Minimal subshifts, Sch utzenberger groups and profinite semigroups - - PowerPoint PPT Presentation

Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp June 29, 2017

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 1 / 40

Outline

Minimal symbolic systems are a special family of symbolic dynamical

• systems. I will present recent results obtained on these systems

involving discrete groups. Profinite algebra is a theory allowing to define and study objects defined as limits of finite structures. I will show how it allows to give an account of the role played by discrete groups in minimal systems.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 2 / 40

Symbolic systems

Consider the set AZ of biinfinite sequences x = (xn)n∈Z with the shift σ : AZ → AZ defined by y = σ(x) if yn = xn+1. A symbolic system (or two-sided subshift) is a set X ⊂ AZ of biinfinite sequences which is

1

closed for the product topology,

2

invariant by the shift, that is σ(X) ⊂ X. Example: the set of two-sided infinite words on A = {a, b} without two consecutive b. Variant: one-sided subshift X ⊂ AN.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 3 / 40

Factorial sets

A set of words on the alphabet A is factorial if it contains A and the factors (or substrings) of its elements. A factorial set F is biextendable if for any w ∈ F there are letters a, b ∈ A such that awb ∈ F.

Proposition

The set of words appearing in the sequences of a symbolic system X is a biextendable set and any biextendable set is obtained in this way.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 4 / 40

Rotations and Sturmian words

Rotation of angle α. R(z) = z + α mod 1. Natural coding: let s(α) = (sn)n∈Z be the sequence sn =

• a

if ⌊z0 + (n + 1)α⌋ = ⌊z0 + nα⌋, b

• therwise

For z0 = 0 and α = (3 − √ 5)/2 this gives the Fibonacci sequence

a b a a b a b a

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

• Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp

Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 5 / 40

Minimal systems

The symbolic system X is minimal if it does not contain properly another nonempty one. An infinite factorial set F is said to be uniformly recurrent if for any word w ∈ F there is an integer n ≥ 1 such that w is a factor of any word of F

• f length n.

Remark that a uniformly recurrent set F is recurrent: for every u, v ∈ F, there is some x such that uxv ∈ F.

Proposition

A system is minimal if and only if the set of its factors is uniformly recurrent.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 6 / 40

Fixed points of morphisms

A morphism (or substitution) ϕ : A∗ → A∗ is primitive if there is an integer n such that any letter a appears in all the ϕn(b) for b ∈ A.

Proposition

Let ϕ : A∗ → A∗ be a primitive morphism and let x be a fixed point of ϕ. The set of factors of x is uniformly recurrent.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 7 / 40

The Fibonacci set

The Fibonacci morphism ϕ : a → ab, b → a is primitive. The set F of factors of its fixed point lim ϕn(a) = abaababa · · · is the Fibonacci set. It is uniformly recurrent. One has F = {a, b, aa, ab, ba, aab, aba, baa, . . .}. Forbidden factors: bb, aaa, babab. A direct way to obtain a two-sided fixed point of ϕ: concatenate the left-infinite word · · · abaabaab

• btained iterating ϕ2n(b) with the Fibonacci word.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 8 / 40

The Thue-Morse set

The Thue-Morse morphism τ : a → ab, b → ba is primitive. The set F of factors of its fixed point lim τ n(a) = abbabaab · · · is the Thue-Morse set. It is uniformly recurrent. One has F = {a, b, aa, ab, ba, bb, aab, aba, abb, baa, bab, bba, . . .}.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 9 / 40

Factor complexity

The factor complexity of a factorial set F on the alphabet A is the sequence pn(F) = Card(F ∩ An). We have p0(F) = 1 and we assume p1(F) = Card(A) for any factorial set. The sets of bounded complexity are the factors of eventually periodic sequences. The binary Sturmian sets are, by definition, those of complexity n + 1 (like the Fibonacci set).

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 10 / 40

Return words

Let F be uniformly recurrent. A return word to x ∈ F is a nonempty word y such that xy is in F and ends with x for the first time. For any x ∈ F, the set R(x) = {y ∈ F | xy ∈ F ∩ A∗x \ A+xA+}.

• f return words to x is finite.

Return words play an important role in the study of minimal systems: they allow to define an induced system.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 11 / 40

Extension graphs

Let F be a factorial set. For a given word w ∈ F, set L(w) = {a ∈ A | aw ∈ F}, left extensions of w E(w) = {(a, b) ∈ A × A | awb ∈ F}, two-sided extensions of w R(w) = {b ∈ A | wb ∈ F} right extensions of w. The extension graph of w in F is the graph on the set of vertices which is the disjoint union of L(w) and R(w) and with edges the set E(w). For example, if A = {a, b} and F ∩ A2 = {aa, ab, ba}, the extension graph

• f ε is

a a b b

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 12 / 40

Tree sets

A factorial set F is a tree set if for any w ∈ F, the extension graph of w is a tree.

Proposition

The factor complexity of a tree set F on k letters is pn(F) = (k − 1)n + 1. Any Sturmian set is a tree set. Thus the Fibonacci set is a tree set. In contrast, the Thue-Morse set is not a tree set since p2(F) = 4.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 13 / 40

The Tribonacci set

The factors of the fixed point of the Tribonacci morphism a → ab, b → ac, c → a is the Tribonacci set. It is a tree set. The graph E(ε) is a a b b c c

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 14 / 40

The return theorem

Theorem (BDDLPRR, Monatsh. Math., 2014)

Let F be a uniformly recurrent set. If F is a tree set, the set of return words to any x ∈ F is a basis of the free group on A. It is not known whether the converse is true. BDDLPRR: Val´ erie Berth´ e, Clelia De Felice, Francesco Dolce, Julien Leroy, Dominique Perrin, Christophe Reutenauer and Giuseppina Rindone.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 15 / 40

Example

Let F be the Fibonacci set. Then, since abaaba, ababaaba ∈ F, we have R(aba) = {aba, ba}. which is a basis of FG(a, b). Indeed, by Nielsen transformations: {aba, ba} → {a, ba} → {a, b}.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 16 / 40

The finite index basis theorem

The previous result is connected with the following one (BDDLPRR, J. Pure Appl. Algebra, 2015)

Theorem

Let F be a uniformly recurrent tree set. Let f : FG(A) → G be a morphism onto a finite group G and let H be a subgroup of G. Then

1

The subgroup h−1(H) has a basis included in F

2

The restriction to F of h is onto. For example, the subgroup formed by the words of even length has the basis X = {aa, ab, ba} included in the Fibonacci set.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 17 / 40

Q&A

Questions: Is there an algebraic structure allowing to concatenate infinite words? What happens to minimal sets? Does it give an interpretation to the group generated by return words? Does it allow to extend a morphism h : FG(A) → G onto a finite group G to infinite words? Answers: Yes: embed A∗ in the free profinite monoid A∗, take the closure of F in A∗. They appear as Sch¨ utzenberger groups of A∗. The morphism extends from a Sh¨ utzenberger group to G

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 18 / 40

Profinite topology

The profinite monoid metric (or pro-M metric) on a monoid M is defined by d(u, v) =

• 2−r(u,v)

if u = v

• therwise

where r(u, v) is the minimal cardinality of a finite monoid N for which there is a morphism ϕ : M → N such that ϕ(u) = ϕ(v). Thus two words are close to each other if a large monoid is needed to separate them. Replacing monoids by groups, one obtains the profinite group metric (or pro-G metric) first introduced by Hall (1950).

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 19 / 40

The free profinite monoid

The completion of the free monoid for the topology induced by the profinite metric is the free profinite monoid, denoted A∗. It is a compact monoid. Its elements are called pseudowords. A sequence xn of pseudowords converges if for any morphism into a finite monoid, the image of the sequence is ultimately constant. For any word x, the sequence xn! converges to a pseudoword denoted xω.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 20 / 40

What do pseudowords look like? They have a well-defined prefix of length n for all n ≥ 1 (like a

• ne-sided infinite word to the right).

They have a well-defined suffix of length n for all n ≥ 1 (idem on the left). They have MANY other things in the middle... They have a length which is a profinite natural integer (to be seen next) which is odd or even. Typical pseudoword: the limit of ϕn!(a) where ϕ is the Fibonacci

• morphism. It begins with the Fibonacci word.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 21 / 40

The free profinite group

Likewise, the completion of the free group FG(A) for the pro-G metric is the free profinite group, denoted FG(A). It is a compact group. In contrast with the monoid case, one has for any x ∈ FG(A), lim xn! = 1. Thus A∗ is dense in FG(A) and the natural projection pG : A∗ → FG(A) is surjective.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 22 / 40

Profinite integers

The profinite monoid on one letter is the monoid of profinite natural integers, denoted N. The profinite group on one letter is the group of profinite integers, denoted

• Z.

The latter can be identified with the set of infinite expansions in the factorial number system formed of the (· · · c3c2c1)! = c1 + c22! + c33! + . . . with digits 0 ≤ ci ≤ i.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 23 / 40

Rauzy graphs

Minimal sets are not in general recognizable by finite monoids. However, the profinite setup allows one to define them by limits of sequences of finite monoids. Let indeed F be a factorial set. For each n, let Fn be the set of words such that all their factors of length n are in F. The set F is the intersection of all Fn. Each Fn is recognized by an automaton Rn(F) called the n-Rauzy graph of F, corresponding to a morphism into a finite monoid. Thus the closure of F in A∗ can be seen as defined by the limit of the graphs Rn(F) or the limit of their monoids.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 24 / 40

Example

Rauzy graphs of order n = 1, 2, 3 of the Fibonacci set. a b a b a aa ab ba b a b a aab aba bab baa a a b a a a

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 25 / 40

Green’s relations

For two elements x, y in a topological monoid M, denote x ≤R y if x ∈ yM (that is x in the right ideal generated by y), x ≤L y if x ∈ My (that is x in the left ideal generated by y), x ≤J y if x ∈ MyM (that is x in the two-sided ideal generated by y). and by R, L, J the equivalences associated to these preorders with H = R ∩ L (all elements in a class generate the same right and left ideal). The classes are, in each case, closed subsets of M.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 26 / 40

The Sch¨ utzenberger group of a J -class

Let J be a J -class of a topological monoid M and let H be an H-class contained in J. The set Γ(H) of translations ρx : y ∈ H → yx ∈ H for all x ∈ M such that Hx = H forms a topological group which depends

• nly of J, called the Sch¨

utzenberger group of J. When H is a group, it is isomorphic with Γ(H).

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 27 / 40

Example

The J -class of aω in A∗ is made of one H-class. The J -class of (ab)ω has four H-classes. The H-classes of (ab)ω and (ba)ω are groups. (ab)ω (ab)ωa b(ab)ω (ba)ω In both cases the Sch¨ utzenberger group is the group ˆ Z of profinite integers.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 28 / 40

The fundamental J -class

Theorem (Almeida 2005)

Let F be a uniformly recurrent set. The set of infinite pseudowords with all their finite factors in F forms a maximal J -class of A∗. The Sch¨ utzenberger group of this J -class is denoted G(F) and called the Sch¨ utzenberger group of F. For example, if F is the Fibonacci set, the J -class J(F) contains the pseudoword ϕω(a) which is the limit of the sequence ϕn!(a).

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 29 / 40

Tree sets and free groups

Theorem (Almeida, Costa, 2015)

Let F be a uniformly recurrent set. If F is a tree set, the group G(F) is a free profinite group. More precisely, the natural map from G(F) into

• FG(A) is an isomorphism.

The proof uses the Return Theorem. The result implies that if f : A∗ → G is a morphism onto a finite group, the restriction of f to G(F) is surjective (an essential part of the Finite Index Basis Theorem).

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 30 / 40

Beyond tree sets

For any uniformly recurrent set F, there is a natural morphism from G(F) into FG(A) since a maximal subgroup of J(F) is included in A∗.

Theorem

Let F be a nonperiodic recurrent set. The following conditions are equivalent. (i) For every x ∈ F, the set RF(x) generates FG(A). (ii) The restriction to any maximal subgroup of J(F) of the natural projection pG : A∗ → FG(A) is surjective.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 31 / 40

Morphisms of finite order

Let f : A∗ → G be a morphism from A∗ into a finite group G and let ϕ : A∗ → A∗ be a morphism. We denote by ϕG the map from G A into itself defined for h ∈ G A and a ∈ A by ϕG(h)(a) = h(ϕ(a)). We say that ϕ has finite f -order if there is an integer n ≥ 1 such that ϕn

G(f ) = f . The

least such integer is called the f -order of ϕ. Any substitution ϕ which is invertible in FG(A) is of finite f -order for any morphism f . Thus the Fibonacci morphism has finite f -order for any f .

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 32 / 40

Fixed points of morphisms

Theorem (Almeida, Costa, 2013)

Let ϕ be a non-periodic primitive substitution over A and let f : A∗ → G be a morphism onto a finite group. The natural morphism from G(F) into G is surjective if and only if ϕ has finite f -order.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 33 / 40

Example 1: trivial image

Let F be the Thue-Morse set and let f : A∗ → S3 be represented below on the left. The f -order of τ is infinite since, f (τ(a)) = f (τ(b)) = (1). The intersection of F with the generating set of the submonoid fixing 1 is represented on the right. 1 2 3 a a a b b b 1 2 3 4 1 1 5 6 8 1 10 1 7 1 9 11 1 a b a b a b b a b b b a a b a a

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 34 / 40

The word aa has rank 3 and image I = {1, 2, 4}. The action on the images 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

Figure : The action on the minimal images

All words with image {1, 2, 4} end with aa. The paths returning for the first time to {1, 2, 4} are labeled by the set RF(aa) = {b2a2, bab2aba2, bab2a2, b2aba2}. Moreover each of the words

• f RF (aa) defines the trivial permutation on the set {1, 2, 4}. Thus the

image of G(F) in S3 is trivial.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 35 / 40

Example 2: full image

Consider again the Thue-Morse substitution τ and the Thue-Morse set F. Let f be the morphism f : a → (123), b → (345) from A∗ onto the alternating group A5. It can be verified that τ has f -order 6. Let Z be the generating set of the submonoid stabilizing 1 and let X = Z ∩ F.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 36 / 40

We represent below the set X keeeping only the nodes with two sons. 1 2 1 3 4 1 5 6 7 1 1 1 8 1 9 10 1 11 1 1 12 1 12 1 1 1 1 a b ab b abba ba a ba aba ba a b2a a τ 2(b) τ 3(a) ba τ 2(a) ba τ 2(a) τ 3(b) τ 2(a) τ 2(bba) τ 2(a) τ 2(b)a τ 2(a) τ 2(b)a

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 37 / 40

The image of τ 4(b) is {1, 3, 4, 9, 10} and thus it is minimal. The action on its image is shown below. The return words to τ 4(b) are τ 4(b), τ 3(a) and τ 5(ab). The permutations on the image of τ 4(b) are the 3 cycles of length 5 indicated in Figure 2. Since they generate the group A5, the image of G(F) in A5 is A5. {1, 3, 4, 9, 10} {1, 2, 7, 8, 12} τ 4(b) | (1, 9, 10, 3, 4) τ 4(a) τ 3(a) | (1, 10, 9, 3, 4) τ 4(b) | (1, 10, 9, 4, 3)

Figure : The action on the minimal images.

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 38 / 40

Conclusion and perspectives

Profinite monoids allow to handle simultaneously all possible morphisms from A∗ into finite groups. They allow to formulate resuts going beyond tree sets Possible next step: replace iterated morphisms by S-adic expansions which are infinite compositions of a family of morphisms?

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 39 / 40

Happy birthday Paul!

Dominique Perrin Groups and Computation, Stevens Institute in the honour of Paul Schupp Minimal subshifts, Sch¨ utzenberger groups and profinite semigroups June 29, 2017 40 / 40