Thomas Fernique
LIRMM - Universit´ e Montpellier II Poncelet Lab. - Independent University of Moscow
DLT’05
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
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
1u α = 12112121121121 · · ·
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 ).
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?
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
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
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 ).
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
3e
1e
2e
3e
2e
12 2
3 3 1
3 2
1 1
1
1 1
3
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
Segment (~ x, i) dual face (~ x, i ∗ ):
e
3e
2e
1e
3e
2e
1e
3e
2e
1Linear 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 ∗ ).
σ : 1 7→ 12, 2 7→ 13, 3 7→ 1:
e3
e2 e1
e3
e2 e1
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
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}.
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.
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
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,β
0.
7→
−1 ∗ −1
(α, β) = (α 0 , β 0 ) growing patches of S α,β :
Θ ∗ (σ) bidimensional substitution.
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
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,εk)