information to a graphic representation of this permutation by writing the dots over the intervals which are flipped. We do this in the preimage as well as in the image, in a top and a bottom row as well. This is just a convention of course, but our reader will see that how this notation becomes useful when we deal with modified Rauzy induction for these maps. A permutation corresponding to a fully flipped interval exchange transformation F FET n has a graphic representation with dots over all the letters in both rows. Definition 7 can be generalized in a natural way to the n-interval exchanges on the circle by fixing a point 0 S 1 that will give a bijection between S 1 and the interval. Note that a corresponding permutation a. depends on the choice of 0 S 1 and b. is usually an element of S n+1 and not of S n since one of the intervals will be cut into two halves, see Figure 7 as an example. 9

10 a. b Figure 7: An example of a non-generic -interval exchange on the circle where one of the singularities is fixed. ( In) this case, if one choses this singularity as a fixed point 0 S 1, a corresponding permutation A Ḃ is. Although the same transformation when 0 S 1 is chosen at the beginning of the second Ḃ A ( ) A B C interval, has a corresponding permutation being C Ȧ B. Generically, the first case doesn t occur and an n-iet on the circle corresponds to an (n + 1)-IET on the interval. On this Figure the circle S 1 is represented as an interval with 0 as a fixed chosen point on the circle. 3. Modified Rauzy induction for IET with flips. The Rauzy induction is a powerful tool in the study of interval exchange transformations. It was introduced in [4] by Gérard Rauzy. For the study of IETs with flips the standard Rauzy induction procedure has to be modified. This was first done by Arnaldo Nogueira in [3]. Interval exchange transformations with at least one flip have a very different behavior from IETs without flips. For example, the IET of two intervals is a rotation. Almost all -IETs are irrational rotations and hence minimal. On the contrary, an IET of two intervals with at least one of the intervals flipped is completely periodic [17]. This dychotomy holds for a bigger number of intervals. Indeed, almost any irreducible n-iet is minimal although almost any n-iet with flips is not. Theorem 1 ([3]). Fix some combinatorics (σ, k) for a map in IETF n. Here σ S n and k has at least one component equal to 1. Then for almost any map F IETF n (in the space of parameters of lengths of permuted intervals (λ i ) n i=1 ) with such combinatorics the following holds: a. the corresponding modified Rauzy induction stops, b. there exists an interval I [0, 1) such that the first return map on this interval is a flip, Such a map F has an interval of periodic orbits and is obviously not minimal. Remark. The conclusion of this theorem was recently sharpened by Skripchenko and Troubetzkoy who proved that the set of parameters with minimal dynamics does not have maximal Hausdorff dimension [6]. We will use a tool of modified Rauzy induction defined by A. Nogueira to understand in more detail the subset CET n τ of the set of fully flipped IETs on the circle. Let us remind the reader about the modified Rauzy induction for the maps in IETF n. We use here the notations analogue to those chosen by Vincent Delecroix in his Lecture notes on IETs, [11] where he follows [19]. We adjust these notations to the case of IETs with flips. Let A be an alphabet of cardinality n. Let the maps σ top and σ bot, σ top, σ bot : A {1,,..., n}, 10

11 be two bijections. These two bijections can be considered as two different orders on the alphabet A. We denote σ a generalized permutation on the alphabet A corresponding to this unified data, σ top and σ bot together: ( ) (σ top 1 (1)... (σ top 1 (n) σ = ( ) σ bot 1 ( ) (1)... σ bot 1. (n) Now let λ := (λ i ) i A be a vector of positive real numbers, let λ := i A λ i. j [[0, n]] the following quantities: α top j := i:σ top (i) j λ i, α bot j := i:σ bot (i) j λ i. Set for every These quantities define the coordinates of the points which give two partitions of the interval [0, λ ]. Namely, we set for each i A: ( ) ( ) I top i := α top σ top (i) 1, αtop σ top (i), Ii bot := α bot σ bot (i) 1, αbot σ bot (i). Let us also fix a vector k = (k i ) i A, k i { 1, 1}. One can associate an IET (with flips) F IETF n to the data (σ, λ, k) such that F ( I top i in a following way. ) = I bot i F in restriction to each interval I top i is either a translation (if k i = 1) or a translation in composition with a flip (if k i = 1). This defines F completely on [0, λ ] outside the extremities of the intervals I top i, i [[1, n]]. We call α top i, i = 1,..., n 1 top singularities and αi bot, i = 1,..., n 1 bottom singularities. Any IET with flips is represented in such a way, and such a representation is unique. In the following, we will identify F IETF n and the corresponding data (σ, λ, k). In the following, we encode the combinatorics of F (given by a generalized permutation σ and a vector k) in one graphic representation. We represent the permutation σ and put the dots over the letters of the alphabet which correspond to the flipped intervals (to the indices i such that k i = 1). See Figure 8 for illustration by example. Definition 9. The generalized permutation σ is irreducible if l, l [[1, n 1]] such that ( σ top ) 1 ({1,,..., l}) = ( σ bot ) 1 ({1,,..., l}). If σ is not irreducible, the corresponding IET is reduced. Now let us introduce the modified Rauzy induction (MRI). This is an iterative (possibly infinite) process, on each step of which a given IET with flips is transformed to a new one following a simple rule. Let us describe one step of this induction. We consider the two "most right" intervals in the interval [0, λ ] and in its image F [0, λ ], i.e. intervals with labels (σ top ) 1 (n) and (σ bot ) 1 (n). The Rauzy induction consists in considering a first return map on the interval [ 0, max ( αn 1, top αn 1)] bot. One can easily see than an induced map is also an IET with new data (σ, λ, k ). The longest interval between the two "most right" intervals is called the winner and the shortest the loser for this step of Rauzy induction. Note that a new permutation σ is defined on the same set of labels. The winner is cut into two non-empty intervals by one of the extremities of the loser. The left of this intervals is relabeled into the label of the winner as well as its image (if the winner was on top) or pre-image (if the winner was on bottom). The label of the loser 11

12 A B C D α top 0 α top 1 α top α top 3 α top 4 D C A B α0 bot α1 bot α bot α3 bot α4 bot Figure 8: Interval exchange transformation with flips defined by the data (σ, λ, k). Interval on the top is cut according to the partition ( ) ( ) I top i while the interval on bottom is according to I bot i. The data (σ, k) is ( united in one matrix n with some entries that can have a dot above. In this example, A Ḃ (σ, k) = Ċ D ) D Ċ A Ḃ. is assigned to the part of the initial winner interval which doesn t have a label in one of the lines (on top or on bottom, if the winner was on bottom or on top, correspondingly). In another line the label of the interval which was a loser doesn t change. Note that the lenghts λ of all the intervals except the winner don t change. The winner s length is diminished by the loser s length. In the exceptional case α top n 1 = αn 1, bot the Rauzy induction stops. In Table 1 one can see some examples of modified Rauzy induction. In the case of IETs with flips, one step of Rauzy induction represents richer combinatorial possibilities than in the case of IETs without flips. Instead of two possibilities for a standard Rauzy induction, we have eight possibilities, depending on the values of the entries of the vector k. The set of couples {(σ, k) σ S n, k {1, 1} n } is called the set of Rauzy classes of interval exchange transformations with flips. These classes form the vertices of an oriented (not necessarily connected) graph whose edges correspond to the top and bottom inductions. This graph is called a modified Rauzy diagram. The properties of the path of (σ, λ, k) in the Rauzy diagram correspond to the dynamics of F. Let us give some notations in order to study Rauzy induction along the paths. Suppose λ = (λ i ) i A is a vector of interval lengths corresponding to some map F IETF n, A = n. Suppose that the corresponding MRI path γ (possibly infinite) in the Rauzy graph has edges e 1,..., e m,.... To each edge e one can associate a linear non-negative matrix A e corresponding to the inverse transformation of the lengths in a following way. If the edge e corresponds to the induction step where λ (m) i > λ (m) j, i j with i, j A then A e := E + E ij. Here E ij is a matrix with all elements equal to zero except one in the i-th row and j-th column which is equal to 1. Let us define A (m) := A e1... A em. Then the lengths of the intervals of continuity for an IET corresponding to the m-step of the Rauzy induction are given by λ (m) = A 1 (m) λ since λ(m) = A 1 e m... A 1 e 1 λ by definition. 1

13 (k B, k D ) top induction, case λ D > λ B bottom induction, case λ D < λ B (1, 1) (1, 1) ( 1, 1) ( 1, 1) λ D = λ D λ B λ B = λ B λ ( ) ) ( ) ( D ) A B (Ȧ Ċ D B Ċ D A B D A Ċ B Ċ D A B D D B A Ċ D A Ċ B Ċ D A Ċ B ( ) ) ( ) ( ) A B (Ȧ Ċ Ḋ Ḃ Ḋ A Ċ B Ċ D A B Ċ Ḋ A B Ḃ Ḋ A Ċ Ḋ A Ċ B Ḋ Ċ D A Ċ B ( ) ) ( ) ( ) A (Ȧ Ḃ Ċ D Ḃ D A Ċ Ḃ Ċ D A Ḃ Ċ D A D Ḃ A Ċ D A Ċ Ḃ Ḋ Ḃ Ċ Ḋ A Ċ Ḃ ( ) ) ( ) ( ) A (Ȧ Ḃ Ċ Ḋ B Ċ Ḋ A Ċ Ḃ Ḋ A Ḃ Ċ Ḋ A D B Ḋ A Ċ Ḋ A Ċ Ḃ Ḃ Ċ D A Ċ Ḃ Table 1: Here are described eight cases of Rauzy induction for interval exchange transformation with flips depending on the relation between the lengths of two end intervals (as in standard Rauzy induction) but also on the fact which of these two intervals are flipped or not. 4 Necessary condition for minimality for an element of CET 3 τ: τ = 1. In this Section we will prove the following Theorem. If F CET 3 τ is minimal then τ = 1. We will see in the following that this theorem gives a necessary condition for non-linearly escaping behavior in triangle tiling billiards. A non-linearly escaping trajectory has to pass through a circumcenter of the first triangle (and, hence, all of the others). First, minimality can be described in terms of modified Rauzy induction. Lemma 1. [17, 3] The map F IETF n is minimal if and only if the modified Rauzy induction never stops, and the vector of the lengths of the intervals λ (m) obtained after m iterations of the MRI, tends to zero: λ (m) = max i A λ(m) i m 0. What can be the combinatorics of F CET 3 τ as an inteval exchange transformation? First note that if I j = F (I j ) I j for some j {a, b, c}, F is obviously not minimal since there exists an interval of -periodic points. Hence if F is minimal then 0 F (I b ). In this case F has a corresponding 4-IET of the interval with the following combinatorics: ( A σ := Ḃ Ċ Ḋ ) Ḃ Ḋ A Ċ, (1) see Figure 9. We will now study the connected component of this permutation σ in a modified Rauzy graph. We will prove that the values of parameters (l j, τ) C S 1 corresponding to infinite 13

14 F A B C 0 F 1 (0) 1 B D A C 0 τ 1 D 0 A B C D F 1 (0) A C D τ B 1 1 F Figure 9: On the left : the representation of the map F CET 3 τ in the case when 0 F (I b ) and hence F as an IET (with flips) of the interval has the combinatorics of the permutation σ described by (1). Here I b is the union of intervals with labels B and C. On the right : the representation of the map F CET 3 τ in the case when 0 F (I a ), where I a is the union of intervals with labels A and B. In this case the map f is obviously not minimal since it has an interval (labelled by A) of periodic points. paths in this graph all belong to the hyperplane {τ = 1}. From now on, we will be working in homogeneous coordinates [λ A, λ B, λ C, λ D ] RP 3 which correspond to the lengths of the four intervals of an IET on [0, 1] to which the circle IET is reduced. One can easily see that for F CET 3 τ defined by a quadruple of parameters (l 1, l, l 3, τ) the corresponding lengths (λ A, λ B, λ C, λ D ) are given by a linear transformation λ A l λ B λ C λ D = Let us note that the equation τ = 1 for F CET3 τ in homogeneous coordinates [l 1 : l : l 3 : τ] can be written as τ = l 1 + l + l 3 and it corresponds to the equation λ A + λ C = λ B + λ D for a 4-IET that F defines in homogeneous coordinates [λ A : λ B : λ C : λ D ]. We will denote the corresponding hyperspace H := { λ RP 3 : λ A + λ C = λ B + λ D } and its orthogonal vector v := (1, 1, 1, 1) T. The following "miracle" happens. Lemma. Fix any vertex in the component of the Rauzy graph of the permutation σ defined by (1). Take a path finite path γ = e 1... e m in the graph leading to this vertex. Then the vector A T γ v := A T (m) v doesn t depend on the path taken and is then an invariant of this vertex. l l 3 τ. Proof. The proof is an explicit verification by computer, see Figure 10. We think that the vector v and its images by inverse Rauzy induction have a geometric interpretation, probably related to the invariants of connected components of the Rauzy graphs. One can make the following remarks based on experimental data: for the vertices in the connected component of σ that belong to the cycles in the Rauzy graph and their assigned vectors A T v, the non-flipped letters correspond to the coordinates equal to 0, and the flipped letters to coordinates equal to ±1. We hope to explore this in the future work. On the Figure 11 we represent a complete strongly connected component in the part of the Rauzy graph that comes from the vertex σ modulo some natural equivalence in order to simplify the understanding of the graph. One can see on this Figure a complete description of the images A T γ v of the vector v corresponding to different permutations. Now we are ready to prove Theorem 4. Proof. Let A = {A, B, C, D}. Consider a connected component of the vertex σ defined by (1) in the Rauzy graph of IET with flips. Suppose that the vector of lengths λ = (λ i ) i A is such that the corresponding MRI doesn t stop. We will prove that this implies λ H. 14

15 Figure 10: The part of the connected component of σ in the Rauzy graph. One can see that to each vertex (σ, k) corresponds an invariant which is an image of vector v under the compositions of matrices A T e leading to this vertex. The vector k is represented by the colors : red colored letters correspond to the components equal to 1 and green colored letters to the components equal to 1. DBAC ACDB A<B C<B B<C DBCA ACDB DBAC ABCD B<A C<D DBCA ACDB DBCA ABCD A<B DABC ACDB B<C DBCA BCAD ABCD BDAC A<D C<D B<D DABC BCAD ABCD BCDA C<A C<D A<C D<C D<A B<A D<A DCAB BCAD DABC BCDA C<A C<A DCAB BCDA A<B DCAB ABCD D<B B<D DCAB ADBC C<D DBCA ABCD B<C A<D D<A DBCA ADBC DABC ABCD DBCA ADBC Figure 11: This is one of nontrivial strongly connected components of the Rauzy graph of IETF 4. Any vertex in this component is connected by some path to the vertex σ which is also represented on this graph. The color of the letters in this representation corresponds to the values of the coordindates of vectors ṽ = (v A, v B, v C, v D ). These vectors are the images of the vector v by the matrices A T γ. By Lemma, these vectors are the invariants of permutations. The red color corresponds to the value 1, the blue one to the value 1 and the green one to the value 0. The initial permutation σ has its corresponding vector v = (1, 1, 1, 1) and hence the letters A and C are colored in red and letters B and D are colored in blue. Image provided by Paul Mercat. 15

16 Suppose that the MRI follows the path γ = e 1... e m.... Since the Rauzy graph is finite, there exists a vertex σ in it such that the path γ passes by σ an infinite number of times. Suppose that the corresponding numbers of steps are (k i ) i=1. In other words, for any m N all the paths e 1... e km end up at σ. Then λ (km) = A 1 (k λ for A m) k m = A e1... A ekm. By Lemma 1 we have λ (km) 0 when k m. Let ṽ j := A T (k j )v = A T e k1... A T e kj be the image of v corresponding to the pre-cycle e k1... e kj. By Lemma this vector ṽj doesn t depend on j N. We denote it by ṽ. Also denote H := A 1 (k 1 ) H. The orthogonality of v and H is equivalent to the orthogonality of ṽ and H. Now let us define the products of matrices that correspond to the cycles in a Rauzy graph followed by the Rauzy path γ that start and end at the vertex σ. Let for any m N, Then, we calculate the scalar products: Ã (m) := A ekm A e km. () < ṽ, λ k 1 >=< (Ã 1 (m) )T ṽ, λ k 1 >=< ṽ, Ã 1 (m) λ(k 1) >=< ṽ, λ (km) > 0, m. From this we get λ (k 1) ṽ which is equivalent to λ H. 5 Generic integrability in CET n τ : yes for n = 3, 4 and no for n Definitions: simple and integrable interval exchange transformations with flips. Each interval exchange transformation with flips F IETF n is defined by the data (σ, λ, k) as discussed in Subsection 3.. The generalized permutation σ encodes in itself two orders on alphabet A that correspond to two words ω top, ω bot A n. Definition 10. The interval exchange transformation with flips F IETF n defined by the data (σ, λ, k) is called simple if the corresponding words ω top and ω bot can be decomposed as a concatenation of shorter words in a following way. There exists some p [[1, n]] such that ω top = ω top 1... ωp top and ω bot = ω1 bot... ωp bot with ωj bot, ω top j being finite words in the alphabet A such that for each j [[1, p]] one of the three possibilities holds: 1. (periodic cylinders : flipped or not flipped) ω top j = ω bot j,. (cylinders of rotation) there exist two different words x = x j, y = y j in the alphabet A such that ω top j = xy and ωj bot = yx, and the corresponding coordinates of k are equal to 1, 3. (cylinders of rotation with a marked singularity) there exist three different words x = x j, y = y j, z = z j in the alphabet A such that ω top j = xyz, ωj bot = zyx; ω top j = xyz, ωj bot = zxy or = xyz, ωj bot = yzx, and the corresponding coordinates of k are equal to 1. ω top j 16

17 Of course, the lengths of ωj bot and ω top j coincide and these words consist of the same sets of letters. Also note that the simplicity of an IET with flips doesn t depend on the lengths of the intervals but only on combinatorial data. Remark. Simplicity can also be defined in purely geometric terms. Indeed, for F FET n which is simple a corresponding to T = F translation surface can be cut along the lines of the flow into the union of the invariant cylinders and tori. The flow preserves a foliation into periodic circles on the cylinders, and the invariant tori come with a linear foliation. Example. The following combinatorial data gives three examples of simple IET with flips. ( A B Ċ Ḋ ) B A Ċ Ḋ with ω top 1 = AB, ω1 bot = BA, ω top = ω bot = CD, k = (1, 1, 1, 1); ( ) B A C D Ė C B A D Ė with ω top 1 = BAC, ω1 bot = CBA, ω top = ω bot = DE, k = (1, 1, 1, 1, 1); ( ) A D E B Ċ B A D E Ċ with ω top 1 = xy, ω1 bot = yx, x = ADE, y = B, ω top = ω bot = C, k = (1, 1, 1, 1, 1). Definition 11. An interval exchange with flips F : I I is integrable if there exists a Poincaré section of F such that the dynamics of the first return map on this section is a simple IET with flips. A circle interval exchange with flips is integrable if a corresponding interval exchange (after cutting a circle in some point) is integrable. One can easily check that this definition doesn t depend on the cutting point. The dynamics of ntegrable IETs with flips can be well understood. Indeed, the integrability condition for some map F gives a very strong topological restriction on a corresponding non-orientable flat surface. This surface can be cut along the lines of the flow into a union of cylinders (on which a first-return map is a flip) and tori (on which a first-return map is a rotation, possibly identical). 5. Integrability of all 3-interval exchange transformations with flips Let us first remark that the following holds. Proposition ([17]). All IETF are integrable and completely periodic Proof. The study of modified Rauzy diagram gives a proof of this Theorem, see Figure 1. If all the intervals are flipped, F is -periodic and the Rauzy induction stops after one iteration. If only one of the intervals is flipped then the number of steps for the induction grows as the ratio λ A λb. Complete periodicity of -IET with flips has been proven by Keane [17] in 1975, four years before the invention of even the standard Rauzy induction. We will prove now an analogous proposition for 3-interval exchange transformations with flips IETF 3. Proposition 3. For any F IETF 3 the modified Rauzy induction stops after a final (depending on F ) number of iterations. Moreover, any F IETF 3 is integrable. 17

18 B > A (Ȧ ) B B A A > B (Ḃ ) A Ḃ A (Ȧ ) Ḃ Ḃ Ȧ A > B ( ) B A B A ( ) A Ḃ Ḃ A A > B B > A B > A ( ) A Ḃ A Ḃ ( A A Ḃ ) Ḃ Figure 1: Full modified Rauzy diagram for the class of maps from IETF with σ top defining the lexicographic order on the set {A, B}. Proof. The proof is an explicit study of a modified Rauzy diagram that can be simplified by noting that one can look at equivalence classes in the Rauzy diagram (as we already did on Figure 1 for n = ). Different vertices in the modified Rauzy diagram correspond to different couples (σ, k) where σ is a permutation on three letters in the alphabet A = {A, B, C} and k is a 3-vector with coordinates ±1. To each vertex (σ, k) in the diagram we associate its equivalence class [(σ, k)] of vertices that can be obtained from (σ, k) by permutation of letters in A. The vector k is changed accordingly. By understanding the connected cmponents of these representatives in the graph we get the understanding of the full graph, since other components have the same combinatorial structure. We prove that Rauzy induction stops after a final number of steps. This and the Proposition suffice to prove integrability. An important remark is that the only cycles that can survive infinitely in the MRI are those in which each one of the labels wins at least one time. On the Rauzy diagram for 3-IETs with flips these cycles simply do not exist. Indeed, the only possible cycles in this graph are one of two types (modulo the renaming of the letters): 1. The cycle of edges (C > A), (C > B) (up to relabelling). Two cycles of type 1. (C > A), (C > B) and (B > A), (B > C) that meet up in one vertex σ : one can get in the vertex by following B > A or C > A and get out of the vertex by following B > C or C > B. These two cycles can t be repeated infinitely, and there are no other cycles in this graph. Hence the MRI stops, and one easily verifies that it stops at a simple permutation. The modified Rauzy diagram illustrating this Proposition analogical to Figure 1 can be drawn explicitely but we don t do it( here to save the space. We only represent a part of the connected A component of the permutation Ḃ Ċ ) Ċ Ȧ Ḃ which consists of irreducible permutations as an example, see Figure 13. One can see that the cycles in this connected component are of type. 5.3 Generic integrability of all 4-interval exchange transformations with flips. Proposition 4. Suppose that for F IETF 4 the corresponding Rauzy induction stops after a finite number of steps. Then F is integrable and the corresponding Poincaré section such that the first return 18

19 CBA BAC ABC CAB A<C B<C B<C CAB BAC ABC BCA C<B A<B C<A CAB BCA ( A Figure 13: Connected component of the permutation Ḃ Ċ ) Ċ Ȧ Ḃ in the Rauzy graph of IETF 3. Image provided by Paul Mercat. The colors correspond to the values of the vector k: red for 1 and green for 1. map of F on this section is simple can be chosen as an interval with a left extremity equal to 0. Proof. If Rauzy induction stopped for F IETF 4 then there exists α [0, 1] and two segments I 1 I [0, 1] sharing a common right extremity such that I = [0, α] is a Poincaré section for F and the restriction of F on I 1 is either an identity map or a flip. If I 1 is not flipped, it means that F is reduced (see Definition 9). In this case, α = 1 and F in restriction to I \ I 1 is a map G IETF 3. By Proposition 3 the restriction of F on I \ I 1 is integrable and hence the initial map is integrable and the corresponding Poincaré section is the union of I 1 and the Poincaré section of the restriction G (union of two segments). If I 1 is flipped, then the first return map of F on I \ I 1 if either a map G IETF 3 or G is a 3-interval exchange transformation without flips. The first case is treated as before. In the second case the proof is also finished since any 3-IET corresponds to a rotation with a marked singularity. Theorem 3. Almost any F IETF 4 is integrable. Proof. This is a direct corollary of Proposition 4 and Theorem Integrability of CET n τ. Proposition 5. Suppose that F = F (l 1, l, l 3, τ) CET 3 τ has two following properties: l j, j = 1,, 3 are independent over Q and the modified Rauzy induction stops for F. Then the following two conditions hold: a. F is integrable. b. Any interval on a corresponding Poincaré section that is mapped to itself by the first-return map, is flipped. Proof. The map F can be interpreted as a map in IETF 4 and by Proposition 4, F is integrable. Denote σ a permutation corresponding to a simple IET defined on a Poincaré section of F provided by Proposition 4. In a Rauzy graph, the vertex corresponding to σ doesn t have any outgoing edges and there is a path in the graph connecting an initial permutation corresponding to F IETF 4 to σ. The Rauzy induction stops at σ if there exists a letter X A = {A, B, C, D} such that (σ top ) 1 (4) = ( σ bot) 1 (4) = X A. One simply notices that k X 1 and hence k X = 1. Indeed, on an arrow going into σ the interval with label X was the longest hence if it was not flipped, it couldn t be placed last after the cutting. 19

20 Suppose now that there exists an invariant interval on a Poincaré section different from the one labeled by X. This can happen only if (σ top ) 1 (j) = ( σ bot) 1 (j) = Y A for j = 1 either j = 3, Y X, Y A = {A, B, C, D}. By following backwards the Rauzy induction, one easily shows that in these two cases σ with property mentionned above can t be connected to any of the three possible combinatorial permutations for F : ( A Ḃ Ċ Ḋ ) Ḃ Ḋ A Ċ ( ; A Ḃ Ċ Ḋ A Ċ Ḋ Ḃ ) ( A ; Ḃ Ċ Ḋ ) Ċ Ȧ Ḃ Ḋ. (3) Indeed, if j = 3 then by following backward Rauzy induction, we obtain the following : ( ) ( Y Ẋ X>Y Ẋ σ = Ẏ )? =: σ 1 Y Ẋ Ẏ Ẋ and here σ 1 doesn t have any entering edges in the Rauzy graph and at the same moment it is not equal to any of the permutations in (3). The case j = 1 is treated analogously. Indeed, any letter moves to the left in the line compared to its position only if it is a loser in a step of Rauzy induction. Hence the situation with j = 1 can occur only if the size of the permutation is or if the permutation has been already reduced with a non-flipped cylinder on the left to start with which is not the case: σ = ( Y Ẋ Y Ẋ ) X>Y (Ẏ Ẋ Ẋ Ẏ ) =: σ?. Theorem 4. Any F CET 3 τ with τ 1 is integrable (and hence, not minimal). Moreover, if the lengths of the exchanged intervals l j, j = 1,, 3 are independent over Q then any periodic point of F belongs to an interval which is flipped on itself by the first return map. Proof. It is a direct corollary of Theorem and Proposition 5. We have seen that the families IETF, IETF 3 consist of only integrable transformations, and the family IETF 4 (as well as CET 3 τ) has almost all of its transformations integrable. Let us remark that starting from n = 5, the statements analogue to those for n 4 are false. Indeed, the stop of the MRI doesn t necessarily imply integrability of F IETF n with n 5. Proposition 6. For any n 5 there exists an open set of non-integrable maps in IETF n. Proof. Indeed, consider an open set of fully flipped 5-interval exchange with the following combinatorics: ( A Ḃ Ċ Ḋ Ė ) Ḃ Ċ Ḋ Ė A and the following restrictions on the lengths : λ A max{λ B, λ C, λ D, λ E }. For any map in this set, four steps of MRI will follow a path A > E, A > D, A > C, A > B (an informal notation corresponding to the fact that A would be a winner for all four steps). The consequent permutations in the Rauzy path will change in a following way before the stop: ( A Ḃ Ċ Ḋ Ė ) Ḃ Ċ Ḋ Ė Ȧ A>E ( E Ȧ Ḃ Ċ Ḋ ) Ḃ Ċ Ḋ E A 0 A>D... A>B ( ) E D C B A B C D E A.

21 An IET on which Rauzy induction stops restricted to some Poincaré section is a map defined on the union of two invariant intervals I 1 and I. The first return map on I is a flip but, in restriction to I 1 it s a 4-IET which has the combinatorics defined by ( ) E D C B. (4) B C D E and is not simple. This argument can be generalized for any value of n 5. Although a family IETF 5 has an open set of non-integrable interval exchange transformations in it, by restricting to a smaller (but still quite large) subset, one finds integrable behavior. Theorem 5. Consider the subset F IETF 5 that is the image of CET 4 τ under a natural inclusion defined in Remark 3.1. Take any F F such that l j that are independent over Q and the Rauzy induction stops. Then the following two conditions hold: a. F is integrable. b. Any periodic interval on a Poincaré section is flipped. Proof. Let us list the possible combinatorics of the maps in F. Note that modulo the orientation of the interval, we can restrict ourselves to the cases when the image of the first or second interval contains 0. The combinatorics are the following : (Ȧ Ḃ Ċ Ḋ Ė ) Ȧ Ċ Ḋ Ė Ḃ (5) in the case if the image of the first interval contains 0 (in this case the first interval is split into two intervals labeled A and B), or (Ȧ Ḃ Ċ Ḋ Ė ) Ḃ Ȧ Ḋ Ė Ċ (6) in the case if the image of the second interval (union of intervals label B and C) contains 0, see Figure 15. In the case 5, the interval [0, 1] = [0, λ A ] [λ A, 1] splits into the union of two invariant segments and the restriction of the map on the second interval is an element of IETF 4. In this case, F is integrable by Proposition 4 and has all of its periodic intervals flipped (the argument is analogical to the proof of Proposition 5). In the case (6), additional work is needed to prove that the Rauzy induction can t stop at some non-integrable IET as it happens in Proposition 6. Let us remark that the only possibility for a IET of 5 intervals to be non-integrable is exactly the case when Rauzy induction leads to the combinatorics of (4), up to renaming the letters. Indeed, one can continue the modified Rauzy induction while there exist flipped intervals. Once all the intervals are non-flipped, the dynamics represented in (4) is the only dynamics on four intervals that gives the behavior that can t be decomposed into rotations on a finite union of intervals with marked singularities. The end of the proof is computer assisted. We calculate explicitely the component of the Rauzy graph corresponding to the combinatorics of the permutation (6) and we verify that the Rauzy induction never stops at the vertices of Rauzy graph of type (4) which are not simple. The number of the vertices which have outcoming edges in this component is equal to 906, We do not draw the entire graph here but a part of it, see Figure 14. The argument for a non-existence of invariant non-flipped intervals is analogical to the proof in Propoisiton 5. 1

22 Figure 14: A part of the connected component of the permutation (6) in the modified Rauzy graph for 5 interval exchange transformations with flips. The colors of the letters correspond to values of the vector k: k i = 1 is equivalent to the color of the letter i A is blue or red. Green color is equivalent to k i = 1. A B C D E B D E τ A C 1 Figure 15: The corresponding map from F to an element of CET 4 τ. Here the lengths of the intervals and the parameter τ are sych that the image of the second interval contains 0. The second interval is hence cut into two intervals with labels B and C.

23 We have seen that almost all of the maps in CET 3 τ as well as in CET 4 τ are integrable. It turns out that starting from n = 5, the dynamics of CET n τ starts to become more complicated. Proposition 7. There exists an open subset U in the set of parameters P = {(l 1,..., l n ) [0, 1] n l l n = 1} { τ τ S 1} for the maps CET n τ, n 5 such that the dynamics of corresponding maps is non-integrable. Proof. Indeed, take U = {(l 1,..., l n, τ) P l 1 > 1 τ > max(l,..., l n )}. In this case the image of the first interval contains 0 and the combinatorics of a correponding flipped IET is the following: ( ) I 1 I+ 1 I... In 1 In I3... In. I 1 I For F CET n τ such that (l 1,..., l m τ) U, the corresponding MRI will follow the path: ( ) ( ) I 1 I+ 1 I... In 1 In I + 1 >In,...,I+ 1 >I I I3... In 1 I n I n 1... I I+ 1 I 1 I I 3... I n. I 1 I I+ 1 Thus, for n 5, the dynamics of the first return map has two invariant cylinders, and a cylinder on which the dynamics is an (n 1)-IET which is not simple. The dynamics of the initial map is non-integrable. 6 The properties of the orbits in triangular tiling billiards. Now we get back to the properties of triangle tiling billiards and we formulate the results that come as consequences of the theorems we have proven in previous sections. 6.1 Symbolic dynamics. Fix a triangular tiling by some triangle. One can associate to any bi-infinite non-singular trajectory a unique bi-infinte word in the alphabet A = {a, b, c}. Here the letters correspond to the sides of. A word corresponds to a trajectory if and only if it represents the sequence of the sides crossed by this trajectory. Sometimes we will also look at the words with a fixed zero position symbol. This corresponds to fixing the starting triangle in the trajectory. Example. On Figure 6 the first depicted trajectory is a closed trajectory corresponding to the sequence ω where ω = bcababcaba and the names a, b, c are given to the sides in the decreasing order of lengts. One can see that by fixing the starting triangle (a grey one), one fixes the zero position symbol of the word ω and we have ω 0 = b. First note that not all the words in A Z correspond to the trajectories: for example, a trajectory can t pass by the same side consequently (the words do not contain subwords aa, bb, cc). The other example of a forbidden subwords are the words cbc and cac with c the shortest side of, see [8]. Denote the set of the words ω A Z such that there exists a trajectory (possibly singular, i.e. passing through the vertices) of some triangular tiling billiard with such a symbolic coding Ω and the subset of this set corresponding to a fixed triangle by Ω. Note that trajectories passing by the vertices have two different codings (wtih common past or future, depending on the orientation). The set Ω is a "small" set : the corresponding dynamics has zero topological entropy and linear complexity. This stays an interesting open question to understand precisely the sets Ω and Ω. The symbolic sequence corresponding to a trajectory characterizes how the trajectory is seen with a naked eye. We can t measure exactly the angles that the trajectory makes with each side it crosses but we can see what side is crossed. 3 I+ 1 I+ 1

24 6. Hooper-Weiss curve. All along the article we study simultaneously the properties of an IET with flips F CET 3 τ and its square which is an IET without flips T = F : S 1 S 1. Note that an accelerated path (making two steps instead of one) of the billiard trajectory corresponding to an orbit of T finishes each step in a triange with the same orientation as a starting one. One can define the following subsets (which are either empty or intervals) of the circle S 1 coded by two-letter words: I ω0 ω 1 := { x S 1 x I ω0, F (x) I ω1 }, ω0, ω 1 A. (7) These subsets are the segments of the continuity of T which is a circle interval exchange of these intervals. Since F has three singular points, T has at most 6 singular points hence at least 3 of the sets defined by (7) are empty. Definition 1. Fix a triangle on the plane with a positive orientation (the labeling of the vertices A, B, C is done in counter-clockwise order) and fix its barycenter that we mark as the origin (0, 0) R. Consider a trajectory of a tiling billiard starting in this triangle and a symbolic sequence associated to this trajectory. A Hooper-Weiss curve is a piecewise linear curve that starts in (0, 0) and connects (0, 0) to the barycenter of a next positively oriented triangle that the chosen trajectory passes through. The Hooper-Weiss curve is an oriented curve with respect to the positive iterations of a tiling billiard map. The terminology we propose here comes from the fact that in their article [16] P. Hooper and B. Weiss while studying rel leaves of translation surfaces, define the object which is exactly one of the curves we define here but only for some very specific angles of the triangle, see more in Subsection 7.. Of course, this curve is the abelianization of the curves studied and defined by C. McMullen (see, for example, []) and P. Arnoux (see, for example, [, 1]). Each trajectory defines its own Hooper-Weiss curve. Note that the segments in a Hooper-Weiss curve are parallel to the sides of the triangle in the plane and the symbolic displacement ω 0 ω 1 corresponds to the displacement along the vector Ω 0 Ω 1, where Ω j is a corresponding vertex, Ω j {A, B, C} (vertex A is a vertex in front of the side a etc.). Example. A Hooper-Weiss curve for the first trajectory of Figure 6 is a union of segments connecting the consecutive results of the sums of vectors BC + AB + AB + CA + BA. Lemma 3. A Hooper-Weiss curve is closed (drift-periodic, linearly escaping) if and only if the corresponding trajectory is closed (drift-periodic, linearly escaping). Proof. This follows obviously from Proposition 1. Understanding Hooper-Weiss curves is analogue to understanding the symbolic dynamics of tiling billiards. 6.3 Properties of periodic orbits. Closed and drift periodic trajectories both correspond to periodic words in symbolic dynamics. Although by looking more precisely at a periodic word, one can tell a difference between these two cases. Indeed, for an infinite periodic word w there exists a finite word ω such that w = ω and the length L of ω is even (since periodic trajectories have an even number of segments in it). The corresponding portion of a Hooper-Weiss curve of length L is a union of segments connecting the consecutive results in the sum ω 0 ω 1 + ω ω ω L 1 ω L. Obviously, 4

25 Proposition 8. A trajectory in a triangle tiling billiard is closed (drift-periodic) if and only if a corresponding symbolic word w is periodic with period ω, w = ω and the sum p := ω 0 ω 1 + ω ω ω L 1 ω L is equal (not equal) to zero. Example. A drift-periodic trajectory on Figure 6 has the coding w = ω with ω = bababc, and p = BA + BA + BC 0. Proposition 9. Suppose that the symbolic coding of some trajectory in a triangle tiling billiard is periodic, i.e. there exists a word ω such that w = ω. Suppose additionally that ω is odd. Then this trajectory is closed. Proof. Indeed, w = ω = ω where ω = ω 0... ω l 1, ω = l, where l is odd. It suffices to show that p l := l 1 k=0 ω k ω k+1 = 0, where and ω j = ω j+l j [[0, l 1]]. This is equivalent to the fact that the corresponding Hooper-Weiss curve closes up, and by Lemma 3, this will finish the proof. We will prove this statement by induction. This involves only very simple linear algebra. For l = 1 we have p 1 = ω 0 ω 0 = 0. Suppose the statement is proven for some odd l and p l = 0 for any word ω. Let us prove it for l +. Now we have the vector p l+ and we want to prove that it is equal to zero. This is easy to prove since the difference p l+ = p l is a sum of a finite, non-depending on l number of vectors. Indeed, now we have ω = ω 0... ω l+1 ω l+... ω l+1 with ω j = ω j+l+ j [[0, l + 1]]. Then we have l+1 p l+1 = ω k ω k+1 = = k=0 l 3 ω k ω k+1 + ω l 1 ω l + ω l+1 ω 0 + l 3 k=0 ω k ω k+1 + ω l 1 ω 0 + k=0 l k= l+3 l k= l+3 ω k ω k+1 + ω l+ ω l+3 = ω k ω k+1 ω l 1 ω 0 + ω l 1 ω l + ω l+1 ω 0 + ω l+ ω l+3 = = ω l+1 ω 0 + ω 0 ω l 1 + ω l 1 ω l + ω l ω l+1 = Generic behavior of trajectories in triangle tiling billiards. In Section 5 we studied the properties of the class CET 3 τ closely connected to triangle tiling billiards. The following theorem is a consequence of Theorem 4 translated to the language of tiling billiards from the language of interval exchange transformations with flips. Theorem 6. Let C = {(l 1, l, l 3 ) R 3 l j 0, l 1 + l + l 3 = 1} be a simplex of normalized angles of triangles, l j = α j. Consider a subset of C of lengths which are independent over Q. Consider the π set of all trajectories of corresponding tiling billiards except for those that pass through circumcenters of the crossed triangles. Then for this set of orbits the following holds: 1. There exist no drift-periodic orbits.. All closed orbits have a symboling coding ω for some word ω in the alphabet a, b, c of odd length. Consequently, their periods are equal to 4n + for some n N. 3. All escaping trajectories are linearly escaping. Their symbolic coding is described by an infinite word ω that can be represented as an infinite concatenation of the words ω 1 and ω that can be put one after the other in some order, where ω 1 and ω are some finite words in the alphabet A. 5

26 Proof. Assertion 1. has been proven in [8]: drift-periodic behavior can appear only if l j are dependent over Q. For any other triangle tiling billiard with some direction of the trajectory τ 1 by Theorem the Rauzy induction for a corresponding F CET 3 τ stops. By Theorem 4 then F is integrable, i.e. it has a Poincaré section with a simple first return map. A point x S 1 has a closed orbit if and only if it is contained in a flipped interval of a Poincaré section or in one of its non-intersecting images. The period of such a point is equal to the period of its image x P on the Poincaré section which belongs to a flipped interval I. Since I maps onto itself by the first return map which, in restriction to I is equal to F d, d N. But the interval is flipped hence d has to be odd. This proves. Moreover, the orbit in the restriction to the flipped interval will close up after d iterations. Since d is odd, d = n + 1 for some n, and the period is equal to 4n +. The escaping trajectories correspond to the orbits of the points on non-flipped intervals for an IET with flips. The non-flipped intervals correspond to the rotations for the Poincaré map. The symbolic coding of the first return to the Poincaré section defines the words ω j, j = 1,. In Subsection 6.3 we have proven that the odd length codings correspond to the periodic orbits. The converse is true as well. Here is not a generic but a general result about closed periodic orbits. Theorem 7. Consider any closed trajectory of a triangle tiling billiard. Then the minimal period of its symbolic coding sequence has an odd length. Consequently, the period of any closed orbit is equal to 4n + for some n N. Proof. The existence of a periodic trajectory for some parameters (l 1, l, l 3, τ) means that the Rauzy induction stops for the corresponding F CET 3 τ, by Lemma 1. If the Rauzy induction stops then either one is in the conditions of Theorem 6 and in this case the symbolic coding of this periodic trajectory has odd length, or l j are rationally dependent. In this case, we perturb the parameters l j in such a way that the dependence is destroyed. If the perturbation is small enough, the symbolic coding (as well as the length) of the periodic trajectory will stay fixed, by Proposition 1. One can then apply Theorem 6. Remark. The periods of drift-periodic orbits are not necessarily of length 4n +. For example, take a triangle with lengths of the sides a, b, c equal to 5, 7 and 8 and τ 1/. One can verify that the orbit passing through the middle of the side b will be drift-periodic with a drift period 4. By looking at the simulations, one can suggest a much stronger property of closed orbits than that of Theorem 7 that will be discussed in the next Section. 6.5 Tree conjecture. Conjecture 1 (Tree conjecture). [8] Let Λ be as a union of all vertices and edges of all drawn triangles in a triangle tiling. Fix some periodic closed trajectory of a corresponding triangle tiling billiard. It incloses some bounded domain U R in the plane and U Λ is an embedding of some graph in the plane. Then this graph is a tree. This conjecture was first formulated in [8] where some progress was made towards its proof. Even though we strongly believe this is true we have not found a complete proof of this fact. This conjecture has an even stronger form corresponding to the fact that any trajectory (not necessarily closed) fills in the subset of the plane that it occupies. Let us precise this more general form. Denote V a set of vertices v in Λ such that the trajectory intersects at least one edge with v as an extremity. We will color the vertices of this set in two colors, black and white : V = B W. We will construct the collections B and W by following the trajectory. Here is the algorithm. Choose 6

27 some edge e crossed by a trajectory with extremities b 0, w 0. Add b 0 B, w 0 W. Then we continue the procedure by adding, after each step of a tiling billiard reflection, the points b j, w j being the extremities of the sides crossed by the trajectory to the collection B W by assigning them colors. The colors are assigned in such a way that the edges of b j b j+1 and w j w j+1 are not crossed by the trajectory and, on the contrary, the edges b j w j+1 and w j b j+1 are crossed by the trajectory. Note that first, some of these edges degenerate into vertices (at each step, either b j = b j+1 or w j = w j+1 ). And second, it may also happen that b j = b k, k < j 1. See Figure 17 for an example. Take two graphs Γ B and Γ W in the plane with vertices being correspondingly the sets V B := B and V W := W and the edges connecting two vertices with consequent indices of the same color. If b j = b j+1, we do not add any loop. Then Conjecture 1 can be generalized to have the following form: Conjecture (Density conjecture). A trajectory is not closed if and only if both of the corresponding graphs Γ B and Γ W are trees. A trajectory is closed if and only if one of the corresponding graphs is a tree (corresponding to the vertices inside the trajectory) and another of these graphs(corresponding to the vertices outside of the trajectory) has a unique cycle in it. Note that for a corresponding system of reflections in a circumcircle, the triangles obtained in a trajectory are exactly those all of whose vertices are colored (belong to V). By coloring the images of black and white vertices in the tiling on the fixed circle in the system (III) as well, one can note that black vertices lay on one of side of the chord defined by a trajectory and white vertices lay on the other side. Proposition 10. Fix some trajectory of a triangle tiling billiard (system (I)). Take a parameter τ corresponding to the direction of this trajectory and the sets Γ B and Γ W defined by this trajectory. Then for the corresponding system of reflections in a circle (system (III)) with the same τ and a corresponding initial condition, all of the images of the vertices of Γ B will be on one side of the chord defined by τ and all of the vertices of Γ W will be on the other side. Proof. This is a very simple geometry argument that uses implicitely the results of Proposition 1, see Figure 16. This remark helps to prove that a trajectory can t make a "tour" around a triangle without getting inside this triangle. Proposition 11. There is no trajectory of triangle tiling billiard not passing through some triangle but at the same moment passing through all of its neighboring triangles (the three triangles a, b, c sharing the sides with ). Proof. Indeed, if such a trajectory exists the vertices A, B, C of are colored in the same color and the other three vertices A, B, C of a, b, c not belonging to are colored in an opposite color. By looking at the system in a circumcircle, one remarks that the image of in the circle lies on one side of the chord corresponding to the trajectory. In this case, its reflection with respect to at least one of its sides also lies on the same side of this chord. This reflection is an image of one of the triangles a, b, c. This forces one of the vertices A, B, C to have the same color as the vertices of. We obtain a contradiction. The proof of this Proposition is simple but we failed to generalize it in order to prove Conjectures 1 and. In our defense, these conjectures may be quiet difficult to prove though since they can be connected to the density theorems proven in [18]. 7

28 Figure 16: The edge that has two different colored vertices is crossed by the trajectory. One can easily see that all the vertices of the same color end up after folding on the same side from the line of trajectory in the circumcircle system. Figure 17: An example of the coloration of the vertices in V for some closed trajectory of a tiling billiard : here the graph Γ W is a tree. 8

29 7 Geometric interpretation of our results and the locus of minimal behavior in the parametric space. As we have seen above, the minimal behavior of CET 3 τ is possible only for τ = 1. In this Section we describe more precisely the set of parameters (l 1, l, l 3 ) C for which the dynamics of the corresponding interval exchange on the circle F CET 3 τ with flips is minimal. 7.1 Squares of fully flipped interval exchange transformations and SAF invariant. The set of fully flipped interval exchange transformations is of a particular interest for us. Obviously, for any element F FET n its square T = F is an interval exchange transformation, T IET n 1. Let F := { T IET n 1 F FET n : F = T }. The set of all (n 1)-IETs is n -dimensional although the set F of IETs that come from flipped n-iets is n 1-dimensional. So F is the set of half a dimension. We still do not understand very well what exactly is this subset F in the set of all interval exchange transformations F IET n 1. What is clear is that those IETs are very special. We will prove here that SAF-invariant of the maps from F is equal to zero. Let us remind the definition and basic properties of the SAF-invariant. Definition 13. Take T is an interval exchange transformation, T IET m such that T [ai 1,a i ) (x) = x + t i, 1 i m, [a i 1, a i ) being the intervals of continuity. Let l i = a i a i 1, i [[1, m]] be their lengths. Then Sah-Arnoux-Fathi invariant of T is a Q-bilinear form which is given by SAF(T ) = m l i Q t i (8) i=1 The SAF invariant of an IET was defined by P. Arnoux in his thesis [3] where he proved the following. Theorem 8. [3] Consider T : [0, 1) [0, 1) an interval exchange transformation. Then its SAFinvariant has the following properties. 1. If T is periodic then SAF(T ) = 0.. Suppose T α is a circle rotation by angle α. Then SAF(T α ) = 1 α α 1. And, consequently, T α is periodic (equivalently, α Q) if only if SAF (T α ) = Suppose that T 1 and T are first return maps of a directional flow on some translation surface M with transversals τ 1, τ M correspondingly. Then SAF(T 1 ) = SAF (T ). In other words, SAF is an invariant of the flow. 4. SAF(T ) is invariant by Rauzy induction. 9

30 A SAF invariant is a very powerful tool for working with interval exchange transformations and translation surfaces. For example, McMullen ([],[1]), Boshernitzan and Arnoux have proven that if SAF = 0 for a translation flow on a surface of small genus (g = ) then this flow is not minimal. Starting from a higher genus, this is not true. There exist examplesof IET with zero SAF-invariant and minimal dynamics corresponding to surfaces of genus 3. The most famous one of them is a so-called Arnoux-Yoccoz example, it was constructed in []. Now we are ready to prove the following Proposition 1. For any map F FET n its square T = F is an interval exchange transformation for which Sah Arnoux Fathi invariant is zero : SAF(T ) = 0. Proof. The interval [0, 1) is a union of m intervals I j = [a j 1, a j ), j [[1, m]]. On each interval I j the restriction F Ij = x + τ j is a flip, τ j R. Let us take a subdivision of the division I j into the subsets I j,k (which are actually intervals or empty sets) defined in a following way: I j,k = {x I j F (x) I k }. (9) Here j, k [[1, m]]. It is possible that I j,k =. Note that m k=1 I j,k = I j and m j=1i j,k = F 1 (I k ). Since F is a bijection (up to a finite number of discontinuity points) we have that not only k I j,k = λ j but also j I j,k = λ k. Then one can calculate explicitely that j, k [[1, m]] the restriction T Ij,k (x) = ( x + τ j ) + τ k = x τ j + τ k. By using the subdivision of [0, 1) by the sets {I j,k } j,k [[1,m]], one can finally calculate m SAF(T ) = I j,k (τ k τ j ) = ( ) I j,k τ j + ( ) I j,k τ k = j,k=1 j k k j = ( ) I j,k τ j + ( ) I j,k τ k = λ j τ j + j k j j k k λ k τ k = 0. A version of this Proposition in the context of flows on translation surfaces has been proven in [5]. There B. Strenner shows that any pseudo-anosov map that is a lift of a pseudo-anosov homeomorphism of a nonorientable surface has a vanishing SAF invariant. He also gives an explicit construction for the Arnoux-Yoccoz interval exchange. His construction can be applied for any F CET 3 1 that gives a pseudo-anosov map. 7. Description of CET 3 τ for τ = 1 and Arnoux-Rauzy family. Even though, thanks to Theorem 4, we understand generic behavior of the systems from CET 3 τ, the behavior of specific ones can be quite surprising. Indeed, the authors in [8] remark that for τ = 1, there exists a vector of lengths (l 1, l, l 3 ) such that the symbolic behavior of some specific (singular) trajectory of a corresponding map F CET 3 τ ressembles to a Rauzy fractal. The form obtained is a Hooper-Weiss fractal defined in [16]. In the paper [16] Patrick Hooper and Barak Weiss define their fractal and also suggest that this fractal, up to rescaling and a uniform affine coordinate change, converges to Rauzy s fractal in the Hausdorff topology. This is still not proven precisely, remaining an open question. The Hooper-Weiss fractal is an example of Hooper-Weiss curve that we defined in Subsection

31 The vector of lengths (proportional to the vector of the angles of ) for which the corresponding triangle tiling billiard gives a Hooper-Weiss fractal is ( 1 x 3 (l 1, l, l 3 ) =, 1 x, 1 x ), (10) where x is an only real solution of x + x + x 3 = 1. Definition 14 ([], [4]). Arnoux-Rauzy family is a 3-parametric family of 6-interval exchange transformations of the circle of length 1 defined in a following way. The circle that is identified with the interval [0, 1) is cut into 6 intervals : [0, 1) = [0, x 1 ) [x 1, x 1 ) [x 1, x 1 + x ) [x 1 + x, x 1 + x ) [1 x 3, 1 x 3 ) [1 x 3, 1). These six intervals have lengths x 1, x 1, x, x, x 3, x 3 where x j 0, j x j = 1. Each of the three lengths x j is represented by two neighbouring intervals. Each transformation from Arnoux-Rauzy family is a composition of two maps. First, the intervals of equal lengths are exchanged. Second, the circle is rotated by 1. Note that the two maps in this definition are both involutions but they do not commute. Proposition 13. Let R be a subset of CET 3 1 corresponding to the parameters l 1, l, l 3 < 1. Then the set {F F R} is exactly Arnoux-Rauzy family of interval exchange transformations on the circle. Proof. The proof is a direct computation. We suppose here that l 3 l l 1. This can be done without loss of generality, by changing the initial orientation of the circle as well as the position of a marked point 0 on it since rotation by τ is the same as the rotation by 1 τ in another direction. We subdivide each one of the intervals I 1 = (0, l 1 ), I = (l 1, l 1 + l ) and I 3 = (l + l 1, 1) into two intervals: I 1 = (0, 1 l ) ( 1 l, l 1 ) =: J + J 3 l = (l 1, 1 + l 1 l 3 ) ( 1 + l 1 l 3, l 1 + l ) =: J + 3 J 1 l 3 = (l 1 + l, l + 1 ) (l + 1, 1) =: J + 1 J. One can see that F is an interval exchange transformation with flips of the six intervals J +, J 3, J + 3, J 1, J + 1, J. Note that one iteration of F for these six intervals is indeed an interval exchange transformation and not a cercle exchange transformation in the sense that 0 is an image of the ends of the intervals. The intervals are exchanged following the permutation ( ) J + J + 3 J 3 J J + 3 J + 1 Applying once more the map F to the image of the interval, one obtains the final result : T = F is a 6-interval exchange of the circle (and 7-interval exchange of the interval). The order of permutation of 6 intervals on the circle can be precised. The cyclical order is initially the following: (J + J 3 J + 3 J 1 J + 1 J ). J 1 J 3 J + 1 J + J J 1. 31

32 I 0 1 l l 1 l + l 1 1 l 3 + l F (I) l 1 + l 1 l 3 l 1 1 l 1 + l 1 + l 1 F (I) 1 l 3 l 1 1 l 1 + l 1 + l 1 Figure 18: The Figure shows how the map F and its square T = F act on the subdivision of S 1 by six intervals J ± j, j = 1,, 3. The three levels of this Figure correspond to the iterations of F (F 0 = Id, F 1 = F, F = T ). One can see that on the interval one would need a 7-interval exchange by dividing the interval J3 into two intervals J3 = ( 1 l, l 3 ) (l 3, l 1 ). The vertical bar corresponds to the place one has to cut this interval in order to pass from the CET on 6 intervals to an IET on 7 intervals. Note that the intervals of the same length in the subdivision neigbour each other on the first level as well as on the last level, after applying T but their order in a couple is reversed. This shows that the squares F define the maps in Arnoux-Rauzy family. After applying F this cyclical order is changed to a new one: (J 3 J + 1 J 1 J + J J + 3 ).The two iterations of the map F are represented on the Figure 18. Note that one naturally defines a symbolic dynamics of the map F in the alphabet {1,, 3} with the help of subdivision into the intervals I j, j = 1,, 3 on the circle S 1. Let us define I j,k as in (9) for j, k = 1,, 3 as subsets that correspond to the points on the circle whose symbolic sequence has jk as a prefix. Then one can see that exactly J 1 = I,3, J + 1 = I 3,1 ; J = I 3,1, J + = I 1,3 ; J 3 = I 1,, J + 3 = I,1 and that these intervals come by couples of equal length : I 1, = I,1 = 1 l 3, I 1,3 = I 3,1 = 1 l, I 3, = I,3 = 1 l 1. From the representation in Figure 18 and the relationships of the lengths of the intervals J ± j we see that F is exactly Arnoux-Rauzy map from Definition 14. The parameters x j are given by the affine change from the lengths of flipped intervals I j, j = 1,, 3: Here x 1 x x 3. x j = 1 l j, j = 1,, 3. This Proposition shows that the study of the dynamics of trajectories of acute triangle tiling billiards passing throught the circumcenter is equivalent to the study of the dynamics of Arnoux- Rauzy family. It is well known that a corresponding F CET 3 1 is minimal if and only if the triple (l 1, l, l 3 ) belongs to the measure zero set of the triangle {τ = 1 } in the space of parameters which is 3

33 known as the Rauzy gasket. The definition of Rauzy gasket and the studies related to its properties can be found in [4, 6, 7]. Let us note that for any F CET 3 1 corresponding to an obtuse triangle (one of l j > 1) the dynamics is completely periodic. Proposition 14. Take an F CET 3 1 completely periodic. with corresponding parameters l j such that l 1 > 1. Then F is Proof. One can easily note that the intervals I 1 := (0, l 1 1) and I := ( 1, l 1) are invariant and consist of periodic orbits of length. By restricting the dynamics of F on the interval [0, 1] \ (I 1 I ) one can see that this dynamics is equivalent to the dynamics of an 3-IET with flips with combinatorics (Ȧ Ḃ Ċ ) Ḃ Ċ A and such a system is integrable by Proposition 3 and, as can be explicitely verified in this case, completely periodic. Hence it now becomes clear that triangle tiling billiards corresponding to fractal behavior of the trajectories are exactly those that correspond to the parameters of the angles α j that belong to the Rauzy gasket.as far as we know, Arnoux-Rauzy family of 6-IETs was widely studied before but it was never noticed that this family is a family of squares of fully flipped 3-IETs. Theorem 9. A trajectory of a triangle tiling billiard is non-linearly escaping if and only if a. the triangle is acute, b. the trajectory passes through the circumcenters of all the triangles that it crosses, c. the angles of the triangle α j define a point (l 1, l, l 3 ) := ( α 1 gasket. π, α π, α 3 π ) that belongs to the Rauzy Proof. The existence of the non-linearly escaping trajectory implies that τ = 1 for a corresponding F CET 3 τ by Theorem 6. Moreover, the triangle in the tiling has to be acute by Proposition 14. The point c. follows from [4] where it was proven that the only IETs from Arnoux-Rauzy family that are minimal are exactly those that correspond to Rauzy gasket parameters. 7.3 Hooper-Weiss curve for minimal triangle tiling billiards. In [16] Hooper and Weiss study the growing fractal curve which corresponds exactly to the Hooper- Weiss curve of a singular trajectory of a triangle tiling with parameters (10). We would like to note that the Hooper-Weiss curve is interesting to study for any parameters in C S 1 that correspond to non-linearly escaping trajectories (exactly those described by Theorem 9). In terms of the work [16], the change of the parameter τ of a triangle tiling billiard corresponds to the change of the vertical (relative) distance between the singularities on the translation surface and the change of l j corresponds to the horizontal distance change. In their work [16], only the parameter τ is changed and the points l j are fixed as in (10). We think that it can be interesting to study the general behavior of all Hooper-Weiss curves. To some extent, this work was already done in [6]. 33

34 7.4 Geometric interpretation of the invariance In this Subsection we explain why the hyperspace τ = 1 is invariant under Rauzy induction in the families of CET n τ. Let F CET n τ acting on S 1 and X F be the cylinder S 1 ( 1, 1) with the following identifications on the horizontal boundaries: (x, 1) is identified with (F (x), 1). In plain terms, on each interval of continuity of F, an interval and its image are identified by flip. X F is a non orientable compact surface. The vertical flow is well-defined on X F. It means that we consider an orientable foliation on a non orientable surface. By definition, the first return time on S 1 {1} of this flow is the map F. When τ = 1/, the surface X F possesses an additional symmetry. It is invariant by the antiholomorphic map φ : (x, y) (x + 1/[1], y). An elementary calculation shows that the quotient is a projective plane (see Figure 7.4). A B C B C A Figure 19: The surface X F and its quotient by the involution φ. The vertical flow descends to the quotient as a foliation. Since the action of Rauzy induction on X F is realized as a cut and paste of rectangles with horizontal and vertical sides, it preserves the symmetry φ. Therefore, Rauzy induction preserves the hypersurface τ = 1/. 7.5 Minimality in the family CET n τ for n 4. We would like to formulate a general Conjecture 3. If F CET n τ is minimal on S 1 then τ = 1. By using the computer-assisted proof and analogously applying it as in Section 4 for n = 3 we were able to obtain Theorem 10. Conjecture 3 is true for n = 4. Proof. Proof is analogical to the case when n = 3, and also uses an analogue of Lemma whose proof is computer assisted. In future work we hope to investigate the relationship of Conjecture 3 with Novikov s problem [1, 13] concerning the plane sections of 3-periodic surfaces. 34

35 8 Quadrilateral tilings. Let us note that the system (II) of reflections in a circumcircle can be generalized to any inscribed n-polygon. The same is applied to the system (III) and the generalization of the system (IV ) has been already given in Definition 8 and discussed a lot above. But, going back to the tilings, not any polygon tiles the plane. It is a well-known fact that tiling by congruent polygones in the plane can be achieved only for n = 3, 4, 5 and 6. For n = 5 and 6, the tiles have to come in special families and we won t study these problems here although we find them very interesting and hope to study them in the future work. Although, for the case of quadrilaterals, we would like to note that any quadrilateral tiles a plane in a way analogous to the triangle tiling (two neighbouring congruent quadrilaterals are symmetric one to each other with respect to the center symmetry in the middle of their common edge). We restrict ourselves here for the case of quadrilaterals that are inscribed in circles (cyclic quadrilaterais) since for them the analogue of Proposition 1 holds. Then by applying Theorems 10 and 5, analogously to the proof of Theorem 6, give: Theorem 11. Consider the set of all cyclic quadrilateral tiling billiards. Take a subset of this set in which the sides l j, j = 1,, 3, 4 of such quadrilaterals are rationally independent. Consider the set of all the trajectories of corresponding tiling billiards except for those that pass through circumcenters. Then for this set of orbits the following holds: 1. there exist no drift-periodic orbits,. all closed orbits have a symboling coding ω for some word ω in the alphabet of sides A := {a, b, c, d} of odd length and, consequently, have periods 4n + for some n N. 3. all escaping trajectories are linearly escaping. Their symbolic coding is described by an infinite word ω that can be represented as an infinite concatenation of the words ω 1 and ω that can be put one after the other in some order, where ω 1 and ω are some finite words in the alphabet A. The tree conjecture (as well as density conjecture) still seems to hold experimentaly modulo minor changes in the formulation, see Figure 8. These minor changes are the following : now, if trajectory comes from one edge to another, it may be possible that the two vertices of the same color can not be connected by the edge in a tiling (in the case when the trajectory crosses two edges having a common vertex, for example). In this case one includes also the last vertex and one connects this vertex to its neighbours and one colors them all in the same color. In this way, all the crossed quadrilaterals will also have all of their vertices colored. The Figure 8 gives some simulations of trajectories for inscribed quadrilaterals. As we have seen in Section 5, starting from n = 5 (reflections of pentagons), the system CET n τ exhibits some nonintegrable behavior in open sets. Maybe this is related to the fact, that most of the inscribed pentagons never tile the plane?... 35

36 Figure 0: The generalisation of the Tree Conjecture from Subsection 6.5 for cyclic quadrilaterals seems to also hold experimentally. This Figure represents a part of the trajectory (in blue) in a billiard with a graph Γ B drawn completely for this part of trajectory and the graph Γ W drawn in part (to be able to show the progression of the growth of the graphs). Acknowledgements. This article wouldn t exist without the help of Paul Mercat who wrote the program that drew modified Rauzy graphs for the maps in CET 3 τ and CET 4 τ. Thanks to his program, we could experimentally verify our intuition that the parameter τ = 1 was a special one for this system. The proof of Lemma which is crucial for our work, is for now computer assisted. The needed calculations were done by the program that Paul wrote. Olga Paris-Romaskevich thanks Ilya Schurov for the collaboration on the program of trajectories in quadrilateral tilings. We would like to thank Pat Hooper and Alexander St Laurent for their program that draws tilling billiard trajectories, accessible on-line [15], and Shigeki Akiyama for suggesting us a new representation of tiling billiards as the systems of tangent reflections. We are very grateful to Diana Davis for her enthousiastic talk in February 017 in CIRM that motivated us to study this system as well as for her elegant idea of a graphic representation of IET s with flips (by drawing the geometric forms over the intervals as in Figure 5) that we use throughout this article. References [1] P. Arnoux, J. Bernat, and X. Bressaud, Geometrical models for substitutions, Exp. Math. 0 (011), [] P. Arnoux, Un exemple de semi-conjugaison entre un échange d intervalles et une translation sur le tore, Bull. Soc. Math. France 116(1988),

37 Figure 1: Cyclic (!) quadrilateral tiling billiards qualitatively have the same behavior as triangle tiling billiards. Here we give an example of a closed trajectory and of a linearly escaping trajectory in such tilings. A program simulating quadrilateral tiling billiards dynamics was written in collaboration with Ilya Schurov. 37