ERGODIC and SPECTRAL PROPERTIES an Interval Exchange

Home , Ergodic theory, Topological conjugacy

STRUCTURE OF THREE-INTERVAL EXCHANGE TRANSFORMATIONS III: ERGODIC AND SPECTRAL PROPERTIES

SEBASTIEN´ FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI

ABSTRACT. In this paper we present a detailed study of the spectral/ergodic properties of three- interval exchange transformations. Our approach is mostly combinatorial, and relies on the diophantine results in Part I and the combinatorial description in Part II. We define a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measure- theoretic point of view and which in turn gives an explicit characterization of the eigenvalues. We obtain necessary and sufficient conditions for weak mixing which, in addition to unifying all previ- ously known examples, allow us to exhibit new examples of weakly mixing three-interval exchanges. Finally we give affirmative answers to two questions posed by W.A. Veech on the existence of three- interval exchanges having irrational eigenvalues and discrete spectrum.

1. INTRODUCTION

An interval exchange transformation is given by a probability vector (α1, α2, . . . , αk) together with a permutation π of {1, 2, . . . , k}. The unit interval [0, 1) is partitioned into k sub-intervals of lengths α1, α2, . . . , αk which are then rearranged according to the permutation π. Katok and Stepin [20] used interval exchanges to exhibit a class of systems having simple continuous spectrum. Interval exchange transformations have been extensively studied by several people including Keane [21] [22], Keynes and Newton [23], Veech [31] to [36], Rauzy [28], Masur [24], Del Junco [14], Boshernitzan [5, 6], Nogueira and Rudolph [27], Boshernitzan and Carroll [7], Berthe,´ Chekhova, and Ferenczi [4], Chaves and Nogueira [10] and Boshernitzan and Nogueira [8]. While most articles in the area aim at establishing generic results for general interval exchange transformations, a few papers provide a detailed analysis of the dynamical behaviour/structure of specific families of interval exchanges. For instance, [14] describes the combinatorial structure of the symbolic sub-shifts associated to a restricted class of three-interval exchange transformations. In [8], Boshernitzan and Nogueira further widen the class of weak mixing examples (see §5), while in [4] there are examples of three-interval exchanges which do not have discrete spectrum. In this paper we give a detailed analysis of the spectral and ergodic properties of the three- interval exchange transformation T with probability vector (α, β, 1 − (α + β)), α, β > 0, and permutation (3, 2, 1) 1 defined by  x + 1 − α if x ∈ [0, α)  (1) T x = x + 1 − 2α − β if x ∈ [α, α + β) x − α − β if x ∈ [α + β, 1). Throughout the paper, T denotes the transformation on X = [0, 1) defined in (1).

Date: October 6, 2003. 1991 Mathematics Subject Classiﬁcation. Primary 37A25; Secondary 37B10. Partially supported by NSF grant INT-9726708. 1All other nontrivial permutations on three letters reduce the transformation to an exchange of two intervals. 1 2 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

Our approach is mostly combinatorial and relies on the arithmetic results in [18] and the combinatorial description in [19] of return words (with respect to the natural coding) to a special family of intervals. [18, 19] develop a theory for three-interval exchange transformations analogous to that developed by Morse-Hedlund [26], Coven-Hedlund [13], and Arnoux-Rauzy [3] which links together the diophantine properties of an irrational number α, the ergodic/dynamical properties of a circle rotation by angle α, and the combinatorial/symbolic properties of a class of binary sequences known as the Sturmian inﬁnite words: this is achieved through a vectorial algorithm of simultane- ous approximation, and a description of a class of sub-shifts of block complexity p(n) = 2n + 1 which generalize Sturmian words. In the present paper we apply this description to prove spectral properties for T . In a forthcoming fourth part, we apply our theory to the study of joinings of T .

It is well known that each transformation T is induced by a rotation on the circle, and some properties of T are readily traced back to the underlying rotation. For instance, under the assump- tion that T satisfies the infinite distinct orbit condition of Keane [21], the system is known to be both minimal and uniquely ergodic. Also, in the case of three intervals, the associated surface (obtained by suspending an interval exchange transformation via the so-called ‘zippered rectan- gles’ [33]) is nothing more than a torus devoid of singularities (see also [2]). We recall that, in the general case of interval exchanges, a celebrated result proved independently by Veech [33] and Masur [24] states that, if the permutation π is irreducible (π{1, . . . , l} = {1, . . . , l} only if l = k), k for Lebesgue almost every λ = (λ1, λ2, . . . , λk) in the set Λk = {λ ∈ R , λi > 0, 1 ≤ i ≤ k} the λ interval exchange transformation defined by the probability vector k and the permutation π is Σi=1λi uniquely ergodic. On the other hand, other properties of T appear not to be directly linked to the underlying rotation. These include for instance the existence and characterization of the eigenvalues of the associated unitary operator (in particular the weak mixing) and joinings (minimal self-joining and simplicity). In [20] Katok and Stepin prove that, for Lebesgue almost every (α, β) in the set {α > 0, β > 0, α + β < 1}, T is weakly mixing (see §5.1 below for the full result). This was later extended by Veech in [34]: if π is irreducible and (1,..., 1) is not in the orthogonal complement k in R of the kernel of the operator L defined by the matrix ((Lij)) where Lij = 1 if π(i) > π(j), 0 otherwise (this is true in particular for the permutation (k, k − 1, k − 2,..., 1) if k is odd), for Lebesgue almost every λ ∈ Λk, the interval exchange transformation defined by the probability λ vector k and the permutation π is weakly mixing. In [27] Nogueira and Rudolph prove that if Σi=1λi π is irreducible and not of rotation class (there is no l such that π(i) = i+l−1 (mod k+1) for all i), for Lebesgue almost every λ ∈ Λk, the interval exchange transformation defined by the probability λ vector k and the permutation π has no non-constant somewhere continuous eigenfunctions Σi=1λi and hence is topologically weakly mixing. More information on the behaviour of eigenfunctions and new proofs of weak mixing properties can be found in [10].

In this paper we obtain necessary and sufﬁcient conditions on α and β for T to be weak mixing. These conditions show that the weak mixing comes from the presence of either a spacer above a column of positive measure (like for Chacon’s map [9]), or of an isolated spacer above a column of small measure (like for del Junco-Rudolph’s map [15]). In addition, we exhibit new examples of weak mixing T . The conditions stem from a combinatorial recursive construction for generating three sequences of nested Rokhlin stacks which describe the system from a measure-theoretic point STRUCTURE OF 3-IETS III 3

of view, and which combined with a result of Choksi and Nadkarni [11] in the class of rank one systems, provide an explicit computation of the eigenvalues. While it is known that, under the infinite distinct orbit condition, T is always topologically weakly mixing (see for instance [27]), in [30] Veech2 proved the surprising existence of transformations T with eigenvalue λ = −1.3 This was later extended by Stewart [29] who showed that 2πpi p q for all rational numbers q there exists a transformation T with eigenvalue e . In this paper we give a simple combinatorial process for constructing the transformations of Veech and Stewart. In addition we exhibit examples of T having a p-adic odometer as factor. However the question concerning the existence of eigenvalues of the form e2πiγ, where γ is irrational, remained unsolved, in spite of some partial results due to Merrill [25] and Parreau (unpublished), see the discussion in §6. In [34] Veech asks, for T satisfying the infinite distinct orbit condition: Question 1.1 (Veech, [34], 1.9, (1984)). Do there exist α and β such that T has an element of its point spectrum which is not a root of unity? Is it possible for T to have pure point spectrum? In this paper we give affirmative answers to both questions:

Theorem 1.2. Let 0 < γ < 1 be an irrational number, [0; y1, y2 ...] its usual continued fraction expansion, and qk, k ≥ 1 the denominators of its convergents, given by qk+1 = yk+1qk + qk−1. If +∞ X qk < +∞, yk+1 k=1 then there exists a T satisfying the i.d.o.c. condition, which is measure-theoretically isomorphic to the rotation of angle γ, and hence has discrete (pure point) spectrum. We also show Theorem 1.3. For every quadratic irrational number γ there exists a T satisfying the i.d.o.c. condition, with eigenvalue e2πiγ. Theorems 1.2 and 1.3 are extremes of one another in that in one case the partial quotients tend to inﬁnity very quickly, while in the other they are eventually periodic. We do not know whether every complex number of modulus 1 is an eigenvalue of some T . Theorems 1.2 and 1.3 suggest that not all properties of T can be traced back to the underlying rotation: the irrational rotation by angle γ of Theorem 1.2 has no connection with the underlying rotation inducing T , and in the case of Theorem 1.3 a factor of T is a rotation with a quadratic angle, while the angle of the inducing rotation is a Liouville number.

ACKNOWLEDGEMENTS The authors were supported in part by a Cooperative Research Travel Grant jointly sponsored by the N.S.F. and C.N.R.S.. The third author was also supported in part by a grant from the Texas Advanced Research Program. We are very grateful to E. Lesigne and A. Nogueira for numerous e-mail exchanges and many fruitful conversations.

2Although the result was actually established in [30], it was only ﬁrst stated in the language of interval exchange transformations in [23], see also [34]. 3To keep in line with the existing litterature, in this introduction we denote eigenvalues multiplicatively. However, in the core of the paper, we shall use an additive notation, see the beginning of §3 below. 4 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

2. DESCRIPTIONOFTHREE-INTERVAL EXCHANGE TRANSFORMATIONS 2.1. Preliminaries. T depends only on the two parameters 0 < α < 1 and 0 < β < 1 − α. We note that T is continuous except at the points α and α + β. Set 1 − α 1 A(α, β) = and B(α, β) = . 1 + β 1 + β Then T is induced by a rotation on the circle by angle A(α, β). More precisely, T is obtained from the 2-interval exchange R on [0, 1) given by ( x + A(α, β) if x ∈ [0, 1 − A(α, β)) (2) Rx = x + A(α, β) − 1 if x ∈ [1 − A(α, β), 1). by inducing (according to the first return map) on the subinterval [0,B(α, β)), and then renormal- izing by scaling by 1 + β. We say T satisfies the infinite distinct orbit condition (or i.d.o.c. for short) of Keane [21] if the −n −n two negative trajectories {T (α)}n≥0 and {T (α + β)}n≥0 of the discontinuities are infinite disjoint sets. Under this hypothesis, T is both minimal and uniquely ergodic; the unique invariant probability measure is the Lebesgue measure µ on [0, 1) (and hence (X, T, µ) is an ergodic system). Because of the connection with the rotation R,T does not satisfy the i.d.o.c. condition if and only if one of the following holds: • A(α, β) is rational, or equivalently pα + qβ = p − q, • B(α, β) = pA(α, β) − q, or equivalently pα + qβ = p − q − 1, • B(α, β) = −pA(α, β) + q, or equivalently pα + qβ = p − q + 1 for some nonnegative integer p, and positive integer q. In the first case, T is not minimal; in the second and third cases, there is an immediate semi-topological conjugacy (hence a measure- theoretic isomorphism) between T and an irrational rotation. 2.2. Properties of the arithmetic algorithm. This subsection summarizes some results from [18]. 2 Let I denote the open interval (0, 1),D0 ⊂ R the simplex bounded by the lines y = 0, x = 0, and 1 x + y = 1, and D the triangular region bounded by the lines x = 2 , x + y = 1, and 2x + y = 1. We define two mappings on I × I, 2x − 1 y F (x, y) = , and G(x, y) = (1 − x − y, y) . x x

We check that if (α, β) ∈ D0 is not in D and is not on any of the rational lines pα + qβ = p − q, pα + qβ = p − q + 1, pα + qβ = p − q − 1 then there exists a unique ﬁnite sequence of integers l0, −1 t l0 l1 l1,..., lk such that (α, β) is in H D where H is a composition of the form G ◦ F ◦ G ◦ F ◦ G... ◦ G ◦ F lk ◦ Gs, s, t ∈ {0, 1}. The function H(α, β) is computed recursively as follows: we start with α(0) = α, β(0) = β. Then, given (α(k), β(k)), we have three mutually exclusive possibilities: if (α(k), β(k)) is in D, the (k) 1 (k) (k) algorithm stops; if α < 2 , we apply G; if 2α + β < 1, we apply F .

Associated to each point (α, β) ∈ D0 there is a sequence (nk, mk, k+1)k≥1, where nk and mk are positive integers, and k+1 is ±1. This sequence we call the three-interval expansion of (α, β), is a variation of the negative slope expansion deﬁned in [18]; it is constructed as follows: STRUCTURE OF 3-IETS III 5

• For (α, β) in D we put 1 − α − β 1 − 2α x = and y = 0 1 − α 0 1 − α and deﬁne for k ≥ 0 n o n o yk xk , if xk + yk > 1  (xk+yk)−1 (xk+yk)−1 (xk+1, yk+1) = n o n o 1−yk 1−xk , if xk + yk < 1  1−(xk+yk) 1−(xk+yk)  yk xk b c, b c if xk + yk > 1  (xk+yk)−1 (xk+yk)−1 (nk+1, mk+1) = 1−yk 1−xk b c, b c if xk + yk < 1  1−(xk+yk) 1−(xk+yk) where {a} and bac denote the fractional and integer part of a respectively. For k ≥ 0 set

k+1 = sgn(xk + yk − 1).

We note that 1 is always −1, hence we ignore it in the expansion. • For (α, β) ∈/ D we let H be the function above for which (α, β) ∈ H−1D and put α,¯ β¯ = H(α, β), ¯ and define (nk, mk, k+1) as in the previous case, starting from α,¯ β ∈ D. The following proposition sums up what we need of [18]; when (α, β) is in D, it is a translation (taking into account the fact that the initial conditions are slightly different) of results in [18]; in the general case, it comes from these results and the definition of (¯α, β¯). Proposition 2.1. (1) If T satisfies the i.d.o.c. condition, then the three-interval expansion (nk, mk, k+1) of (α, β) is infinite. (2) An infinite sequence (nk, mk, k+1) is the expansion of at least one pair (α, β) defining a T satisfying the i.d.o.c. condition, if and only if nk and mk are positive integers, k+1 = ±1, (nk, k+1) 6= (1, +1) for infinitely many k and (mk, k+1) 6= (1, +1) for infinitely many k. (3) Each infinite sequence (nk, mk, k+1) satisfying the conditions in (2) is the three-interval expansion of a countable family of couples (α, β), with exactly one couple in each of the disjoint triangles H−1D, where H has any of the possible forms defined earlier in this section, including the identity. 1 (4) A(¯α, β¯) = 1 2 + 2 m1 + n1 − 3 m2 + n2 − m3 + n3 − ... (5) A(α, β) has bounded partial quotients (in the usual continued fraction expansion) if and only if in the three-interval expansion of (α, β) the nk + mk are bounded, as well as the lengths of strings of consecutive (1, 1, +1). (6) The following two properties are equivalent – in the three-interval expansion of (α, β), there exists c > 0 such that for every M > 0 there is a k such that mk + nk > M and

mk − nk c < < 1 − c. mk + nk 6 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

– there exists c0 > 0 such that for every > 0, there exist integers p, q such that

p A(α, β) − ≤ q q2 and for any integer r 0 r c B(α, β) − > . q q 2.3. Combinatorial properties. This subsection summarizes some results from [19]. We deﬁne the natural partition

P1 = [0, α),

P2 = [α, α + β),

P3 = [α + β, 1). n For every point x in [0, 1), we define an infinite sequence (xn)n∈N by putting xn = i if T x ∈ Pi, i = 1, 2, 3. This sequence, also denoted by x, is called the trajectory of x. If T satisfies the i.d.o.c. condition, the minimality of the system implies that all trajectories contain the same finite words as factors. 0 n−1 −i 0 Let I be a set of the form ∩i=0 T Pki ; we say I has a name of length n given by k0 . . . , kn−1; 0 note that I is necessarily an interval and k0, . . . , kn−1 is the common beginning of trajectories of all points in I0. For each interval J, it is known (see for example [12]) that the induced map of T on J is an exchange of three or four intervals. More precisely, there exists a partition Ji, 1 ≤ i ≤ t of J into hi j subintervals (with t = 3 or t = 4), and t integers hi, such that T Ji ⊂ J, and {T Ji}, 1 ≤ i ≤ t, 0 ≤ j ≤ hi − 1, is a partition of [0, 1) into intervals: this is the partition into Rokhlin stacks associated to T with respect to J. The intervals Ji have names of length hi, called return words to J. In [19] we give an explicit construction of the trajectories of the points 1 − α and 1 − α − β. As a consequence we have the following structure theorem [19].

Theorem 2.2. Let T satisfy the i.d.o.c. condition, and let (nk, mk, k+1)k≥1, be the three-interval expansion of (α, β). Then there exists an inﬁnite sequence of nested intervals Jk, k ≥ 1, which have name wk and exactly three return words, Ak, Bk and Ck, given recursively for k ≥ 1 by the following formulas nk−1 mk−1 Ak = Ak−1 Ck−1Bk−1 Ak−1,

nk−1 mk Bk = Ak−1 Ck−1Bk−1,

nk−1 mk−1 Ck = Ak−1 Ck−1Bk−1 if k+1 = +1, and nk−1 mk Ak = Ak−1 Ck−1Bk−1,

nk−1 mk−1 Bk = Ak−1 Ck−1Bk−1 Ak−1,

nk−1 mk Ck = Ak−1 Ck−1Bk−1Ak−1 if k+1 = −1.

For k = 0, the words A0, B0, C0 are deﬁned as follows. STRUCTURE OF 3-IETS III 7

bk−1 T Jk2

ak−1 T Jk1

ck−1 T Jk3

TJk1 TJk2 TJk3

Jk1 Jk2 Jk3

FIGURE 1. The three Rokhlin stacks

Proposition 2.3. When (α, β) is in D, we have A0 = 13, B0 = 2, C0 = 12. When (α, β) is not in ¯ D, we deﬁne H and (¯α, β) as in the previous subsection, and two substitutions, σF by 1 → 1 2 → 21 3 → 31 and σG by 1 → 3 2 → 2 3 → 1 and we deﬁne σH by replacing, in the expression of H, F by σF and G by σG. Then we have A0 = σH (13), B0 = σH (2), C0 = σH (12). In all cases the lengths of A0 and B0 differ by ±1.

Though we shall not use this fact in the sequel, if (α, β) is in D then Jk is an interval of continuity n n of T for some n such that 1−α−β is in Jk and α is in T Jk. Equivalently the wk are the bispecial factors (in the minimal subshift associated with T , see [19]) beginning in 1 and ending in 2. When ¯ 0 ¯ (α, β) is not in D, let Jk be, for the three-interval exchange T deﬁned by (¯α, β), the intervals of 0n ¯ ¯ 0n ¯ continuity of T for some n, such that 1 − α¯ − β is in Jk and α¯ is in T Jk; let w¯k be the name of ¯ Jk. Then Jk is the interval whose name is σH (w ¯k).

2.4. Rokhlin stacks. Theorem 2.2 gives an explicit construction of the Rokhlin stacks mentioned in the previous subsection. Let wk be the name of Jk. Every trajectory under T is a concatenation of words Ak, Bk, Ck, which we call the k-words. We say that a k-word occurs at its legal k-place in a trajectory if it is immediately followed by the word wk. A concatenation of k-words occurs at its legal k-place if each of its k-words occur at their legal k-place. 8 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

We define FAk to be set of x ∈ X such that (in the trajectory of x) x0 is the first letter of the k-word Ak in its legal k-place, and similarly FBk and FCk. Note that FAk ∪ FBk ∪ FCk = Jk, and these are exactly the three intervals of continuity of the induced map of T on Jk. Let ak, bk, ck be the lengths of Ak, Bk and Ck; it follows from Theorem 2.2 and Proposition 2.3 that |ak − bk| = |a0 − b0| = 1 and ck is either ak − ak−1 = bk − bk−1 or ak + ak−1 = bk + bk−1. In j particular, we have ck ≤ 2ak. Then X = [0, 1) is the disjoint union of T FAk, 0 ≤ j ≤ ak − 1, j j j T FBk, 0 ≤ j ≤ bk − 1, T FCk, 0 ≤ j ≤ ck − 1; we denote by τAk the disjoint union of T FAk, 0 ≤ j ≤ ak − 1, and define similarly τBk and τCk; τAk, τBk and τCk are called the k-stacks, and j the T FAk, 0 ≤ j ≤ ak − 1 are the levels of the stack τAk, and similarly for B and C. The levels are intervals of small diameter, as they have names of arbitrarily large length; hence any integrable function f can be approximated (in L1 for example) by functions fk which are constant on each level of each k-stack. Hence, in the language of finite rank systems, see [16], the Theorem 2.2 implies that

Corollary 2.4. T is of rank at most three, generated by the stacks τAk, τBk, τCk; the recursion formulas in the Theorem 2.2 give an explicit construction by cutting and stacking of these stacks.

The rank at most three was known at least since [28] (while, by the above remark, the rank at most four was in the folklore and is proved in [12]), but the explicit construction of Theorem 2.2 is new, and from it comes all our knowledge of the system. The fact that Ak, Bk, Ck are return words of the same word is useful additional information, but it may be deduced from the explicit construction. The ﬁnite rank structure is particularly relevant in the measure-theoretic study of the system (X, T, µ), and in this framework, the following lemma will be useful:

Lemma 2.5. Let T satisfy the i.d.o.c. condition. For any trajectory x, the Lebesgue measure 1 µ(τAk) is the limit when n goes to inﬁnity of n times the total number of indices 0 ≤ i ≤ n − 1 such that xi belongs to a word Ak in its legal k-place. Similarly for Bk and Ck.

Proof. This follows from the unique ergodicity of T .

Remark An immediate consequence of the ﬁnite rank structure is that T is completely known, up to measure-theoretic and topologic isomorphisms, from the associated three-interval expansion (nk, mk, k+1)k≥1, and the initial lengths a0, b0, c0. Two three-interval exchange transformations T 0 and T with the same expansion (nk, mk, k+1)k≥1 need not be measure-theoretically isomorphic, see Proposition 6.2 below, but are related by induction on intervals: that is, there exist intervals I00 and I0 such that the induced maps of T on I00 and of T 0 on I0 are the same. Also, it is noted in [19] that, if we apply our recursion formulas, but starting from A0 = B0, the systems we get are just irrational rotations.

3. WEAKMIXINGOF T We recall two deﬁnitions of measure-theoretic ergodic theory: if (X, S, ν) is a measure-theoretic dynamical system, we say that a real number 0 ≤ γ < 1 is an eigenvalue of S (denoted additively) if there exists a nonconstant f in L2(X, R/Z) such that f ◦ S = f + γ (in L2(X, R/Z))); f is then an eigenfunction for the eigenvalue γ. As constants are not eigenfunctions, γ = 0 is not an eigenvalue if S is ergodic. S is weak mixing if it has no eigenvalue. STRUCTURE OF 3-IETS III 9

3.1. The fundamental lemma. In this subsection, we use the fact that there are two Rokhlin stacks of heights differing by one to kill eigenvalues: this is a trick inspired by Chacon’s map [9]. We ﬁrst use it to give a short proof of a classical result of topological dynamics: the transformation T cannot have continuous eigenfunctions. Then we tackle the general case of measurable eigenfunctions; the proofs of Proposition 3.1 and Lemma 3.2 below give some clues about the difference between continuous and non-continuous eigenfunctions: the presence of two nonempty stacks of heights differing by one is enough to prevent the existence of continuous eigenfunctions. However, to show that there are no measurable eigenfunctions, we must also know that the measure of these stacks is not too small. In the sequel ||x|| denotes the distance of x to the nearest integer. Proposition 3.1. Every T satisfying the i.d.o.c. condition is topologically weakly mixing: it has no continuous eigenfunction. Proof. Let γ be an eigenvalue with a continuous eigenfunction f; then, for given , if k is large ak enough, |f(z) − f(y)| < (in R/Z) if z and y are in Jk. We take x ∈ FAk ⊂ Jk; then T x ∈ Jk, hence, ak ||γak|| = |f(T x) − f(x)| ≤ ; 0 taking a point x in FBk, we get the same relation with bk = ak ± 1, hence γ = 0, which is not an eigenvalue as T is ergodic. Remark This result was known since [27]. The topology we use here is the one of the interval [0, 1); but the same proof works if we look at T as the shift on the symbolic trajectories, equipped with the product topology on {1, 2, 3}N, as the bases of the k-stacks are cylinders associated to arbitrarily long words. Note that for a three-interval exchange transformation with a permutation such as (132), there are continuous eigenfunctions, as T is trivially conjugate to a rotation. In [1] Arnoux exhibits an example of an interval-exchange transformation (on seven intervals) having a nontrivial continuous eigenfunction.

Lemma 3.2. If γ is an eigenvalue of T , there exists a sequence δk → 0 such that for all k

δk ||γak|| < , µ(τAk)

δk ||γbk|| < , µ(τBk) δk ||γck|| < . µ(τCk) R 0 Proof. Suppose f is an eigenfunction for T for the eigenvalue γ; then we have |f − fk| < δk, 0 with δk → 0 and fk is constant on every level of the stack τAk, τBk or τCk. 0 To prove the first inequality, we define a new word Dk, and an integer k (k) ≤ k − 1 such that • k0(k) tends to infinity when k tends to infinity, 0 • Dk is a concatenation of k -words, 0 • Dk in its legal k -place is a prefix of Ak, ak • the length of Dk is dk ≥ 3 , • in any trajectory x, each occurrence of a k-word in its legal k-place is followed by an 0 occurrence of Dk in its legal k -place.

The word Dk is deﬁned as follows: 10 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

nk−1 mk−1 • if nk > 1 or mk > 1, we put Dk = Ak−1 Ck−1Bk−1 , the common prefix of the three ak 0 k-words; thus dk ≥ 2 , and we set k = k − 1; 0 • if mk = nk = 1, let k (k) be the largest l < k such that (nl, ml, l+1) 6= (1, 1, +1); then the k−k0 k−k0 recursion formulas give either Ak = Ck0 Ak0 , Bk = Ck0 Bk0 , Ck = Ck0 (if k+1 = +1), k−k0 k−k0 k−k0 k−k0−1 or Ak = Ck0 Bk0 , Bk = Ck0 Ak0 , Ck = Ck0 Bk0 Ck0 Ak0 (if k+1 = −1); and 0 ak0 k−k ak 0 ck > 2 . Then we take Dk = Ck0 , from which it follows that dk ≥ 3 . 0 Because (nk, mk, k+1) cannot be (1, 1, +1) ultimately (Proposition 2.1), k (k) → +∞ when k → +∞. The other properties of Dk are clear from its construction. 0 0 ak We call τk the union of the first (or lower) dk levels of τAk; thus, for x ∈ τk, x and T x are in the same level of the same k0-stack. 0 ak ak Now if x ∈ τk, fk0 (T x) = fk0 (x) while f(T x) = γak + f(x); we have Z Z µ(τA ) ak k |T fk0 − γak − fk0 | = |γak| ≥ ||γak|| 0 0 3 τk τk and Z Z Z ak ak ak 0 |T fk0 − γak − fk0 | ≤ |T fk0 − T f| + |fk0 − f| < 2δk0 0 0 0 τk τk τk δ0 k0 which gives ||γak|| < 6 , hence the first inequality. µ(τAk) The second inequality is proved in the same way, as is the third, except that we replace Dk by 0 Ck and τk by τCk in the case mk = nk = 1.

Remark on the proof. In the proof of Lemma 3.2, we can alternatively use wk itself instead of building Dk (which is a preﬁx of wk), and make similar computations, replacing dk by the length 0 0 0 dk of wk, and τk by the union of the ﬁrst (or lower) dk levels of τAk. In this case, the results follow i 0 0 from the fact that T Jk is still an interval for 0 ≤ i ≤ dk −1, and that dk ≥ ak −2 ([19]). However, we prefer to prove Lemma 3.2 directly from the recursion formulas.

Corollary 3.3. If for a constant c and inﬁnitely many k we have both µ(τAk) ≥ c and µ(τBk) ≥ c, then T is weakly mixing.

Proof. The first two inequalities of Lemma 3.2, together with the relation bk = ak ± 1 give ||γ|| < 2cδk, hence γ = 0. 3.2. Study of possible non weakly mixing cases. Corollary 3.3 gives some sufficient conditions for T to be weakly mixing. We shall now focus on the cases when these conditions are not fulfilled, and show that this may happen only for very special sequences (nk, mk, k+1), namely those with a subsequence on which mk is much bigger (or smaller) than nk while outside this subsequence we must have (nk, k+1) = (1, +1) or (mk, k+1) = (1, +1) (or else (nk, k+1) = (1, −1) or (mk, k+1) = (1, −1), these last two cases being allowed only one step before mk is much bigger or smaller than nk).

Lemma 3.4. If T is not weakly mixing, then for every integer M there exists an integer k0(M) such that for every k > k0(M), one of the following two assertions is satisﬁed:

• there exists l ≥ k + 1 such that ml ≥ Mnl, k+2 = ... = l−1 = +1 if l > k + 2 and, if l > k + 1, nk+1 = ... = nl−1 = 1 if l = +1, mk+1 = ... = ml−1 = 1 if l = −1; • there exists l ≥ k + 1 such that nl ≥ Mml, k+2 = ... = l−1 = +1 if l > k + 2 and, if l > k + 1, mk+1 = ... = ml−1 = 1 if l = +1, nk+1 = ... = nl−1 = 1 if l = −1. STRUCTURE OF 3-IETS III 11

Proof. By Corollary 3.3, if T is not weakly mixing, then for every δ and k sufﬁciently large, either µ(τAk) < δ or µ(τBk) < δ. We estimate µ(τAk), using Lemma 2.5 and Theorem 2.2. If nk+1 > 1, words Ak in their legal nk+1−1 k-place cover a proportion at least of each k + 1-word (recall that ck < 2ak and that, nk+1+mk+1+1 k being large enough, bk = ak ± 1 can be treated as ak) and hence

nk+1 − 1 µ(τAk) ≥ . nk+1 + mk+1 + 1

If nk+1 = 1, words Ak in their right k-place do not appear in both Ak+1 and Bk+1, and we have 1 µ(τAk) ≥ µ(τAk+1) mk+1 + 2

if k+2 = +1, 1 1 µ(τAk) ≥ µ(τBk+1) + µ(τCk+1) mk+1 + 2 mk+1 + 3 if k+2 = −1. Hence, if µ(τAk) is small, then

• either mk+1 is much bigger than nk+1, 1 • or nk+1 = 1, k+2 = +1, and µ(τAk+1) is small, mk+1+2 1 1 • or nk+1 = 1, k+2 = −1 and both µ(τBk+1) and µ(τCk+1) are small. mk+1+2 mk+1+3 In the third case, we may have mk+2 = 1; but in that case, as k+2 = −1, we have ck+1 ≥ ak+1, 1 and µ(τCk+1) is at least because this is the minimal proportion of each k + 2-word which mk+2+2 it covers. Hence, if we are in the third case and not in the first, we must have nk+2 much bigger than mk+2. Note that in the second case, however, µ(τAk) may be small even with the ml and nl are bounded for l ≥ k + 1, provided there is a long string of nl = 1 with l+1 = +1 ahead. More precisely, the fact that µ(τAk) is small implies that 0 • either from l = k + 1 to some k ≥ l, there is a (possibly empty), string of nl = 1, with l+1 = +1 except maybe the last one, followed by an nk0 much smaller than mk0 , (or followed by an mk0 much smaller than nk0 if the last l+1 is a −1); 0 • or else that from l = k + 1 to some l there is a very long string of nl = 1, with l+1 = +1. If T is not weakly mixing, the hypotheses of Corollary 3.3 are not satisfied for k large enough. In view of the computations above, for every integer M there exists k0(M) such that, for every k > k0(M), at least one of the following four assertions is satisfied: • the first assertion of this lemma; • the second assertion of this lemma; • nk+1 = ... = nk+M = 1, k+2 = ... = k+M+1 = +1; • mk+1 = ... = mk+M = 1, k+2 = ... = k+M+1 = +1. To prove the lemma, it remains to prove that the last two assertions alone are not sufficient to prevent weak mixing. The main argument here is that if the sequence (nk, mk, k+1) is made only of long strings (nk, k+1) = (1, +1) or (mk, k+1) = (1, +1), then transitions between strings of (nk, k+1) = (1, +1) and of (mk, k+1) = (1, +1) must occur infinitely often, and these transitions will always produce weak mixing. But, to prove this, we need some involved computations (using, as in Lemma 3.2, differences of 1 between lengths of words) in the case where these transitions are made of strings of (nk, mk, k+1) = (1, 1, +1). 12 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

Namely, we suppose that there are arbitrarily large k which do not satisfy the ﬁrst and second assertion. Take such a k and suppose for example that nk+1 = ... = nl = 1, k+2 = ... = l+1 = +1, l − k > M; as we cannot have (nk, k+1) = (1, +1) ultimately, we can choose l such that either nl+1 > 1 or l+2 = −1. If nl+1 > 1, we cannot have ml+1 > Mnl+1 as k would satisfy 1 the ﬁrst assertion, hence either nl+1 > Mml+1 or ml+1 = 1, and this implies that µ(τAl) ≥ 4 . If nl+1 = 1, then l+2 = −1 and we still cannot have ml+1 > Mnl+1; then either nl+2 > Mml+2, but that is excluded as k would satisfy the second assertion, or ml+1 = 1 and ml+2 > Mnl+2, and 1 this implies again µ(τAl) ≥ 4 . But also l − M + 3 must satisfy one of our four assertions; because of our assumptions on k and nl+1, this can only happen if ml+3−M = ... = ml = 1, while still nk+1 = ... = nl = 1, k+2 = ... = l+1 = +1.

Suppose ﬁrst that there exists some k + 1 ≤ k1 ≤ l + 2 − M such that mk1 > 1, and take the

largest possible such k1. Then nk1 = 1, mk1 = m > 1, k1+1 = +1. Then τBk1−1 has measure at 1 least 4 , and by Lemma 3.2 we have ||γbk1−1|| < δ, with δ small if k is large. Now, A = (C )M 0 A , with M 0 ≥ M − 4, and C = C Bm−1, hence c > 1 a and l k1 k1 k1 k1−1 k1−1 k1 2 k1 1 1 µ(τCk1 ) ≥ 2 µ(τAl) ≥ 8 . Hence by Lemma 3.2 it follows that ||γck1 || < δ. 0 We need to show also that ||γM ck1 || < 20δ; this is done by an argument similar to the proof 00 M of Lemma 3.2: we take M = [ 4 ], and τ the set of those x for which x0 lies in a word Ck1 00 in its legal k1-place, followed by at least M words Ck1 in their legal k1-places, and estimate 00 1 1 R M ck1 00 τ |T fk1 − γM ck1 − fk1 |; as µ(τ) ≥ 2 µ(τCk1 ) ≥ 16 , we get the desired estimate for 00 0 γM ck1 and hence for γM ck1 because of the last result.

Hence, as γal is close to 0 (modulo 1), we get that both γck1 and γak1 are close to 0. But m−1 m−1 Ck1 = Ck1−1(Bk1−1) and Ak1 = Ck1−1(Bk1−1) Ak1−1, hence we get that ||γak1−1|| < 100δ, and hence ||γ|| < 101δ. Suppose now that mk+1 = ... = ml+2−M = 1. Then we cannot have nl+1 > Mml+1 or l+2 = −1, nl+1 = 1, ml+2 > Mnl+2 as in both cases k would satisfy the first or second assertion, hence nl+1 > 1, ml+1 = 1. And l cannot satisfy the first or second assertion as k would then satisfy it, nor the third one as nl+1 > 1. Hence we must have ml+1 = ... = ml+M = 1, l+2 = ... = l+M+1 = 0, and we can start the same reasoning with l replacing k, and the m and n exchanged; but we know that nl+1 > 0, hence we are in the case where this implies that γ is close to 0. Hence this situation cannot occur infinitely many times if T is not weakly mixing.

3.3. Sufﬁcient conditions for weak mixing. Lemma 3.4 gives a necessary condition for T not to be weakly mixing; our aim now is to reduce the set of possibly non-weakly mixing T to an explicit set of rank-one (see below) transformations, ready to be studied by classical methods. The main improvement over Lemma 3.4 is that, on the sequence k(j) where mk is much smaller mk(j)∧nk(j) or bigger than nk, the ratios must form a converging series, otherwise there is weak mk(j)∨nk(j) mixing by a Borel-Cantelli argument. Some technical conditions are also added to deal with the cases where mk(j) ∧ nk(j) = 1. Note that the condition depends only on the expansion (nk, mk, k+1)k≥1.

Theorem 3.5. Let T satisfy the i.d.o.c. condition , and let (nk, mk, k+1)k≥1, be the three-interval expansion of (α, β). Then, if T is not weakly mixing, there exists a strictly increasing sequence of integers k(j), j ∈ N, such that STRUCTURE OF 3-IETS III 13

• (Ma) +∞ X mk(j) ∧ nk(j) < +∞; m ∨ n j=1 k(j) k(j)

• (Mb) mk(j) 6= nk(j); • (Mc) if mk(j) > nk(j), then for every k(j − 1) < k < k(j) − 1, k+1 = +1; and for every k(j − 1) < k < k(j), nk = 1 if k(j) = +1, mk = 1 if k(j) = −1; • (Md) if nk(j) = 1, then mk(j+1) > nk(j+1) if k(j)+1k(j+1) = −1 or if k(j + 1) = k(j) + 1 and k(j)+1 = −1, nk(j+1) > mk(j+1) otherwise; • (Mc’) if nk(j) > mk(j), then for every k(j − 1) < k < k(j) − 1, k+1 = +1; and for every k(j − 1) < k < k(j), mk = 1 if k(j) = +1, nk = 1 if k(j) = −1; • (Md’) if mk(j) = 1, then nk(j+1) > mk(j+1) if k(j)+1k(j+1) = −1 or if k(j + 1) = k(j) + 1 and k(j)+1 = −1, mk(j+1) > nk(j+1) otherwise. Proof. As T is not weakly mixing, by Lemma 3.4 we can find a sequence k(j) such that m ∧ n k(j) k(j) → 0 mk(j) ∨ nk(j) when j tends to infinity, and such that Mc and Mc’ are satisfied. We can of course choose the k(j) so that also mk(j) 6= nk(j). We modify this sequence to get Md and Md’: if k(j) is such that nk(j) = 1 but Md is not 0 satisfied, or that mk(j) = 1 and Md’ is not satisfied, then we look at the first j > j such that nk(j0) ∧ 0 mk(j0) > 1 or nk(j0) = 1 and Md is satisfied, or mk(j0) = 1 and Md’ is satisfied. Such a j does exist, as otherwise we would have either (nk, k+1) = (1, +1) ultimately, or (mk, k+1) = (1, +1) ultimately, or else we would have infinitely many situations similar to the following: nk = 1, m m mk = m, k+1 = 1, mk+1 = 1, nk+1 = n, so for example Ak = Ck−1Bk−1, Ck = Ck−1Bk−1Ak−1, and m n−1 m m Ak+1 = (Ck−1Bk−1) Ck−1Bk−1Ak−1Ck−1Bk−1.

Then, as in the proof of Lemma 3.4, we show (possibly by going to the next mk0 > 1) that γbk−1, γak and γak+1 are all close to 0 (modulo 1), and by substraction we get that γak−1 is also close to 0, and this cannot happen if k is large enough. And, if we delete k(j), k(j + 1), ..., k(j0 − 1) from the sequence k(n), the modiﬁed sequence k(n) keep all its other properties: this is clear for Ma and Mb, but we have to check that the 0 k(j − 1) < k < k(j) − 1 do satisfy Mc and Mc’. The part of Mc and Mc’ concerning mk and 0 nk follows from the properties of k(j), k(j + 1), ..., k(j − 1) , so we just have to check the k+1; 0 now, if k(j − 1) + 1 < k < k(j ) − 1 and k+1 = −1, we check that we get again a weakly mixing situation, and conclude that this cannot happen if k is large enough. We have now to prove condition Ma. We estimate the measures of τAk(j)−1 and τBk(j)−1. Suppose for example mk(j) > nk(j); then µ(τBk(j)−1) > 1 − δ if j is large enough. If nk(j) > 1, then nk(j) − 1 nk(j) nk(j) µ(τAk(j)−1) ≥ ≥ ≥ . nk(j) + mk(j) + 1 2nk(j) + 2mk(j) + 2 2mk(j)

Suppose now nk(j) = 1, mk(j) = m, and suppose for example that k(j)+1 = k(j+1) = +1. Because of Md and Md’, we must have nk(j+1) > mk(j+1). Let k2 be the ﬁrst k > k(j) such that

nk2 , mk2 , k2+1 6= (1, 1, +1) ; then either nk2 is much bigger than mk2 , or nk2 > 1 and mk2 = 1, or 14 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

nk2 = mk2 = 1, k2 = −1 which is excluded because of the hypothesis on k(j+1). In each case, 1 we have µ(τAk2−1) ≥ 4 . If k2 = k(j) + 1, we get 1 1 nk(j) µ(τAk(j)−1) ≥ µ(τAk(j)) ≥ ≥ . mk(j) + 1 4mk(j) + 4 5mk(j)

If k2 > k(j) + 1, as k(j)+1 = +1, we conclude like in the proof of Lemma 3.4 that both γak(j) and γck(j) are close to 0 (modulo 1), which together imply that γak(j)−1 is close to 0, and that cannot happen if j is large enough. Similar reasonings take care of the other cases for nk(j+1), mk(j+1), k(j)+1, and k(j+1), except that, in the case k2 > k(j) + 1 and k(j)+1 = −1, we use the fact that 1 µ(τCk(j)) ≥ 2 and 1 1 nk(j) µ(τAk(j)−1) ≥ µ(τCk(j)) ≥ ≥ . mk(j) + 3 2mk(j) + 6 3mk(j) Suppose now that +∞ X mk(j) ∧ nk(j) = +∞. m ∨ n j=1 k(j) k(j) Then at least one of the four series • X nk(j) , mk(j) j;mk(j)>nk(j),ak(j)−1−bk(j)−1=1 • X nk(j) , mk(j) j;mk(j)>nk(j),ak(j)−1−bk(j)−1=−1 • X mk(j) , nk(j) j;nk(j)>mk(j),ak(j)−1−bk(j)−1=1 • X mk(j) , nk(j) j;nk(j)>mk(j),ak(j)−1−bk(j)−1=−1 diverges. Suppose for example the ﬁrst one diverges and call l(j) the (inﬁnite) sequence of those k(j) for which mk(j) > nk(j) and ak(j)−1 − bk(j)−1 = 1.

We fix some j0; let Uj = ∪j0≤i≤jτAl(i)−1. We have µ(Uj) → 1 when j → +∞ by Borel- Cantelli. Uj \ Uj−1 is made of those x whose coordinate x0 lies in a word Al(j)−1 (in its legal l(j)−1-place) but not in a word Al(i)−1 (in its legal l(i)−1-place) for any j0 ≤ i ≤ j −1; for these i, an l(i) − 1-word (in its legal l(i) − 1 -place) which is not an Al(i)−1 is either a Cl(i)−1, or a Bl(i)−1 ml(i)−1 ml(i) in a string (Bl(i)−1) or (Bl(i)−1) inside an l(i)-word, or a single (Bl(i)−1) at the beginning of an l(i)-word. Let Zj be the set of those x whose coordinate x0 lies in a word Al(j)−1 (in its legal l(j) − 1-place) but, for any j0 ≤ i ≤ j − 1, not in a word Al(i)−1 (in its legal l(i) − 1-place) nor in a word Cl(i)−1 (in its legal l(i) − 1-place), nor in a single (as defined above) word Bl(i)−1, nor ml(i)−1 ml(i) in the first word Bl(i)−1 of a string (Bl(i)−1) or (Bl(i)−1) . Then, as the ml(i) are large, 1 µ(Zj) > 2 if j0 is large enough. 0 Let Zj = Zj \ Zj−1. Let dl(j)−1 be defined as in the proof of Lemma 3.2 and Yj the set of 0 the x in Zj whose coordinate x0 lies in the prefix of length dl(j)−1 of its word Al(j)−1. Then 1 0 1 al(j)−1 µ(Yj) ≥ 3 µ(Zj), and µ(∪j≥j0 Yj) ≥ 6 . The Yj are disjoint; and also all the T Yj are disjoint, al(j)−1 0 as T Yj is made of x whose coordinate x0 lies in the same l(i)-words than the x in Yj for STRUCTURE OF 3-IETS III 15

j0 ≤ i ≤ j − 1, but in an l(j) − 1-word just after an Al(j)−1, and this has to be either an Al(j)−1, or a Cl(j)−1, or a single Bl(j)−1, or the ﬁrst Bl(j)−1 of a string. 0 Let γ be an eigenvalue, f its eigenfunction, fk its approximation as in Lemma 3.2, kj = l(j)−1. R We ﬁx δ, and choose j such that for all j ≥ j we have |f − f 0 | < δ and ||γb || ≤ δ. As in 0 0 kj l(j)−1 a Lemma 3.2, if x ∈ Y , f 0 (T l(j)−1 x) = x. Then j kj Z X Z X Z ||γ|| ≤ 6 |γ| = 6 |γ| < δ + 6 |γ(bl(j)−1 + 1)| = ∪ Yj Yj Yj j≥j0 j≥j0 j≥j0 X Z δ + 6 |γal(j)−1|. Yj j≥j0 And Z Z X X a |γa | = |T l(j)−1 f 0 − γa − f 0 | ≤ l(j)−1 kj l(j)−1 kj Yj Yj j≥j0 j≥j0 Z Z X a a X |T l(j)−1 f 0 − T l(j)−1 f| + |f 0 − f| = kj kj Yj Yj j≥j0 j≥j0 X Z X Z |f 0 − f| + |f 0 − f| = kj kj al(j)−1 T Yj Yj j≥j0 j≥j0 Z Z |f 0 − f| + |f 0 − f| ≤ 2δ. kj kj a ∪ l(j)−1 ∪j≥j0Y j≥j0T Yj j Hence, if the series in condition Ma diverges, T is weakly mixing.

4. CHARACTERIZATION OF THE EIGENVALUES Let T satisfy the i.d.o.c. condition. We are now ready to give a necessary and sufficient condition for a number to be an eigenvalue of T : either T has no eigenvalue, or, by Theorem 3.5, T has an explicit construction as a rank one transformation, in which case we can apply a criterion due to Choksi and Nadkarni [11], which translates into a condition on the (nk, mk, k+1) and (a0, b0, c0), to find the eigenvalues (which may still not exist). Theorem 4.1. Let T satisfy the i.d.o.c. condition and conditions Ma to Md’ with the sequence k(j). Then γ is an eigenvalue of T if and only if both the following conditions are satisfied:

• (Eγ) +∞ X pk(j)||hjγ|| < +∞, j=1

pk(j) = mk(j) ∨ nk(j), hj is the length of Ak(j)−1 if nk(j) > mk(j) and the length of Bk(j)−1 otherwise; • (Fγ) +∞ X Qj(γ) < +∞, j=1 16 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

where |Q0 | Q (γ) = 1 − k(j+1)−1 j t0(k(j + 1) − 1) with the following deﬁnitions: 0 0 • Qk(j) = 1 and t (k(j)) = 1 (hence Qj = 0 whenever k(j + 1) = k(j) + 1); • if k(j) + 2 ≤ k(j + 1), and nk(j+1) > mk(j+1) with k(j+1) = +1, or mk(j+1) > nk(j+1) with k(j+1) = −1 e−2πink(j)+1ak(j)γ − 1 0 2πik(j)+1ak(j)−1γ −2πiak(j)γ Qk(j)+1 = 1 + e e , e−2πiak(j)γ − 1 0 t (k(j) + 1) = nk(j)+1 + 1;

• if k(j) + 2 ≤ k(j + 1), and mk(j+1) > nk(j+1) with k(j+1) = +1, or nk(j+1) > mk(j+1) with k(j+1) = −1 e−2πimk(j)+1bk(j)γ − 1 0 2πik(j)+1bk(j)−1γ −2πimk(j)+1bk(j)γ Qk(j)+1 = e e + , e−2πibk(j)γ − 1 0 t (k(j) + 1) = mk(j)+1 + 1;

• if k(j) + 1 < k ≤ k(j + 1) − 1 and nk(j+1) > mk(j+1) with k(j+1) = +1, or mk(j+1) > nk(j+1) with k(j+1) = −1, e−2πinkak−1γ − 1 0 −2πi(ak−1−ak−2)γ 0 0 −2πi(ak−1−ak−2)γ 0 Qk = e Qk−1 + Qk−1 − e Qk−2, e−2πiak−1γ − 1 0 0 0 t (k) = (nk + 1)t (k − 1) − t (k − 2);

• if k(j) + 1 < k ≤ k(j + 1) − 1 and mk(j+1) > nk(j+1) with k(j+1) = +1, or nk(j+1) > mk(j+1) with k(j+1) = −1, e−2πimkbk−1γ − 1 0 0 −2πi(mkbk1 −bk−2)γ 0 0 Qk = Qk−1 + e (Qk−1 − Qk−2), e−2πibk−1γ − 1 0 0 0 t (k) = (mk + 1)t (k − 1) − t (k − 2).

Proof. We suppose Ma to Md’ are satisﬁed. If nk(j) > mk(j), we put Hj = Ak(j)−1, hj = ak(j)−1, 0 0 Hj = Bk(j)−1. The total proportion of words Hj in every k(j)-word, and hence in Hj+1, is at most mk(j) mk(j) ≤ and the total proportion of words Ck(j)−1 in every k(j)-word, and hence in mk(j)+nk(j)−1 nk(j) 1 2mk(j) Hj+1, is at most ≤ ; the corresponding relations hold for nk(j) < mk(j), with mk(j)+nk(j)−2 nk(j) 0 Hj = Bk(j)−1, hj = bk(j)−1, Hj = Ak(j)−1, and the m and n exchanged.

The system (X, T, µ) is then of rank one as the sequence of stacks τHj generate the whole space, see for example [16] for precise deﬁnitions. The formulas in Theorem 2.2 allow us to write Hj under the form Sj,0Hj−1Sj,1Hj−1 ...Sj,t(j)−1Hj−1Sj,t(j) 0 where Sj,r is either empty or a concatenation of words Hj−1 and Ck(j−1)−1. Let sj,r be the length of Sj,r. Because of the computations above, the total proportion of words Sj,r in Hj is t(j) 1 X mk(j−1) ∧ nk(j−1) s ≤ 3 h j,r m ∨ n j r=0 k(j−1) k(j−1) STRUCTURE OF 3-IETS III 17

and is the general term of a convergent series.

We build a rank one system (X0,T 0, µ0) by cutting and stacking in the following way, see [16]: we start from a set E with the required measure, which is cut into H0 equal parts to make the first stack. To get the j-stack, we cut the j − 1-stack into t(j) columns, add sj,r spacers (that is c pieces of E with the required measure) above the r-th column, and also sj,0 spacers under the first column, and stack these columns by putting the t(j)-th above the t(j) − 1-th . . . above the first. T 0 is the transformation that sends each point in a stack, except those in the top level, to the point just above. We build a measure-theoretic isomorphic between (X, T, µ) and (X0,T 0, µ0), by sending 0 the j-th level of τHk to the j-th level of the k-th stack for T : it is consistent by construction, and is 1 Pt(j) defined almost everywhere because of the condition on sj,r. Henceforth we shall identify hj r=0 (X, T, µ) and (X0,T 0, µ0).

The eigenvalues of such a system are computed in [11]. Theorem 4 of [11] states that, provided sj,0 = 0 and sj,t(j) = 0 for all j, γ is an eigenvalue if and only if

+∞ X 1 1 − |P (e2πiγ)|2 < +∞, t(j)2 j j=1 where t(j)−1 X −vj,r Pj(z) = z r=0 and r X vj,r = rhj−1 + sj,t(j)−r0 . r0=1 We shall now translate this criterion in our setting. First, as indicated in [11], we can change the system so that sj,t(j) = 0 for all j: in the construction, we replace sj,t(j) by 0, and, for 1 ≤ r < t(j), 0 Pj−1 sj,r by sj,r = sj,r + j0=1 sj0,t(j0) (sj,0 is not changed). The new system is still, by the same trick as Pj−1 above, measure-theoretically isomorphic to T , and has j-stacks of height hj = hj − j0=1 sj0,t(j0). 0 We check then that Choksi-Nadkarni’s criterion written with the sj,r and hj is exactly the same as the one written with the sj,r and hj, as the t(j) and the vj,r are not changed, and hence we can just suppress the condition sj,t(j) = 0. Similarly, we can suppress the condition sj,0 = 0 by changing 0 0 Pj−1 the sj,r to sj,r + j0=1 sj0,0 for 1 ≤ r ≤ t(j) − 1, and keep the same formula. Suppose for example Hj and Hj+1 are A-words, and k(j+1) = +1. To get our parameters, we write ﬁrst the formulas nk(j)−1 mk(j)−1 Ak(j) = Ak(j)−1 Ck(j)−1Bk(j)−1 Ak(j)−1,

nk(j)−1 mk(j)−1 Ck(j) = Ak(j)−1 Ck(j)−1Bk(j)−1 , or nk(j)−1 mk(j) Ak(j) = Ak(j)−1 Ck(j)−1Bk(j)−1,

nk(j)−1 mk(j) Ck(j) = Ak(j)−1 Ck(j)−1Bk(j)−1Ak(j)−1, 18 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

and replace in them the Bk(j)−1 and Ck(j)−1 by strings of spacers (of the same length as the word they replace). Without changing the system up to measure-theoretic isomorphism (as the pro- m portion we change is smaller than k(j) , which is the general term of a convergent series), we nk(j) replace also the word Ak(j)−1 by a string of spacers, when this word Ak(j)−1 appears at the end of a k(j)-word (this is just a cosmetic modification to make the formulas a little nicer). Then, either nk−1 k(j) = k(j + 1) − 1, or, for every k(j) < k ≤ k(j + 1) − 1, we have Ak = Ak−1 Ck−1Ak−1; Ck−1 is written above for k − 1 = k(j), while, for k − 1 > k(j) (if there are such k), Ck−1 is just Ak−1 deprived of its last word Ak−2; we use these formulas, taking into account the modifications we made to Ak(j and Ck(j) and without further modifications. Applying the definitions, we get that

−2πi(nk(j)−1)hj γ 2πiγ e − 1 0 Pj+1(e ) = Qk(j+1)−1 e−2πihj γ − 1 and 0 t(j + 1) = (nk(j) − 1)t (k(j + 1) − 1), with the claimed values of the Q0 and t0. A similar reasoning takes care of the other possibilities for Hj, Hj+1 and k(j+1) (when Hj is a B-word, we replace by spacers the Ak(j)−1 and Ck(j)−1, mk(j) and also, for cosmetic reasons, one word Bk(j)−1 in each string (Bk(j)−1) appearing in a k(j)- word).

Let Qj be as claimed and

−2πi(n −1)hj γ 1 e k(j) − 1 Rj = 1 − −2πih γ . nk(j) − 1 e j − 1 2 2 P+∞ If γ is an eigenvalue, then 1 − (1 − Qj) (1 − Rj) = zj where j=1 zj < +∞. As both Qj and Rj are real numbers between zero and one, this implies Qj and Rj are close to zero, and zj is −2πihj γ −2πi(n −1)hj γ equivalent to 2Rj +2Qj. Since Rj is close to zero, we deduce that both e and e k(j) are close to one (of course, this can also be proved directly as in Lemma 3.2 and Lemma 3.4) and ||(n −1)h γ|| that k(j) j is close to one. Hence, by developing the exponentials at order 2 we get that (nk(j)−1)||hj γ|| 1 Rj is equivalent to 2 nj||hjγ||. Hence, if γ is an eigenvalue, the two series stated in this theorem converge. Conversely, if nj||hjγ|| and Qj are close to zero, we get that Rj is close to zero and 1 2 2 equivalent to 2 (nk(j) −1)||hjγ||, and that 1−(1−Qj) (1−Rj) is equivalent to 2(Qj +Rj), hence, if the two series in this theorem converge, the series in Choksi-Nadkarni’s criterion converges and γ is an eigenvalue. Corollary 4.2. T is weakly mixing if and only if, either Theorem 3.5 is not satisfied, or Theorem 3.5 is satisfied with a sequence k(j) and, for that sequence and every γ,Eγ or Fγ is not satisfied.

Proof. Immediate. The condition F γ being a little cumbersome, we note that it implies a simpler necessary condition, which will be used to give new examples of weakly mixing T , or a simpler sufﬁcient condition, which will be used to give examples of T with nontrivial eigenvalues.

Corollary 4.3. Suppose T satisﬁes conditions Ma to Md’ with the sequence k(j); let pk = mk ∨nk; let gk be the length of Ak−1 if k = k(j) and nk(j) > mk(j), or if k(j) < k < k(j + 1) and nk(j+1) > mk(j+1) with k(j+1) = +1, or if k(j) < k < k(j + 1) and mk(j+1) > nk(j+1 with STRUCTURE OF 3-IETS III 19

k(j+1) = −1, and the length of Bk−1 otherwise.

If 0 < γ < 1 is an eigenvalue of T , then Eγ is satisﬁed and

pk||gkγ|| → 0 when k → +∞.

If 0 < γ < 1 and +∞ X pk||gkγ|| < +∞, k=1 γ is an eigenvalue of T .

Proof. The ﬁrst assertion comes from the fact that, if γ is an eigenvalue, pkγ is close to 0 modulo 1, as in the proof of Lemma 3.2 and Lemma 3.4. Conversely, if the second assertion is true, Eγ −2πi(n +2)a γ holds, as does Fγ because, after equivalence arguments on quantities such as e k k−1 −1 , we e−2πiak−1γ −1 0 Pk(j+1)−2 can bound by recursion the Qk, while Qj is bounded by 2 k=k(j)−1 pk||gkγ||.

5. EXAMPLES OF WEAKLY MIXING T In every example of weakly mixing T which we know, our criterion can be used to re-prove the weak mixing in a very quick way. And it allows us to ﬁnd new examples.

5.1. Katok - Stepin’s examples. Katok and Stepin [20] showed that if there exists c > 0 such that for every > 0, there exist integers p, q with

p A(α, β) − ≤ q q2 and for any integer r

r c B(α, β) − > , q q then T is weakly mixing. Now, by assertion 6 of Proposition 2.1, and by the computations at the beginning of the proof of Lemma 3.4, the weak mixing in these cases follows from Corollary 3.3.

5.2. If A(α, β) has bounded partial quotients. In the unpublished [8], it is shown that whenever A(α, β) has bounded partial quotients, T is weakly mixing. This follows from assertion 5 of Proposition 2.1 and Lemma 3.4. This category includes the examples with nondiscrete spectrum in [4], which are in fact weakly mixing.

The examples in §5.1 and §5.2 can be said to be weakly mixing a` la Chacon, i.e. the eigenvalues are killed by the occurrence of a shift of one (equivalently, a spacer) above a column of positive measure, like in Chacon’s map [9]. Because of the above examples, we have that, for every irrational α0, there exists a set M(α0) of measure one such that, if β0 is in M(α0), the transformation T such that A(α, β) = α0 and B(α, β) = β0 is weakly mixing. Consequently, for a set of (α, β) of measure one, the associated T is weakly mixing, as was known since [20]. 20 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

1 1 2λ 5.3. Del Junco’s examples. Starting with an irrational 0 < λ < 1, we put α = 3 , β = 3 − 3 1 2λ 1 if λ < 2 , β = 1 − 3 if λ > 2 . Then it is shown in [14] that the three-interval exchange transformation T is weakly mixing. 1 This can also be shown by computing the three-interval expansion of (α, β): if λ > 2 , (α, β) is in the preferred triangle D, and we check that x1 − y1 = 1, hence m1 = n1 + 1, x2 = y2, and 1 mk = nk for every k ≥ 2. The case λ < 2 splits into a countable number of subintervals according ¯ to the value of (¯α, β), and we check that in each case x1 − y1 is an integer and mk = nk for every k ≥ 2. Hence the weak mixing comes from Lemma 3.4. 3 Note that in all cases B(α, β) = 2 A(α, β), hence we are never in Katok - Stepin’s situation. 5.4. New examples not satisfying Lemma 3.4. Consider an expansion made of unbounded strings of (1, 1, +1) separated by isolated (1, 2, +1) and (2, 1, +1), each of them inﬁnitely many times. Then the weak mixing comes from the presence of an isolated spacer above a column of small measure. This can be called weak mixing a` la del Junco - Rudolph [15]; thus the examples of del Junco are, according to the continued fraction expansion of λ, weakly mixing either a` la Chacon or a` la del Junco - Rudolph.

5.5. New examples not satisfying conditions Ma to Md’. Consider an expansion with mk = 2, nk = k for every k, k+1 being arbitrary. Then the weak mixing comes from a Borel-Cantelli argument which seems completely new.

5.6. New examples not satisfying condition Eγ for any γ. Consider an expansion and (a0, b0, c0) 2 such that mk = ak−1, nk > k mk, k+1 = +1; then conditions Ma to Md’ are satisﬁed with k(j) = j, but as ak+1 = (mk+1 + nk+1)ak − ak−1 − (mk+1 − 1)ek, where ek = ak − bk = 1 as all the k+1 are +1,Eγ implies that γ = 0, and T is weakly mixing. 5.7. New examples not satisfying condition Eγ or Fγ for any γ. Consider an expansion such that a0 and c0 are even (for example, take (α, β) ∈ D), and m2k+1 = n2k+1 = 1, k+1 = +1, 2 m2k = 2, n2k = k . Then this satisﬁes Ma to Md’ with k(j) = 2j, and we have k2−1 A2k = A2k−1C2k−1B2k−1A2k−1, k2−1 k2−1 A2k+1 = A2k−1C2k−1B2k−1A2k−1C2k−1B2k−1A2k−1,

k2−1 C2k+1 = A2k−1C2k−1B2k−1, 2 a2k+1 = (2k + 2)a2k−1 − 2a2k−3 − 2. 1 Hence Eγ implies that 2γ is an integer, but as a2k is odd, γ = 2 is excluded by Fγ which implies that ||a2kγ|| → 0, hence we have weak mixing.

6. EXAMPLESOF T WITH EIGENVALUES The existence of non weakly mixing T (except in the trivial case when it does not satisfy the i.d.o.c. condition) follows from [30], see the discussion below. Here we re-prove in a few lines the results of Veech, and give examples of T (satisfying the i.d.o.c. condition) with irrational eigenvalues. The idea is that, when conditions Ma to Md’ are satisfied, the system is of rank one, and generated by the stacks (for example) τAk, and the information on the eigenvalues is given by their heights ak. Using the latitude we have in choosing the (mk, nk, k+1), while ensuring that Ma to Md’ are satisfied, we can build examples with specific properties. In Proposition 6.1 the ak are all multiples of an integer p; in Theorem 6.3 the ak are the convergents qk of some irrational γ (in STRUCTURE OF 3-IETS III 21

a very particular class, as it has very large partial quotients), and this gives, by identiﬁcation of the generating Rokhlin stacks, a measure-theoretic isomorphism between T and the rotation of angle γ; in Proposition 6.4 and Theorem 6.5 the ak (or else, alternatively ak and bk) are a subsequence of the convergents qk of some irrational γ (in wider classes, in particular the partial quotients may be periodic), and this gives γ as an eigenvalue of T .

6.1. Rational eigenvalues. Proposition 6.1. For every rational number 0 < γ < 1, there exists a T satisfying the i.d.o.c. condition, with eigenvalue γ. For every integer p, there exists a T satisfying the i.d.o.c. condition, which has a p-adic odometer (see [16] for a deﬁnition) as a factor. For every rational 0 < γ < 1, for every α0 with unbounded partial quotients, there exists a continuum K(α0, γ) such that whenever β0 ∈ K, the transformation T such that A(α, β) = α0 and B(α, β) = β0 satisﬁes the i.d.o.c. condition and has γ as an eigenvalue.

1 Proof. It is enough to prove the first assertion for γ = p with p an integer. We define a transforma- 2 tion T such that for all k mk > 1 and nk > k mk ; hence Theorem 3.5 is satisfied with k(j) = j for all j. We choose for example (α, β) ∈ D, k+1 = +1 for all k ≥ 2. We have then

ak+1 = (mk+1 + nk+1)ak − ak−1 − mk+1 + 1

with a0 = 2, a1 = 2n1 + m1 + 1; hence we may choose m1, n1, m2, n2 so that a1 and a2 are multiples of p, and then mk+1 such that mk+1 − 1 is a multiple of p for every k ≥ 3; hence ak is a multiple of p for every k large enough. Then we can apply Theorem 4.1, or re-prove it in this particular case by building an eigen- 1 j function for the eigenvalue p : we put φk(x) = p if x lies in the rp + j-th level of τAk, for 0 ≤ j ≤ p − 1, 0 ≤ rp + j ≤ ak − 1. The hypotheses ensure that T has rank one, with an explicit way to build τAk+1 by cutting and stacking from τAk: we cut the k-stack into nk+1 equal columns, stack these columns by putting the nk+1 − 1-th above the nk+1 − 2-th . . . above the ﬁrst, add mk+1ak −ak−1 −mk+1 +1 spacers above the nk+1 −1-th column, and the nk+1-th column above P+∞ mk 2 the spacers. This and the fact that < +∞ ensure that the φk converge in L (X, / ) to k=1 nk R Z 1 a function φ, which satisﬁes T φ = p + φ.

To get a p-adic odometer as a factor, we do the same construction, but choose the mk and nk r such that, for each r, ak is a multiple of p for k larger than some k(r).

1 0 1 We prove the third assertion for 3 ≤ α ≤ 2 ; we expand it in the form 1 α0 = 1 2 + r − 2 1 r − 3 2 ...

with ri ≥ 2 and i+1 = ±1; there are many ways to do this, as at each stage we have two choices, one with ri = r and i = −1 and one with ri = r + 1 and i = +1 (except if r = 1, where the only 22 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

possible choice is ri = 2, i+1 = +1). We ﬁx some integers zk such that +∞ X 1 < +∞ zk k=1

and expand A(α, β) with the following choices: from the beginning, we choose i+1 = +1 until, by this process, we get a string of ri = 2, i+1 = +1 of length at least z1; then (if this happens) we replace the choice just before this string by i+1 = −1, and this allows us to replace the string by an ri ≥ z1, i+1 = +1 (see [18]). Then we continue the choice i+1 = +1 until we get a string of ri = 2, i+1 = +1 of length at least z2, make the same modifications if this happens and continue in the same way. Now, any expansion of α0, which has unbounded partial quotients, must have either unbounded ri, or unbounded strings of ri = 2, i = +1, by the same proof as assertion 5 of Proposition 2.1; it follows that in our expansion there is a sequence k(j) such that rk(j) ≥ zj, and k+1 is always +1 except maybe for k = k(j) − 1. Let the k+1 be as above; we choose now, for each k(j), integers mk(j) ≥ 2 and nk(j) ≥ 2 such nk(j) 1 mk(j) 1 that rk(j) = mk(j) +nk(j) and either ≥ zj or ≥ zj, in such a way that mk(j+1) −nk(j+1) mk(j) 2 nk(j) 2 has the same sign as mk(j) −nk(j) if k(j+1) = +1, the opposite sign otherwise ; for k(j)+1 ≤ k ≤ k(j +1)−1 we put either mk = 1, nk = rk −1, or mk = 1, nk = rk −1, in order to satisfy Mc and Mc’. And we can choose the values of mk(j) and nk(j) to ensure that both k(j)-words used at the next stage (either ak(j) and ck(j) or bk(j) and ck(j)) are multiples of p. Hence conditions Ma to Md’ are satisfied, and the recursion formulas ensure that gk is a multiple of p for all k large enough; 1 hence Corollary 4.3 ensures that p is an eigenvalue of the corresponding T . Now, our expansion being given, we choose a corresponding (α, β) in the preferred triangle D, by assertion 3 of Proposition 2.1. Then, by construction and by assertion 4 of Proposition 2.1 it 0 satisfies A(α, β) = α . As we can choose from a continuum of sequences (nk, mk, k+1), the result is true for a continuum of β0 = B(α, β). To get the result for other values of α0, we choose one of the several possible triangles H−1D 0 compatible with the value of α , and work in this triangle: we choose a suitable (nk, mk, k+1), 0 knowing the values of a0, b0, c0 imposed by the triangle, and all these will give our β . Remarks. The same method allows us to build a T satisfying the i.d.o.c. condition, with a countable family of prescribed rational eigenvalues 1 , or with a countable product of p -adic pn n odometers as a factor. The first and third assertions of this proposition were proved by Veech for p = 2; the result is published in a different form [30]: Veech proved that for every α00 with unbounded partial quotients and β00 in a continuum K00(α00), the map taking value 1 on [0, β00) and −1 on [β00, 1) is a 00 1 coboundary for the rotation of angle α , and this implies immediately that 2 is an eigenvalue for the induced map S of this rotation on [β00, 1). The fact that S is indeed a map T satisfying the i.d.o.c. condition is mentioned first in [23], where the power S2 provides an example of non-uniquely ergodic interval exchange transformation satisfying the i.d.o.c. condition; it is also mentioned in [34], 1.8. The results of Veech were generalized to the case p ≥ 2 by Stewart [29]. A by-product of the proof allows us to distinguish between two three-interval exchange transformations with the same expansion. Proposition 6.2. There exist two three-interval exchange transformations with the same expansion (nk, mk, k+1)k≥1, but with different eigenvalues, and hence they are not measure-theoretically isomorphic. STRUCTURE OF 3-IETS III 23

2 Proof. Choose an expansion such that for every k ≥ 1 k+1 = +1, nk > k mk, mk and nk are odd. If we start from a0 = c0 = 2, b0 = 1, which happens if (α, β) is in D, we get that all the ak are 1 even, and we apply the proof of the ﬁrst assertion of Proposition 6.1 to get that 2 is an eigenvalue. −1 If we start from a0 = c0 = 3, b0 = 2, which happens if (α, β) is in F D, we get that all the a2k 1 1 are odd, and, because E 2 is not satisﬁed, 2 is not an eigenvalue.

6.2. Irrational eigenvalues.

Theorem 6.3. Let 0 < γ < 1 be an irrational number, [0; y1, y2 ...] its usual continued fraction expansion, and qk, k ≥ 1 the denominators of its convergents, given by qk+1 = yk+1qk + qk−1. If

+∞ X qk < +∞, yk+1 k=1 there exists a T , satisfying the i.d.o.c. condition, which is measure-theoretically isomorphic to the rotation of angle γ, and hence has discrete (pure point) spectrum.

Proof. We deﬁne a transformation T such that a1 = q1, a2 = q2, 2 = 3 = +1, and, for all k ≥ 2, k+2 = +1, mk+1 = qk − 2qk−1 + 1, nk+1 = yk+1 − mk+1 + 1. Then we have

ak+1 = (mk+1 + nk+1)ak − ak−1 − mk+1 + 1

and ak = qk for every k. We have P+∞ mk < +∞, hence T has rank one, and we do not change it up to measure- k=1 nk theoretic isomorphism by replacing the last word Ak in Ak+1 by a string of spacers. Hence, to get the k + 1-stack τAk+1 we cut the k-stack into nk+1 − 1 equal columns, stack these columns by putting the nk+1 − 1-th above the nk+1 − 2-th . . . above the ﬁrst, and add (mk+1 + 1)ak − ak−1 − mk+1 + 1 spacers above the nk+1 − 1-th column. For the rotation of angle γ, the standard Sturmian trajectories (see [17] for example) are concate- +∞ q nations of words A0 and C0 with C0 = A0 and A0 = (A0 )yk+1 C0 . As P k−1 < +∞, k k k+1 k k+1 k k k=1 yk+1qk we can make the following standard rank one construction: the k-stack has height qk and, to get the k + 1-stack, we cut the k-stack into yk+1 equal columns, stack these columns by putting the yk+1-th above the yk+1 − 1-th . . . above the ﬁrst, and add qk−1 spacers above the yk+1-th column. Hence if, in the construction of the rotation, we replace the last yk+1 − nk+1 + 1 = qk − 2qk−1 columns of the k-stack by strings of spacers of length qk, we get the construction of T . As P+∞ qk < +∞, the two systems are measure-theoretically isomorphic, as in the proof of k=1 yk+1 Theorem 4.1.

Remarks. Theorem 6.3 provides afﬁrmative answers to both parts of Question 1.1, see the introduction. An alternative method of answering the ﬁrst of the two questions is to build α00 and β00 such that the map taking value 1 on [0, β00) and e2πiγ on [β00, 1) is a coboundary for the rotation of angle α00. This approach was initiated by Merrill in [25] who built quasi-coboundaries (rather than coboundaries) having the required properties, and is being extended by Parreau (work in progress), to give examples of T with irrational eigenvalues. In the last proof, we could also take mk+1 = rk+1qk −2qk−1 +1 and nk+1 = yk+1 −mk+1 +rk+1 P+∞ rk+1qk for any sequence rk such that < +∞, for example any bounded sequence rk. Now, if k=1 yk+1 24 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

we take (α, β) in D, a non-standard continued fraction expansion of A(α, β) is 1 A(α, β) = 1 2 + 1 m + n − 1 1 1 m + n − 2 2 ... 0 by assertion 4 of Proposition 2.1 and mk + nk = yk + rk. We see that for a given α whose partial quotients grow fast enough, there is a continuum of numbers 0 < γ < 1 such that a T induced by the rotation of angle α0 is isomorphic to the rotation of angle γ. In particular, some of the possible γ are rationally independent of α0. All the γ and α0 are Liouville numbers, and hence transcendental.

Proposition 6.4. Let 0 < γ < 1 be an irrational number, [0; y1, y2 ...] its usual continued fraction expansion, qk, k ≥ 1 the denominators of its convergents. If there exists a sequence of numbers zj and a strictly increasing sequence of integers w(j) such that • P+∞ 1 < +∞, j=1 zj 2 • yw(j)+1 ≥ zj , 2 yw(j)+1 2 • zjq ≤ qw(j+1) ≤ q , w(j) zj w(j) there exists a T satisfying the i.d.o.c. condition, which has γ as an eigenvalue. Proof. We deﬁne a transformation T satisfying the conditions in Theorem 3.5 with k(j) = j, and aj = qw(j). We choose all the k+1 to be +1. We must ﬁnd mj+1 and nj+1 such that

qw(j+1) = (mj+1 + nj+1)qw(j) − qw(j−1) − mj+1 + 1.

We write qw(j+1) = Nqw(j) + l, 0 ≤ l ≤ qw(j) − 1, and take mj+1 = 2qw(j) − qw(j−1) − l + 1 and nj+1 = N − mj+1 + 2. Then mj+1 is between qw(j) − qw(j)−1 and 2qw(j); hence mj+1 ≥ 1 and zj the lower bound on qw(j+1) ensures that nj+1 > 2 mj+1. The upper bound on qw(j+1) ensures that qw(j+1) yw(j)+1 nj+1 1 nj+1 is at most ≤ qw(j) and hence is at most . qw(j) zj qw(j)+1 zj Hence we have +∞ X mj+1 < +∞ n j=1 j+1 and +∞ X nj+1 < +∞, q j=1 w(j)+1 while 1 ||qw(j)γ|| ≤ . qw(j)+1 We can then apply Theorem 4.1, or re-prove it in this particular case by building an eigenfunction for the eigenvalue γ: we put φk(x) = pγ if x lies in the p-th level of τAk, for 0 ≤ p ≤ ak − 1. The hypotheses ensure that T has rank one, with an explicit way to build τAk+1 by cutting and stacking 2 from τAk as in the proof of Proposition 6.1, and this ensures that the φk converge in L (X, R/Z) to a function φ, which satisﬁes T φ = γ + φ. STRUCTURE OF 3-IETS III 25

Remarks. The same method also works if we put aj = pjqw(j), where pj is an integer which is small compared to qw(j). In this way, we can build a T having both an irrational eigenvalue (hence an irrational rotation as a factor) and an odometer (or a product of odometers) as a factor. The possible γ we get under the hypothesis of Proposition 6.4 have unbounded partial quotients (with a subsequence going to infinity quickly, but also possibly with bounded infinite subse- quences), although not every γ with unbounded partial quotients satisfies this hypothesis. Theorem 6.5. For every quadratic irrational number 0 < γ < 1 there exists a T satisfying the i.d.o.c. condition, with eigenvalue γ.

Proof. We deﬁne a transformation T satisfying Theorem 3.5 with k(j) = j, and hj = qw(j), where 0 0 hj = aj and hj = bj if mj+1 < nj+1, hj = bj and hj = aj if mj+1 > nj+1. We choose all the k+1 to be −1; then we have 0 0 hj+1 = (pj+1 + pj+1)hj + hj−1 − pj+1ej, 0 0 where ej = hj − hj = ±1, pj+1 + pj+1 = mj+1 + nj+1, and, depending on which words have 0 length hj and hj+1, pj+1 is either mj+1 ∧ nj+1 or (mj+1 ∧ nj+1) − 1. Let yk and qk be the partial quotients and the denominators of the convergents of γ, and t be the period of the sequence (yk). For every k large enough, we check that t−1 qkt+t = Rqkt + (−1) qkt−t, R being a ﬁxed positive integer. We have then, if t is even,

qkt+lt ≡ −qkt−lt mod qkt for 0 ≤ l ≤ k − 1, and, if t is odd, l+1 qkt+lt ≡ (−1) qkt−lt mod qkt 9 for 0 ≤ l ≤ k − 1. Let r = 5 ; we have always

q[kr]t ≡ xkt mod qkt

qk where |xkt| ≤ q kt . We also have that k is close to one when k is large enough, for some constant 4 cν c and some real number ν > 1. We start from w(1) large enough and deﬁne w(j) by w(j) = w0(j)t, w0(j + 1) = [rw0(j)]. We choose mj+1 and nj+1 such that ej is always 1 (this is done by alternating m > n and n > m, so that hj will be alternatively a or b, while the sign of aj −bj alternates because the j+1 are −1). We 0 0 write qw(j+1) = Nqw(j) + l, with |l| ≤ |xw(j)| and take pj+1 = qw(j−1) − l, pj+1 = N − pj+1. Then 2w(j) mj+1 ∧ nj+1 is between qw(j−1) − q w(j) and qw(j−1) + q w(j) , hence between 1 and 2q 3w(j) < ν 3 ; 4 4 5 3w(j) 9w(j) and mj+1 ∨ nj+1 is between ν 4 and ν 10 . Hence we have +∞ X mj+1 ∧ nj+1 < +∞ m ∨ n j=1 j+1 j+1 and +∞ X mj+1 ∨ nj+1 < +∞, q j=1 w(j)+1 26 S. FERENCZI, C. HOLTON, AND L.Q. ZAMBONI

while 1 ||qw(j)γ|| ≤ . qw(j)+1 We conclude as in Proposition 6.4. Remark. In the examples of Theorem 6.5, if we take (α, β) in D, the (usual) continued fraction approximation of A(α, β) has partial quotients tending to inﬁnity (they are ultimately mk + nk as k+1 = −1), while γ has periodic partial quotients, and for example may be the golden ratio number.

Open questions • Can any irrational number be an eigenvalue of a transformation T satisfying the i.d.o.c. condition? • Can a transformation T satisfying the i.d.o.c. condition have two rationally independent irrational eigenvalues? • Can a transformation T satisfying the i.d.o.c. condition be measure-theoretically isomorphic to the Cartesian product of two irrational rotations?

REFERENCES [1] P. ARNOUX: Un exemple de semi–conjugaison entre un echange´ d’intervalles et une translation sur le tore, (in French) Bull. Soc. Math. France 116 (1988), p. 489–500. [2] P. ARNOUX, A.M. FISCHER: The scenery flow for geometric structures on the torus: the linear setting, Chinese Ann. Math. Ser. B 22 (2001), p. 427–470. [3] P. ARNOUX, G. RAUZY: Representation´ geom´ etrique´ de suites de complexite´ 2n + 1, Bull. Soc. Math. France 119 (1991), p. 199–215. [4] V. BERTHE,´ N. CHEKHOVA, S. FERENCZI: Covering numbers: arithmetics and dynamics for rotations and interval exchanges, J. Analyse Math. 79 (1999), p. 1–31. [5] M. BOSHERNITZAN: A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J. 52 (1985), p. 723–752. [6] M. BOSHE RNITZAN: Rank two interval exchange maps, Ergodic Th. Dyn. Sys.8 (1988), p. 379–394. [7] M. BOSHERNITZAN, C.R. CARROLL: An extension of Lagrange’s Theorem to interval exchange transformations over quadratic fields, J. Analyse Math. 72 (1997), p. 21–44. [8] M. BOSHERNITZAN, A. NOGUEIRA: Mixing properties of interval exchange transformations, in preparation. [9] R.V. CHACON: A geometric construction of measure–preserving transformations, Proc. of the Fifth Berkeley Symposium on Mathematical Statistics and Probability , Univ. of California Press (1965), p. 335–360. [10] J. CHAVES, A. NOGUEIRA: Spectral properties of interval exchange maps, Monatsh. Math. 134 (2001), p. 89–102. [11] J.R. CHOKSI, M.G. NADKARNI: The group of eigenvalues of a rank–one transformation, Canad. Math. Bull. 38 (1995), p. 42–54. [12] I.P CORNFELD, S.V. FOMIN, Ya.G. SINAI : Ergodic Theory, Springer–Verlag (1982). [13] E.M. COVEN, G.A. HEDLUND: Sequences with minimal block growth, Math. Systems Theory 7 (1972), p. 138–153. [14] A. del JUNCO: A family of counterexamples in ergodic theory, Israel˝ J. Math. 44 (1983), p. 160–188. [15] A. del JUNCO, D.J. RUDOLPH: A rank–one, rigid, simple, prime map, Ergodic Th. Dyn. Syst. 7 (1987), p. 229–247. [16] S. FERENCZI: Systems of finite rank, Colloq. Math. 73 (1997), p. 35–65. [17] S. FERENCZI, C. MAUDUIT: Transcendence of numbers with a low complexity expansion, J. Number Theory 67 (1997), p. 146–161. [18] S. FERENCZI, C. HOLTON, L. ZAMBONI: The structure of three–interval exchange transformations I: an arithmetic study, Ann. Inst. Fourier (Grenoble) 51 (2001), p. 861–901. STRUCTURE OF 3-IETS III 27

[19] S. FERENCZI, C. HOLTON, L. ZAMBONI: The structure of three–interval exchange transformations II: a combinatorial description of the trajectories, J. Anal. Math. 89 (2003), p. 239–276. [20] A.B. KATOK, A.M. STEPIN: Approximations in ergodic theory, Usp. Math. Nauk. 22 (1967), p. 81–106 (in Russian), translated in Russian Math. Surveys 22 (1967), p. 76–102. [21] M.S. KEANE: Interval exchange transformations, Math. Zeitsch. 141 (1975), p. 25–31. [22] M.S. KEANE: Non–ergodic interval exchange transformations, Israel˝ J. Math. 26 (1977), p. 188–196. [23] H. KEYNES, D. NEWTON: A “minimal” non–uniquely ergodic interval exchange transformation, Math. Zeitsch. 148 (1976), p. 101–106. [24] H. MASUR: Interval exchange transformations and measured foliations, Ann. of Math. 115 (1982), p. 169–200. [25] K.D. MERRILL: Cohomology of step functions under irrational rotations, Israel˝ J. Math. 52 (1985), p. 320–340. [26] M. MORSE, G.A HEDLUND: Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), p. 1–42. [27] A. NOGUEIRA, D. RUDOLPH: Topological weak–mixing of interval exchange maps, Ergodic Th. Dyn. Syst. 17 (1997), p. 1183–1209. [28] G. RAUZY: Echanges´ d’intervalles et transformations induites, (in French), Acta Arith. 34 (1979), p. 315–328. [29] M. STEWART: Irregularities of uniform distribution, Acta Math. Acad. Sci. Hungar. 37 (1981), p. 185–221. [30] W.A. VEECH: Strict ergodicity in zero dimensional dynamical systems and the Kronecker–Weyl theorem mod 2, Trans. Amer. Math. Soc. 140 (1969), p. 1–33. [31] W.A. VEECH: Interval exchange transformations, J. Analyse Math. 33 (1978), p. 222–272. [32] W.A. VEECH: Projective Swiss cheeses and uniquely ergodic interval exchange transformation, Progr. Math. 10 (1981), Birkhauser,˝ Boston , p. 113–193. [33] W.A. VEECH: Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. 115 (1982), p. 201–242. [34] W.A. VEECH: The metric theory of interval exchange transformations I: Generic spectral properties, Amer. J. Math. 106 (1984), p. 1331–1359. [35] W.A. VEECH: The metric theory of interval exchange transformations II: Approximation by primitive interval exchanges, Amer. J. Math. 106 (1984), p. 1361–1387. [36] W.A. VEECH: The metric theory of interval exchange transformations III: The Sah-Arnoux-Fathi invariant, Amer. J. Math. 106 (1984), p. 1389–1422.

INSTITUTDE MATHEMATIQUES´ DE LUMINY, CNRS, UPR 9016, 163 AV. DE LUMINY, F13288 MARSEILLE CEDEX 9(FRANCE), AND FED´ ERATION´ DE RECHERCHEDES UNITESDE´ MATHEMATIQUES´ DE MARSEILLE, CNRS - FR 2291 E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS RLM 8.100, UNIVERSITYOF TEXAS AT AUSTIN, 26THAND SPEEDWAY, AUSTIN, TX 78712-1082 (USA) E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS,UNIVERSITYOF NORTH TEXAS,DENTON, TX 76203-5116 (USA) E-mail address: [email protected]