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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
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
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
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)
I The letter 3 appear only in the image of 2,behind a prefix 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)
I The letter 2 appear only in the image of 1,behind a prefix 1.
T (2) = hT (1) + πP(1)
I The letter 3 appear only in the image of 2,behind a prefix 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 .
How to define location of tiles ?
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
}.
How to define location of tiles ?
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