Acta Polytechnica


doi:10.14311/AP.2013.53.0444
Acta Polytechnica 53(5):444–449, 2013 © Czech Technical University in Prague, 2013

available online at http://ojs.cvut.cz/ojs/index.php/ap

ITINERARIES INDUCED BY EXCHANGE OF TWO INTERVALS

Zuzana Masáková, Edita Pelantová∗

Department of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00 Praha 2,
Czech Republic

∗ corresponding author: edita.pelantova@fjfi.cvut.cz

Abstract. We focus on the exchange T of two intervals with an irrational slope α. For a general
subinterval I of the domain of T , the first return time to I takes three values. We describe the structure
of the set of return itineraries to I. In particular, we show that it is equal to {R1,R2,R1R2,Q} where
Q is amicable with R1, R2 or R1R2.

Keywords: interval exchange, first return map, return time.

Submitted: 27 March 2013. Accepted: 22 April 2013.

1. Introduction
We study the symbolic dynamical system given by
the transformation T of the unit interval, T : [0, 1) →
[0, 1),

T(x) =
{
x−α for x ∈ [α, 1),
x + 1 −α for x ∈ [0,α),

(1)

where α is a fixed number in [0, 1). Transformation
T has only one discontinuity point, such a dynamical
system is the simplest dynamical system with discon-
tinuous transformation. Dynamical systems defined
by continuous transformations F : J → J have a
number of nice properties, for example, there exists
a fixed point ρ ∈ J, F(ρ) = ρ. The famous theorem
of Sharkovskii [1] describes the structure of periodic
points, i.e., fixed points of Fk for some k ∈ N.
If one chooses the parameter α in (1) irrational,

the map T has no periodic point, in other words,
the orbit {ρ,T(ρ),T 2(ρ), . . .} is infinite for every ρ ∈
[0, 1). Nevertheless, T has a weaker property, namely
that although Tk(ρ) 6= ρ for any k ∈ N, one can get
arbitrarily close to a point ρ with some of its iterations.
More precisely,

∀ε > 0 ∃n ∈ N, n ≥ 1 :
∣∣Tn(ρ) −ρ∣∣ < ε. (2)

Moreover, property (2) holds for every ρ ∈ [0, 1).
It is well known that every point ρ ∈ [0, 1) can

be uniquely represented using the infinite string of 0
and 1, which constitutes the binary expansion of the
number ρ. The mapping T of (1) allows another type
of representation of ρ, namely by the coding of the
orbit of ρ under T. Denote J0 = [0,α), J1 = [α, 1)
and set

un =
{

0 if Tn(ρ) ∈ J0,
1 if Tn(ρ) ∈ J1.

Knowledge of the infinite word uρ := (un)∞n=0 allows
one to determine the number ρ, i.e., the mapping
ρ 7→ uρ is one-to-one. The above defined infinite words

uρ appear naturally in diverse mathematical problems;
they were discovered and re-discovered several times
and given different names. We will call the infinite
word uρ a Sturmian word with slope α and intercept
ρ.

Let us point out one important difference between
binary expansion of numbers and their representation
by Sturmian words with a fixed slope α. Every string
of length n of letters 0 and 1 appears in the binary
expansion of some real number ρ ∈ [0, 1). The number
of such strings is obviously 2n. By contrast, the list
of all strings of length n appearing in the represen-
tation uρ of all ρ ∈ [0, 1) has exactly n + 1 elements.
Nevertheless, one can still represent a continuum of
real numbers ρ. On the other hand, any type of repre-
sentation using at most n strings of 0 and 1 of length
n would allow representation of only countably many
numbers. In that sense, Sturmian words represent
real numbers in the most economical way.

Sturmian words have many other remarkable prop-
erties, for a review, see [2]. Generalizations of Stur-
mian words are treated in [3].

The property (2) expresses the fact that iterations
Tn(ρ) return arbitrarily close to ρ. This allows one to
define, for a subinterval I ⊂ [0, 1) of positive length,
the so-called return time r : I → N by

r(ρ) := min
{
n ∈ N, n ≥ 1, : Tn(ρ) ∈ I

}
.

The return time represents the number of iterations
needed for a point ρ to come back to the interval where
it comes from. The movement of point ρ on its path
from I back to I is recorded by the so-called I-itinerary
of ρ, which we denote by R(ρ). It is defined as the
finite word w0w1 · · ·wn−1 in the alphabet A = {0, 1}
of length n = r(ρ) such that

wi = a, if Ti(ρ) ∈ Ja, a ∈A.

Equivalently, the I-itinerary R(ρ) of ρ is a prefix of
the infinite word uρ of length r(ρ). In our consider-
ations, the interval I is fixed. Thus, for simplicity

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vol. 53 no. 5/2013 Itineraries Induced by Exchange of Two Intervals

of notation, we avoid marking the dependence on I
of the first return time and return itinerary, i.e., we
write r(x), R(x) instead of rI(x), RI(x), respectively.
The position of the point ρ ∈ I after its return the
interval I defines a new transformation TI : I → I by

TI(ρ) = Tr(ρ)(ρ), (3)

which is usually called the first return map or induced
map.

The I-itineraries for a special type of interval I were
studied in diverse contexts:

• If the boundary points of the interval I are
neighbouring elements of the set {α,T−1(α), . . . ,
T−n(α)} for some n ∈ N, then the set of I-
itineraries R(ρ) for ρ ∈ I consists of only two words.
This reformulates the result of Vuillon [4] about
the existence of exactly two return words to a fixed
factor of a Sturmian word.

• If the Sturmian word uρ is invariant under a sub-
stitution 0 7→ ϕ(0), 1 7→ ϕ(1), then there exists
an interval I ⊂ [0, 1), ρ ∈ I, such that the in-
duced map TI is homothetic to T, and the finite
words ϕ(0), ϕ(1) are the I-itineraries. Invariance
of Sturmian words under substitutions was studied
by Yasutomi [5].

• An Abelian return word to a factor of a Sturmian
word is an I-itinerary for I = [0,β) or I = [β, 1)
for some β ∈ [0, 1), see [6]. As follows from the
result of [7], the intervals of the mentioned form
have at most three itineraries R1, R2, R3 and for
their length one has |R3| = |R1| + |R2|. In [8], we
have shown that a stronger statement holds, namely
that the word R3 is a concatenation of words R1
and R2.

The aim of this paper is to describe the structure
of the set of I-itineraries for a general position and
length of the subinterval I ⊂ [0, 1). The set of all
I-itineraries R(x) for x ∈ I is denoted by ItI. For the
description, we will use the notion of word amicability.
We say that two finite words w and v over the alphabet
{0, 1} are amicable, if there exist words p,q ∈{0, 1}∗
such that w = p01q and v = p10q or w = p10q and
v = p01q. In other words, v is obtained from w
by interchanging the order of letters 0 and 1 at two
neighbouring positions i− 1, i.
It follows from [9] that for every interval I there

exist at most four I-itineraries, i.e., #ItI ≤ 4. We
will show the following theorem.

Theorem 1.1. Let T be the transformation (1) for
some irrational α ∈ (0, 1) and let I ⊂ [0, 1) be an
interval. Then there exist words R1,R2 ∈{0, 1}∗ such
that for the set ItI of all I-itineraries one has

ItI ⊂{R1,R2,R1R2,Q},

where Q is amicable with R1, R2 or R1R2.

From the proof of Theorem 1.1 (at the end of Sec-
tion 2) one can see that in the generic case,

ItI = {R1,R2,R1R2,Q}.

In Section 3 we discuss the possibilities for Q if #ItI =
4 and determine the cases for which the set ItI has
less than 4 elements.

2. Interval Exchange
Transformations

First, let us recall the definition and certain properties
of k-interval exchange maps, which we use for k = 2
and 3.

Definition 2.1. Let J0∪J1∪·· ·∪Jk−1 be a partition
of the interval J, where Ji are intervals closed from the
left and open from the right for every i = 0, . . . ,k−1.
The transformation T : J → J is called a k-interval
exchange if there exist constants c0,c1, . . . ,ck−1 ∈ R
such that

T(x) = x + cj, x ∈ Jj,

and T is a bijection on J.

Since T is a bijection, intervals T(Ji) for j =
0, 1, . . . ,k − 1 form a partition of J. The order of
indices j which determines the ordering of intervals
T(Ji) in J is usually expressed by a permutation π. A
trivial example of a k-interval exchange is the choice
cj = 0 for j = 0, . . . ,k−1. Then T is the identity map
and π is the identity permutation. The transformation
T of (1) is a 2-interval exchange with permutation
(21).

Example 2.2. Consider a,b ∈ (0, 1), a < b. Put

I0 = [0,a), I1 = [a,b), I2 = [b, 1).

Then the transformation T : [0, 1) → [0, 1) given by

T(x) =



x + 1 −a if x ∈ [0,a),
x + 1 −a− b if x ∈ [a,b),
x− b if x ∈ [b, 1),

(4)

is a 3-interval exchange with permutation π = (321),
see Figure 1.

From now on, we focus on the exchange T of two
intervals given by the prescription (1) with an irra-
tional slope α. We will study the first return map TI
defined by (3) to the subinterval I ⊂ [0, 1).

In [10] it is shown how TI depends on the length of
the interval I. For an irrational α ∈ (0, 1) with the
continued fraction α = [0,a1,a2, . . . ] and convergents
pn
qn

set

δk,s :=
∣∣(s− 1)(pk −αqk) + pk−1 −αqk−1∣∣,

for k ≥ 0, 1 ≤ s ≤ ak+1. (5)

For the numbers δk,s one has δk,s > δk′,s′ if and only
if k′ > k or k′ = k and s′ > s.

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Z. Masáková, E. Pelantová Acta Polytechnica

I0︷ ︸︸ ︷ I1︷ ︸︸ ︷ I2︷ ︸︸ ︷

︸ ︷︷ ︸
T (I2)

︸ ︷︷ ︸
T (I1)

︸ ︷︷ ︸
T (I0)

Figure 1. Exchange of three intervals.

In [10], we study infinite words associated to cut-
and-project sequences which we show to be exactly
codings of exchanges of two or three intervals. The
following proposition is a reformulation of statements
of Theorem 4.1 and Proposition 4.5 of [10] in the
framework of interval exchanges.

Proposition 2.3. Let T : [0, 1) → [0, 1) be an ex-
change of two intervals with irrational slope α and let
I = [c,d) ⊂ [0, 1). For the induced map TI one has
(1.) If d− c = δk,s for some k,s, defined in (5), then
TI is an exchange of two intervals.

(2.) Otherwise, TI is an exchange of three intervals
with permutation (321). Moreover, the lengths
of intervals I0, I1, I2 forming the partition of I
depend only on d−c and for the return time r(x0),
r(x1), r(x2) of points x0 ∈ I0, x1 ∈ I1, x2 ∈ I2,
x0 < x1 < x2, one has r(x1) = r(x0) + r(x2).

Remark 2.4. Proposition 4.5 of [10] also allows to
determine the exact two or three values of return time
r(x) to I. In fact, if d− c = δk,s, then — keeping the
notation of (5) — r(x) takes two values{

r(x) : x ∈ I
}

=
{
qk,sqk + qk−1

}
.

If d − c is between δk,s and its successor in the de-
creasing sequence (δk,s), then r(x) takes three values{
r(x) : x ∈ I

}
=
{
qk,sqk + qk−1, (s + 1)qk + qk−1

}
.

The values of return time are connected to the so-
called three-distance theorem [11, 12]. Another point
of view on return time in Sturmian words is presented
in [13].

Although the return time r(x) to a given interval I
can take only three values, the set ItI of I-itineraries
can have more than three elements. The following
statement can be extracted from the proof of the
Theorem in [9, §2]. It is convenient to provide the
demonstration here.

Proposition 2.5. Let T : [0, 1) → [0, 1) be an ex-
change of two intervals with irrational slope α and let
I = [c,d) ⊂ [0, 1). Then ItI has at most 4 elements.

Proof. Choose x ∈ I. Denote R(x) its I-itinerary and
r = r(x) its return time. Let H ⊂ I be the maximal
interval containing x such that for every x′ ∈ H one
has R(x) = R(x′). For H, it holds that
(1.) Ti(H) ⊂ [0,α) or Ti(H) ⊂ [α, 1) for i =

0, 1, . . . ,r − 1;
(2.) Ti(H) ∩ I = ∅ for i = 1, . . . ,r − 1;
(3.) Tr(H) ⊂ I.
The theorem will be established by showing that

there are only four candidates for the left end-point
of the interval H = [c̃, d̃). Obviously, one of them
is c̃ = c. If it is not the case, maximality of H and
properties (1.), (2.), and (3.) imply that c < c̃ < d̃ ≤ d
and there exists
(a) l̃, r − 1 ≥ l̃ ≥ 1 such that T l̃(c̃) = d; or
(b) ñ, r − 1 ≥ ñ ≥ 0 such that T ñ(c̃) = α; or
(c) m̃, r − 1 ≥ m̃ ≥ 1 such that Tm̃(c̃) = c.

Suppose that possibility (a) happened. Let us mention
that it is possible only if d < 1. Denote

l = min
{
k ∈ Z, k ≥ 1 : T−k(d) ∈ I

}
. (6)

Since T−l̃(d) = c̃ ∈ H ⊂ I, we have by definition of l
that l̃ ≥ l. We will show by contradiction that l̃ = l.
If l̃ > l, then T l̃−l(c̃) = T−l

(
T l̃(c̃)

)
= T−l(d) ∈ I, and

by definition of return time r = r(c̃) ≤ l̃ − l. This
contradicts the fact that l̃ ≤ r − 1.

Similar discussion for possibilities (b) and (c) shows
that the left end-point of the interval H is equal either
to T−l(d) where l is defined by (6), or T−n(α), where

n = min
{
k ∈ Z, k ≥ 0 : T−k(α) ∈ I

}
, (7)

or T−m(c), where

m = min
{
k ∈ Z, k ≥ 1 : T−k(c) ∈ I

}
. (8)

This means that I is divided by the three (not neces-
sarily distinct) points T−l(d), T−n(α), T−m(c) into
at most 4 subintervals H on which the I-itinerary is
constant.

Proposition 2.6. Let ItI be the set of I-itineraries
for the interval I = [c,d) ⊂ [0, 1) under an exchange
of two intervals with irrational slope α. There exist
neighbourhoods Hc and Hd of c, d, respectively, such
that for every c̃ ∈ Hc and d̃ ∈ Hd, 0 ≤ c̃ < d̃ ≤ 1 one
has

ItĨ ⊇ ItI, where Ĩ = [c̃, d̃).

Proof. Let ItI = {R1, . . . ,Rp}. Proposition 2.5 im-
plies that p ≤ 4 and for every 1 ≤ i ≤ p the elements x
such that R(x) = Ri form an interval, say Ii. Choose
xi ∈ Ii such that for q with 0 ≤ q ≤ r(xi) − 1 =
|Ri| − 1 one has Tq(xi) /∈ {c,d,α}, (it suffices to
choose xi /∈ Z[c,d,α]). Denote M = {c,d,α} and
N = {Tq(xi) : i = 1, . . . ,p, 0 ≤ q ≤ r(xi) − 1}. Put

ε := min
{
|a− b| : a ∈ M,b ∈ N

}
.

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vol. 53 no. 5/2013 Itineraries Induced by Exchange of Two Intervals

Then for every c̃ ∈ (c−ε,c+ε) and d̃ ∈ (d−ε,d+ε), the
I-itineraries R(x1), . . . , R(xp) are also Ĩ-itineraries,
where Ĩ = [c̃, d̃).

Proof of Theorem 1.1. If I = [c,d) where c = 0 or
d = 1, then by Theorem 4.5 of [8], the set ItI of I-
itineraries is of the form ItI ⊂{R,R′,RR′}. Without
loss of generality, we can therefore assume that c 6= 0
and d 6= 1.
If c, d, or d− c belongs to Z[α] (which is dense in

R), we can always use Proposition 2.6 to find Ĩ =
[c̃, d̃) such that ItĨ ⊇ ItI. Therefore, without loss of
generality we assume c,d,d− c /∈ Z[α]. In particular,
d− c 6= δk,s. From the proof of Proposition 2.5, the
interval I is divided into at most four subintervals
with constant I-itinerary by points

λ = T−l(d), l = min
{
k ≥ 1 : T−k(d) ∈ I

}
,

µ = T−m(c), m = min
{
k ≥ 1 : T−k(c) ∈ I

}
,

ν = T−n(α), n = min
{
k ≥ 0 : T−k(α) ∈ I

}
.

Moreover, λ and µ separate intervals with different
return times. In particular, for sufficiently small ε,
one has

l = r(λ−ε) < r(λ + ε),
m = r(µ + ε) < r(µ−ε).

(9)

By Proposition 2.3, the induced map TI is an exchange
of three intervals with permutation (321). Let I = I0∪
I1 ∪ I2 be the corresponding partition of I, where for
every x0 ∈ I0, x1 ∈ I1, x2 ∈ I2 one has x0 < x1 < x2.
By the same proposition r(x1) = r(x0) + r(x2), which
together with inequalities (9) implies that the right
end-point of I0 is equal to λ, the left end-point of I2
is equal to µ, and r(x1) = l + m.
Since c,d,d− c /∈ Z[α], we also have λ /∈ Z[α], and

thus one can choose ε sufficiently small, so that the
interval [λ − ε,λ + ε] does not contain any of the
points T−j(α) for 0 ≤ j ≤ l + m. This implies that
Tj
(
[λ−ε,λ + ε]

)
is an interval not containing α for

any j = 0, 1, . . . , l + m − 1, and consequently, the
prefix of length l + m of the infinite word uρ is the
same for any ρ ∈ [λ−ε,λ + ε]. We have

T l(λ−ε) = d−ε ∈ I,
T l(λ + ε) = d + ε /∈ I.

For the corresponding I-itineraries, we thus have

R(λ + ε) = R(λ−ε)R(d−ε).

We can set R1 = R(λ−ε), R2 = R(d−ε), to have

ItI ⊃{R1,R2,R1R2}.

By Proposition 2.5, the set ItI may have four el-
ements. Let us determine the fourth element Q.
Consider the point ν = T−n(α), n = min{k ≥ 0 :
T−k(α) ∈ I}, which, by the proof of Proposition 2.5
splits one of the intervals I0, I1, I2, into two, so that

the I-itinerary on the new partition is constant. By
the assumption that c,d /∈ Z[α], we have ν 6= λ, ν 6= µ.

Consider the points ν−ε, ν +ε for sufficiently small
ε. Obviously, their return time coincides, r(ν −ε) =
r(ν+ε) = r(ν), thus the I-itineraries R(ν−ε), R(ν+ε)
are of the same length r(ν). Since Tn(ν) = α, we have
Tn+1(ν) = 0 /∈ I, and thus r(ν) ≥ n + 1. We can see
that

Tn+1(ν + ε) = ε, Tn+2(ν + ε) = 1 −α + ε,
Tn+1(ν −ε) = 1 −ε, Tn+2(ν −ε) = 1 −α−ε,

which implies that

R(ν −ε) = u0 · · ·un−101un+2 · · ·ur(ν)−1,
R(ν + ε) = u0 · · ·un−110un+2 · · ·ur(ν)−1.

Necessarily, R(ν−ε) and R(ν +ε) are amicable words.
One of them is Q, the other one is equal to R1, R2
or R1R2, according to whether the point ν belongs to
I0, I1 or I2.

3. Case study
Let us give several examples illustrating the possible
outcomes for the set ItI of I-itineraries for general
subinterval I = [c,d) ⊂ [0, 1). According to our main
Theorem 1.1, we have

ItI ⊂{R1,R2,R1R2,Q},

where Q is a word amicable with one of R1,R2,R1R2.
In fact, as we see in the following examples, we can
have all possibilities.
For simplicity in the examples, we always keep

α = σ, where σ = 12 (
√

5 − 1) is the reciprocal of
the golden ratio. In calculations, we use the relation
σ2 = σ + 1.
First, we choose the most generic cases, namely

examples where #ItI = 4. Let I = [c,d) where
d − c = σ3 + σ6. Since d − c 6= δk,s for any k,s, by
Proposition 2.3, the induced map TI is an exchange
of three intervals with permutation (321), and, more-
over, the lengths of exchanged intervals I0, I1, I2 do
not depend on the position of the interval I. In the
notation introduced in the proof of Theorem 1.1,

λ = c + σ4, µ = c + σ3.

Hence, in particular,

I0 = [c,c + σ4),
I1 = [c + σ4,c + σ3),
I2 = [c + σ3,c + σ3 + σ6).

Independently on c, the return time r(x) to the inter-
val I satisfies

r(x) =




3 if x ∈ I0,
5 if x ∈ I1,
2 if x ∈ I2.

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Z. Masáková, E. Pelantová Acta Polytechnica

(In fact, for any subinterval I ⊂ [0, 1) the return time
takes two or three values, for α = σ always equal to
two or three consecutive Fibonacci numbers.)
We consider several examples of positions of the

interval I.

Example 3.1. Let c = σ4. Then ν = T−1(α) = σ3 ∈
I0 splits the interval I0 into I0 = IL0 ∪ IR0 , where

IL0 = [σ
4,σ3), IR0 = [σ

3,σ3 + σ6).

The I-itinerary satisfies

R(x) =




001 if x ∈ IL0 ,
010 if x ∈ IR0 ,
01001 if x ∈ I1,
01 if x ∈ I2.

We put

R1 = 01, R2 = 001, R1R2 = 01001, Q = 010,

where Q is amicable with R2. Note that we have
another choice for notation,

R1 = 010, R2 = 01, R1R2 = 01001, Q = 001,

where Q is amicable with R1.

Example 3.2. Let c = σ6. Then ν = T−1(α) = σ3 ∈
I1 splits the interval I1 into I1 = IL1 ∪ IR1 , where

IL1 = [σ
4 + σ6,σ3), IR1 = [σ

3,σ3 + σ6).

The I-itinerary satisfies

R(x) =




001 if x ∈ I0,
00101 if x ∈ IL1 ,
01001 if x ∈ IR1 ,
01 if x ∈ I2.

We put

R1 = 001, R2 = 01, R1R2 = 00101, Q = 01001,

where Q = R2R1 is amicable with R1R2.

Example 3.3. Let c = σ3 + σ5 + σ7. Then ν =
T 0(α) = σ ∈ I2 splits the interval I2 into I2 = IL2 ∪IR2 ,
where

IL2 = [σ
2 + σ4 + σ6 + σ9,σ), IR2 = [σ,σ + σ

7).

The I-itinerary satisfies

R(x) =




010 if x ∈ I0,
01010 if x ∈ I1,
01 if x ∈ IL2 ,
10 if x ∈ IR2 .

We put

R1 = 01, R2 = 010, R1R2 = 01010, Q = 10,

where Q is amicable with R1, or

R1 = 010, R2 = 10, R1R2 = 01010, Q = 01,

where Q is amicable with R2.

Let us discuss the cases for which #ItI < 4. This
can happen if d−c 6= δk,s, (i.e., TI is still an exchange
of three intervals), but ν ∈{c,λ,µ}. It can be derived
from the proof of Theorem 1.1, that, in this case, the
set of I-itineraries is of the form

ItI = {R1,R2,R1R2}.

Note that c = 0 is a special case of such situation.
For, we have c = 0 = T(α), whence µ = T−m(0) =
T−m+1(α) = ν. Similarly, the case d = 1 corresponds
to λ = ν.

Example 3.4. Let c = σ2, d − c = σ3 + σ6. Then
ν = T 0(α) = σ = µ. The I-itinerary satisfies

R(x) =




010 if x ∈ I0,
01010 if x ∈ I1,
10 if x ∈ I2.

With R1 = 010, R2 = 10, we have ItI =
{R1,R2,R1R2}.

Consider the situation that d− c = δk,s for some
k,s as defined in (5). By Proposition 2.3, the induced
map TI is an exchange of two intervals, since λ = µ.
The set of I-itineraries is then either ItI = {R1,R2},
which happens if ν ∈ {c,λ}, or ItI = {R1,R2,Q},
where Q is amicable with R1 or with R2, according
to the position of ν in the interval I.

4. Conclusions
Notions such as return time, return itinerary, first re-
turn map, etc. for the exchange of two intervals have
been studied by many authors. For an overview, see
for example [14]. This notion occurs in various con-
texts such as return words, Abelian return words, or
substitution invariance of the corresponding codings,
i.e., Sturmian words. The many equivalent definitions
of Sturmian words allow one to combine different
points of view which contributes substantially to the
solution of such problems.
A detailed solution of analogous questions for ex-

changes of more than two intervals is still unknown.
We believe that at least for exchanges of three inter-
vals one can obtain an explicit description of return
times and return itineraries, since the corresponding
codings are geometrically representable by cut-and-
project sequences, in a similar way that Sturmian
words are identified with mechanical words.

Acknowledgements
The results presented in this paper, as well as other results
about exchange of intervals, have been obtained with the
use of geometric representation of the associated codings
in the frame of cut-and-project scheme, see [10]. We were
lead to this topic from the study of mathematical mod-
els of quasicrystals, based on the rich collaboration of
our institute with Jiří Patera from Centre de Recherches
Mathématiques in Montréal. This collaboration was initi-
ated, encouraged and is still maintained by prof. Miloslav

448



vol. 53 no. 5/2013 Itineraries Induced by Exchange of Two Intervals

Havlíček, to whose honour this issue of Acta Polytechnica
is published.

We acknowledge financial support from Czech Science
Foundation grant 13-03538S.

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	Acta Polytechnica 53(5):444–449, 2013
	1 Introduction
	2 Interval Exchange Transformations
	3 Case study
	4 Conclusions
	Acknowledgements
	References