— We prove a conjecture of Gérard Rauzy related to the structure of billiard trajectories in the cube : let us associate to any such trajectory the sequence with values in {1,2,3} given by coding 1 (resp. 2, 3) any time the particle rebounds on a frontal (resp. lateral, horizontal) side of the cube. We show that, if the direction is totally irrational, the number of distinct finite words of length n appearing in this sequence is exactly n2 + n + 1. 1. Statement of the result We consider billiard problems with elastic reflexion on the boundary ; the simplest of these problems is the billiard in the square. It is natural to code an orbit for this billiard problem with initial direction (α, β) by the sequence of the sides it meets, coding 1 for vertical sides and 2 for (*) Texte reçu le 14 janvier 1992. P. ARNOUX, Laboratoire de Mathématiques Discrètes, UPR 9016, Faculté des Sciences de Luminy, Case 930, 163, avenue de Luminy, 13288 Marseille CEDEX 9 France. Ch. MAUDUIT, Laboratoire de Logique, Mathématiques Discrètes et Informatique, Université Lyon I, 43 Bd du 11 novembre 1918, 69622 Villeurbanne CEDEX France. I. SHIOKAWA, Department of Mathematics, Keio University, Hiyoshi, Yokohama, 223 Japan. J.-I. TAMURA, Faculty of General Education, International Junior College, Ekoda 415-1, Nakano-ku, Tokyo, 165 Japan. AMS classification : 58F03, 05A15, 05B45. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037–9484/1994/1/$ 5.00 c © Société mathématique de France 2 P. ARNOUX, C. MAUDUIT, I. SHIOKAWA AND J.-I. TAMURA horizontal sides ; one can then show, if α and β are rationally independant, that the number of words of length n appearing in this sequence is equal to n+1 ; this is the minimal number of words for a non-periodic sequence, and caracterizes so-called Sturmian sequences (cf. [HM]). This problem is analogous to the study of the intersection of a line of slope β/α in the xy-plane with the net formed by the lines x = n and y = n (n ∈ N). An immediate generalization of this problem is the billiard in the 3dimensional cube. Let us denote by I the unit cube of R, I = { (x1, x2, x3) | 0 ≤ xi ≤ 1, i = 1, 2, 3 } . The billiard flow in I is the geodesic flow with respect to the natural euclidian metric on the unit tangent bundle, with elastic reflexions on the boundary. This means that, on the face x1 = 0, we identify the two points (0, x2, x3, α1, α2, α3) and (0, x2, x3,−α1, α2, α3) of the unit tangent bundle I × S, and similarly for the other faces. It is clear that the set I3×(±α1,±α2,±α3) is invariant by the flow, and it is classical that, if α1, α2 and α3 are rationally independent (we say that the vector (α1, α2, α3) is totally irrational), the restriction of the flow to this set is minimal and uniquely ergodic. In fact, it can then be reduced to a flow on the threedimensional torus ; we will use this fact later. We can code an orbit by a sequence (un)n∈N ∈ {1,2,3}N, un = i if the n-th face met by the orbit is xi = 0 or xi = 1. DEFINITION. — Let u = (un)n∈N be a sequence with values in a finite set A. The language associated to u is the set L(u) of finite words appearing in (un)n∈N, i.e. the set of words uiui+1 . . . ui+k, (i, k) ∈ N. The complexity of the sequence u is the function : p(n) = #L(u) ∩An. In this paper, we prove the following : THEOREM. — The complexity of a sequence generated by the cubic billiard with totally irrational initial direction is p(n) = n + n+ 1. This theorem (cf. [R2], [R3]) was first conjectured by Gérard RAUZY in 1981, and proved independently some months ago by the french and the japanese authors of this paper. To prove the theorem, we will first, in part 2 and 3, reduce the problem to the following : given a certain partition of the torus T in three parallelograms and a translation of T, find the number of connected components of the intersection of the partition with its first n images by the translation ; we finish the proof by a simple combinatorial argument in part 4. In the last parts, we make a few remarks about possible extensions. TOME 122 — 1994 — N◦ 1 COMPLEXITY OF SEQUENCES DEFINED BY BILLIARDS IN THE CUBE 3 2. Symbolic dynamics associated to the cubic billiard Along the trajectory, the velocity will be (±α1,±α2,±α3), the signs being determined by the parity of the number of reflexions in the corresponding faces. Since we denote the two parallel faces in each pair by the same symbol, the coding of a trajectory is not changed if we make a symmetry with respect to one of the three planes xi = 12 , i = 1, 2, 3. So for each point with given direction, we can find by symmetry seven other points with the same coding, and exactly one of these eight points has a direction with all components positive ; if we always choose this point for representant, we get the flow with slope (α1, α2, α3) on the torus T = R/Z. This flow has the same coding as the billard, when we code a trajectory by the crossing of the projections of the faces on the torus. If we go to the universal cover, this gives a very simple definition of the billiard sequences : take a line with direction v, and code the order in which it meets the planes xi = n, n ∈ Z. REMARK. — Strictly speaking, we have defined the flow only if the particle meets the interior of a face ; we can define the flow when it meets the edge intersection of the faces xi = ni, xj = nj, (ni, nj) ∈ {0, 1}2 as a succession of reflexions on the two faces ; since these reflexions commute, this is well defined. There is no natural way to define a unique coding for this point, so we will accept the two sequences ij and ji on this edge. This does not cause problems in the totally irrational case, because then the orbit can meet at most once an edge of type ij, otherwise it should be contained in a rational plane. In particular, the orbit starting from (0, 0, 0) never meets an edge (In the rational case, things are more complicated ; in particular, if the direction is a multiple of an integer vector, each orbit is periodic ; if we allow arbitrary coding for meeting of the edges, we could obtain a non-periodic sequence, and in this case, we have to choose a consistent set of rules for the coding). We can extend the flow to the vertices in the same way, accepting the six words ijk to code the vertices. Let us recall that the shift σ on the set of sequences with value in {1,2,3} is defined by : σ((un)n∈N) = (vn)n∈N with vn = un+1. The coding we have defined is not injective, because we are working with a flow, and not with a map (two points on a segment of orbit that does not meet the faces have same associated sequences) ; we can restrict to points on the faces, and it is clear that the first return map of the flow on this set is conjugate to the shift on the admissible sequences. LEMMA. — Let v be a totally irrational direction, and Ωv be the set of all admissible sequences for the billiard with initial direction v. The BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 4 P. ARNOUX, C. MAUDUIT, I. SHIOKAWA AND J.-I. TAMURA set Ωv is a closed subset of {1,2,3}N, invariant by σ, and the dynamical system (Ωv, σ) is minimal (i.e., every orbit is dense). Proof. — Let us consider the map φ which associates to an admissible sequence u the point x in the union of faces whose orbit has coding u. This map is well defined, because, by minimality of an irrational flow on the torus, no two points have the same coding ; it is continuous, but not injective, and it conjugates the shift to the first return map of the flow. The inverse map φ−1, which associates to a point the set of coding sequences (1,2,4 or 6 depending whether the orbit meets an edge or a vertex) is injective, not continuous (this is impossible for topological reasons), but it has the following property : if the points xn converge to x, and if we have associated sequences u(xn) that converge to a sequence u, then u is one of the coding sequences for x. Here, we must take care that each points may have several coding sequences, and also that, if the orbit of x goes through an edge or a vertex, we can find xn converging to x such that the associated coding sequences do not converge (but all convergent subsequences will go to coding sequences for x). This implies immediately that Ωv is invariant by the shift, and closed : if admissible sequences un converge to a sequence u, we can suppose, by compacity, that the points xn = φ(un) converge to x, and by the above property, u is one of the coding sequences associated to x. The minimality of (Ωv, σ) is an immediate consequence of the minimality of the first return map. This implies that, while the sequence depends on the initial position, the set of finite words appearing in this sequence depends only on the initial direction ; in particular, all sequences with same initial direction have the same complexity. Instead of counting the number of different words in a given sequence, one can find the complexity by computing the number of different initial segments of length n ; more precisely, we can restrict this to trajectories that do not meet an edge before the n-th crossing. Our proof will be based on this fact. 3. Reduction to a translation on T We use the description of admissible sequences as generated by a line in R ; we need some definitions. We note the canonical basis of R by e1, e2, e3 ; we define the set of integral points of height n by : P̃n = { ae1 + be2 + ce3 | (a, b, c) ∈ Z, a+ b+ c = n } . We callH the diagonal planeH = {(x, y, z), x+y+z = 0}, we denote by TOME 122 — 1994 — N◦ 1 COMPLEXITY OF SEQUENCES DEFINED BY BILLIARDS IN THE CUBE 5 π the projection of R onto H along the direction (α1, α2, α3), and define fi = π(ei) ; remark that P̃0 is a lattice in H , generated by fi−fj = ei−ej. In what follows, Σ̃ (or P̃ , F̃ , . . .) will always be some object in R, Σ̄ its projection on H ; this projection will be invariant by the lattice P̃0, and Σ will be the quotient by P̃0, subset of the torus H/P̃0 (on some occasions, Σ will also denote a set of represe