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Groups, bifix codes and minimal sets
Dominique Perrin
25 janvier 2016
Dominique Perrin Groups, bifix codes and minimal sets
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Outline
Uniformly recurrent setsGroup of a bifix codeExamples
Dominique Perrin Groups, bifix codes and minimal sets
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Uniformly recurrent sets
A set F of finite words on A is called factorial if it contains thealphabet A and all the factors of its elements.It is called extendable if for any word w in F , there are letters a, bsuch that awb ∈ F .A factorial set F 6= {ε} is recurrent (or irreducible) if for anyu, v ∈ F , there is w ∈ F such that uwv ∈ F .An infinite factorial set is said to be uniformly recurrent (orminimal) if for any w ∈ S there is an integer n such that w is afactor of any word of F of length n.
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Pseudowords
We consider the free profinite monoid A∗.Let F be a uniformly recurrent set of finite words on the alphabetA. The closure F of F in A∗ is also factorial.As seen before, all the infinite pseudowords in the closure F of Fare are J -equivalent. We denote by J(F ) their J -class.
Example
The Fibonacci morphism ϕ is primitive. The set F of factors of thewords ϕn(a) for n ≥ 1 is called the Fibonacci set. It contains theinfinitely recurrent pseudowords ϕω(a) and ϕω(b).
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Return words
A return word to x ∈ F is a nonempty word w ∈ F which beginsand ends by x but no internal factor of w has the same property.We denote by RF (x) the set of return words to x .
Example
Let F be the Fibonacci set. The set of return words to a isRF (a) = {a, ba}. Similarly, RF (b) = {ab, aab}.
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Multiplicity
Let F be a factorial set on the alphabet A. For w ∈ F , we denote
LF (w) = {a ∈ A | aw ∈ F},
RF (w) = {a ∈ A | wa ∈ F},
EF (w) = {(a, b) ∈ A× A | awb ∈ F}
and further
ℓF (w) = Card(LF (w)), rF (w) = Card(RF (w)), eF (w) = Card(EF (w))
For w ∈ F , we define the multiplicity of w as
mF (w) = eF (w)− ℓF (w)− rF (w) + 1.
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Neutral sets
A word w is called neutral if mF (w) = 0. A factorial set F isneutral if every word in F is neutral. By a result of (Balkova,Pelantova, Steiner, 2008), in a uniformly recurrent neutral set, onehas
Card(RF (x)) = Card(A) (1)
for every x ∈ F .
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Return words
Let F be neutral set. For x ∈ J(F ), a return word to x is the limitof a sequence of return words to xn for a sequence (xn) of words ofF converging to x .
Example
Let ϕ be the Fibonacci morphism and F be the Fibonacci set. It isa neutral set. The set of return words to a is RF (a) = {a, ba}.Accordingly, the set of return words to ϕω(a) is {ϕω(a), ϕω(ba)}.Actually, any set of return words has two elements.
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Schutzenberger groups
Let x be an elementof a semigroup S and H be its J -class. Wedenote by G (x) the Schutzenberger group of x . It is, by definition,the group of translations ρ(z) : y ∈ H 7→ yz ∈ H for all z suchthat Hz = z . When S is a topological semigroup, it is a topologicalgroup.When H is a group, it is isomorphic to H.
Theorem (Almeida, Costa, 2013)
Let F be a uniformly recurrent neutral set. For any x ∈ J (F ), thegroup G (x) is the closure of the subgroup generated by any set of
return words to x.
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Example
Let ϕ be the Fibonacci morphism and let F be the Fibonacci set.Let x = ϕω(a) and y = ϕω(b). The group G (x) is the closure ofthe group generated by x and yx , that is, isomorphic to the free
profinite group FG (A).
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Tree sets
Let F be a factorial set of words. For w ∈ F , we consider the setEF (w) as an undirected graph on the set of vertices which is thedisjoint union of LF (w) and RF (w) with edges the pairs(a, b) ∈ EF (w). This graph is called the extension graph of w .A factorial set is a tree set is for every x ∈ F , the graph EF (x) is atree. A tree set is neutral.
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Example 1
The Fibonacci set is a tree set. This follows from the fact that it isa Sturmian set and that every Sturmian set is a tree set.
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Example 2
The Tribonacci set is the set of factors of the fixed point of themorphism ϕ : a 7→ ab, b 7→ ac , c 7→ a. It is a also a tree set. Thegraph E (ε) is represented below.
a
b
c a
b
c
Figure: The extension graph of ε in the Tribonacci set.
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The Return Theorem
Theorem (Berthe, De Felice, Dolce, Leroy, Perrin, Reutenauer,Rindone, 2013)
Let F be a uniformly recurrent tree set. For any x ∈ F , the set
RF (x) is a basis of the free group FG (A).
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Example
Let F be the Tribonacci set on A = {a, b, c}. ThenRF (a) = {a, ba, ca}, which is easily seen to be a basis of FG (A).
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We denote by G (F ) the Schutzenberger group of J(F ) (that is theSchutzenberger group of any element of J(F )).
Theorem (Ameida, Costa, 2015)
Let F be a nonperiodic uniformly recurrent tree set on the
alphabet A. The group G (F ) is the free profinite group on A.
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Example
Let F be the Fibonacci set. We have seen that G (F ) is the freeprofinite group on A (Example 3).
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Let ϕ : A∗ → A∗ be a primitive substitution and let F (ϕ) be theset of factors of a fixed point of ϕ. We denote by J(ϕ) the closureof F (ϕ) and by G (ϕ) the Schutzenberger group of J(ϕ).A connexion for ϕ is a word ba with b, a ∈ A such that ba ∈ F (ϕ),the first letter of ϕω(a) is a and the last letter of ϕω(b) is b. Everyprimitive substitution has a connexion. A connective power of ϕ isa finite power ϕ of ϕ such that the first letter of ϕ(a) is a, the lastletter of ϕ(b) is b. We denote Xϕ(a, b) = aRF (ba)a
−1. The setXϕ(a, b) is a code.A substitution ϕ over A is proper if there are letters a, b ∈ A suchthat for every d ∈ A, ϕ(d) starts with a and ends with b.
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Presentation of the Schutzenberger group
Theorem (Almeida, Costa, 2013)
Let ϕ be a non-periodic proper primitive substitution over a finite
alphabet A. Then G (ϕ) admits the presentation
〈A | ϕω
G (a) = a, a ∈ A〉G .
H H
A∗ A∗
ϕ
ϕω+1
ϕω ϕω
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Example
Let A = {a, b} and let ϕ : a 7→ ab, b 7→ a3b. The Schutzenbergergroup of J(ϕ) has the presentation 〈a, b | ϕω
G(a) = a, ϕω
G(b) = b〉.
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Bifix codes
Let F be a recurrent set. Given a bifix code X ⊂ F , a parse of aword w is a triple (s, x , p) such that
(i) w = sxp,ii) s has no suffix in X ,
(iii) x ∈ X ∗,(iv) p has no prefix in X .
The F -degree of X is the maximal number of parses of a word inF . It is finite for any finite F -maximal bifix code X .
Example
For any n ≥ 1, the set F ∩ An is a finite F -maximal bifix code. Ithas F -degree n.
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Cardinality Theorem
The following result shows that in a recurrent tree set, thecardinality of an F -maximal bifix code depends only of itsF -degree.
Theorem (Berthe, De Felice, Dolce, Leroy, P., Reutenauer,Rindone, J. Pure Appl. Algebra, 2015)
Let F be a recurrent neutral set. For any finite F -maximal bifix
code X , one has Card(X ) = dX (F )(Card(A)− 1) + 1.
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The Finite Index Basis Theorem
Theorem (Berthe, De Felice, Dolce, Leroy, P., Reutenauer,Rindone, J. Pure Appl. Algebra, 2015)
Let F be a minimal tree set. A finite bifix code X ⊂ F is
F -maximal of F -degree d if and only if it is a basis of subgroup of
finite index d of the free group on A.
This implies for tree sets the Cardinality Theorem because, bySchreier’s Formula, a subgroup of index d of a free group of rank k
has rank d(k − 1) + 1.The proof uses the Return Theorem.
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Examples
Let F be the Fibonacci set. The 3 F -maximal bifix codes ofF -degree 2 are
aa, ab, ba,
a, bab, baab,
aa, aba, b
corresponding to the three subgroups of index 2
1 2
a, b
a, b
1 2
a ab
b
1 2
b ba
a
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The F -degree of a prefix code
Let F be a factorial set and let A be a finite deterministicautomaton. The rank of a word w is the number of states reachedby w . The F -minimal rank of A is the minimal value of the rank ofthe words in F .An automaton of F -minimal rank 1 is called F -synchronized.Let X ⊂ F be a finite F -maximal prefix code. The F -degree of X ,denoted dX (F ), is the F -minimal rank of the minimal automatonof X ∗.
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Example
Let F be the Fibonacci set. The X be the F -maximal prefix coderepresented below has F -degree 3. It is not bifix and it generatesthe free group.
1
2
3
4
5
1
6
7
8
4
9
1
1
6
3
8
1
1
1
a
b
a
b
a
b
a
a
a
b
a
b
b
a
a
a
a
b
Figure: A prefix code of F -degree 3
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The set Im(a2) of states reachable by a2 is Im(a2) = {1, 2, 4}. Theaction on the 3-element sets of states of the automaton is shownbelow (right).
1
2
3
4
5
1
6
7
8
4
9
1
1
6
3
8
1
1
1
a
b
a
b
a
b
a
a
a
b
a
b
b
a
a
a
a
b 1, 2, 4 3, 5, 6 1, 7, 8
1, 3, 91, 2, 3
b a
b
a
a
a
Thus dX (F ) = 3.
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The F -group of a prefix code
Let F be a recurrent set and let A be a deterministic automaton.The F -group of the automaton is obtained by choosing a wordw ∈ F of minimal rank. It is generated by the permutations onI = Im(w) realized by the return words to w (in terms ofsemigroup theory, it is the group of the D-class of the elements ofF of minimal rank).Let X be a finite prefix code. The F -group of X , denoted GX (F ) isthe F -group of the minimal automaton of X ∗.
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The following is proved for a Sturmian set in (Berstel,De Felice, P.,Reutenauer, Rindone, J. Algebra, 2012).
Theorem
Let F be a uniformly recurrent tree set and let X be a finite
F -maximal bifix code. The group GX (F ) is equivalent to the
representation of the free group FA on the cosets of the group
generated by X .
We conjecture that for a finite F -maximal prefix code X , the groupGX (F ) is transitive.
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Example 0
Let F be the Fibonacci set and let X = {aa, ab, ba}. The minimalautomaton of X ∗ is represented on the left.
3 1 2
a
a, bb
a
1, 2 1, 3
a
b
a
The word a has 2 parses and its image is the set {1, 2}. The actionof the return words to a on the minimal images is indicated on theright. The word a defines the permutation (12) and the word ba
the identity.
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Example 1 : the alternating group A5
Let F be the set of factors of a fixed point of ϕ : a 7→ ab, b 7→ a3b.We consider the morphim h : A∗ → A5 from A∗ onto thealternating group of degree 5 defined by h : a 7→ (123), b 7→ (345).We denote by ϕG the map from GA into itself defined for f ∈ GA
and a ∈ A by ϕG (f )(a) = f (ϕ(a)). We say that ϕ has finitef -order if there is an integer n ≥ 1 such that ϕn
G(f ) = f . The least
such integer is called the h-order of ϕ.The morphism ϕ has h-order 12 and thus, by a result of(Almeida,Costa, 2013), h induces a surjective map from anymaximal subgroup of J(ϕ) onto A5.
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Let Z be the bifix code generating the submonoid stabilizing 1 andlet X = Z ∩ F . The F -maximal bifix code X has 8 elements. It isrepresented below with the states of the minimal automatonindicated on its prefixes.
1
2
1
3
4
1
5
6
7
1
8
9
10
11
12
9
14
15
16
11
17
1
14
14
15
15
17
17
1
1
15
15
1
1
a
b
a
b
ab
a
a
ab
b
a
a
a
b
a
b
a
a
a
a
b
a
b
b
a
a
a
a
b
b
a
a
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The F -minimal D-class
1, 2, 3, 16, 17 1, 4, 5, 14, 15 1, 2, 6, 7, 17 1, 4, 8, 9, 15 1, 2, 6, 10,
1/2, 4/3, 6, 15/ a3
a3b a
3ba * a
3bab
8/9
1/2/11, 17/6 ba3 *
7, 9, 10
1/3, 6, 15/9, 14 aba3 *
5, 8/4
1/11, 17/7, 16 * baba3
bab
3, 6/2
1/2, 4/9, 14 *
3, 6, 15/5, 12
1/2, 4/3, 6, 15 *
10/11
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The word a3 has rank 5 and RF (a3) = {babaaa, babababaaa}. The
corresponding permutations defined on the image {1, 2, 3, 16, 17}of a3 are respectively
(1, 2, 16, 3, 17) (1, 17, 16, 2, 3)
which generate A5.
Dominique Perrin Groups, bifix codes and minimal sets
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Example 2 : a trivial group
Let F be the Thue-Morse set and let A be the automatonrepresented below on the left.
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2
3
4
11
5
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1
10 1
7
1
9 11 1
a
b
ab
ab
ba
b
b b
aa
ba a
0 1
2
a
aab b
b
Dominique Perrin Groups, bifix codes and minimal sets
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The action on the minimal images
The word aa has rank 3 and image I = {1, 2, 4}. The action on theimages accessible from I is given below.
1, 2, 4 1, 3, 6 1, 3, 5 1, 2, 7 1, 3, 9 1, 2, 11
1, 2, 8 1, 3, 10
b b a b a
a
a
b
b
a
All words with image {1, 2, 4} end with aa. The paths returning forthe first time to {1, 2, 4} are labeled by the setRF (aa) = {b2a2, bab2aba2, bab2a2, b2aba2}. Thus rankA(F ) = 3.Moreover each of the words of RF (a
2) defines the trivialpermutation on the set {1, 2, 4}. Thus GA(F ) is trivial.
Dominique Perrin Groups, bifix codes and minimal sets
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Example 3
Consider again the Thue-Morse substitution τ and the Thue-Morseset F . Let h be the morphism h : a 7→ (123), b 7→ (345) from A∗
onto the alternating group A5 (already used in Example 1). Onemay verify that τ has h-order 6 and thus, h extends to a surjectivecontinuous morphim from any maximal subgroup of J(ϕ) onto A5.Let Z be the group code generating the submonoid stabilizing 1and let X = Z ∩ F .
Dominique Perrin Groups, bifix codes and minimal sets
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The F -maximal bifix code X
We represent only the nodes corresponding to right special words,that is, vertices with two sons.
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1
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1
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a
b
ab
b
abba
ba
a
ba
aba
ba
a
b2a
a
τ2(b)
τ3(a)
ba
τ2(a)
ba
τ2(a)
τ3(b)
τ2(a)
τ2(bba)
τ2(a)
τ2(b)a
τ2(a)
τ2(b)a
Dominique Perrin Groups, bifix codes and minimal sets
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The image of τ4(b) is {1, 3, 4, 9, 10} and thus it is minimal. Theaction on its image is shown below. The return words to τ4(b) areτ4(b), τ3(a) and τ5(ab). The permutations on the image of τ4(b)are the 3 cycles of length 5 indicated below. Since they generatethe group A5, we have GX (F ) = A5.
{1, 3, 4, 9, 10} {1, 2, 7, 8, 12}τ4(b) | (1, 9, 10, 3, 4)τ4(a)
τ3(a) | (1, 10, 9, 3, 4)
τ4(b) | (1, 10, 9, 4, 3)
Dominique Perrin Groups, bifix codes and minimal sets