Toward an Algebraic Characterization of Substitutive Multidimensional Sturmian Sequences

Thomas Fernique

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

SSAO’05

2 Hyperplane sequences and generalized substitutions Stepped hyperplanes and associated sequences Generalized substitutions

3 Toward a characterization for hyperplane sequences

Multidimensional continued fractions

1 Characterizations for words and tilings

Substitutions, continued fractions and return words Pseudo-self-similarity and derived Vorono¨ı tilings

2 Hyperplane sequences and generalized substitutions Stepped hyperplanes and associated sequences Generalized substitutions

3 Toward a characterization for hyperplane sequences

Multidimensional continued fractions

Alphabet A words A ^∗ and sequences A ^ω .

Repetitive sequence: bounded gaps between occurences of a factor.

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

σ : 1 7→ 12, 2 7→ 1:

1 → 12 → 121 → 12112 → 12112121 → 1211212112112 → . . .

Fixed-point: u ∈ A ^ω | u = σ(u).

Substitutive sequence: morphic image of a fixed-point.

Alphabet A words A ^∗ and sequences A ^ω .

Repetitive sequence: bounded gaps between occurences of a factor.

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

σ : 1 7→ 12, 2 7→ 1:

1 → 12 → 121 → 12112 → 12112121 → 1211212112112 → . . .

Fixed-point: u ∈ A ^ω | u = σ(u).

Substitutive sequence: morphic image of a fixed-point.

Alphabet A words A ^∗ and sequences A ^ω .

Repetitive sequence: bounded gaps between occurences of a factor.

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

σ : 1 7→ 12, 2 7→ 1:

1 → 12 → 121 → 12112 → 12112121 → 1211212112112 → . . .

Fixed-point: u ∈ A ^ω | u = σ(u).

Substitutive sequence: morphic image of a fixed-point.

Line of R ² with a slope α / ∈ Q Sturmian sequence u α ∈ {1, 2} ^ω .

α = 1 + √

5

2 = 1 + 1

1 + ¹ ...

u α = 12112121121121 · · ·

Theorem (Algebraic characterization)

The Sturmian sequence u _α is a substitutive if and only if α has a

ultimately periodic continued fraction expansion.

Line of R ² with a slope α / ∈ Q Sturmian sequence u α ∈ {1, 2} ^ω .

α = 1 + √

5

2 = 1 + 1

1 + ¹ ...

u α = 12112121121121 · · ·

Theorem (Algebraic characterization)

The Sturmian sequence u _α is a substitutive if and only if α has a

ultimately periodic continued fraction expansion.

Repetitive sequence u locator sets L _n (u) ⊂ N, return words R n (u) ⊂ A ^∗ and derived sequences DV n (u) ∈ {1 . . . |R _n (u)|} ^ω .

u = 12112121121121211212112 · · ·

Repetitive sequence u locator sets L _n (u) ⊂ N, return words R n (u) ⊂ A ^∗ and derived sequences DV n (u) ∈ {1 . . . |R _n (u)|} ^ω .

u = ˆ 12112ˆ 121ˆ 12112ˆ 12112ˆ 121ˆ 12 · · ·

Repetitive sequence u locator sets L _n (u) ⊂ N, return words R n (u) ⊂ A ^∗ and derived sequences DV n (u) ∈ {1 . . . |R _n (u)|} ^ω .

u = ˆ 12112 ˆ 121 ˆ 12112ˆ 12112 ˆ 121 ˆ 12 · · ·

Repetitive sequence u locator sets L _n (u) ⊂ N, return words R n (u) ⊂ A ^∗ and derived sequences DV n (u) ∈ {1 . . . |R _n (u)|} ^ω .

u = ˆ 12112 ˆ 121 ˆ 12112ˆ 12112 ˆ 121 ˆ 12 · · · DV 4 (u) = 121121 · · ·

Repetitive sequence u locator sets L _n (u) ⊂ N , return words R _n (u) ⊂ A ^∗ and derived sequences DV _n (u) ∈ {1 . . . |R _n (u)|} ^ω .

u = ˆ 12112 ˆ 121 ˆ 12112ˆ 12112 ˆ 121 ˆ 12 · · · DV ₄ (u) = 121121 · · ·

Theorem (Durand)

A sequence u is substitutive if and only if {DV _n (u), n ∈ N } is finite.

1 Characterizations for words and tilings

Substitutions, continued fractions and return words Pseudo-self-similarity and derived Vorono¨ı tilings

2 Hyperplane sequences and generalized substitutions Stepped hyperplanes and associated sequences Generalized substitutions

3 Toward a characterization for hyperplane sequences

Multidimensional continued fractions

Multidimensional generalization of sequences tilings of R ⁿ . Repetitive tiling: bounded distance between occurences of a patch.

Expansion map: linear map Φ s.t. |Φ(x)| = λ|x|, λ > 1.

Pseudo-self-similar tiling: T locally derivable from Φ(T ).

Repetitive tiling T locator sets L _r (T ) ⊂ R ⁿ and derived Vorono¨ı tiling DV _r (T ).

(dessin)

Theorem (Solomyak-Priebe)

A non-periodic repetitive tiling T is pseudo-self-similar if and only

if {DV _r (T ), r ∈ R ⁺ }/{Φ ⁿ , n ∈ N } is finite for an expansion map Φ.

Repetitive tiling T locator sets L _r (T ) ⊂ R ⁿ and derived Vorono¨ı tiling DV _r (T ).

(dessin)

Theorem (Solomyak-Priebe)

A non-periodic repetitive tiling T is pseudo-self-similar if and only

if {DV _r (T ), r ∈ R ⁺ }/{Φ ⁿ , n ∈ N } is finite for an expansion map Φ.

1 Characterizations for words and tilings

Substitutions, continued fractions and return words Pseudo-self-similarity and derived Vorono¨ı tilings

2 Hyperplane sequences and generalized substitutions Stepped hyperplanes and associated sequences Generalized substitutions

3 Toward a characterization for hyperplane sequences

Multidimensional continued fractions

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Definition (Stepped hyperplane)

~

α ∈ R ⁿ S _{~ α} = {(~ x, i ^∗ ) | 0 ≤ h~ x, ~ αi < h~ e _i , ~ αi}.

e

e

e

e

e

e

e

e

e

Definition (Stepped hyperplane)

~

α ∈ R ⁿ S _{~ α} = {(~ x, i ^∗ ) | 0 ≤ h~ x, ~ αi < h~ e _i , ~ αi}.

Theorem (Hyperplane sequences)

One can bijectively map the faces of the stepped hyperplane S _{α ~} to the letters of a (n − 1)-dimensional sequence U _{~ α} over {1, . . . , n}.

Coordinates of α ~ linearly independent over Q ⇒ U _{~ α} Sturmian.

↔

1 Characterizations for words and tilings

Substitutions, continued fractions and return words Pseudo-self-similarity and derived Vorono¨ı tilings

2 Hyperplane sequences and generalized substitutions Stepped hyperplanes and associated sequences Generalized substitutions

3 Toward a characterization for hyperplane sequences

Multidimensional continued fractions

Θ(σ)(~ x, i ^∗ ) = M _σ ⁻¹ ~ x + X

j∈A

X

s|σ(j)=p·i·s

( ~ f (s ), j ^∗ ),

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

e₃

e₂

e3

e2 e₁

~

α ⁰ ∝ ^t M σ α ~ ⇒ Θ ^∗ (σ)(S _{~ α} ) = S _{~ α}

.

7→

1 Characterizations for words and tilings

Substitutions, continued fractions and return words Pseudo-self-similarity and derived Vorono¨ı tilings

2 Hyperplane sequences and generalized substitutions Stepped hyperplanes and associated sequences Generalized substitutions

3 Toward a characterization for hyperplane sequences

Multidimensional continued fractions

Modified Jacobi-Perron for ~ α ∈ [0, 1) ⁿ :

~

α = [(a 1 , ε 1 ), . . . , (a _k , ε _k ), [~ α _k ]]

= [(a 1 , ε 1 ), (a 2 , ε 2 ), . . .]

where a _i ∈ N , ε _i ∈ {1, . . . , n} and α ~ _k ∈ [0, 1) ⁿ . Matrix viewpoint:

~

α k−1 = η k t M σ

k)

~ α k ,

where η _k ∈ R and σ _{(a ,ε )} unimodular substitution on {1, . . . , n}.

Modified Jacobi-Perron for ~ α ∈ [0, 1) ⁿ :

~

α = [(a 1 , ε 1 ), . . . , (a _k , ε _k ), [~ α _k ]]

= [(a 1 , ε 1 ), (a 2 , ε 2 ), . . .]

where a _i ∈ N , ε _i ∈ {1, . . . , n} and α ~ _k ∈ [0, 1) ⁿ . Matrix viewpoint:

~

α k−1 = η k t M σ

k)

~ α k ,

where η _k ∈ R and σ _{(a ,ε )} unimodular substitution on {1, . . . , n}.

By theorem (Action of Θ(σ)):

Θ(σ _(a

_,ε

₎ )(S _{~ α}

) = S _{~ α}

.

Then, Θ(σσ ⁰ ) = Θ(σ ⁰ )Θ(σ) yields:

Θ(σ _(a

_,ε

₎ . . . σ _(a

_,ε

₎ )(S _{~ α}

) = S _{α ~} .

1 Characterizations for words and tilings

Substitutions, continued fractions and return words Pseudo-self-similarity and derived Vorono¨ı tilings

2 Hyperplane sequences and generalized substitutions Stepped hyperplanes and associated sequences Generalized substitutions

3 Toward a characterization for hyperplane sequences

Multidimensional continued fractions

~

α _p = ~ α periodic expansion ~ α = [(a ₁ , ε ₁ ), . . . , (a _p , ε _p ), [~ α]]:

Θ(σ _(a

_,ε

₎ . . . σ _(a

_,ε

₎ )( S _{~ α}

|{z}

S

) = S _{α ~} .

Theorem

The multidimensional Sturmian sequence U _{α ~} is a fixed-point if α ~

has a periodic multidimensional continued fraction expansion.

~

α _p = ~ α periodic expansion ~ α = [(a ₁ , ε ₁ ), . . . , (a _p , ε _p ), [~ α]]:

Θ(σ _(a

_,ε

₎ . . . σ _(a

_,ε

₎ )( S _{~ α}

|{z}

S

) = S _{α ~} .

Theorem

The multidimensional Sturmian sequence U _{α ~} is a fixed-point if α ~

has a periodic multidimensional continued fraction expansion.