Interval exchange transformations
(Intercambios de intervalos)
Part I: Rotations (rotaciones)
Vincent Delecroix
(Vicente de la Cruz)
November 2015, Salta
(Noviembre 2015 )
A rotation is a 2-interval exchange transformation
A rotation is a 2-interval exchange transformation
The rotation of angle α is the map Tα : [0, 1] → [0, 1] defined by
Tα (x) = (x + α)
mod 1.
A rotation is a 2-interval exchange transformation
The rotation of angle α is the map Tα : [0, 1] → [0, 1] defined by
Tα (x) = (x + α)
mod 1.
It can be seen as a 2-interval exchange transformation
A rotation is a 2-interval exchange transformation
The rotation of angle α is the map Tα : [0, 1] → [0, 1] defined by
Tα (x) = (x + α)
mod 1.
It can be seen as a 2-interval exchange transformation
1−α
α
A rotation is a 2-interval exchange transformation
The rotation of angle α is the map Tα : [0, 1] → [0, 1] defined by
Tα (x) = (x + α)
mod 1.
It can be seen as a 2-interval exchange transformation
1−α
α
+α
+α − 1
A rotation is a 2-interval exchange transformation
The rotation of angle α is the map Tα : [0, 1] → [0, 1] defined by
Tα (x) = (x + α)
mod 1.
It can be seen as a 2-interval exchange transformation
1−α
α
+α
+α − 1
We will study the dynamics of interval exchange transformations
from both the topological and measurable viewpoints.
A rotation is a 2-interval exchange transformation
We will study the dynamics of interval exchange transformations
from both the topological and measurable viewpoints.
A rotation is a 2-interval exchange transformation
We will study the dynamics of interval exchange transformations
from both the topological and measurable viewpoints.
I
A rotation preserves the Lebesgue measure.
A rotation is a 2-interval exchange transformation
We will study the dynamics of interval exchange transformations
from both the topological and measurable viewpoints.
I
A rotation preserves the Lebesgue measure.
I
Warning: a rotation is not continuous.
A rotation is a 2-interval exchange transformation
We will study the dynamics of interval exchange transformations
from both the topological and measurable viewpoints.
I
A rotation preserves the Lebesgue measure.
I
Warning: a rotation is not continuous.
But we can build a continuous map Tb : Xα → Xα where Xα is a
Cantor set, Tb is an homeomorphism and there is a projection
p : Xα → I that commutes with the dynamics.
A rotation is a 2-interval exchange transformation
We will study the dynamics of interval exchange transformations
from both the topological and measurable viewpoints.
I
A rotation preserves the Lebesgue measure.
I
Warning: a rotation is not continuous.
But we can build a continuous map Tb : Xα → Xα where Xα is a
Cantor set, Tb is an homeomorphism and there is a projection
p : Xα → I that commutes with the dynamics. (we say that T is a
factor of Tb )
Coding
A
aba
B
Coding
A
x0
A
B
Coding
A
x0
B
x1
AA
Coding
A
x0
B
x1
AAB
x2
Coding
x0
A
x3
B
x1
AABA
x2
Coding
x0
A
x3
B
x1
AABAB
x4
x2
Coding
x5 x0
A
x3
B
x1
AABABA
x4
x2
Coding
x5 x0
A
x3
B
x6 x1
AABABAA
x4
x2
Coding
x5 x0
A
x3
x6 x1
x4
AABABAAB. . .
B
x7 x2
Coding
x5 x0
A
x3
x6 x1
x4
ba. AABABAAB. . .
B
x7 x2
Coding
x5 x0
A
x3
x6 x1
B
x4 x−1x7 x2
B. AABABAAB. . .
Coding
x5 x0
A
x3 x−2x6 x1
B
x4 x−1x7 x2
AB. AABABAAB. . .
Coding
x5 x0
A
x3 x−2x6 x1
B
x4 x−1x7 x2 x−3
BAB. AABABAAB. . .
Coding
x5 x0
A
B
x3 x−2x6 x1 x−4x4 x−1x7 x2 x−3
. . . ABAB. AABABAAB. . .
Coding
x5 x0
A
B
x3 x−2x6 x1 x−4x4 x−1x7 x2 x−3
. . . ABAB. AABABAAB. . .
To each point x ∈ [0, 1] that is not singular we associate a
biinfinite sequence that is the coding of x (for rotations these are
called Sturmian sequences).
Coding
x5 x0
A
B
x3 x−2x6 x1 x−4x4 x−1x7 x2 x−3
. . . ABAB. AABABAAB. . .
To each point x ∈ [0, 1] that is not singular we associate a
biinfinite sequence that is the coding of x (for rotations these are
called Sturmian sequences).
Taking the closure of the set of codings, we obtain a shift
Xα ⊂ {A, B}Z .
Coding
Theorem
If α is irrational, there is a unique continuous map p : Xα → [0, 1]
so that the coding of p(w ) is w . All points have exactly one
preimage except the singular orbits that have two.
Coding
Theorem
If α is irrational, there is a unique continuous map p : Xα → [0, 1]
so that the coding of p(w ) is w . All points have exactly one
preimage except the singular orbits that have two.
There is only one singular point for T −1 (i.e. α) that has a well
defined future orbit with code ω− ∈ {A, B}{0,1,2,3,...} .
Coding
Theorem
If α is irrational, there is a unique continuous map p : Xα → [0, 1]
so that the coding of p(w ) is w . All points have exactly one
preimage except the singular orbits that have two.
There is only one singular point for T −1 (i.e. α) that has a well
defined future orbit with code ω− ∈ {A, B}{0,1,2,3,...} .
There is exactly one singular point for T (i.e. 1 − α) that has a
well defined past orbit ω+ ∈ {A, B}{...,−2,−1,0} .
Coding
Theorem
If α is irrational, there is a unique continuous map p : Xα → [0, 1]
so that the coding of p(w ) is w . All points have exactly one
preimage except the singular orbits that have two.
There is only one singular point for T −1 (i.e. α) that has a well
defined future orbit with code ω− ∈ {A, B}{0,1,2,3,...} .
There is exactly one singular point for T (i.e. 1 − α) that has a
well defined past orbit ω+ ∈ {A, B}{...,−2,−1,0} .
The points in Xα that projects to the same point then correspond
to the orbit of ω+ ABω− and ω+ BAω− .
Dynamical results
Theorem
Let α be irrational, and Xα be the Sturmian shift associated to the
rotation Tα . Then:
I
pXα (n) = n + 1, in particular Xα has 0 entropy;
Dynamical results
Theorem
Let α be irrational, and Xα be the Sturmian shift associated to the
rotation Tα . Then:
I
pXα (n) = n + 1, in particular Xα has 0 entropy;
I
the shift Xα is minimal (all orbits are dense);
Dynamical results
Theorem
Let α be irrational, and Xα be the Sturmian shift associated to the
rotation Tα . Then:
I
pXα (n) = n + 1, in particular Xα has 0 entropy;
I
the shift Xα is minimal (all orbits are dense);
I
(Hecke (1922), Ostrowski (1922)) any clopen Y ⊂ Xα has
bounded remainder: there exists µY and CY so that
n
X
∀x ∈ Xα , ∀n ≥ 0, (χY (Tαn x) − µY ) ≤ CY .
k=0
In particular, the shift Xα is uniquely ergodic.
Dynamical results
Theorem
Let α be irrational, and Xα be the Sturmian shift associated to the
rotation Tα . Then:
I
pXα (n) = n + 1, in particular Xα has 0 entropy;
I
the shift Xα is minimal (all orbits are dense);
I
(Hecke (1922), Ostrowski (1922)) any clopen Y ⊂ Xα has
bounded remainder: there exists µY and CY so that
n
X
∀x ∈ Xα , ∀n ≥ 0, (χY (Tαn x) − µY ) ≤ CY .
k=0
In particular, the shift Xα is uniquely ergodic.
remark: for the clopens Y = [A] or Y = [B] we can pick CY = 1
(1-balancedness).
Complexity and minimality
Proof.
Irrationality!
Rauzy induction and continued fractions
For a pair of positive real numbers λ = (λA , λB ) we consider the
map Tλ : [0, |λ|] → [0, |λ|] given by
Tλ (x) = x 7→ (x + λB )
mod (λA + λB ).
Rauzy induction and continued fractions
For a pair of positive real numbers λ = (λA , λB ) we consider the
map Tλ : [0, |λ|] → [0, |λ|] given by
Tλ (x) = x 7→ (x + λB )
mod (λA + λB ).
The map Tλ is a rescaling of the rotation with α = λB /(λA + λB ).
Rauzy induction and continued fractions
For a pair of positive real numbers λ = (λA , λB ) we consider the
map Tλ : [0, |λ|] → [0, |λ|] given by
Tλ (x) = x 7→ (x + λB )
mod (λA + λB ).
The map Tλ is a rescaling of the rotation with α = λB /(λA + λB ).
The Rauzy induction is the procedure which associates to the map
Tλ the induced map on [0, min(λA , λB )].
Rauzy induction and continued fractions
top induction
case λB > λA
A
bot induction
case λB < λA
B
B
A
A
B
B
A
Rauzy induction and continued fractions
top induction
case λB > λA
A
bot induction
case λB < λA
B
B
A
A
B
B
A
Rauzy induction and continued fractions
top induction
case λB > λA
A
bot induction
case λB < λA
B
B
A
A
A
B
B
A
B
B
A
Rauzy induction and continued fractions
top induction
case λB > λA
A
bot induction
case λB < λA
B
B
A
A
A
B
B
A
B
B
A
B
A
B
A
Rauzy induction and continued fractions
top induction
case λB > λA
A
bot induction
case λB < λA
B
B
A
A
A
B
B
A
(λA , λB ) 7→ (λA , λB − λA )
B
B
A
B
A
B
A
(λA , λB ) 7→ (λA − λB , λB )
Rauzy induction and Rohlin towers
Rauzy induction and Rohlin towers
Rauzy induction describes a sequence of two Rohlin towers of a
rotation.
Rauzy induction and Rohlin towers
Rauzy induction describes a sequence of two Rohlin towers of a
rotation.
(. . . Sage . . . )
Hecke-Ostrowski theorem
Theorem
I
(Hecke (1922), Ostrowski (1922)) any clopen Y ⊂ Xα has
bounded remainder: there exists µY and CY so that
n
X
∀x ∈ Xα , ∀n ≥ 0, (χY (Tαn x) − µY ) ≤ CY .
k=0
In particular, the shift Xα is uniquely ergodic.
Proof.
Hecke-Ostrowski theorem
Theorem
I
(Hecke (1922), Ostrowski (1922)) any clopen Y ⊂ Xα has
bounded remainder: there exists µY and CY so that
n
X
∀x ∈ Xα , ∀n ≥ 0, (χY (Tαn x) − µY ) ≤ CY .
k=0
In particular, the shift Xα is uniquely ergodic.
Proof.
We proceed in several steps:
Hecke-Ostrowski theorem
Theorem
I
(Hecke (1922), Ostrowski (1922)) any clopen Y ⊂ Xα has
bounded remainder: there exists µY and CY so that
n
X
∀x ∈ Xα , ∀n ≥ 0, (χY (Tαn x) − µY ) ≤ CY .
k=0
In particular, the shift Xα is uniquely ergodic.
Proof.
We proceed in several steps:
I
It is enough to do it for cylinders Y = [u],
Hecke-Ostrowski theorem
Theorem
I
(Hecke (1922), Ostrowski (1922)) any clopen Y ⊂ Xα has
bounded remainder: there exists µY and CY so that
n
X
∀x ∈ Xα , ∀n ≥ 0, (χY (Tαn x) − µY ) ≤ CY .
k=0
In particular, the shift Xα is uniquely ergodic.
Proof.
We proceed in several steps:
I
It is enough to do it for cylinders Y = [u],
I
Letters have uniform frequencies,
Hecke-Ostrowski theorem
Theorem
I
(Hecke (1922), Ostrowski (1922)) any clopen Y ⊂ Xα has
bounded remainder: there exists µY and CY so that
n
X
∀x ∈ Xα , ∀n ≥ 0, (χY (Tαn x) − µY ) ≤ CY .
k=0
In particular, the shift Xα is uniquely ergodic.
Proof.
We proceed in several steps:
I
It is enough to do it for cylinders Y = [u],
I
Letters have uniform frequencies,
I
From letters to all cylinders (. . . Sage . . . ).
Rauzy induction and continued fractions
Let
A(λ) =
where
Atop =
Abot
Atop
1 0
1 1
if λA > λB ,
if λA < λB .
Abot =
1 1
.
0 1
Rauzy induction and continued fractions
Let
A(λ) =
where
Atop =
Abot
Atop
1 0
1 1
if λA > λB ,
if λA < λB .
Abot =
1 1
.
0 1
Then the Rauzy induction can be written R(λ) = A(λ)−1 λ (it is
piecewise linear).
Rauzy induction and continued fractions
Let
A(λ) =
where
Atop =
Abot
Atop
1 0
1 1
if λA > λB ,
if λA < λB .
Abot =
1 1
.
0 1
Then the Rauzy induction can be written R(λ) = A(λ)−1 λ (it is
piecewise linear). And its powers is a matrix product:
R n (λ) = (An (λ))−1 λ where
An (λ) = A(λ)A(Rλ) . . . A(R n−1 λ).
Rauzy induction and continued fractions
From this description it is possible to show that (λA , λB ) can be
written as
1
λB
= a0 +
.
λA
1
a1 +
1
a2 +
...
Rauzy induction and continued fractions
From this description it is possible to show that (λA , λB ) can be
written as
1
λB
= a0 +
.
λA
1
a1 +
1
a2 +
...
This is called the continued fraction of λB /λA .
Rauzy induction and continued fractions
From this description it is possible to show that (λA , λB ) can be
written as
1
λB
= a0 +
.
λA
1
a1 +
1
a2 +
...
This is called the continued fraction of λB /λA .
(. . . exercise . . . )