General theory of unitary Pisot substitution · Applications Low complexity and sturmian sequences Discrete lines in the plane Generalizations: Hyperplanes, lines and Pisot substitutions

Background: Low complexityFree groups automorphisms and tilings: An example

Applications

Low complexity and sturmian sequencesDiscrete lines in the planeGeneralizations: Hyperplanes, lines and Pisot substitutionsGeneralizations

General theory of unitary Pisot substitution

I The case of Pisot substitution in now well understood.

I Crowning the work of several people, a recent paper (Bargeand Kwapisz) shows that, if the substitution satisfies theso-called ”strong coincidence condition”, the structure of theassociated system is completely understood:

I discrete lines and planes

I Rauzy fractal

I symbolic dynamics

I All known Pisot substitutions satisfy the strong coincidencecondition

Pierre Arnoux Pavages, substitutions et automorphismes de groupe


Applications






I Rauzy fractal

I symbolic dynamics




Applications






I Rauzy fractal

I symbolic dynamics




Applications






I Rauzy fractal

I symbolic dynamics




Applications






I Rauzy fractal

I symbolic dynamics




Applications






I Rauzy fractal

I symbolic dynamics




Applications


Discrete 2-planes in R4

??



Applications


Discrete 2-planes in R4

??



Applications


Possible generalizations

Unitary Pisot substitutions

I Non-unitary substitutions:p-adic component (Siegel)

I From substitutions to Pisotautomorphisms of free groups:recent results(A-Berthe-Hilion-Siegel)

I Non Pisot case: the firstexample



Applications









Applications









Applications









Applications


A remark on entropy

I The entropy of a toral automorphism is the sum of thelogarithms of the eigenvalues larger than 1.

I The entropy of the subshift of finite type associated to asubstitution is the logarithm of the largest eigenvalue.

I These coincide only in the Pisot case

I This hints to the necessity of considering exterior powers ofthe matrix A



Applications


A remark on entropy







Applications


A remark on entropy







Applications


A remark on entropy







Applications

NotationsExtensions of free group automorphismsPlane tiling

A free group automorphism

σ automorphism of the free group F4:

1 7→ 2

2 7→ 3

3 7→ 4

4 7→ 41−1

Matrix M =

0 0 0 −11 0 0 00 1 0 00 0 1 1



Applications



Characteristic polynomial X 4 − X 3 + 1Eigenvalues 1.01891± 0.602565i , −0.518913± 0.66661iNon Pisot!Expanding plane Pe ≡ CContracting plane Pc ≡ CAssociated projections πe , πc



Applications






Applications






Applications






Applications


Projections of the canonical basis

-0.2 0.2 0.4 0.6 0.8 1 1.2

0.25

0.5

0.75

1

1.25

1.5

!e!e1"

!e!e2"

!e!e3"!e!e4"

-0.5-0.25 0.25 0.5 0.75 1

-0.75

-0.5

-0.25

0.25

0.5

0.75

!c!e1"

!c!e2"

!c!e3"

!c!e4"



Applications


Geometric extensions of free group automorphisms I

I Words as discrete lines in R4

I σ acts naturally on discrete lines in R4

I Map E1(σ) defined on space G1 of weighted sum of discretelines

I This is still 1-dimensional!



Applications









Applications









Applications









Applications


Geometric extensions of free group automorphisms II

I i : s 7→ i(s) and j : t 7→ j(t)segments; define the orientedface i ∧ j as the oriented surface(s, t) 7→ i(s) + j(t).

I σ acts in a natural way on facesby taking i ∧ j toE1(σ)(i) ∧ E1(σ)(j)

I Map E2(σ) defined on space G2

of weighted sum of discretefaces

I The matrix of E2(σ) is positive!

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 2)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 3)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 4)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 3)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4)

1



Applications








2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (E2(σ)(0, 2 ∧ 4))

1



Applications








2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 2)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 3)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 3)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4) +πe (−e1 − e4, 1 ∧ 2)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 3)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4) +πe (−e1 + e4, 1 ∧ 3)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (−e1 + e4, 1 ∧ 4)

1



Applications








2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 2)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 3)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 3)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4) +πe (−e1 − e4, 1 ∧ 2)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 3)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4) +πe (−e1 + e4, 1 ∧ 3)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (−e1 + e4, 1 ∧ 4)

1



Applications


Geometric extensions III: formalism

I We define the space of formal finite sums of weighted 2-faces(x, i ∧ j), with x ∈ Z4.

I The 2-dimensional extension E2(σ) is defined on this space by:E2(σ)(x, i ∧ j) :=∑li

m=1

∑ljn=1

(A(x) + f (P

(i)m ) + f (P

(j)m ), W

(i)m ∧W

(j)n

)I The matrix associated to E2(σ) is the exterior square of M:

0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0



Applications





m=1

∑ljn=1

(A(x) + f (P

(i)m ) + f (P

(j)m ), W

(i)m ∧W

(j)n


0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0



Applications





m=1

∑ljn=1

(A(x) + f (P

(i)m ) + f (P

(j)m ), W

(i)m ∧W

(j)n


0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0



Applications



I The matrix associated to E2(σ) is the exterior square of M:

0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0

I This matrix is positive!

I This is the correct generalization of the notion of substitution(seen as a positive free group endomorphism) in the non-Pisotcase



Applications




0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0





Applications




0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0





Applications


A new substitution tiling of the plane

I by projection πe :

I A substitution rule on theexpanding plane

I That generates asubstitution polygonaltiling



Applications






2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 2)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 3)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 3)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 1 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4) +πe (−e1 − e4, 1 ∧ 2)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 3)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 2 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4) +πe (−e1 + e4, 1 ∧ 3)

2

2

Πee1

Πee2

Πee3

Πee4

πe (0, 3 ∧ 4)

E2(σ)→

2

2

Πee1

Πee2

Πee3

Πee4

πe (−e1 + e4, 1 ∧ 4)

1



Applications






-4 4

-4

49 step

Πee1Πee2Πee3

Πee4

-4 4

-4

412 step

Πee1Πee2Πee3

Πee4

-4 4

-4

421 step

Πee1Πee2Πee3

Πee4

-4 4

-4

46 step

Πee1Πee2Πee3

Πee4

-4 4

-4

47 step

Πee1

Πee2Πee3

Πee4

-4 4

-4

48 step

Πee1Πee2Πee3

Πee4

-4 4

-4

43 step

Πee1Πee2Πee3

Πee4

-4 4

-4

44 step

Πee1

Πee2Πee3

Πee4

-4 4

-4

45 step

Πee1

Πee2Πee3

Πee4

-4 4

-4

40 step

Πee1Πee2Πee3

Πee4

-4 4

-4

41 step

Πee1Πee2Πee3

Πee4

-4 4

-4

42 step

Πee1Πee2Πee3

Πee4



Applications




Applications


An exact substitution tiling

I By replacing each face by the limit of its renormalization, oneobtains an exactly self-similar tiling, with fractal tiles.

I The fractal tiles are solutions of a graph-directed IFS given bythe substitution rule.



Applications


An exact substitution tiling

I By replacing each face by the limit of its renormalization, oneobtains an exactly self-similar tiling, with fractal tiles.

I The fractal tiles are solutions of a graph-directed IFS given bythe substitution rule.



Applications


5/09/06 1:40Nautilus_fract_patch_03.gif 932x738 pixels

Page 1 sur 1file://localhost/Users/pierrearnoux/Desktop/Nautilus_fract_patch_03.gif



Applications




Applications




Applications


A discrete surface in R4

I The tiling lifts to a unique discrete surface in R4

I Discrete approximation of the expanding plane



Applications


A discrete surface in R4

I The tiling lifts to a unique discrete surface in R4

I Discrete approximation of the expanding plane



Applications


The discrete surface



Applications


Duality

We can do exactly the same for the contracting plane:Define the dual map E 2(σ).It is also positive.Get dual substitution tiling and a dual self-similar tiling.



Applications




Applications




Applications


5/09/06 1:41Conch_fract_patch_03.gif 926x734 pixels

Page 1 sur 1file://localhost/Users/pierrearnoux/Desktop/Conch_fract_patch_03.gif



Applications




Applications

Quasi-crystals and Rauzy fractalsA complex number systemSymbolic dynamicsOpen problems

Cut-and-project tiling

I The fractal tiles of the expanding tiling are solution of a GIFS.

I The vertices of the contracting tiling are solution of a GIFS.

I After projection on the expanding space, we can observe avery much curious phenomenon:

I The second IFS is the opposite of the first!

I These polygonal tilings are cut-and-project tilings.



Applications










Applications










Applications










Applications










Applications


Generalized Rauzy fractals

The window for the tiling of the expanding plane is the contractingRauzy fractal X c = ∪X c

i∧j .It can be obtained by projecting on the contracting plane thevertices of the discrete approximation to the expanding plane.It can also be obtained by renormalization of the projection of theimage of a patch of faces by the action of the dual map:

X c = lim M−n(πc(E∗2 (σ)n(U)))

the same property is true for the expanding Rauzy fractal.



Applications


The window

-1 -0.5 0.5 1

-1

-0.5

0.5 Πce1

Πce2

Πce3

Πce4

Sc (2 ∧ 1)

-1 -0.5 0.5 1

-1

-0.5

0.5 Πce1

Πce2

Πce3

Πce4

Sc (1 ∧ 3)

-1 -0.5 0.5 1

-1

-0.5

0.5 Πce1

Πce2

Πce3

Πce4

Sc (4 ∧ 1)

-1 -0.5 0.5 1

-1

-0.5

0.5 Πce1

Πce2

Πce3

Πce4

Sc (3 ∧ 2)

-1 -0.5 0.5 1

-1

-0.5

0.5 Πce1

Πce2

Πce3

Πce4

Sc (2 ∧ 4)

-1 -0.5 0.5 1

-1

-0.5

0.5 Πce1

Πce2

Πce3

Πce4

Sc (4 ∧ 3)

-1 -0.5 0.5 1

-1

-0.5

0.5 Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

⋃i∧j∈Vc

Sc (i ∧ j)

Figure 21: Sc (i ∧ j)

29



Applications


renormalization and projection

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (1)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (2)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (2)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (3)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (3)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (4)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (4)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (5)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (6)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (6)c

-5 -4-3.5-3 2.5-2-1.5-1-0.50.511.522.533.54 5

-6

-5

-4

-3

-2-1.5

-1-0.5

0.51

1.52

2.53

4

5

E29 (θ)U (1)

c

-5 -4-3.5-3 2.5-2-1.5-1-0.50.511.522.533.54 5

-6

-5

-4

-3

-2-1.5

-1-0.5

0.51

1.52

2.53

4

5

E28 (θ)U (8)

c

-4.5-4-3.5-3 2.5-2-1.5-1-0.5 0.511.522.533.5

-3.5-3

-2.5-2

-1.5-1

-0.5

0.51

1.52

2.53

3.5

E27 (θ)U (7)

c

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

6

E210 (θ)U (2)

c

32



Applications



-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

E211 (θ)U (3)

c

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

-7

-6

-5-4

-3-2

-1

1

2

3

4

5

6

E212 (θ)U (4)

c

-8 7 8

-8

78910

E213 (θ)U (5)

c

Figure 24:

-1 -0.5 0.5 1x

-1

-0.5

0.5

1y

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4

Πce1

Πce2

Πce3

Πce4! -1 -0.5 0.5 1

x

-1

-0.5

0.5

1y

-1 -0.5 0.5 1x

-1

-0.5

0.5

1y

Figure 25: 2 hexagonal patches of U (6)c

33



Applications



-1 -0.5 0.5 1

-1

-0.5

0.5

1

Πce1

Πce2

Πce3

Πce4

Xc (2 ∧ 1)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Πce1

Πce2

Πce3

Πce4

Xc (1 ∧ 3)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Πce1

Πce2

Πce3

Πce4

Xc (4 ∧ 1)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Πce1

Πce2

Πce3

Πce4

Xc (3 ∧ 2)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Πce1

Πce2

Πce3

Πce4

Xc (2 ∧ 4)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Πce1

Πce2

Πce3

Πce4

Xc (4 ∧ 3)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Πce1

Πce2

Πce3

Πce4

⋃i∧j∈Vc

Xc (i ∧ 1)

Figure 10: Xc (i ∧ j)

12



Applications


The other window

-1 -0.5 0.5

-0.5

0.5Πee1

Πee2

Πee3

Πee4

Se (ϕ (4 ∧ 3))

-1 -0.5 0.5

-0.5

0.5Πee1

Πee2

Πee3

Πee4

Se (ϕ (2 ∧ 4))

-1 -0.5 0.5

-0.5

0.5Πee1

Πee2

Πee3

Πee4

Se (ϕ (3 ∧ 2))

-1 -0.5 0.5

-0.5

0.5Πee1

Πee2

Πee3

Πee4

Se (ϕ (4 ∧ 1))

-1 -0.5 0.5

-0.5

0.5Πee1

Πee2

Πee3

Πee4

Se (ϕ (1 ∧ 3))

-1 -0.5 0.5

-0.5

0.5Πee1

Πee2

Πee3

Πee4

Se (ϕ (2 ∧ 1))

-1 -0.5 0.5

-0.5

0.5Πee1

Πee2

Πee3

Πee4

Πee1

Πee2

Πee3

Πee4

Πee1

Πee2

Πee3

Πee4

Πee1

Πee2

Πee3

Πee4

Πee1

Πee2

Πee3

Πee4

Πee1

Πee2

Πee3

Πee4

Πee1

Πee2

Πee3

Πee4

⋃i∧j∈Ve

Se (ϕ (i ∧ j))

Figure 20: Se (ϕ (i ∧ j))

27



Applications


Arithmetics: complex β-expansion

There is an associated complex β-expansion, whose domain is thegeneralized Rauzy fractal.In this expansion, any complex number can be written in a(almost) unique way:

∞∑n=N0

εnµn

where µ is the small complex eigenvalue of the matrix M, and εnbelongs to a finite set {0, f1, f2} and satisfies a Markov condition,related to the GIFS.



Applications


Symbolic dynamics

By taking the product of the corresponding Rauzy fractals:

X ci∧j × X e

k∧l

one obtains a partition of the torus T4.This partition gives a symbolic dynamics for the action of thematrix A which is a subshift of finite type.This is the first known explicit Markov partition for a non-Pisotirreducible automorphism of the torus.It is the natural extension of the β-expansion.



Applications


Transversal dynamics

I Study the transversal flow of these tilings.

I Find a good symbolic dynamics for this R2-action

I Meaning unclear:

I pseudo-group of translations?



Applications





I Meaning unclear:




Applications





I Meaning unclear:




Applications





I Meaning unclear:




Applications



-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (1)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (2)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (2)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (3)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (3)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (4)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (4)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (5)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

U (6)c

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

E2 (θ)U (6)c

-5 -4-3.5-3 2.5-2-1.5-1-0.50.511.522.533.54 5

-6

-5

-4

-3

-2-1.5

-1-0.5

0.51

1.52

2.53

4

5

E29 (θ)U (1)

c

-5 -4-3.5-3 2.5-2-1.5-1-0.50.511.522.533.54 5

-6

-5

-4

-3

-2-1.5

-1-0.5

0.51

1.52

2.53

4

5

E28 (θ)U (8)

c

-4.5-4-3.5-3 2.5-2-1.5-1-0.5 0.511.522.533.5

-3.5-3

-2.5-2

-1.5-1

-0.5

0.51

1.52

2.53

3.5

E27 (θ)U (7)

c

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

6

E210 (θ)U (2)

c

32



Applications


Classification problems

I Which automorphisms have positive E2(σ)?

I Do all of them define a fractal tiling?

I What of those with positive matrix A: can we get rid ofcancellations?

I In the general case, can we obtain results by grouping tiles(blocking)?



Applications









Applications









Applications









Applications


Arithmetic problems

I Can we get completely real examples?

I What can be said on the number system?

I Can we obtain good approximation in this way?

I Can we go beyond the algebraic case (generalized continuedfractions)?



Applications


Arithmetic problems







Applications


Arithmetic problems







Applications


Arithmetic problems







Applications


Word problems

I Can we define a good structure of 2-dimensional wordscorresponding to the tilings (underlying lattice)?

I Can we define a reasonable notion of complexity?

I Can we define discrete dynamical systems (transversaldynamics)?

I Can we find an equivalent of sturmian sequences in thissetting?



Applications


Word problems







Applications


Word problems







Applications


Word problems






Documents

General theory of unitary Pisot substitution · Applications Low complexity and sturmian sequences Discrete lines in the plane Generalizations: Hyperplanes, lines and Pisot substitutions