Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at...

Tree substitutions and Rauzy fractals

Thierry Coulbois

with Milton Minervino

December 5, 2017

Tree-bonacci

� : a 7! ab, b 7! ac , c 7! a

Tree-bonacci

� : a 7! ac , b 7! ab, c 7! b

Tree-bonacci

� : a 7! ab, b 7! c , c 7! a

Summary

• Substitutive subshifts

? Singular words, asymptotic pairs, infinite special words? Share-a-half relation and its quotient? Trees associated with free group automorphisms

• Self-similarity

? Prefix-su�x expansion and automaton? Self-similarity of the shift and of the tree? Prefix-su�x expansion of singular words

• Tree substitution

• Rauzy fractal

? Self-similarity of the Rauzy fractal? Global picture? Tree substitution embedded in the Rauzy fractal? Pruning, covering

• Interval exchange on the circle

? Contour of the tree? Arnoux-Yoccoz example

Attracting shift

Tribonacci substitution

� : a 7! ab, b 7! ac , c 7! a

�1(a) = abacabaabacababacabaabac · · ·

Attracting shift

X� = set of bi-infinite indexed words in AZ

whose factors are factors of �n(i) for i 2 A and n � 1.Cantor set.

Shift map S : X� ! X� (shifting the index).

�-invariant: � : X� ! X�

Homeomorphisms.

Singular words

Attracting shift

·abacabaabacabab · · · = �1(a)

wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac

a 7! abb 7! acc 7! a

7! abac 7! abacaba7! aba 7! abacab7! ab 7! abac

wa = limn!1 �3n(a) · �n(a)

wb = limn!1 �3n(b) · �n(a)

wc = limn!1 �3n(c) · �n(a)

wa, wb and wc are singular words. Their right part is an infinite leftspecial word. They form asymptotic pairs.

Singular words

Attracting shift

wa = limn!1 �3n(a) · �n(a)

wb = limn!1 �3n(b) · �n(a)

wc = limn!1 �3n(c) · �n(a)

Singular words

Attracting shift

7! abac

7! abacaba7! aba 7! abacab7! ab 7! abac

wa = limn!1 �3n(a) · �n(a)

wb = limn!1 �3n(b) · �n(a)

wc = limn!1 �3n(c) · �n(a)

Singular words

Attracting shift

7! abac 7! abacaba

7! aba 7! abacab7! ab 7! abac

wa = limn!1 �3n(a) · �n(a)

wb = limn!1 �3n(b) · �n(a)

wc = limn!1 �3n(c) · �n(a)

Singular words

Attracting shift

wa = limn!1 �3n(a) · �n(a)

wb = limn!1 �3n(b) · �n(a)

wc = limn!1 �3n(c) · �n(a)

Singular words

Attracting shift

wa = limn!1 �3n(a) · �n(a)

wb = limn!1 �3n(b) · �n(a)

wc = limn!1 �3n(c) · �n(a)

Singular words

Attracting shift

wa = limn!1 �3n(a) · �n(a)

wb = limn!1 �3n(b) · �n(a)

wc = limn!1 �3n(c) · �n(a)

Singular words (continued)

One periodic Nielsen path of period 3:

�!abacaba·

abacab

bacabacaba

e�1(a) = · · · bacabaabacba·abacababacaba · · · = e�31(a) : w 0

bacabaabacaba · · · = e�31(b) : w 0b

cabaabacababa · · · = e�31(c) : w 0c

�̃ : a 7! ba, b 7! ca, c 7! a

�!abacaba·

abacab

bacabacaba

�̃ : a 7! ba, b 7! ca, c 7! a

�!abacaba·

abacab

bacabacaba

�̃ : a 7! ba, b 7! ca, c 7! a

�!abacaba·

abacab

bacabacaba

�̃ : a 7! ba, b 7! ca, c 7! a

Singular words and index

(4� 2) + (4� 2) = 4 = 2⇥ 3� 2

Theorem [Gaboriau-Jaeger-Levitt-Lustig]

� substitution and free group automorphism, fully irreducible (iwip).Then, the index of X� is at most 2N � 2.

(N = card(A)) fully irreducible (iwip) ) primitive.

Singular words and index

(4� 2) + (4� 2) = 4 = 2⇥ 3� 2

� substitution and free group automorphism, fully irreducible (iwip).Then, the index of X� is at most 2N � 2.

(N = card(A)) fully irreducible (iwip) ) primitive.

Share-a-half

� iwip substitution ) index of X� is at most 2N � 2.

If the index is maximal then � is parageometric.Tribonacci is parageometric. Many other examples [Leroy].

Share-a-half relation on bi-infinite words. ⇠ its transitive closure.

X�/ ⇠

Theorem [GJLL, LL, C-Hilion-Lustig, CH, Kapovich-L]

If � is iwip and parageometric then ⌦� is a tree.

⌦� ✓ T��1 : repelling tree

Share-a-half

X�/ ⇠

Share-a-half

X�/ ⇠ (devil staircase)

Share-a-half

⌦� = X�/ ⇠ (devil staircase)

Share-a-half

⌦� = X�/ ⇠ (devil staircase)

Share-a-half

Q : X� ! ⌦� = X�/ ⇠

Repelling tree

Q : X� ! ⌦� ✓ T��1

Repelling tree

T��1 is a metric space, with an action of the free group FA by isometriesand a contracting homothety H.

Q(S(Z )) = Z�10 Q(Z ) Q(�(Z )) = H(Q(Z ))

kukT��1 = limn!1

k��n(u)kA(��1)n

��1 expansion factor of ��1.

� : a 7! abb 7! acc 7! a

��1 : a 7! cb 7! c�1ac 7! c�1b

Self-similarity

Desubstitution [Mosse]

� primitive. Each w 2 X� can be uniquely written as

w = S |p|(�(w 0)), for some w 0 2 X�, p 2 A⇤

Prefix-su�x graph:

w0p,s � w 0

0 () �(w 00) = pw0s

(p,w0, s) 2 A⇤ ⇥ A⇥ A⇤ a

a, ✏

✏, b

✏, ca, ✏

✏, ✏

Iterating we have that each w 2 X� has an infinite desubstitution path:

� : X� ! P, w 7! � = w0p0,s0 � w 0

0p1,s1 � w 00

0p2,s2 � · · ·

� is onto, continuous and 1-to-1 except on the set of periodic points of �.

Self-similarity (continued)

Cylinders

X� bi-inifinite words with prefix-su�x expansion ending with �

Self-similar decomposition (� primitive)

X� =G

|�|=n

X��0 = S |p(�)|(�|�|(X�0))

Self-similarity of the repelling tree (� iwip parageometric)

⌦� = Q(X�) a subtree (i.e. connected)

⌦� =[

|�|=n

⌦� , #(⌦� \ ⌦�0) 1 singular words in X�

⌦��0 = p(�)�1H |�|(⌦�0)

Desubstitution of singular words

Theorem [Que↵elec, Holton-Zamboni]

• A primitive � has finitely many singular bi-infinite words (left-specialor right-special).

• Each singular word w 2 X� has an eventually periodic desubstitutionpath �.

wa,wb,wc : a ✏

w 0a: a a a b

✏ ✏ ✏

w 0b: b a a a b

a ✏ ✏ ✏

w 0c : c b a a a b

a a ✏ ✏ ✏

Abstract tree substitution

• Decomposition of ⌦� in subtrees ⌦�

• ⌦� and ⌦�0 translate of one-another if they start at the same letter

• #(⌦� \ ⌦�0) 1, gluing points given by singular words.

• Finitely many gluing points in each prototile ⌦a

⌦a 7! ⌦b

⌦c 7!

Initial patch

Iterating the tree substitution

Theorem 1 [C.-Minervino]

Let � be a primitive substitution on a finite alphabet A and aparageometric iwip automorphism of the free group FA.There exists a tree substitution ⌧ such that the iterations of ⌧ on theinitial tree renormalized by the ratio of the contracting homothetyconverge to the repelling tree of the automorphism �.We provide an algorithm to construct this tree substitution.

Tree substitution ⌧ [Jullian, Bressaud-Jullian]

• Prototiles (Wa)a2A: trees with gluing points

• Initial patch W =FWa/ ⇠ some gluing points are identified

• ⌧(Wa) union of copies of the prototiles with some gluing pointsidentified

• ⌧ maps gluing points of Wa to gluing points of the tiles of ⌧(Wa)

Twelfth iterate for Tribonacci (prunned)

That’s the Rauzy Fractal !?

Rauzy Fractal

• Incidence matrix M� and its characteristic polynomial ��

M� =⇣

1 1 11 0 00 1 0

⌘, ��(x) = x3 � x2 � x � 1

• � primitive () 9n, Mn� > 0

• dominant eigenvalue �� (expansion factor)

• positive Perron-Frobenius eigenvector u

• �� is a Pisot number, its Galois conjugates are < 1 in modulus

• � irreducible Pisot

? �� is irreducible and �� is Pisot? Contracting hyperplane Ec , RA = Ec � Ru

Rauzy Fractal and the map '

For any infinite path in the prefix-su�xautomaton:

� = a0p0,s0 � a1

p1,s1 � a2p2,s2 � · · ·

We have a converging series:

'(�) =1X

⇡c(Mi� `(pi )) 2 Ec

a, ✏

✏, b

✏, ca, ✏

✏, ✏

P: infinite prefix-su�xexpansions

Rauzy fractal [Rauzy, Arnoux-Ito]

R� = '(P) ✓ Ec , ' : P ! R� continuous

R� = {'(�0) | �0 ends with �} R� =G

|�|=n

� = a0p0,s0 � a1

p1,s1 � a2p2,s2 � · · ·

'(�) =1X

⇡c(Mi� `(pi )) 2 Ec

a, ✏

✏, b

✏, ca, ✏

✏, ✏

R� = {'(�0) | �0 ends with �} R� =G

|�|=n

� = a0p0,s0 � a1

p1,s1 � a2p2,s2 � · · ·

'(�) =1X

⇡c(Mi� `(pi )) 2 Ec

a, ✏

✏, b

✏, ca, ✏

✏, ✏

R� = {'(�0) | �0 ends with �} R� =G

|�|=n

Global picture

� irreducible Pisot and parageometric iwip

AZ ◆ X�� //

��

⌦� ⇢ T��1

R� ⇢ Ec

(⌦�) is a tree inside the Rauzy fractal (Peano tree?)

Global picture

AZ ◆ X�� //

��

⌥⌥'

⌦� ⇢ T��1

R� ⇢ Ec

Global picture

AZ ◆ X�� //

��

⌥⌥'

⌦� ⇢ T��1

R� ⇢ Ec

Global picture

AZ ◆ X�� //

��

⌥⌥'

⌦� ⇢ T��1

⌘⌘R� ⇢ Ec

Global picture

AZ ◆ X�� //

��

⌥⌥'

⌦� ⇢ T��1

⌘⌘R� ⇢ Ec

Fractal tree inside the Rauzy fractal

• (⌦�) is a tree inside the Rauzy fractal (Peano tree?)

• ⌦� is approximated by the tree substitution

• The tree substitution is given by gluing points

• Gluing points are described by singular words and their prefix-su�xexpansions

• We know the map ' : P ! R�

� irreducible Pisot and parageometric iwip.Then, the tree substitution ⌧ can be realized inside the contracting spaceEc of � and the renormalized iterated images Mn

�⌧n(W ) converge to the

Rauzy fractal R�.In all our examples we provide a covering tree substitution which yieldstrees inside the contracting space.

Pruning and covering

The problem is the non-injectivity of the map .

Pruning and covering (continued)

⌦a 7! ⌦b

⌦c 7!

7! 7! 7!

Bigger alphabet (more prototiles), but now its injective in the Rauzyfractal.

Conjecture

Up to covering the initial tree substitution, it is always possible to embedit in the Rauzy fractal.

⌦a 7! ⌦b

⌦c 7!

7! 7! 7!

Conjecture

⌦a 7! ⌦b

⌦c 7!

7! 7! 7!

Conjecture

⌦a 7! ⌦b

⌦c 7!

Conjecture

⌦a 7! ⌦b

⌦c 7!

Conjecture

⌦a 7! ⌦b

⌦c 7!

7! 7! 7!

Conjecture

⌦a 7! ⌦b

⌦c 7!

7! 7! 7!

Conjecture

Inside the dual substitution

7!7! 7!

Question: Adjacency of the tiles

If the tiles ⌦� and ⌦�0 have a common point, is it true that thecorresponding tiles of the dual substitution have a common point?

Inside the dual substitution

Question: Adjacency of the tiles

If the tiles ⌦� and ⌦�0 have a common point, is it true that thecorresponding tiles of the dual substitution have a common point?

And more

Application [Bressaud-Jullian, C.-Minervino]

Contour of the tree: piecewise exchange on the circle

Perspectives

• Non parageometric iwip substitutions

• Parageometric ) Rauzy fractal arcwise connected. Disk-like?

• Spectrum of the contour IET versus spectrum of X�?

• Pisot conjecture?

Thank You !

And more

Application [Bressaud-Jullian, C.-Minervino]

Contour of the tree: piecewise exchange on the circle

Perspectives

• Non parageometric iwip substitutions

• Parageometric ) Rauzy fractal arcwise connected. Disk-like?

• Spectrum of the contour IET versus spectrum of X�?

• Pisot conjecture?

Thank You !

Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at...

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