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Bidimensional Sturmian Sequences and Substitutions

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Thomas Fernique

LIRMM - Universit´ e Montpellier II Poncelet Lab. - Independent University of Moscow

DLT’05

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The linear map Θ(σ) The dual map Θ (σ)

3 Bidimensional Sturmian sequences

Stepped planes and associated sequences The action of Θ

4 Algebraic characterization

Bidimensional continued fractions

The case of periodic expansions

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α ∈ R \ Q sturmian word u α :

1 2

1 1

1

1 1

1 1

1

2 2

2

2

α 1

Here: α = 1+

√ 5

2 = 1 + 1+ 1

1

u α = 12112121121121 · · ·

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Substitution: morphism σ of A s.t. |σ n (i)| → ∞ for i ∈ A.

σ : 1 7→ 12, 2 7→ 1:

1 → 12 → 121 → 12112 → 12112121 → . . . σ extended to A ω fixed-point: u ∈ A ω | u = σ(u).

n→∞ lim σ n (1) = 12112121121121 · · · = u α = σ(u α ).

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Theorem (Algebraic Characterization I)

The Sturmian word u α is a fixed-point if and only if α has a periodic continued fraction expansion.

Generalization of this algebraic characterization?

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1 Prelude: Sturmian words and substitutions

2 Generalized substitutions The linear map Θ(σ) The dual map Θ (σ)

3 Bidimensional Sturmian sequences

Stepped planes and associated sequences The action of Θ

4 Algebraic characterization

Bidimensional continued fractions

The case of periodic expansions

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A = {1, 2, 3} and (~ e 1 , ~ e 2 , ~ e 3 ) canonical basis of R 3 .

u ∈ A broken line of segments [~ x, ~ x + ~ e i ] = (~ x, i), ~ x ∈ N 3 :

e 3

e 1

e 2 2 2

3

3

1

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σ on A linear map Θ(σ) on segments:

Θ(σ) : (~ x, i ) 7→ M σ ~ x + X

p|σ(i)=p·j ·s

( ~ f (p), j ),

where (M σ ) i,j = |σ(j )| i and ~ f (u) = t (|u| 1 , |u| 2 , |u| 3 ).

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σ :

1 7→ 12 2 7→ 13 3 7→ 1

, M σ =

1 1 1 1 0 0 0 1 0

 , σ(22331) = 13131112.

e

3

e

1

e

2

e

3

e

2

e

1

2 2

3 3 1

3 2

1 1

1

1 1

3

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1 Prelude: Sturmian words and substitutions

2 Generalized substitutions The linear map Θ(σ) The dual map Θ (σ)

3 Bidimensional Sturmian sequences

Stepped planes and associated sequences The action of Θ

4 Algebraic characterization

Bidimensional continued fractions

The case of periodic expansions

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Segment (~ x, i) dual face (~ x, i ):

e

3

e

2

e

1

e

3

e

2

e

1

e

3

e

2

e

1

Linear map Θ(σ) dual map Θ (σ):

Θ (σ)(~ x, i ) = M σ −1 ~ x + X

j ∈A

X

s|σ(j )=p·i·s

( ~ f (s), j ).

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σ : 1 7→ 12, 2 7→ 13, 3 7→ 1:

e3

e2 e1

e3

e2 e1

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1 Prelude: Sturmian words and substitutions

2 Generalized substitutions The linear map Θ(σ) The dual map Θ (σ)

3 Bidimensional Sturmian sequences

Stepped planes and associated sequences The action of Θ

4 Algebraic characterization

Bidimensional continued fractions

The case of periodic expansions

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α, β ∈ [0, 1) 2 : P α,β = {~ x ∈ R 3 | h~ x, t (1, α, β )i = 0}.

Definition (Stepped plane)

S α,β = {(~ x, i ) | 0 ≤ h~ x, t (1, α, β)i < h~ e i , t (1, α, β)i}.

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Theorem

One can bijectively map the faces of the stepped plane S α,β to the letters of a bidimensional sequence U α,β over A = {1, 2, 3}.

1, α and β linearly independent over Q ⇒ U α,β Sturmian.

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1 Prelude: Sturmian words and substitutions

2 Generalized substitutions The linear map Θ(σ) The dual map Θ (σ)

3 Bidimensional Sturmian sequences

Stepped planes and associated sequences The action of Θ

4 Algebraic characterization

Bidimensional continued fractions

The case of periodic expansions

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Theorem (Action of Θ )

t (1, α 0 , β 0 ) ∝ t M σ t (1, α, β) ⇒ Θ (σ)(S α,β ) = S α

0

0

.

7→

−1 ∗ −1

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(α, β) = (α 0 , β 0 ) growing patches of S α,β :

Θ (σ) bidimensional substitution.

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1 Prelude: Sturmian words and substitutions

2 Generalized substitutions The linear map Θ(σ) The dual map Θ (σ)

3 Bidimensional Sturmian sequences

Stepped planes and associated sequences The action of Θ

4 Algebraic characterization

Bidimensional continued fractions

The case of periodic expansions

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Modified Jacobi-Perron:

(α, β) = [(a 1 , ε 1 ), . . . , (a k , ε k ), [(α k , β k )]]

= [(a 1 , ε 1 ), (a 2 , ε 2 ), . . .]

where a i ∈ N and ε i ∈ {0, 1}.

Matrix viewpoint:

t (1, α k−1 , β k−1 ) = η k t M σ

(ak

k)

t (1, α k , β k ),

where η k ∈ R and σ (a

k

k

) substitution on A = {1, 2, 3}.

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By theorem (Action of Θ ):

Θ (a

k

k

) )(S α

k

k

) = S α

k−1

k−1

.

Then, Θ (σσ 0 ) = Θ 0 (σ) yields:

Θ (a

k

k

) . . . σ (a

1

1

) )(S α

k

k

) = S α,β .

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1 Prelude: Sturmian words and substitutions

2 Generalized substitutions The linear map Θ(σ) The dual map Θ (σ)

3 Bidimensional Sturmian sequences

Stepped planes and associated sequences The action of Θ

4 Algebraic characterization

Bidimensional continued fractions

The case of periodic expansions

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p , β p ) = (α, β) periodic expansion:

(α, β) = [(a 1 , ε 1 ), . . . , (a p , ε p ), [(α, β)]].

Then, S α,β fixed-point:

Θ (a

p

p

) . . . σ (a

1

1

) )(S α

p

p

| {z }

S

α,β

) = S α,β .

(α, β) = [(1, 1), (1, 1), (1, 0), (1, 1), (1, 1), (1, 0), (1, 1), . . .]

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Theorem (Algebraic Characterization II)

The bidimensional Sturmian sequence U α,β is a fixed-point if (α, β) has a periodic bidimensional continued fraction expansion.

What about the “only if” part?

Références

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