F Éd

Ération De

Institut Camille Jordan

Keywords: Mathematics Subject Classification, Primary 37A05; Secondary 37A25, 37B10, 1

EIGENVALUES AND SIMPLICITY OF INTERVAL EXCHANGE TRANSFORMATIONS

S ÉBASTIEN FERENCZI AND LUCA Q. ZAMBONI ABSTRACT. For a class of d-interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first nontrivial examples with eigenvalues (rational or irrational), the first ever example of an exchange on more than three intervals satisfying Veech's simplicity (though this weakening of the notion of minimal self-joinings was designed in 1982 to be satisfied by interval exchange transformations), and an unexpected example which is non uniquely ergodic, weakly mixing for one invariant ergodic measure but has rational eigenvalues for the other invariant ergodic measure.

PRELIMINARIES

Interval exchange transformations have been introduced by Oseledec [START_REF] Oseledec | The spectrum of ergodic automorphisms[END_REF], following an idea of Arnold [START_REF] Arnold | Small denominators and problems of stability of motion in classical and celestial mechanics[END_REF]; an exchange of d intervals is defined by a probability vector of d lengths and a permutation on d letters; the unit interval is then partitioned according to the vector of lengths, and T exchanges the intervals according to the permutation, see Sections 1.1 and 1.2 below for all definitions. Katok and Stepin [START_REF] Katok | Approximations in ergodic theory[END_REF] used these transformations to exhibit a class of systems with simple continuous spectrum. Then Keane [START_REF] Keane | Interval exchange transformations[END_REF] defined a condition called i.d.o.c. ensuring minimality, and was the first to use the idea of induction, which was later formalized by Rauzy [START_REF] Rauzy | Échanges d'intervalles et transformations induites[END_REF], as a generalization of the continued fraction algorithm. These tools formed the basis for future studies of various ergodic and spectral properties for these dynamical systems. For general properties of interval exchange transformations, the reader can consult the courses of Viana [START_REF] Viana | Dynamics of interval exchange maps and Teichmuller flows[END_REF] and Yoccoz [START_REF] Yoccoz | Échanges d'intervalles[END_REF] [START_REF] Yoccoz | Continued fraction algorithms for interval exchange maps: an introduction[END_REF].

In this paper we study d-interval exchange transformations T , defined by a vector (α 1 , . . . α d ) of lengths and the symmetric permutation πi = d + 1i, 1 ≤ i ≤ d; we call I the set of (λ 1 , . . . , λ d ) in R +d for which T , defined by the vector ( λ 1 λ 1 +...λ d , . . . , λ d λ 1 +...λ d ), satisfies the i.d.o.c. condition; henceforth we shall consider only transformations satisfying this condition: let U, resp. M ′ , M, N , S be the subset of I for which T is uniquely ergodic, resp. topologically weakly mixing, resp. weakly mixing for at least one invariant measure, resp. not weakly mixing for at least one invariant measure, resp. simple for at least one invariant measure. A great part of the history of this area is made by the difficult results about these sets. After Keane proved m(R d+ \I) = 0 for the Lebesgue measure m on R d and the surprising result that (for d = 4) U c (for X ∈ {U, M ′ , M, N , S} we call X c its complement in I) is not empty [START_REF] Keane | Non-ergodic interval exchange transformations[END_REF], he conjectured that m(U c ) = 0. This conjecture was proved by Masur [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF] and Veech [START_REF] Veech | Gauss measures for transformations on the space of interval exchange maps[END_REF], see also Boshernitzan [6] for a combinatorial proof closer to the spirit of the present paper. Then Veech [START_REF] Veech | The metric theory of interval exchange transformations I: Generic spectral properties[END_REF] proved that m(M c ) = 0 for some permutations, not including the symmetric one for d = 4; it took quite a long time to have, for all permutations outside the rotation class, first m(M ′ c ) = 0 (Nogueira-Rudolph [START_REF] Nogueira | Topological weak-mixing of interval exchange maps[END_REF]), then at last m(M c ) = m(N ) = 0 (Avila-Forni [START_REF] Avila | Weak mixing for interval exchange maps and translation flows[END_REF]); whether m(S c ) = 0 is still an open question asked by Veech [START_REF] Veech | A criterion for a process to be prime[END_REF]; note that the result on weak mixing in [START_REF] Avila | Weak mixing for interval exchange maps and translation flows[END_REF] is valid both for one invariant measure and all invariant measures because m(U c ) = 0.

While all these extremely powerful articles establish generic results for general interval exchange transformations, here we aim to provide a detailed analysis of the dynamical behaviour of specific families of interval exchanges; more precisely, we want to address problems concerning relations between the sets defined above, nothing of which was known until recently for d > 3, except obvious relations as M ′ ⊂ M, U ∩ N ∩ M = ∅ and (U ∩ N ) ∪ (U ∩ M) = U. It was not known whether N is nonempty or even that S, which is likely to have full measure (indeed, the whole notion of simplicity has been devised for that, and Veech's question has been much investigated), is nonempty; we can also ask about the non-emptiness of some intersections such as U c ∩ M or (more difficult as these are two small sets) U c ∩ N . Another problem is to find explicit examples (in the sense that maybe the vector of lengths is not given, but it can be computed by an explicit algorithm), and not only existence theorems; very few of them were known: for d = 4, explicit elements of U c are given by Keane [START_REF] Keane | Non-ergodic interval exchange transformations[END_REF] while explicit elements of U can be deduced from the same paper, or built from substitutions, or pseudo-Anosov maps, by a classical construction; but none were known in other sets, even in the bigger ones, until, during the preparation of the present paper, Sinai and Ulcigrai [START_REF] Ya | Weak mixing in interval exchange transformations of periodic type[END_REF] found explicit elements of M, while Yoccoz [START_REF] Yoccoz | Échanges d'intervalles[END_REF] built explicit elements of U c for every k; other related results [START_REF] Hmili | Non topologically weakly mixing interval exchanges[END_REF] [START_REF] Chaika | There exists a topologically mixing IET[END_REF] were derived after preliminary versions of the present paper were circulated, see the discussion in Section 6 below. Similar questions have been addressed for the (by unanimous consent much easier) case d = 3, by Veech [START_REF] Veech | Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2[END_REF], del Junco [START_REF] Junco | A family of counterexamples in ergodic theory[END_REF], and the present authors plus Holton [START_REF] Ferenczi | The structure of three-interval exchange transformations I: an arithmetic study[END_REF][16] [START_REF] Ferenczi | The structure of three-interval exchange transformations III: ergodic and spectral properties[END_REF] [START_REF] Ferenczi | Joinings of three-interval exchange transformations[END_REF]; the methods of these papers have had to be considerably upgraded to tackle the next case, d = 4. Thus we have introduced a new notion of induction, beside the classical ones due to Rauzy [START_REF] Rauzy | Échanges d'intervalles et transformations induites[END_REF], Zorich [START_REF] Zorich | Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents[END_REF], and more recently Yoccoz ( [START_REF] Marmi | The cohomological equation for Roth-type interval exchange maps[END_REF] where a good survey of all these notions can also be found). This self-dual induction, studied in more details in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF], is a variant of the less well-known induction of da Rocha [START_REF] Lopes | Invariant measures for Gauss maps associated with interval exchange maps[END_REF] [START_REF] Cruz | A generalization of the Gauss map and some classical theorems on continued fractions[END_REF], and for d = 3 its measure-theoretic properties and self-duality are studied in [START_REF] Ferenczi | A self-dual induction for three-interval exchange transformations[END_REF]. We present it in Section 2 below, and use it in Sections 3 and 4 to build families of explicit examples of four-interval exchanges; each example is described by four families of Rokhlin towers, depending on partial quotients of our induction algorithm. After a good choice of these partial quotients, our transformation will have the required properties through a measure-theoretic isomorphism with a rank one system. Whether and why this new induction was necessary to answer the questions we addressed will be discussed at the end of Section 6 below.

What we obtain in the end is some groups of examples for d = 4: two in U ∩ M ′ ∩ M c , one having rational eigenvalues and the other being measure-theoretically isomorphic to an irrational rotation, one in U ∩ M ′ ∩ M ∩ S, and one in U c ∩ M ′ ∩ M ∩ N . We find also elements of U ∩ M which are measure-theoretically isomorphic to some of the so-called Arnoux-Rauzy systems. All the examples we produce come from expansions having (very) unbounded partial quotients in our induction algorithm. That makes our elements of M a priori different from Sinai-Ulcigrai's ones, these being obtained from periodic examples relative to a different induction algorithm; in particular, our examples are all rigid, and completely new; their existence was not unexpected, but the existence of an example with irrational eigenvalues for the simpler case d = 3 was the object of a question of [START_REF] Veech | The metric theory of interval exchange transformations I: Generic spectral properties[END_REF] which was solved only in [17] (2004); our examples prove also that Avila-Forni's result is strictly stronger than Nogueira-Rudolph's. The first example of an exchange on more than three intervals which is simple is not surprising, but this resisted the efforts of specialists during 25 years, and constitutes a first step towards Veech's open question. As for our last example, which is weakly mixing for one of the two invariant ergodic measure but has rational eigenvalues for the other, it came as a surprise even for the authors.

For generalizations (to other permutations and values of d), see Section 6 below.

Acknowledgments: the authors wish to thank J. Cassaigne, C. Mauduit, and J. Rivat for their help in arithmetics, T. Monteil for drawing some of the pictures. The second author was partially supported by grant no. 090038011 from the Icelandic Research Fund.

= d + 1 -i, 1 ≤ i ≤ d defined by T x = x + π -1 j<π -1 i α j - j<i α j .
when x is in the interval

∆ i = j<i α j , j≤i α j .
We denote by

β i , 1 ≤ i ≤ d -1, the i-th discontinuity of T -1 , namely β i = d j=d+1-i α j , while γ i is the i-th discontinuity of T , namely γ i = i j=1 α j = 1 -β d-j . Then ∆ 1 = [0, γ 1 [, ∆ i = [γ i-1 , γ i [, 2 ≤ i ≤ d -1 and ∆ d = [γ d-1 , 1[. Definition 1.2.
T satisfies the infinite distinct orbit condition (or i.d.o.c. for short) of Keane [START_REF] Keane | Interval exchange transformations[END_REF]

if the d -1 negative trajectories {T -n (γ i )} n≥0 , 1 ≤ i ≤ d -1 of the discontinuities of T are infinite disjoint sets.
The i.d.o.c. condition for T is (strictly) weaker than the total irrationality condition on the lengths, where the only rational relation between

α i , 1 ≤ i ≤ d, is d i=1 α i = 1.
As here π is primitive, the i.d.o.c. condition implies that T is minimal (every orbit is dense) [START_REF] Keane | Interval exchange transformations[END_REF].

A few notions from ergodic theory.

A general reference for this section is [START_REF] Cornfeld | Ergodic Theory[END_REF]. Definition 1.3. A system (X, T ) is uniquely ergodic if it admits only one invariant probability measure. Definition 1.4. Let (X, T, µ) be a finite measure-preserving dynamical system.

A real number 0 ≤ γ < 1 is an eigenvalue of T (denoted additively) if there exists a nonconstant f in L 2 (X, R/Z) such that f • T = f + γ in L 2 (X, R/Z); f is then an eigenfunction for the eigenvalue γ. As, following [START_REF] Cornfeld | Ergodic Theory[END_REF], we consider only non-constant eigenfunctions, γ = 0 is not an eigenvalue if T is ergodic. T is weakly mixing if it has no eigenvalue. Definition 1.5. (X, T ) is topologically weakly mixing if it has no continuous (non-constant) eigenfunction.

In the particular case of interval-exchange transformations, the topology we use here is the standard one (induced by the Lebesgue measure) on the interval [0, 1[ (though T itself is not continuous), but the proofs in the present paper work in the same way if we view T as the shift on the symbolic trajectories, equipped with the product topology on {1, ..., d} N ; the two topologies are not equivalent, and it does not seem to be known whether a continuous eigenfunction for one has to be continuous for the other. Definition 1.6. (X, T, µ) is rigid if there exists a sequence s n → ∞ such that for any measurable set A µ(T sn A∆A) → 0. Definition 1.7. In (X, T ), a (Rokhlin) tower of base F is a collection of disjoint measurable sets called levels F , T F , . . . , T h-1 F . If X is equipped with a partition P such that each level T r F is contained in one atom P w(r) , the name of the tower is the word w(0) . . . w(h -1).

We shall use also the notion of rank one, for various definitions see [START_REF] Choksi | The group of eigenvalues of a rank-one transformation[END_REF] [14] [START_REF] Ornstein | Equivalence of measure preserving transformations[END_REF]. Here we need only the definition of a particular class of rank one systems; they come equipped with a partition and an invariant measure; we use the same notation for a tower and its name, and s (for "spacers") is the name of one atom of the partition, corresponding to levels added after the initial stage: Definition 1.8. Let x k and y k be two sequences positive integers, and let the concatenation of two strings of letters v and w be denoted multiplicatively by vw, while v k is a concatenation of k times the string v.

The rank one system defined by the word H 0 and the towers

H k+1 = s y k+1 H x k+1 k s z k+1
, where, if h 0 is the length of H 0 and h k+1 = x k+1 h k +y k+1 +z k+1 the length of H k+1 , we have +∞ k=1 y k+1 +z k+1

x k+1 h k < +∞, is the system (X, T, µ) built by cutting and stacking in the following way: we start from a set E of measure ξ, which is cut into H 0 equal parts to make the first tower. To get the j + 1-tower, we cut the j-tower into x j+1 columns, stack these columns by putting the x j+1 -th above the x j+1 -1th . . . above the first, and add z j+1 spacer levels (that is, pieces of E c with equal measure ) one above the other above the top, and y j+1 spacer levels one above the other under the bottom. T is the transformation that sends each point in a tower, except those in the top level, to the point just above.

The number ξ and the common measure ρ j of the spacer levels in the j-tower are defined uniquely so that µ is a probability preserved by T , and X is partitioned so that H 0 is the name of the 0-tower, while E c is the atom named s.

A standard argument proves that Proposition 1.1. The rank one systems defined above are rigid.

The following necessary condition for any θ to be an eigenvalue of a rank one transformation was originally deduced (in [START_REF] Ferenczi | The structure of three-interval exchange transformations III: ergodic and spectral properties[END_REF]) from a condition of Choksi and Nadkarni [START_REF] Choksi | The group of eigenvalues of a rank-one transformation[END_REF]; we give it here with a new direct proof adapted from [START_REF] Cassaigne | Weak mixing and eigenvalues of Arnoux-Rauzy systems[END_REF]: Proposition 1.2. If θ is an eigenvalue for the rank one system defined above by the word H 0 and the towers H k+1 = s y k+1 H x k+1 k s z k+1 , then x k+1 ||h k θ|| → 0 when k → +∞, where || || denotes the distance to the nearest integer.

Proof

Let f be an eigenfunction for the eigenvalue θ; the σ-algebra generated by the levels of the k-tower converges to the full σ-algebra when k tends to infinity, thus for each ε > 0 there exists N(ε) such that for all k > N(ε) there exists f k , which satisfies ||ff k ||dµ < ε and is constant on each level of the k-tower.

Let j be any integer with 0 ≤ j ≤ x k +1

2

. Let τ k be the union of the levels of the k-tower between the y k + 1-th and y k + [ x k 2 ]h k-1 -th levels; by construction, for any point ω in τ k , T jh k-1 ω is in the same level of the k -1-tower as ω. Thus for µ-almost every

ω in τ k , f k (T jh k-1 ω) = f k (ω) while f (T jh k-1 ω) = θjh k-1 + f (ω); we have τ k ||f k • T jh k-1 -jθh k-1 -f k ||dµ = τ k ||jθh k-1 ||dµ = ||jθh k-1 ||µ(τ k ) and τ k ||f k • T jh k-1 -jθh k-1 -f k ||dµ ≤ τ k ||f k • T jh k-1 -f • T jh k-1 ||dµ + τ k ||f k -f ||dµ < 2ε.
As µ(τ k ) ≥ 1 3 for k large enough, the above estimates imply ||jθh k-1 || < 6ε, for any integer

0 ≤ j ≤ x k +1 2 . Thus ||jθh k-1 || < 12ε for any integer 0 ≤ j ≤ x k . Let ε < 1 40 , and suppose ||x k θh k-1 || = x k ||θh k-1 ||: let i be the smallest 0 ≤ j ≤ x k such that ||jθh k-1 || = j||θh k-1 ||, then i ≥ 2 and ||(i -1)θh k-1 || = (i -1)||θh k-1 ||, thus i||θh k-1 || = (i - 1)||θh k-1 ||+||θh k-1 || = ||(i-1)θh k-1 ||+||θh k-1 || < 18ε < 1 2 thus ||iθh k-1 || = ||(i||θh k-1 ||)|| = i||θh k-1 ||, contradiction. Thus we get x k ||θh k-1 || < 12ε.
Definition 1.9. A self-joining (of order two) of a system (X, T, µ) is any measure ν on X × X, invariant under T × T , for which both marginals are µ.

An ergodic system (X, T, µ) is simple (of order two) if any ergodic self-joining of order two ν is either the product measure µ × µ or a measure defined by ν(A × B) = µ(A ∩ U -1 B) for some measurable transformation U commuting with T .

THE SELF-DUAL INDUCTION

In the remainder of this paper (except for one example in Section 2.2), we call transformation T a symmetric d-interval exchange transformation satisfying the i.d.o.c. condition and the condition of alternate discontinuities:

β 1 < γ 1 < β 2 < γ 2 < ...β d-1 < γ d-1 .
The condition of alternate discontinuities avoids introducing a lot of particular cases in the first steps of our induction; the way it can be dispensed with is discussed in Section 6 below.

2.1. Castles and induction: definitions. Our transformation T is now fixed, on the interval [0, 1[. We consider its induced maps: an induced map of T on a set Y is the map y → T r(y) y where, for y ∈ Y , r(y) is the smallest r ≥ 1 such that T r y is in Y (when such an r exists, which will be true in all cases considered in this paper).

In classical inductions, Y is generally an interval; here we consider disjoint unions of d -1 intervals; and as for any induction, there is a canonical way to build towers; following [START_REF] Cruz | A generalization of the Gauss map and some classical theorems on continued fractions[END_REF], we say that a union of towers is a castle (the Ornstein school used the words stacks and gadgets instead of towers and castles).

Definition 2.1. Given d -1 disjoint intervals E i , 1 ≤ i ≤ d -1, let S be the induced map of T on E 1 ∪ ...E d-1 . The induction castle of the E i is the unique partition of X into levels T r I i,t , 1 ≤ i ≤ d -1, 1 ≤ t ≤ k i , 0 ≤ r ≤ h i,t -1, where • each interval E i is partitioned into k i subintervals I i,t , 1 ≤ t ≤ k i ,
• SI i,t is a subinterval of E j i,t , and on I i,t S = T h i,t .

A castle is indeed a union of Rokhlin towers, each tower being made with the levels T r I i,t , 0 ≤ r ≤ h i,t -1. Note that the k i are finite by compactness, but that in general each of the d -1 intervals could be partitioned in many subintervals; only for interval exchange transformations and the type of induction chosen shall we be able to bound these numbers.

We define now a new induction operation, as a way to associate d -

1 new intervals E ′ i to d -1 intervals E i , 1 ≤ i ≤ d -1.
It was primarily motivated by considerations from word combinatorics, the d -1 families of subintervals corresponding to the bispecial factors of the associated language, which implies that their endpoints are the points where the orbit of any discontinuity of T comes close to any discontinuity of T -1 ; this in turn implies an interesting geometric property of the natural extension of our induction, studied in [START_REF] Ferenczi | A self-dual induction for three-interval exchange transformations[END_REF] for d = 3, which prompted us to call our induction self-dual.

The process is discussed and described in full generality in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF]; we give here a self-contained and slightly different description, adapted to our present (mainly ergodic) aims: indeed, the result we use in the present paper is the explicit description of the induction castles, which appears only as a by-product in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF]. Our intervals will be built so that the induction castles have always a nice structure: namely, the intervals at the initial stage are the ∆ i , 1 ≤ i ≤ d -1, and, as we shall see in Lemma 2.2 below, their induction castle is binary:

Definition 2.2. A castle is binary if for each 1 ≤ i ≤ d -1 k i = 2 and there are exactly two j l,t , 1 ≤ l ≤ d -1, t = 1, 2 which are equal to i.
When a castle is binary, we denote by E i,m and E i,p the left and right subintervals among the two I i,t , by E i,-and E i,+ the left and right subintervals among the two SI l,t which are in E i . Also, we denote by p(i), resp. m(i), the j such that E j contains SE i,p , resp. SE i,m . Finally, we denote by l i , resp. r i , the length of E i,-, resp. E i,+ for 1 ≤ i ≤ d -1.

It seems likely that for all binary castles we have SE i,m = E m(i),+ and SE i,p = E p(i),-, but we have not been able to find a direct proof using the i.d.o.c. condition. Indeed, we do not know any example of a binary castle other than those built by our induction, or small variants of it (see also Section 2.2 below), and for them the above properties are true by construction, implying that p and m are bijections.

One of our aims is to keep all induction castles binary throughout the process; to achieve that, we use an auxiliary property, which at the initial stage is satisfied with s being the identity:

Definition 2.3. A binary castle is symmetric if it is endowed with a bijection s on {1, ..., d -1} such that s -1 = psp = msm = s
and that for all i, we have the relations

• l s(i) + r s(i) = l i + r i • l i = l ps(i) , • r i = r ms(i) .
The relations above are studied in depth in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF] where (in contrast with the present paper) they are used as the basic tool to define the induction.

Definition 2.4. A relation is called

trivial if it is l s(i) + r s(i) = l i + r i with s(i) = i, or l i = l ps(i)
with ps(i) = i, or r i = r ms(i) with ms(i) = i, non-trivial otherwise. We may have s = Id, the identity; in that case all the relations l s(i) + r s(i) = l i + r i are trivial, and the only non-trivial relations are l i = l p(i) for p(i) = i and r i = r m(i) for m(i) = i; this is what happens (for d = 4 intervals) in the first stage of the example in Section 2.2 just below, where it turns out that there are only two different non-trivial relations. It can also happen that s has a cycle of length two, as in the second stage of the example in Section 2.2; then when s(i) = i, the nontrivial relation l s(i) + r s(i) = l i + r i expresses that the intervals E i and E s(i) have the same length, but there are also non-trivial relations l i = l ps(i) or r i = r ms(i) . Indeed in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF] it is proved that in all binary symmetric castles used in the induction, there are exactly d -2 different non-trivial relations, and we shall check this for d = 4 in Lemma 3.1 below. Note that in a symmetric binary castle p and m are bijections.

Binary symmetric castles are conveniently described by the following object:

Definition 2.5. The castle graph of a binary symmetric castle is the oriented graph G whose vertices are the two-letters words is(i), 1 ≤ i ≤ d -1, and for each i there is a positive edge from is(i) to p(i)sp(i) and a negative edge from is(i) to m(i)sm(i).

The induction associates to d - [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF] for details; the choice of E i,m or E i,p is then made to ensure that β i is in E ′ i . The choices of cutting or not cutting E i are made so that the induction castle of the E ′ i remains binary symmetric, this will be the difficult part and this last property is the crucial one for the sequel. Definition 2.6. We call self-dual induction the following process: suppose

1 intervals E i containing β i , 1 ≤ i ≤ d -1, a new family of intervals E ′ i . For a given 1 ≤ i ≤ d -1, either E ′ i = E i , or E ′ i = E i,m , or E ′ i = E i,p , with the notations of Definition 2.2. When E i is cut, it is cut by the point separating E i,m and E i,p , which is indeed the first point T -s γ j , s > 0, 1 ≤ j ≤ d -1, to fall in the interior of E i , see
E i = [β i -l i , β i + r i [, 1 ≤ i ≤ d -1 are d -1 disjoint
subintervals such that their induction castle is binary symmetric and has a castle graph G with bijections p, m, s, and that for every i l ir s(i) = l s(i)r i = 0; we define the instruction ι by the sign (+ or -) of this last quantity, ιi = ιs(i) = sgn(l ir s(i) ) = sgn(l s(i)r i ); let C be the maximal union of same-sign circuits of G using only the edges starting from is(i) and of sign ιi,

1 ≤ i ≤ d -1; then we define d -1 new disjoint intervals by • if is(i) ∈ C and ιi = +, E ′ i = E i,p , • if is(i) ∈ C and ιi = -, E ′ i = E i,m , • if is(i) ∈ C, E ′ i = E i . 2.

Castles and induction: examples.

It is time now to look at castles and castle graphs in concrete situations. We look first at what happens for d = 4 intervals, at the first stage of the induction, see Lemma 2.2 below. To draw the pictures, we assume, together with the condition of alternate discontinuities, that β 1 , resp.

β 2 , β 3 , is to the left of T -1 γ 3 , resp. T -1 γ 2 , T -1 γ 1 .
Figure 1 shows the induction castle of the intervals

E 1 = ∆ 1 = [0, γ 1 [, E 2 = ∆ 2 = [γ 1 , γ 2 [, E 3 = ∆ 3 = [γ 2 , γ 3 [; it
is made of three towers, which we draw separately because we choose to forget that E 1 , E 2 and E 3 are adjacent, as it happens at this stage only; to save space, we denote by γ

(j) i the point T -j γ i .
The picture shows that the castle is binary, with

E 1,-= [0, β 1 [, E 1,+ = [β 1 , γ 1 [, E 1,m = [0, T -1 γ 3 [, E 1,p = [T -1 γ 3 , γ 1 [, [γ 3 , 1[= T E 1,p , E 2,-= [γ 1 , β 2 [,
and so on. The labels give the

4 1 0 β 1 γ (1) 3 γ 1 γ 3 1 2 γ 1 β 2 γ (1) 2 γ 2 3 γ 2 β 3 γ (1) 1 γ 3 FIGURE 1. First stage of towers 1, 2 & 3.
names of the towers: on the first tower they indicate that E 1 is a subinterval of ∆ 1 and T E 1,p is a subinterval of ∆ 4 , thus when we read them from bottom to top, we get the M 1 and P 1 of Lemma 2.4 below. We see also that S = T 2 on E 1,p , S = T everywhere else. Some more information we have not yet written is that

SE 1,m ⊂ E 3 thus m(1) = 3; indeed we have SE 1,m = E 3,+ , and similarly SE 1,p = E 1,-, SE 2,m = E 2,+ , SE 2,p = E 3,-, SE 3,m = E 1,+ , SE 3,p = E 2,-, thus p(1) = 1, m(2) = 2, p(2) = 3, m(3) = 1, p(3) = 2.
Moreover we check that the castle is indeed symmetric for s = Id: this means checking p 2 = m 2 = Id and the nine relations on lengths in Definition 2.3: five of them are trivial (l i +r i = l i +r i for i = 1, 2, 3, l 1 = l 1 , r 2 = r 2 ), the non-trivial ones are r 1 = r 3 and l 2 = l 3 , each of them appearing for two values of i.

Thus the information which was not in the picture of the castle is conveniently summarized by the castle graph on the left of Figure 2 We look now at what happens at the next stage, assuming the condition of alternate discontinuities, and the respective positions of β i and T -1 γ j from the first stage.

Applying Definition 2.6, we see that at the first stage the instruction is ι 1 = ι 2 = ι 3 = -. In the castle graph there is acircuit with the vertices 11 and 33, and aloop around the vertex 22, thus C = {11, 22, 33}, and for i = 1, 2, 3 the E i at second stage is the E i,m of first stage.

Thus we can draw the induction castle of the new E 1 , E 2 , E 3 ; to position the points, we make the extra assumption that β 1 , resp.

β 2 , β 3 , is to the left of T -2 γ 1 , resp. T -2 γ 2 , T -2 γ 3 . 3 1 0 β 1 γ (1) 3 γ (2) 1 γ (1) 1 γ 3 2 2 γ 1 β 2 γ (1) 2 γ (2) 2 γ (1) 2 γ 2 1 3 4
γ 2 β 3 γ (1) 1 γ (2) 3 γ (1) 3 γ 1 γ 3 1 FIGURE 3. Second stage of towers 1, 2 & 3.
The reader can now decipher this picture as in the previous stage. The extra information is that now m A non-symmetric binary castle can be found in Section 5.1 of [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF], for a 4-interval exchange with permutation π1 = 4, π2 = 3, π3 = 1, π4 = 2. In the initial stage, the castle of

(1) = 3, p(1) = 2, m(2) = 2, p(2) = 3, m(3) = 1, p(3) = 1,
E 1 , E 2 , E 3 is binary with p(1) = 3, m(1) = 2, p(2) = 1, m(2) = 3 p(3) = 2, m(2) = 1,
and the relations between the parameters are l i,n + r i,n = l p(i),n + r m(i),n , i = 1, 2, 3 (these hold also for the symmetric castles considered in the present paper, see the proof of Proposition 2.1 below) but they do not yield the relations of Definition 2.3, and indeed in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF] we choose the parameters so that no relation l i = l j , r i = r j or l i + r i = l j + r j holds for i = j. Thus for no choice of s can the relations in Definition 2.3 be satisfied.

With the symmetric permutation π1 = 4, π2 = 3, π3 = 2, π4 = 1, we get non-symmetric castles when we induce against the rules of Definition 2.6, for example if at the first stage above we choose the new E 1 to be the full old E 1 instead of the old E 1,m , but such castles are not binary either.

As one of the referees pointed out, for a binary castle where m and p are bijections, the reciprocal map is also a binary castle, whose combinatorics is given by m -1 and p -1 , and, for a symmetric binary castle, the reciprocal castle has the same combinatorics up to a permutation of names by an involution. This last condition, however, is in general weaker than the symmetry we define, as it is satisfied by the non-symmetric castle defined above for a non-symmetric permutation π, with the involution s(1) = 1, s(2) = 3, s(3) = 2; what are missing there are the non-trivial relations between the lengths, for example l 2 + r 2 = l 3 + r 3 . Still, it is quite possible that when the permutation π is the symmetric one every binary castle is symmetric, see the remark after Definition 2.2. We do not know whether a castle can have the same combinatorics as its reciprocal castle up to a permutation of names which is not an involution.

2.3. Castles and induction: results. The following proposition describes how the induction works, and gives conditions ensuring that it can be iterated.

Proposition 2.1. If E i , 1 ≤ i ≤ d -1, is a set of disjoint intervals such that (1) E i = [β i -l i , β i + r i [, l i > 0, r i > 0, (2
) SE i,p = E p(i),-= [β p(i) -l p(i) , β p(i) [, ( 4
) SE i,m = E m(i),+ = [β m(i) , β m(i) + r m(i) [. 5
Then we can apply the self-dual induction to the E i , and the new E ′ i satisfy ( 1) to [START_REF] Boshernitzan | A unique ergodicity of minimal symbolic flows with linear block growth[END_REF], with new parameters l ′ i , r ′ i , and bijections p ′ , m ′ , s ′ given by the following rules

• if is(i) ∈ C and ιi = +, l ′ i = l i -r s(i) = l s(i) -r i , r ′ i = r i , s ′ (i) = sp(i), p ′ (i) = p(i), m ′ (i) = mp(i), • if is(i) ∈ C and ιi = -, l ′ i = l i , r ′ i = r i -l s(i) = r s(i) -l i , s ′ (i) = sm(i), p ′ (i) = pm(i), m ′ (i) = m(i), • if is(i) ∈ C, l ′ i = l i , r ′ i = r i , s ′ (i) = s(i), p ′ (i) = p(i), m ′ (i) = m(i). Proof We know that E i,-= [β i -l i , β i [ and E i,+ = [β i , β i + r i [; the
symmetry of the castle implies the relations of Definition 2.3. We know also that E i,p is the right subinterval of E i with the same length as E p(i),-, namely

E i,p = [β i -l p(i) + r i , β i + r i [. Similarly E i,m = [β i -l i , β i -l i + r m(i) [.
This implies the train-track equalities (see [START_REF] Penner | Combinatorics of train tracks[END_REF] for example) l i + r i = l p(i) + r m(i) , which is another way of stating the above relations (the equivalence of the set of the train-track equalities and the set of relations in Definition 2.3 is shown in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF], it is not used in the present paper). This implies also that l ir s(i) = l s(i)r i = 0, as otherwise β i would be the left endpoint of E i,p , hence its image by S would be the left endpoint of E p(i) , which is impossible because of (2) and the i.d.o.c. condition.

Thus we can apply the self-dual induction, with C as in the definition. Let S ′ be the induced map of

T on E ′ 1 ∪ ...E ′ d-1 . If is(i) ∈ C,
with ιi = +: we say that E i has been cut on the left; because of the relation l p(i) = l s(i) , we have l p(i)r i > 0, and thus

β i ∈ E i,p = E ′ i . If is(i) ∈ C with ιi = -, where E i is cut on the right, we use the relation r m(i) = r s(i) to prove that β i ∈ E i,m = E ′ i . If is(i) ∈ C, E i is not cut and β i ∈ E i = E ′ i .
Thus in each case we can define l ′ i and r ′ i , they are given by the claimed expression. Moreover the endpoints of the E ′ i have the required form.

β k β j β i M j P i E k,+ E j,- E i,- E i,+ E j,m E i,m E i,p T -a γ T -b γ ′ FIGURE 4. Evolution of Rokhlin towers.
We look at the action of S ′ on E ′ i . Suppose is(i) ∈ C, with ιi = +. Then the situation is completely described in Figure 4, where j = p(i) and k = m(j), while the P i and M i are the names of the towers, to be discussed in Section 2.5 below.

As p(i)sp(i) is also in C, E p(i),-intersects both E p(i),p and E p(i),m . The new

E ′ p(i) is E p(i),p = [β p(i) -l pp(i) + r p(i) , β p(i) + r p(i) [. The image of E ′ i by S is the interval E p(i),-, which is not included in the new E ′ p(i) : it is made of X i = E p(i),m , the left subinterval of E p(i) which is not in E ′ p(i)
, and

Y i = E p(i),-∩ E p(i),p .
This creates a partition of E ′ i : the right subinterval of E ′ i with the same length as Y i , which we denote by E ′ i,p , is sent by S on Y i ⊂ E ′ p(i) , and on this interval S ′ = S. The left subinterval of E ′ i with the same length as X i , which we denote by

E ′ i,m , is sent by S on X i ⊂ E p(i) -E ′ p(i) ; then X i = E p(i),m is sent by S on E mp(i),+ ⊂ E mp(i) . As p(i)sp(i) is on the same positive circuit in C as is(i), it cannot be on a negative circuit in C, hence neither can mp(i)smp(i); hence mp(i)smp(i)
is either on a positive circuit in C, or not in C, hence either E mp(i) has been cut on the left or not cut; thus E mp(i),+ ⊂ E ′ mp(i) . Hence on E ′ i,m we have S ′ = S 2 , and

S ′ sends E ′ i,m onto a subinterval of E mp(i) .
Thus, in the new castle, E ′ i is indeed partitioned into two subintervals, and we can define p

′ (i) = p(i) and m ′ (i) = mp(i). S ′ E ′ i,p is the interval Y i , which is the left subinterval of E ′ p ′ (i) of length l ′ p ′ (i) , hence we can call it E ′ p ′ (i),-= [β p ′ (i) -l ′ p ′ (i) , β p ′ (i)
[. And, whether E mp(i) has been cut on the left or not cut,

S ′ E ′ i,m is the interval E ′ m ′ (i),+ = [β m ′ (i) , β m ′ (i) + r ′ m ′ (i) [.
A similar reasoning takes care of the case is(i) ∈ C with ιi = -, where we can define p

′ (i), m ′ (i), E ′ p ′ (i),-and E ′ m ′ (i),+ by the claimed formulas. If is(i) ∈ C, E i is not cut and S sends E ′ i,p = E i,p on E p(i),-, which is still in E ′ p(i)
as p(i)sp(i) cannot be on a positive circuit of C and hence E p(i) has not been cut on the left; similarly S sends

E ′ i,m on E m(i),+ ⊂ E ′ m(i) , thus we define p ′ (i), m ′ (i), E ′ p ′ (i),
-and E ′ m ′ (i),+ by the claimed formulas.

As p ′ and m ′ are bijections, the new induction castle is indeed binary, and our p ′ and m ′ are its defining bijections. We define now s ′ by s ′ = sp on C ∩ {ι = +}, s ′ = sm on C ∩ {ι = -}, s ′ = s on C c . It is then straightforward to check that the castle is symmetric: for example if is(i) ∈ C and ιi = +, s(i)i is also in C (as ιs(i) = ιi), hence ms(i)sms(i) cannot be on a negative circuit in C and hence r ′ ms(i) = r ms(i) , while

r ′ i = r i and m ′ s ′ (i) = mpsp(i) = ms(i); hence the relation r ′ i = r ′ m ′ s ′ (i)
is satisfied, and similarly for the other cases and relations. Thus we have proved (1) to [START_REF] Boshernitzan | A unique ergodicity of minimal symbolic flows with linear block growth[END_REF] for the E ′ i . 1) to ( 5), and their castle graph G 0 is defined by the bijections s(i

Lemma 2.2. If we put

E i = ∆ i , 1 ≤ i ≤ d -1, they satisfy (
) = i, 1 ≤ i ≤ d -1, m(i) = d -i, 1 ≤ i ≤ d -1, p(i) = d + 1 -i, 2 ≤ i ≤ d -1, p(1) = 1.

Proof

The proof consists of a simple verification, using the relative positions of the β i and γ i which are assumed in the condition of alternate discontinuities.

The graph of graphs.

As for the classical inductions, the self-dual induction is represented by paths in a graph; each vertex of this graph is not a permutation as in the case of the Rauzy induction, but a castle graph: Definition 2.7. Given a castle graph G with bijections p, m, s, an instruction on G is a map from the set of vertices of G to {-, +} d-1 such that ι • s = ι; the castle graph J ι G is the castle graph defined by the bijections p ′ , m ′ , s ′ described in Proposition 2.1.

Let G 0 be as in Lemma 2.2, let G(G 0 ) be the smallest set of castle graphs which contains G 0 and is stable by the map J ι for all possible instructions ι. The graph of graphs Γ d is the oriented graph whose vertices are the elements of G(G 0 ), with an edge labeled by ι from G to J ι (G).

If E i are intervals satisfying [START_REF] Arnold | Small denominators and problems of stability of motion in classical and celestial mechanics[END_REF] to [START_REF] Boshernitzan | A unique ergodicity of minimal symbolic flows with linear block growth[END_REF], and their castle graph is a vertex a of the graph of graphs; if we apply the self-dual induction, the castle graph of the intervals E ′ i is the vertex b such that from a to b there is an edge labeled by the instruction ι of Proposition 2.1. Definition 2.8. Let Γ be an infinite path in the graph of graphs; let G n , n ∈ N be its vertices; for each n, let ι n be the instruction labeling the edge from G n to G n+1 , let s n , p n , m n be the bijections defining the castle graph G n , let C n be the maximal union of same-sign circuits of G n using only the edges starting from is n (i) and of sign

ι n i, 1 ≤ i ≤ 3. Γ is admissible if • G 0 is as in Lemma 2.2, • if i ∈ C n , ι n+1 i = ι n i,
• for each i, ι n i = + for infinitely many n,

• for each i, ι n i =for infinitely many n.

The following theorem is proved in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF]; the proof uses elaborate combinatorial tools; in the next section we give a simpler proof for d = 4, to make the present paper self-contained.

Theorem 2.3. Every transformation T defines an admissible infinite path in the graph of graphs.

Every admissible infinite path in the graph of graphs is the path of at least one transformation T .

2.5.

Names. The self-dual induction gives a way to generate any transformation T by 2d -2 families of Rokhlin towers; when we know the path of T in the graph of graphs, we know how to build these towers recursively, or, equivalently, how to build their names for the partition of [0, 1[

into ∆ i , 1 ≤ i ≤ d.
In the initial castle, and hence in all the castles we consider, each level T r E i,p is contained in one interval ∆ w(r,i,p) , w(r, i, p) ∈ {1, ...d -1}, and the same holds if we replace p by m. Thus we can define the names of our towers as in Definition 1.7; there are 2d -2 names, we denote by P i and M i the names of the towers of bases E i,p and E i,m . Lemma 2.4. In the initial castle,

P 1 = 1d, M i = i, 1 ≤ i ≤ d -1, P i = i, 2 ≤ i ≤ d -1.
If we apply the self-dual induction to a castle with names P i and M i , the new names

P ′ i and M ′ i are given by • if is(i) ∈ C and ιi = +, P ′ i = P i , M ′ i = P i M p(i) ; • if is(i) ∈ C and ιi = -, P ′ i = M i P m(i) , M ′ i = M i ; • if is(i) ∈ C, P ′ i = P i , M ′ i = M i .

Proof

The proof can be obtained by following the steps of the proof of Proposition 2.1, adding the names M i and P i of the towers as in Figure 4.

In classical inductions, we generate T by only d families of Rokhlin towers; this is possible also for the self-dual induction, by inducing T further on one of our d -1 subintervals; but, as will be seen in Lemma 4.2, this requires the knowledge of the path in the graph of graphs some way beyond the stage we are considering, thus we shall do it only for some particular families of examples; more generally, the reasoning of Lemma 4.2 and the result in its corollary can be repeated for any given infinite path in the graph of graphs.

STRUCTURE OF SYMMETRIC 4-INTERVAL EXCHANGE TRANSFORMATIONS

Throughout the remainder of this paper, we restrict ourselves to d = 4.

= r 2 , l 1 = l 3 , III s = (123), p = (213), m = (132), with r 2 = r 3 , l 2 = l 1 , IV s = (321), p = (231), m = (321), with l 1 + r 1 = l 3 + r 3 , l 2 = l 3 , V s = (213), p = (231), m = (213), with l 1 + r 1 = l 2 + r 2 , l 1 = l 3 , VI s = (132), p = (231), m = (132), with l 2 + r 2 = l 3 + r 3 , l 2 = l 1 , VII s = (132), p = (132), m = (312), with l 2 + r 2 = l 3 + r 3 , r 1 = r 2 , VIII s = (321), p = (321), m = (312), with l 1 + r 1 = l 3 + r 3 , r 1 = r 3 ,
IX s = (213), p = (213), m = (312), with l 1 + r 1 = l 2 + r 2 , r 2 = r 3 , and whose edges, labeled by instructions (ι1, ι2, ι3), are the following from I (-, +, -) and (-, -, -) to IV , (-, +, +) and (+, +, +) to V II, (-, -, +), (+, -, -), (+, -, +), (+, +, -) to I, from II (-, -, +) and (-, -, -) to V , (+, -, +) and (+, +, +) to V III, (-, +, -), (-, +, +), (+, -, -), (+, +, -) to II, from III (+, -, -) and (-, -, -) to V I, (+, +, -) and (+, +, +) to IX, (-, -, +), (-, +, -), (-, +, +), (+, -, +) to III, from IV (+, +, +) to V , (-, +, -) and (-, -, -) to I, (+, -, +) to IV , from V (+, +, +) to V I, (-, -, +) and (-, -, -) to II, (+, +, -) to V , from V I (+, +, +) to IV , (+, -, -) and (-, -, -) to III, (-, +, +) to V I, from V II (-, -, -) to IX, (-, +, +) and (+, +, +) to I, (+, -, -) to V II, from V III (-, -, -) to V II, (+, -, +) and (+, +, +) to II, (-, +, -) to V III, from IX (-, -, -) to V III, (+, +, -) and (+, +, +) to III, (-, -, +) to IX.

Proof

The proof follows from straightforward computations, applying Definition 2.6. In each case, the knowledge of m, p and s allows us to write the set of non-trivial relations of Definition 2.4.

A simplified graph of graphs is shown in Figure 5: we have omitted the edges going from one vertex to itself, an edge +.+ denotes two edges, (+, -, +) and (+, +, +), and similarly for other edges labeled with points.

--- --- --- • -- • -- --• --• -• - -• - + • + + • + + + • + + • • + + • + + + + + + + + + + + I II III IV V V I V II V III IX FIGURE 5.
The graph of graphs.

We prove now Theorem 2.3 for d = 4. Only Proposition 3.4 and Lemma 3.3, restricted to some of the particular cases studied in the proof of the lemma, are necessary for the sequel, but we wish to give the reader the complete recipe to make his own examples.

Proposition 3.2. Every transformation T defines an admissible infinite path in the graph of graphs.

Proof

Given T , we start from E i = ∆ i and apply the self-dual induction recursively; we get an infinite path in the graph of graphs (though only admissibility will prove that it does not become stationary). The first condition of admissibility (Definition 2.8 above) is satisfied because of Lemma 2.2 and the second one because of Definition 2.6 and Proposition 2.1.

Let E i,n be the interval E i at stage n; let us prove first that whenever E i,n is cut for infinitely many n, then it is cut on the right for infinitely many n and cut on the left for infinitely many n. Indeed, by construction, the left and right endpoints of E i,n are respectively T a(n) γ b(n) and T a ′ (n) γ b ′ (n) , for integers a(n) ≤ 1 and a ′ (n) ≤ 1, and there is no point

T x γ b(n) or T x ′ γ b ′ (n) inside E i,n for a(n) ≤ x ≤ 1 and a ′ (n) ≤ x ′ ≤ 1. If E i,n is cut infinitely often, a(n) → -∞ or a ′ (n) → -∞,
and thus there exists j and c(n) → -∞ such that E i,n does not contain any T x γ j for c(n) ≤ x ≤ 1. But this contradicts minimality if E i,n is ultimately not cut to the right (resp. left).

We prove now that each E i,n is indeed cut for infinitely many n; this is done by looking precisely at the possible paths in the graph of graph. There are 27 cases to consider, we look at two of the most significant.

Suppose for some N G N is vertex I, and let us show that E 1,n will be cut at least once for n ≥ N. If ι N 1 = +, 1s N (1) = 11 is in C N , because there is a + loop around 11 in the castle graph I, and we are done for n = N.

We suppose now that ι N 1 = -. If ι N 3 = -, we are done as 11 is in C N , because of thecircuit {11, 33}. For all n ≥ N if E 1,n is never cut we have ι n 1 =and we can go only from I to V II, I to I, V II to I, or V II to V II; hence ι n 3 = + for all n ≥ N, as otherwise 1s n (1) is in C n , either because we are in I with acircuit {11, 33}, or because we are in V II with acircuit {11, 23, 32} and ι n 2 = ι n 3 because s n (2) = 3. Then, if there exists N ′ such that for all n ≥ N ′ E 3,n is not cut, we have ι n 2 =for n ≥ N ′ , otherwise 3s n (3) is in C n , as both in I and V II there is a + circuit {2s(2), 3s(3)}. Then for all n ≥ N ′ G n is vertex I (if it was vertex V II we would have ι n 2 = ι n 3), and 2s n (2) = 22 is in C n because there is aloop around 22, thus E 2,n is cut infinitely often but ultimately only on the right, which as we just proved is impossible. So E 3,n has to be cut infinitely often, hence infinitely often on the right, and, for some n > N, ι n 3 = -, contradiction.

Suppose for some N G N is vertex IV , and let us show that E 2,n will be cut at least once for n ≥ N. As there is aloop around 22 in the castle graph IV , this implies that ι n 2 = + for all n ≥ N. As there is a + circuit [START_REF] Junco | A rank-one, rigid, simple, prime map, Ergodic[END_REF][START_REF] Hmili | Non topologically weakly mixing interval exchanges[END_REF][START_REF] Ornstein | Equivalence of measure preserving transformations[END_REF] in the castle graph IV , this implies in turn that ι n 1 = ι n 3 =for all n ≥ N such that we are in IV . As E 2,n is never cut, we can only go from IV to I, from I to IV , and from I to I (but not from IV to IV ); this implies that we are in I for infinitely many n, and that ι n 3 =also for all n ≥ N such that we are in I, because of the + circuit [START_REF] Hmili | Non topologically weakly mixing interval exchanges[END_REF][START_REF] Penner | Combinatorics of train tracks[END_REF] in I. Let N ′ be an n for which we are in I; if E 3,n was never cut for n ≥ N ′ , this would mean ι n 1 = + for all n ≥ N ′ , we would stay always in I, and E 1,n would be cut infinitely often (thanks to the loop around 11 in I) but only on the left, and this is impossible. Thus E 3,n is cut for some n ≥ N ′ , thus for infinitely many n, thus E 3,n is cut on the left for infinitely many n, and this contradicts the assumption on ι n 3.

The same reasoning applies, mutatis mutandis, for any E i,N when G N is any vertex, and we have proved the last two conditions of admissibility. Definition 3.1. We say that i takes +, resp -, at stage n, if ι n i = +, resp -, and is n (i) ∈ C n .

We say that ij takes +, resp -, at stage n if i takes +, resp -, at stage n and s n (i) = j.

Lemma 3.3. Let G 0 , . . . G n , . . . be a given admissible path in the graph of graphs. For any 1 ≤ i ≤ 3, 1 ≤ j ≤ 3 and any pair of signs e ∈ {-, +}, e ′ ∈ {-, +}, there exist a positive integer t and a finite sequence 1 ≤ j s ≤ 3, 0 ≤ s ≤ t, such that • j 0 = i, e 0 = e, j t = j, (-1) t e = e ′ ,

• for all 1 ≤ s ≤ t, j s-1 j s takes (-1) s e at infinitely many stages.

Proof

The result is clearly true for paths where each ij takes + andat infinitely many stages; admissibility implies each i takes + andat infinitely many stages, but it is not always true for each ij, and we must prove the lemma individually for paths where each of the 18 possibilities does not occur.

Note that if an admissible path visits all vertices ultimately, to allow the transitions each ij has to take + andinfinitely often, and the lemma is proved. Now we look at admissible paths who do not visit all vertices. An admissible path cannot visit only one vertex ultimately, as, when we go from I to I, 3 cannot take + nor -, and similarly for the other vertices. An admissible path cannot visit ultimately only two adjacent vertices: if they are I and IV , 2 and 3 cannot take + ultimately, and all the other possibilities are similar.

We look now at a path which ultimately visits only the vertices I, IV and V II: to allow the transitions 22, 33, 23, 32 take + infinitely often, 11, 33, 13, 31 takeinfinitely often and, to ensure admissibility, 11 takes + infinitely often and 22 takesinfinitely often, and this is enough to satisfy the lemma: for example, take i = 1, e = -; then by putting j 1 = 1, j 2 = 3 we get the result for (j, e ′ ) = (1, +) and (j, e ′ ) = (3, -); by putting j 3 = j 4 = 2, we get the result for (j, e ′ ) = (2, +) and (j, e ′ ) = (2, -), while by putting j 3 = 3 we get the result for (j, e ′ ) = (3, +); and a similar computation works for other (i, e).

For a path which ultimately visits only the vertices IV , V and V I: then, to allow the transitions, 13, 22, 31, 11, 23, 32, 12, 21, 33 take + infinitely often, and, to ensure admissibility, 11, 22 and 33 takeinfinitely often, and we check this is again enough to satisfy the lemma. Let us now take a path which ultimately visits only the vertices I, IV , V , II, V III, V II, I, and always in that circular order; then, to allow the transitions, 13, 31, 22, 11, 33, 23, 32 take + infinitely often, 11, 33, 12, 21, 13, 31, 22 takeinfinitely often, and again this is enough to satisfy the lemma.

Other cases are similar to one of these or contain more possibilities.

Proposition 3.4. Every admissible infinite path in the graph of graphs is the path of at least one transformation T .

Proof The proof (in contrast with the general proof in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF] which uses word combinatorics) follows the strategy of [START_REF] Keane | Non-ergodic interval exchange transformations[END_REF]: we find the coordinates l 1,0 , r 1,0 , l 2,0 , r 2,0 , l 3,0 , r 3,0 defining our transformation by showing that some intersection of open cones is nonempty, though here we have to take its further intersection with a subspace of dimension 4 because of the nontrivial relations defined in Definition 2.4 and expressed in Lemma 3.1.

Let G 0 , . . . G n , . . . be a given admissible path. Let ι n and C n be the associated instructions and unions of same-sign circuits.

We need to find a sequence of strictly positive vectors v n = (l 1,n , r 1,n , l 2,n , r 2,n , l 3,n , r 3,n ) such that for each n,

• the coordinates of v n satisfy the two non-trivial relations corresponding to G n as stated in Lemma 3.1,

• v n+1 = U n v n ,
where the linear operator U n from R 6 to R 6 is defined by

U n (l 1 , r 1 , l 2 , r 2 , l 3 , r 3 ) = (l ′ 1 , r ′ 1 , l ′ 2 , r ′ 2 , l ′ 3 , r ′ 3 ) with • if is n (i) ∈ C n and ι n i = +, l ′ i = l i -r sn(i) , r ′ i = r i , • if is n (i) ∈ C n and ι n = -, l ′ i = l i , r ′ i = r i -l sn(i) , • if is n (i) ∈ C n , l ′ i = l i , r ′ i = r i .
A direct consequence of the formulas is that U n is invertible and the matrix of U -1 n has nonnegative entries. What we shall show now is that for any k, any n large enough, W k,n = U -1 k . . . U -1 n has a matrix whose all entries are strictly positive. We look at how v n is deduced from v n+1 ; if is m (i) ∈ C m and ι m i = +, we have r i,m = r i,m+1 and l i,m = l i,m+1 + r sm(i),m = l i,m+1 + r sm(i),m+1

as s m (i)i is in the same positive circuit in C m as is m (i). Similarly, if is m (i) ∈ C m and ι m i = -, we have l i,m = l i,m+1 and r i,m = r i,m+1 + l sm(i),m+1 , and if is m (i) ∈ C m we have l i,m = l i,m+1
and r i,m = r i,m+1 . Hence l i,m+1 appears always in the expression of l i,m , and hence in every l i,p for p ≤ m; it appears also in the expression of r j,m when ij takesat stage m, and if there exists q ≤ m such that ij has takenat stage q, it appears in every r i,p for p ≤ q.

Let k > 0 be fixed. We take i and j and two signs e and e ′ , and choose j 1 , ..., j t as in Lemma 3.3. As j s-1 j s takes (-1) s e at infinitely many stages, we can find k < k 1 < ...k t such that j s-1 j s takes (-1) s e at stage k s for all 1 ≤ s ≤ t. And if n > k t , this implies that l j,n if e ′ = -, resp. r j,n if e ′ = +, appears in the coordinate l i,k of v k if e = +, resp. r i,k if e = -. By doing the same for every choice of 1 ≤ i ≤ 3, 1 ≤ j ≤ 3, e ∈ {-, +}, e ′ ∈ {-, +} and taking n larger than all the corresponding k t , we get our assertion on W k,n .

We write now the reasoning of [START_REF] Keane | Non-ergodic interval exchange transformations[END_REF], in a little more explicit way; let

Ω = {l i > 0, r i > 0, i = 1, 2, 3} be the open positive cone in R 6 , Ω = {l i ≥ 0, r i ≥ 0, i = 1, 2, 3} its closure, K n = W 1,n Ω, K n = W 1,n Ω, K ′ n = K n \ {0}; we have K n ⊂ K ′ n ⊂ K n .
The condition on the matrices ensures that for all k and n > k, if v is in Ω with at least one strictly positive coordinate, then

W k,n v is in Ω, thus ∩ n≥1 K n = ∩ n≥1 K n \ {0} = ∩ n≥1 K ′ n .
The last part of Keane's reasoning (which will not be used here but imitated) says that each K ′ n is invariant by v → λv for any scalar λ, thus the K ′ n are decreasing compact sets in a projective space, thus their infinite intersection is non-empty; thus ∩ n≥1 K n is non-empty.

We introduce now the relations: let Ξ n be the subset of R 6 made of vectors (l 1 , r 1 , l 2 , r 2 , l 3 , r 3 ) whose coordinates satisfy the two non-trivial relations corresponding to G n in Lemma 3.1; in particular Ξ 0 = {r 1 = r 3 , l 2 = l 3 } as G 0 is vertex I. It follows from Proposition 2.1, and can also be re-checked by direct computation, that

Ξ n+1 = U n Ξ n .

Now, the above considerations imply that

Ξ 0 ∩ ∩ n≥1 K n = Ξ 0 ∩ ∩ n≥1 K ′ n .
We look at the intersections of the K ′ n with the space Ξ 0 = {r 1 = r 3 , l 2 = l 3 }: they are nonempty as, because of the expression of the relations in Lemma 3.1, each (Ω ∩ Ξ n+1 ) is non-empty, thus also its image by W 1,n , and we have

W 1,n Ω ∩ Ξ 0 = W 1,n (Ω ∩ Ξ n+1 ) = Ξ 0 ∩ K n ⊂ Ξ 0 ∩ K ′ n . Each Ξ 0 ∩ K ′
n is invariant by v → λv for any scalar λ, thus the Ξ 0 ∩ K ′ n are decreasing compact sets in a projective space, thus their infinite intersection is non-empty. Thus the infinite intersection

∩ +∞ n=1 (Ξ 0 ∩ W 1,n Ω) = ∩ +∞ n=1 W 1,n (Ω ∩ Ξ n+1
) is non-empty. A vector v 0 in this latter set is such that v n has strictly positive coordinates for all n, and satisfies the required relations for all n. After normalization by l 1,0 + r 1,0 + l 2,0 + r 2,0 + l 3,0 + r 3,0 + l 1,0 = 1, we define a symmetric 4-interval exchange transformation by α 1 = l 1,0 + r 1,0 , α 2 = l 2,0 + r 2,0 , α 3 = l 3,0 + r 3,0 , α 4 = l 1,0 , and the required inequalities on the β i and γ j are satisfied.

By construction the self-dual induction is iterated infinitely, defining the path G 0 , . . . , G n , . . . and by admissibility each E i is cut infinitely often on the left and on the right; thus the height of each tower tends to +∞; as the negative orbits of the discontinuities of T appear as the endpoints of levels in the castles, while the negative orbits of the discontinuities of T -1 appear in the interiors of these levels, the i.d.o.c. condition is satisfied.

UNIQUELY ERGODIC EXAMPLES

In this section, we define a family of examples depending on three sequences of integers m k , n k , p k , which we call the partial quotients for the self-dual induction: m k (resp n k , p k ) is the number of consecutive times when 22 (resp. 33, 11) takes -, theedge from 22 (resp. 33, 11) being a loop in the castle graph.

Definition 4.1. Given m = {m k , k ∈ N}, n = {n k , k ∈ N ⋆ }, p = {p k , k ∈ N ⋆ }, let Γ(m, n, p)
be the admissible path defined as follows, which starts from I, then makes infinitely many circuits through vertices IV , V , V I: laps are numbered from k = 0; before lap 0, we go from I to IV by (-, -, -); for all k ≥ 0, at the beginning of lap k we are in IV ; we apply instruction (+, -, +) m k times if k > 0, resp. m 0 -1 times if k = 0, staying in IV , then go to V by (+, +, +), then apply instruction (+, +, -) n k+1 times, staying in V , then go to V I by (+, +, +), then apply instruction (-, +, +) p k+1 times, staying in V I, then go to IV by (+, +, +).

All transformations T in this section are such that their path in the graph of graphs is a Γ(m, n, p).

Note that in Definition 4.1, and hence in Lemma 4.1 below, when we look at what happens between vertex IV in lap k and vertex IV in lap k + 1, we have chosen to use m k , n k+1 and p k+1 . This is intentional, because the fundamental Corollary 4.3 below, which depends on what happens between just after vertex IV in lap k and just after vertex IV in lap k + 1, will thus depend on n k+1 , p k+1 and m k+1 , and that corollary will be used extensively in the sequel. The case k = 0 is special, as 22 takeswhen we go from the initial state to vertex IV in lap 0, thus 22 has to take only m 0 -1 times in the latter situation.

Lemma 4.1. The names of towers P i (k) and M i (k) when we are in vertex IV at the beginning of lap k are given by the following rules:

• P 1 (k + 1) = (P 1 (k)M 2 (k) m k P 2 (k)M 3 (k)) p k+1 P 1 (k) • P 2 (k + 1) = M 2 (k) m k P 2 (k) • P 3 (k + 1) = (P 3 (k)M 1 (k)) n k+1 P 3 (k), • M 1 (k + 1) = (P 1 (k)M 2 (k) m k P 2 (k)M 3 (k)) p k+1 P 1 (k)M 2 (k) m k P 2 (k)P 3 (k)M 1 (k), • M 2 (k + 1) = M 2 (k) m k P 2 (k)(P 3 (k)M 1 (k)) n k+1 P 3 (k)P 1 (k)M 2 (k), • M 3 (k + 1) = (P 3 (k)M 1 (k)) n k+1 P 3 (k)P 1 (k)M 2 (k) m k P 2 (k)M 3 (k);
with m k replaced by m 0 -1 if k = 0, and initial values P 1 (0) = 13, P 2 (0) = 22, P 3 (0) = 314,

M 1 (0) = 1, M 2 (0) = 2, M 3 (0) = 3.

Proof

The proof comes from applying Lemma 2.4 at each stage.

As was announced in Section 2.5, we replace the six towers by four: Lemma 4.2. Let E 1 (k) be the interval E 1 when we are in vertex IV at the beginning of lap k; its induction castle is made of four towers, whose names are

• A k = M 1 (k)P 3 (k), • B k = P 1 (k)M 2 (k) m k P 2 (k)M 3 (k), • C k = P 1 (k)M 2 (k) m k P 2 (k)P 3 (k), • D k = P 1 (k)M 2 (k) m k +1 P 2 (k)P 3 (k), with all m k replaced by m 0 -1 if k = 0. Proof The induced map of T on E 1 (k) is an induced map of the induced map of T on E 1 (k) ∪ E 2 (k) ∪ E 3 (k)
, whose castle is vertex IV . To find the castle we want, we look at concatenations of towers starting from E 1 (k) and coming back to it, and this corresponds to paths in the castle graph IV : starting from 13, we can go to 31 by M 1 and come back to 13 by M 3 or P 3 , or else go to 22 by P 1 , make an unknown number of times the loop M 2 around 22, then go to 31 by P 2 and come back to 13 by M 3 or P 3 ; thus the possible names of our concatenations of towers are M 1 (k)P 3 (k), M 1 (k)M 3 (k), P 1 (k)M 2 (k) s P 2 (k)M 3 (k), and P 1 (k)M 2 (k) t P 2 (k)P 3 (k) for (a priori) any positive integers s and t. But the same formulas hold with k replaced by k + 1, while concatenations of towers starting from E 1 (k + 1) and coming back to it are also concatenations of the above concatenations starting from E 1 (k) and coming back to it. Taking into account the formulas of Lemma 4.1, we see that M 1 (k)M 3 (k) does not occur, and that there are only two possible values for t, t = m k and t = m k + 1, and one possible value for s, s = m k (with the usual modification for k = 0).

Corollary 4.3. The above names are given by the formulas

• A k+1 = B p k+1 k C k A n k+1 +1 k , • B k+1 = B p k+1 k C k (A n k+1 k D k ) m k+1 A n k+1 k B k , • C k+1 = B p k+1 k C k (A n k+1 k D k ) m k+1 A n k+1 k • D k+1 = B p k+1 k C k (A n k+1 k D k ) m k+1 +1 A n k+1 k . with initial values A 0 = 1314, B 0 = 132 m 0 -1 223, C 0 = 132 m 0 -1 22314, D 0 = 132 m 0 22314.
In all the sequel we denote by a k , b k , c k , d k the lengths of the names A k , B k , C k , D k ; these are also the heights of the corresponding towers, which we denote by tower A k , tower B k , tower C k , tower D k , each of these being a k-tower.

By minimality, for each ǫ, if k is large enough, the lengths of the intervals are all less than ǫ; hence any integrable function f can be approximated (in L 1 for example) by functions f k which are constant on each level of each k-tower. Thus the above formulas give a complete description of T as a system of rank at most four by intervals (the original reference on finite rank is [START_REF] Ornstein | Equivalence of measure preserving transformations[END_REF], but finite rank by intervals was not defined in print before [START_REF] Ferenczi | Systems of finite rank[END_REF]). From these formulas, T is determined up to measure-theoretic and topological isomorphisms. Now, the secret for building interesting examples is to play on our partial quotients; we shall first ensure that our system is of rank one, the tower A k being the only one which is not of very small measure (for any invariant measure µ, but this fact by itself ensures that µ is unique). Moreover, this is a rank one system as in Definition 1.8, and all its properties come from the values of a k . In Theorem 4.6 we ensure that the a k are the denominators of the convergents (for the Euclid algorithm) of an irrational θ, and thus T has θ as an eigenvalue, and even is measure-theoretically isomorphic to the irrational rotation of angle θ. In Theorem 4.7, each a k will be a multiple of an integer N, and T has 1

N as an eigenvalue. In both cases, as the tower B k is not negligible from the topological point of view, a relation between a k and b k will ensure topological weak mixing.

′ k , b ′ k such that a ′ k a k -b ′ k b k = 1, and we have n k+1 > a ′ k , p k+1 > b ′ k ;
then the transformation T is topologically weakly mixing. Proof Recall that the union of the bases of the towers A k , B k , C k , D k is the interval E 1 (k), and, by minimality, for each ǫ, if k is large enough, the lengths of the intervals are all less than ǫ. Let θ be an eigenvalue with a continuous eigenfunction f ; then, for given

ǫ, if k is large enough, |f (z) -f (y)| < ǫ (in R/Z) if z and y are in E 1 (k).
Because in the formulas of Corollary 4.3 A n k+1 k occurs in (for example) A k+1 , there exists x in the basis of the tower

A k such that T a ′ k a k x is again in E 1 (k) hence ||θa ′ k a k || = |f (T a ′ k a k x) -f (
k + c k + d k n k+1 a k < +∞,
then T is uniquely ergodic and (X, T, µ) is measure-theoretically isomorphic to the rank one system (X ′ , T ′ , µ ′ ) defined (as in Definition 1.8) by the word A 0 and the towers Let µ be any invariant probability for T : each level in a given tower has the same measure, hence the above condition ensures that the towers B k , C k , D k , have measure at most ǫ k , the k-th term in the above series, in each tower A k+1 , B k+1 , C k+1 , D k+1 , hence in the whole space, where +∞ k=0 ǫ k < +∞. The system (X, T, µ) is then of rank one by intervals as the sequence of towers A k generate the whole space, see for example [START_REF] Ferenczi | Systems of finite rank[END_REF] for precise definitions. We build a measuretheoretic isomorphism between (X, T, µ) and (X ′ , T ′ , µ ′ ), by sending the j-th level of the tower A k to the j-th level of the tower A ′ k for T ′ : it is consistent by construction, as the length of A ′ k is a k , and is defined almost everywhere because of the condition on ǫ k . The unique ergodicity is a consequence of the rank one by intervals: as is mentioned in Definition 1.8, the definition of µ ′ ensures that it is the unique invariant probability measure on (X ′ , T ′ ), and any invariant measure ν = µ on (X, T ) would define an invariant measure ν ′ = µ ′ on (X ′ , T ′ ) through the above isomorphism.

A ′ k+1 = s a k+1 -(n k+1 +1)a k (A ′ k ) n k+1 +1 . Proof Note that
Theorem 4.6. One can construct recursively sequences m, n, p such that the corresponding transformation T is uniquely ergodic, topologically weakly mixing, and measure-theoretically isomorphic to an irrational rotation on T 1 .

Proof

We build the partial quotients for the self-dual induction recursively as follows: we choose m 0 such that a 0 and b 0 are coprime, and we have b 0 > a .

At stage k, we assume a k and b k are coprime, and b k > a k ; by Bezout's identity we can find positive integers

a ′ k and b ′ k such that a ′ k a k -b ′ k b k = 1. We choose first p k+1 , such that p k+1 > b ′ k and p k+1 b k + c k ≡ a k-1 mod a k ; this is possible as b k is invertible modulo a k ; then we choose n k+1 large enough for n k+1 a k > 2 k ((p k+1 + 1)b k + c k + d k ), n k+1 > a ′
k , and such that (n k+1 + 1)a k + p k+1 b k + c k is coprime with b ka k ; this is possible as a k is invertible modulo b ka k ; finally we choose m k+1 = t k+1 a k+1 for some t k+1 ∈ N ⋆ .

As by Corollary 4.

3 b k+1 -a k+1 = b k -a k + m k+1 (n k+1 a k + d k ), we have b k+1 -a k+1 ≡ b k -a k modulo a k+1
by choice of m k+1 as in the previous equation; as b ka k is invertible modulo a k+1 , so is b k+1a k+1 , and thus a k+1 and b k+1 are again coprime, and b k+1 > a k+1 .

Our transformation T satisfies the hypothesis of Proposition 4.5, thus is uniquely ergodic and measure-theoretically isomorphic to the rank one system T ′ . T is topologically weakly mixing by Proposition 4.4.

Because of the second equation in the choice of p k+1 above a k+1 = y k+1 a k + a k-1 for positive integers y k+1 . We choose the irrational θ whose partial quotients (for the Euclid algorithm) are y 0 , y 1 , . . . so that the a k are the denominators of its convergents. For the rotation of angle θ, the standard Sturmian trajectories (see [START_REF] Ferenczi | Transcendence of numbers with a low complexity expansion[END_REF] for example) are concatenations of words A" k and C" k with C" k+1 = A" k and A" k+1 = C" k (A" k ) y k+1 . As +∞ k=1 a k-1 y k+1 a k < +∞ because the hypothesis of Proposition 4.5 is satisfied, this rotation is measure-theoretically isomorphic to the rank one system defined by the word A" 0 and the towers A" k+1 = s a k-1 (A" k ) y k+1 , by the same proof as in Proposition 4.5.

And T ′ and T " are measure-theoretically isomorphic as in the proof of Proposition 4.5, as build an isomorphism between T " and T ′ by sending some A" k to strings of spacers of length a k , on a part of the space of measure ǫ k with +∞ k=1 ǫ k < +∞.

k + c k + d k ), n k+1 > a ′
k , and such that (n k+1 + 1)a k + p k+1 b k + c k is coprime with b ka k ; this is possible as a k is invertible modulo b ka k ; finally we choose m k+1 = t k+1 a k+1 for some t k+1 ∈ N ⋆ , hence a k+1 and b k+1 are again coprime, and b k+1 > a k+1 , while a k+1 is a multiple of N.

At the initial stage, if N = 2 or N = 4 we can choose m 0 such that b 0 is coprime with a 0 , and our assumptions are satisfied at stage 0, so we begin the above process at k = 0. Otherwise, our assumptions will be satisfied at stage 1, in the following way: we choose m 0 such that m 0 and m 0 + 4 are both coprime with 4N (let 4N = s i=0 π α i i be the decomposition of 4N into prime factors, with π 0 = 2 < π 1 < ...; for 0 ≤ i ≤ s, let Ψ i be the set of 0 < m < 4N such that m and m + 4 are coprime with π 0 , ..., π i : we have #Ψ 0 = 2N, and, by the Chinese remainder theorem,

#Ψ i+1 = #Ψ i (1 -1 2π i+1 ), thus #Ψ s = s i=0 π α i -1 i s i=1 (π i -2) > 0,
and any m 0 in Ψ s is convenient). Thus m 0 is coprime with N and with 4 and m 0 + 4 is coprime with m 0 N, and for any n 1 and any element x of Z/m 0 NZ we can find p 1 such that a 1 = (m 0 + 4)p 1 + 4n 1 + m 0 + 10 ≡ x mod Nm 0 . Hence we choose any n 1 , and then p 1 such that a 1 is a multiple of N and coprime with m 0 = b 0a 0 , then, with m 1 = t 1 a 1 for some t 1 ∈ N ⋆ , we get that a 1 and b 1 are coprime.

Our transformation T satisfies the hypothesis of Proposition 4.5, thus is uniquely ergodic and measure-theoretically isomorphic to the rank one system T ′ in Proposition 4.5. T is topologically weakly mixing by Proposition 4.4.

On (X ′ , T ′ ), for k ≥ 1 we put φ k (x) = j N if x lies in the pN + j-th level of the tower A ′ k , for integers 0 ≤ p ≤ a k N -1, 0 ≤ j ≤ N -1. Because a k is a multiple of N, this is consistent and the φ k converge in L 2 (X, R/Z) to a function φ, which satisfies T ′ φ = 1 N + φ. Thus T ′ and T have the required eigenvalue.

We can also build such a transformation T with both rational and irrational eigenvalues, by building a θ such that the a k are the denominators of its convergents, multiplied by N.

We turn now to weakly mixing examples; the first one imitates the famous rank one system of del Junco-Rudolph [START_REF] Junco | A rank-one, rigid, simple, prime map, Ergodic[END_REF] by ensuring a recurrence relation a k+1 = y k+1 a k + 1. Theorem 4.8. One can construct recursively sequences m, n, p such that the corresponding transformation T is uniquely ergodic, weakly mixing, and simple (of order two).

Proof

We build the partial quotients for the self-dual induction recursively as follows: we choose m 0 such that a 0 and b 0 are coprime, and we have b 0 > a 0 .

At stage k, we assume a k and b k are coprime, and b k > a k ; we choose p k+1 such that

p k+1 b k + c k ≡ 1 mod a k ;
this is possible as b k is invertible modulo a k ; then we choose n k+1 large enough for n k+1 a k > 2 k ((p k+1 + 1)b k + c k + d k ), and such that (n k+1 + 1)a k + p k+1 b k + c k is coprime with b ka k , and m k+1 = t k+1 a k+1 for some t k+1 ∈ N ⋆ . Thus a k+1 and b k+1 are again coprime, and b k+1 > a k+1 . Our transformation T satisfies the hypothesis of Proposition 4.5, thus is uniquely ergodic and measure-theoretically isomorphic to the rank one system T ′ .

By construction a k+1 = y k+1 a k + 1 for positive integers y k+1 > 2 k+1 . Thus T ′ is measuretheoretically isomorphic to the rank one system T " defined by the word A 0 and the towers A" k+1 = s(A" k ) y k+1 , as we build an isomorphism between T " and T ′ by sending some A" k to strings of spacers of length a k , on a part of the space of measure ǫ k with +∞ k=1 ǫ k < +∞. This last system is weakly mixing and simple exactly in the same way as del Junco -Rudolph's map [START_REF] Junco | A rank-one, rigid, simple, prime map, Ergodic[END_REF], which is the rank one system defined by some H 0 and the towers H k+1 = H 2 k k sH 2 k k (this defines a transformation by an appropriate modification of Definition 1.8); the main (and quite involved) argument in Theorem 1 of [START_REF] Junco | A rank-one, rigid, simple, prime map, Ergodic[END_REF] uses only the fact that there are isolated spacers between long concatenations of the same tower.

Note that we deduce from [START_REF] Junco | A rank-one, rigid, simple, prime map, Ergodic[END_REF] that this system is also prime (it has no nontrivial invariant sub-σ -algebra) and rigid.

Of course, as most transformations T are weakly mixing, we may expect to find many more examples with this property. Indeed, we can build a lot of them by adapting to the family of transformations T in the present section the method described in the proof of Theorem 5.5 below.

Another unexpected way is to use the so-called Arnoux-Rauzy systems [START_REF] Arnoux | Représentation géométrique de suites de complexité 2n + 1[END_REF]. These are symbolic systems defined by three names X k , Y k , Z k , build recursively by using a sequence of combinatorial rules; by rule 1,

X k+1 = X k , Y k+1 = Y k X k , Z k+1 = Z k X k ; by rule 2, X k+1 = X k Y k , Y k+1 = Y k , Z k+1 = Z k Y k ; by rule 3, X k+1 = X k Z k , Y k+1 = Y k Z k , Z k+1 = Z k . At the beginning, X 0 = 1, Y 0 = 12, Z 0 = 13.
Here we restrict ourselves to a paréar class of Arnoux-Rauzy systems, built by applying successively rule 1 q 3l+1 times, rule 2 q 3l+2 times, rule 3 q 3l+3 times, then rule 1 q 3l+4 times and so on, starting from l = 0; this gives a uniquely ergodic (by Boshernitzan's result using complexity [START_REF] Boshernitzan | A unique ergodicity of minimal symbolic flows with linear block growth[END_REF]) system (Y, S), and, when the q k grow to infinity fast enough, as a straightforward consequence of the definition, this system is measure-theoretically isomorphic to a rank one system defined by the word H 0 and the towers H k+1 = s t k (H k ) q k+1 , where, for k = 3l+1 (resp. k = 3l+2, k = 3l + 3) H k has name Y q 1 +...q k (resp. Z, X), and t k is the length of Z q 1 +...q k (resp. X, Y ). These systems are proved to be weakly mixing in [START_REF] Cassaigne | Weak mixing and eigenvalues of Arnoux-Rauzy systems[END_REF]. Proposition 4.9. One can construct recursively sequences m, n, p such that the corresponding transformation T is uniquely ergodic, weakly mixing, and measure-theoretically isomorphic to an Arnoux-Rauzy system.

Proof

We build simultaneously m k , n k , p k defining our transformation T and q k defining our Arnoux-Rauzy system.

At each stage, a k and b k are coprime, b k > a k , and a k = h k , h k being the length of H k . At the beginning, we choose the first parameters so that the assumptions are satisfied at stage 1. At stage k choose first p k+1 , such that, if t k is defined above from q 1 , ... ,q k and the rules defining an Arnoux-Rauzy system, as the length of Z q 1 +...q k , resp. X, Y according to the class of k modulo 3,

p k+1 b k + c k ≡ t k mod a k ;
then we choose n k+1 large enough for satisfying the hypothesis of Proposition 4.5 and such that (n k+1 + 1)a k + p k+1 b k + c k is coprime with b ka k , then m k+1 = u k+1 a k+1 , for a positive integer u k+1 , so that a k+1 and b k+1 are again coprime, and b k+1 > a k+1 . Then we choose q k+1 so that h k+1 = a k+1 . We conclude as in the proof of Theorem 4.6.

Note that all the examples in this section are rigid by Proposition 1.1.

NON UNIQUELY ERGODIC EXAMPLES

Definition 5.1. Given m = {m k , k ∈ N}, n = {n k , k ∈ N ⋆ }, with n k+1 > m k > n k , let Γ(m, n)
be the admissible path defined as follows, which starts from I and then follows infinitely many times a path

IV -I -V II -I -IV : let f 0 = n 1 , e k = m k -f k-1 and f k = n k+1 -e k for k ≥ 1, thus e k > 0 and f k > 0.
At the beginning of step k we are in IV ; we go to I by (-, -, -), then apply instruction (+, -, +) e k -1 times, staying in I, then then go to V II by (+, +, +), then go to I by (+, +, +), apply instruction (+, -, -) f k -1 times, staying in I, then go back to IV by (-, -, -). Before step 1, starting from I we apply instruction (+, -, -) f 0 -1 times, staying in I, then go to IV by (-, -, -).

All transformations T in this section are such that their path in the graph of graphs is a Γ(m, n).

Indeed, in this definition m k is the number of consecutive times when 22 takes -, theedge from 22 being a loop in the castle graph; 22 does takewhen we are in I or IV but not when we are in V II, so m k counts also the number of times we are consecutively in I, IV , and I again, between two passages in V II. Similarly n k is the number of consecutive times when 11 takes +, the + edge from 11 being a loop in the castle graph, and that happens when we are in I or V II.

The e k and f k can be seen as auxiliary quantities with m k = e k + f k-1 and n k+1 = e k + f k ; the indexing has been chosen so that Lemma 5.1 depends on e k , f k , and Corollary 5.3 will thus depend on n k+1 and m k+1 , and only that corollary will be used in the sequel.

In the same way as in the previous section we prove Lemma 5.1. The names of towers P i (k) and M i (k) when we are in vertex IV at the beginning of step k are given by the following rules:

•

P 1 (k + 1) = (M 1 (k)P 3 (k)) e k +f k M 1 (k)M 3 (k)P 1 (k), • P 2 (k + 1) = (M 2 (k) e k P 2 (k)M 3 (k)P 1 (k)M 2 (k)) f k M 2 (k) e k P 2 (k), • P 3 (k + 1) = M 3 (k)P 1 (k)M 2 (k) e k P 2 (k)M 3 (k)M 1 (k)P 3 (k), • M 1 (k + 1) = (M 1 (k)P 3 (k)) e k +f k M 1 , • M 2 (k + 1) = M 2 (k) e k P 2 (k)M 3 (k)P 1 (k)M 2 (k), • M 3 (k + 1) = M 3 (k)P 1 (k)M 2 (k) e k P 2 (k)M 3 (k); with initial values P 1 (0) = (14) f 0 -1 13, P 2 (0) = 2 f 0 +1 , P 3 (0) = 314, M 1 (0) = (14) f 0 -1 1, M 2 (0) = 2, M 3 (0) = 3.
Note that P 2 (k + 1) does indeed contain M 2 (k + 1), and even M 2 (k + 1) f k , as a strict prefix, as the last instruction is (---) from I, and the instruction for 2 has beenf k times. Lemma 5.2. Let E 1 (k) be the interval E 1 when we are in vertex IV at the beginning of step k; its induction castle is made of four towers, whose names are

• A k = M 1 (k)P 3 (k), • B k = M 1 (k)M 3 (k), • C k = P 1 (k)M 2 (k) e k P 2 (k)M 3 (k), • D k = P 1 (k)M 2 (k) e k +1 P 2 (k)M 3 (k).
Corollary 5.3. The above names are given by the formulas

• A k+1 = A n k+1 k B k C k A k , • B k+1 = A n k+1 k B k C k , • C k+1 = A n k+1 k B k C k D m k+1 k C k , • D k+1 = A n k+1 k B k C k D m k+1 +1 k C k . with initial values A 1 = (14) n 1 1314, B 1 = (14) n 1 -1 13, C 1 = (14) n 1 -1 132 m 1 +1 3, D 1 = (14) n 1 -1 132 m 1 +2 3.
Towers and lengths are denoted as in the previous section. Now we shall fix our partial quotients so that the towers A k and the towers D k behave like independent systems, so that the transformation T has two ergodic invariant measures, one mainly concentrated on the towers A k and giving a rank one system with this family of towers, and the other doing the same with the towers D k . By ensuring the a k are even, we get an eigenvalue 1 2 for the first system, while the lengths d k will ensure the second one is weakly mixing by contradicting the criterion in Proposition 1.2.

Proposition 5.4. If

+∞ k=1 b k + c k n k+1 a k < 1 4 , +∞ k=1 n k+1 a k + b k + 2c k m k+1 d k < 1 4 ,
then T has exactly two ergodic invariant probability measures µ 1 and µ 2 ; (X, T, µ 1 ) is measuretheoretically isomorphic to the rank one system defined by the word A 1 and the towers

A ′ k+1 = (A ′ k ) n k+1 s a k+1 -n k+1 a k ; (X, T, µ 2
) is measure-theoretically isomorphic to the rank one system defined by the word D 1 and the towers

D ′ k+1 = s d k+1 -(m k+1 +1)d k -c k (D ′ ) m k+1 +1 k s c k .
Proof Let ǫ k and η k be respectively the k-th term of the first and second series above. Let µ be any invariant probability for T : each level in a given tower has the same measure, hence the above conditions and the formulas in Corollary 5.3 ensure that the tower A k has measure at least 1ǫ k in the tower A k+1 and the tower D k has measure at least 1η k in the tower D k+1 , while the towers B k , C k have measure at most ǫ k + η k in each tower A k+1 , B k+1 , C k+1 , D k+1 , hence in the whole space.

Thus we can build a measure-theoretic isomorphism between the rank one system (X ′ , T ′ , µ ′ ) with towers A ′ k and (X, T ) equipped with some invariant probability measure µ 1 which we retrieve from µ ′ , and µ 1 is ergodic as µ ′ is. We do the same for the rank one system with towers D ′ k , defining an ergodic µ 2 . Then the tower A 1 has measure greater than 1 2 for µ 1 and smaller than 1 2 for µ 2 , thus they are different, and it is known [START_REF] Katok | Invariant measures of flows on orientable surfaces[END_REF] [37] that T has at most two invariant ergodic probabilities.

Note that the two convergent series conditions are exactly the one needed in the definition of rank one systems. Theorem 5.5. One can construct recursively sequences m, n such that the corresponding transformation T is not uniquely ergodic, topologically weakly mixing, weakly mixing for one of its invariant ergodic measures, while for the other one it has 1 2 as an eigenvalue. Proof We fix M > 5 such that for all y ≥ M there exist a prime number between 6y/10 and 9y/10 and a prime number between 11y/10 and 14y/10. This is possible as a consequence of the prime numbers theorem.

At the beginning, note that a 1 , b 1 are even; we choose n 1 and m 1 such that c 1 is even, d 1 is odd, Thus our assumptions are satisfied for k + 1 (note that m k+1 has to be even).

We explain now how to choose the p k . When m k+1 and n k+1 are fixed, for any 0 < p < d k there is at most one integer 0 < l < d k+1 such that p d k -l d k+1 < 1 2Md k+1 .

We call this integer l = φ k+1 (p), when it exists. Now, our choice of partial quotients ensures that Starting from k 1 = 1, we define inductively a sequence of integers k j ; at stage j we put p k j = 1; then p k j +1 = φ k j +1 (2) if it exists, otherwise p k j +1 = φ k j +1 (3) if it exists, and so on. . . If no φ k j +1 (p) exists anymore, we put k j+1 = k j + 1, otherwise p k j +1 will be some φ k j +1 (p), and for p k j +2 we try first φ k j +2 φ k j +1 (p + 1) if it exists, then φ k j +2 φ k j +1 (p + 2) and so on. . . If no φ k j +2 φ k j +1 (q) exists anymore, we put k j+1 = k j + 1, otherwise p k j +2 is some φ k j +2 φ k j +1 (q), and for p k j +3 we try first φ k j +3 φ k j +2 φ k j +1 (q + 1) if it exists, and so on. . . After at most d k j -1 steps, we have defined k j+1 and ensured that for any 0 < p < d k j , there exists t ≤ k j+1 such that φ t . . . φ k j +1 (p) does not exist; then we start again from p k j+1 = 1.

p k d k+1 ≡ p k u k + p k (c k -a k ) mod d k ,
We apply Proposition 5.4 to get µ 1 and µ 2 . As a k is always even, (X, T, µ 1 ) has an eigenvalue 1 2 as in Theorem 4.7. Now, let θ be an eigenvalue for (X, T, µ 2 ): by Proposition 1.2 we must have m k+1 ||d k θ|| < 1 4M for k large enough, which implies |θ -t k d k | < 1 3M d k+1 for k large enough and some 0 < t k < d k ; this implies that for all k large enough φ k+1 (t k ) exists and is t k+1 , and this is impossible by the choice of p k . Thus (X, T, µ 2 ) is weakly mixing, which implies the topological weak mixing of T (which we could also have ensured directly as in Proposition 4.4).

As in [START_REF] Keane | Non-ergodic interval exchange transformations[END_REF] we can choose the vector of lengths (among a segment of possible solutions), so that µ 1 is the Lebesgue measure, or so that µ 2 is the Lebesgue measure, or so that neither µ 1 nor µ 2 is the Lebesgue measure. Note that (X, T, µ 1 ) and (X, T, µ 2 ) are rigid by Proposition 1.1.

QUESTIONS AND COMMENTS

Among examples we would have liked to build are transformations T with two (or more) rationally independent irrational eigenvalues; a similar result has been claimed by Parreau and Guenais (still unpublished) for d = 3 intervals, by very different methods which do not generalize to d > 3; the methods of the present paper being based on rank one, what we would need is an explicit rank one construction for rotations of T 2 , and this in itself is an interesting open problem.

Very interesting also would be a transformation T with a continuous eigenfunction; this does not exist for d = 3 intervals [START_REF] Nogueira | Topological weak-mixing of interval exchange maps[END_REF]; for every d ≥ 4 nontrivial examples have been derived by Hmili [START_REF] Hmili | Non topologically weakly mixing interval exchanges[END_REF] (in answer to a question asked in a preliminary version of the present paper): these examples are semi-conjugate, in a rather straightforward way, to rotations of T 1 . Older examples have been built by Arnoux and Yoccoz [START_REF] Arnoux | Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore[END_REF] for some permutation on d = 7 intervals: they are semi-conjugate, in a non-straightforward way, to rotations of T 2 . No example we know of has total irrationality.

The condition of alternate discontinuities simplifies the situation but can be dispensed with, see [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF]. The generalization of our methods to build examples on d intervals should not introduce any fundamental difficulty but the computations become horrendous; as for other permutations than the symmetric one, while our self-dual induction is not defined in the general case, it can be made to work on classes of examples as in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems, to appear[END_REF]; but the case d = 4 for one non-trivial permutation is representative of the whole problem, as happens for Keane's [START_REF] Keane | Non-ergodic interval exchange transformations[END_REF] non-uniquely ergodic examples which were not extended beyond that until a recent course of Yoccoz [START_REF] Yoccoz | Échanges d'intervalles[END_REF].

We recall that Veech's question on simplicity is far from solved; another question is to define a set M" as in the introduction by requiring T to be topologically strongly mixing, that is, for every Borelian A and B, T n A ∩ B is nonempty for n large enough. Boshernitzan (unpublished) has proved that M" is empty for d = 3 intervals, but, after many computer simulations, conjectures that M" is of full measure for d ≥ 4 intervals. Again during the process of refereeing the present paper, this question has been mostly solved for d = 4 by Chaika [START_REF] Chaika | There exists a topologically mixing IET[END_REF] (he gets a residual set, though not necessarily of full measure).

As for the specificity of the self-dual induction: it is possible that these or similar examples could have been obtained via other well-known induction methods, by first building a parametrized family of examples, and then manipulating the parameters. Indeed this approach was used by

1. 1 .

 1 The main definitions. Definition 1.1. A symmetric d-interval exchange transformation is a d-interval exchange transformation T with probability vector (α 1 , ..., α d ), and permutation πi

FIGURE 2 .

 2 FIGURE 2. The castle graphs at first and second stage.

  and the new castle is symmetric with the involution s(1) = 3, s(2) = 2, s(3) = 1; this involves checking psp = msm = s and the non-trivial relations r 1 + l 1 = r 3 + l 3 (the intervals E 1 and E 3 have the same length), l 2 = l 3 . Thus the new castle graph is shown on the right of Figure2; it is vertex IV of the graph of graphs Γ 4 , see Lemma 3.1 below.

  ) all their endpoints are of the form T a γ b for a ≤ 1 and b = 1, 2, 3, (3) their induction castle is binary symmetric, with bijections p, m, s, (

Lemma 3 . 1 .

 31 The graph of graphs Γ 4 is the graph whose vertices are I s = (123), p = (132), m = (321), with nontrivial relations r 1 = r 3 , l 2 = l 3 , II s = (123), p = (321), m = (213), with r 1

Proposition 4 . 4 .

 44 If for infinitely many k, there exist positive integers a

  x)| ≤ ǫ; similarly there exists y in the basis of the tower B k such that T b ′ k b k y is again in E 1 (k), and we get ||θ(a ′ k a kb ′ k b k )|| < ǫ, hence θ = 0, which is not possible as T is minimal and the existence of a continuous non-constant eigenfunction for θ = 0 would imply the existence of a non-trivial closed invariant subset. Proposition 4.5. If +∞ k=1 (p k+1 + 1)b

  the above condition uses d k and not m k+1 d k as it is enough since both A n k+1 k and D k have their lengths multiplied by m k+1 in the formulas of Corollary 4.3.

Theorem 4 . 7 . 1 N

 471 For any integer N ≥ 2, one can construct recursively sequences m, n, p such that the corresponding transformation T is uniquely ergodic, topologically weakly mixing, and has as an eigenvalue.ProofWe build the partial quotients for the self-dual induction as follows: at stage k ≥ 1, we assume a k and b k are coprime, b k > a k and a k is a multiple of N; by Bezout's identity we can find positive integers a ′ k and b ′ k such that a ′ k a kb ′ k b k = 1. We choose first p k+1 , such that p k+1 > b ′ k and p k+1 b k + c k ≡ 0 mod N; this is possible as b k is invertible modulo a k hence modulo N; then we choose n k+1 large enough for n k+1 a k > 2 k ((p k+1 + 1)b

a 1 and d 1

 1 are coprime. Given a k , b k , c k , d k , some p k to be specified later, and the assumptions that a k and d k are coprime, a k , b k , c k are even and d k is odd, we choose the next partial quotients as follows. Let z k be the greatest common divisor of p k and d k , withd k = d ′ k z k , p k = p ′ k z k . Note that d ′ k > 2 as d k is odd. Let p" k be an inverse of p ′ k mod d k . We choose first a unit u ′ k of Z/d ′ k Z such that u ′ k ≡ p ′ k (a kc k ) + t ′ mod d ′ k for anyd ′ k /2M < t ′ < d ′ k /2M. This is possible: if d ′ k ≥ M,we take the class modulo d ′ k of one of the two prime numbers defined above for y = d ′ k (the first one if the class of p ′ k (a kc k ) is between 0 and d ′ k /2, the second one otherwise), while if 2 < d ′ k < M this forbids at most one unit. We choose now a unit u kof Z/d k Z such that u k ≡ p" k u ′ k mod d ′ k(this is possible as, to be a unit, u k has just to be coprime with the prime factors of d k which are not factors of d ′ k ). Now we choose n k+1 large enough for the first condition of Proposition 5.4 and such thatn k+1 a k ≡ u ka kb kc k mod d k ,thus a k+1 is coprime with d k ; and we choose then m k+1 large enough for the second condition of Proposition 5.4 and such that (m k+1 + 1)d k + c ka k is invertible modulo a k+1 .

  and p k u k ≡ p k (a kc k ) + t mod d k for any -d k /2M < t < d k /2M (by multiplying by z k the relation satisfied by u ′ k , and p ′ k u k , mod d ′ k ), thus p k d k+1 ≡ t mod d k for any -d k /2M < t < d k /2M; this means exactly that φ k+1 (p k ) does not exist.

Chaika to answer a related question [START_REF] Chaika | There exists a topologically mixing IET[END_REF], starting from the family of examples in [START_REF] Keane | Non-ergodic interval exchange transformations[END_REF] which was built by using a variant of the Rauzy induction (actually anterior to Rauzy).

However, the authors found the self-dual induction developed herein to be well suited for this task. In particular, one can stress the role of the quantities we call partial quotients, which appear naturally as the number of consecutive times a given loop is followed in a castle graph, and which share some of the arithmetic properties of the usual partial quotients in the Euclid algorithm; indeed, in the simpler case of d = 3 intervals, they are used to define a multiplicative self-dual induction [START_REF] Ferenczi | A self-dual induction for three-interval exchange transformations[END_REF], though this is less obvious for d = 4.