S Ébastien Ferenczi

DYNAMICAL GENERALIZATIONS OF THE LAGRANGE SPECTRUM

Keywords: Mathematics Subject Classification. 37B10, 11J06, 68R15

We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's ne n , where e n is the smallest measure of a cylinder of length n, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we get for the family of rotations with the upper limit of 1 nen .

The Lagrange spectrum is the set of finite values of L(α) for all irrational numbers α, where L(α) is the largest constant c such that |α -p q | ≤ 1 cq 2 for infinitely many integers p and q (a variant is known as the Markov spectrum, see Section 1.3 below). It was recently remarked that this arithmetic definition can be replaced by a dynamical definition involving the irrational rotations of angle α, through their natural coding by the partition {[0, 1α[, [1 -α, 1[}. Namely, as we prove in Theorem 2.4 below which was never written before, L(α) is also the upper limit of the inverse of the so-called Boshernitzan's ne n , where e n is the smallest (Lebesgue) measure of the nonempty cylinders of length n.

Thus, for any symbolic dynamical system, it is interesting to compute two new invariants of topological conjugacy, lim sup n→+∞ 1 nen and lim inf n→+∞ 1 nen . Moreover, for a given family of systems, the set of all values of these invariants can be called the upper, resp. lower BL (for Boshernitzan and Lagrange) spectrum. In this paper, we compute these spectra for three families of systems: the irrational rotations (seen as two-interval exchanges), the three-interval exchanges, both coded by the natural partition of the interval generated by the discontinuities, and the Arnoux-Rauzy systems. In each of these cases, we use an induction (or renormalization) process, which is respectively a variant of the Euclid algoritm, the selfdual induction of [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems[END_REF], and the natural one defined in [START_REF] Arnoux | Représentation géométrique de suites de complexité 2n + 1[END_REF]. A multiplicative form of the process yields explicit formulas for our invariants, and these formulas are then exploited in each case by using the underlying algorithm of approximation of real numbers by rationals, which is respectively the classical continued fraction expansion, an extension of a semi-regular continued fraction expansion, and the algorithm which motivated the study of Arnoux-Rauzy systems.

What we get in the end is a first partial description of the five new sets we introduced beside the classical Lagrange spectrum. For rotations, the lower BL spectrum is a compact set starting with 1 and an interval (at least) as far as 1, 03..., ending at 1, 38..., with gaps, above an accumulation point at 1, 23.... For three-interval exchanges, the upper BL spectrum looks, perhaps deceptively, like two times the Lagrange spectrum, starting at 2 √ 5 with gaps and an accumulation point at 6, and ending with an interval (at least) from 14, 8... to infinity; the lower BL spectrum is fully determined and is none other than the interval [2, +∞]. For Arnoux-Rauzy systems, we deal with cubic numbers and our knowledge is only embryonic: the upper BL spectrum starts at 8, 44..., with gaps, and ends at infinity, the lower BL spectrum starts at 2, ends at infinity, and contains at least all the integers as accumulation points.

As a consequence, we get new uniquely ergodic systems for which ne n does not tend to zero when n tends to infinity, showing that Boshernitzan's criterion (see Section 1.4) is not a necessary condition; their existence in the family of three-interval exchanges was known but a proof was never written, while the examples in the family of Arnoux-Rauzy systems are new, and surprising as these systems are often thought to behave like rotations, see the discussion at the end of Section 4. An interesting open problem would be to compute the BL spectra of the family of all uniquely ergodic symbolic systems, see Section 5.

1. Preliminaries 1.1. Languages. Definition 1.1. We look at finite words on a finite alphabet A. A word with r letters, w 1 ...w r , is of length r. The concatenation of two words w and w is denoted by ww . The empty word is the unique word of length zero. A word w = w 1 ...w r occurs at place i in a word v = v 1 ...v s or an infinite sequence v = v 1 v 2 ... if w 1 = v i , ..., w r = v i+r-1 . We say that w is a factor of v. The empty word is a factor of any v. Prefixes and suffixes are defined in the usual way. A language L is a set of words such if w is in L, all its factors are in L, aw is in L for at least one letter a of A, and wb is in L for at least one letter b of A. A language L is uniformly recurrent if for each w in L there exists n such that w occurs in each word of length n of L.

A language L is now fixed. Definition 1.2. A word w is right special, resp. left special if there exist at least two different letters x such that xw, resp. wx, is in L. If w is both right special and left special, w is bispecial. The complexity of L is the function p L which to each positive integer n associates the number of different words of length n in L. The Rauzy graph of length n of L is the graph whose vertices are the words of length n in L, with an edge w → w if there exists a word v of length n -1 such that w = av, w = vb, and avb ∈ L.

Symbolic dynamics.

Definition 1.3. The symbolic dynamical system associated to a language L is the one-sided shift S(x 0 x 1 x 2 ...) = x 1 x 2 ... on the subset X L of A IN made with the infinite sequences such that for every r < s, x r ...x s is in L.

For a word w = w 1 ...w r in L, the cylinder [w] is the set {x ∈ X L ; x 0 = w 1 , ...x r-1 = w r }. (X L , S) is minimal if L is uniformly recurrent. (X L , S
) is uniquely ergodic if there is one S-invariant probability measure µ; then the frequency of the word w is the measure µ[w].

Starting from a (in general, geometric in origin) topological dynamical system (X, T ), we can get a symbolic dynamical system: Definition 1.4. For a transformation T defined on a set X, partitioned into X 1 , ... X r , and a point x in X, its trajectory is the infinite sequence (x n ) n∈IN defined by

x n = i if T n x falls into X i , 1 ≤ i ≤ r.
The language L(T ) is the set of all finite factors of its trajectories. The coding of (X, T ) by the partition {X 1 , ...X r } is the symbolic dynamical system (X L(T ) , S).

We shall often assimilate a dynamical system (X, T ) with its coding by a partition. Note that the word w 1 ...w n has length n; in most cases, the cylinder [w 1 ...w n ] is assimilated to an interval J, on which µ is the Lebesgue measure; the (geometrical) length of J is also the Lebesgue measure of [w 1 ...w n ], and thus the frequency of the word w 1 ...w n , which should not be mistaken with its (symbolic) length.

If the transformation T is minimal (i.e. every orbit is dense), all its trajectories have the same finite factors, and the language L(T ) is uniformly recurrent; the special words depend on the language and not on the individual trajectories; thus they are defined by any trajectory of T . If there is no periodic orbit, every word w is a factor of a bispecial word; hence the bispecial words determine the finite factors of the trajectories, and thus the symbolic dynamical system (X L(T ) , S).

1.3. Continued fractions. In Section 3 below, we shall use the generalized, or semi-regular continued fraction expansion 1

a 1 + ε 1 1 a 2 + ε 2 1 a 3 + • • • for a i positive integers, ε i = -or +, ε i = + if a i = 1. We denote the above expression by [0, a 1 * ε 1 , a 2 * ε 2 , ...].
We write a n with no sign if the expansion stops at a n . In Section 2 we shall deal only with classic (Euclid) continued fraction expansions, and write only a i to denote a i * +. The periodic sequence

a 1 * ε 1 , ..., a n * ε n , a 1 * ε 1 , ..., a n * ε n , , ... is denoted by (a 1 * ε 1 , ..., a n * ε n ) ω .
It is shown in [START_REF] Ferenczi | The structure of three-interval exchange transformations I: an arithmetic study[END_REF] that [0, a 1 * ε 1 , ...]. has bounded partial quotients (for the Euclid algorithm) if and only if the a i are bounded and the number of consecutive 2 * -is bounded.

For a full study of the Lagrange spectrum, we refer the reader to the monograph [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF]; the following definition is equivalent to the one given in the introduction of the present paper.

Definition 1.5. The Lagrange spectrum is the set of all finite values of lim sup

k→+∞ 1 q k |q k α -p k | ,
for α irrational, the p k q k being the convergents of α for the Euclid algorithm.

Let us just recall that the Lagrange spectrum is a closed set, its lowest elements are √ 5, then 2 √ 2, and discrete values (which, including these first two, are called the Lagrange numbers) up to a first accumulation point at 3; above 3, its structure is more complicated and not yet fully known, but it contains every real number above a value (which is known to be optimal) near 4, 52... Note that to get the Markov spectrum, we replace the upper limit by a supremum in the above definition; the Markov spectrum will not be used in the present paper.

1.4. Boshernitzan's ne n . In [START_REF] Boshernitzan | A condition for minimal interval exchange maps to be uniquely ergodic[END_REF] M. Boshernitzan introduced the following quantity: Definition 1.6. Let (X L , S) be a minimal symbolic system. If µ is an S-invariant probability measure, for each natural integer n, we denote by e n (µ) the smallest positive frequency of the words of length n of L. If µ is the only invariant probability measure, e n (µ) is simply denoted by e n .

After partial results in [START_REF] Boshernitzan | A condition for minimal interval exchange maps to be uniquely ergodic[END_REF] and [START_REF] Veech | Boshernitzan's criterion for unique ergodicity of an interval exchange transformation[END_REF], it was proved in [START_REF] Boshernitezan | A condition for unique ergodicity of minimal symbolic flows[END_REF] (see also the survey [START_REF] Ferenczi | Infinite words with uniform frequencies, and invariant measures[END_REF]) that whenever, for some invariant probability measure µ, ne n (µ) does not tend to 0 when n tends to +∞, then the system (X L , S) is uniquely ergodic. This sufficient condition for unique ergodicity has been known since [START_REF] Veech | Boshernitzan's criterion for unique ergodicity of an interval exchange transformation[END_REF] as Boshernitzan's citerion.

In the present paper, all systems considered are uniquely ergodic, and we consider the quantity ne n for its own sake. Thus we define Definition 1.7.

B = lim sup n→+∞ 1 ne n , B = lim inf n→+∞ 1 ne n .
Proposition 1.1. B and B are invariants of topological conjugacy among uniquely ergodic symbolic dynamical systems.

Proof For such a conjugacy φ between two symbolic systems, the i-th coordinate of φ((x n ) n∈IN ) depends only on x i , ..., x i+r for a fixed integer r, see for example Lemma 5.1.14 of [START_REF] Fogg | Substitutions in dynamics, arithmetics and combinatorics[END_REF], and the image of the unique invariant probability measure on the first system is the one on the second system.

A first crude estimate can be given using the complexity function,

Lemma 1.2. B ≥ lim sup n→+∞ p L (n) n , B ≥ lim inf n→+∞ p L (n) n . Proof
As the total measure of the space is 1, we have e n ≤ 1 p L (n) . This is enough to show that Boshernitzan's criterion is not a necessary condition: there are uniquely ergodic symbolic systems of exponential complexity [START_REF] Grillenberger | Constructions of strictly ergodic systems. I. Given entropy[END_REF], and thus with ne n → 0, see also the discussion at the end of Section 4. But of course the above lemma implies that the study of these invariants is interesting only for systems of sub-linear complexity, for which the question of necessity can be asked again.

In view of Theorem 2.5 below, we are led to define the following sets: Definition 1.8. For a family of uniquely ergodic symbolic dynamical systems (X a , S), a ∈ F, the upper BL spectrum is the set of all values of B taken by the systems in this family, and the lower BL spectrum is the set of all values of B taken by the systems in this family.

Rotations and the dynamical definition of the Lagrange spectrum

Surely there is nothing new to find about irrational rotations? The computation of B in Thorem 2.4 below, and the subsequent Theorem 2.5, which was the main motivation for the present paper, were known to P. Hubert and T. Monteil (private communications), but never written to our knowledge. The quantity 1 B was indeed computed in [START_REF] Chekhova | Covering numbers of rotations[END_REF] (see also [START_REF] Berth É | Covering numbers: arithmetics and dynamics for rotations and interval exchanges[END_REF]) as, for irrational rotations, it is equal to another invariant of topological conjugacy, the covering number by intervals [START_REF] Chekhova | Covering numbers of rotations[END_REF], which involves covering the space by Rokhlin towers; the spectrum of its possible values is the object of a question in [START_REF] Chekhova | Covering numbers of rotations[END_REF] and in [START_REF] Cassaigne | Limit values of the recurrence quotient of Sturmian sequences[END_REF], to which Theorem 2.6 below gives a first (to our knowledge), though belated and partial, answer.

Let α < 1 2 be an irrational number; the rotations with α > 1 2 are treated in a similar way and all the results in this section from Theorem 2.4 onwards remain valid; the rotation of angle α, is also the two-interval exchange defined by

T x = x + α if x ∈ X 1 = [0, 1 -α[ x -1 + α if x ∈ X 2 = [1 -α, 1[.
With this definition, a rotation admits a natural coding, by the partition of X = [0, 1[ into X 1 and X 2 . Then L(T ) has complexity n + 1 and the trajectories are called Sturmian sequences. Irrational rotations are minimal and uniquely ergodic.

To get Theorem 2.4 below, we rely on a computation of both frequencies and lengths of factors of Sturmian sequences, which was done in [START_REF] Berth | Fréquences des facteurs des suites sturmiennes. (French) [Frequencies of the factors of Sturmian sequences[END_REF], but which we provide again here by using a different version of the classic Euclid algorithm, making the computations quicker and ready to be generalized. This algorithm is the self-dual induction of [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems[END_REF] in the particular case of two intervals; all what we need to know is contained in the following proposition, which can also be proved directly without difficulty. Proposition 2.1. If we build inductively real numbers l n and r n and words w n , M n , P n in the following way:

l 1 = α, r 1 = 1 -2α, w 1 = 1, M 1 = 1, P 1 = 21. Then • whenever l n > r n , l n+1 = l n -r n , r n+1 = r n , w n+1 = w n P n , P n+1 = P n , M n+1 = M n P n ; • whenever r n > l n , l n+1 = l n , r n+1 = r n -l n , w n+1 = w n M n , P n+1 = P n M n , M n+1 = M n .
Then the w n are all the nonempty bispecial words of L(T ), w n+1 being the shortest bispecial word beginning with w n . The cylinder [w n ] is the interval [α -l n , α + r n [; its left and right subintervals separated by α are respectively T [2w n ] and T [1w n ], thus l n , resp. r n , is the frequency of the word 2w n , resp. 1w n . Also,

[w n 1] = [w n M n ] and [w n 2] = [w n P n ].
The parameters l n and r n govern the process; the irrationality of α ensures that always l n = r n . Two fundamental relations come as direct consequences of the above formulas, as we check easily the considered quantities are not changed from n to n + 1: for all n, (1)

l n |P n | + r n |M n | = 1, (2)
|M n | + |P n | -|w n | = 2.
We define now the multiplicative form of this algorithm:

Corollary 2.2. We define l 0 = α, r 0 = 1 -α, M 0 = 1, P 0 = 2, w 0 being the empty word. Then we define a 1 , a 2 , ... such that r n > l n for 0 ≤ n ≤ a 1 -1, r n < l n for a 1 ≤ n ≤ a 1 + a 2 -1, and so on. Let

o k = a 1 + ...a k , o 0 = 0; let α k = r o k and q k = |M o k | if k is even, α k = l o k and q k = |P o k | if k is odd.
Then for all k ≥ 0, if q -1 = q 0 = 1,

α k = a k+1 α k+1 + α k+2 , q k+1 = a k+1 q k + q k-1 . If k is even α k+1 = l o k and q k-1 = |P o k |, while if k is odd α k+1 = r o k and q k-1 = |M o k |.
The Euclid continued fraction expansion of α is [0, a 1 + 1, a 2 , ...], and the q k , k ≥ 0, are the denominators of the convergents of α.

Proof

The written formulas are straightforward consequences of Proposition 2.1. They imply that

α k+1 α k = 1 a k+1 + α k+2 α k+1
and thus α k+1 α k has the continued fraction expansion [0, a k+1 , a k+2 , ...]. Going to k = 0, α 1 α 0 = α 1-α has the continued fraction expansion [0, a 1 , a 2 , ...], which implies the last assertions.

The relations ( 1) and ( 2)

imply |w o k | = q k + q k-1 -2 and q k α k + q k-1 α k+1 = 1.
We can now compute the frequencies as in [START_REF] Berth | Fréquences des facteurs des suites sturmiennes. (French) [Frequencies of the factors of Sturmian sequences[END_REF]:

Lemma 2.3. For |w n | + 2 ≤ s ≤ |w n+1 |,
the words of length s have three possible frequencies which are l n+1 , r n+1 , and l n+1 + r n+1 ; for s = |w n | + 1, the words of length s have two possible frequencies which are l n and r n .

Proof

As in [START_REF] Berth | Fréquences des facteurs des suites sturmiennes. (French) [Frequencies of the factors of Sturmian sequences[END_REF], we build the Rauzy graphs. If w is not right special, w is not left special, and w → w , then w and w have the same frequency.

In the case of Sturmian sequences, for each s, there is one right special word of length s and there is one left special word G s of length s; there are at most three frequencies, which are for example those of G s , 1G s and 2G s , and these are exactly l m + r m , l m and r m for the smallest m such that |w m | ≥ s.

But we need to check whether each of these frequencies is indeed the frequency of some word. We take an n such that r n > l n , and let p be the length of w n . The possible frequencies of words of length p are l n , r n , l n + r n , each one is indeed the frequency of a word of length p, as we take respectively 2w n deprived of its last letter, 1w n deprived of its last letter, and w n . Then the words of length p + 1 have possible frequencies l n+1 , r n+1 , l n+1 + r n+1 , which are respectively l n , r n -l n , r n ; each of the two old frequencies l n , r n is the frequency of a word of length p + 1, as we take respectively 2w n and 1w n , but the new one r n -l n is not the frequency of any word of length p + 1. The words of length p + 2 have possible frequencies l n+1 , r n+1 , l n+1 + r n+1 , and each one is the frequency of a word of length p + 2, as we take respectively 2w n 1, 1w n 1, and the prefix of length p + 2 of w n M n ; and this remains true (by extending the considered words to the right following w n M n ) for p + 3,... until we reach the length of w n+1 . And the reasoning is similar if

l n > r n . Theorem 2.4. For a rotation of irrational angle α = [0, b 1 , .., .b n , ...], if we define v k = [0, b k , b k-1 , ...b 1 ] and t k = [0, b k+1 , b k+2 , ...] then B = lim sup k→+∞ 1 v k + t k = lim sup k→+∞ (b k + v k-1 + t k ) , B = lim inf k→+∞ (1 + t k v k ) . Proof Suppose k is even. Then for o k ≤ m ≤ o k+1 -1, l m = l o k and l m < r m ; thus the smallest frequency of a word of length n is l o k for |w o k -1 | + 2 ≤ n ≤ |w o k+1 -1 | + 1.
We know that l o k = α k+1 and |w o k | = q k + q k-1 -2, and we check

|w o k -1 | = q k -2. Thus the minimal value of ne n for |w o k -1 | + 2 ≤ n ≤ |w o k+1 -1 | + 1 is q k α k+1 ; the maximal value of ne n for |w o k -1 | + 2 ≤ n ≤ |w o k+1 -1 | + 1 is q k+1 α k+1 ; all this is still true if k is odd.
In Corollary 2.2, we identify b 1 with a 1 + 1 and b i with a i for i ≥ 2. Thus we get α k+1 α k = t k , while by construction of the q k we get q k-1

q k = v k . Then lim sup n→∞ 1 nen = lim sup k→∞ 1 q k α k+1 = lim sup k→∞ q k+1 α k+1 +q k α k+2 q k α k+1 = lim sup k→∞ ( q k+1 q k + α k+2 α k+1 ), which is the first formula at stage k + 1. Similarly, lim inf n→∞ 1 nen = lim inf k→∞ 1 q k+1 α k+1 = lim inf k→∞ q k+1 α k+1 +q k α k+2 q k+1 α k+1 = lim inf k→∞ (1 + α k+2 α k+1 q k q k+1
), which is the second formula at stage k + 1.

Theorem 2.5. The upper BL spectrum of the family of rotations is the union of the Lagrange spectrum and +∞.

Proof

We check that

p k α k + p k-1 α k+1 = α and |p k q k-1 -p k-1 q k | = 1, thus lim inf k→∞ q k |q k α -p k | is equal to lim inf k→∞ q k α k+1 .
As for the lower LB spectrum, it seems to have never been studied to our knowledge, and its study looks to be of the same level of difficulty as for the Lagrange spectrum. We give now some of the first results about it. Note that B is not the lower limit of 1 q k |q k α-p k | and thus is not directly linked to the quality of the approximation of α by rationals.

Theorem 2.6. The lower BL spectrum of the family of rotations has 1 as its smallest element, with B = 1 if and only if the angle has unbounded partial quotients. It is a closed set. Its two largest elements are 5-

√ 5 2
= 1, 38196... and 3 -√ 3 = 1, 26794..., and there is no other element above 5 4 .

It contains an accumulation point equal to

√ 5 -1 = 1, 2360... It contains the interval [1, 1 + 4 83+18 √ 2 = 1, 03688...].

Proof

The first two assertions are straightforward consequences of the formula giving B . The spectrum is closed by the standard reasoning of [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF], Chapter 1, Corollary to Lemma 6, as every value greater than one is reached by an angle with bounded partial quotients, and for any sequence of angles such that the corresponding B converge to a number different from one, we can take the partial quotients uniformly bounded.

Let α = [0, b 1 , .., .b n , ...] be the angle of a rotation.

Suppose that there are infinitely many b k ≥ 4; then for these k, t k v k ≤ 1 4 . If there are only 1, 2 or 3 with infinitely many 3, we have t k ≤ [0, [START_REF] Chekhova | Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci[END_REF]

ω ]. If b k = 3, we have v k ≤ [0, 3(31) ω ] and we get t k v k ≤ √ 21-3 √ 21+15 = 0,
= b k+1 = 2 we get t k ≤ [0, 2(21) ω ] and v k ≤ [0, 2(21) ω ], thus t k v k ≤ 12-6 √ 3 9
= 0, 1786... If there are only 1 and 2 with no 22, infinitely many 2 and infinitely many 11

, if b k = 2, b k+1 = b k+2 = 1 we get v k ≤ [0, 21(12) ω ], t k ≤ [0, 11(12) ω ] and t k v k ≤ 6- √ 3 11 3- √ 3 2
= 0, 2459... Thus the two highest values are reached for the angles [0, ( 21) ω ] and [0, (1) ω ], and the above estimates give a bound on the gap below them.

If we take α = [0, (1 j 2) ω ], for a j ≥ 3, the smallest value of t n v n is reached when b k = 2 or b k+1 = 2, and when j tends to infinity, this value tends to [0, ( 1) ω ][0, 2( 1) ω ] = √ 5 -2. Thus we get an accumulation point, approached from above if j is odd.

To prove the last assertion, we use Theorem 3.2 of [START_REF] Hall | On the sum and product of continued fractions[END_REF]: any real number s > 1 can be written s = (r + [0, a 1 , a 2 , ...])(r + [0, a 1 , a 2 , ...]) with all the a i and a i taking values 1, 2, 3 or 4. Moreover, from the proof of this theorem we get ( [START_REF] Hall | On the sum and product of continued fractions[END_REF]

p. 974) that if s is at least n 2 +( √ 2- 1)n + 3-2 √ 2 4
we can take both r and r greater or equal to n.

Thus if s is at least 83+18 √ 2 4
, we write s in that way, with r and r at least 5. We choose now a sequence k n → +∞, and take the rotation of angle [0,

a k 1 , a k 1 -1 , .., a 1 , r, r , a 1 , ...a k 1 , a k 2 , a k 2 -1 , .., a 1 , r, r , a 1 , ...a k 2 , ...];
then the minimal values of t k v k are taken when a k = r, and their lower limit is exactly 1 s .

The third highest number in this spectrum is 16-4 √ 6 5

= 1, 2404..., as can be seen with longer computations; the point √ 5 -1 is the highest accumulation point, but to prove it requires a machinery similar to the one used to prove Theorem 5 in Chapter 1 of [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF].

Three-interval exchanges

3.1. The transformations. Definition 3.1. Given two numbers 0 < α, 0 < β with α + β < 1, we define a three-interval exchange on X = [0, 1[ by

T x =      x + 1 -α if x ∈ X 1 = [0, α[ x + 1 -2α -β if x ∈ X 2 = [α, α + β[ x -α -β if x ∈ X 3 = [α + β, 1[.
Throughout this section, we ask that α and β satisfy the i.d.o.c condition of Keane, which means in that case that they do not satisfy any rational relation of the forms pα +qβ = p-q, pα + qβ = p -q + 1, or pα + qβ = p -q -1, for p and q integers.

The points α and α + β are the discontinuities of T , while β 1 = 1 -α -β and β 2 = 1 -α are the discontinuities of T -1 . The i.d.o.c. condition ensures that the negative orbits of the discontinuities of T are infinite and have an empty intersection (it is its original definition; see [START_REF] Ferenczi | The structure of three-interval exchange transformations II: a combinatorial description of the trajectories[END_REF] for the equivalence with the one stated here).

A three-interval exchange admits a natural coding, by the partition of X into X 1 , X 2 , X 3 . Under the i.d.o.c. condition, (X, T ) is minimal and uniquely ergodic and L(T ) has complexity 2n + 1.

We recall that the induced, or first return, map S of a transformation T on a set E is defined on E by Sx = T g(x) x, where g(x) is the smallest integer r > 0 such that T r x is in E (a finite g(x) does indeed exist in all cases occurring in the present paper).

A three-interval exchange defined as above is always the induced map of the (irrational) rotation of angle 1-α 1+β on the interval [0, 1 1+β [.

Throughout this section, we add the conditions 0 < α < 1 2 , and 2α + β > 1; they ensure that the induction process described below does not have an irregular behaviour in the early stages: as is shown in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems[END_REF], their absence modifies only a finite number of stages, and all the results in this section from Theorem 3.10 onwards remain valid without these extra conditions.

3.2. The self-dual induction for 3 intervals. We state now some results from [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems[END_REF]. Definition 3.2. We define an operation called self-dual induction, which builds numbers l i,n and r i,n , i = 1, 2, through an infinite sequences of states in the following way:

State I is defined by the relation

r 1,n = r 2,n . In Substate Ia l 1,n > r 1,n , l 2,n > r 2,n . Then for i = 1, 2 we put l i,n+1 = l i,n -r i,n , r i,n+1 = r i,n .
For n + 1 we are again in state I.

In Substate Ib l 1,n < r 1,n , l 2,n > r 2,n . Then l 1.n+1 = l 1,n , r 1,n+1 = r 1,n , l 2,n+1 = l 2,n -r 2,n , r 2,n+1 = r 2,n .
For n + 1 we are again in state I. In Substate Ic l 1,n > r 1,n , l 2,n < r 2,n . This is deduced from Ib by exchanging 1 and 2, and for n + 1 we are again in state I. In Substate Id l 1,n < r 1,n , l 2,n < r 2,n . Then for i = 1, 2, l i,n+1 = l i,n , r i,n+1 = r i,n -l i,n . For n + 1 we are in state II described just below.

State II is defined by the relation l 1,n + r 1,n = l 2,n + r 2,n . Note that in this state l 1,n > r 2,n if and only if l 2,n > r 1,n . In Substate IIa l 1,n > r 2,n , l 2,n > r 1,n . Then for i = 1, 2, l i.n+1 = l i,n -r 3-i,n , r i,n+1 = r i,n .

For n + 1 we are in state III described below. In Substate IIb l 1,n < r 2,n , l 2,n < r 1,n . Then for i = 1, 2, r i.n+1 = r i,n -l 3-i,n , l i,n+1 = l i,n .

For n + 1 we are in state I.

State III is symmetrical to state I, with left and right exchanged, and the relation l 1,n = l 2,n ; there are four substates, IIIa to IIId, and the induction goes either to state III or to state II. Proposition 3.1. For a given three-interval exchange, we build inductively real numbers l i,n and r i,n and words w i,n , M i,n , P i,n , i = 1, 2, in the following way: • whenever l i,n+1 < l i,n , then w i,n+1 = w i,n P snpn(i),n , P i,n+1 = P i,n , M i,n+1 = M pn(i),n P i,n ; • whenever r i,n+1 < r i,n , then w i,n+1 = w i,n M snmn(i),n , P i,n+1 = P mn(i),n M i,n , M i,n+1 = M i,n ; • otherwise w i,n+1 = w i,n , P i,n+1 = P i,n ,, M i,n+1 = M i,n . Then the w i,n , n = 1, 2, are all the nonempty bispecial words of L(T ); either w i,n+1 = w i,n or w i,n+1 is the shortest bispecial word beginning with w i,n . The cylinder [w i,n ] is the interval [β i -l i,n , β i + r i,n [; its left and right subintervals separated by β i are respectively T [bw i,n ] and T [aw i,n ] if bw i,n and aw i,n , a < b, are the two left extensions of w i,n ; thus indeed l 1,n , r 1,n , l 2,n , r 2,n are respectively the frequencies of the words 3w 1,n , 2w 1,n , 2w 2,n , 1w 2,n . Also, [w i,n M snmn(i),n ] = [w i,n c] and [w i,n P snpn(i),n ] = [w i,n d], if w i,n c and w i,n d, d > c, are the two right extensions of w i,n .

l 1,1 = 1 -α -β, r 1,1 = 2α + β -1 = r 2,1 , l 2,1 = 1 -2α, w i,1 = i and M i,1 = i for i = 1, 2, P 1,1 = 31, P 2,1 = 2.
A three-interval exchange defines an infinite sequence of states labelled Ia to Id, IIa, IIb, IIIa to IIId, following the rules of Definition 3.2, and such that each one of the four parameters l 1,n , r 1,n , l 2,n , r 2,n is modified infinitely often. Conversely, every such sequence of states defines a three-interval exchange which generates it as described above.

Note that the i.d.o.c. condition ensures that we dont have l 1,n = r 1,n or similar equalities. We check from the formulas that the following bilinear form is invariant by the induction, and thus for all n

(3) l 1,n |P 1,n | + r 1,n |M 1,n | + l 2,n |P 2,n | + r 2,n |M 2,n | = 1.
The other relations are more complicated than in the case of rotations, we state them now but they are a straightforward consequence of Corollary 3.2 below:

• whenever for n we are in II,

(4) |P 2,n | + |M 1,n | = |M 2,n | + |P 1,n | + 1 = |w 2,n | + 2 = |w 1,n | + 2;
• whenever for n we are in I,

|P 1,n | + |M 1,n | = |w 1,n | + 2, |P 2,n | + |M 2,n | = |w 2,n | + 1, |P 1,n | = |P 2,n | + 1;
• whenever for n we are in III,

|P 1,n | + |M 1,n | = |w 1,n | + 1, |P 2,n | + |M 2,n | = |w 2,n | + 2, |M 2,n | = |M 1,n | + 1.
We use now the self-dual induction to retrieve the frequencies and the lengths of the bispecial words, and this involves a multiplicative form of the algorithm; this has been done in [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] but in a form which is more complicated and less explicit (the frequencies of words are somewhat hidden) than the one which we deduce now from [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems[END_REF]; the results in this section are new, but could be deduced from [START_REF] Ferenczi | The structure of three-interval exchange transformations I: an arithmetic study[END_REF] and [START_REF] Ferenczi | The structure of three-interval exchange transformations II: a combinatorial description of the trajectories[END_REF], see [START_REF] Ferenczi | A self-dual induction for three-interval exchange transformations[END_REF] for the equivalence between the two forms of the algorithm. Thus the present paper is independent of all the previous ones except [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems[END_REF], and all the information we need from the latter are in Definition 3.2 and Proposition 3.1 above. Corollary 3.2. For a given 3-iet, we define o k , k = 1, 2... to be the sequence of n ≥ 1 such that for n we are in state II; we define also o 0 = 0. Then for k ≥ 1, we define two positive integers, n k , resp. m k , as the number of o k-1 < n < o k such that w 1,n+1 = w 1,n , resp. w 2,n+1 = w 2,n . Finally we define a sequence η k , which is -, resp. +, whenever for n = o k -1 we are in state I, resp. III.

We define also parameters for n = 0, namely, we are in state II, l 1,0 = 1 -α -β, r 1,0 = β, l 2,0 = 1 -2α, r 2,0 = α, w 1,0 , w 2,0 and P 1,0 are the empty word, P 2,0 = 3, M 1,0 = 1, M 2,0 = 2, η 0 = +.

Then o 0 = 0, η 1 = -. In the case

η k+1 = -, for 0 ≤ a ≤ n k+1 -1, 0 ≤ b ≤ m k+1 -1, r 1,o k +1+a = r 1,o k -l 2,o k = r 2,o k -l 1,o k , l 1,o k +1+a = l 1,o k -a((r 1,o k -l 2,o k ), r 2,o k +1+b = r 1,o k -l 2,o k = r 2,o k -l 1,o k , l 2,o k +1+b = l 2,o k -b(r 1,o k -l 2,o k ), w 1,o k +1+a = w 1,o k M 1,o k (P 2,o k M 1,o k ) a , w 2,o k +1+b = w 2,o k M 2,o k (P 1,o k M 2,o k ) b , M 1,o k +1+a = M 1,o k (P 2,o k M 1,o k ) a , P 1,o k +1+a = P 2,o k M 1,o k ; M 2,o k +1+b = M 2,o k (P 1,o k M 2,o k ) b , P 2,o k +1+b = P 1,o k M 2,o k .
Then

o k+1 = max{o k + 1 + n k+1 , o k + 1 + m k+1 }; between min{o k + n k+1 , o k + m k+1 } and max{o k + n k+1 , o k + m k+1 }
one of the intervals is not modified, and

r 1,o k+1 = n k+1 (r 1,o k -l 2,o k ) -l 1,o k , l 1,o k+1 = l 1,o k -(n k+1 -1)((r 1,o k -l 2,o k ), r 2,o k+1 = m k+1 (r 1,o k -l 2,o k ) -l 2,o k , l 2,o k+1 = l 2,o k -(m k+1 -1)((r 1,o k -l 2,o k ), w 1,o k+1 = w 1,o k M 1,o k (P 2,o k M 1,o k ) n k+1 -1 M 2,o k (P 1,o k M 2,o k ) m k+1 -1 , w 2,o k+1 = w 2,o k M 2,o k (P 1,o k M 2,o k ) m k+1 -1 M 1,o k (P 2,o k M 1,o k ) n k+1 -1 , P 1,o k+1 = P 1,o k M 2,o k M 1,o k (P 2,o k M 1,o k ) n k+1 -1 , M 1,o k+1 = M 1,o k (P 2,o k ,1 M 1,o k ) n k+1 -1 , P 2,o k+1 = P 2,o k M 1,o k M 2,o k (P 1,o k M 2,o k ) m k+1 -1 , M 2,o k+1 = M 2,o k (P 1,o k M 2,o k ) m k+1 -1 .
We get the case η k+1 = + by exchanging the l and r, or the M and P , in both sides of all the above equalities.

A three-interval exchange defines an infinite sequence

(n k , m k , η k ), k ≥ 1, n k ≥ 1, m k ≥ 1, η k = ±,
such that we do not have ultimately n k = 1 and η k constant, or m k = 1 and η k constant. Conversely, every such sequence defines a three-interval exchange which generates it as described above.

Proof Through all the substates of state I we have r i,n+1 = r i,n , and the l i,n are decreased by a fixed quantity, so after a finite number of steps we are in Id and proceed to II, and similarly throughout stage III. Thus we are infinitely many times in state II and we can define o k ; n k and m k are at least one as for n = o k + 1 we are in state I or III; we check that our parameters for n = 0 are compatible with the ones already defined for n = 1. Then we apply recursively Definition 3.2 and Proposition 3.1.

In the sequel, for a given three-interval exchange T , we shall use either the sequence (n k , m k , η k ) or (n k , m k , -η k η k+1 ), which we both call the expansion of T ; an expansion is admissible if we do not have ultimately n k = 1 and η k constant, or m k = 1 and η k constant. Note that [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] [19] use the expansion (n k , m k , k+1 ) where k+1 = η k η k+1 .

We make now some further computations on the formulas above.

Lemma 3.3. For all k ≥ 1, (5)   r 1,o k-1 + r 2,o k-1 r 1,o k-1 -r 2,o k-1 l 1,o k-1 + r 1,o k-1   =   η k 0 m k + n k -2η k 0 1 (n k -m k )η k η k 0 m k + n k -η k     r 1,o k + r 2,o k r 1,o k -r 2,o k l 1,o k + r 1,o k   . Let ∆ k = l 1,o k + r 1,o k = l 2,o k + r 2,o k , t k = ∆ k+1 ∆ k . Then (6) r 1,o k + r 2,o k -∆ k = -η k+1 ∆ k+1 , (7)
t k = [0, (m k+1 + n k+1 ) * -η k+1 η k+2 , (m k+2 + n k+2 ) * -η k+2 η k+3 , ...] (8) 1 -α 1 + β = [0, 2 * +, (m 1 + n 1 ) * -η 1 η 2 , (m 2 + n 2 ) * -η 2 η 3 , ...]
Proof (5) comes directly from the formulas in Corollary 3.2. These imply also [START_REF] Berth É | Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly, Algebraic and topological dynamics[END_REF], and

t k = -η k+1 r 1,o k +r 2,o k -∆ k ∆ k , thus we get t k = 1 m k+1 +n k+1 - η k+1 η k+2 t k+1
, and t k is given by the semi-regular continued fraction expansion [START_REF] Boshernitzan | A unique ergodicity of minimal symbolic flows with linear block growth[END_REF]. Going to k = 0 we get a semi-regular continued fraction expansion of α+β 1-α , which can be put in the form [START_REF] Boshernitzan | A condition for minimal interval exchange maps to be uniquely ergodic[END_REF].

Lemma 3.4. For all k ≥ 1, (9)
  |P 2,o k | + |P 1,o k | |P 1,o k | -|P 2,o k | |M 2,o k | + |P 1,o k |   = V k +   η k 0 m k + n k -2η k 0 1 n k -m k η k 0 m k + n k -η k     |P 2,o k-1 | + |P 1,o k-1 | |P 1,o k-1 | -|P 2,o k-1 | |M 2,o k-1 | + |P 1,o k-1 |   .
where

V k =   n k + 1 n k -1 n k   if η k = -, V k =   m k -1 1 -m k m k   if η k = +.
We define

z k = |M 2,o k | + |P 1,o k |, v k = q k-1 q k , where q k+1 = (m k+1 + n k+1 )q k -η k η k+1 q k-1 , q -1 = 0, q 0 = 1. Then (10) |M 2,o k | + |M 1,o k | = z k + η k z k-1 + 1 + η k . (11) lim k→+∞ (z k ∆ k -η k η k+1 z k-1 ∆ k+1 ) = 1. ( 12
) v k = [0, (m k + n k ) * -η k-1 η k , (m k-1 + n k-1 ) * -η k-2 η k-1 , ..., (m 2 + n 2 ) * -η 1 η 2 , m 1 + n 1 ] (13) lim k→+∞ z k-1 z k -v k = 0.
Proof (9) comes directly from the formulas in Corollary 3.2. This implies in turn [START_REF] Cassaigne | Limit values of the recurrence quotient of Sturmian sequences[END_REF].

|P 2,o k | + |P 1,o k | = z k -η k z k-1 -η k and thus
We write now the relation (3 [START_REF] Cassaigne | Limit values of the recurrence quotient of Sturmian sequences[END_REF]. Because of ( 6), we get finally that

) for n = o k + 1. Suppose η k+1 = -1. We notice that r 1,o k +1 = r 2,o k +1 = r 1,o k -l 2,o k = r 2,o k -l 1,o k is also equal to ∆ k+1 , and thus ∆ k+1 (|M 2,o k | + |M 1,o k |)+l 1,o k (|P 2,o k |+|M 1,o k |)+l 2,o k (|M 2,o k |+|P 1,o k |) = 1. We recall that |M 2,o k |+|P 1,o k | = z k and |P 2,o k | + |M 1,o k | = z k + 1, while |M 2,o k | + |M 1,o k | is given by
z k ∆ k + η k z k-1 ∆ k+1 + (1 + η k )∆ k+1 + l 1,o k = 1. Similar computations when η k+1 = +1 give z k ∆ k -η k z k-1 ∆ k+1 -η k ∆ k+1 + r 2,o k = 1.
In both cases we get [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF]. ( 12) is a straightforward consequence of the definition of q k . We show now that z k-1 z k -q k-1 q k tends to 0 when k tends to +∞. Indeed, from the matrix equality with B k and [START_REF] Cassaigne | Limit values of the recurrence quotient of Sturmian sequences[END_REF]

we get z k+1 = (m k+1 + n k+1 )z k -η k η k+1 z k-1 -η k η k+1 + θ k+1 , where θ k = n k if η k = -1, θ k = m k if η k = +1. If Φ k = q k z k-1 -q k 1 z k , then Φ k+1 = -η k η k+1 Φ k -q k (θ k+1 + η k η k+1 )
; we see that Φ k grows more slowly than q k , and this implies [START_REF] Chekhova | Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci[END_REF] as z k tends to infinity with k.

Note that the matrices in Lemmas 3.3 and 3.4 are the same except for one entry, the self-duality of the induction works but not as perfectly as for the rotations.

Though this will not be used in the sequel, the q k are indeed present in the computation of the lengths of the intervals, as we check that the third line of the matrix A 1 ...A k is made with the entries η k q k-1 , 0, q k -η k q k-1 , and this implies that

q k ∆ k -η k η k+1 q k-1 ∆ k+1 = 1 -α.
Thus from ( 11), [START_REF] Chekhova | Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci[END_REF], and the last equality, we deduce that q k z k → 1 -α when k tends to infinity.

From now on, when we study quantities involving large k, we shall always discard higher order terms such as

|w 1,o k | -z k or z k ∆ k -z k-1 ∆ k+1 -1, and replace z k-1 z k by v k .
3.3. General formulas for B and B .

Lemma 3.5. The frequencies of words of length n take at most six different values, which are l i,q(i,n) , r i,q(i,n) , and l i,q(i,n) + r i,q(i,n) , i = 1, 2, where w i,q(i,n) is the shortest bispecial word (in the families built by the self-dual induction) of length at least n.

Proof

We check that if w = w 1 ...w r is in L(T ), so is w = w r ...w 1 , with the same frequency as w: this is done by induction on the length of w, as the frequencies of the words of length n are deduced from the frequencies of words of length n -1 and the expression of the words of length n, see for example [START_REF] Boshernitzan | A condition for minimal interval exchange maps to be uniquely ergodic[END_REF] or Proposition 4 of [START_REF] Ferenczi | Languages of k-interval exchange transformations[END_REF].

Then we build the Rauzy graphs, as was done in [START_REF] Boshernitzan | A condition for minimal interval exchange maps to be uniquely ergodic[END_REF] without naming them. Given w, there exists at least one (left or right) special word linked by arrows with w, otherwise there is a periodic orbit and this contradicts the i.d.o.c. condition; thus w has the same frequency as v, where v is a word with n letters, and either v is left special, or v → v where v is left special, or v is right special, or v → v where v is right special; by looking also at w, we can suppose v is left special or v → v where v is left special.

There are two left special words with n letters, v i,n beginning with i; let L i,n be the frequency of bv i,n , R i,n the frequency of av i,n , for the two left extensions of v i,n , b > a. Then the frequency of v i,n is L i,n + R i,n , and the frequency of a v such that v → v i,n is either L i,n or R i,n . Moreover, let W i,n be the shortest bispecial word having v i,n as a prefix, which is also the shortest bispecial word beginning with i and with length at least n; then W i,n is w i,q for q = q(i, n), L i,n = l i,q and R i,n = r i,q . Lemma 3.5 was proved in [START_REF] Ferenczi | Structure of k-interval exchange transformations: induction, trajectories, and distance theorems[END_REF] and can also be deduced from [START_REF] Boshernitzan | A condition for minimal interval exchange maps to be uniquely ergodic[END_REF], we reprove it for sake of completeness.

Lemma 3.6. Suppose r i,n+1 < r i,n ; then each of the frequencies l i,n+1 and l i,n+1 + r i,n+1 is the frequency of a word of length s for every |w i,n | + 1 ≤ s ≤ |w i,n+1 |; the frequency r i,n+1 is the frequency of a word of length s for every |w i, |+2 ≤ s ≤ |w i,n+1 |, but not for s = |w i,n |+1. The same is true with l and r exchanged.

Proof This is the same as the last part of the proof of Lemma 2.3, mutatis mutandis: in particular, while the word w n could be preceded by two letters 2 > 1, and followed by two letters 1 < 2, here the word w i,n can be preceded by two letters b > a, and followed by two letters c < d, and 1w n 2 is replaced by aw i,n d and so on. Proposition 3.7. Let T be a three-interval exchange with expansion (n k , m k , -η k η k+1 ); let t k be as in Lemma 3.3, v k as in Lemma 3.4; let

t k,l = l 2,o k l 1,o k t k,r = r 2,o k r 1,o k , v k,M = (m k -1)z k-1 + |M 2,o k-1 | (n k -1)z k-1 + |M 1,o k-1 | , v k,M m k z k-1 + |M 2,o k-1 | n k z k-1 + |M 1,o k-1 | , B k = 1 1 -η k η k+1 t k + v k 1 -v k , B k = 1 1 + η k η k+1 t k - v k v k + 1 .
Then B is the largest of the upper limits, when k → ∞, of the following quantities

(14) 1 v k -η k η k+1 t k , (15) (1
+ t k,l ) (1 + v k,M ) B k , (16) 1
+ 1 t k,l 1 + 1 v k,M B k , (17)
(1 + t k,r ) 1 + v k,M B k , (18) 1
+ 1 t k,r 1 + 1 v k,M B k ,
where in formulas [START_REF] Ferenczi | The structure of three-interval exchange transformations I: an arithmetic study[END_REF] to [START_REF] Ferenczi | Infinite words with uniform frequencies, and invariant measures[END_REF], when η k = +, t k,l and t k,r have to be exchanged, and v k,M and v k,M have to be replaced by v k,P and v k,P , where the M are replaced by P .

Proof We look at the possible frequencies of words from

n = o k-1 + 1 to n = o k ; suppose η k = -1. Then r 1,n is ∆ k from n = o k-1 + 1 to n = o k-1 + n k , then r 1,n is r 1,o k from n = o k-1 + n k + 1 to n = o k ; r 2,n is ∆ k from n = o k-1 + 1 to n = o k-1 + m k , then r 2,n is r 2,o k from n = o k-1 + m k to n = o k ; l 1,n takes values which are (by definition of the induction) larger than r 1,n from n = o k-1 + 1 to n = o k-1 + n k -1, then l 1,n is l 1,o k from n = o k-1 + n k to n = o k . l 2,n
takes values which are (by definition of the induction) larger than r 2,n from n

= o k-1 + 1 to n = o k-1 + m k -1, then l 2,n is l 2,o k from n = o k-1 + m k to n = o k .
By Lemmas 3.5 and 3.6, we get that the smallest value of ne n between n = |w 1,o k-1 | + 2 and n = |w 1,o k + 1| is reached by one of five quantities:

• pr 1,n = pr 2,n for p = |w 1,o k-1 | + 2, n = o k-1 + 1, • pl 1,n for p = |w 1,o k-1 +n k -1 | + 2, n = o k , • pl 2,n for p = |w 2,o k-1 +m k -1 | + 2, n = o k , • pr 1,n for p = |w 1,o k-1 +n k | + 2, n = o k , • pr 2,n for p = |w 2,o k-1 +m k | + 2, n = o k .
Note that if n k = 1, the first quantity is smaller than the second, and could be dispensed with, but it is easier to use all the five quantities than to make many special subcases.

The first one is z k-1 ∆ k , which gives formula [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF] in view of [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF].

The other ones are

l 1,o k |w 1,o k-1 +n k -1 |, l 2,o k |w 2,o k-1 +m k -1 |, r 1,o k |w 1,o k-1 +n k |, r 2,o k |w 2,o k-1 +m k |.
We deal first with the third and fourth ones: we write (3) for n = o k -1; we get that (l 6) and imputting the values of the lengths of the bispecial words, this gives [START_REF] Ferenczi | The structure of three-interval exchange transformations III: ergodic and spectral properties[END_REF], and ( 18) is similar.

1,o k + l 2,o k )z k-1 + ∆ k (|w 1,o k-1 +n k | + |w 2,o k-1 +n k | -2z k-1 ), which is equal to ∆ k (|w 1,o k-1 +n k | + |w 2,o k-1 +m k |) -(r 1,o k + r 2,o k )z k-1 , is close to 1; we write r 1,o k |w 1,o k-1 +n k | = (r 1,o k +r 2,o k )(|w 1,o k-1 +n k |+|w 2,o k-1 +m k |) r 1,o k + r 2,o k r 1,o k |w 1,o k-1 +n k | + |w 2,o k-1 +m k | |w 1,o k-1 +n k | , and notice that |w 1,o k-1 +n k | + |w 2,o k-1 +m k | is close to z k + z k-1 ; after deducing ∆ k r 1,o k +r 2,o k from (
The expression (3 15) and ( 16).

) for n = o k -1 is also close to (l 1,o k + l 2,o k )z k-1 + ∆ k (|w 1,o k-1 +n k -1 | + |w 2,o k-1 +m k -1 |), which is thus close to 1; as |w 1,o k-1 +n k -1 | + |w 2,o k-1 +m k -1 | is close to z k -z k-1 , this gives (
And the case η k = 1 is deduced as usual.

To some extent, the quantities B k and B k measure the quality of approximation of the angle of the inducing rotation by the semi-regular continued fraction [START_REF] Boshernitzan | A condition for minimal interval exchange maps to be uniquely ergodic[END_REF], which is best if t k or v k is close to 0 or 1, see Proposition 5.1 of [START_REF] Ferenczi | The structure of three-interval exchange transformations I: an arithmetic study[END_REF].

The value of B is not changed if we change a finite number of parameters of the expansion; this will be understated in the sequel when not recalled explicitely.

We remark that v k depends only on m j + n j for j ≤ k and -η j η j+1 for j ≤ k -1; η k η k+1 t k depends only on m j + n j for j ≥ k + 1 and -η j η j+1 for j ≥ k; t k,l and t k,r depend only on (n j , m j ) for j ≥ k + 1 and -η j η j+1 for j ≥ k; the v k,. and v k,. depend only on (n j , m j ) for j ≤ k and -η j η j+1 for j ≤ k -1. Proposition 3.8. B is the smallest of the lower limits, when k → ∞, of the following quantities

• when η k = -and |w 2,o k-1 +m k -1 | ≤ |w 2,o k-1 +m k | ≤ |w 1,o k-1 +n k -1 | ≤ |w 1,o k-1 +n k |: 1 ∆ k |w 2,o k-1 +m k -1 | , 1 l 2,o k |w 2,o k-1 +m k | , 1 min(l 2,o k ,r 2,o k )|w 1,o k-1 +n k -1 | , 1 min(l 1,o k ,l 2,o k ,r 2,o k )|w 1,o k-1 +n k | , 1 min(l 1,o k ,l 2,o k ,r 1,o k ,r 2,o k )z k ; • when η k = -and |w 2,o k-1 +m k -1 | ≤ |w 1,o k-1 +n k -1 | ≤ |w 2,o k-1 +m k | ≤ |w 1,o k-1 +n k |: 1 ∆ k |w 2,o k-1 +m k -1 | , 1 l 2,o k |w 1,o k-1 +n k -1 | , 1 min(l 1,o k ,l 2,o k )|w 2,o k-1 +m k | , 1 min(l 1,o k ,l 2,o k ,r 2,o k )|w 1,o k-1 +n k | , 1 min(l 1,o k ,l 2,o k ,r 1,o k ,r 2,o k )z k ;
• as in the first two cases with 1 and 2 exchanged, m and n exchanged;

• as in the first four cases with r and l exchanged and η k = +.

Proof

It is a straightforward consequence of the proof of Proposition 3.7.

We show how to compute effectively B for a particular family of cases: Proposition 3.9. Let n ≥ m ≥ 1 be two integers; we consider a three-interval exchange such that for all k n k = n, m k = m, η k η k+1 = -1. Then t is the smaller root of the polynomial X 2 + (m + n)X -1, and B is the largest of the three quantities

t + 1 t , (m + n) 2 m 2 1 1 + t + t 1 -t , 2(1 -t) 2 (1 -t) 2 + (m -n)t 2(1 + t) 2 (1 + t) 2 + (n -m)t 1 1 + t + t 1 -t .
Proof Then for every k we have

t k = v k = t. Thus r 1,o k + r 2,o k = ∆ k (1 -η k+1 )t, while r 1,o k -r 2,o k = ∆ k η k+1 u,
for some constant u; by [START_REF] Berth É | Covering numbers: arithmetics and dynamics for rotations and interval exchanges[END_REF] we get u = (n-m)t 1+t . Similarly, up to higher order terms,

z k-1 = tz k , and |M 1,o k | -|M 2,o k | = z k u with u = (n-m)t
1-t . Without loss of generality, we suppose η k = -1, and compute the various quantities in formulas ( 14) to [START_REF] Ferenczi | Infinite words with uniform frequencies, and invariant measures[END_REF]. Formula ( 14) becomes t + 1 t , and

B k = B k = 1 1+t + t 1-t .
Note also that formula [START_REF] Ferenczi | The structure of three-interval exchange transformations III: ergodic and spectral properties[END_REF] gives a smaller estimate than [START_REF] Ferenczi | Infinite words with uniform frequencies, and invariant measures[END_REF] and thus will not be used.

We look now at 1 + 1 t k,r ; inputting the above values, we get that it is equal to

2-2t 2
1-t 2 +(m-n)t ; but, using the equation defining t, we check that it is just equal to m+n m . Similarly, we get

1 + t k,l = 2(1+t) 2 (1+t) 2 +(m-n)t and 1 + 1 t k,l = 2(1+t) 2 (1+t) 2 +(n-m)t
, and these do not seem to admit simpler expressions.

In the same way, we get that 1

+ 1 v k,M
is equal to

2+ 2 t 2m+1+t+(m-n) t 1-t
; this last quantity turns out to be also equal to m+n m . Then 1 + v k,M is equal to

-2+ 2 t 2n-1+t+(n-m) t 1-t
, which, by using the equation defining t, is also equal to

2(1-t) 2
(1-t) 2 +(n-m)t . Finally 1 + 1 v k,M can be recovered from the last quantity and is equal to

2(1-t) 2
(1-t) 2 +(m-n)t . But at this stage we notice that formula [START_REF] Ferenczi | The structure of three-interval exchange transformations I: an arithmetic study[END_REF] gives a smaller estimate than [START_REF] Ferenczi | The structure of three-interval exchange transformations II: a combinatorial description of the trajectories[END_REF] and thus will not be used. Thus we have the three claimed formulas.

Proof

We look at formulas [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF] to [START_REF] Ferenczi | Infinite words with uniform frequencies, and invariant measures[END_REF]. The maximum of ( 15) and ( 16) is at least 4B k , while the maximum of ( 17) and ( 18) is at least 4B k .

We take a three-interval exchange with expansion (n k , m k , η k ) and look at the sequence (a k , ε k ) where

a k = m k + n k , ε k = -η k η k+1 .
If there are infinitely many a k ≥ 6, or (a k , ε k ) = (5, +), then for infinitely many k formula [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF] gives a bound greater or equal to 5. If there are only a k ≤ 4 or (a k , ε k ) = (5, -), and for infinitely many k we have ε k = -; then we have always t k ≥ [0, (5 * -) ω ] = 5- [START_REF] Ferenczi | Infinite words with uniform frequencies, and invariant measures[END_REF] gives at least 15+7 There remains only the case (a k , ε k ) = (4, +) and m k = n k , where B has the claimed value; note that each of the formulas ( 14) to [START_REF] Ferenczi | Infinite words with uniform frequencies, and invariant measures[END_REF] gives the same result.

v k,M = 3z k-1 +|M 1,o k-1 | z k-1 +|M 2,o k-1 | ≥ 3z k-1 z k-1 +|M 1,o k-1 |+|M 2,o k-1 | = 3z k-1 2z k-1 +z k-2 ; as z k-2 = v k-1 z k-1 , formula
If we take (a k , ε k ) to be ((2, +) 2 , (4, +) j ) ω , and m k = n k , the value of B is reached (among others) by 4B k for a k = 2 and a k-1 = 4; when j tends to infinity, these values of v k and t k both tend to [0, 2 * +(4 * +) ω ] = 1 √ 5 , and thus B tends to 6. The spectrum is closed by the standard reasoning of [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF], Chapter 1, Corollary to Lemma 6, as every finite value is reached with bounded m k and n k , and for any sequence of three-interval exchanges such that the corresponding B converge to a finite number, we can take the m k and n k uniformly bounded (and +∞ is in the spectrum, see Proposition 3.11 below). 6 is indeed an accumulation point, approached from below if j is even.

The above computations illustrate how B is easiest to compute when m k = n k for all k; when m k = n k , the situation becomes much more complicated; indeed, some of the bounds in the above computations may look quite crude, as for example if m k + n k = 3 we cannot have m k = n k and the bounds 4B k and 4B k are never reached, but in general they are not easy to improve.

In view of the considerations above, the values of B between 2 √ 5 and 6 were found only by trial and error; the second value is very likely to be 4 Thus the first, second, third, fourth smallest element we found in the upper BL spectrum of three-interval exchange transformations is respectively twice the first, second, sixth and twelfth Lagrange number. Though of course we might have missed some values, it seems likely that the upper BL spectrum of three-interval exchanges below 6 is strictly included in twice the Lagrange spectrum below 3; thus we conjecture that 6 is the lowest accumulation point of our spectrum.

When (n k , m k , -η k η k+1 ) is (2, 2, +) ω , the values of α = β and the angle of the inducing rotation can be deduced from (8); we get α = β = 3- √ 5

2 , and the inducing rotation is of angle 1 √ 5 ; as in that case [START_REF] Boshernitzan | A condition for minimal interval exchange maps to be uniquely ergodic[END_REF] gives a classical continued fraction expansion, the B of the rotation is computed by Theorem 2.4, and we check that the B of the interval exchange is the one of the inducing rotation. By similar computations, for (1, 1, +) ω , the B of the interval exchange is twice the one of the inducing rotation. For ((1, 1, +) 2 , (2, 2, +) j ) ω with j large, the B of the interval exchange is close to 6, while the one of the inducing rotation is between 4 and 5.

A partial analogy with the Lagrange spectrum is that in the latter the sequence increasing to 3 is given by angles whose Euclid continued fraction expansion is [0, (2 2 1 2j ) ω ], while in our upper BL spectrum the sequence increasing to 6 is given by quantities t k whose Euclid continued fraction expansion is [0, (2 2 4 2j ) ω ]; but it fails for the angles whose classic continued fraction expansion is [0, (2 4 1 2 ) ω ], which gives a rotation with B = 2, 9992...., while the threeinterval exchange for which (n k , m k , -η k η k+1 ) is ((1, 1, +) 4 , (2, 2, +) 2 ) ω has B = 6, 06..... All the values of B we have found below 6 are reached by the estimates 4B k or 4B k , though when (n k , m k , -η k η k+1 ) is either (2, 2, +) ω or (3, 3, -) ω they are also reached by formula [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF]. The smallest value we found without m k = n k for all k large enough is 5 The upper BL spectrum above 6 seems to become quite complicated, maybe more than the Lagrange spectrum above 3. However, we can find enough values given by formula [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF]

Proof

We follow the proof of Theorem 2 in Chapter 4 of [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF], which in turns follows [START_REF] Hall | On the sum and product of continued fractions[END_REF] . We can build G as a Cantor set in the following way: we start form the

• if m = r + 1 = s, let q = n i-1 and suppose q < m -2; then θ q+1 ≥ θ m-1 + θ m ≥ 4θ m , and θ q ≥ θ m-2 ≥ 2(θ m-1 + θ m ) ≥ 8θ m , we get B ≥ 12, • if m = r + 1 = s and k m ≥ 4; then θ r+1 ≥ θ s+1 , and θ r ≥ 4θ s + 4θ s+1 , we get B ≥ 9. Hence either B ≥ 9, or k n takes only the values 1, 2, 3, with 3 possible only if r = m -1 and q = m -2 in the notations above. In this last case θ m-1 ≥ 3(θ m + θ m+1 ), and θ m-2 ≥ θ m-1 + θ m + θ m+1 ≥ 4(θ m + θ m+1 ). If k m+1 = 3, we have θ m-1 ≥ 9θ m+1 and, computing it from m -1, we get B ≥ 9. If k m+1 ≤ 2, we have θ m ≤ 2θ m+1 + 2θ m+2 + θ m+3 while θ m+2 ≤ θ m+1 -θ m+3 , hence θ m ≤ 4θ m+1 , and, computing it from m-2, we get B ≥ 35 4 = 8, 75. Thus we can suppose now that the k n take only the values 1 and 2; then we can improve the estimates on the ratios of lengths, with

2|Jn i | Sn i ≥ 1
3 , and, as the estimates from [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF] ; the lower bounds on the ratios θ m+1 θm are also improved. Thus we can prove that if for infinitely many m either k m = k m+1 = 2, or k m = 2 when m is not an n i , or m is not an n i and m + 1 is not an n i , B ≥ 181 21 Similarly, further improvement of the estimates, and extensive computations, allow us to eliminate all the k m = 2 and all the m which are not an n i .

Theorem 4.4. The smallest element in the lower BL spectrum of the family of Arnoux-Rauzy systems is 2, and the largest is +∞. Every integer greater or equal to 2 is in the lower BL spectrum, and is an accumulation point, as is +∞.

Proof

The smallest element is at least 2 by Lemma 1.2. We take n i = 2i, with k 2n+1 = 1 and a sequence k 2n growing to +∞. k 2n+4 , both θ 2n+1 θ 2n+3 and θ 2n θ 2n+3 are smaller than 2k 2n+2 (1 + n ), where n → 0.

Putting everything together, we get that B is at most 2, and thus equal to 2.

We take now n i = i, with a sequence k n tending to +∞. Then for all n we have

|J n | < |G n | < |H n |, S n+1 = (2k n+1 -1)|H n | + |G n | + |J n | < (2k n+1 + 1)|H n |, θ n ≥ k n+1 θ n+1 ≥ k n+1 k n+2 θ n+2
and thus B is at least the lower limit of 2k n+1 k n+2 2k n+1 +1 , which gives +∞.

If we replace k 2n+1 = 1 by k 2n+1 = m in the first construction of the proof above, we get B = m + 1, for any integer m ≥ 1.

If we replace the variable k n or k 2n in the constructions above by a constant k and let k tend to infinity, we get sequences in the lower spectrum tending to +∞ or to m + 1, for any integer m ≥ 1.

our new condition for being natural codings, and thus we need new ideas to find natural codings of these rotations.

Questions

N. Pytheas Fogg (private communication) asked what should be the BL spectra of the family of all uniquely ergodic symbolic systems. The upper BL spectrum contains the union of the Lagrange spectrum and +∞, and it is quite possible that it is just that. The lower BL spectrum contains the lower BL spectrum of rotations, and the interval [2, +∞], but also at least the point 3 2 , which is the value of B for the so-called period-doubling symbolic system, whose language is generated by the fixed point of the substitution a → ab, b → aa (N. Pytheas Fogg, unpublished).

N. Pytheas Fogg has also started to investigate the spectrum of the joint values of (B, B ) for rotations, see https://www.lirmm.fr/ monteil/hebergement/pytheas-fogg/BL spectrum.pdf.

  l i,n and r i,n are built by Definition 3.2, starting from n = 1, for which we are in State I. For each n > 0 ; let s n (1) = 1, s n (2) = 2 if for n we are in I or III, s n (1) = 2, s n (2) = 1 if for n we are in II; p n (1) = 1, p n (2) = 2 if for n we are in I, p n (1) = 2, p n (2) = 1 if for n we are in II or III; m n (1) = 1, m n (2) = 2 if for n we are in III, m n (1) = 2, m n (2) = 1 if for n we are in I or II. Then for 1 ≤ i ≤ 2

3. 4 . 13 = 4 ,

 4134 The upper BL spectrum: smallest elements. Theorem 3.10. The smallest element in the upper BL spectrum of three-interval exchange transformations, and the only one below 12+29 √ 3 786..., is 2 √ 5 = 4, 47.... It is reached if and only if for all k large enough m k = n k = 2 and η k η k+1 = -. The spectrum is a closed set and contains an accumulation point equal to 6.

√ 21 2 4 √ 21 3 = 6 , 6 - 2 . 1 √ 6 5 = 5 , 4 √ 3 -6 3 , 4 √ 3 -3 13 and thus 4B k is at least 12+29 √ 3 13 = 4 ,

 2421366216554334313134 and also v k ≥ [0, (5 * -) ω ]. Then if ε k = -, 4B k is at least 11... If there are only (a k , ε k ) = (3, +), (a k , ε k ) = (4, +) or (a k , ε k ) = (2, +), with infinitely many (2, +); then we have always v k ≥ [0, (2 * +, 4 * +) ω ] = √ If (a k+1 , e k+1 ) = (2, +), then t k ≥ [0, 2 * +(2 * +, 4 * +) ω ] = and thus 4B k is at least 8+8 √ 6 519.... If there are only (a k , ε k ) = (3, +) or (a k , ε k ) = (4, +), with infinitely many (3, +); then we have always v k ≥ [0, (3 * +, 4 * +) ω ] = If (a k+1 , e k+1 ) = (3, +), then t k ≥ [0, 3 * +(3 * +, 4 * +) ω ] = 786.... If there are only (a k , ε k ) = (4, +), with for infinitely many k m k = n k ; then t k = v k = √ 5-2. We take a k such that m k = n k ; without loss of generality we take η k = -. If m k = 1 and n k = 3, then r 1,o k > r 2,o k , and 1

√ 5 4 = 7 ,

 47 663... If m k = 3 and n k = 1, formula (17) gives the same estimate.

√ 2 = 5 , 26 17 = 5 , 4 √ 209306 305 = 5 ,

 252617543055 65..., which is reached when (n k , m k , -η k η k+1 ) is either (1, 1, +) ω or (3, 3, -) ω ; the latter gives the minimal value for m k and n k constant and η k η k+1 = +1. The third value we found is 20 √ 9988..., which is reached for ((1, 1, -) 2 , (2, 2, -) 2 ) ω and the fourth one is 999996..., which is reached for ((1, 1, -) 2 , (2, 2, -) 4 ) ω .

√ 29 4 = 6 , 5 . 3 . 5 .Proposition 3 . 11 .

 46535311 73..., which is reached when (n k , m k , -η k η k+1 ) is (3, 2, +) ω ; for (3, 1, +) ω we get B = 8 √ The upper BL spectrum: largest elements. For a three-interval exchange, B = +∞ if and only if the angle of the inducing rotation has unbounded partial quotients for the Euclid algorithm. Proof By the formulas in Proposition 3.7, B = +∞ if and only if either m k + n k is unbounded, or t k or v k takes values arbitrarily close to one. The last two possibilities are equivalent to the existence of unbounded strings of m k + n k = 2, -η k η k+1 = -, and this gives the result by (8) and Section 1.3.

2 √

 2 (this happens when all m k = n k and some of them are large enough) to fill an interval [C, +∞[. This involves adapting to semi-regular continued fraction expansions the famous Theorem 3.1 of [23]: Lemma 3.12. Let G be the set of numbers [0, a 1 * ε 1 , ..., a n * ε n , ....] where all the a i are 2, 4, 6 or 8, the ε i are -or +, and there are never two consecutive 2 * -; then G + G is the interval [ 23108..., 1, 30949...], and G -G is the interval [ 53920..., 0, 53920...].

-12 3 ,

 3 . The largest number in G is [0, (2 * -, 2 * +, 8 * +) ω ] = √ 195and the smallest one is [0, (8 * +, 2 * -, 2 * +) ω ] = √ 195-12 17

  Then we have |J 2n | < |G 2n | < |H 2n |. |H 2n+2 | = k 2n+2 (|H 2n | + |G 2n |) + |H 2n |, |G 2n+2 | = k 2n+2 (|H 2n | + |G 2n |) + |H 2n | + |J 2n |, |J 2n+2 | = |H 2n | + |G 2n | . When n tends to +∞, |G 2n | |H 2n | → 1 and |J 2n | |H 2n | → 0, thus |J 2n | |S 2n+2 | → 0 and both |H 2n | |S 2n+2 | and |G 2n | |S 2n+2 | are equivalent to 1 4k 2n+2 .As for the ratios of frequencies,θ 2n+1 = k 2n+2 (θ 2n+2 + θ 2n+3 ) + θ 2n+4 = k 2n+2 (2θ 2n+3 + θ 2n+4 + θ 2n+5 ) + θ 2n+4 ≤ k 2n+2 (2θ 2n+3 + 2θ 2n+4 ) + θ 2n+4 , and θ 2n = θ 2n+1 + θ 2n+2 + θ 2n+3 ≤ (k 2n+2 + 1)(2θ 2n+3 + 2θ 2n+4 ) + θ 2n+4 . As θ 2n+4 ≤ θ 2n+3

  2424... If there are only 1 and 2 with infinitely many 22, if b k

  imply that either |S n i | ≤ 2|G n i | -|J n i | or |S n i | ≤ 2|H n i | -|J n i |, either

	2|Hn i | Sn i	or	2|Gn i | Sn i	are greater
	than 8 7

Institut de Mathématiques de Luminy, CNRS -UMR 6206, Case 907, 163 av. de Luminy, F13288 Marseille Cedex 9 (France), and Fédération de Recherche des Unités de Mathématiques de Marseille, CNRS -FR 2291 E-mail address: ferenczi@iml.univ-mrs.fr

We check that this dissection can be done in successive stages, where each time we delete the interval A 2 from the interval A 1 ∪ A 2 ∪ A 3 , and the length of A 2 is smaller than the length of A 1 and than the length of A 3 ; then Lemmas 2 to 4 of [START_REF] Cusick | The Markoff and Lagrange spectra[END_REF] give the result. The same reasoning works for G+(-G). Proof By Proposition 3.11, the upper BL spectrum contains +∞. Let s be a real number larger than 12 + 2 √ 2; by Lemma 3.12, we can write

where r is an even positive integer, all the a i and a i are 2, 4, 6 or 8, the ε i , ε i , ε and ε are -or +, and there are never two consecutive 2 * -in the a n * ε n and a n * ε n . We choose some increasing sequence k n such that a kn * ε kn is not 2 * -.

We define the expansion (

.. We take now the three-interval exchange for which m k + n k and -η k η k+1 are defined in that way, and m k = n k for all k; it exists as this gives an admissible expansion. By Proposition 3.7 B is the maximum of the upper limits of 1 v k -η k η k+1 t k , and of 4B k and 4B k . As r is at least 14, the first of these upper limits can be taken on those k for which m k + n k = r, and is exactly s.

We look now at the other two upper limits, for which we need an upper bound for t k and v k . If we compute t k when (m k + n k , -η k η k+1 ) is inside a string of (a n , ε n ), we do not see two consecutive (2, -) in that string, while, when we start inside a string of (a n , ε n ), in that string we do not see (a, -) followed by (2, -) followed by (2, f ) for any a and f , and the two cases are exchanged if we compute v k ; in both cases starting from a (r, ε ) or near a junction of two different strings does not change the bounds. Thus either t k is at most [0, 2 * -, 2] = 2 3 , and

We input these values in the quantities we want to bound, together with t k > 0, v k > 0; the worst case happens when (m k + n k , -η k η k+1 ) = (2, -), and we get that the last two upper limits are not larger than 12 + 2 √ 2 ≤ s.

The lower bound on the interval we give should certainly be improved, but let us point that this would need a different method, as, to improve significantly the bound in Lemma 3.12, we would need to replace G by a set where the expansions do not contain 2 * -, and such a set does not contain elements below 1 2 . 3.6. The lower BL spectrum. Proposition 3.8 allows us to compute B for individual interval exchanges, by the usual trick of replacing the 1 in the numerators by the quantity in relation [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF], thus we have to estimate ratios of lengths and frequencies; for example, for

we check that these three quantitites are all equal and that B = 5 -√ 5 = 2, 763... But if we want to compute B for a general three-interval exchange, these formulas are complicated and do not reduce to simpler expressions as for B. However, the lower BL spectrum for the family is fully known, and quite different from the lower BL spectrum of rotations.

Theorem 3.14. The lower BL spectrum of the family of three-interval exchanges is the interval [2, +∞].

Proof

No number smaller than 2 can be in the lower BL spectrum because of Lemma 1.2. We take 2 ≤ s ≤ +∞, and choose integers m k < n k such that m k and n k grow to infinity with k, and m k +n k m k → s when k goes to infinity; we require n k -m k → +∞, which is an additional condition when s = 2; we choose η k = -for all k.

Then by Lemmas 3.3 and 3.4, we have

while, by the computations in the proof of Proposition 3.7

, thus the first lower limit is a limit and is equal to s. The same is true for the second one as 1 ∆ k+1 z k and ∆ k ∆ k+1 are equivalent to m k+1 + n k+1 , and, because r

Thus for some uniquely ergodic three-interval exchange transformations we have ne n → 0 when n tends to infinity; this result, and its consequence that Boshernitzan's criterion is not a necessary condition in this family of systems, are stated without proof in [START_REF] Veech | Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form[END_REF]. Note that the covering number by intervals (see the opening of Section 2 above) of a three-interval exchange is shown in [START_REF] Berth É | Covering numbers: arithmetics and dynamics for rotations and interval exchanges[END_REF] to be the same as for the inducing rotation, and thus is not equal to 1 B , in contrast with the case of rotations.

Arnoux-Rauzy systems

The Arnoux-Rauzy systems are defined in [START_REF] Arnoux | Représentation géométrique de suites de complexité 2n + 1[END_REF] as the minimal symbolic systems on the alphabet {1, 2, 3} such that the complexity of the language is 2n + 1 for all n, and, for all n, there are one right special and one left special word. Then [START_REF] Arnoux | Représentation géométrique de suites de complexité 2n + 1[END_REF] proceeds to give a constructive (additive) algorithm to generate them with three families of words, built with three rules denoted by a, b and c; [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF] gives a multiplicative version of this construction, which we take here as a definition, valid up to permutations of {1, 2, 3}: the k n are the number of consecutive times a given rule is used, while the n i > 1 mark the times where three consecutive rules are all different, such as, up to permutations of {a, b, c}, rule a used k n i -1 times, then rule b used k n i times, then rule c used k n i +1 times. Definition 4.1. Given two infinite sequences of integers k n ≥ 1, n ≥ 1, and n 1 < n 2 ... < n i < ... the Arnoux-Rauzy system (X L , S) defined by them is the symbolic system associated to the language L of all factors of (H n ) n∈IN , where the three words H n , G n , J n are built from H 0 = 1, G 0 = 2, J 0 = 3 by two families of rules:

Every Arnoux-Rauzy system is minimal [START_REF] Arnoux | Représentation géométrique de suites de complexité 2n + 1[END_REF] and uniquely ergodic (by [START_REF] Boshernitzan | A unique ergodicity of minimal symbolic flows with linear block growth[END_REF] because the complexity is 2n + 1). Though they are defined as symbolic systems, they have also geometric models, see [START_REF] Arnoux | Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore[END_REF] [2] [3] [START_REF] Rauzy | Nombres algébriques et substitutions[END_REF]: every Arnoux-Rauzy system is a coding of a six-interval exchange on the circle, and some of them are codings of rotations of the 2-torus. Proposition 4.1. Let o n = k 1 + ...k n , n > 0, and o 0 = 0. All the bispecial words of the language are the w n , where w 0 is the empty word and for 1 ≤ j ≤ k n+1 , w on+j = w on H j n . For 1 ≤ j ≤ k n+1 , let θ n,j , resp. γ n,j , ι n,j , be the frequency of w on+j x where x is the first letter of H n+1 , resp. G n+1 , J n+1 ; we denote θ n,k n+1 , resp. γ n,k n+1 , ι n,k n+1 , by θ n+1 , resp. γ n+1 , ι n+1 , then

Proof This is a straightforward consequence of [START_REF] Arnoux | Représentation géométrique de suites de complexité 2n + 1[END_REF], taking into account the multiplicative definition, as in Proposition 9 of [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF].

We check that we have the relations [START_REF] Ferenczi | A self-dual induction for three-interval exchange transformations[END_REF] |H

We shall use also Lemma 7 of [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF], with the more precise (and non-trivial) estimate used in its proof: namely, for all n, if (t 1,n , t 2,n , t 3,n ) is the triple

Proof On the Rauzy graph of length m, there are at most four frequencies, which are for example those of w on+j 1, w on+j 2, w on+j 3, and w on+j , thus θ n,j , γ n,j , ι n,j and θ n,j + γ n,j + ι n,j , for the For all n, the formulas imply that θ n ≥ γ n and θ n ≥ ι n We have always θ n+1 = γ n and either γ n+1 = ι n or ι n+1 = ι n , thus always γ n ≥ ι n . Then for 1 ≤ j ≤ k n+1 -1, we have either

Taking into account that ι n+1 = ι n if and only if n + 1 is not an n i , we get that for every i ≥ 1, e r is equal to

We have . We now input the relation [START_REF] Ferenczi | A self-dual induction for three-interval exchange transformations[END_REF], and the fact that for any m, γ m = θ m+1 and, if m is the smallest

Note that in the formula giving B, the ratios of lengths of words depend only on the n j for j ≤ i and the k t for t ≤ n i , while the ratios of frequencies depend only on the n j for j > i and the k t for t > n i ; in the formula giving B , there is no such dichotomy. To every Arnoux-Rauzy system is associated an algoritm of simultaneous approximation of two irrationals, see [3] [11]; this involves the θ n and the |H n |, and the approximation is best when the k n are large; this algorithm is hidden in all the computations of this section, though only B is linked to the quality of the approximation, and only in a loose way.

Proposition 4.3. The upper BL spectrum of the family of Arnoux-Rauzy systems contains +∞, which is reached if and only if the k n , n ∈ IN, or the n i+1 -n i , i ≥ 1, are unbounded. Its smallest element, and the only one below 181 21 = 8, 619..., is reached for the Tribonacci system where n i = i for all i ≥ 1 and k n = 1 for all n ≥ 1; for this system, if y = 1, 8392... is the root bigger than 1 of the polynomial X 3 -X 2 -X -1, then B = 2y 2 + 4y y 2 +1 = 8, 4445.... Proof We use the estimates from [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF]; if n is an n i , the smallest of the three lengths is |J n |, and thus either

are between 1 and 2, while

Thus the finiteness of B depends on the two ratios θn i θ n i+1 +1 and

θ n i+1 +1 , and this gives the assertion on the highest value. For the Tribonacci system, we get Let us now sketch the proof that in all other cases B ≥ 181 21 . We suppose first that k m ≥ 3 for infinitely many m; we take such an m; let r = n i be the largest n i < m, s = n i+1 .

• if m = r + 1 < s: then θ r+1 ≥ θ s+1 , and θ r ≥ 3θ r+1 + θ r+2 + 3θ s+1 ≥ 3(θ r+2 + θ r+3 ) + θ r+2 + 3θ s+1 ≥ 10θ s+1 ; putting all our estimates together, we get B ≥ 11,

Thus we have a new, and very simple, family of counter-examples to the necessity of Boshernitzan's criterion.

Of course, there are other values in the lower spectrum than those in Theorem 4.4, and we conjecture that the lower BL spectrum of the family of Arnoux-Rauzy systems is the interval [2, +∞].

We look at some individual values of B and B , for three examples which are codings of rotations of the 2-torus [START_REF] Arnoux | Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore[END_REF] [START_REF] Arnoux | ITO: Pisot substitutions and Rauzy fractals[END_REF].

For the Tribonacci system B = 2y + 4 y 2 +1 = 4, 5911.... In [START_REF] Chekhova | Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci[END_REF], various constants are computed for the Tribonacci system, including one covering number, though not by intervals (see Section 2 above), and a measure of the quality of simultaneous approximation of ( 1y , 1 y 2 ) by rationals; none of these is equal to B or B .

For

, and when n is large H n+1 is close to y 0 H n while θ n is close to y 0 θ n+1 , if y 0 = 2, 83... is the root bigger than 1 of the polynomial X 3 -2X 2 -2X -1. We get B = 2y 2 0 + 4y 0 y 2 0 +y 0 +1 = 16, 96..., B = 2y 0 + 4 y 2 0 +y 0 +1 = 5, 99... For n i = 2i and k n = 1 for all n, we get

y 1 y 2 +y 1 +1 + 1 y 1 ) = 11, 61..., which is a candidate for the second smallest value in the upper spectrum, and B = B y 1 y 2 = 3, 61... The Arnoux-Rauzy systems raise questions about rotations of the 2-torus, and we may ask what could be the BL spectra for that family of systems, but the problem is that at this time we do not know any coding which may be called natural. In Section 3 of [START_REF] Berth É | Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly, Algebraic and topological dynamics[END_REF], there is a discussion about what should be the properties of such a coding; in view of the present paper, by analogy with the rotations of the 1-torus, it becomes reasonable to add to these properties the boundedness of the lower BL spectrum these natural codings would define for the family of rotations of the 2-torus. Now, if we code a rotation of the 2-torus with the Cartesian product of two partitions of the 1-torus, then the complexity is quadratic and all B and B are infinite by Lemma 1.2, which gives another trivial counter-example to the necessity of Boshernitzan's criterion, but this coding has never been considered as natural. The Arnoux-Rauzy systems were devised to provide codings with sub-linear complexity for rotations of the 2-torus, but this was succesful only in a limited number of cases. Still, if we consider these cases, the tentative lower BL spectrum of the family of rotations of the 2-torus seems to be quite different from the lower BL spectrum of rotations of the 1-torus: if we take an Arnoux-Rauzy system with n i = i and constant k n = k, it is a coding of a rotation of the 2-torus by [START_REF] Arnoux | ITO: Pisot substitutions and Rauzy fractals[END_REF], and these give arbitrarily high values for B ; if n i = i and k n grows slowly (for example k n ≤ 1 15 n), we get an infinite B while the Arnoux-Rauzy system is shown in [START_REF] Cassaigne | Weak mixing and eigenvalues for Arnoux-Rauzy sequences[END_REF] to have two continuous eigenfunctions, and is still conjectured to be a coding of a rotation of the 2-torus. Thus it seems that even those Arnoux-Rauzy which do code rotations of the 2-torus fail to satisfy