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 A Z of biinfinite sequences x = ( x n ) n ∈ Z with the shift σ : A Z → A Z defined by y = σ ( x ) if y n = x n +1 . A symbolic system (or two-sided subshift) is a set X ⊂ A Z of biinfinite sequences which is closed for the product topology, 1 invariant by the shift, that is σ ( X ) ⊂ X . 2 Example: the set of two-sided infinite words on A = { a , b } without two consecutive b . Variant: one-sided subshift X ⊂ A N . 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 ) = ( s n ) n ∈ Z be the sequence � a if ⌊ z 0 + ( n + 1) α ⌋ = ⌊ z 0 + n α ⌋ , s n = b otherwise √ For z 0 = 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 of 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 ϕ n ( 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 ϕ n ( 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 obtained iterating ϕ 2 n ( 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 p n ( F ) = Card( F ∩ A n ). We have p 0 ( F ) = 1 and we assume p 1 ( 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 + } . of 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 ∩ A 2 = { aa , ab , ba } , the extension graph of ε is b 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 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 p n ( 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 p 2 ( 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 c c b 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 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 The subgroup h − 1 ( H ) has a basis included in F 1 The restriction to F of h is onto. 2 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
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