Word combinatorics, tilings and discrete geometry V. Berth e - PowerPoint PPT Presentation

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Word combinatorics, tilings and discrete geometry V. Berth´ e LIRMM-CNRS-Montpellier-France berthe@lirmm.fr http://www.lirmm.fr/˜berthe Journ´ ees Montoises Rennes, 2006

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces From discrete surfaces to multidimensional words... ...via tilings and word combinatorics

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Some objects... • discrete lines, planes, surfaces, rotations ...and some transformations acting on them • substitutions ∗ Θ σ ∗ ∗ Θ σ Θ σ ∗ Θ σ • flips s

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Arithmetic discrete lines The arithmetic discrete line D ( a , b , µ, ω ) is defined as D ( a , b , c , µ, ω ) = { ( x , y ) ∈ Z 2 | 0 ≤ ax + by + µ < ω } • µ is the translation parameter of D ( a , b , µ, ω ) • ω is the width of D ( a , b , µ, ω ) • If ω = max {| a | , | b |} , then D ( a , b , µ, ω ) is said naive • If ω = | a | + | b | , then D ( a , b , c , µ, ω ) is said standard

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Arithmetic discrete lines The arithmetic discrete line D ( a , b , µ, ω ) is defined as D ( a , b , c , µ, ω ) = { ( x , y ) ∈ Z 2 | 0 ≤ ax + by + µ < ω } • µ is the translation parameter of D ( a , b , µ, ω ) • ω is the width of D ( a , b , µ, ω ) • If ω = max {| a | , | b |} , then D ( a , b , µ, ω ) is said naive • If ω = | a | + | b | , then D ( a , b , c , µ, ω ) is said standard Reveill` es’91, Fran¸ con, Andres, Debled-Renesson, Jacob-Dacol, Kiselman, Vittone, Chassery, G´ erard, Buzer, Brimkov, Barneva, Rosenfeld, Klette...

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Arithmetic discrete lines The arithmetic discrete line D ( a , b , µ, ω ) is defined as D ( a , b , c , µ, ω ) = { ( x , y ) ∈ Z 2 | 0 ≤ ax + by + µ < ω } • µ is the translation parameter of D ( a , b , µ, ω ) • ω is the width of D ( a , b , µ, ω ) • If ω = max {| a | , | b |} , then D ( a , b , µ, ω ) is said naive • If ω = | a | + | b | , then D ( a , b , c , µ, ω ) is said standard

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Discrete lines and Sturmian words One can code such a discrete line (Freeman code) over the two-letter alphabet { 0 , 1 } . One gets a Stumian word ( u n ) n ∈ N ∈ { 0 , 1 } N 0100101001001010010100100101

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Discrete lines and Sturmian words Let R α : R / Z → R / Z , x �→ x + α mod 1. Sturmian words [Morse-Hedlund] Let ( u n ) n ∈ N ∈ { 0 , 1 } N be a Sturmian word. There exist α ∈ (0 , 1), α �∈ Q , x ∈ R such that ⇒ R n ∀ n ∈ N , u n = i ⇐ α ( x ) = n α + x ∈ I i (mod 1) , with I 0 = [0 , 1 − α [ , I 1 = [1 − α, 1[ or I 0 =]0 , 1 − α ] , I 1 =]1 − α, 1] .

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Discrete lines and Sturmian words Let R α : R / Z → R / Z , x �→ x + α mod 1. Sturmian words are codings of irrational rotations of T 1 = R / Z with respect either to the partition { I 0 = [0 , 1 − α [ , I 1 = [1 − α, 1[ } or to { I 0 =]0 , 1 − α ] , I 1 =]1 − α, 1] } .

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces A key lemma Let I 0 = [0 , 1 − α [, I 1 = [1 − α, 1[. Let R α : x �→ x + α mod 1. Lemma The word w = w 1 · · · w n over the alphabet { 0 , 1 } is a factor the Sturmian word u iff I w 1 ∩ R − 1 α I w 2 ∩ · · · R − n +1 I w n � = ∅ . α

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces A key lemma Let I 0 = [0 , 1 − α [, I 1 = [1 − α, 1[. Let R α : x �→ x + α mod 1. Lemma The word w = w 1 · · · w n over the alphabet { 0 , 1 } is a factor the Sturmian word u iff I w 1 ∩ R − 1 α I w 2 ∩ · · · R − n +1 I w n � = ∅ . α Proof: One has ∀ i ∈ N , u n = i ⇐ ⇒ n α + x ∈ I i (mod 1) . • One first notes that u k u k +1 · · · u n + k − 1 = w 1 · · · w n iff 8 k α + x ∈ I w 1 > > ( k + 1) α + x ∈ I w 2 < ... > > ( k + n − 1) α + x ∈ I w n : • One then applies the density of ( k α ) ∈ N in R / Z .

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces A key lemma Let I 0 = [0 , 1 − α [, I 1 = [1 − α, 1[. Let R α : x �→ x + α mod 1. Lemma The word w = w 1 · · · w n over the alphabet { 0 , 1 } is a factor the Sturmian word u iff I w 1 ∩ R − 1 α I w 2 ∩ · · · R − n +1 I w n � = ∅ . α Application: one deduces combinatorial properties on the • number of factors of given length/ enumeration of local configurations • densities of factors/ statistical properties of local configurations • powers of factors, repetitions, palindromes/symmetries

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces A key lemma Let I 0 = [0 , 1 − α [, I 1 = [1 − α, 1[. Let R α : x �→ x + α mod 1. Lemma The word w = w 1 · · · w n over the alphabet { 0 , 1 } is a factor the Sturmian word u iff I w 1 ∩ R − 1 α I w 2 ∩ · · · R − n +1 I w n � = ∅ . α Remark: The sets I w 1 ∩ R − 1 α I w 2 ∩ · · · R − n +1 I w n are intervals of T = R / Z . α The factors of u are in one-to-one correspondence with the n + 1 intervals of T whose end-points are given by − k α mod 1 , for 0 ≤ k ≤ n . Sturmian words [Coven-Hedlund] A word u ∈ { 0 , 1 } N is Sturmian iff it admits eactly n + 1 factors of length n .

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Tilings of the line • By projecting the vertices of the discrete line, one gets a tiling of the line. • This corresponds to a cut-and-project scheme in quasicrystallography. • If one modifies the width of the selection window, then one gets a tiling of the line by two or three tiles. • The corresponding codings are three-interval exchange words [Guimond-Mas´ akov´ a-Pelantov´ a]. • They have complexity n + cste , or 2 n + 1.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces To summarize... We have used • A coding as an infinite binary word • A dynamical system: the rotation of R / Z , R α : x �→ x + α • The key lemma: bijection between intervals and factors

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces To summarize... We have used • A coding as an infinite binary word • A dynamical system: the rotation of R / Z , R α : x �→ x + α • The key lemma: bijection between intervals and factors Dynamical system A dynamical system ( X , T ) is defined as the action of a continuous and onto map T on a compact space X .

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces To summarize... We have used • A coding as an infinite binary word • A dynamical system: the rotation of R / Z , R α : x �→ x + α • The key lemma: bijection between intervals and factors We have considered so far naive and standard lines. • Can we work with more general widths ω ? • Can we go to higher dimensions?

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Arithmetic discrete planes The arithmetic discrete plane P ( a , b , c , µ, ω ) is defined as P ( a , b , c , µ, ω ) = { ( x , y , z ) ∈ Z 3 | 0 ≤ ax + by + cz + µ < ω } . • µ is the translation parameter of P ( a , b , c , µ, ω ) • ω is the width of P ( a , b , c , µ, ω ) • If ω = max {| a | , | b | , | c |} , then P ( a , b , c , µ, ω ) is said naive • If ω = | a | + | b | + | c | , then P ( a , b , c , µ, ω ) is said standard

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces From a discrete plane to a tiling by projection.... 3 3 1 2 1 2 1 1 2 1 2 3 3 3 2 1 2 1 2 1 2 1 1 3 3 2 1 2 1 1 2 1 2 1 3 3 3 1 2 1 2 1 2 1 1 2 3 3 1 2 1 1 2 1 2 1 ....and from a tiling by lozenges to a ternary coding

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Discrete planes and two-dimensional Sturmian words Definition Such a coding is called a 2D Sturmian word.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces Discrete planes and two-dimensional Sturmian words Definition Such a coding is called a 2D Sturmian word. Theorem [B.-Vuillon] Let ( U m , n ) ( m , n ) ∈ Z 2 ∈ { 1 , 2 , 3 } Z 2 be a 2D Sturmian word. Then there exist x ∈ R , and α , β ∈ R such that 1 , α, β are Q -linearly independent and α + β < 1 such that ∀ ( m , n ) ∈ Z 2 , Um , n = i ⇐ ⇒ R m α R n β ( x ) = x + n α + m β ∈ I i (mod 1) , with I 1 = [0 , α [ , I 2 = [ α, α + β [ , I 3 = [ α + β, 1[ or I 1 =]0 , α ] , I 2 =] α, α + β ] , I 3 =] α + β, 1] .