Pavages, substitutions et automorphismes de groupe ... · Pavages, substitutions et automorphismes...

Background: Low complexityFree groups automorphisms and tilings: An example

Applications

Pavages, substitutions et automorphismes de groupeTilings, substitutions and group automorphisms

Pierre Arnoux

15 mars 2007, Montreal

Joint work with

Maki FurukadoEdmund O. Harriss

Shunji Ito

Pierre Arnoux Pavages, substitutions et automorphismes de groupe

Applications

Aim of the talkWe hint at a new way to generalize substitutive words, beyond thePisot case, and going to 2-dimensional words.After giving some background, we consider a particular example ofan automorphism of the free group on 4 generators.We present an explicit way to build:

I A discrete approximation for the stable and unstable planesfor the related matrix

I A substitution polygonal tiling of the stable and unstablespaces

I A related numeration system

I A self-similar tiling with fractal tiles of the stable and unstablespaces

The polygonal tilings can be seen as 2-dimensional words of lowcomplexity.

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

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

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

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

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these pictures, and many other tilings, can be found on the site:

http://saturn.math.uni-bielefeld.de/tilings/index

maintained by E. Harriss and D. Frettloeh

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

Language complexity

One can see the language theory as divided in 2 domains:

I High complexity (positive entropy, most rational languages)

I Low complexity (substitutions...)

I These domains are linked (adic constructions)

I The emblematic example of low complexity is given bysturmian words

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Language complexity

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Language complexity

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Language complexity

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Language complexity

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Sturmian words

I Smallest possible complexity for an aperiodic word:p(n) = n + 1.

I Many remarkable properties

I The substitutive case is of special interest

I A nice way to understand their properties is the geometricrepresentation

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Sturmian words

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Sturmian words

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Sturmian words

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Discrete lines in the plane

Discrete lines:

I Sturmian sequences(Hedlund-Morse)

I Sliding squares

I Cut-and project method:sturmian tilings

I Renormalization andcontinued fractions

I The quadratic (orsubstitutive) case

An example: The Fibonacci sequence

01001010010010100101 . . .

fixed point of the substitution

0 7→ 01

1 7→ 0

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Discrete lines:

I Sliding squares

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Discrete lines:

I Sliding squares

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Discrete lines:

I Sliding squares

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Discrete lines:

I Sliding squares

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Discrete lines:

I Sliding squares

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Sturmian sequences and geometry

I For a given direction, thespace of pointed sturmiantilings of the line is a torus

I This space has a naturalpartition in rectangles

I If the direction isquadratic, renormalizationgives rise to a torusendomorphism

I The natural partition is aMarkov partition(Well-known sinceAdler-Weiss)

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Sturmian sequences: the dual viewpoint

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Discrete hyperplanes

I Codimension 1

I Sliding cubes

I Lozenge tilings

I Generalized continuedfractions (A-Berthe-Ito)

I 2-dimensional words

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I Codimension 1

I Sliding cubes

I Lozenge tilings

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I Codimension 1

I Sliding cubes

I Lozenge tilings

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I Codimension 1

I Sliding cubes

I Lozenge tilings

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I Codimension 1

I Sliding cubes

I Lozenge tilings

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discrete lines in R3

I No canonical choice for the window

I Sliding cubes are not good (high complexity)

I Possible in the algebraic case:

I breakthrough in 1982 (G. Rauzy, Nombres algebriques etsubstitutions)

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Unitary Pisot substitutions

I Fixed point as discreteline in space

I Projection along theeigenline

I Compact window=Rauzyfractal

I Dual substitution: tilingof the contracting plane(A-Ito)

Substitution:

1 7→ 12

2 7→ 13

3 7→ 1

Fixed point:12131211213...

Applications

Formalism I: geometric models of words and weightedpaths

For W ∈ A∗:|W |i = number of occurences of i in W .Abelianization map (Parykh map)f : A∗ → Zd .

f (W ) = (|W |1, |W |2, . . . , |W |d)

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f (W ) = (|W |1, |W |2, . . . , |W |d)

Example: f (21121211) = (5, 3).

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f (W ) = (|W |1, |W |2, . . . , |W |d)

for x ∈ Zd , i ∈ A:(x , i) represents the segment (x , x + ei)We denote by G1 the set of functions fromZd ×A to R with compact supportG1 is the set of formal sums of weightedpaths

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f (W ) = (|W |1, |W |2, . . . , |W |d)

This weighted path is((0, 0), 1) + ((1, 0), 1) + ((2, 0), 2),representation of the word 112

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Formalism II: the geometric model for the substitution

σ substitution on A alphabet of cardinal dFor i ∈ A:

σ(i) =W (i)

=W(i)1 . . .W

=P(i)k W

(i)k S

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σ substitution on A alphabet of cardinal dFor i ∈ A:

σ(i) =W (i)

=W(i)1 . . .W

=P(i)k W

(i)k S

Abelianization of σ: matrix A such that:f (σ(W )) = A.f (W ).A is given by: ai,j = number of occurences of i in σ(j).

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We want to associate to the substitution σa geometric map E1(σ)

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σ(i) =W (i)

=W(i)1 . . .W

=P(i)k W

(i)k S

1-dimensional extension E1(σ): To eachsegment of type i, we associate the pathσ(i)

Applications

σ(i) =W (i)

=W(i)1 . . .W

=P(i)k W

(i)k S

1-dimensional extension E1(σ): To eachsegment of type i, we associate the pathσ(i) Example:

1 7→ 112

2 7→ 12

Applications

σ(i) =W (i)

=W(i)1 . . .W

=P(i)k W

(i)k S

1 7→ 112

2 7→ 12

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σ(i) =W (i)

=W(i)1 . . .W

=P(i)k W

(i)k S

1 7→ 112

2 7→ 12

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σ(i) =W (i)

=W(i)1 . . .W

=P(i)k W

(i)k S

1 7→ 112

2 7→ 12

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σ(i) =W (i)

=W(i)1 . . .W

=P(i)k W

(i)k S

1-dimensional extension E1(σ): formaldefinition:E1(σ)(x, i) =∑li

(A(x) + f (P

(i)n ),W

The shift of the origin, from x to A.x , isneeded to obtain the connexity of theimage of a connected path.

Applications

Formalism III: the dual map for the substitution

Since E1(σ) is a linear map on a vector space, we can defineformally its dual map.If the matrix A is invertible, this dual map is easily computed:

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Since E1(σ) is a linear map on a vector space, we can defineformally its dual map.If the matrix A is invertible, this dual map is easily computed:

E ∗1 (σ)(x, i∗) =∑

W(j)n =i

(A−1

(x− f (P

(j)n )

), j∗

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We can give a geometricinterpretation of this map,representing (x, i∗) by the upper faceof the unit cube at x orthogonal to(x, i).Example:1 7→ 122 7→ 133 7→ 1

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We can give a geometricinterpretation of this map,representing (x, i∗) by the upper faceof the unit cube at x orthogonal to(x, i).Example:1 7→ 122 7→ 133 7→ 1In the case of sturmian substitutions,this turns out to be exactly the dualsubstitution defined yesterday in thelecture of V. Berthe

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