Introduction to Rauzy fractals - IRISA · Introduction to Rauzy fractals Guanghzhou, 2010. Several questions I Geometry Eﬃcient way to produce a pixelized line? A pixelized plane?

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Introduction to Rauzy fractals

Guanghzhou, 2010

Several questions

I Geometry Efficient way to produce a pixelized line ? A pixelized plane ?

I Number theory Compute good rational approximations of a vector ?

I Physics Appropriate location for atoms to enhance conductivity propertiesand build frying pans ?

I Mathematics Provide codings for toral translations ?

Behind all these questions, a similar process : self-induced objects

Common reformulation

Do fractal shapes generate a tiling of a plane ?

Purpose of the lecture : Origin of the main properties of this fractal shape

Substitution / periodic point

Alphabet A = {1, . . . , n}.

Substitution : replacement rule on Aσ(1) = 12, σ(2) = 13, σ(3) = 1

Periodic point

I Iterate σ : eventually, the image of a letter starts by the same letter.

σd(a) = a....

I Iterate σd : Increasing set of words with the same beginnings.

11212 1312 13 12 112 13 12 1 12 13 1212 13 12 1 12 13 12 12 13 12 1 12 13

I The limit is a periodic point of the substitution

u = σ∞(a) σk(u) = u

Substitution / periodic point

Alphabet A = {1, . . . , n}.

Substitution : replacement rule on Aσ(1) = 12, σ(2) = 13, σ(3) = 1

Periodic point

I Iterate σ : eventually, the image of a letter starts by the same letter.

σd(a) = a....

I Iterate σd : Increasing set of words with the same beginnings.

11212 1312 13 12 112 13 12 1 12 13 1212 13 12 1 12 13 12 12 13 12 1 12 13

I The limit is a periodic point of the substitution

u = σ∞(a) σk(u) = u

Primitivity

Definition A substitution σ is primitive if there exists k such that all lettersappears in the image of all letters through σk .

Checking ?

I Incidence matrix M : abelianization of the substitution

I σ primitive iff Mk has only positive coefficients for one k.

Examples

σ(1) = 12, σ(2) = 13, σ(3) = 1

M =

0@1 1 11 0 00 1 0

1APrimitive (M3 > 0)

σ(1) = 12, σ(2) = 32, σ(3) = 23

M =

0@1 0 01 1 10 1 1

1AMn always has zeros in the first line.

Not primitive

Main interest The matrix M has a simple dominant eigenvalue

Primitivity

Definition A substitution σ is primitive if there exists k such that all lettersappears in the image of all letters through σk .

Checking ?

I Incidence matrix M : abelianization of the substitution

I σ primitive iff Mk has only positive coefficients for one k.

Examples

σ(1) = 12, σ(2) = 13, σ(3) = 1

M =

0@1 1 11 0 00 1 0

1APrimitive (M3 > 0)

σ(1) = 12, σ(2) = 32, σ(3) = 23

M =

0@1 0 01 1 10 1 1

1AMn always has zeros in the first line.

Not primitive

Main interest The matrix M has a simple dominant eigenvalue

Existence of periodic points

Theorem Every substitution on a n-letter alphabet has at least one (and atmost n) periodic points.

Proof

I if w is a periodic point, then σk(u0) starts with u0.

I This means that there exists a loop starting from u0 in the graph :a → b iff σ(a) starts with b

I The graph is finite and every node have an exiting edge.It contains at least one loop !

Represent a word ?

How to see the fixed point of a substitution on a n-letters alphabet ?

Abelianization map Linearly map letters to n independent vectors e1, . . . en

P : w1 . . . wk ∈ An 7→ ew1 + · · ·+ ewk ∈ Rn.

1211212112

12 13 12 1 12 13 12 12 13 12 1 12 13

Commutation formula applying σ to a word is equivalent to applying M toits linearized vector.

P(σ(W )) = MP(W )

Represent a word ?

How to see the fixed point of a substitution on a n-letters alphabet ?

Abelianization map Linearly map letters to n independent vectors e1, . . . en

P : w1 . . . wk ∈ An 7→ ew1 + · · ·+ ewk ∈ Rn.

1211212112

12 13 12 1 12 13 12 12 13 12 1 12 13

Commutation formula applying σ to a word is equivalent to applying M toits linearized vector.

P(σ(W )) = MP(W )

Pisot Case

Specific case ? The stair remains close to a line ?

Substitution of Pisot type The dominant eigenvalue of its incidence matrix isa unit Pisot number

I the determinant is ±1

I all the roots of the characteristic polynomial have a modulus less than orequal to 1.

In other words The matrix M has one expanding eigenvalue β and all othereigenvalues β(i) are contracting.

Pisot Case

Specific case ? The stair remains close to a line ?

Substitution of Pisot type The dominant eigenvalue of its incidence matrix isa unit Pisot number

I the determinant is ±1

I all the roots of the characteristic polynomial have a modulus less than orequal to 1.

In other words The matrix M has one expanding eigenvalue β and all othereigenvalues β(i) are contracting.

Projecting a stair on an appropriate plane

Hypothesis σ substitution unit of Pisot type on a n letter-alphabet.

Projections

I He expanding line of M.

I Hc contracting hyperplane of M.

I π : Rn → Hc projection along He .

Commutation relation : h = πM is a contraction on the hyperplane Hc .

Application The projection of P(σ(W )) is smaller than P(W ).

Projecting a stair on an appropriate plane

Hypothesis σ substitution unit of Pisot type on a n letter-alphabet.

Projections

I He expanding line of M.

I Hc contracting hyperplane of M.

I π : Rn → Hc projection along He .

Commutation relation : h = πM is a contraction on the hyperplane Hc .

Application The projection of P(σ(W )) is smaller than P(W ).

Distance property

Proposition If σ is unit of Pisot type, then the stair of any periodic pointremains at a finite distance from the expanding line.

Proof Node in the stair : P(u0 . . . uK ).

I Embrace u0 . . . uk between σd(u0 . . . ul) and σd(u0 . . . ul+1)

I There exists a part P0 of σ(ul+1) such that

u0 . . . uk = σd(u0 . . . ul)P0

I Iterate this decompositionu0 . . . uk = σdN(PN)σd(N−1)(PN1) . . . σd(P1)P0

I LinearizeπP(u0 . . . uk) = hdNP(PN) + hd(N−1)P(PN1) · · ·+ hσdP(P1) + P(P0)

I h is a contraction and the pk ’s are in finite number

The projections of the stair along the expanding line are bounded

Distance property





u0 . . . uk = σd(u0 . . . ul)P0





Distance property





u0 . . . uk = σd(u0 . . . ul)P0





Rauzy fractals

Zoom on the remaining part...

Rauzy fractalProject the nodes of the stair and take the closure.

T := {π(P(u0 . . . uN−1)) | N ∈ N}.

Subtiles Look at the last edge to be read.T (i) := {π (P(u0 . . . uk−1)) | N ∈ N, uN = i}.

Rauzy fractals

Zoom on the remaining part...

Rauzy fractalProject the nodes of the stair and take the closure.

T := {π(P(u0 . . . uN−1)) | N ∈ N}.

Subtiles Look at the last edge to be read.T (i) := {π (P(u0 . . . uk−1)) | N ∈ N, uN = i}.

Decomposition

σ(1) = 12, σ(2) = 13, σ(3) = 1.

h is a contracting similarity of angle ' 2/3

I The letter 1 appears in the images of 1, 2 and3, with no prefix behind.

T (1) = hT (1) ∪ hT (2) ∪ hT (3)

I The letter 2 appear only in the image of 1,behind a prefix 1.

T (2) = hT (1) + πP(1)


T (3) = hT (2) + πP(1)

General decomposition rule

T (i) =[

σ(j)=pis

hT (j) + πP(p).

Decomposition

σ(1) = 12, σ(2) = 13, σ(3) = 1.

h is a contracting similarity of angle ' 2/3

I The letter 1 appears in the images of 1, 2 and3, with no prefix behind.

T (1) = hT (1) ∪ hT (2) ∪ hT (3)


T (2) = hT (1) + πP(1)


T (3) = hT (2) + πP(1)

General decomposition rule

T (i) =[

σ(j)=pis

hT (j) + πP(p).

Measure of tiles

Theorem The measures of the subtiles are proportional to the expandingeigenvector of the incidence matrix.

Proof

I Decomposition rule

T (i) =[

σ(j)=pis

hT (j) + πP(p).

I h is a contraction with ratio 1/β.

µ(T (i)) ≤ 1/βP

σ(j)=pis µ(T (j)) (µ(T (i)))i ≤ 1/βM(µ(T (i)))i

I Perron-Frobenius theorem. Let M a matrix with positive entries and β beits dominant eigenvalue. Let X be a vector with positive entries. Then wehave MX ≤ βX. The equality holds only when X is a dominanteigenvector of M.

Measure of tiles

Theorem The measures of the subtiles are proportional to the expandingeigenvector of the incidence matrix.

Proof

I Decomposition rule

T (i) =[

σ(j)=pis

hT (j) + πP(p).

I h is a contraction with ratio 1/β.

µ(T (i)) ≤ 1/βP

σ(j)=pis µ(T (j)) (µ(T (i)))i ≤ 1/βM(µ(T (i)))i

I Perron-Frobenius theorem. Let M a matrix with positive entries and β beits dominant eigenvalue. Let X be a vector with positive entries. Then wehave MX ≤ βX. The equality holds only when X is a dominanteigenvector of M.

GIFS : telling which tile to put in the tile

T (i) =[

σ(j)=pis

hT (j) + πP(p).

Graph directed iterated function system (G , {τe}e∈E )

I Finite directed graph G with no stranding vertices

I With each edge e of the graph, is associated a contractive mappingτe : Rn → Rn.

GIFS attractors unique compact sets Ki such that

Ki =S

ie−→j

τe(Kj)

Prefix-suffix graph : defined naturally from the decomposition formula

i(p,i,s)−−−→ j iff σ(j) = pis

GIFS : telling which tile to put in the tile

T (i) =[

σ(j)=pis

hT (j) + πP(p).

Graph directed iterated function system (G , {τe}e∈E )

I Finite directed graph G with no stranding vertices

I With each edge e of the graph, is associated a contractive mappingτe : Rn → Rn.

GIFS attractors unique compact sets Ki such that

Ki =S

ie−→j

τe(Kj)

Prefix-suffix graph : defined naturally from the decomposition formula

i(p,i,s)−−−→ j iff σ(j) = pis

GIFS property

Theorem(Arnoux&Ito’01, Sirvent&Wang’02) Let σ be a primitive unit Pisotsubstitution over the alphabet A of n letters.The subtiles of T are solutions of the GIFS :

∀i ∈ A, T (i) =[j∈A,

i(p,i,s)−−−→j

hT (j) + πP(p).

The pieces are disjoint in the decomposition of each subtile

Self-similar ? h must have eigenvalues with the same modulus.

First consequence : dependency to the fixed point ?

Theorem A Rauzy fractal is compact and it is independant from the periodicpoint that was chosen.

Proof Same GIFS equation whenever the periodic point.

Second consequence : condition for disjointness

Are the subtiles disjoint ?

Strong coincidence condition For every pair of letters (j1, j2) ∈ A2, we shallcheck that σk(j1) = p1is1 and σk(j2) = p2is2 with P(p1) = P(p2) orP(s1) = P(s2).

Theorem(Arnoux&Ito’01) If σ satisfies the strong coincidence condition, thenthe subtiles T (i) of the central tile T have disjoint interiors.

Proof : hT (j1) + P(p1) and hT (j2) + P(p2) both appear in the GIFSdecomposition of T (i). Therefore they are disjoint.

Second consequence : condition for disjointness

Are the subtiles disjoint ?

Strong coincidence condition For every pair of letters (j1, j2) ∈ A2, we shallcheck that σk(j1) = p1is1 and σk(j2) = p2is2 with P(p1) = P(p2) orP(s1) = P(s2).

Theorem(Arnoux&Ito’01) If σ satisfies the strong coincidence condition, thenthe subtiles T (i) of the central tile T have disjoint interiors.

Proof : hT (j1) + P(p1) and hT (j2) + P(p2) both appear in the GIFSdecomposition of T (i). Therefore they are disjoint.

Inflating the fractal ?

We have a Rauzy fractal and a decomposition rule.

Can we produce a tiling from these pieces ?

Formal definition

Multiple tilings Let Ki , i ∈ A be a finite collection of compact sets of aEuclidean space H.

A multiple tiling of the space H by the compact sets Ki is given by atranslation set Γ ⊂ H× A such that

(A) Covering property

H =[

(γ,i)∈Γ

Ki + γ

(B) Uniform covering property almost all points in H are covered exactly ptimes for some positive integer p.

(C) Local “sparsity” Each compact subset of H intersects a finite number oftiles.

Tiling property If p = 1, then the multiple tiling is called a tiling.

How to define location of tiles ?

Face of type i located in x orthogonal to the i-th axis of a translate of theunit cube located at x.

Stepped hyperplane (Arnoux&Berthe&Ito’02) Union of faces of unit cubeswhich cross the contracting plane

x is above Hc and x− ei is below Hc .


Translation set Γ ⊂ H|{z}locationof a tile

× A|{z}name ofthe tileto draw

?

DefinitionThe Self-replicating translation set is the projection of the stepped surface onthe contracting plane.

Γsr = {[γ, i∗] ∈ π(Zn)×A | γ = π(x), x ∈ Zn| {z }projection on Hc

, x above Hc , x− ei below Hc| {z }stepped surface

}.


Translation set Γ ⊂ H|{z}locationof a tile

× A|{z}name ofthe tileto draw

?

DefinitionThe Self-replicating translation set is the projection of the stepped surface onthe contracting plane.

Γsr = {[γ, i∗] ∈ π(Zn)×A | γ = π(x), x ∈ Zn| {z }projection on Hc

, x above Hc , x− ei below Hc| {z }stepped surface

}.

From covering to Rauzy fractals

Covering property The tiles T (i) + γ for all [γ, i ] in the translation set coverthe full plane Rn−1.

Proof

I Zn is covered by translated copies of the stair projection along the steppedsurface.

I From Kronecker’s theorem, every point of the plane is approximated byprojection of points in Zn.

Application to Rauzy fractal The Rauzy fractal has a nonempty interior.

From covering to Rauzy fractals

Covering property The tiles T (i) + γ for all [γ, i ] in the translation set coverthe full plane Rn−1.

Proof

I Zn is covered by translated copies of the stair projection along the steppedsurface.

I From Kronecker’s theorem, every point of the plane is approximated byprojection of points in Zn.

Application to Rauzy fractal The Rauzy fractal has a nonempty interior.

Self-replicating ?

Substitution on faces E∗1 is defined on the subsets of π(Zn)× A :

E1∗{[γ, i∗]} =

[j∈A, σ(j)=pis

{[h−1(γ + πP(p)), j∗]},

Similar to the rules of decomposition of the Rauzy fractal !

h−1T (1) = T (1) ∪ T (2) ∪ T (3)h−1T (2) = T (1) + π(e3)h−1T (3) = T (2) + π(e3)

[0, 1∗] 7→ [0, 1∗] ∪ [0, 2∗] ∪ [0, 3∗][0, 2∗] 7→ [π(e3), 1

∗][0, 3∗] 7→ [π(e3), 2

∗]

→ → →

The translation set is a fixed point(Arnoux&Ito’01) E∗1 (Γsr ) = Γsr

The substitution E∗1 maps two different tips on disjoint patches

Self-replicating ?

Substitution on faces E∗1 is defined on the subsets of π(Zn)× A :

E1∗{[γ, i∗]} =

[j∈A, σ(j)=pis

{[h−1(γ + πP(p)), j∗]},

Similar to the rules of decomposition of the Rauzy fractal !

h−1T (1) = T (1) ∪ T (2) ∪ T (3)h−1T (2) = T (1) + π(e3)h−1T (3) = T (2) + π(e3)

[0, 1∗] 7→ [0, 1∗] ∪ [0, 2∗] ∪ [0, 3∗][0, 2∗] 7→ [π(e3), 1

∗][0, 3∗] 7→ [π(e3), 2

∗]

→ → →

The translation set is a fixed point(Arnoux&Ito’01) E∗1 (Γsr ) = Γsr

The substitution E∗1 maps two different tips on disjoint patches

Application to the boundary of the Rauzy fractal

Proposition The boundary of the Rauzy fractal has a zero measure and theRauzy fractal is the closure of its interior.

I From the GIFS equation, either all subtiles have zero measure boundariesor none.

I Play with the Perron-Frobenius formula

I Deeply use that patches in iterations of E∗1 are disjoint and correspondingRauzy pieces are measure disjoint.

Consequence : multiple tiling

Proposition The self-replicating translation set Γsr is repetitive and satisfies

Hc =[

[γ,i∗]∈Γsr

(T (i) + γ)

where almost all points of Hc are covered p times (p is a positive integer).

In other words : we have defined aSelf-replicating multiple tiling

Proof for multiple tiling

Locally finite deduced from compactness.

Repetitivity Kronecker’s theorem + relatively dense.

Multiple tiling Boundary with zero measure + repetitivity.

Conclusion : quite general proofs !

Framework

I GIFS equation with a primitive graph

I information on location of pieces (=properties of E∗1 )

Conclusions

I Compact set with nonempty interior.

I Closure of its interior.

I Boundary with zero measure.

I Covering with a almost constant degree.

How valid are these proofs for all GIFS’s ? ?

No general topological properties

Zero inner point, connectivity, disklikeness are not always satisfied.

Relations to other talks in the conference

I Arnoux : Geometric representation of a symbolic dynamical system.

I Adamczewski : Rauzy fractal represent integers in non-real expansionsbases.

I Berthe : Example of local rules to generate combinatorial tilings.

I Akiyama : The tiling property is very close to the Pisot conjecture.

I Harriss, Bressaud : can we exhibit matching rules for the Rauzy tiling ?

I Sellami, Jolivet : properties of families of Rauzy fractals....

I Other talks : good chance that Rauzy fractals are hidden somewhere

To be continued... Rauzy fractal represent self-induced actions, thereforethey lie in many mathematical objects

Documents

Introduction to Rauzy fractals - IRISA · Introduction to Rauzy fractals Guanghzhou, 2010. Several questions I Geometry Eﬃcient way to produce a pixelized line? A pixelized plane?