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 dio- phantine 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 combi- natorial 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 spec- tral 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 ro- tation. 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 transfor- mations 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 (un- published), 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.