Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at...
Transcript of Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at...
![Page 1: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/1.jpg)
Tree substitutions and Rauzy fractals
Thierry Coulbois
with Milton Minervino
December 5, 2017
![Page 2: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/2.jpg)
Tree-bonacci
� : a 7! ab, b 7! ac , c 7! a
![Page 3: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/3.jpg)
Tree-bonacci
� : a 7! ac , b 7! ab, c 7! b
![Page 4: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/4.jpg)
Tree-bonacci
� : a 7! ab, b 7! c , c 7! a
![Page 5: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/5.jpg)
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
![Page 6: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/6.jpg)
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.
![Page 7: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/7.jpg)
Singular words
Attracting shift
X� = set of bi-infinite indexed words in AZ
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
Tribonacci substitution
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.
![Page 8: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/8.jpg)
Singular words
Attracting shift
X� = set of bi-infinite indexed words in AZ
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
Tribonacci substitution
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.
![Page 9: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/9.jpg)
Singular words
Attracting shift
X� = set of bi-infinite indexed words in AZ
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
Tribonacci substitution
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.
![Page 10: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/10.jpg)
Singular words
Attracting shift
X� = set of bi-infinite indexed words in AZ
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
Tribonacci substitution
a 7! abb 7! acc 7! a
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)
wa, wb and wc are singular words. Their right part is an infinite leftspecial word. They form asymptotic pairs.
![Page 11: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/11.jpg)
Singular words
Attracting shift
X� = set of bi-infinite indexed words in AZ
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
Tribonacci substitution
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.
![Page 12: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/12.jpg)
Singular words
Attracting shift
X� = set of bi-infinite indexed words in AZ
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
Tribonacci substitution
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.
![Page 13: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/13.jpg)
Singular words
Attracting shift
X� = set of bi-infinite indexed words in AZ
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
Tribonacci substitution
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.
![Page 14: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/14.jpg)
Singular words (continued)
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
One periodic Nielsen path of period 3:
ab
baca
�3
�!abacaba·
abacab
bacabacaba
e�1(a) = · · · bacabaabacba·abacababacaba · · · = e�31(a) : w 0
a
bacabaabacaba · · · = e�31(b) : w 0b
cabaabacababa · · · = e�31(c) : w 0c
�̃ : a 7! ba, b 7! ca, c 7! a
![Page 15: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/15.jpg)
Singular words (continued)
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
One periodic Nielsen path of period 3:
ab
baca
�3
�!abacaba·
abacab
bacabacaba
e�1(a) = · · · bacabaabacba·abacababacaba · · · = e�31(a) : w 0
a
bacabaabacaba · · · = e�31(b) : w 0b
cabaabacababa · · · = e�31(c) : w 0c
�̃ : a 7! ba, b 7! ca, c 7! a
![Page 16: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/16.jpg)
Singular words (continued)
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
One periodic Nielsen path of period 3:
ab
baca
�3
�!abacaba·
abacab
bacabacaba
e�1(a) = · · · bacabaabacba·abacababacaba · · · = e�31(a) : w 0
a
bacabaabacaba · · · = e�31(b) : w 0b
cabaabacababa · · · = e�31(c) : w 0c
�̃ : a 7! ba, b 7! ca, c 7! a
![Page 17: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/17.jpg)
Singular words (continued)
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
One periodic Nielsen path of period 3:
ab
baca
�3
�!abacaba·
abacab
bacabacaba
e�1(a) = · · · bacabaabacba·abacababacaba · · · = e�31(a) : w 0
a
bacabaabacaba · · · = e�31(b) : w 0b
cabaabacababa · · · = e�31(c) : w 0c
�̃ : a 7! ba, b 7! ca, c 7! a
![Page 18: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/18.jpg)
Singular words and index
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
e�1(a) = · · · bacabaabacba·abacababacaba · · · = e�31(a) : w 0
a
bacabaabacaba · · · = e�31(b) : w 0b
cabaabacababa · · · = e�31(c) : w 0c
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.
![Page 19: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/19.jpg)
Singular words and index
·abacabaabacabab · · · = �1(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �31(c) = · · · ababacabaabac
e�1(a) = · · · bacabaabacba·abacababacaba · · · = e�31(a) : w 0
a
bacabaabacaba · · · = e�31(b) : w 0b
cabaabacababa · · · = e�31(c) : w 0c
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.
![Page 20: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/20.jpg)
Share-a-half
Theorem [Gaboriau-Jaeger-Levitt-Lustig]
� 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
![Page 21: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/21.jpg)
Share-a-half
Theorem [Gaboriau-Jaeger-Levitt-Lustig]
� 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
![Page 22: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/22.jpg)
Share-a-half
Theorem [Gaboriau-Jaeger-Levitt-Lustig]
� 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�/ ⇠ (devil staircase)
Theorem [GJLL, LL, C-Hilion-Lustig, CH, Kapovich-L]
If � is iwip and parageometric then ⌦� is a tree.
⌦� ✓ T��1 : repelling tree
![Page 23: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/23.jpg)
Share-a-half
Theorem [Gaboriau-Jaeger-Levitt-Lustig]
� 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�/ ⇠ (devil staircase)
Theorem [GJLL, LL, C-Hilion-Lustig, CH, Kapovich-L]
If � is iwip and parageometric then ⌦� is a tree.
⌦� ✓ T��1 : repelling tree
![Page 24: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/24.jpg)
Share-a-half
Theorem [Gaboriau-Jaeger-Levitt-Lustig]
� 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�/ ⇠ (devil staircase)
Theorem [GJLL, LL, C-Hilion-Lustig, CH, Kapovich-L]
If � is iwip and parageometric then ⌦� is a tree.
⌦� ✓ T��1 : repelling tree
![Page 25: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/25.jpg)
Share-a-half
Theorem [Gaboriau-Jaeger-Levitt-Lustig]
� 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.
Q : X� ! ⌦� = X�/ ⇠
Theorem [GJLL, LL, C-Hilion-Lustig, CH, Kapovich-L]
If � is iwip and parageometric then ⌦� is a tree.
⌦� ✓ T��1 : repelling tree
![Page 26: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/26.jpg)
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
![Page 27: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/27.jpg)
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
b
c
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 �.
![Page 28: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/28.jpg)
Self-similarity (continued)
Cylinders
X� bi-inifinite words with prefix-su�x expansion ending with �
Self-similar decomposition (� primitive)
X� =G
|�|=n
X�
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)
![Page 29: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/29.jpg)
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 �.
![Page 30: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/30.jpg)
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 �.
·abacabaabacabab · · · = �31(a)
wa : �31(a) = · · · abacababacabawb : �31(b) = · · · abacabaabacabwc : �1(c) = · · · ababacabaabac
e�31(a) = · · · bacabaabacba·abacababacaba · · · = e�31(a) : w 0
a
bacabaabacaba · · · = e�31(b) : w 0b
cabaabacababa · · · = e�31(c) : w 0c
![Page 31: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/31.jpg)
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
✏ ✏ ✏
a
w 0b: b a a a b
a ✏ ✏ ✏
a
w 0c : c b a a a b
a a ✏ ✏ ✏
a
![Page 32: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/32.jpg)
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
7!
⌦c 7!
Initial patch
7!
![Page 33: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/33.jpg)
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)
![Page 34: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/34.jpg)
Twelfth iterate for Tribonacci (prunned)
![Page 35: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/35.jpg)
Twelfth iterate for Tribonacci (prunned)
That’s the Rauzy Fractal !?
![Page 36: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/36.jpg)
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
![Page 37: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/37.jpg)
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
i=0
⇡c(Mi� `(pi )) 2 Ec
Prefix-su�x graph:
a
b
c
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
R�
![Page 38: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/38.jpg)
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
i=0
⇡c(Mi� `(pi )) 2 Ec
Prefix-su�x graph:
a
b
c
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
R�
![Page 39: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/39.jpg)
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
i=0
⇡c(Mi� `(pi )) 2 Ec
Prefix-su�x graph:
a
b
c
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
R�
![Page 40: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/40.jpg)
Global picture
� irreducible Pisot and parageometric iwip
AZ ◆ X�� //
Q
��
P
'
ss
⌦� ⇢ T��1
R� ⇢ Ec
(⌦�) is a tree inside the Rauzy fractal (Peano tree?)
![Page 41: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/41.jpg)
Global picture
� irreducible Pisot and parageometric iwip
AZ ◆ X�� //
Q
��
PQ
⌥⌥'
ss
⌦� ⇢ T��1
R� ⇢ Ec
(⌦�) is a tree inside the Rauzy fractal (Peano tree?)
![Page 42: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/42.jpg)
Global picture
� irreducible Pisot and parageometric iwip
AZ ◆ X�� //
Q
��
'
PQ
⌥⌥'
ss
⌦� ⇢ T��1
R� ⇢ Ec
(⌦�) is a tree inside the Rauzy fractal (Peano tree?)
![Page 43: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/43.jpg)
Global picture
� irreducible Pisot and parageometric iwip
AZ ◆ X�� //
Q
��
'
PQ
⌥⌥'
ss
⌦� ⇢ T��1
⌘⌘R� ⇢ Ec
(⌦�) is a tree inside the Rauzy fractal (Peano tree?)
![Page 44: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/44.jpg)
Global picture
� irreducible Pisot and parageometric iwip
AZ ◆ X�� //
Q
��
'
PQ
⌥⌥'
ss
⌦� ⇢ T��1
⌘⌘R� ⇢ Ec
(⌦�) is a tree inside the Rauzy fractal (Peano tree?)
![Page 45: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/45.jpg)
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�
Theorem 2 [C.-Minervino]
� 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.
![Page 46: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/46.jpg)
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�
Theorem 2 [C.-Minervino]
� 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.
![Page 47: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/47.jpg)
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�
Theorem 2 [C.-Minervino]
� 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.
![Page 48: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/48.jpg)
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�
Theorem 2 [C.-Minervino]
� 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.
![Page 49: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/49.jpg)
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�
Theorem 2 [C.-Minervino]
� 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.
![Page 50: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/50.jpg)
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�
Theorem 2 [C.-Minervino]
� 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.
![Page 51: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/51.jpg)
Pruning and covering
�
The problem is the non-injectivity of the map .
![Page 52: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/52.jpg)
Pruning and covering (continued)
⌦a 7! ⌦b
7!
⌦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.
![Page 53: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/53.jpg)
Pruning and covering (continued)
⌦a 7! ⌦b
7!
⌦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.
![Page 54: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/54.jpg)
Pruning and covering (continued)
⌦a 7! ⌦b
7!
⌦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.
![Page 55: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/55.jpg)
Pruning and covering (continued)
⌦a 7! ⌦b
7!
⌦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.
![Page 56: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/56.jpg)
Pruning and covering (continued)
⌦a 7! ⌦b
7!
⌦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.
![Page 57: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/57.jpg)
Pruning and covering (continued)
⌦a 7! ⌦b
7!
⌦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.
![Page 58: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/58.jpg)
Pruning and covering (continued)
⌦a 7! ⌦b
7!
⌦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.
![Page 59: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/59.jpg)
Inside the dual substitution
7!7! 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?
![Page 60: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/60.jpg)
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?
![Page 61: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/61.jpg)
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 !
![Page 62: Thierry Coulbois with Milton Minervino December 5, 2017 · iwip substitution ) index of X is at most 2N 2. If the index is maximal then is parageometric. Tribonacci is parageometric.](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e86ff3b10ba3f6db139f676/html5/thumbnails/62.jpg)
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 !