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# Sturmian Sequences

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Thomas Fernique

LIRMM - Universit´e Montpellier II

12/19/2005

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2 Sturmian sequences Cut and project Substitutions Continued fractions

3 Minimal complexity 1D sequences nD sequences

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## function of quasiperiodicity/recurrence

Definition

SetA fA :R+→R+ s.t. :

∃x ∈A |P ⊂A∩B(x,r) ⇒ ∀y ∈A, P ⊂A∩B(y,fA(r)).

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## Generation : substitutions

Tiling : partition into a finite number (up to translation/rotation) of polygonal tiles.

Substitution : inflation of tiles + dissection into new tiles.

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## Cut & Project (Meyer, 1972, Katz et Duneau, 1985)

directive spaceD, projective space P s.t. Rd =P×D; window Ω⊂P (bounded open set) ;

strip S(Ω) =D×Ω ; projection π onP alongD,

set π(Zd∩S(Ω)) or tiling projecting the faces ofZd∩S(Ω)).

Penrose :d = 5, dim(P) = 2.

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1 Aperiodic order

2 Sturmian sequences Cut and project Substitutions Continued fractions

3 Minimal complexity 1D sequences nD sequences

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## Cut & Project in codimension 1 : stepped plane

~

α∈Rn+ S~α.

Entries of~α linearly independent over Q: Sturmian plane.

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## Stepped plane tiling sequence

7→

23 3 1 23 23 3 1 23 23 3 23 3 1 2 3

2 3 3 1 2 3

2 3

1 23 2

3 3 1 23 2

3 3 23 3 1 23 23 3 1 2 3

23 3 2

3 3 1 2 3

2 3 3 2

3 3 1 23 23 3 1 23 23 3 1 23 3 1 2 3

23 3 1 2 3 3 1 2 3

2 3 3 1 23 2

3 3 1 23 23 3 23 3

1 23 23 3 23 3 1 2 3

2 3 3 1 2 3

2 3 3 23 3 1 23 23

3 23 3 1 23 23 3 1 2 3

23 3 2

3 3 1 23 2

3 3 1 23

3 1 23 23 3 1 23 23 3 1 23 3 1 2 3

2 3 3 1 2 3

2 3 3

2 3 3 1 23 2

3 3 1 23 23 3 23 3 1 2 3

23 3 1 2 3

2 3

1 2 3

2 3 3 1 2 3

2 3 3 23 3 1 23 23 3 1 23 23 3 2

3 3 1 2 3

23 3 2

3 3 1 23 2

3 3 1 23 23 3 23 3 1 2 3

23 3 23 3 1 2 3

2 3 3 1 2 3

2 3 3 1 23 3 1 23 23 3

1 23 3 1 23 23 3 1 2 3

23 3 1 2 3

2 3 3 2

3 3 1 23 23

3 23 3 1 23 23 3 1 23 23 3 2

3 3 1 2 3

2 3 3 1 23

3 1 23 2

3 3 1 23 23 3 23 3 1 2 3

23 3 1 2 3

2 3 3

2 3 3 1 2 3

2 3 3 1 23 3 1 23 23 3 1 23 23 3 1 2 3 23

In dimension 1 : classic Sturmian sequences.

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1 Aperiodic order

2 Sturmian sequences Cut and project Substitutions Continued fractions

3 Minimal complexity 1D sequences nD sequences

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Substitution : morphismσ of A s.t. |σn(i)| → ∞ for i ∈ A.

σ: 17→12,27→1 :

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

n→∞lim σn(1) = 12112121121121· · ·=uα =σ(uα).

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A={1,2,3}and (~e1, ~e2, ~e3) canonical basis ofR3.

u∈ A broken line of segments [~x, ~x+~ei] = (~x,i),~x∈N3 :

e3

e1

e2 2 2

3

3 1

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σ onA linear map Θ(σ) on segments : Θ(σ) : (~x,i)7→Mσ~x+ X

p|σ(i)=p·j·s

(~f(p),j),

where (Mσ)i,j =|σ(j)|i and~f(u) = (|u|1, . . . ,|u|n).

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σ :

17→12 27→13 37→1

, Mσ =

1 1 1 1 0 0 0 1 0

, σ(22331) = 13131112.

e3

e1

e2

e3

e2 e1

2 2

3 3 1

3 2

1 1

1

1 1

3

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

e3

e2 e1

e3

e2 e1

e3

e2 e1

If det(Mσ) =±1 : linear map Θ(σ) dual map Θ(σ) : Θ(σ)(~x,i) =Mσ−1~x+X

j∈A

X

s|σ(j)=p·i·s

(~f(s),j).

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σ: 17→12,27→13,37→1 :

e3

e2 e1

e3

e2 e1

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Theorem

Θ(σ)(S~α) =StMσ~α.

7→

Mσ−1Pα,β =Pα00 Θ(σ) “discretization” ofMσ−1.

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1 Aperiodic order

2 Sturmian sequences Cut and project Substitutions Continued fractions

3 Minimal complexity 1D sequences nD sequences

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## A necessary condition to be substitutive

Proposition

IfSα~ is a fixed point of a substitution Θ(σ), then the entries ofα~ belong toQ(λ), where λis an algebraic number of d ≤n.

Θ(σ)(Sα~) =Sα~ ⇒ StMσ~α=S~α

tMσ~α=λ~α.

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## A sufficient condition to be substitutive

If there exists unimodular matricesMi s.t. :

~

α=λM1×M2×. . .×Mp~α, thenS~α is substitutive.

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## The case n = 2 : continued fractions

Gauss map :T :α 7→ α11

α

= α1 −a.

Matrix viewpoint : α

a 1 1 0

1 T(α)

= 1

α

.

So, if (Tn(α))n is periodic, thenS(1,α) is substitutif.

Lagrange :α quadratic has a periodic expansion (Tn(α))n. Thus : Theorem

S(1,α) substitutif iffα has a periodic continued fraction expansion.

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## The case n > 2 : Brun’s expansion

~

α= (α1, . . . , αn)∈[0,1)n\{0}: T(α1, . . . , αn) =

α1

αi, . . . ,αi−1

αi , 1 αi

1 αi

i+1

αi , . . . ,αn

αi

,

whereαi = maxαj. a(~α) =

1 maxαj

and ε(~α) = min{i |αi = max

j αj}.

Brun expansion (an, εn) = (a(Tn(~α)), ε(Tn(~α)))

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Matrix viewpoint :αεAa,εt(1,T(~α)) =t(1, ~α), where :

Aa,ε =

a 1

Iε−1

1 0

In−ε

 ,

Theorem

If~α has a periodic Brun expansion, thenSt(1,~α) is substitutif.

Lagrange ?

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1 Aperiodic order

2 Sturmian sequences Cut and project Substitutions Continued fractions

3 Minimal complexity 1D sequences nD sequences

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Foru= (ui)i∈Z,pu(n) : number of different factor of length n inu.

Theorem

pu bounded iff ∃n |pu(n)≤n iff u is periodic.

∀n, pu(n) =n+ 1 :u aperiodic sequence of minimal complexity.

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Theorem

Let u s.t.∀n, pu(n) =n+ 1. Three cases :

1 u is sturmian,

2 ∃σ ∈<l,r,e > |u =σ(ω010ω),

3 ∃σ ∈<l,r,e > |u =σ(ω01ω), where :

l :

07→0

17→01 , r :

07→0

17→10 , e :

07→1 17→0 .

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## Geometric viewpoint

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1 Aperiodic order

2 Sturmian sequences Cut and project Substitutions Continued fractions

3 Minimal complexity 1D sequences nD sequences

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nD sequence :u = (ui1,...,in)ijZ.

A⊂Zn A-complexity :pu(A) denotes the number of different factors of shapeA in u.

rectangular complexity :A={(i,j) |1≤i ≤m, 1≤j ≤n}. One writespu(m,n).

Question :u periodic iffpu(A)≤?

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nD Sequenceu :r-periodic, 0≤r ≤n.

proposition

u n-periodic iff pu(m1, . . . ,mn) bounded.

Limit 1-periodicity/aperiodicity ?

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Conjecture (Nivat, 1997) For a 2D sequenceu :

∃(m0,n0) |pu(m0,n0)≤m0n0 ⇒ u periodic.

sequences of complexitymn+ 1 : characterized 2D sequences over{0,1};

proved for pu(m0,n0)≤ 161 m0n0;

converse is false (rectangle = bad shape ?) ; extensions for d >2 do not hold.

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Conjecture (Nivat, 1997) For a 2D sequenceu :

∃(m0,n0) |pu(m0,n0)≤m0n0 ⇒ u periodic.

sequences of complexitymn+ 1 : characterized 2D sequences over{0,1};

proved for pu(m0,n0)≤ 161 m0n0;

converse is false (rectangle = bad shape ?) ; extensions for d >2 do not hold.

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Conjecture (Nivat, 1997) For a 2D sequenceu :

∃(m0,n0) |pu(m0,n0)≤m0n0 ⇒ u periodic.

sequences of complexitymn+ 1 : characterized 2D sequences over{0,1};

proved for pu(m0,n0)≤ 161 m0n0;

converse is false (rectangle = bad shape ?) ; extensions for d >2 do not hold.

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## Geometric viewpoint

sequence 1D over{0,1} “curve” of the plane.

sequence 2D over{0,1} ?

“stepped surface” can be encoded by 2D sequences over{1,2,3}

1 1

3 1 2 1 3 2 1

2 2 1 3 2 1

2 2 1 3 2 1

3 2 3 2

1 2 1 2 1

3 2 1

1 3 2 1

3 2 1

3 1 3 2 1

2 3 2

3 3

2 1 3

1 1 3

1 2 1 3 2 1

3 3

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Stepped planes are stepped surfaces. So : Conjecture

Sturmian planes = aperiodic nD sequences of minimal complexity amongstepped surfaces ?

Proposition

If this conjecture holds, then the Nivat conjecture also holds.

Converse ?

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Proposition

Ifu is a 2D sequence which encodes a sturmian stepped plane, thenpu(m,n) =mn+m+n.

Characterization of the 2D sequence of complexitymn+m+n? Notice that Θ(σ) acts on stepped surfaces as well.

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