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STRUCTURE OF K -INTERVAL

EXCHANGE TRANSFORMATIONS:

INDUCTION, TRAJECTORIES, AND DISTANCE THEOREMS

By

SEBASTIEN FERENCZI AND LUCA Q. ZAMBONI

Abstract. We define a new induction algorithm for k-interval exchange trans-

formations associated to the “symmetric” permutation i 7→ k − i + 1. Acting as a

multi-dimensional continued fraction algorithm, it defines a sequence of general-

ized partial quotients given by an infinite path in a graph whose vertices, or states,

are certain trees we call trees of relations. This induction is self-dual for the duality

between the usual Rauzy induction and the da Rocha induction. We use it to de-scribe those words obtained by coding orbits of points under a symmetric interval

exchange, in terms of the generalized partial quotients associated with the vector

of lengths of the k intervals. As a consequence, we improve a bound of Bosher-

nitzan in a generalization of the three-distances theorem for rotations. However,

a variant of our algorithm, applied to a class of interval exchange transformations

with a different permutation, shows that the former bound is optimal outside the

hyperelliptic class of permutations.

1 Preliminaries

Interval exchange transformations were originally introduced by Oseledec [27],

following an idea of Arnold [2]; see also Katok and Stepin [20]. An exchange of

k intervals, denoted throughout this paper by I, is given by a probability vector of

k lengths (α1, . . . , αk) together with a permutation π on k letters. The unit interval

is partitioned into k subintervals of lengths α1, . . . , αk, which are rearranged by I

according to π. It was Rauzy [29] who first saw interval exchange transformations

as a possible framework for generalizing the well-known interaction between cir-

cle rotations on one hand and Sturmian sequences on the other via the continued

fraction algorithm (see, for example, the survey [7]). He defined an algorithm of

renormalization for interval exchange transformations, now called Rauzy induc-

tion, which generalizes the Euclidean algorithm and coincides with it for k = 2.

The Rauzy induction, further developed by Veech [31], and modified by Zorich

[35] and more recently by Marmi, Moussa and Yoccoz [25], had a tremendous

JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 112 (2010)

DOI 10.1007/s11854-010-0031-2

289

290 SEBASTIEN FERENCZI AND LUCA Q. ZAMBONI

success in solving many problems associated with interval exchange transforma-

tions, from unique ergodicity [26][33] to weak mixing [4]. But, in contrast with

the Sturmian case, though this induction gives access to the symbolic dynamics

of the trajectories, it is difficult to describe them explicitly and indeed the Rauzy

induction was never actually used for that purpose. A partial description of the

trajectories was given in [24], and it aroused interest by defining a different kind

of induction; this da Rocha induction was then considered to be dual to the Rauzy

induction: the Rauzy induction builds the points where the orbits of the disconti-

nuities of I approximate 0, while the da Rocha induction builds the points where

the orbit of 0 approximates the discontinuities of I−1. Note that (non-constructive)

combinatorial characterizations of the language of trajectories were given recently

in [16] and [5].

These inductions are also a fundamental tool in the study of the space of mod-

uli of Riemann surfaces, and the various strata of its unit tangent bundle, through

what has been a basic object of interest for the last 25 years, the Teichmuller flow

on a stratum; in the case of interval exchanges, we consider the Teichmuller flow

for the Riemann surface obtained by glueing parallel opposite sides of a polygon.

The Rauzy induction chooses an initial segment of a horizontal separatrix and fol-

lows its vertical separatrix till it intersects this initial sgement, and then considers

shorter and shorter initial segments. This approach led to the proof of many re-

sults, but a basic flaw is that we only consider one horizontal separatrix; the da

Rocha induction considers all the horizontal separatrices and one vertical sepa-

ratrix, and its duality with the Rauzy induction appears in the natural extension

of the induction process. The trouble with both procedures is that they destroy

the symmetry of the geometrical situation, by giving a special role to one of the

separatrices; because of that, each foliation admits several descriptions, and the

relative position of the separatrices is not taken into account.

Our aim is to describe a new induction algorithm for interval exchange trans-

formations, which gives information that the existing inductions do not give: our

prime motivation is to describe completely the trajectories of points, and to relate

both the combinatorial and dynamical properties of the underlying system to the

number-theoretic properties of an associated multi-dimensional continued fraction

algorithm. But this leads us to a self-dual induction (for the duality mentioned

above), which assigns the same role to each horizontal and vertical separatrix.

In this paper, carrying over the results of earlier work of the authors together

with C. Holton [13] from k = 3 to the much more involved case of 4 intervals

and more, we accomplish what we wanted for the class of interval exchange trans-

formations where the permutation is the “symmetric” one σ(i) = k + 1 − i for all

STRUCTURE OF K -IETS 291

1 ≤ i ≤ k, and where the discontinuities βi of I−1 and γi of I satisfy the inequal-

ities β1 < γ1 < β2 < γ2 < · · · < βk < γk (in this case, we say I is a symmetric

k-interval exchange transformation with alternate discontinuities); this corre-

sponds to a special case of the hyperelliptic stratum. For this class, we define

a new induction algorithm, where at each stage we induce I on a disjoint union

of k − 1 intervals, each one containing a discontinuity βi and having its extremi-

ties on orbits of the discontinuities γ j . This ensures the self-duality (see [14] for

k = 3), and also that each of our intervals is the cylinder of a so-called bispecial

factor of the trajectories. Our induction algorithm defines a multi-dimensional

continued fraction algorithm generated by the 2k − 2 non-independent parameters

corresponding to the lengths of each side of βi in our k − 1 intervals. It constitutes

a non-trivial generalization of the Euclid algorithm and, unlike other such algo-

rithms arising in the context of symbolic dynamics [3, 18], it is defined on a set of

full measure. As in the case of the other inductions, it is described by an infinite

path in a finite graph, whose vertices are the states of the inductions.

But the self-duality implies that, in contrast to the Rauzy and da Rocha in-

ductions, where only one parameter is changed at each step, the induction will

be neither unique nor straightforward to implement; as there is no canonical order

between the parameters to be changed, there are choices to make (see Section 2.2).

The problem we have to solve is the same as the problem of induction of a train

track [28], which is not solved in the general case. Here we solve it by choosing

carefully our set of states. For the Rauzy induction and its multiplicative acceler-

ations, the states are defined by permutations; in our case, as in da Rocha’s, each

state corresponds to a set of relations between the parameters, suitably described

in terms of an abstract object called a tree of relations. Their properties allow

us to make the self-dual induction work and get a deterministic and completely

symmetric process.

Each tree of relations describes the induced map of I in the corresponding state

of the induction, and the sequence of trees of relations defines inductively the bi-

special words of the trajectories; thus the path of our induction through the states

in the graph of graphs defines directly the bispecial factors, together with the cor-

responding intervals, in the same way as the usual coding of Sturmian sequences

stems from the sequence of partial quotients of the slope.

Our induction is more difficult to describe for interval exchanges given by ar-

bitrary permutations π; we give an example in a particular case in Section 5. How-

ever, the alternating condition on the discontinuities plays a less critical role and is

removed in Section 3.1, by changing a finite number of initial steps of the induc-

tion and using composite trees of relations.


Our induction algorithm is used to obtain several new results of both a com-

binatorial and dynamical nature: we give an S-adic presentation of symmetric

interval exchange transformations and characterize those whose trajectories are

self-similar and generated by a primitive substitution rule. Then we answer a

question of Boshernitzan on the repetition of words in the trajectories, and finally

we prove a generalization of the well-known 3-distances theorem for rotations

(see [1] for a survey). For k-interval exchange transformations in the hyperelliptic

class, we prove that the (k − 1)n + 1 subintervals created by the n first iterates

of the discontinuities have at most 2k − 1 different lengths, improving the bound

3k − 3 given by Boshernitzan [8]. It is commonly thought that the transformation

I behaves “better” when π is in the hyperelliptic class, in the sense that typically

results are easier to prove. However here, for the first time to our knowledge, we

show an actual difference of behaviour, by showing that Boshernitzan’s bound is

optimal outside the hyperelliptic class. Another major application of our induction

is to obtain, in [15], new ergodic results concerning 4-interval exchanges, solving

some longstanding questions on eigenvalues and the property of simplicity in the

sense of Veech [32].

1.1 Interval exchanges.

Definition 1.1. A k-interval exchange transformation I with proba-

bility vector (α1, α2, . . . , αk) and permutation π is defined by

(1) Ix = x +∑

π( j )<π(i)

α j −∑

j<i

α j

when x is in the interval[

∑

j<i

α j ,∑

j≤i

α j

[

,

and this interval is denoted by 1i .

We denote by βi , 1 ≤ i ≤ k − 1, the i-th discontinuity of I−1, namely βi =∑

π( j)≤π(i) α j , while γi is the i-th discontinuity of I, namely γi =∑

j≤i α j . Then

1i is the interval [γi−1, γi[ if 2 ≤ i ≤ k−1, while11 = [0, γ1[ and1k = [γk−1, 1[.

Definition 1.2. I satisfies the infinite distinct orbit condition or i.d.o.c. of

Keane [21] if the k − 1 negative orbits {I−nγi}n≥0, 1 ≤ i ≤ k − 1, of the disconti-

nuities of I are infinite disjoint sets.

The i.d.o.c. condition implies that I has no periodic orbit and is minimal: ev-

ery orbit is dense. If π is primitive, that is, π({1, . . . , j}) 6= {1, . . . , j} for every

1 ≤ j ≤ k − 1, the i.d.o.c. condition is (strictly) weaker than total irrationality,

where the only rational relation satisfied by αi , 1 ≤ i ≤ k − 1, is∑k

i =1 αi = 1.


Definition 1.3. For every point x in [0, 1[, its trajectory is the infinite se-

quence (xn)n∈IN defined by xn = i if Inx falls into 1i , 1 ≤ i ≤ k.

Definition 1.4. A symmetric k-interval exchange transformation is a

k-interval exchange transformation I with permutation π = σ, where σ j = k+1− j ,

1 ≤ i ≤ k.

Definition 1.5. The induced map of any transformation 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 ry is in Y

(in all cases considered in this paper, r(y) is finite).

1.2 Word combinatorics. We look at finite words on a finite alphabetA =

{1, . . . , k}. A word with r letters, w1 . . .wr , 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.

Definition 1.6. A word w = w1 . . . wr occurs at place i in a word v =

v1 . . . vs or an infinite sequence v = v1v2 . . . if w1 = vi , . . .wr = vi+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.

Definition 1.7. A language L over K is a set of words such that 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 minimal if for each w in L, there exists n such that w occurs

in each word of length n of L.

The language L(u) of an infinite sequence u is the set of its finite factors.

Definition 1.8. A language L being fixed, for a word w, we call the set of all

letters x such that xw is in L the arrival set of w and denote it by A(w). We call

the set of all letters x such that wx is in L the departure set of w and denote it

by D(w).

A word w is called right special, resp. left special, if #D(w) > 1, resp.,

#A(w) > 1. If w is both right special and left special, then w is called bispecial.

Definition 1.9. Given a wordw = w1w2 . . .wr , letw denote the retrograde

word of w, that is w = wrwr−1 . . .w1.

A language is symmetric if it is is invariant by the operation w → w.

Definition 1.10. The symbolic dynamical system associated to a lan-

guage L is the one-sided shift S(x0x1x2 . . .) = x1x2 . . . on the subset XL of AIN

made with the infinite sequences such that for every r < s, xr . . . xr+s−1 is in L.


For a word w = w1 . . . wr in L, the cylinder [w] is the set

{x ∈ XL; x0 = w1, . . . , xr−1 = wr}.

If the transformation I is minimal, all its trajectories have the same finite fac-

tors, whose set is called the language L(I), and this language is minimal. The

arrival sets, departure sets, and special words depend on the language and not on

the individual trajectories. In the sequel, a transformation I being fixed, all these

objects are those defined by any trajectory of I. 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 XL(I).

2 Induction in the symmetric case with alternate dis-continuities

2.1 Alternate discontinuities.

Definition 2.1. Let I be a k-interval exchange transformation; it has alter-

nate discontinuities if βi < γi for each 1 ≤ i ≤ k − 1 and γi < βi+1 for each

1 ≤ i ≤ k − 2.

Let I be a k-interval exchange transformation with alternate discontinuities; we

denote by Oi the trajectory of βi , 1 ≤ i ≤ k − 1.

Proposition 2.1. The left (resp. right) special words of length 1 are i,

1 ≤ i ≤ k − 1. For every n and every 1 ≤ i ≤ k − 1, there is one left spe-

cial word w of length n beginning with i, which is a prefix of Oi , and [w] is a

subinterval of 1i containing βi .

Ifw = w1 . . .ws is a bispecial word,w is the longest of all the possible words v

such that [v] = [w]; if w′ is the shortest bispecial word having w as a strict prefix,

[w′] is the intersection of [w] and the interval [0, x[ or [x, 1[, where x = I−sγ j , for

some j , is in the interior of [w] and no I−tγ j ′ , for 1 ≤ j ′ ≤ k − 1, 0 ≤ t < s, or

j ′ 6= j , t = s, is in that interior.

Proof. The special words of length 1 come from the condition of alternate

discontinuities.

A word w is left special if and only [w] intersects two intervals I1 j and I1 j ′ ,

thus if and only if [w] contains a βi in its interior; thus the left special words are

all the prefixes of Oi that begin with i, 1 ≤ i ≤ k − 1.

If a word w is not right (resp., left) special, there exists a unique letter i such

that [wi] = [w] (resp. [iw] = [w]), while if w is right (resp., left) special, [wi] 6=

[w] (resp., [iw] 6= [w]) for every letter i.


If w is right special, then [w] contains in its interior x = I−sγ j for some j .

If w is also left special, because of the i.d.o.c. condition, there is a unique letter

u such that [wu] contains βi ; wu is then the shortest left special word extending

w to the right, and, if w′ is the shortest bispecial word extending w to the right,

[w′] = [wu] is either the part of [w] which lies left of x or the part which lies right

of x. Moreover, because of the definition of a cylinder, the interior of [w] cannot

contain any I−tγ j for 0 ≤ t < s, while, because of the alternate discontinuities, it

cannot contain more than one I−sγ j , 1 ≤ j ≤ k − 1. �

Thus, if w is bispecial, the endpoints of the interval [w] are in the union of all

the negative orbits of the γ j , 1 ≤ j ≤ k − 1, except that the endpoints are 0 or a γi

for a finite number of the shortest bispecial words.

2.2 The self-dual induction(s). Starting from an interval exchange trans-

formation with alternate discontinuities (this just makes things easier at the initial

stages), we aim to build the points where the negative orbits of the discontinuities

of I approximate the discontinuities of I−1. The motivation is that, by approximat-

ing the discontinuities of I−1 from the right and the left, we build small intervals

around them, and by Proposition 2.1, these correspond to the bispecial words of

L(I). At the same time, this allows us to realize the self-duality mentioned in the

introduction, with its potential value for the study of the Teichmuller flow on the

associated surface [31]. As already mentioned in the introduction, at some points

there are choices to be made.

Indeed, we want to build k − 1 nested families of subintervals Ei,n =

[βi − li,n, βi + ri,n[, 1 ≤ i ≤ k − 1, starting from Ei,0 = 1i = [i], so that the

Ei,n are the intervals containing βi , and whose endpoints are the successive I−mγ j

which fall closest to βi . The notion of successive is unambiguous: as in the proof

of Proposition 2.1, for any j and j ′, I−m′

γ j ′ is after I−mγ j if m′ > m.

Throughout this paper, γi,n is the first element I−mγ j , m > 0, 1 ≤ j ≤ k − 1,

which falls in the interior of Ei,n; it exists by minimality.

Thus we could tentatively define Ei,n+1 by partitioning Ei,n into two subintervals

[βi − li,n, γi,n[ and [γi,n, βi + ri,n[, and setting Ei,n+1 to be the one of these two

subinbtervals which contains βi .

However, it may very well happen that, for example, γ1,n = I−5γ1 while

γ2,n = I−2γ1 and γ2,n+1 = I

−3γ1, which creates a desynchronization between the

tentatives E1,n+1 and E2,n+1; then it seems natural, and turns out to be necessary, to

wait before cutting E1,n; that is, to put E1,n+1 = E1,n.

Thus, for each i and n, we decide either to put Ei,n+1 = Ei,n or to define Ei,n+1

as the subinterval [βi − li,n, γi,n[ or [γi,n, βi + ri,n[ which contains βi .


Throughout this paper, we say that Ei,n is cut on the right if Ei,n+1 =

[βi − li,n, γi,n[, Ei,n is cut on the left if Ei,n+1 = [γi,n, βi + ri,n[, Ei,n is not cut if

Ei,n+1 = Ei,n.

There are many possible choices of cutting or not at each stage, and thus a priori

many self-dual inductions. However, as we have seen above (see also Lemma 2.3

below), for a given interval Ei,n, the first strict subinterval to be generated by any

self-dual induction is the same, independent of the choices. Thus, all sequences

of choices for which Ei,n+1 6= Ei,n for infinitely many n yield the same sequences

of different intervals, though not numbered in the same way; in that sense there is

only one self-dual induction. To define one sequence of choices which makes the

induction work is the non-trivial aim of Section 2: this will be defined inductively

by using combinatorial considerations.

Thus we define an algorithm which is one of the possible ways to build the

sequence of intervals we want; it is indeed an induction algorithm because the

intervals Ei,n+1 are built from the Ei,n by using Sn where, throughout this paper, Sn

denotes the induced map of I on the set⋃k−1

i =1 Ei,n. Indeed, all the information about

I is given by the Sn, n ≥ 0. These maps Sn, defined on disjoint unions of intervals

are the effective object of our study, and we can completely forget the interval

exchange I, which can always be retrieved from the initial S0 when needed.

2.3 The case of three-interval exchanges. Exchanges of two intervals

are just rotations of angle α. Our induction for two intervals finds the iterates of

the point 1 − α which approximate the point α, from the right or from the left; as

there is only one interval to cut at each stage, there is no choice involved, and what

we retrieve is just the additive Euclidean algorithm.

We look now at the next case: throughout this section, I is a symmetric 3-

interval exchange transformation with alternate discontinuities, satisfying the

i.d.o.c. condition.

For each n, we build two disjoint intervals Ei,n as explained in Section 2.2, in

the following way. Let Sn be the induced map of I on E1,n ∪ E2,n; its description

will determine the state of the induction at stage n.

State I. For n = 0, we recall that E1,0 = [0, α1[ and E2.0 = [α1, α1 + α2[. We

check (using the condition of alternate discontinuities) that for n = 0, Sn deter-

mines a partition of each Ei,n, i = 1, 2, into two new subintervals separated by a

point γi,n, such that Sn sends by a translation,

• [β1 − l1,n, γ1,n[ onto [β2, β2 + r2,n[,

• [γ1,n, β1 + r1,n[ onto [β1 − l1,n, β1[,


• [β2 − l2,n, γ2,n[ onto [β1, β1 + r1,n[,

• [γ2,n, β2 + r2,n[ onto [β2 − l2,n, β2].

As I preserves the lengths on its continuity intervals, for i = 1, 2, we have

γi,n = βi + ri,n − li,n, and the relation

r1,n = r2,n.

Whenever for some n the above description of Sn is satisfied, we say that at stage

n, we are in state I , and the construction will be the same as for n = 1.

Now, for i = 1, 2, Sn is continuous on each of the new subintervals of Ei,n,

which implies that γi,n is indeed the first element I−mγ j , m > 0, j = 1, 2, which

falls in the interior of Ei,n. Thus the tentative Ei,n+1 (if we decide to cut) is the one

of the two new subintervals which contains βi ; therefore, it is the right one if βi

is to the right of γi,n; that is, if li,n > ri,n, the left one if li,n < ri,n, while the case

where βi = γi,n, li,n = ri,n, is excluded by the i.d.o.c. condition.

Substate Ia. l1,n > r1,n, l2,n > r2,n. We decide to cut E1,n and E2,n.

Then E1,n+1 = [γ1,n, β1 + r1,n[ and, E2,n+1 = [γ2,n, β2 + r2,n[ and, for i = 1, 2,

li.n+1 = li,n − ri,n, ri,n+1 = ri,n. We see that the new intervals satisfy r1,n+1 = r2,n+1.

As for Sn+1, it is also the induced map of Sn on E1,n+1 ∪ E2,n+1, and we can

describe it by using the description of Sn and the definition of the Ei,n+1, as will be

done in full generality in the proof of Proposition 2.4 and Figure 1 below; and we

check that Sn+1 has the same description as Sn above, with n replaced by n + 1. At

stage n + 1 we are again in state I , and this will be true whenever at stage n we are

in Ia.

Substate Ib. l1,n < r1,n, l2,n > r2,n. We decide to cut only E2,n. This decison

will be explained after the full description of the induction.

Then E1,n+1 = E1,n while E2,n+1 = [γ2,n, β2 + r2,n[; l2.n+1 = l2,n − r2,n and the

other parameters are unchanged; the description of Sn+1 implies that we are again

in state I .

Substate Ic. l1,n > r1,n, l2,n < r2,n. We decide to cut only E1,n.

This is deduced from Ib by exchanging 1 and 2, and at stage n + 1 we are again

in state I .

Substate Id. l1,n < r1,n, l2,n < r2,n. We decide to cut E1,n and E2,n.

Then E1,n+1 = [β1− l1,n, γ1,n[ and E2,n+1 = [β2− l2,n, γ2,n[. The same reasoning

as above shows that at stage n + 1 we are in state II described just below.

State II. This state corresponds to the following description of Sn: it deter-

mines a partition of each Ei,n, i = 1, 2, into two new subintervals separated by a

point γi,n, such that Sn sends by a translation,


• [β1 − l1,n, γ1,n[ onto [β2, β2 + r2,n[,

• [γ1,n, β1 + r1,n[ onto [β2 − l2,n, β2[,

• [β2 − l2,n, γ2,n[ onto [β1, β1 + r1,n[,

• [γ2,n, β2 + r2,n[ onto [β1 − l1,n, β1[.

This implies that for i = 1, 2, we have γi,n = βi + ri,n − l3−i,n, the relation

l1,n + r1,n = l2,n + r2,n,

and the tentative Ei,n+1 (if we decide to cut) is the one of the two new subintervals

which contains βi , the right one if li,n > r3−i,n, the left one if li,n < r3−i,n, while

the case li,n = r3−i,n is excluded by the i.d.o.c. condition. Note that in this state

l1,n > r2,n if and only if l2,n > r1,n, hence there will be only two substates.

Substate IIa. l1,n > r2,n, l2,n > r1,n. We decide to cut E1,n and E2,n.

Then E1,n+1 = [γ1,n, β1 +r1,n[, E2,n+1 = [γ2,n, β2 +r2,n[; the new intervals satisfy

l1,n+1 = l2,n+1, and we are in state III described below.

Substate IIb. l1,n < r2,n, l2,n < r1,n. We decide to cut E1,n and E2,n.

Then E1,n+1 = [β1 − l1,n, γ1,n[, E2,n+1 = [β2 − l2,n, γ2,n[; the new intervals

satisfy r1,n+1 = r2,n+1, and we are in state I .

State III. This is symmetrical to state I, with left and right exchanged, and

the relation

l1,n = l2,n;

there are four substates, IIIa to IIId, and the induction goes either to state III or to

state II.

Now is the time to explain our decisions of cutting or not cutting. Given that

it seems natural to decide to cut, only the decision of not cutting in Ib needs to be

discussed (Ic is similar). Thus we decided not to cut E1,n; we did it because:

• This does not block the process. If after the induction we have r2,n+1 > l2,n+1,

we are in Id and we can proceed; otherwise, we are still in Ib and we take

the same decision again, decreasing l2,n+1 by r2,n+1 = r2,n, and so on. After a

finite number of steps, we are in Id and we can proceed.

• This allows us to keep the process in a simple set of states. The other possible

decisions would be (except to cut nothing, which would of course block the

process) either to cut both intervals, or to cut only E1,n. In either case, this

creates a new state with the relation l1,n+1 + r1,n+1 = r2,n+1. The reader can

check that the description of Sn+1 is more complicated and that, moreover,

γ1,n+1 is not apparent from the description of Sn+1; γ1,n+1 will be known when

we know Sn′ for some n′ > n + 1 and there is no way to cut E1,n+1.


Because we have ensured that the process is never blocked, our induction does

indeed cut each Ei,n for infinitely many values of n, and thus does indeed generate

the nested intervals we wanted to build.

2.4 The general case: getting started. Throughout the remainder of

Section 2, I is a symmetric k-interval exchange transformation with alternate dis-

continuities, satisfying the i.d.o.c. condition.

For each n, the disjoint intervals Ei,n = [βi − li,n, βi + ri,n[ and the map Sn are

as in Section 2.2. The states are again described by relations between the 2k − 2

non-independent parameters li,n and ri,n, a given set of relations determining the

full description of Sn in that state. The non-trivial part of the work is again to make

choices at each stage so that we keep the process in a simple set of states, and build

explicitly the bispecial words.

For n = 0, we recall that Ei,0 = 1i . We check (using the condition of alternate

discontinuities) that S0 determines a partition of each Ei,0, 1 ≤ i ≤ k − 1, into

two new subintervals such that S0 sends, by a translation, the left subinterval Ei,m,0

onto [βm(i), βm(i) + rm(i),0[ and the right subinterval Ei,p,0 onto [βp(i) − lp(i),0, βp(i)[,

where m(i) = k − i, 1 ≤ i ≤ k − 1, p(i) = k + 1 − i, 2 ≤ i ≤ k − 1, p(1) = 1.

The preservation of lengths implies the relations ri,0 = rk−i,0, 1 ≤ i ≤ k − 1,

li,0 = lk+1−i,0, 2 ≤ i ≤ k − 1. We prefer to write these relations in the order

r1,0 = rk−1,0, lk−1,0 = l2,0, r2,0 = rk−2,0,. . . . Thus 1 is related to k − 1 by a relation

on the r which we choose to call a − relation; then k−1 is related to 2 by a relation

on the l which we call a + relation, and so on. We symbolize this chain of relations

by the tree G0 = 1−(k − 1)+2−(k − 2)+3−(k − 3) · · · .

2.5 Trees of relations. The elements of the set of states in which we shall

suceed in keeping the process share with the initial state the fundamental property

that each interval is cut in two pieces, namely, Sn sends a right part of Ei,n onto

a left part of some interval Ep(i),n, and a left part of Ei,n onto a right part of some

interval Em(i),n; this is crucial to ensure that the γi,n of Section 2.2 are explicit from

Sn. Not all maps m and p are possible, however, and we need an auxiliary (at that

point of the paper) map s, which is the identity in the initial state. These maps can

be deduced from the set of relations between the parameters, which may also be

symbolized by a tree, though as in the case k = 3, we have to allow three types of

relations, li,n = l j,n, ri,n = r j,n, li,n + ri,n = l j,n + r j,n. Then, for example, l1,n = l2,n,

l2,n + r2,n = l3,n + r3,n, r3,n = r4,n is symbolized by 1+2 =3−4; we consider only

sets of relations which may be symbolized in this way. Note that our trees may

have more than one branch, a bifurcation being, for example, 1+2 =3 and 2−4, or


l1,n = l2,n, l2,n + r2,n = l3,n + r3,n, r2,n = r4,n.

Thus we introduce abstract objects which we call trees of relations; these will

constitute the states of the induction, in the same way as permutations constitute

the states of the classical Rauzy or Zorich induction.

Definition 2.2. A tree of relations G on a finite set K is a (non-oriented)

graph with #K vertices i, i ∈ K , and #K − 1 edges labelled +, −, or =, such that

G is connected, and two adjacent edges never have the same label.

Each tree can be written in several ways; for example, 1−2 =3 and 3 =2−1

define the same object.

Definition 2.3. Let G be a tree of relations; its bijections are the three maps

s, p,m defined on K in the following way:

• s(i) is the only j such that there is a = edge between i and j , or s(i) = i if

there is no such edge,

• p(i) is the only j such that there is a + edge between s(i) and j , or p(i) = s(i)

if there is no such edge,

• m(i) is the only j such that there is a − edge between s(i) and j , or m(i) =

s(i) if there is no such edge.

Throughout Section 2, we take K = {1, . . . , k − 1}. The bijections always

satisfy s−1 = psp = msm = s; however, not all bijections satisfying these equalities

are defined by a tree of relations (take, for example, K = {1, 2, 3}, s = (213),

p = (231), m = (312)). The reason for the a priori strange definition of p and m

through s is explained in the discussion after Definition 2.4 below.

Note also that, conversely, the three bijections determine the tree of relations

G.

We now use the tree of relations to describe the states of our induction.

Definition 2.4. Let G be a tree of relations. We say that at stage n we are in

state G if the parameters satisfy the following set of relations:

• l j,n + r j,n = li,n + ri,n whenever there is an edge i = j in G,

• li,n = l j,n whenever there is an edge i+ j in G,

• ri,n = r j,n whenever there is an edge i− j in G,

and Sn determines a partition of each Ei,n, 1 ≤ i ≤ k −1, into two new subintervals

Ei,m,n (left) and Ei,p,n (right) such that Sn sends by a translation the left one onto

[βm(i), βm(i) + rm(i),n[, and the right one onto [βp(i) − lp(i),n, βp(i)[, where m and p

are the third and second bijection of G.


Because of Section 2.4, at stage 0 we are in state G0, and we have seen how G0

symbolizes the set of relations in that state; a similar symbolization can be done in

state G, as explained in the beginning of the present section.

If at stage n we are in state G, each Ei,n has two decompositions: according

to its images by Sn, into Ei,m,n and Ei,p,n; and according to its images by S−1n , into

Ei,−,n = [βi − li,n, βi[ and Ei,+,n = [βi, βi + ri,n[. Then the preservation of lengths

by Sn implies the train-track equalities [28], for 1 ≤ i ≤ k − 1,

li,n + ri,n = lp(i),n + rm(i),n.

This family of equalities is indeed equivalent to the three families in Definition 2.4,

which can be written also, by using Definition 2.3 and adding trivial equalities, for

1 ≤ i ≤ k − 1,

ls(i),n + rs(i),n = li,n + ri,n, ls(i),n = lp(i),n, rs(i),n = rm(i),n.

This equivalence is non-trivial; it can be checked immediately for n = 0 and car-

ried to every n by the induction defined below, or proved for every n by using the

tree structure of Gn (thus the relations in Definition 2.4 are a consequence of the

structure of Sn, but at that stage it is simpler to put them in the definition). So

the obvious choice for defining p(i) and m(i), as the numbers of the two intervals

where Sn takes Ei,n, implies that we have relations such as lp(i),n = ls(i),n, and not

lp(i),n = li,n. As our symbolization led us to define the trees with an edge i+ j

if li,n = l j,n, this explains why the definition of p in Definition 2.3 looks a little

unnatural.

The first bijection s of G appears as yet only in the relations. It can be the

identity, as in the initial state G0; when s(i) 6= i, it implies that the two different

intervals Ei,n and Es(i),n have the same length. The full significance of s will be-

come apparent when we study the bispecial words wi,n such that Ei,n = [wi,n], as

from Theorem 2.8 we get that wi,n and ws(i),n are retrograde of each other; thus, if

s(i) = i, wi,n is a palindrome, which is indeed the case at the initial stage where

wi,0 = i.

The reader can now check from Section 2.3 that for k = 3, states I, II, III

correspond respectively to the trees of relations 1−2, 1 =2, 1+2.

2.6 Induction using trees of relations. We now prove the fundamental

result that, if at stage n we are in state G, there is a way to make the induction so

that at stage n + 1 we are in state G′, where G′ is still a tree of relations.

For this, we build a candidate for the new tree G′; thus we define an operation

on trees of relations.


Definition 2.5. Let G be a tree of relations. A positive induction branch

is a maximal connected subtree of G without − edges. A negative induction

branch is a maximal connected subtree of G without + edges.

Definition 2.6. Let G be a tree of relations, (s, p,m) its bijections. An in-

struction on G is an application ι from K to {−,+} such that ιs(i) = ιi for every

i ∈ K .

An accepted induction branch for the instruction ι is a positive induction

branch for which ιi = + on all vertices, or a negative induction branch for which

ιi = − on all vertices.

Let B the set of the vertices of all the accepted induction branches for ι. The

tree of relations Jι(G) is defined by the vertices i, i ∈ K , and the following edges

• if i ∈ B , j ∈ B , ιi = ι j = +, and i+ j in G, then s(i) =s( j ) in Jι(G);

• if i ∈ B , j ∈ B , ιi = ι j = −, and i− j in G, then s(i) =s( j ) in Jι(G);

• if i ∈ B , j ∈ B , ιi = ι j = +, and i = j in G, then i+ j in Jι(G);

• if i ∈ B , j ∈ B , ιi = ι j = −, and i = j in G, then i− j in Jι(G);

• if i 6∈ B or j 6∈ B , the edge between i and j in G stays in Jι(G) with the same

label.

Note that Jι(G) is indeed a tree of relations, as the number of vertices and edges

and the condition on adjacent edges are preserved.

Lemma 2.2. The bijections s′, p′,m′ of Jι(G) are given by:

• if i ∈ B and ιi = +, s′(i) = sp(i), p′(i) = p(i), m′(i) = mp(i);

• if i ∈ B and ιi = −, s′(i) = sm(i), p′(i) = pm(i), m′(i) = m(i);

• if i 6∈ B, s′(i) = s(i), p′(i) = p(i), m′(i) = m(i).

Proof. Straightforward. �

Going back to the induction, if at stage n we are in state G, the parameters

l j,n and r j,n determine whether each βi is in the right or left new subinterval, and

thus whether Ei,n, if we decide to cut it, will be cut on the left or right; this is

summarized by an instruction ι defined on G. But then we have to make choices,

of cutting or not in the sense of Section 2.2. Most of these choices would lead to

very complicated sets of states and induced maps, much worse than the state with

the relation l1,n + r1,n = r2,n we mentioned at the end of Section 2.3, and would not

allow a straightforward definition of the next stage. The key to avoiding that is to

cut intervals in such a way that, if we cut Ei,n on the left then Ep(i),n is also cut on

the left, and similarly if we cut Ei,n on the right then Em(i),n is also cut on the right.

The aim of ensuring this property led naturally to the use of trees of relations as

good describers of the states, and what we do ensure is that at stage n + 1 we shall

be in state Jι(G).


Lemma 2.3. If at stage n we are in state G, the point separating Ei,m,n and

Ei,p,n is the point γi,n defined in Section 2.2.

If s is the first bijection of G, then li,n − rs(i),n = ls(i),n − ri,n 6= 0 for all i.

In any form of the self-dual induction defined in Section 2.2, the tentative Ei,n+1

(if we decide to cut) is Ei,p,n if li,n − rs(i),n > 0, Ei,m,n if li,n − rs(i),n < 0.

Proof. For 1 ≤ i ≤ k − 1, Sn is continuous on each of the intervals Ei,m,n and

Ei,p,nl hence the first element I−mγ j , m > 0, 1 ≤ j ≤ k − 1, which falls in the

interior of Ei,n, is the point separating them, and this is γi,n by definition.

We have the relation ls(i),n + rs(i),n = li,n + ri,n; and if li,n − rs(i),n = ls(i),n − ri,n =0,

βi = γi,n is the left endpoint of Ei,p,n, hence its image by Sn is the left endpoint of

Ep(i),n, which is impossible because of the i.d.o.c. condition.

The tentative Ei,n+1 (if we decide to cut) is the one of the two new subintervals

which contains βi ; therefore, it is the right one if βi is to the right of γi,n, the left

one if βi is to the left of γi,n, while βi = γi,n has just been excluded.

If ls(i),n − ri,n > 0, because of the relation lp(i),n = ls(i),n, lp(i),n − ri,n > 0, and

βi is to the right of γi,n, while otherwise, because of the relation rm(i),n = rs(i),n, βi

is to the left of γi,n, hence the last conclusions. �

Definition 2.7. The choices defining our induction are the following: if at

stage n we are in the state G with bijections s, p, m, we define the instruction ι

by the sign (+ or −)

ιi = ιs(i) = sgn(li,n − rs(i),n) = sgn(ls(i),n − ri,n).

Let B be the set of vertices of the accepted induction branches for ι; then we decide

to cut Ei,n for every i ∈ B , and not to cut Ei,n for i 6∈ B .

Before going further, we check that this is coherent with Section 2.3: in sub-

state Ia the instruction is (+,+), the accepted induction branches are 1 and 2,

B = {1, 2}, thus we cut E1,n and E2,n; in substate Ib the instruction is (−,+), the

only accepted induction branch is 2, B = {2}, thus we cut only E2,n; in substate Id

the instruction is (−,−), the accepted induction branch is 1−2, B = {1, 2}, thus

we cut E1,n and E2,n; the other cases are similar.

Proposition 2.4. If at stage n we are in state G, if ι and B are as in Definition

2.7, then at stage n + 1 we are in state JιG. The new parameters are given by the

following rules

• if i ∈ B and ιi = +, li,n+1 = li,n − rs(i),n = ls(i),n − ri,n, ri,n+1 = ri,n,

• if i ∈ B and ιi = −, li,n+1 = li,n, ri,n+1 = ri,n − ls(i),n = rs(i),n − li,n,

• if i 6∈ B, li,n+1 = li,n, ri,n+1 = ri,n.


Proof. The expressions of the new parameters come from Definition 2.7 and

Lemma 2.3.

βmp(i)

βp(i)

βi

Sn

Sn

Emp(i),+,n

Ep(i),−,n

Ei,−,n Ei,+,n

Ep(i),m,n

Ei,m,n Ei,p,n

γi,n

γp(i),n

Figure 1. The induced map Sn+1.

We look at the action of Sn+1 on Ei,n+1. Suppose i ∈ B , with ιi = +. Then the

situation is completely described in Figure 1. As p(i) is also in B because of the

definition of the accepted induction branches, Ep(i),−,n intersects both Ep(i),p,n and

Ep(i),m,n, and Ep(i),n+1 = Ep(i),p,n.

If we start from Ei,n+1 by Sn, we arrive on the interval Ep(i),−,n, which contains

Ep(i),m,n and intersects Ep(i),p,n.

This creates a partition of Ei,n+1: the right subinterval of Ei,n+1 with the

same length as Ep(i),−,n ∩ Ep(i),p,n, which is indeed Ei,p,n+1, is sent by Sn on

Ep(i),−,n ∩ Ep(i),p,n which is a subset of Ep(i),n+1, and on this interval Sn+1 = Sn.


The left subinterval of Ei,n+1 with the same length as Ep(i),m,n, which is Ei,m,n+1, is

sent by Sn on Ep(i),m,n ⊂ Ep(i)n \ Ep(i),n+1; then Ep(i),m,n is sent by Sn on Emp(i),+,n ⊂

Emp(i),n. As p(i) is on the same positive induction branch in B as i, it cannot be on

a negative induction branch in B , hence neither can mp(i); hence mp(i) is either

on a positive induction branch in B , or not in B , hence Emp(i),n is either cut on the

left or not cut, thus Emp(i),+,n ⊂ Emp(i),n+1. Hence on Ei,m,n+1 we have Sn+1 = S2n ,

and Sn+1 sends Ei,m,n+1 onto a subinterval of Emp(i),n.

Thus Ei,n+1 is indeed partitioned into two subintervals Ei,m,n+1 and Ei,p,n+1 by

a new point γi,n+1 = S−1n γi,p(n), and we can define p′(i) = p(i) and m′(i) = mp(i).

Sn+1Ei,p,n+1 is indeed [βp′(i) − lp′(i),n+1, βp′(i)[. And whether Emp(i),n is cut on the left

or not cut, Sn+1Ei,m,n+1 is the interval [βm′(i), βm′(i) + rm′(i),n+1[.

A similar reasoning takes care of the case i ∈ B with ιi = − with p′(i) = pm(i),

m′(i) = m(i).

If i 6∈ B , Ei,n is not cut and Sn sends Ei,p,n+1 = Ei,p,n on Ep(i),−,n, which is still

in Ep(i),n+1 as p(i) cannot be on a positive induction branch in B and hence Ep(i),n

is not cut on the left. Similarly, Sn sends Ei,m,n+1 on Em(i),+,n ⊂ Em(i),n+1, thus we

define p′(i) = p(i), m′(i) = m(i).

We define now s′ by s′ = sp on B ∩ {ι = +}, s′ = sm on B ∩ {ι = −}, s′ = s on

Bc; by Lemma 2.2, s′, p′,m′ are the bijections of Jι(G). �

Thus we can iterate the self-dual induction, and are sure to stay among states

defined by trees of relations. It remains to prove, as in Section 2.3, that our choices

do not block the process, namely that each Ei,n is cut for infinitely many n. We

shall show a little more, but for this we need an abstract lemma which makes a

fundamental use of the tree structures:

Lemma 2.5. Let Gn = Jιn−1. . . Jι0 G0 be an infinite sequence of trees of rela-

tions on {1, . . . , k − 1} with instructions ιn, and Bn the set of the vertices of all the

accepted induction branches in Gn for ιn. If

• for every i which is not in Bn, ιn+1i = ιni,

• for all i which are in Bn for infinitely many n, ιni = + for infinitely many n,

and ιni = − for infinitely many n,

then for every 1 ≤ i ≤ k − 1, ιni = + for infinitely many n, and ιni = − for

infinitely many n.

Proof. We prove the lemma by recursion on k; for k = 1, any tree of relations

is reduced to a vertex 1, and 1 is always in Bn whatever the instruction, thus the

conclusion is satisfied.

We have to show that each i is in Bn for infinitely many n. Let now i be a vertex

which is not in Bn for all n > N ; then ιni is constant, for example +, for n > N . Let


Cn be the positive induction branch containing i; if j ∈ Cn, let (i, x1, . . . , xr, j ) be

the path connecting i with j in Cn. We define C ′n to be the set of those j ∈ Cn for

which ιx1 = · · · = ιxr = ι j = +. For n > N , all j in C ′n are out of Bn as otherwise i

would be in Bn, thus j is not modified by the operations J , and C ′n ⊂ C ′

n+1. If, for

some n, C ′n = {1, . . . , k − 1}, then i would be in Bn; thus, for n > N ′, there exists

x ∈ C ′n, with an edge x+y, and ιny = −.

For n > N ′, let Dn be the tree Gn deprived of the edge x+y, and D ′n the con-

nected component of Dn containing y. All the operations J we make on the Gn do

not change the edge x+y, thus they project as operations J on the trees of relations

D ′n, with a sequence of instructions (the restrictions of ιn) satisfying the hypothesis

of the present proposition. As D ′n has at most k − 1 elements, by our recursion

hypothesis these instructions must take the values + and − infinitely many times

on all vertices of D ′n, which contradicts the assumption that ιny = − ultimately. �

Proposition 2.6. If, starting from the initial parameters li,0, ri,0, 1 ≤ i ≤

k − 1, we iterate infinitely many times the self-dual induction defined in Definition

2.7 and described in Proposition 2.4, then for each 1 ≤ i ≤ k − 1, Ei,n is cut on

the left (resp. right) for infinitely many n.

Proof. Let us prove that if i is such that Ei (n) is cut for infinitely many n, then

ιni = + and ιni = − for infinitely many n. Indeed, by construction, the left and

right endpoints of Ei(n) are respectively Ia(n)γb(n) and I

a′(n)γb′(n), and there is no

point Ixγ j or Ix′

γ j ′ inside Ei (n) for a(n) ≤ x ≤ 1 and a′(n) ≤ x′ ≤ 1. If Ei (n) is cut

infinitely often, a(n) → −∞ or a′(n) → −∞, and thus there exists c(n) → −∞

such that Ei(n) does not contain any Ixγ j for c(n) ≤ x ≤ 1. But this contradicts

minimality if Ei,n is ultimately not cut on the right (resp. left).

Thus our sequence of states Gn satisfies the second hypothesis of Lemma 2.5,

while the first one is satisfied because of the definition of the instructions; and the

conclusion of Lemma 2.5 implies the proposition. �

Thus our algorithm is indeed able to build the nested intervals defined in Sec-

tion 2.2. Among the several possible ways to build them, we think it is the sim-

plest, but also, as we shall see in Section 2.8 below, it has the extra advantage of

giving an explicit construction of the bispecial words wi,n such that Ei,n = [wi,n].

2.7 The graph of graphs. What is proved in Section 2.6 can be summa-

rized in

Proposition 2.7. Through the self-dual induction, I defines an infinite se-

quence of trees of relations (Gn, n ∈ IN), starting from G0. The instructions ιn


and Bn, the set of the vertices of all the accepted induction branches in Gn for ιn,

satisfy:

• if i is not in Bn, ιn+1i = ιni,

• for all i, ιni = + for infinitely many n, and ιni = − for infinitely many n.

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 tree of relations.

Definition 2.8. For a fixed k, the graph of graphs Ŵk is the oriented graph

whose vertices are all the possible trees of relations on {1, . . . , k − 1} which can

be reached from G0 = 1−(k − 1)+2−(k − 2)+3−(k − 3) . . . by a finite sequence of

inductions Jιn . . . Jι0 where ιn+1i = ιni for i 6∈ Bn, with an edge labelled by ι from

G to JιG. The sequence (Gn, n ∈ IN) of Proposition 2.7 is called the induction

path of I in Ŵk.

Ŵ2 is made with one tree of relations, 1.

As we have seen when revisiting Section 2.3, the vertices of Ŵ3 are (I)1−2, (II)

1 =2, (III) 1+2. We have J−,−I =II, J−,−II =I, J+,+II =III, J+,+III =II, J−,+I = J+,−I =

J+,+I =I, J−,+III = J+,−III = J−,−III =III.

Ŵ4 is made with nine trees of relations: 1−3+2, 2−1+3, 3−2+1, 1 =3+2,

2 =1+3, 3 =2+1, 1−3 =2, 2−1 =3, 3−2 =1. Its edges are described in [15].

A general combinatorial study of trees of relations is carried out in [11]. We

show that a tree of relations G is in Ŵk if and only if its bijections satisfy pms(i) =

i + 1, 1 ≤ i ≤ k − 2, pms(k − 1) = 1 (as a consequence, trees of relations of every

shape exist in Ŵk and, if in the definition of Ŵk we replace G0 by another tree of

relations on k letters, Ŵk remains the same up to a renaming of the letters), and that

the number of vertices of Ŵk is 3(2k − 2)!/((k + 1)!(k − 2)!).

2.8 Description of the trajectories. We get now an explicit description

of the trajectories of a given symmetric k-interval exchange transformation. Let

us stress that this is possible also by using the existing induction algorithms such

as the Rauzy induction and its accelerations, or the da Rocha induction, but, to our

knowledge, this has been done explicitly only for the last one [24].

Theorem 2.8. The bispecial words of L(I) are generated in the following way,

where the three bijections of the tree of relations Gn are denoted by sn, pn, mn:

(i) For all n ≥ 0, 1 ≤ i ≤ k − 1, we build words wi,n, Pi,n, Mi,n such that

• Ei,n = [wi,n],

• wi,n is a bispecial word beginning with i and ending with sn(i),


• wi,n is not a factor of w j,n for any i 6= j ,

• Pi,n is a word beginning with k + 1 − pn(i) and ending with i,

• Mi,n is a word beginning with k − mn(i) and ending with i,

• wsn(i),n = wi,n,

• wi,nPsnpn(i),n = Pi,nwpn(i),n; moreover, this word contains no occurrence

of any wt,n except the prefix wi,n and the suffix wpn(i),n,

• wi,nMsnmn(i),n = Mi,nwmn(i),n; moreover, this word contains no occur-

rence of any wt,n except the prefix wi,n and the suffix wmn(i),n.

(ii) The words are built in the following way:

For 1 ≤ i ≤ k −1, wi,0 = i, 1 ≤ i ≤ k −1, P1,0 = k1, Pi,0 = i, 2 ≤ i ≤ k −1,

Mi,0 = i, 1 ≤ i ≤ k − 1.

For each n > 0 and 1 ≤ i ≤ k − 1:

• ιn(i) = + if the wordwi,n(k +1−sn(i)) is left special, ιni = − if the word

wi,n(k − sn(i)) is left special,

• if i ∈ Bn and ιni = +, wi,n+1 = wi,nPsnpn(i),n, Pi,n+1 = Pi,n, Mi,n+1 =

Mpn(i),nPi,n,

• if i ∈ Bn and ιna = −, wi,n+1 = wi,nMsnmn(i),n, Pi,n+1 = Pmn(i),nMi,n,

Mi,n+1 = Mi,n,

• if i 6∈ Bn, wi,n+1 = wi,n, Pi,n+1 = Pi,n, Mi,n+1 = Mi,n.

(iii) When wi+1,n 6= wi,n, it is the shortest bispecial word which admits wi,n as a

strict prefix. The bispecial words of L(I) are the empty word and the wi,n,

n ≥ 0, 1 ≤ i ≤ k − 1.

Proof. By construction, the interval Ei,n is a cylinder. Thus it can be written

as [v] for several different words v ; as Ei,n contains βi , each of these v is left

special. Let wi,n be the longest of these v ; then wi,n must be also right special.

We check that all assumptions in (i) are satisfied for n = 0 and suppose they

are still satisfied for n.

The wordwi,n can be followed by two possible letters, k+1−sn(i) and k−sn(i);

hence we identify Ei,n,p as the cylinder [wi,n(k +1−sn(i)] and Ei,n,m as the cylinder

[wi,n(k − sn(i))], which implies the claimed value for ιn. Similarly, wi,n can be

preceded by two possible letters, k + 1 − i and k − i, and we identify Ei,n,+ as

I[(k − i)wi,n] and Ei,n,− as I[(k + 1 − i)wi,n].

Suppose i ∈ Bn and ιn(i) = +. Then, by construction, Ei,n+1 is the cylinder

[wi,n(k + 1 − sn(i)], this word is left special, it can only be extended to the right in

one way, and so are its successors until we arrive at wi,nPsnpn(i),n; the latter is still

left special, and Ei,n+1 is also the cylinder [wi,nPsnpn(i),n].

But snpn(i) is on the same positive induction branch as i, hence it is also in Bn

with ιn(snpn(i)) = +. Therefore βsnpn(i) is to the right of the point γsnpn(i),n, and thus


the word (k+1−snpn(i))wsnpn(i),n, such that Esnpn(i),n,− = I[(k+1−snpn(i))wsnpn(i),n]

is right special. By extending it to the left, we get again right special words, until

we reach Pi,nwpn(i),n, which is still right special, and thuswi,nPsnpn(i),n, which is the

same word, is right special. Hence we have identified wi,n+1 as the bispecial word

wi,nPsnpn(i),n, the longest v such that Ei,n+1 = [v], and the shortest bispecial word

extending wi,n to the right.

A similar reasoning applies when i is in Bn and ιn(i) = −, and there is nothing

to prove when i is not in Bn.

We show now that nowl,n+1 is a factor ofwi,n+1: we take the case whenwi,n+1 =

wi,nPsnpn(i),n. Then, by the hypotheses (i) at stage n, w j,n is a suffix of wi,n+1 for

j = pn(i), and the only wt,n occurring in wi,n+1 are its prefix wi,n and suffix w j,n.

But for 1 ≤ t ≤ k − 1, wt,n is a prefix of wt,n+1, strict for t = j because j is on the

same positive induction branch as i; thus no wt,n+1 occurs in wi,n+1 for k 6= i. A

similar reasoning takes care of the case when wi,n+1 = wi,nMsnmn(i),n, and the case

wi,n+1 = wi,n is immediate.

The computation of the new P and M is similar to the proof of Proposition 2.4,

and it implies the remaining assertions of (i) at stage n+1. (iii) is straightforward.�

Note that a consequence of Theorem 2.8 is that the language L(I) is symmetric.

This could be proved without the induction algorithm, as it is due to the fact that

the symmetric permutation satisfies σ−1 = σ, and thus I−1 is conjugate to I. But

here we have an explicit realization of the symmetry through the first bijections sn

of the trees of relations.

The P and M words are also the names of the Rokhlin towers. Namely, if

the interval Ei,n is cut into intervals Ei,n,m and Ei,n,p as in Proposition 2.4, each

level IrEi,n,p which does not intersect Ei,n is contained in one interval 1w(r,i,n,p),

w(r, i, n, p) ∈ {1, . . . , k}, and the same if we replace p by m. This gives two words

w(0, i, n, p) . . .w(h1, i, n, p) and w(0, i, n, p) . . .w(h2, i, n, p), and we check that

they are respectively Pi,n and M i,n. This is indeed why we have chosen to number

the words so that Pi,n and Mi,n end with i.

Thus our algorithm gives also a way to generate any transformation I by six

families of Rokhlin towers. When we know the path of I 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 1i , i = 1, . . . , k; this is proved more directly

and used extensively in [15].

2.9 Structure theorem.

Theorem 2.9. Any infinite path (Gn, n ∈ IN) in Ŵk, starting from G0, where


the instructions ιn and Bn, the set of the vertices of all the accepted induction

branches in Gn for ιn, satisfy

• if i is not in Bn, ιn+1i = ιni,

• for all i, ιni = + for infinitely many n, and ιni = − for infinitely many n,

is the induction path of at least one symmetric k-interval exchange transformation

with alternate discontinuities, satisfying the i.d.o.c. condition.

Proof. With the graphs Gn and the instructions ιn, we build words wi,n, 1 ≤

i ≤ k − 1, n ≥ 0, by the rules in (ii) of Theorem 2.8. Let L be the set of all their

finite factors. We check by induction that wsn(i),n = wi,n, thus L is symmetric; L is

indeed a language, as every word of L can be extended to the right by construction,

and hence to the left because L is symmetric.

Let us show that the wi,n are actually bispecial words of L. The word wi,n can

be followed by either Psnpn(i),n or Msnmn(i),n. Suppose that wi,n+1 = wi,nPsnpn(i),n;

then wi,nPsnpn(i),n is in L. Let j = snmn(i), and r be the first integer q ≥ n such

that j is in Bq and ιq( j ) = ιq(sq( j )) = − (it exists by the hypothesis). Then, by the

formulas in (ii) of Theorem 2.8, for every n ≤ q ≤ r, there exists a concatenation

Qq of P words such that Msnmn(i),nQq is the unique M word at stage q which has j

as its last letter. At stage r + 1 there is a bispecial word wt,r+1 = wt,rMsnmn(i),nQr

and wt,r has the same last letter as wi,n, thus wi,n is one of its suffixes; thus wt,r+1

has wi,nMsnmn(i),n as a factor. Hence wi,nMsnmn(i),n is actually in L, and wi,n is right

special. A similar reasoning takes care of the case wi,n+1 = wi,nMsnmn(i),n. And, as

wi,n is also right special, wi,n is bispecial.

As in the proof of Theorem 2.8, we check by induction that all the assertions

(except the first one) of (i) of Theorem 2.8 are satisfied, and thus the bispecial

words of L are the empty word and the wi,n, n ≥ 0, 1 ≤ i ≤ k − 1.

Now, Proposition 7 of [16] states that a language L is the language of a sym-

metric k-interval exchange transformation with alternate discontinuities, satisfy-

ing the i.d.o.c. condition, if (and only if) it is minimal, the set of words of length

2 is {1k, 1(k − 1), 2(k − 1), 2(k − 2), . . . , (k − 1)2, (k − 1)1, k1}, and, if w is

a (non-empty) bispecial word, there exist a and b such that A(w) = {a, a + 1},

D(w) = {b, b + 1}, there are three words of the form xwy for letters x, y and these

are (a + 1)wb, aw(b + 1) and either awb or (a + 1)w(b + 1).

Our language L satisfies the condition on words of length 2 by construction.

If w is a non-empty bispecial word of L, w is one of the wi,n. By construc-

tion, it satisfies the requirements on A(w) and D(w), and, with the above notation,

(a + 1)wb and aw(b + 1) are in L. As one and only one of wb and w(b + 1) is left

special, there are three words xwy in L; thus L satisfies the condition on bispecial

words.


The non-trivial condition to satisfy is minimality. The words of L are the factors

of the wi,n, or of the k − 1 infinite sequences Oi which begin with wi,n for all n.

Let w be a word of L: w must be a suffix of infinitely many words of L, because

every word of L can be extended infinitely many times to the left, thus w occurs

infinitely often in at least one Oi . Hence all the prefixes of Oi occur infinitely often

in at least one O j , and if j 6= i, then we can drop Oi and use only the other O j

to generate L. Thus L is the union of the L(ui) for d ≤ k − 1 one-sided infinite

sequences ui which are recurrent: each factor of ui occurs at infinitely many places

in ui .

Let now L′ =⋃d

i =2 L(ui). Suppose there exist words w1 in L(u1) \ L′ and w′ in

L′ \ L(u1). Suppose w′ is in L(ur): we check that L(u1) and L(ur) have at least a

one-letter word in common, otherwise this contradicts the condition on the words

of length 2 of L. Take a non-empty word w0 in L(u1) ∩ L(ur). By the recurrence

property, we can find a word w1v1w0v′1w1 in L1 ⊂ L, and a word w′v2w0v

′2w

′ in

L′ ⊂ L; but no word w1vw2 or w2vw1 is in L, as such a word is neither in L(u1)

nor in L′, and this implies that there is a bispecial word of L, w3, such that two

of the four possible words xw3y are not in L, and this contradicts the condition on

bispecial words. Hence either L′ ⊂ L(u1), in which case L = L(u1), or L(u1) ⊂ L′,

and then we can drop u1. By iterating the process, we get that L = L(u) for one

recurrent infinite sequence u.

Let now w be a factor of u. We look at the possible words v such that wv

is right special. If two different such words v and v ′ have a common suffix s,

then there exist two different words such that v1s and v2s are right special, and we

find a bispecial word w4 with four possible xw4y, which contradicts the condition

on bispecial words. Thus there is at most one such possible v with last letter i,

1 ≤ i ≤ k − 1 (and none with last letter k), and thus there are at most k ways of

going from one occurrence of w in u to the next one. Hence any factor w occurs

in u at infinitely many places with bounded gaps, and L = L(u) is minimal.

Thus we have found I such that L = L(I) and, by following again the proof of

Theorem 2.8, we check that its path by the self-dual induction is the right one. �

Given an infinite path, it follows from the above proof and [16] that a vector of

length for an associated I is given by (µ[1], . . . , µ[k]) for any invariant probability

µ on the symbolic dynamical sytem XL. Thus for one path there may be several

corresponding symmetric k-interval exchange transformations I. The solution I is

unique if and only if I is uniquely ergodic (it has a unique invariant probability

measure). A famous result of Veech [33] and Masur [26] states that the set of

(α1, . . . , αk) in IR+k for which I, defined by the vector ( α1

α1+···+αk, . . . , αk

α1+···+αk), is

uniquely ergodic has full Lebesgue measure. A mainly combinatorial proof of this


result, quite in the spirit of the present paper, can be deduced from [8] and [9].

When the solutions are non-uniquely ergodic, the analysis has been conducted in

[22]: our induction path corresponds to any symmetric k-interval exchange whose

vector of lengths lies in a convex set S, with extremal points v1, . . . , vd , d ≤[

k2

]

[19], [31]. Each vi defines, by µi [ j ] = (vi) j , an ergodic invariant measure µi for

every symmetric I with vector of lengths v ∈ S; for such an I, all its invariant

ergodic probabilities are the µi . For a given solution I of vector of lengths v , the

Lebesgue measure is such that µ[ j ] = v j ; the Lebesgue measure will be ergodic

if and only if v is one of the vi .

3 Induction in the symmetric and hyperelliptic cases

3.1 Induction using composite tree of relations. In this section, I is a

symmetric k-interval exchange transformation, satisfying the i.d.o.c. condition. If

the condition of alternate discontinuities is not satisfied, the first bispecial words

have a less regular structure (for example, a short word may be extended to the

right in more than two different ways), and our self-dual induction will be more

complicated — though only for a finite number of initial stages. We define now an

appropriate generalization of the trees of relations, and conduct simultaneously the

construction in Proposition 2.4 and Theorem 2.8. We do not give explicit proofs

as they are basically the same as for the above results.

Definition 3.1. A composite tree of relations G on a finite set K is the

union of a tree of relations G on a subset K ′ of K and, for each element i of

Q = K \ K ′, an oriented edge from i to one element j = h(i) of K ′, such that

h−1( j ) is an integer interval; we denote these edges by j>i or i< j .

The maps s, p,m, h of G are the bijections s, p,m of G, defined on K ′ and the

map h from Q to K ′.

In the sequel we take either K = {1, . . . , k − 1} or K = {2, . . . , k}.

Definition 3.2. Let G, K , K ′, h, s be as in Definition 3.1. An instruction ι

on G associates to each i ∈ H a subsequence of the sequence (−, x, x+1, . . . , y,+),

where [x, y] = h−1(i).

For a sequence of numbers kn which increases to k − 1, we build intervals Ei,n

and bispecial words wi,n, 1 ≤ i ≤ kn, such that Ei,n = [wi,n].

At stage 0, there are k0 ≤ k − 1 bispecial words of one letter, j1 ≤ j2 ≤ jk0,

and k0 intervals Ei,0 = [ ji ]; we have either j1 = 1 or jk0= k, but not both. Let K

be {1, . . . , k − 1} if j1 = 1, {2, . . . , k} otherwise. We define K0 ⊂ K as the set

{ j1, . . . , jk0}, and Q0 = K \ K0.


We define a tree of relation G0 and a composite tree of relations G0; G0 is

j1− jh+ j2− jh−1 . . . if j1 = 1, jh+ j1− jh−1+ j2 . . . otherwise. The extra edges are

j>t, for any j ′ < t < j ′′, if j ′− j + j ′′. At the ends of G0:

• if there is an end j1− jh, then j1>t for any jh < t < k,

• if there is an end jh+ j1, then jh>t for any 1 < t < j2,

• if there is an end jr + j ′ for r 6= h, then jr>t for any jr < t < j ′,

• if there is an end jr− j ′ for r 6= 1, then jr>t for any j ′ < t < jr .

For i ∈ K0, we definewi,0 = i; Pi,0 = i unless if there is an end j1− jh, in which

case P j1= k j1; Mi,0 = i unless if there is an end jh+ j1, in which case M jh

= 1 jh.

For i ∈ Q0, we define Ui = ih(i).

At stage n, we have defined Ei,n, and a composite tree of relations Gn, defined

by Kn ⊂ K , a tree of relations Gn, maps sn, pn,mn, hn; Ei,n is partitioned into

intervals I[xwi,n], x ∈ A(wi,n), by points β j , and into [wi,nx], x ∈ D(wi,n), by

points γ j,n in the negative orbits of the γi . We denote by li,n the distance between

the left end of Ei,n and the smallest of its β j , and by ri,n the distance between the

right end of Ei,n and the largest of its β j , and by ui,1,n, . . . , ui,z(i,n),n the distances

between consecutive β j in Ei,n, ordered from right to left (vi,1,n is the distance

between the last β j to the right and the last but one, and so on). For i ∈ Q0, ui is

the length of the interval 1i = [i].

The edges of Gn correspond to relations between these parameters; i+ j means

again li,n = l j,n; i− j means again ri,n = r j,n; i = j means now that

li,n + ui,1,n + · · · + ui,z(i,n),n + ri,n = l j,n + u j,1,n + · · · + u j,z( j,n),n + r j,n,

and also z(i, n) = z( j, n); i< j means ui = u j,t,n for the t corresponding to the

interval I[iw j,n].

Thus each Ei,n is partitioned (by the cylinders I[xwi,n]) into a left subinterval

Ei,n,− of length li,n, a right subinterval Ei,n,+ of length ri,n, and intermediate subin-

tervals Ei,n,A,t of length vi,t,n, t decreasing from z(i, n) to 1 when we go from left to

right. Each Ei,n is also partitioned (by the cylinders [wi,nx]) into a left subinterval

Ei,n,m, a right subinterval Ei,n,p and intermediate subintervals Ei,n,D,t, t increasing

from 1 to z(i, n) when we go from left to right.

Furthermore, if sn, pn,mn, hn are the maps of Gn, Sn the induced map of I on⋃hn

i =1 Ei,n, then SnEi,n,p = Epn(i),n,− SnEi,n,m = Emn(i),n,+, SnEi,n,D,t = Esn(i),n,A,z(i,n)+1−t .

To each i in Kn we associate a bispecial word wi,n beginning with i and ending

with sn(i), words Pi,n beginning with max D(pn(i)) and ending with i, Mi,n begin-

ning with min D(mn(i)) and ending with i. As for Qn = K \ Hn, it is a subset of

Qn−1 ⊂ · · · ⊂ Q0 = K \ H0, and for each i ∈ Qn there is a word Ui,n beginning

with i and ending with snhn(i). Moreover, we have, for all i ∈ Hn, wsn(i),n = wi,n;


wi,nPsnpn(i),n = Pi,nwpn(i),n, and this word contains no occurrence of anywt,n except

the initial wi,n and the final wpn(i),n; wi,nMsnmn(i),n = Mi,nwmn(i),n, and this word

contains no occurrence of any wt,n except the initial wi,n and the final wmn(i),n. For

all i ∈ Qn, wsnhn(i),nUi,n = Ui,nwhn(i),n, and this word contains no occurrence of any

wt,n except the initial wsnhn(i),n and the final whn(i),n.

We show now how to go from stage n to stage n + 1.

For each i ∈ Kn, we define the instruction ιni: let D(wi,n) = [x, y]. It is still

true (as at the initial stage) that if they exist, x + 1,. . . , y − 1 are in h−1n (i). We put

− in ιn(i) if wi,nx is left special; for x + 1 ≤ z ≤ y − 1, we put z in ιn(i) if wi,nz is

left special, and we put + in ιn(i) if wi,ny is left special. Note that ιn consists (if we

replace + by y and − by x) of all the intersections of two consecutive D(twi,n).

The choice of which Ei,n are cut is also more complicated than in Section 2.6.

We define first, as in Section 2.6, the + and − induction branches of Gn (which

do not depend on ιn). Then, knowing ιn, we define also mixed induction branches:

whenever, for i = j , ιni contains + and − while ιn j contains neither + nor −, we cut

the + induction branch of i = j , keeping the half-branch ending in j and containing

i. We do the same for the − induction branch, and we glue these two half-branches

together, i = j being the glueing points. Then a +, resp. − induction branch is

accepted if ιni contains +, resp. −, for all its vertices. A mixed induction branch

is accepted if ιni contains + for all the vertices (except possibly the glueing points)

on its + half-branches and ιni contains − for all the vertices (except possibly the

glueing points) on its − half-branches (note that the parts of + and − induction

branches we have discarded when creating mixed induction branches cannot be

accepted for this induction). And Bn is again the union of vertices of all accepted

induction branches.

If i is not in Bn, we put wi,n+1 = wi,n and if i is in Bn, with instruction

( j1, . . . , jr), we create r successors of wi,n: they are the wi,nWl , where, if j l ∈

h−1n (i), Wl is the word U j l,n and if j1 = −, W1 is Msnmn(i),n and if jr = +, Wr is

Psnpn(i),n.

We name the new bispecial words: wi,nU j,n = w j,n+1 if j ∈ Qn; wi,nMsnmn(i),n =

wi,n+1 if jr 6= +; wi,nPsnpn(i),n = wi,n+1 if j1 6= −; if both wi,nMsnmn(i),n and

wi,nPsnpn(i),n are among the new bispecial words, then there is an i = j in Gn with

no w j,nPt,n and no w j,nMt,n among the new bispecial words, and we decide, for

example, that wi,nMsnmn(i),n = w j,n+1 and wi,nPsnpn(i),n = wi,n+1 (this last choice is

arbitrary; we could exchange P and M ). Thus the new bispecial words are still

associated to elements of K ′ and Kn+1 ⊃ Kn (possibly Kn+1 = Kn). The cylin-

ders E j,n+1 = [w j,n+1] are identified as Ei,n,m if wi,nMsnmn(i),n = w j,n+1, Ei,n,p if

wi,nPsnpn(i),n = w j,n+1, Ei,n,D, j if wi,nU j,n = w j,n+1.


We can now describe the new composite tree of relations Gn+1 by the following

rules: all edges in Gn which do not touch points in Bn are unchanged; all edges in

Gn which do not touch glueing points of mixed inductions branches are changed

by the rules of Definition 2.6; edges i> j , where i ∈ Bn and j is not in ιni, be-

come edges i ′> j where i ′ is one of the successors of i (i.e., wi,n is the prefix of

wi ′,n+1) in such a way that, for each successor i ′ of i, the j for which i ′> j in the

new tree form either an interval or an empty set; if i is in Bn and has no edge

i =t in Gn, all the edges i> j where j ∈ ιni are replaced by i− j1+ j2− j3 . . . or

i+ j1− j2+ j3 . . . for some order on the j (depending on the instruction through the

description of the sets D(xwi,n)); if i and j are in Bn and i = j , but this is not a

glueing point of a mixed induction branch, the edges i = j , i>t where t ∈ ιni, and

j>t where t ∈ ιn j are replaced by i−t1+t2−t3 . . . −ts− j or i+t1+t2−t3 . . . −ts+ j

for some order on the t depending on the instructions; finally, if i = j is a glueing

point of an accepted mixed induction branch, where ιi contains + and −, then t−i

becomes sn(t) = j , t− j is unchanged, t+ j becomes t+i, t+i becomes sn(t) =i, while

i = j and all the edges i>t, where t ∈ ιni, and j>t where t ∈ ιn j are replaced by

i−t1+t2−t3 . . . −ts+ j for some order on the t depending on the instructions. Thus

Gn+1 is indeed a composite tree of relations.

The new words P,M,U are built in the following way: if j ∈ Kn+1 \ Kn, there

is no U j,n+1; if wi,n is changed, its successors being wi,nWl where W1 is an M

word and no Wl is a P word, then the U j,n, j ∈ h−1n (i), such that wsn(i),nU j,n is not

a new bispecial word, are replaced by U j,nW1, and Psn(i),n is replaced by several

successors, all the Psn(i),nWl named in the same way as the new w j,n+1; if wi,n

is changed, its successors being wi,nWl where Wr is a P-word and no Wl is an

M word, then the U j,n, j ∈ h−1n (i), such that wsn(i),nU j,n is not a new bispecial

word, are replaced by U j,nWr , and Psn(i),n is replaced by several successors, all the

Psn(i),nWl named in the same way as the new w j,n+1. Every other P, M or U word

is unchanged.

By adapting the proof of Proposition 2.5, we check that every wi,n is extended

infinitely many times; thus, because of minimality, there exists N such that KN =

K , and Gn = Gn is a tree of relations for all n ≥ N . For n ≥ N , the k − 1

bispecial words wi,n begin with the k − 1 different wi,N , i ∈ K , and end by the

wsn(i),N = wsN sn(i),N . The language is still symmetric.

3.2 Structure theorem. We did not give a full formal definition of the

induction as it is rather technical, but since Gn+1 depends only on Gn and ιn, we

can define JιG.


Definition 3.3. A sequence of instructions ιn on K is admissible if:

• there exists a composite tree of relations G0 on K such that, if Gn =

Jιn−1Jιn−2

. . . Jι0G0, its maps are sn, pn,mn, hn; ιn is an instruction on Gn,

• ιni contains + iff ιnsn(i) does not contain −,

• if h−1n (i) is empty, ιni is − or +,

• if h−1n (i) = [x, y] and h−1

n (sn(i)) is empty, either ιni = ιnsn(i) = −, or ιni =

ιnsn(i) = +, or ιni = z ∈ [x, y] and ιnsn(i) = (−,+),

• if h−1n (i) = [x + 1, x + r] and h−1

n (sn(i)) = [x′ + 1, x′ + r ′]; let x1, . . . , xt be

the elements of ιni which are in [x + 1, x + r], we add x0 = x if ιni contains

− and xt+1 = x + r + 1 if ιni contains +; let x′1, . . . , x′

t′ be the elements of

ιnsn(i) which are in [x′ + 1, x′ + r ′], we add x′0 = x′ if ιnsn(i) contains − and

x′t′+1 = x′ + r ′ + 1 if ιnsni contains +.

We build inductively integer intervals Dl , 0 ≤ l ≤ r ′ + 1, by putting Dr ′+1 =

[x0, x0] if ιnsn(i) does not contain +, Dr ′+1 = [x, x1] if ιnsn(i) contains +; then,

if max Dl = x j , we put Dl−1 = [x j , x j ] if ιnsn(i) does not contain x′ + l − 1

(or − if l = 1) or there is no x j+1, Dl−1 = [x j , x j+1] if x j+1 exists and ιnsn(i)

contains x′ + l − 1 (or − if l = 1). We build D ′l similarly by exchanging the ′

signs, and we require

x + l ∈ Dl′ iff x′ + l′ ∈ D ′l for 0 ≤ l ≤ r + 1, 0 ≤ l′ ≤ r ′ + 1,

• if i is not in Bn, the set of vertices of all accepted induction branches in Gn

for ιn, then ιn+1i = ιni,

• for each i, ιni = + for infinitely many n, ιn− = − for infinitely many n,

• for each i ∈ Q0 (the set of vertices in G0 which have only < edges), there

exist n and j such that ιn j ⊃ i.

The complicated second to fifth conditions are just a translation of Figure 2

and the symmetry.

[wx0] [wx1] [wx2]

. . .

. . .

[w]

I[x′r′+1

w] I[x′r′w] I[x′

r′−1w]

Figure 2. Bispecial intervals


Theorem 3.1. (i) A symmetric k-interval exchange transformation, satis-

fying the i.d.o.c. condition, defines as its induction path a unique sequence

Gn = Jιn Jιn−1. . . Jι0G0 of composite trees of relations, where the sequence ιn

is admissible. The bispecial words of L(I) are the wi,n constructed in Section

3.1, n ∈ N , i ∈ Hn. For n large enough, Gn is a tree of relations and Hn = K .

(ii) Any such sequence is the induction path of at least one such transformation.

Proof. (i) is the construction above.

The proof of (ii) is similar to the proof of Theorem 2.9, with Proposition 7 of

[16] replaced by the more general Theorem 2 of the same paper. �

3.3 Substitutions.

Definition 3.4. A substitution τ is an application from an alphabet A into

the set A⋆ of finite words on A; it extends naturally to a morphism of A⋆.

It is primitive if there exists k such that a occurs in τkb for any a ∈ A, b ∈ A.

A fixed point of τ is an infinite sequence u with τu = u.

A sequence x in AIN is primitive substitutive if x = ψ(u), where u is a fixed

point of a primitive substitution on an alphabet A0, and ψ is a map from A0 to A⋆.

A symbolic system XL is a primitive substitutive system if L = L(x) for a

primitive substitutive sequence x.

Substitutions enter the picture if, instead of generating all factors of the trajec-

tories by the bispecial wordswi,n, we generate them by the words Pi,n and Mi,n for

n large enough. The advantage is that, at least for n large enough, the Pi,n+1 and

Mi,n+1 are given as concatenations of the P j,n and M j,n, which can be described by

a substitution τn on 2k − 2 symbols mi and pi : we put

• if i ∈ Bn and ιni = +, τnpi = pi , τnmi = mpn(i)pi ;

• if i ∈ Bn and ιni = −, τnpi = pmn(i)mi , τnmi = mi ;

• if i 6∈ Bn, τnpi = pi , τnmi = mi .

Proposition 3.2. The symbolic system XL(I) is the shift on all sequences x

such that, for every s < t, there exists n such that xs . . . xt is a factor of any Pi,n.

Proof. This comes from the minimality of the system. �

This last result, together with the formulas above, can be interpreted as an

explicit presentation of XL(I) as an adic system; see, for example, [34]. We can

also say that it is generated by the substitutions τn. As τn of any symbol is a word

of length 1 or 2, they form a finite family of substitutions, and thus define XL(I) as

an S-adic system; see [12].


It is clear that this result is not optimal, as it is possible to use substitutions on k

symbols. This can be done from the present algorithm, by computing return words

as in [13] (in all cases for k = 3) or [15] (in examples for k = 4), but, though it is

possible in each given example, a general formula seems out of reach for k > 3;

from the classical algorithms, no general formula is available either.

The following result is “in the folklore” and holds if we replace our induction

algorithm by any classical one, but we state it for sake of completeness.

Proposition 3.3. Let I be a symmetric k-interval exchange transformation,

satisfying the i.d.o.c. condition. Then the symbolic system XL(I) is a primitive

substitutive system if and only if its induction path is ultimately periodic.

Proof. If the induction path is ultimately periodic, let q be its period and n

a fixed integer chosen large enough. The rules defining the Pi,n+q and Mi,n+q, as

a function of the Mi,n and Pi,n define one substitution on 2k − 2 letters, primitive

because of minimality and with a fixed point u. Then, if φ associates to each

symbol the corresponding Pi,n or Mi,n, Proposition 3.2 implies the “if” part.

If the coding is a primitive substitutive system, Theorem 7.2 of [17] implies that

the number of induced systems on the union of cylinders is finite (up to a letter-to-

letter bijection). As the algorithm defined in Section 2.6 or 3.1 is an induction on

the union of cylinders, this means that it is ultimately periodic, which means the

induction path is ultimately periodic. �

A difficult open question is to prove that a k-interval exchange transformation

corresponds to a substitutive system if and only if the lengths of the intervals are

algebraic of degree at most k. The “if” part is known, and could be deduced from

the above result by using substitutions on k letters, but the “only if” part has been

proved only for k = 3 [13], and in that case the degree always falls to k − 1 = 2.

3.4 The hyperelliptic class. The Rauzy class of a permutation π is de-

fined, for example, in [25] to which we refer the reader; π′ is in the class of π if

and only if some interval exchange with permutation π′ is related by the Rauzy

induction to an interval exchange with permutation π. The Rauzy class of the

permutation σ is called the hyperelliptic class, and I is hyperelliptic if its per-

mutation is hyperelliptic. The class is completely described in [29], though its

name was given later; see [23], for example. A fundamental result of [29] is that

any hyerelliptic interval exchange transformation, satisfying the i.d.o.c. condition,

induces a symmetric interval exchange on a suitable subinterval; equivalent results

are not known for other Rauzy classes.


By inducing in the reverse direction, we can deduce the bispecial factors of

a hyperelliptic k-interval exchange transformation from those of a symmetric k-

interval exchange transformation by a substitution, and thus we can use all the

above results to describe them. However, their generation by a composite tree of

relations does not seem to generalize directly.

4 Applications of the induction in the hyperelliptic case

4.1 Repetitions of words. The following result answers a question of

Boshernitzan to the first author (2008). It illustrates how the self-dual induction

gives a good knowledge of the word combinatorics of the langugae.

Proposition 4.1. The language of a hyperelliptic interval exchange transfor-

mation, satisfying the i.d.o.c. condition, contains the word ww for infinitely many

different words w.

Proof. We suppose first that I is symmetric with alternate discontinuities. By

Theorem 2.8, Msn(i),n, resp. Psn(i),n, have a common suffix with wi,n. We check

inductively that this suffix is the full M , resp. P, word, unless the P word is equal

to the initial P1,0 and this does not happen for n large enough. Thus both the words

Msn(i),nMmnsn(i),n and Psn(i),nPpnsn(i),n exist for all n large enough.

We use now the tree structure of the Gn. In any of them, there exists at least

one end, namely one vertex i which is the end of only one edge, going to a vertex

j which is the end of at most two edges (otherwise Gn is not a tree). If i− j , then

sn(i) = pn(i) = i and Pi,nPi,n is in the language; if i+ j , then sn(i) = mn(i) = i and

Mi,nMi,n is in the language. If i = j− j ′, when N is the smallest n′ ≥ n for which

i ∈ Bn′ and ιn′ i = +, at stage N + 1 the only edge touching i is i+ j (though j may

have other neighbours) and, as above, Mi,N +1Mi,N +1 is in the language. If i = j + j ′,

when N is the smallest n′ ≥ n for which i ∈ Bn′ and ιn′ i = −, at stage N + 1 the

only edge touching i is i− j and, as above, Pi,N +1Pi,N +1 is in the language. Thus

Pi,nPi,n or Mi,nMi,n are in the language for abitrarily large n and these are words

whose length tends to infinity.

The same reasoning works if I is symmetric, by taking n large enough for Gn

to be a tree of relation. And if I is hyperelliptic, the ww in the language of an

induced symmetric interval exchange yield similar words in the language of I. �

4.2 The 2k − 1 distances theorem.

Theorem 4.2. Let I be a hyperelliptic k-interval exchange transformation,

satisfying the i.d.o.c. condition. Then the (k − 1)n + 1 intervals of the partition of


[0, 1[ by the points I− jγi, 0 ≤ j ≤ n − 1, 1 ≤ i ≤ k − 1 have at most 2k − 1

different lengths.

Proof. The intervals we consider are the cylinders [w], for all the words w

with n letters (we write this, and not that n is the length of w, to avoid confusion

with the length of the interval).

We suppose first that I is symmetric and with alternate discontinuities. We

check that the intervals [w] and [w] have the same length: this is done by induction

on the number of letters of w, as the lengths of the cylinders with n letters are

known if we know the lenghts of the cylinders with n − 1 letters and all the words

with n letters; see, for example, [8] or Proposition 4 of [16].

Then we follow Boshernitzan [8] by building what is now called the Rauzy

graph (see also [30] and [3]): if w and w′ are words of n letters in L(I), we write

w → w′ if there exists a word v with n − 1 letters such that w = av , w′ = vb,

and avb ∈ L(I). If w is not right special, w′ is not left special, and w → w′, then

the intervals [w] and [w′] have the same length. 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 length 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 k − 1 left special words with n letters, vi,n beginning with i. Let Li,n

be the length of [vi,n] ∩ [0, βi[, Ri,n the length of [vi,n] ∩ [βi, 1[. Then the length of

[vi,n] is Li,n + Ri,n, and the length of a [v] such that v → vi,n is either Li,n or Ri,n.

Moreover, let Wi,n be the shortest bispecial word having vi,n as a prefix, which is

also the shortest bispecial word beginning with i and having at least n letters; then

Wi,n is some wi,q of Theorem 2.8 and, with the notation of Section 2.6, Li,n = li,q

and Ri,n = ri,q.

Thus we have proved, as in Boshernitzan [8] but in a particular case, that the

cylinders with n letters have at most 3k − 3 different lengths; and these are Li,n,

Ri,n, and Li,n + Ri,n. We shall now improve this estimate by studying the evolution

of the Wi,n, which are the wi,n with a different numbering, varying with i.

Here Wi,n is the word wi,q for some (maybe several) q depending on n and i;

the word Wi,n is of length at least n, and its last letter is dn(i) = sq(i) for any q

such that Wi,n = wi,q. Suppose Wi,n 6= Wi,n+1; then Wi,n has n letters. In this case

(though not necessarily in general) its retrograde W i,n is Wdn(i),n and dndn(i) = i.

Then Wi,n+1 is computed from any wi,q such that Wi,n = wi,q, by taking the first

wi,q′ 6= wi,q for q′ > q.


At the beginning Wi,1 = wi,0 for all 1 ≤ i ≤ k − 1. We have k − 2 different

non-trivial relations Ri,1 = Rk−i,1, 1 ≤ i ≤ [ k−12

], corresponding to edges (k − i)−i

in G0, and Li,1 = Lk+1−i,1, 2 ≤ i ≤ [ k2], corresponding to edges (k + 1 − i)+i in G0.

Suppose that for i 6= j we have Ri,n = R j,n, and an edge i− j in Gq for some

q such that Wi,n = wi,q and W j,n = w j,q. Let r be the smallest q′ ≥ q such that

Ei,q′ is cut on the right; it is also the smallest q′ ≥ q such that E j,q′ is cut on the

right. If l = sr(i), l′ = sr( j ), we have l 6= l′ and wl′,r+1 is the retrograde of wl,r+1.

Let n1, resp. n2, be the number of letters of wi,r , resp. w j,r ; then Wi,n1= wi,r and

W j,n2= w j,r .

For every n ≤ n′ ≤ n1 ∧ n2 we have Ri,n′ = R j,n′ = Ri,n = R j,n. Then:

• If n1 = n2, the formulas in Section 2.6 give the new relation

Ll,n1+1 + Rl,n1+1 = Ll′,n1+1 + Rl′,n1+1

and we have an edge l =l′ in Gr+1, with Wl,n1+1 = wl,r+1 and Wl′,n1+1 = wl′,r+1.

• If n1 < n2, Wl,n1+1 = wl,r+1, while W j,n1+1 is a prefix of w j,r , and we have the

new relation

R j,n′ = Ll,n′ + Rl,n′

for every n1 + 1 ≤ n′ ≤ n2. Then Wl′,n2+1 = wl′,r+1. Let n3 be the number

of letters of wl′,r+1 or wl,r+1: we have n2 + 1 ≤ n3, and Wl,n′ = wl,r+1 for all

n1 + 1 ≤ n′ ≤ n3, hence Wl,n2+1 = wl,r+1. Thus we have the relation

Ll,n2+1 + Rl,n2+1 = Ll′,n2+1 + Rl′,n2+1

and an edge l =l′ in Gr+1, with Wl,n2+1 = wl,r+1 and Wl′,n2+1 = wl′,r+1.

• The case n1 > n2 is similar.

Finally, if we have a non-trivial relation Li,n + Ri,n = L j,n + R j,n, and an edge

i = j in Gq for q such that Wi,n = wi,q, and W j,n = w j,q, and let r be the smallest

q′ such that Ei,q′ and E j,q′ are cut, and n4 the number of letters of wi,r or w j,r , then

the relation holds for n ≤ n′ ≤ n4, and either Ri,n4+1 = R j,n4+1, and there is an edge

i− j in Gr+1 with Wi,n4+1 = wi,r+1 and W j,n4+1 = w j,r+1, or Li,n4+1 = L j,n4+1, and

there is an edge i+ j in Gr+1 with Wi,n4+1 = wi,r+1 and W j,n4+1 = w j,r+1.

Similar reasonings apply, mutatis mutandis, for the evolution of relations Li,n =

L j,n with an edge i+ j in Gq for some q such that Wi,n = wi,q and W j,n = w j,q.

Hence we have proved that there are k −2 non-trivial relations between the Li,n

and Ri,n and that they are always of one of the five forms:

• Ri,n = R j,n,

• Li,n = L j,n,

• Li,n + Ri,n = L j,n + R j,n,


• Li,n + Ri,n = R j,n,

• Li,n + Ri,n = L j,n.

Hence the 3k − 3 quantities considered at the beginning of this proof take at

most 2k − 1 different values.

When I is symmetric but not with alternate discontinuities, we use the con-

struction in Section 3.1 to get that we have at most 2k − 1 values, and the same

relations, for n large enough. For the first values of n, there are less left special

words, and we can bound the number of different lengths by 2h − 1 + k − h for

some h ≥ k, thus our result is still true.

And the result extends from I to its Rauzy induced map I′, as the I

′−nγ′i , n =

0, 1, 2, . . . , are exactly the I−mγi , m = −1, 0, 1, . . . , which fall into the induction

interval (including its right end). Thus our theorem is true also in the hyperelliptic

class. �

The above proof gives also a way to compute recursively the expression of

each of the 2k − 1 lengths. The theorem is still true if we replace the lengths of

the intervals by their measures for a given I-invariant probability, as they follow

the same rules.

For general (including non-hyperelliptic) k-interval exchange transformations,

the number of lengths is bounded by 3k − 3 [8]; this generalizes the famous three

distances theorem for rotations, corresponding to k = 2; see [1] for further refer-

ences. The bound 2k − 1 in the symmetric case is of course the same for k = 2;

it can be deduced from [1] for k = 3, and is new for k ≥ 4; there may be less

than 2k − 1 lengths, either because there are “accidental” relations which do not

contradict the i.d.o.c. property, or because some Li,n or Ri,n is not the length of any

cylinder with n letters, as in the Rauzy graph there is a v → v ′ where v is right

special and v ′ is left special. The latter phenomenon is known to happen for k = 2,

where for infinitely many n there are two distances only; see [6], for example. But

we can prove

Proposition 4.3. If I is symmetric and totally irrational, there exists a subset

of IN of density 1 for which there are exactly 2k − 1 different lengths in Theorem

4.2.

Proof. For n large enough, there are k − 1 left special words of length n, and

3k−3 quantities Li,n, Ri,n, Li,n +Ri,n of the previous proof. If these 3k−3 quantities

take strictly less than 2k − 1 different values, this creates an extra relation which,

going back to the initial state, contradicts the hypothesis of total irrationality. If

any of these quantities is not a length, this means (see, for example, [10]) that n−1

is the number of letters of a bispecial word. But then, by construction, for each


i, the difference between the numbers of letters of Wi,n+1 and Wi,n is 0 or tn, with

tn → +∞ with n. Thus, for n in a set of density one, each of the 2k − 1 above

quantities is indeed a length in Theorem 4.2. �

5 Induction in the non-hyperelliptic case

The above induction is not restricted to the symmetric case, though the underlying

symmetry of the language is used many times, explicitly or implicitly. Indeed,

for general permutations π the self-dual induction(s) can be defined as in Section

2.2, and in the present section we make it work in a particular case with k = 4

and the permutation π1 = 4, π2 = 3, π3 = 1, π4 = 2. This permutation is not

in the hyperelliptic class but in the Rauzy class of the circular permutation, and

hence dynamically the system is just a rotation. However, as the coding is not

the usual one, this does not give useful information on the word combinatorics,

and we shall see in Theorem 5.2 below that in a sense it is more complicated than

for the hyperelliptic class. For this permutation the parameters, even in the initial

state, do not satisfy relations which can be symbolized by a tree of relations as in

Section 2.5, and in general we have not been able to find choices which allow a

self-dual induction to stay, even ultimately, in the “natural” case where each Ei,n is

cut into two parts at each stage. This makes the induction rather unwieldy (under

the condition of alternate discontinuities, the authors could not find less than 164

states for this permutation, to compare with the 9 states of the symmetric case for

k = 4), preventing us from getting a structure theorem as in Section 2.9. Still, it

works in particular cases.

5.1 A family of examples. In this section the permutation is π1 = 4,

π2 = 3, π3 = 1, π4 = 2. We define a self-dual induction process as in Section 2.6,

in a particular case.

Again, we give only the construction, as all the proofs are basically the same

as for Proposition 2.4 and Theorem 2.8. The only difference is that here there is

no symmetry, so the bijections s and sn are replaced by the identity. The equalities

wi,nPsnpn(i),n = Pi,nwpn(i),n are weakened to wi,nPpn(i),n = P′i,nwpn(i),n, for another

word P′i,n, and similarly for the M words, but these are sufficient to carry out the

proofs.

We build unions of k − 1 disjoint intervals Ei,n = [βi − li,n, βi + r,n[,

i = 1, . . . , k − 1, defined as in Section 2.2, such that for bijections p and m,

each Ei,n is partitioned into a left subinterval Ei,m,n and a right subinterval Ei,p,n

with SEi,p,n = [βp(i) − lp(i),n, βp(i)[ and SEi,m,n = [βm(i), βm(i) + rm(i),n[.


Here the relations between parameters are not given by a tree of relations, but

have the more general form

li,n + ri,n = lp(i),n + rm(i),n,

i = 1, 2, 3. We go through 3 different states. In state G0, p(1) = 3, m(1) = 2,

p(2) = 1, m(2) = 3, p(3) = 2, m(2) = 1; in state G1, p(1) = 1, m(1) = 2, p(2) = 2,

m(2) = 3, p(3) = 3, m(2) = 1; in state G2, p(1) = 2, m(1) = 2, p(2) = 3, m(2) = 3

p(3) = 1, m(2) = 1.

We define our choices only in the particular cases which will be used for our

examples

(i) If at stage n we are in state G0, G1 or G2 and ri,n − lp(i),n = rm(i),n − li,n > 0

for all i, then for all i we put Ei,n+1 = Ei,m,n, and we get li,n+1 = li,n, ri,n+1 =

ri,n − lp(i),n.

(ii) If at stage n we are in state G1 and ri,n − lp(i),n = rm(i),n − li,n is (strictly)

positive for i = 1, then for i = 1 we put Ei,n+1 = Ei,m,n, and we get li,n+1 =

li,n, ri,n+1 = ri,n − lp(i),n, while for i = 2, 3 we put Ei,n+1 = Ei,n.

(iii) If at stage n we are in state G1 and ri,n − lp(i),n = rm(i),n − li,n is negative for

i = 1 and (strictly) positive for i = 2, then for i = 2 we put Ei,n+1 = Ei,m,n,

and we get li,n+1 = li,n, ri,n+1 = ri,n − lp(i),n, while for i = 1, 3 we put

Ei,n+1 = Ei,n.

(iv) If at stage n we are in state G1 and ri,n − lp(i,n = rm(i),n − li,n is negative for

i = 1, 2 and (strictly) positive for i = 3, then for i = 3 we put Ei,n+1 = Ei,m,n,

and we get li,n+1 = li,n, ri,n+1 = ri,n − lp(i),n, while for i = 1, 3 we put

Ei,n+1 = Ei,n.

We check that if at stage n we are in state G0, resp. G1, resp. G2 and in case

(i), at stage n + 1 we are in state G1, resp. G2, resp. G0; if at stage n we are in state

G1 and in case (ii), (iii) or (iv), at stage n + 1 we are in state G1.

Our examples will satisfy the condition of alternate discontinuities βi < γi for

each 1 ≤ i ≤ k − 1 and γi < βi+1 for each 1 ≤ i ≤ k − 2.We start from Ei,0 = 1i ,

i = 1, 2, 3 and check that we are in state G0. Then we define the following path,

defined by three sequences of integers an, bn, cn, and we start from q0 = 0:

• at stage qn, we are in state G0 and case (i),

• at stages qn + 1 to qn + 1 + an, we are in state G1 and case (ii),

• at stages qn + 2 + an to qn + 2 + an + bn, we are in state G1 and case (iii),

• at stages qn + 2 + an + bn to qn + 2 + an + bn + cn, we are in state G1 and case

(iv),

• at stage qn + 3 + an + bn + cn, we are in state G1 and case (i),

• at stage qn + 4 + an + bn + cn, we are in state G2 and case (i),


• qn + 5 + an + bn + cn = qn+1.

The bispecial words wi,n such that Ei,n = [wi,n] are built as in Theorem 2.8,

for 1 ≤ i ≤ 3, wi,0 = i, 1 ≤ i ≤ 3, P1,0 = 41, Pi,0 = i, 2 ≤ i ≤ 3, Mi,0 = i,

1 ≤ i ≤ 3.

At stage n > 0 and for 1 ≤ i ≤ k − 1:

• if Ei,n has been cut on the left, wi,n+1 = wi,nPpn(i),n, Pi,n+1 = Pi,n, Mpn(i),n+1 =

Mi,nPpn(i),n;

• if Ei,n has been cut on the right, wi,n+1 = wi,nMmn(i),n, Mi,n+1 = Mi,n,

Pmn(i),n+1 = Pi,nMpn(i),n;

• if Ei,n has not been cut, wi,n+1 = wi,n, Pi,n+1 = Pi,n, Mi,n+1 = Mi,n.

And there exist words P′i,n and M ′

i,n such that:

• wi,nPpn(i),n = P′i,nwpn(i),n; moreover, this word contains no occurrence of any

wt,n except the prefix wi,n and suffix wpn(i),n.

• wi,nMmn(i),n = M ′i,nwmn(i),n); moreover, this word contains no occurrence of

any wt,n except the prefix wi,n and suffix wmn(i),n.

Proposition 5.1. For any choice of an > 0, bn > 0, cn > 0, there exists an

interval exchange I, with permutation π as above, satisfying the i.d.o.c. condition

and having the above path for the self-dual induction.

Proof. We could use the same kind of proof as for Theorem 2.9 but here, for

the purposes of Theorem 5.2 below, we rather use the reasoning of [22]. We note

that if at stage n we are in state G1, the relations simplify to r1,n = r2,n = r3,n;

hence, at each stage qn + 1 we take as independent parameters the parameters

l1,qn+1, l2,qn+1, l3,qn+1 and r1,qn+1 = r2,qn+1 = r3,qn+1 = rqn+1. If we find ri,0 and

li,0, i = 1, 2, 3, such that l1,qn+1, l2,qn+1, l3,qn+1 and rqn+1 are strictly positive for

every n, we have won. Now we check that l1,qn+1 = l1,qn+1+1 + anrqn+1+1, l2,qn+1 =

l2,qn+1+1 + bnrqn+1+1, l3,qn+1 = l3,qn+1+1 + cnrqn+1+1, rqn+1 = l1,qn+1+1 + l2,qn+1+1 + l3,qn+1+1 +

(an + bn + cn + 1)rqn+1+1, thus when we go from qn+2 + 1 to qn + 1 the matrix is

strictly positive. Hence the argument (using a compactness argument for an infinite

intersection of cones in projective space) in [22], p. 191, allows us to find l1,q0+1,

l2,q0+1, l3,q0+1 and rq0+1, and then l1,0 = l1,1, l2,0 = l2,1, l3,0 = l3,1, r1,0 = l3,1 + r1,

r2,0 = l1,1 + r1, r3,0 = l2,1 + r1 give the required vector. Next, we normalize by

l1,0 + r1,0 + · · · + l3,0 + r3,0 + l1,0 = 1 and obtain the αi of I. We then retrieve the

permutation from the initial state.

We check in the usual way that the wi,n are indeed the bispecial words of L(I),

and this in turn implies that I satisfies the i.d.o.c. condition, as otherwise there

would be a bispecial word w with only two xwy in the language. �


5.2 Application. We can now show that, without the hypothesis of hyper-

ellipticty, we have more than 2k − 1 lengths in Theorem 4.2, and that indeed the

bound 3k − 3 of Boshernitzan cannot be improved.

Theorem 5.2. There exist sequences an, bn, cn such that, if I is defined as in

the previous section, then for a sequence tn → +∞, the 3tn + 1 intervals of the

partition of [0, 1[ by the points I− jγi , 0 ≤ j ≤ tn − 1, 1 ≤ i ≤ 3 have exactly 9

different lengths.

Proof. We check by induction that for n ≥ 1, w1,qn= M ′′

2,qnP1,qn

where

M2,qn= 24M ′′

2,qn, w2,qn

= M ′′3,qn

P2,qnwhere M3,qn

= 3M ′′3,qn

, w3,qn= M ′′

1,qnP3,qn

where M1,qn= 1M ′′

1,qn, and then

w1,qn+1= M ′′

2,qn(P1,qn

M2,qn)an+1M3,qn

(P2,qnM3,qn

)bnM1,qn(P3,qn

M1,qn)cn,

w2,qn+1= M ′′

3,qn(P2,qn

M3,qn)bn+1M1,qn

(P3,qnM1,qn

)cnM2,qn(P1,qn

M2,qn)an,

w3,qn+1= M ′′

1,qn(P3,qn

M1,qn)cn+1M2,qn

(P1,qnM2,qn

)anM3,qn(P2,qn

M3,qn)bn .

Thus the number of letters of wi,qnis the same for i = 1, 2, 3; we denote it by

tn.

As in the proof of Theorem 4.2, li,qn, ri,qn

, li,qn+ ri,qn

are the different possible

lengths. We have

l1,qn= l1,qn+1,

l2,qn= l2,qn+1,

l3,qn= l3,qn+1,

r1,qn= l3,qn+1 + rqn+1,

r2,qn= l1,qn+1 + rqn+1,

r3,qn= l2,qn+1 + rqn+1

for the four independent parameters defined above.

There are two ways to ensure that the nine lengths thus defined are indeed

different:

• the first one is to imitate the reasoning of Theorem 6 of [22] to get that l1,1,

l2,1, l3,1 and r1 have no rational relations except the normalization;

• the other one is to take an, bn and cn constant, with a1, b1 and c1 all different.

Then for all n, l1,qn+1, l2,qn+1, l3,qn+1 and rqn+1 are proportional to l1,1, l2,1,

l3,1 and r1, which is the Perron–Frobenius eigenvector of a given primitive

matrix, and we check that this eigenvector satisfies no relations implying

equalities among the 9 lengths.

�


Acknowledgements. The authors thank T. Monteil for drawing the pictures.

REFERENCES

[1] P. Alessandri and V. Berthe Three distance theorem and combinatorics on words, Enseign. Math.

(2) 44 (1998), 103–132.

[2] V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial

mechanics, Uspekhi Mat. Nauk. 18 (1963), 91–192; translated in Russian Math. Surveys 18

(1963), 86–194.

[3] P. Arnoux and G. Rauzy, Representation geometrique de suites de complexite 2n + 1, Bull. Soc.Math. France 119 (1991), 199–215.

[4] A. Avila and G. Forni, Weak mixing for interval exchange maps and translation flows, Ann. of

Math. (2) 165 (2007), 637–664.

[5] A. Ya. Belov and A. L. Chernyatev, Describing the set of words generated by interval exchange

transformation, Comm. Algebra 208 (2010), 2588–2605.

[6] V. Berthe, Frequences des facteurs des suites sturmiennes, Theoret. Comput. Sci. 165 (1996),

295–309.

[7] V. Berthe, S. Ferenczi, and L. Zamboni, Interactions between dynamics, arithmetics and

combinatorics: the good, the bad, and the ugly, Algebraic and Topological Dynamics,Amer. Math. Soc., Providence, RI, 2005, pp. 333–364.

[8] M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke

Math. J. 52 (1985), 723–752.

[9] M. Boshernitzan, A unique ergodicity of minimal symbolic flows with linear block growth,

J. Analyse Math. 44 (1984/85), 77–96.

[10] J. Cassaigne, Limit values of the recurrence quotient of Sturmian sequences, Theoret.

Comput. Sci. 218 (1999), 3–12.

[11] J. Cassaigne, S. Ferenczi, and L. Zzmboni, Trees of relations, preprint (2007), http://iml.univ-mrs.fr/ ferenczi/cfz2.pdf.

[12] S. Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16 (1996),

663–682.

[13] S. Ferenczi, C. Holton, and L. Zamboni, The structure of three–interval exchange transformations

II: a combinatorial description of the trajectories, J. Analyse Math. 89 (2003), 239–276.

[14] S. Ferenczi and L. F. C. da Rocha, A self-dual induction for three-interval exchange transforma-

tions, Dynamical Systems, Dyn. Syst. 24 (2009), 393–412.

[15] S. Ferenczi, L. Zamboni, Eigenvalues and simplicity of interval exchange transformations,

preprint, http://iml.univ-mrs.fr/ ferenczi/fz2.pdf.

[16] S. Ferenczi and L. Zamboni, Languages of k-interval exchange transformations, Bull. London

Math. Soc. 40 (2008), 705–714.

[17] C. Holton and L. Zamboni, Descendants of primitive substitutions, Theory Comput. Syst. 32

(1999), 133–157.

[18] P. Hubert and A. Messaoudi, Best simultaneous diophantine approximations of Pisot numbers

and Rauzy fractals, Acta Arith. 124 (2006), 1–15.

[19] A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR 211

(1973), 775–778.

[20] A. B. Katok and A. M. Stepin Approximations in ergodic theory, Uspekhi Mat. Nauk 22 (1967),

81–106; translated in Russian Math. Surveys 22 (1967), 76–102.

[21] M. S. Keane, Interval exchange transformations, Math. Z. 141 (1975), 25–31.


[22] M. S. Keane, Non–ergodic interval exchange transformations, Israel J. Math. 26 (1977),

188–196.

[23] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differen-

tials with prescribed singularities, Invent. Math. 153 (2003), 631–678.

[24] A. O. Lopes and L .F. C. da Rocha, Invariant measures for Gauss maps associated with interval

exchange maps, Indiana Univ. Math. J. 43 (1994), 1399–1438.

[25] S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval ex-

change maps, J. Amer. Math. Soc. 18 (2005), 823–872 (electronic).

[26] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115

(1982), pp. 169–200.

[27] V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR 168 (1966),

1009–1011.

[28] R. C. Penner and J. L. Harer, Combinatorics of Train Tracks, Princeton University Press, Prince-

ton, NJ, 1992.

[29] G. Rauzy, Echanges d’intervalles et transformations induites, Acta Arith. 34 (1979), 315–328.

[30] G. Rauzy, Suites termes dans un alphabet fini in Seminar on Number Theory, Exp. No. 25, Univ.

Bordeaux I, Talence, 1984.

[31] W. A. Veech, Interval exchange transformations, J. Analyse Math. 33 (1978), 222–272.

[32] W. A. Veech, A criterion for a process to be prime, Monatsh. Math. 94 (1982), 335–341.

[33] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann.

of Math. (2) 115 (1982), 201–242.

[34] A. M. Vershik and A. N. Livshits, Adic models of ergodic transformations, spectral theory, substi-

tutions, and related topics, in Representation Theory and Dynamical Systems, Amer. Math. Soc.,

Providence, RI, 1992, pp. 185–204.

[35] A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov

exponents, Ann. Inst. Fourier (Grenoble) 46 (1996), 325–370.

Sebastien Ferenczi

INSTITUT DE MATHEMATIQUES DE LUMINY

CNRS - UMR 6206 CASE 907,

163 AV. DE LUMINY

F13288 MARSEILLE CEDEX 9, FRANCE

and

FEDERATION DE RECHERCHE DES UNITES DE MATHEMATIQUES DE MARSEILLE

CNRS - FR 2291

email: [email protected]

Luca Q. Zamboni

INSTITUT CAMILLE JORDAN

UNIVERSITE CLAUDE BERNARD LYON 1

43 BOULEVARD DU 11 NOVEMBRE 1918

F69622 VILLEURBANNE CEDEX, FRANCE

and

REYKJAVIK UNIVERSITY, SCHOOL OF COMPUTER SCIENCE

KRINGLAN 1

103 REYKJAVIK, ICELAND

email: [email protected]

(Received May 4, 2009)

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