Rauzy fractals associated with
cubic real number fields
Timo Jolivet
LIAFA, Paris, France and FUNDIM, Turku, Finland
Joint work with Valérie Berthé and Anne Siegel
Algebra Seminar
Technische Universität Wien
Freitag, den 2. Dezember 2011
The central tool: substitutions
1 7→ 12
2 7→ 1
The central tool: substitutions
1 7→ 12
2 7→ 1
1
The central tool: substitutions
1 7→ 12
2 7→ 1
12
The central tool: substitutions
1 7→ 12
2 7→ 1
121
The central tool: substitutions
1 7→ 12
2 7→ 1
12112
The central tool: substitutions
1 7→ 12
2 7→ 1
12112121
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112
The central tool: substitutions
1 7→ 12
2 7→ 1
121121211211212112121
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112121121211211
The central tool: substitutions
1 7→ 12
2 7→ 1
a →
7
aabb
b →
7
aabb
0 7→ 10
1 7→ 01
x →
7
xxyz
y →
7
yz
z →
7
xyz
1
2
3
4
7→
7
→
7→
7
→
12
13
14
1
1211212112112121121211211212112112121121211211
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112121121211211
a →
7
aabb
b →
7
aabb
a
0 7→ 10
1 7→ 01
0
x →
7
xxyz
y →
7
yz
z →
7
xyz
x
1
2
3
4
7→
7
→
7→
7
→
12
13
14
1
1234
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112121121211211
a →
7
aabb
b →
7
aabb
aabb
0 7→ 10
1 7→ 01
10
x →
7
xxyz
y →
7
yz
z →
7
xyz
xxyz
1
2
3
4
7→
7
→
7→
7
→
12
13
14
1
1213141
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112121121211211
a →
7
aabb
b →
7
aabb
aabbaabbaabbaabb
0 7→ 10
1 7→ 01
0110
x →
7
xxyz
y →
7
yz
z →
7
xyz
xxyzxxyzyzxyz
1
2
3
4
7→
7
→
7→
7
→
12
13
14
1
1213121412112
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112121121211211
a →
7
aabb
b →
7
aabb
aabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaab
0 7→ 10
1 7→ 01
10010110
x →
7
xxyz
y →
7
yz
z →
7
xyz
xxyzxxyzyzxyzxxyzxxyzyzxyzyzxyzxxyzyzxyz
1
2
3
4
7→
7
→
7→
7
→
12
13
14
1
1213121412131211213121213
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112121121211211
a →
7
aabb
b →
7
aabb
aabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaab
0 7→ 10
1 7→ 01
0110100110010110
x →
7
xxyz
y →
7
yz
z →
7
xyz
xxyzxxyzyzxyzxxyzxxyzyzxyzyzxyzxxyzyzxyzxx
1
2
3
4
7→
7
→
7→
7
→
12
13
14
1
1213121412131211213121412131212131214121312131
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112121121211211
a →
7
aabb
b →
7
aabb
aabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaab
0 7→ 10
1 7→ 01
10010110011010010110100110010110
x →
7
xxyz
y →
7
yz
z →
7
xyz
xxyzxxyzyzxyzxxyzxxyzyzxyzyzxyzxxyzyzxyzxx
1
2
3
4
7→
7
→
7→
7
→
12
13
14
1
1213121412131211213121412131212131214121312112
The central tool: substitutions
1 7→ 12
2 7→ 1
1211212112112121121211211212112112121121211211
a →
7
aabb
b →
7
aabb
aabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaab
0 7→ 10
1 7→ 01
0110100110010110100101100110100110010110011010
x →
7
xxyz
y →
7
yz
z →
7
xyz
xxyzxxyzyzxyzxxyzxxyzyzxyzyzxyzxxyzyzxyzxx
1
2
3
4
7→
7
→
7→
7
→
12
13
14
1
1213121412131211213121412131212131214121312112
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1


1 1 1


Mσ = 1 0 0
0 1 0
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1
1


1 1 1


Mσ = 1 0 0
0 1 0
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1
12


1 1 1


Mσ = 1 0 0
0 1 0
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1
1213


1 1 1


Mσ = 1 0 0
0 1 0
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1
12131213


1 1 1


Mσ = 1 0 0
0 1 0
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1
121312131213121


1 1 1


Mσ = 1 0 0
0 1 0
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1
1213121312131211213121121312


1 1 1


Mσ = 1 0 0
0 1 0
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1


1 1 1


Mσ = 1 0 0
0 1 0
1213121312131211213121121312121312112131212131211213 · · ·
The central tool: substitutions
We focus on substitutions σ:
I
over the alphabet {1, 2, 3};
I
that are unimodular: det(Mσ ) = ±1;
that are Pisot irreducible:
I
I
I
|β| > 1 and |β 0 |, |β 00 | < 1, where sp(Mσ ) = {β, β 0 , β 00 };
the characteristic polynomial of Mσ is irreducible.
Example: the Tribonacci substitution:


 1 7→
12
2 7→ 13
σ:


3 7→ 1


1 1 1


Mσ = 1 0 0
0 1 0
1213121312131211213121121312121312112131212131211213 · · ·
å Symbolic dynamical system (Xσ , S). (Subshift of {1, 2, 3}Z .)
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Rauzy fractals
To a substitution σ : {1, 2, 3}∗ → {1, 2, 3}∗ , we can associate a
Rauzy fractal (definition later).
I
Compact subset of R2 with fractal boundary.
I
Uses: geometric realizations of substitutions (originally),
many applications (dynamical systems, number theory,
discrete geometry).
First example [Rauzy ’82]:
The Tribonacci substitution


 1 7→
12
2 7→ 13
σ:

 3 7→ 1
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 2
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 21
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 213
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 2131
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 21312
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 213121
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 2131212
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 21312121
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 213121211
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 2131212112
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 213121211212131
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 21312121121213121213
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: 2131212112121312121312121
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: · · · 2131212112121312121312121· · · ∈ Xσ ⊆ {1, 2, 3}Z
Dynamics of σ : 1 7→ 12, 2 7→ 1312, 3 7→ 112
Orbit: · · · 2131212112121312121312121· · · ∈ Xσ ⊆ {1, 2, 3}Z
Xσ , shift
, domain exchange
Rauzy fractals: tilings of the plane
Self-similar (aperiodic) tiling:
Periodic tiling:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Shift:
· · · 2131212112 · · · ∈ Xσ
Translation on the torus:
Dynamics of σ, continued
Domain exchange:
Translation on the torus:
Shift:
· · · 2131212112 · · · ∈ Xσ
Three directions,
same translation.
Dynamics of σ, continued
Domain exchange:
Translation on the torus:
Shift:
· · · 2131212112 · · · ∈ Xσ
Three directions,
same translation.
Dynamics of σ, continued
Domain exchange:
Translation on the torus:
Shift:
· · · 2131212112 · · · ∈ Xσ
Three directions,
same translation.
Dynamics of σ, summary
Xσ , shift
, domain exchange
T2 , translation
Dynamics of σ, summary
Xσ , shift
, domain exchange
Are these nice properties always true?
T2 , translation
Dynamics of σ, summary
Xσ , shift
, domain exchange
Are these nice properties always true?
Pisot conjecture
Yes, if σ is unimodular Pisot irreducible.
T2 , translation
Dynamics of σ, summary
Xσ , shift
, domain exchange
T2 , translation
Are these nice properties always true?
Pisot conjecture
Yes, if σ is unimodular Pisot irreducible.
Equivalent formulations:
I
The tiles don’t overlap in the tilings.
I
Pure discreteness of the spectrum of (Xσ , S).
I
Coincidence conditions on σ.
I
Geometric combinatorial conditions (cf. later).
I
Many criteria, from many different viewpoints.
Today we would like to prove. . .
Main result
Theorem [Berthé-J.-Siegel 2011]
Let K be a cubic real extension of Q.
There exist α, β ∈ K and an unimodular Pisot irreducible
substitution σ such that:
1. K = Q(α, β).
2. (T2 , Tα,β ) (Xσ , S).
Main result
Theorem [Berthé-J.-Siegel 2011]
Let K be a cubic real extension of Q.
There exist α, β ∈ K and an unimodular Pisot irreducible
substitution σ such that:
1. K = Q(α, β).
2. (T2 , Tα,β ) (Xσ , S).
In this talk, we will:
1. Explain where (α, β) and σ come from.
(Jacobi-Perron algorithm and Dubois-Paysant Leroux’s result)
Main result
Theorem [Berthé-J.-Siegel 2011]
Let K be a cubic real extension of Q.
There exist α, β ∈ K and an unimodular Pisot irreducible
substitution σ such that:
1. K = Q(α, β).
2. (T2 , Tα,β ) (Xσ , S).
In this talk, we will:
1. Explain where (α, β) and σ come from.
(Jacobi-Perron algorithm and Dubois-Paysant Leroux’s result)
2. Prove the isomorphism.
(geometric combinatorial methods)
1. Find α, β.
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Jacobi-Perron algorithm
A result of Dubois and
Paysant-Le Roux
Jacobi-Perron algorithm
Let v = (a, b, c) ∈ R3 be Q-linearly independent, 0 < a, b 6 c.
JP algorithm
v = (a, b, c)
JP
7−→
v1 = (b − bb/aca, c − bc/aca, a)
Jacobi-Perron algorithm
Let v = (a, b, c) ∈ R3 be Q-linearly independent, 0 < a, b 6 c.
JP algorithm
v = (a, b, c)
JP
7−→
v1 = (b − bb/aca, c − bc/aca, a)
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Generalizes continued fraction expansions (Euclid’s algorithm).
(Continued fraction of α ⇐⇒ JP expansion of (1, α).)
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Yields simultaneous rationnal approximations of a, b, c.
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Good convergence properties.
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Many other generalizations exist! (Brun, Poincaré, Selmer,
Arnoux-Rauzy, Tamura-Yasutomi, . . . )
Example
v = (1,
√
2,
√
√
√
π) 7→ ( 2 − 1, π − 1, 1)
√
√
B1 = b
2
1 c
=1
C1 = b
π
1 c
=1
Example
v = (1,
√
2,
√
√
√
π) →
7
( 2 − 1,
√ π − 1,√1) √
√
7→ ( π − 2, 3 − 2 2, 2 − 1)
√
B1 = b √12 c = 1
B2 = b √π−1
c=1
2−1
√
C1 = b 1π c = 1
1
C2 = b √2−1
c=2
Example
v = (1,
√
2,
√
√
√
π) →
7
( 2 − 1,
√ π − 1,√1) √
√
7→ ( π − 2, 3 − 2 2, 2 − 1)
7→ · · ·
√
B1 = b √12 c = 1
B2 = b √π−1
c=1
2−1
√
√
C1 = b 1π c = 1
1
C2 = b √2−1
c=2
√
2−1
3−2 √2
√ c=1
c = 0 C3 = b √π−
B3 = b √
π− 2
2
Example
v = (1,
√
2,
√
√
√
π) →
7
( 2 − 1,
√ π − 1,√1) √
√
7→ ( π − 2, 3 − 2 2, 2 − 1)
7→ · · ·
√
B1 = b √12 c = 1
B2 = b √π−1
c=1
2−1
√
√
C1 = b 1π c = 1
1
C2 = b √2−1
c=2
√
2−1
3−2 √2
√ c=1
c = 0 C3 = b √π−
B3 = b √
π− 2
2
Expansion (Bn , Cn )n>1 :
(1, 1), (1, 2), (0, 1), (0, 2), (0, 3), (0, 3), (2, 4), (0, 1), (0, 7),
(0, 1), (29, 36), (5, 5), (4, 19), (1, 1), (2, 3), (0, 2), (6, 8), (0, 1), . . .
Periodic expansions
Open problem
In 1D: Theorem: The continued fraction of α ∈ R is periodic
⇐⇒ α is quadratic.
In 2D: Which (1, α, β) have a periodic JP expansion?
Periodic expansions in cubic fields
Theorem [Dubois and Paysant-Le Roux 1975]
Let K be a cubic real extension of Q.
There exist α, β ∈ K such that:
1. K = Q(α, β).
2. The JP expansion of (1, α, β) is periodic.
1 bis. Find σ.
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Matrix formulation of JP
Jacobi-Perron matrices
Classical formulation of JP
v = (a, b, c)
JP
7−→
v1 = (b − bb/aca, c − bc/aca, a)
Jacobi-Perron matrices
Classical formulation of JP
v = (a, b, c)
JP
7−→
v1 = (b − bb/aca, c − bc/aca, a)
Formulation with matrices
v = MB,C v1 , where
MB,C =
0 0 1 10B
01C
B = bb/ac
C = bc/ac
Jacobi-Perron matrices
Classical formulation of JP
v = (a, b, c)
JP
7−→
v1 = (b − bb/aca, c − bc/aca, a)
Formulation with matrices
v = MB,C v1 , where
MB,C =
0 0 1 10B
01C
B = bb/ac
C = bc/ac
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v = MB1 ,C1 · · · MBn ,Cn vn .
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MB1 ,C1 · · · MBn ,Cn u converges to v for all u.
Jacobi-Perron substitutions


 1 7→
3
2 7→ 13B

 3 7→ 23C
å The incidence matrix of σB,C is t MB,C .
å σB,C is unimodular Pisot irreducible.
We choose σB,C :
Jacobi-Perron substitutions


 1 7→
3
2 7→ 13B

 3 7→ 23C
å The incidence matrix of σB,C is t MB,C .
å σB,C is unimodular Pisot irreducible.
We choose σB,C :
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Let (B1 , C1 ), . . . , (B` , C` ) = periodic expansion of (1, α, β).
Jacobi-Perron substitutions


 1 7→
3
2 7→ 13B

 3 7→ 23C
å The incidence matrix of σB,C is t MB,C .
å σB,C is unimodular Pisot irreducible.
We choose σB,C :
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Let (B1 , C1 ), . . . , (B` , C` ) = periodic expansion of (1, α, β).
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Let σ = σB1 ,C1 · · · σB` ,C` .
Jacobi-Perron substitutions


 1 7→
3
2 7→ 13B

 3 7→ 23C
å The incidence matrix of σB,C is t MB,C .
å σB,C is unimodular Pisot irreducible.
We choose σB,C :
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Let (B1 , C1 ), . . . , (B` , C` ) = periodic expansion of (1, α, β).
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Let σ = σB1 ,C1 · · · σB` ,C` .
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(1, α, β) is the eigenvector of t MB,C associated with the
largest eigenvalue, so we are done:
The toral translation associated with (Xσ , S) is (T2 , Tα,β ).
2. Prove (T2, Tα,β ) (Xσ , S).
Combinatorics:
discrete surfaces
multidimensional substitutions
a definition of Rauzy fractals
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I
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Unit faces
A unit face [x, i]∗ consists of:
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a position x ∈ Z3 ;
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a type i ∈ {1, 2, 3}.
e3
e1
e2
[(−1, 1, 0), 1]∗
[(−3, 0, −1), 3]∗
Unit faces
A unit face [x, i]∗ consists of:
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a position x ∈ Z3 ;
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a type i ∈ {1, 2, 3}.
e3
e1
e2
[(−1, 1, 0), 1]∗
[(−3, 0, −1), 3]∗
[x, 1]∗ = {x + λe2 + µe3 : λ, µ ∈ [0, 1]} =
[x, 2]∗ = {x + λe1 + µe3 : λ, µ ∈ [0, 1]} =
[x, 3]∗ = {x + λe1 + µe2 : λ, µ ∈ [0, 1]} =
Discrete planes
Let v ∈ R3>0 . The discrete plane Γv of normal vector v is
the discrete surface that “intersects” the plane Pv .
(Formally: Γv = {[x, i]∗ : 0 6 hx, vi < hei , vi}.)
Γ(1,1,1)
Discrete planes
Let v ∈ R3>0 . The discrete plane Γv of normal vector v is
the discrete surface that “intersects” the plane Pv .
(Formally: Γv = {[x, i]∗ : 0 6 hx, vi < hei , vi}.)
Γ(1, √2, √17)
Dual substitutions E∗1 (σ)
Definition [Arnoux-Ito 2001]
Let σ : {1, 2, 3}∗ → {1, 2, 3}∗ such that det(Mσ ) = ±1.
E∗1 (σ)([x, i]∗ ) = Mσ −1 x +
[
[
k=1,2,3 s|σ(k)=pis
where ` : {1, 2, 3}∗ → Z3+ , w 7→ (|w|1 , |w|2 , |w|3 ).
[`(s), k]∗ ,
Dual substitutions E∗1 (σ)
Definition [Arnoux-Ito 2001]
Let σ : {1, 2, 3}∗ → {1, 2, 3}∗ such that det(Mσ ) = ±1.
E∗1 (σ)([x, i]∗ ) = Mσ −1 x +
[
[`(s), k]∗ ,
[
k=1,2,3 s|σ(k)=pis
where ` : {1, 2, 3}∗ → Z3+ , w 7→ (|w|1 , |w|2 , |w|3 ).
Example: E∗1 (σ) for σ : 1 7→ 12, 2 7→ 13, 3 7→ 1
[x, 1]∗ →
7
Mσ −1 x + [(1, 0, −1), 1]∗ ∪ [(0, 1, −1), 2]∗ ∪ [(0, 0, 0), 3]∗
∗
[x, 2] →
7
Mσ −1 x + [(0, 0, 0), 1]∗
[x, 3]∗ →
7
Mσ −1 x + [(0, 0, 0), 2]∗
M−1
σ x
x
7−→
x
7−→
x
M−1
σ x
7−→
M−1
σ x
Jacobi-Perron E∗1 substitutions




[0, 1]∗






7→ [(B, 0, 0), 2]∗
[0, 2]∗ 7→ [(C, 0, 0), 3]∗






∗


 [0, 3]





















7→
7→
7→
7→ [0, 1]∗ ∪
B−1
[
C−1
[
k=0
`=0
[(k, 0, 0), 2]∗ ∪
[(`, 0, 0), 3]∗
B
C
(here: B = 3, C = 5)
E∗1 (σ)(1 7→ 12, 2 7→ 13, 3 7→ 1)
E∗1 (σ)(1 7→ 12, 2 7→ 13, 3 7→ 1)
E∗1 (σ)(
)
E∗1 (σ)(1 7→ 12, 2 7→ 13, 3 7→ 1)
E∗1 (σ)2 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 13, 3 7→ 1)
E∗1 (σ)3 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 13, 3 7→ 1)
E∗1 (σ)4 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 13, 3 7→ 1)
E∗1 (σ)5 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 13, 3 7→ 1)
E∗1 (σ)6 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 13, 3 7→ 1)
E∗1 (σ)7 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 1312, 3 7→ 112)
E∗1 (σ)(1 7→ 12, 2 7→ 1312, 3 7→ 112)
E∗1 (σ)(
)
E∗1 (σ)(1 7→ 12, 2 7→ 1312, 3 7→ 112)
E∗1 (σ)2 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 1312, 3 7→ 112)
E∗1 (σ)3 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 1312, 3 7→ 112)
E∗1 (σ)4 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 1312, 3 7→ 112)
E∗1 (σ)5 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 31, 3 7→ 1)
E∗1 (σ)(1 7→ 12, 2 7→ 31, 3 7→ 1)
E∗1 (σ)(
)
E∗1 (σ)(1 7→ 12, 2 7→ 31, 3 7→ 1)
E∗1 (σ)2 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 31, 3 7→ 1)
E∗1 (σ)3 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 31, 3 7→ 1)
E∗1 (σ)4 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 31, 3 7→ 1)
E∗1 (σ)5 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 31, 3 7→ 1)
E∗1 (σ)6 (
)
E∗1 (σ)(1 7→ 12, 2 7→ 31, 3 7→ 1)
E∗1 (σ)7 (
)
E∗1 (σ) and discrete planes
Theorem [Arnoux-Ito 2001, Fernique 2007]
E∗1 (σ)n (
) ⊆ a discrete plane.
E∗1 (σ) and discrete planes
Theorem [Arnoux-Ito 2001, Fernique 2007]
E∗1 (σ)n (
) ⊆ a discrete plane.
Even better: E∗1 (σ)(Γv ) = Γt Mσ v .
E∗1 (σ) and discrete planes
Theorem [Arnoux-Ito 2001, Fernique 2007]
E∗1 (σ)n (
) ⊆ a discrete plane.
Even better: E∗1 (σ)(Γv ) = Γt Mσ v .
Theorem [Arnoux-Ito 2001]
[x, i]∗ , [y, j]∗ ∈ Γv
=⇒
E∗1 (σ)([x, i]∗ ) and E∗1 (σ)([y, j]∗ ) are disjoint.
Definition of Rauzy fractals using E∗1 (σ)
Idea: renormalize E∗1 (σ)n (U) with n → ∞.
U
Definition of Rauzy fractals using E∗1 (σ)
Idea: renormalize E∗1 (σ)n (U) with n → ∞.
E∗1 (U)
Definition of Rauzy fractals using E∗1 (σ)
Idea: renormalize E∗1 (σ)n (U) with n → ∞.
E∗1 2 (U)
Definition of Rauzy fractals using E∗1 (σ)
Idea: renormalize E∗1 (σ)n (U) with n → ∞.
E∗1 3 (U)
Definition of Rauzy fractals using E∗1 (σ)
Idea: renormalize E∗1 (σ)n (U) with n → ∞.
E∗1 4 (U)
Definition of Rauzy fractals using E∗1 (σ)
Idea: renormalize E∗1 (σ)n (U) with n → ∞.
E∗1 5 (U)
Definition of Rauzy fractals using E∗1 (σ)
Idea: renormalize E∗1 (σ)n (U) with n → ∞.
E∗1 6 (U)
Definition of Rauzy fractals using E∗1 (σ)
Idea: renormalize E∗1 (σ)n (U) with n → ∞.
E∗1 7 (U)
Definition of Rauzy fractals using E∗1 (σ)
σ is Pisot:
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One expanding direction of Mσ (Pisot eigenvalue |β| > 1).
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Two contracting directions of Mσ (eigenvalues |β 0 |, |β 00 | < 1.).
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Let Pc be the contracting plane of Mσ spanned by vβ 0 , vβ 00 .
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Let π : R3 → Pc be the projection on Pc along vβ .
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So:
Renormalization = Mσ ◦ π
Definition of Rauzy fractals using E∗1 (σ)
U
π(U)
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)(U)
Mσ π(E∗1 (σ)(U))
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)2 (U)
Mσ 2 π(E∗1 (σ)2 (U))
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)3 (U)
Mσ 3 π(E∗1 (σ)3 (U))
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)4 (U)
Mσ 4 π(E∗1 (σ)4 (U))
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)5 (U)
Mσ 5 π(E∗1 (σ)5 (U))
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)6 (U)
Mσ 6 π(E∗1 (σ)6 (U))
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)7 (U)
Mσ 7 π(E∗1 (σ)7 (U))
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)7 (U)
Mσ ∞ π(E∗1 (σ)∞ (U))
Definition of Rauzy fractals using E∗1 (σ)
E∗1 (σ)7 (U)
Mσ ∞ π(E∗1 (σ)∞ (U))
Definition [Arnoux-Ito 2001]
The Rauzy fractal of σ is the Hausdorf limit of
Mσ n π(E∗1 (σ)n (U)) as n → ∞.
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 1312, 3 7→ 112
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Rauzy fractal of 1 7→ 12, 2 7→ 31, 3 7→ 1
Back to our original goal:
(T2, Tα,β ) (Xσ , S)
with σ = σB1,C1 · · · σB`,C` .
Combinatorial criterion
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Many crieteria/techniques exist for a given single σ, but:
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We must deal with the infinite family of all the
σB1 ,C1 · · · σB` ,C` that can arise from α, β.
Combinatorial criterion
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Many crieteria/techniques exist for a given single σ, but:
I
We must deal with the infinite family of all the
σB1 ,C1 · · · σB` ,C` that can arise from α, β.
I
We will use the following condition:
Theorem [Ito-Rao 2006]
The isomorphism holds if and only if E∗1 (σ)n ([0, i]∗ )
contains arbitrarily large balls as n → ∞, for i ∈ {1, 2, 3}.
Combinatorial criterion
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Many crieteria/techniques exist for a given single σ, but:
I
We must deal with the infinite family of all the
σB1 ,C1 · · · σB` ,C` that can arise from α, β.
I
We will use the following condition:
Theorem [Ito-Rao 2006]
The isomorphism holds if and only if E∗1 (σ)n ([0, i]∗ )
contains arbitrarily large balls as n → ∞, for i ∈ {1, 2, 3}.
å How do we prove that balls grow in E∗1 (σ)n ([0, i]∗ )?
The annulus property
Annulus A of P :
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A is edge-connected,
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A “surrounds” P .
The annulus property
Annulus A of P :
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A is edge-connected,
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A “surrounds” P .
The annulus property: E∗1 (σ)(annulus) = annulus.
The annulus property
Annulus A of P :
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A is edge-connected,
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A “surrounds” P .
E∗1 (σ1,1 )
7−→
The annulus property: E∗1 (σ)(annulus) = annulus.
The annulus property
Annulus A of P :
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A is edge-connected,
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A “surrounds” P .
E∗1 (σ1,1 )
7−→
The annulus property: E∗1 (σ)(annulus) = annulus.
If
1. E∗1 (σ)([0, i]∗ )n contains an annulus for some n.
2. Annulus property for E∗1 (σ).
Then E∗1 (σ)([0, i]∗ )n contains arbitrarily large balls. (induction)
Unfortunately. . .
. . . the annulus property doesn’t hold for σB,C :
Unfortunately. . .
. . . the annulus property doesn’t hold for σB,C :
E∗1 (σ1,4 )
7−→
Unfortunately. . .
. . . the annulus property doesn’t hold for σB,C :
E∗1 (σ1,4 )
7−→
å We have to be more careful.
Unfortunately. . .
. . . the annulus property doesn’t hold for σB,C :
E∗1 (σ1,4 )
7−→
å We have to be more careful.
å Stronger assumptions on A: covering properties.
L-coverings
L =
(
)
P is L-covered if ∀f, g ∈ P , there is an L-path from f to g.
L-covering and Jacobi-Perron
I
We will use LJP = {
,
,
,
,
,
,
,
}.
L-covering and Jacobi-Perron
I
We will use LJP = {
I
Let’s look at our problem again:
,
Σ1,4
,
7−→
,
,
,
,
,
}.
L-covering and Jacobi-Perron
I
We will use LJP = {
I
Let’s look at our problem again:
,
Σ1,4
,
7−→
I
We “see” the problem.
,
,
,
,
,
}.
L-covering and Jacobi-Perron
I
We will use LJP = {
I
Let’s look at our problem again:
,
,
,
,
Σ1,4
7−→
I
We “see” the problem.
I
We must “complete” some patterns.
,
,
,
}.
L-covering and Jacobi-Perron
I
We will use LJP = {
I
Let’s look at our problem again:
,
,
,
,
Σ1,4
7−→
I
We “see” the problem.
I
We must “complete” some patterns.
,
,
,
}.
L-covering and Jacobi-Perron
I
We will use LJP = {
I
Let’s look at our problem again:
,
,
,
,
,
,
,
}.
Σ1,4
7−→
I
We “see” the problem.
I
We must “complete” some patterns.
I
å Strong covering: every two-face edge-connected
pattern is covered.
L-covering and Jacobi-Perron
I
We will use LJP = {
I
Let’s look at our problem again:
,
,
,
,
,
,
,
}.
Σ1,4
7−→
I
We “see” the problem.
I
We must “complete” some patterns.
I
å Strong covering: every two-face edge-connected
pattern is covered.
I
New LJP =
n
,
,
,
,
,
,
,
,
,
o
.
Why LJP = { ,
,
,
,
,
,
,
,
,
Trial and error:
7→
7→
7→
7→
7→
7→
7→
7→
7→
7→
7→
7→
}?
Why LJP = { ,
,
,
,
,
,
,
,
,
Trial and error:
7→
7→
7→
7→
7→
7→
7→
7→
7→
7→
7→
7→
We remove the “bad” patterns. . .
}?
Why LJP = { ,
,
,
,
,
,
,
We get:
7→
7→
7→
7→
7→
7→
7→
7→
,
,
}?
Why LJP = { ,
,
,
,
,
,
,
,
,
We get:
7→
7→
7→
7→
7→
7→
7→
7→
to guarantee strong LJP -covering.)
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(Plus
I
The images are strongly LJP -covered: stability.
and
}?
Finally. . .
Finally. . .
1. Strongly LJP -covered annuli are preserved by E∗1 (σB,C ).
Finally. . .
1. Strongly LJP -covered annuli are preserved by E∗1 (σB,C ).
Proof: long enumeration of cases. (induction step)
Finally. . .
1. Strongly LJP -covered annuli are preserved by E∗1 (σB,C ).
Proof: long enumeration of cases. (induction step)
2. ∃n s.t. (E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) has an annulus.
Finally. . .
1. Strongly LJP -covered annuli are preserved by E∗1 (σB,C ).
Proof: long enumeration of cases. (induction step)
2. ∃n s.t. (E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) has an annulus.
Proof: careful study of the way patterns grow. (base case)
So:
Finally. . .
1. Strongly LJP -covered annuli are preserved by E∗1 (σB,C ).
Proof: long enumeration of cases. (induction step)
2. ∃n s.t. (E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) has an annulus.
Proof: careful study of the way patterns grow. (base case)
So:
Theorem [Ito-Ohtsuki 1994, Berthé-Bourdon-J.-Siegel 2011]
(E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) covers arbitrarily large balls.
Finally. . .
1. Strongly LJP -covered annuli are preserved by E∗1 (σB,C ).
Proof: long enumeration of cases. (induction step)
2. ∃n s.t. (E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) has an annulus.
Proof: careful study of the way patterns grow. (base case)
So:
Theorem [Ito-Ohtsuki 1994, Berthé-Bourdon-J.-Siegel 2011]
(E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) covers arbitrarily large balls.
Moreover:
I
(E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) is simply connected.
Finally. . .
1. Strongly LJP -covered annuli are preserved by E∗1 (σB,C ).
Proof: long enumeration of cases. (induction step)
2. ∃n s.t. (E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) has an annulus.
Proof: careful study of the way patterns grow. (base case)
So:
Theorem [Ito-Ohtsuki 1994, Berthé-Bourdon-J.-Siegel 2011]
(E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) covers arbitrarily large balls.
Moreover:
I
(E∗1 (σB1 ,C1 ) · · · E∗1 (σB` ,C` ))n ([0, i]∗ ) is simply connected.
I
The Rauzy fractal is connected.
Conclusion
We have reached our initial objective:
Corollary
Let K be a cubic real extension of Q.
I
There exist α, β ∈ K with periodic JP expansion
(B1 , C1 ), . . . , (B` , C` ) such that K = Q(α, β).
[Dubois and Paysant-Le Roux 1975]
Conclusion
We have reached our initial objective:
Corollary
Let K be a cubic real extension of Q.
I
There exist α, β ∈ K with periodic JP expansion
(B1 , C1 ), . . . , (B` , C` ) such that K = Q(α, β).
[Dubois and Paysant-Le Roux 1975]
I
For σ = σB1 ,C1 · · · σB` ,C` , we have (T2 , Tα,β ) (Xσ , S).
[Geometrical combinatorial methods]
Thank you for your attention.
Any questions?
Dream case (Arnoux-Rauzy)
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