introduction to rauzy fractals - irisa · introduction to rauzy fractals guanghzhou, 2010. several...
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Introduction to Rauzy fractals
Guanghzhou, 2010
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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
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Common reformulation
Do fractal shapes generate a tiling of a plane ?
Purpose of the lecture : Origin of the main properties of this fractal shape
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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
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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
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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
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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
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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 !
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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 )
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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 )
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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.
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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.
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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 ).
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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 ).
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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
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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
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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
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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}.
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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}.
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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).
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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).
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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.
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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.
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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
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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
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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.
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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.
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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.
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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.
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Inflating the fractal ?
We have a Rauzy fractal and a decomposition rule.
Can we produce a tiling from these pieces ?
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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.
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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 .
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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
}.
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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
}.
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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.
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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.
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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
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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
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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.
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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
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Proof for multiple tiling
Locally finite deduced from compactness.
Repetitivity Kronecker’s theorem + relatively dense.
Multiple tiling Boundary with zero measure + repetitivity.
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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 ? ?
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No general topological properties
Zero inner point, connectivity, disklikeness are not always satisfied.
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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