Generation of discrete planes P. Arnoux, V. Berth e, A. Siegel - PowerPoint PPT Presentation

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Generation of discrete planes P. Arnoux, V. Berth´ e, A. Siegel LIRMM-CNRS-Montpellier-France berthe@lirmm.fr http://www.lirmm.fr/˜berthe Journ´ ees Montoises Rennes, 2006

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Discrete plane Let P ( a , b , c ) ⊂ R 3 be the plane with equation ax + by + cz = 0. • We suppose that a , b , c > 0. • We want to approximate the plane P by a union of faces of integral cubes.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Discrete plane Let P ( a , b , c ) ⊂ R 3 be the plane with equation ax + by + cz = 0. • We suppose that a , b , c > 0. • We want to approximate the plane P by a union of faces of integral cubes.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Discrete plane Let P ( a , b , c ) ⊂ R 3 be the plane with equation ax + by + cz = 0. • We suppose that a , b , c > 0. • We want to approximate the plane P by a union of faces of integral cubes. Let ( e 1 , e 2 , e 3 ) denote the canonical basis of R 3 . Integral cube We call integral cube any set ( p , q , r ) + C where ( p , q , r ) ∈ Z 3 and C is the fundamental unit cube: C = { λ e 1 + µ e 2 + ν e 3 , ( λ, µ, ν ) ∈ [0 , 1] 2 } .

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Discrete plane Discrete plane The discrete plane associated with P ( a , b , c ) is the boundary of the set of integral cubes that intersect the lower half-space ax + by + cz < 0 . This discrete plane is denoted by P ( a , b , c ) .

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Discrete plane Discrete plane The discrete plane associated with P ( a , b , c ) is the boundary of the set of integral cubes that intersect the lower half-space ax + by + cz < 0 . This discrete plane is denoted by P ( a , b , c ) .

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Discrete plane Discrete plane The discrete plane associated with P ( a , b , c ) is the boundary of the set of integral cubes that intersect the lower half-space ax + by + cz < 0 . This discrete plane is denoted by P ( a , b , c ) . Vertex A vertex of the discrete plane P ( a , b , c ) is an integral point that belongs to the discrete plane. Let V ( a , b , c ) stand for the set of vertices of P ( a , b , c ) . According to Reveill` es’ terminology in discrete geometry, V ( a , b , c ) is a standard discrete plane.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Faces Let F 1 , F 2 , and F 3 be the three following faces: { λ e 2 + µ e 3 , ( λ, µ ) ∈ [0 , 1[ 2 } F 1 = { λ e 1 + µ e 3 , ( λ, µ ) ∈ [0 , 1[ 2 } = F 2 { λ e 1 + µ e 2 , ( λ, µ ) ∈ [0 , 1[ 2 } . F 3 = We call pointed face the set ( p , q , r ) + F i . e 3 0 0 0 e e 2 1

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Faces Let F 1 , F 2 , and F 3 be the three following faces: { λ e 2 + µ e 3 , ( λ, µ ) ∈ [0 , 1[ 2 } F 1 = { λ e 1 + µ e 3 , ( λ, µ ) ∈ [0 , 1[ 2 } = F 2 { λ e 1 + µ e 2 , ( λ, µ ) ∈ [0 , 1[ 2 } . F 3 = We call pointed face the set ( p , q , r ) + F i . e 3 0 0 0 e e 2 1 Distinguished vertex Let F be the set of pointed faces. Let v : F → Z 3 defined by v (( p , q , r ) + F i ) = ( p , q , r ) + e 1 + · · · + e i − 1 , for i ∈ { 1 , 2 , 3 } . The vertex v ( p , q , r ) is called the distinguished vertex of the face ( p , q , r ) + F i .

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Faces Let F 1 , F 2 , and F 3 be the three following faces: { λ e 2 + µ e 3 , ( λ, µ ) ∈ [0 , 1[ 2 } = F 1 { λ e 1 + µ e 3 , ( λ, µ ) ∈ [0 , 1[ 2 } F 2 = { λ e 1 + µ e 2 , ( λ, µ ) ∈ [0 , 1[ 2 } . = F 3 We call pointed face the set ( p , q , r ) + F i . e 3 0 0 0 e e 2 1 Coding A point ( p , q , r ) ∈ Z 3 is the distinguished vertex of a face in P ( a , b , c ) of type • 1 if and only if ap + bq + cr ∈ [0 , a [ • 2 if and only if ap + bq + cr ∈ [ a , a + b [ • 3 if and only if ap + bq + cr ∈ [ a + b , a + b + c [. 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 1 2 1 2 1 2 1

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Two-dimensional coding We thus can code a discrete plane by a two-dimensional sequence in the following way. Projection Let π the orthogonal projection on the diagonal plane x + y + z = 0. This projection sends the lattice Z 3 to a lattice Γ onto the diagonal plane. Fact The restriction of π to the set of vertices V ( a , b , c ) is a bijection onto its image Γ. This allows us to define a two-dimensional word, by associating with any point in Γ, the type of the face having its preimage as a distinguished vertex.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Two-dimensional Sturmian words 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 1 2 1 2 1 2 1 Two-dimensional coding Let ( m , n ) ∈ Z 2 and g = m π ( e 1 ) + n π ( e 2 ) in the lattice Γ. There exists a unique integer U ( m , n ) ∈ { 1 , 2 , 3 } such that π − 1 ( g ) is the distinguished vertex of a face of type U ( m , n ) in the discrete plane P ( a , b , c ) : U ( m , n ) = 1 if ( am + bn ) mod ( a + b + c ) ∈ [0 , a [ , U ( m , n ) = 2 if ( am + bn ) mod ( a + b + c ) ∈ [ a , a + b [ , U ( m , n ) = 3 if ( am + bn ) mod ( a + b + c ) ∈ [ a + b , a + b + c [ . The sequence ( U ( a , b , c ) ( m , n )) Z 2 is called the the two-dimensional coding associated with the plane ax + by + cz = 0.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Incidence matrix Assume A = { 1 , 2 , 3 } and let σ be a substitution over A . The incidence matrix M σ of σ is the 3 × 3 matrix defined by: M σ = ( | σ ( j ) | i ) ( i , j ) ∈{ 1 , 2 , 3 } 2 , where | σ ( j ) | i is the number of occurrences of the letter i in σ ( j ).

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Incidence matrix Assume A = { 1 , 2 , 3 } and let σ be a substitution over A . The incidence matrix M σ of σ is the 3 × 3 matrix defined by: M σ = ( | σ ( j ) | i ) ( i , j ) ∈{ 1 , 2 , 3 } 2 , where | σ ( j ) | i is the number of occurrences of the letter i in σ ( j ). Example Let σ : 1 �→ 13 , 2 �→ 1 , 3 �→ 2. 0 1 1 1 0 M σ = 0 0 1 @ A 1 0 0

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Incidence matrix Assume A = { 1 , 2 , 3 } and let σ be a substitution over A . The incidence matrix M σ of σ is the 3 × 3 matrix defined by: M σ = ( | σ ( j ) | i ) ( i , j ) ∈{ 1 , 2 , 3 } 2 , where | σ ( j ) | i is the number of occurrences of the letter i in σ ( j ). Unimodular substitution A substitution σ is unimodular if det M σ = ± 1. Abelianization Let l : { 1 , 2 , 3 } ⋆ → N 3 be the Parikh mapping: l ( w ) = t ( | w | 1 , | w | 2 , | w | 3 ) .

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Generalized substitution Generalized substitution [Arnoux-Ito] Let σ be a unimodular substitution on three letters. We call generalized substitution the following tranformation acting on a face x + F i defined by: [ [ M − 1 Σ σ ( x + F i ) = ( x + l ( S )) + F k . σ k ∈{ 1 , 2 , 3 } S , σ ( k )= PiS

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Generalized substitution Generalized substitution [Arnoux-Ito] Let σ be a unimodular substitution on three letters. We call generalized substitution the following tranformation acting on a face x + F i defined by: [ [ M − 1 Σ σ ( x + F i ) = ( x + l ( S )) + F k . σ k ∈{ 1 , 2 , 3 } S , σ ( k )= PiS e 3 0 0 0 e e 2 1

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences Generalized substitution Generalized substitution [Arnoux-Ito] Let σ be a unimodular substitution on three letters. We call generalized substitution the following tranformation acting on a face x + F i defined by: [ [ M − 1 Σ σ ( x + F i ) = ( x + l ( S )) + F k . σ k ∈{ 1 , 2 , 3 } S , σ ( k )= PiS Example Let σ : 1 �→ 13 , 2 �→ 1 , 3 �→ 2. 0 1 0 1 1 1 0 0 0 1 M − 1 A . M σ = 0 0 1 and = 1 0 − 1 @ A @ σ 1 0 0 0 1 0 ( x + F 1 ) �→ [( M − 1 σ x + e 1 − e 2 ) + F 1 ] ∪ ( M − 1 σ x + F 2 ) ( x + F 2 ) �→ M − 1 Σ σ : σ x + F 3 ( x + F 3 ) �→ M − 1 σ x + F 1 . ∗ Θ σ ∗ ∗ Θ σ Θ σ ∗ Θ σ