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) ⊂ R3 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) ⊂ R3 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) ⊂ R3 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 (e1, e2, e3) denote the canonical basis of R3.

Integral cube

We call integral cube any set (p, q, r) + C where (p, q, r) ∈ Z3 and C is the fundamental unit cube: C = {λe1 + µe2 + νe3, (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 F1, F2, and F3 be the three following faces: F1 = {λe2 + µe3, (λ, µ) ∈ [0, 1[2} F2 = {λe1 + µe3, (λ, µ) ∈ [0, 1[2} F3 = {λe1 + µe2, (λ, µ) ∈ [0, 1[2}. We call pointed face the set (p, q, r) + Fi.

1

e2 e3 e

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Faces

Let F1, F2, and F3 be the three following faces: F1 = {λe2 + µe3, (λ, µ) ∈ [0, 1[2} F2 = {λe1 + µe3, (λ, µ) ∈ [0, 1[2} F3 = {λe1 + µe2, (λ, µ) ∈ [0, 1[2}. We call pointed face the set (p, q, r) + Fi.

1

e2 e3 e

Distinguished vertex

Let F be the set of pointed faces. Let v : F → Z3 defined by v((p, q, r) + Fi) = (p, q, r) + e1 + · · · + ei−1, for i ∈ {1, 2, 3}. The vertex v(p, q, r) is called the distinguished vertex of the face (p, q, r) + Fi.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Faces

Let F1, F2, and F3 be the three following faces: F1 = {λe2 + µe3, (λ, µ) ∈ [0, 1[2} F2 = {λe1 + µe3, (λ, µ) ∈ [0, 1[2} F3 = {λe1 + µe2, (λ, µ) ∈ [0, 1[2}. We call pointed face the set (p, q, r) + Fi.

1

e2 e3 e

Coding

A point (p, q, r) ∈ Z3 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 1 2 1 1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 1 2 1 1 3 2 2 1 1 2 2 2 1 1 1 2 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 Z3 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 1 2 1 1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 1 2 1 1 3 2 2 1 1 2 2 2 1 1 1 2 2 1

Two-dimensional coding

Let (m, n) ∈ Z2 and g = mπ(e1) + nπ(e2) 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))Z2 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 Mσ of σ is the 3 × 3 matrix defined by: Mσ = (|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 Mσ of σ is the 3 × 3 matrix defined by: Mσ = (|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. Mσ = @ 1 1 1 1 1 A

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 Mσ of σ is the 3 × 3 matrix defined by: Mσ = (|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 Mσ = ±1.

Abelianization

Let l : {1, 2, 3}⋆ → N3 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 + Fi defined by: Σσ(x + Fi) = [

k∈{1,2,3}

[

S, σ(k)=PiS

M−1

σ

(x + l(S)) + Fk.

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 + Fi defined by: Σσ(x + Fi) = [

k∈{1,2,3}

[

S, σ(k)=PiS

M−1

σ

(x + l(S)) + Fk.

1

e2 e3 e

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 + Fi defined by: Σσ(x + Fi) = [

k∈{1,2,3}

[

S, σ(k)=PiS

M−1

σ

(x + l(S)) + Fk.

Example

Let σ : 1 → 13, 2 → 1, 3 → 2. Mσ = @ 1 1 1 1 1 A and M−1

σ

= @ 1 1 −1 1 1 A . Σσ : (x + F1) → [(M−1

σ x + e1 − e2) + F1] ∪ (M−1 σ x + F2)

(x + F2) → M−1

σ x + F3

(x + F3) → M−1

σ x + F1.

Θ

∗

Θ Θσ

∗ σ ∗

Θσ

∗ σ

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Generalized substitution

Property [Arnoux-Ito, Fernique]

• Any unimodular substitution σ over a three-letter alphabet can be extended to a

generalized substitution Σσ acting on faces of any discrete plane P(a,b,c).

• This generalized substitution maps any pattern of P(a,b,c) to a pattern of the

discrete plane P(a1,b1,c1) where

t(a, b, c) =t MΣ t(a1, b1, c1).

• Furthermore the images of two distinct faces do not intersect.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Generalized substitution

Property [Arnoux-Ito, Fernique]

• Any unimodular substitution σ over a three-letter alphabet can be extended to a

generalized substitution Σσ acting on faces of any discrete plane P(a,b,c).

• This generalized substitution maps any pattern of P(a,b,c) to a pattern of the

discrete plane P(a1,b1,c1) where

t(a, b, c) =t MΣ t(a1, b1, c1).

• Furthermore the images of two distinct faces do not intersect.

The unit cube

One easily sees that E1, E2 and E3 all belong to P(a,b,c). We denote by U the unit cube being the union of these three faces.

Property

One has U ⊂ Σσ(U).

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Generation of a discrete plane

Question

Consider a discrete plane associated with a positive eigenvector of Mσ. By considering the iterates Σn

σ(U), are we able to generate the whole discrete plane P(a,b,c)?

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Pisot type substitution

Definition

• A morphism σ on three letters is of Pisot type if its eigenvalues satisfy

α > 1 > |λ1| ≥ |λ2| > 0.

• In particular, the dominant eigenvalue α is a Pisot number, that is, an algebraic

integer whose conjugates are smaller than 1 in modulus.

• Furthermore, its incidence matrix Mσ is primitive, that is, it admits a power with

strictly positive entries.

Property

Let σ be a unimodular substitution of Pisot type, and let P(a,b,c) be the discrete plane approximating the contracting plane of Mσ. Then there exists a finite patch V in P such that P(a,b,c) = ∪n∈NΣn

σ(V).

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Motivation

Problem

We would like to give conditions under which the plane is obtained from the unit cube U by

• iterating a generalized substitution Σσ
• or a sequence of generalized substitutions (S-adic generation).

This has applications to:

• Discrete geometry: generation of discrete planes, recognition issues.
• Tilings: connections with central tiles and their associated tilings in β-numeration

(Akiyama, Bernat, Steiner,Thuswaldner...)

• Symbolic dynamics: sufficient condition for discrete spectrum for Pisot

substitutive dynamical systems

• Diophantine approximation: bounded remainder sets

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Two assumptions

• The surrounding hypothesis
• The generation hypothesis

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Neighborhood of a subset of a discrete plane

How to define a notion of neighborhood? The easiest way is to define a distance.

• This distance could be defined on the lattice after projection of the distinguished

vertices,

• or on the discrete plane itself. This is the option we chose.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Distance for a discrete plane

Path

Let P(a,b,c) be a discrete plane. A finite sequence of faces contained in P(a,b,c) is a path of length n in P(a,b,c) if every pair of consecutive faces share a common edge.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Distance for a discrete plane

Path

Let P(a,b,c) be a discrete plane. A finite sequence of faces contained in P(a,b,c) is a path of length n in P(a,b,c) if every pair of consecutive faces share a common edge.

Distance

• The distance of two faces is the length of the shortest path that joins them.
• The ball of radius n around a face F in the discrete plane P(a,b,c), denoted by

B(a,b,c)(F, n), is the union of faces at distance at most n of F.

• The neighborhood of size n of a patch W, also denoted by B(a,b,c)(W, n), is the

union of the balls of radius n around the faces in W.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

The surrounding hypothesis

We want the image of a neighborhood to be a neighborhood of the image

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

The surrounding hypothesis

We want the image of a neighborhood to be a neighborhood of the image

Surrounding property

A generalized substitution Σσ is said to satisfy the surrounding property if for every discrete plane P(a,b,c), and every face (p, q, r) + Fi ⊂ P(a,b,c), then the image by Σσ of the ball of radius 1 around the face (p, q, r) + Fi contains the neighborhood of size 1 of Σσ((p, q, r) + Fi).

Property

If Σσ satisfies the surrounding property, then the image of the neighborhood of size n

• f any patch W contained in a discrete plane contains a neighborhood of size n of the

image of the set.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

The generation hypothesis

Unit cube

One has U ⊂ P(a,b,c) for all positive (a, b, c). Furthermore, the unit cube is always contained in its image under Σσ. We want the unit cube U, which is always contained in its image, to generate a neighborhood of itself under Σσ.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

The generation hypothesis

Unit cube

One has U ⊂ P(a,b,c) for all positive (a, b, c). Furthermore, the unit cube is always contained in its image under Σσ. We want the unit cube U, which is always contained in its image, to generate a neighborhood of itself under Σσ.

Generation hypothesis

The generalized substitution Σσ satisfies the generation hypothesis if Σσ(U) contains a neighborhood of size 1 of U.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

The ring lemma

Ring lemma

Let (Σσ) be a generalized substitution that satisfies

• the generation hypothesis
• and the surrounding hypothesis.

Then for every n ∈ N and every (a, b, c) positive which is the image of a positive vector by Mn

σ, Σn σ(U) contains the neighborhood of size n of U in P(a,b,c):

B(a,b,c)(U, n) ⊂ Σn

σ(U).

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

The ring lemma

Ring lemma. S-adic version

Let (Σσn)n∈N be a sequence of generalized substitutions that all satisfy

• the generation hypothesis
• and the surrounding hypothesis.

Then for every n ∈ N and every (a, b, c) positive which is the image of a positive vector by Mσ1 . . . Mσn, the composition Σσ1 . . . Σσn(U) contains the neighborhood of size n of U in P(a,b,c): B(a,b,c)(U, n) ⊂ Σσ1 . . . Σσn(U). The proof is by induction.

Generation of the whole discrete plane

Under the hypothesis of the ring lemma, we can generate the whole discrete plane P(a,b,c) by the sequence of generalized substitutions: P(a,b,c) = ∪n∈NΣσ1 . . . Σσn(U).

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

S-adic sequence

Definition

A sequence u is said S-adic if there exist

• a finite set of substitutions S over an alphabet D = {0, ..., d − 1}
• a morphism ϕ from D⋆ to A⋆
• an infinite sequence of substitutions (σn)n≥1 with values in S

such that u = lim

n→+∞ σ1σ2...σn(0).

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

A special type of S-adic system

We consider σ1 : 1 → 1 2 → 21 3 → 31 σ2 : 1 → 12 2 → 2 3 → 32 σ3 : 1 → 13 2 → 23 3 → 3 and generalized substitutions Σσi . They generate Episturmian words (Arnoux,Rauzy,Justin,Pirillo,Richomme)

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

Bounded partial quotients

• We consider the case of bounded partial quotients of order 4: any substitution
• ccurs in any word of length 4.
• Up to renumbering the substitutions, there are six possible sequences

1123, 1213, 1231, 1223, 1232, 1233, giving rise to six substitutions σ1123 and so on. These substitutions all satisfy the generation property.

Discrete planes Generalized substitutions The ring lemma Application to Arnoux-Rauzy sequences

But...

These substitutions all satisfy the generation property... But, they do not satisfy the surrounding property. The image of F2 by the substitution σ1231 The image of F3 by the substitution σ1231