A new algebraic invariant for weak equivalence of sofic subshifts - - PowerPoint PPT Presentation

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A new algebraic invariant for weak equivalence of sofic subshifts

Laura Chaubard

LIAFA, CNRS et Universit´ e Paris 7

Alfredo Costa

Centro de Matem´ atica da Universidade de Coimbra

Symbolic dynamical systems

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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σ : AZ → AZ (xi)i∈Z → (xi+1)i∈Z

A subset X of AZ is a symbolic dynamical system or subshift if:

• X is topologically closed
• σ(X ) ⊆ X
• σ−1(X ) ⊆ X

Symbolic dynamical systems

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 2

σ : AZ → AZ (xi)i∈Z → (xi+1)i∈Z

A subset X of AZ is a symbolic dynamical system or subshift if:

• X is topologically closed
• σ(X ) ⊆ X
• σ−1(X ) ⊆ X

L(X ) = {u ∈ A+ : u = xixi+1 . . . xi+n for some x ∈ X, i ∈ Z, n ≥ 0}.

Symbolic dynamical systems

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 2

σ : AZ → AZ (xi)i∈Z → (xi+1)i∈Z

A subset X of AZ is a symbolic dynamical system or subshift if:

• X is topologically closed
• σ(X ) ⊆ X
• σ−1(X ) ⊆ X

L(X ) = {u ∈ A+ : u = xixi+1 . . . xi+n for some x ∈ X, i ∈ Z, n ≥ 0}.

• L(X ) is factorial and prolongable
• If L is a factorial prolongable language of A+, then there is a unique

subshift of AZ such that L = L(X ):

X ⇄ L(X )

Symbolic dynamical systems

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 2

σ : AZ → AZ (xi)i∈Z → (xi+1)i∈Z

A subset X of AZ is a symbolic dynamical system or subshift if:

• X is topologically closed
• σ(X ) ⊆ X
• σ−1(X ) ⊆ X

L(X ) = {u ∈ A+ : u = xixi+1 . . . xi+n for some x ∈ X, i ∈ Z, n ≥ 0}.

• L(X ) is factorial and prolongable
• If L is a factorial prolongable language of A+, then there is a unique

subshift of AZ such that L = L(X ):

X ⇄ L(X )

A subshift X is called sofic if L(X ) is rational.

Codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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• A code between two subshifts X and Y is a continuous map from X

to Y that respects the shift operation:

X

σ

• f
• X

f

• Y

σ

Y

• Conjugation: bijective code
• Two subshifts are conjugate if there is a conjugation between them

Codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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• A code between two subshifts X and Y is a continuous map from X

to Y that respects the shift operation:

X

σ

• f
• X

f

• Y

σ

Y

• Conjugation: bijective code
• Two subshifts are conjugate if there is a conjugation between them
• Open problem: is conjugation of sofic subshifts decidable?

Sliding block codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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Let x ∈ AZ. Given a map f : Ak → B, we can code x with f:

Sliding block codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 4

Let x ∈ AZ. Given a map f : Ak → B, we can code x with f:

• choose m and n such that k = m + n + 1;

Sliding block codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 4

Let x ∈ AZ. Given a map f : Ak → B, we can code x with f:

• choose m and n such that k = m + n + 1;
• make yi = f(x[i−m,i+n]).

Sliding block codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 4

Let x ∈ AZ. Given a map f : Ak → B, we can code x with f:

• choose m and n such that k = m + n + 1;
• make yi = f(x[i−m,i+n]).

. . . xi−4 xi−3xi−2xi−1xi

f

 

• xi+1xi+2xi+3 . . .

. . . yi−2 yi−1 yiyi+1yi+2 . . .

Sliding block codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 4

Let x ∈ AZ. Given a map f : Ak → B, we can code x with f:

• choose m and n such that k = m + n + 1;
• make yi = f(x[i−m,i+n]).

. . . xi−4xi−3 xi−2xi−1xixi+1

f

 

• xi+2xi+3 . . .

. . . yi−2yi−1 yi yi+1yi+2 . . .

Sliding block codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 4

Let x ∈ AZ. Given a map f : Ak → B, we can code x with f:

• choose m and n such that k = m + n + 1;
• make yi = f(x[i−m,i+n]).

. . . xi−4xi−3xi−2 xi−1xixi+1xi+2

f

 

• xi+3 . . .

. . . yi−2yi−1yi yi+1 yi+2 . . .

Sliding block codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 4

Let x ∈ AZ. Given a map f : Ak → B, we can code x with f:

• choose m and n such that k = m + n + 1;
• make yi = f(x[i−m,i+n]).

. . . xi−4xi−3xi−2 xi−1xixi+1xi+2

f

 

• xi+3 . . .

. . . yi−2yi−1yi yi+1 yi+2 . . .

• the map

F : (xi)i∈Z → (yi)i∈Z

is a code and all codes have this form

Sliding block codes

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 4

Let x ∈ AZ. Given a map f : Ak → B, we can code x with f:

• choose m and n such that k = m + n + 1;
• make yi = f(x[i−m,i+n]).

. . . xi−4xi−3xi−2 xi−1xixi+1xi+2

f

 

• xi+3 . . .

. . . yi−2yi−1yi yi+1 yi+2 . . .

• the map

F : (xi)i∈Z → (yi)i∈Z

is a code and all codes have this form

• we say that f is a block map of F and that F has window size k

Coding of finite words

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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Let u = u1 . . . un ∈ A+, where ui ∈ A. Given a map f : Ak → B, we can use f to code u, through the following map ¯

f:

• if n < k then ¯

f(u) = 1

• if n ≥ k then

¯ f(u) = f(u1 . . . uk)f(u2 . . . uk+1) . . . f(un−k+1 . . . un)

Division of subshifts

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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For an alphabet A, let $ be a letter which is not in A. Denote by A$ the alphabet A ∪ {$}.

Division of subshifts

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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For an alphabet A, let $ be a letter which is not in A. Denote by A$ the alphabet A ∪ {$}.

• A subshift X of AZ is a divisor of a subshift Y of BZ if there is a code

F : AZ → BZ

$ such that X = F −1Y.

Division of subshifts

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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For an alphabet A, let $ be a letter which is not in A. Denote by A$ the alphabet A ∪ {$}.

• A subshift X of AZ is a divisor of a subshift Y of BZ if there is a code

F : AZ → BZ

$ such that X = F −1Y.

• Two subshifts are said to be weak equivalent if they divide each other.

Division of subshifts

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 6

For an alphabet A, let $ be a letter which is not in A. Denote by A$ the alphabet A ∪ {$}.

• A subshift X of AZ is a divisor of a subshift Y of BZ if there is a code

F : AZ → BZ

$ such that X = F −1Y.

• Two subshifts are said to be weak equivalent if they divide each other.

Theorem 1. Two conjugate subshifts are weak equivalent.

Division of subshifts

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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For an alphabet A, let $ be a letter which is not in A. Denote by A$ the alphabet A ∪ {$}.

• A subshift X of AZ is a divisor of a subshift Y of BZ if there is a code

F : AZ → BZ

$ such that X = F −1Y.

• Two subshifts are said to be weak equivalent if they divide each other.

Theorem 1. Two conjugate subshifts are weak equivalent. Theorem 2. (M.-P. B´ eal and D. Perrin) Every two mixing subshifts of finite type are weak equivalent.

Division of subshifts

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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X ⊆ AZ Y ⊆ BZ c d a b e b b c a c d a b b b b c a

The subshifts X and Y are conjugate, but there is not a map f : An → B$ such that

L(X ) ∩ A≥n = ¯ f−1(L(Y)).

Pseudovarieties

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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• A divisor of a semigroup S is a homomorphic image of a

subsemigroup of S

• A pseudovariety of semigroups is a class of finite semigroups closed

under taking divisors and finite direct products.

Pseudovarieties

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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• A divisor of a semigroup S is a homomorphic image of a

subsemigroup of S

• A pseudovariety of semigroups is a class of finite semigroups closed

under taking divisors and finite direct products. Examples:

• The class G of finite groups
• The class A of finite semigroups whose subgroups are trivial
• The class Sl of finite idempotent commutative semigroups
• Given a pseudovariety V, the class LV of semigroups whose

submonoids belong to V

• The class D of semigroups whose idempotents are right zeros

Wreath product

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

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• Let S and T be semigroups. For h ∈ ST and t ∈ T , let th be the

map th(s) = h(st)

• The wreath product of S and T , denoted by S ◦ T is the set ST 1 × T

endowed with the following semigroup operation:

(f, t) · (h, s) = (f · th, t · s)

• The semi-direct product of two pseudovarieties V and W, denoted

by V ∗ W, is the class of the divisors of semigroups of the form

S ◦ T with S ∈ V and T ∈ W

• (V1 ∗ V2) ∗ V3 = V1 ∗ (V2 ∗ V3)
• D ∗ D = D
• LV ∗ D = LV

Wreath product

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 9

• Let S and T be semigroups. For h ∈ ST and t ∈ T , let th be the

map th(s) = h(st)

• The wreath product of S and T , denoted by S ◦ T is the set ST 1 × T

endowed with the following semigroup operation:

(f, t) · (h, s) = (f · th, t · s)

• The semi-direct product of two pseudovarieties V and W, denoted

by V ∗ W, is the class of the divisors of semigroups of the form

S ◦ T with S ∈ V and T ∈ W

• (V1 ∗ V2) ∗ V3 = V1 ∗ (V2 ∗ V3)
• D ∗ D = D
• LV ∗ D = LV

Wreath product

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 9

• Let S and T be semigroups. For h ∈ ST and t ∈ T , let th be the

map th(s) = h(st)

• The wreath product of S and T , denoted by S ◦ T is the set ST 1 × T

endowed with the following semigroup operation:

(f, t) · (h, s) = (f · th, t · s)

• The semi-direct product of two pseudovarieties V and W, denoted

by V ∗ W, is the class of the divisors of semigroups of the form

S ◦ T with S ∈ V and T ∈ W

• (V1 ∗ V2) ∗ V3 = V1 ∗ (V2 ∗ V3)
• D ∗ D = D
• LV ∗ D = LV

Wreath product

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 9

• Let S and T be semigroups. For h ∈ ST and t ∈ T , let th be the

map th(s) = h(st)

• The wreath product of S and T , denoted by S ◦ T is the set ST 1 × T

endowed with the following semigroup operation:

(f, t) · (h, s) = (f · th, t · s)

• The semi-direct product of two pseudovarieties V and W, denoted

by V ∗ W, is the class of the divisors of semigroups of the form

S ◦ T with S ∈ V and T ∈ W

• (V1 ∗ V2) ∗ V3 = V1 ∗ (V2 ∗ V3)
• D ∗ D = D
• LV ∗ D = LV

Wreath product

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 9

• Let S and T be semigroups. For h ∈ ST and t ∈ T , let th be the

map th(s) = h(st)

• The wreath product of S and T , denoted by S ◦ T is the set ST 1 × T

endowed with the following semigroup operation:

(f, t) · (h, s) = (f · th, t · s)

• The semi-direct product of two pseudovarieties V and W, denoted

by V ∗ W, is the class of the divisors of semigroups of the form

S ◦ T with S ∈ V and T ∈ W

• (V1 ∗ V2) ∗ V3 = V1 ∗ (V2 ∗ V3)
• D ∗ D = D
• LV ∗ D = LV

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b a a b a b a b

Transition semigroup denoted by D2

D2 ∈ D

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/f(aab) a/f(bba) a/f(aba) b/f(bab) a/f(baa) b/f(abb) a/f(aaa) b/f(bbb) f : {a, b}3 → {x, y, z}

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z a/z b/y a/x b/z a/y b/x f : {a, b}3 → {x, y, z}

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z a/z b/y a/x b/z a/y b/x f : {a, b}3 → {x, y, z} ¯ f(aababbbb) = · · ·

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z a/z b/y a/x b/z a/y b/x f : {a, b}3 → {x, y, z} ¯ f(aababbbb) = · · ·

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z a/z b/y a/x b/z a/y b/x f : {a, b}3 → {x, y, z} ¯ f(aababbbb) = x · · ·

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z a/z b/y a/x b/z a/y b/x f : {a, b}3 → {x, y, z} ¯ f(aababbbb) = xz · · ·

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z a/z b/y a/x b/z a/y b/x f : {a, b}3 → {x, y, z} ¯ f(aababbbb) = xzy · · ·

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z b/z a/z b/y a/x a/y b/x f : {a, b}3 → {x, y, z} ¯ f(aababbbb) = xzyz · · ·

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z b/z a/z b/y a/x a/y b/x f : {a, b}3 → {x, y, z} ¯ f(aababbbb) = xzyzx · · ·

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z b/z a/z b/y a/x a/y b/x f : {a, b}3 → {x, y, z} ¯ f(aababbbb) = xzyzxx

The De Bruijn automaton and the transducer of a block map

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 10

aa ab ba bb b/x a/z b/z a/z b/y a/x a/y b/x f : {a, b}3 → {x, y, z} F(. . . aababbbb . . .) = . . . xzyzxx . . .,

where F : {a, b}Z → {x, y, z}Z has f as block map

Inverse image

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 11

Theorem 3. Given alphabets A and B and a positive integer k, consider a map f : Ak → B. Let Y be a rational language of B+. Then Synt( ¯

f−1Y ) divides Synt(Y ) ◦ Dk−1.

Inverse image

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 11

Theorem 3. Given alphabets A and B and a positive integer k, consider a map f : Ak → B. Let Y be a rational language of B+. Then Synt( ¯

f−1Y ) divides Synt(Y ) ◦ Dk−1.

Theorem 4. (L. Chaubard and A. Costa) Let X and Y be subshifts of AZ and BZ, respectively. Consider a code F : AZ → BZ with window size k. Then Synt(L(F −1Y)) divides Synt(L(Y)) ◦ Dk−1.

Inverse image

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 11

Theorem 3. Given alphabets A and B and a positive integer k, consider a map f : Ak → B. Let Y be a rational language of B+. Then Synt( ¯

f−1Y ) divides Synt(Y ) ◦ Dk−1.

Theorem 4. (L. Chaubard and A. Costa) Let X and Y be subshifts of AZ and BZ, respectively. Consider a code F : AZ → BZ with window size k. Then Synt(L(F −1Y)) divides Synt(L(Y)) ◦ Dk−1. For a pseudovariety V of semigroups, denote by S(V) the class of subshifts X such that Synt(L(X )) ∈ V.

Inverse image

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 11

Theorem 3. Given alphabets A and B and a positive integer k, consider a map f : Ak → B. Let Y be a rational language of B+. Then Synt( ¯

f−1Y ) divides Synt(Y ) ◦ Dk−1.

Theorem 4. (L. Chaubard and A. Costa) Let X and Y be subshifts of AZ and BZ, respectively. Consider a code F : AZ → BZ with window size k. Then Synt(L(F −1Y)) divides Synt(L(Y)) ◦ Dk−1. For a pseudovariety V of semigroups, denote by S(V) the class of subshifts X such that Synt(L(X )) ∈ V. Corollary 5. If Sl ⊆ V then S(V ∗ D) is closed under taking divisors.

Inverse image

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 11

Theorem 3. Given alphabets A and B and a positive integer k, consider a map f : Ak → B. Let Y be a rational language of B+. Then Synt( ¯

f−1Y ) divides Synt(Y ) ◦ Dk−1.

Theorem 4. (L. Chaubard and A. Costa) Let X and Y be subshifts of AZ and BZ, respectively. Consider a code F : AZ → BZ with window size k. Then Synt(L(F −1Y)) divides Synt(L(Y)) ◦ Dk−1. For a pseudovariety V of semigroups, denote by S(V) the class of subshifts X such that Synt(L(X )) ∈ V. Corollary 5. If Sl ⊆ V then S(V ∗ D) is closed under taking divisors. Proposition 6. (A. Costa) If S(V) is closed under taking conjugate subshifts then LSl ⊆ V and S(V) = S(V ∗ D).

Example

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 12

X Y a a a a c b c b d a a a a c b b c d

Example

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 12

X Y a a a a c b c b d a a a a c b b c d

• V = [[x3 = x2]]

Example

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 12

X Y a a a a c b c b d a a a a c b b c d

• V = [[x3 = x2]]
• Sl ⊆ V

Example

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 12

X Y a a a a c b c b d a a a a c b b c d

• V = [[x3 = x2]]
• Sl ⊆ V
• LV = LV ∗ D

Example

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 12

X Y a a a a c b c b d a a a a c b b c d

• V = [[x3 = x2]]
• Sl ⊆ V
• LV = LV ∗ D
• X /

∈ S(LV), Y ∈ S(LV)

Example

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 12

X Y a a a a c b c b d a a a a c b b c d

• V = [[x3 = x2]]
• Sl ⊆ V
• LV = LV ∗ D
• X /

∈ S(LV), Y ∈ S(LV)

• hence X and Y are not weak equivalent

Classes closed under division

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 13

• (M.-P. B´

eal, F. Fiorenzi, D. Perrin + A. Costa) The almost finite type subshifts are the irreducible members of

L[[xωyω = yωxω]].

Corollary 7. The class of almost finite type subshifts is closed under taking irreducible divisors.

Classes closed under division

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 13

• (M.-P. B´

eal, F. Fiorenzi, D. Perrin + A. Costa) The almost finite type subshifts are the irreducible members of

L[[xωyω = yωxω]].

Corollary 7. The class of almost finite type subshifts is closed under taking irreducible divisors.

• (M.-P. B´

eal, F. Fiorenzi, D. Perrin) The aperiodic subshifts are the irreducible members of A. Corollary 8. The class of aperiodic subshifts is closed under taking irreducible divisors.

How we proved Theorem 4

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 14

ω-semigroups

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 15

An ω-semigroup is a pair S = (S+, Sω) equipped with

• a semigroup structure in S+
• an action S+ × Sω → Sω such that s(tu) = (st)u for all s, t ∈ S+,

u ∈ Sω

• a map π : SN

+ → Sω such that

• π(s0s1 · · · s(i1−1), si1 · · · s(i2−1), · · · ) = π(s0, s1, s2, · · · )
• s π(s0, s1, s2, · · · ) = π(s, s0, s1, s2, · · · )

An ˜

ω-semigroup is a pair S = (S+, S˜

ω) defined similarly, but with

products operating on the left.

ω-semigroups

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 15

An ω-semigroup is a pair S = (S+, Sω) equipped with

• a semigroup structure in S+
• an action S+ × Sω → Sω such that s(tu) = (st)u for all s, t ∈ S+,

u ∈ Sω

• a map π : SN

+ → Sω such that

• π(s0s1 · · · s(i1−1), si1 · · · s(i2−1), · · · ) = π(s0, s1, s2, · · · )
• s π(s0, s1, s2, · · · ) = π(s, s0, s1, s2, · · · )

An ˜

ω-semigroup is a pair S = (S+, S˜

ω) defined similarly, but with

products operating on the left.

☞ Notation:

• π(s0, s1, s2, · · · ) = s0s1s2 · · ·
• sω = sssss . . .
• s˜

ω = . . . sssss

ζ-semigroups

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 16

A ζ-semigroup is a 4-tuple S = (S+, Sω, S˜

ω, Sζ) equipped with a

• an ω-semigroup structure in (S+, Sω)
• an ˜

ω-semigroup structure in (S+, S˜

ω) (the same operation in S+)

• a surjective map · : S˜

ω × Sω → Sζ such that if s ∈ S˜ ω, t ∈ S+, and

u ∈ Sω then s · (tu) = (st) · u.

ζ-semigroups

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 16

A ζ-semigroup is a 4-tuple S = (S+, Sω, S˜

ω, Sζ) equipped with a

• an ω-semigroup structure in (S+, Sω)
• an ˜

ω-semigroup structure in (S+, S˜

ω) (the same operation in S+)

• a surjective map · : S˜

ω × Sω → Sζ such that if s ∈ S˜ ω, t ∈ S+, and

u ∈ Sω then s · (tu) = (st) · u.

Let S and T be ζ-semigroups. A ζ-semigroup morphism from S into T is a map ϕ : S → T such that for all λ ∈ {+, ω, ˜

ω, ζ} one has ϕ(Sλ) ⊆ Tλ and ϕ behaves well with respect to the operations.

The restriction ϕ : Sτ → Tτ is denoted by ϕτ .

ζ-semigroups

Symbolic dynamical systems Codes Sliding block codes Coding of finite words Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division

ω-semigroups ζ-semigroups

slide 16

A ζ-semigroup is a 4-tuple S = (S+, Sω, S˜

ω, Sζ) equipped with a

• an ω-semigroup structure in (S+, Sω)
• an ˜

ω-semigroup structure in (S+, S˜

ω) (the same operation in S+)

• a surjective map · : S˜

ω × Sω → Sζ such that if s ∈ S˜ ω, t ∈ S+, and

u ∈ Sω then s · (tu) = (st) · u.

Let S and T be ζ-semigroups. A ζ-semigroup morphism from S into T is a map ϕ : S → T such that for all λ ∈ {+, ω, ˜

ω, ζ} one has ϕ(Sλ) ⊆ Tλ and ϕ behaves well with respect to the operations.

The restriction ϕ : Sτ → Tτ is denoted by ϕτ . Example 9. Denote by Aζ the quotient of AZ under the equivalence relation:

u ∼σ v ⇔ ∃n | u = σn(v).

The 4-tuple A∞ = (A+, AN, AZ−, Aζ) equipped with the usual concatenation is a ζ-semigroup, called the free ζ-semigroup on A.

The syntactic ζ-semigroup

slide 17

Consider a subset P of Aζ. The syntactic congruence on P is the 4-tuple of equivalence relations (∼+, ∼ω, ∼˜

ω, ∼ζ) defined by

1.

∀s, t ∈ A+, s ∼+ t ⇐ ⇒ 8 > > < > > : ∀x ∈ A˜

ω, ∀y ∈ Aω,

xsy ∈ P ⇔ xty ∈ P ∀x ∈ A˜

ω, ∀y ∈ A+,

x(sy)ω ∈ P ⇔ x(ty)ω ∈ P ∀x ∈ A+, ∀y ∈ Aω, (xs)˜

ωy ∈ P

⇔ (xt)˜

ωy ∈ P

∀x ∈ A+, (xs)ζ ∈ P ⇔ (xt)ζ ∈ P

2.

∀s, t ∈ Aω, s ∼ω t ⇐ ⇒ ˆ ∀x ∈ A˜

ω,

xs ∈ P ⇔ xt ∈ P ˜

3.

∀s, t ∈ A˜

ω,

s ∼˜

ω t ⇐

⇒ ˆ ∀x ∈ Aω, xs ∈ P ⇔ xt ∈ P ˜

4.

∀s, t ∈ Aζ, s ∼ζ t ⇐ ⇒ ˆ s ∈ P ⇔ t ∈ P ˜

The syntactic ζ-semigroup

slide 17

Consider a subset P of Aζ. The syntactic congruence on P is the 4-tuple of equivalence relations (∼+, ∼ω, ∼˜

ω, ∼ζ) defined by

1.

∀s, t ∈ A+, s ∼+ t ⇐ ⇒ 8 > > < > > : ∀x ∈ A˜

ω, ∀y ∈ Aω,

xsy ∈ P ⇔ xty ∈ P ∀x ∈ A˜

ω, ∀y ∈ A+,

x(sy)ω ∈ P ⇔ x(ty)ω ∈ P ∀x ∈ A+, ∀y ∈ Aω, (xs)˜

ωy ∈ P

⇔ (xt)˜

ωy ∈ P

∀x ∈ A+, (xs)ζ ∈ P ⇔ (xt)ζ ∈ P

2.

∀s, t ∈ Aω, s ∼ω t ⇐ ⇒ ˆ ∀x ∈ A˜

ω,

xs ∈ P ⇔ xt ∈ P ˜

3.

∀s, t ∈ A˜

ω,

s ∼˜

ω t ⇐

⇒ ˆ ∀x ∈ Aω, xs ∈ P ⇔ xt ∈ P ˜

4.

∀s, t ∈ Aζ, s ∼ζ t ⇐ ⇒ ˆ s ∈ P ⇔ t ∈ P ˜

☞ Denote by S(P) the 4-tuple (A+/∼+, Aω/∼ω, A˜

ω/∼˜ ω, Aζ/∼ζ).

☞ Denote by πP the quotient map from A∞ to S(P).

Proposition 10. If S(P) is finite then πP defines in S(P) a structure of ζ-semigroup for which πP is a homomorphism of ζ-semigroups. Moreover, πP recognizes P .

The syntactic ζ-semigroup

slide 17

Consider a subset P of Aζ. The syntactic congruence on P is the 4-tuple of equivalence relations (∼+, ∼ω, ∼˜

ω, ∼ζ) defined by

1.

∀s, t ∈ A+, s ∼+ t ⇐ ⇒ 8 > > < > > : ∀x ∈ A˜

ω, ∀y ∈ Aω,

xsy ∈ P ⇔ xty ∈ P ∀x ∈ A˜

ω, ∀y ∈ A+,

x(sy)ω ∈ P ⇔ x(ty)ω ∈ P ∀x ∈ A+, ∀y ∈ Aω, (xs)˜

ωy ∈ P

⇔ (xt)˜

ωy ∈ P

∀x ∈ A+, (xs)ζ ∈ P ⇔ (xt)ζ ∈ P

2.

∀s, t ∈ Aω, s ∼ω t ⇐ ⇒ ˆ ∀x ∈ A˜

ω,

xs ∈ P ⇔ xt ∈ P ˜

3.

∀s, t ∈ A˜

ω,

s ∼˜

ω t ⇐

⇒ ˆ ∀x ∈ Aω, xs ∈ P ⇔ xt ∈ P ˜

4.

∀s, t ∈ Aζ, s ∼ζ t ⇐ ⇒ ˆ s ∈ P ⇔ t ∈ P ˜

☞ Denote by S(P) the 4-tuple (A+/∼+, Aω/∼ω, A˜

ω/∼˜ ω, Aζ/∼ζ).

☞ Denote by πP the quotient map from A∞ to S(P).

Proposition 10. If S(P) is finite then πP defines in S(P) a structure of ζ-semigroup for which πP is a homomorphism of ζ-semigroups. Moreover, πP recognizes P . Proposition 11. Let X be a sofic subshift of AZ. Then S(X ) if finite and the syntactic semigroup of L(X ) is S(X )+.

Wreath product for ζ-semigroups

slide 18

Let S be a finite ζ-semigroup, and T a semigroup from D. Denote by S ◦ T the 4-tuple

• SE(T)

+

× T, SE(T)

ω

, S˜

ω × E(T), Sζ

• equipped with the following structure:

1.

SE(T )

+

× T is the semigroup given by (f1, t1)·(f2, t2)=(f, t1t2) with f(e)=f1(e)f2(et1).

2. for all (f, t) ∈ SE(T )

+

× T and for all g ∈ SE(T )

ω

we have (a) (f, t) · g = h, with h(e) = f(e)g(et), (b) (f, t)ω =h, with h(e) = f ′(e)(f ′(t′)

ω, where (f ′, t′) is the idempotent power of (f, t).

3. for all (s, e) ∈ S˜

ω × E(T) and for all have (f, t) ∈ SE(T ) +

× T

(a) (s, e) · (f, t) = (sf(e), et), (b) (f, t)˜

ω =

• f ′(t′)˜

ω, t′

, where (f ′, t′) is the idempotent power of (f, t). 4. for all (s, e) ∈ S˜

ω × E(T) and for all g ∈ SE(T ) ω

we have (s, e) · g = sg(e).

Wreath product for ζ-semigroups

slide 18

Let S be a finite ζ-semigroup, and T a semigroup from D. Denote by S ◦ T the 4-tuple

• SE(T)

+

× T, SE(T)

ω

, S˜

ω × E(T), Sζ

• equipped with the following structure:

1.

SE(T )

+

× T is the semigroup given by (f1, t1)·(f2, t2)=(f, t1t2) with f(e)=f1(e)f2(et1).

2. for all (f, t) ∈ SE(T )

+

× T and for all g ∈ SE(T )

ω

we have (a) (f, t) · g = h, with h(e) = f(e)g(et), (b) (f, t)ω =h, with h(e) = f ′(e)(f ′(t′)

ω, where (f ′, t′) is the idempotent power of (f, t).

3. for all (s, e) ∈ S˜

ω × E(T) and for all have (f, t) ∈ SE(T ) +

× T

(a) (s, e) · (f, t) = (sf(e), et), (b) (f, t)˜

ω =

• f ′(t′)˜

ω, t′

, where (f ′, t′) is the idempotent power of (f, t). 4. for all (s, e) ∈ S˜

ω × E(T) and for all g ∈ SE(T ) ω

we have (s, e) · g = sg(e).

Proposition 12. S ◦ T is a ζ-semigroup. Remark 13. (S ◦ T)+ is a homomorphic image of S+ ◦ T .

Wreath product for ζ-semigroups

slide 19

Theorem 14. (Chaubard Master’s Thesis) Let F : Aζ → Bζ be a code with window size k and let Y be a subset of Bζ recognized by a finite ζ-semigroup Z. Then F −1Y is recognized by Z ◦ Dk−1.

Wreath product for ζ-semigroups

slide 19

Theorem 14. (Chaubard Master’s Thesis) Let F : Aζ → Bζ be a code with window size k and let Y be a subset of Bζ recognized by a finite ζ-semigroup Z. Then F −1Y is recognized by Z ◦ Dk−1. Theorem 15. Let F : AZ → BZ be a code with window size k and let Y be a sofic subshift of BZ. Then Synt(L(F −1Y)) divides Synt(L(Y)) ◦ Dk−1.

• Proof. Let Z be the syntactic ζ-semigroup of Y. By Theorem 14 there is a ζ-semigroup homomorphim

ψ : A∞ → Z ◦ Dk−1 such that F −1(Y) = ψ−1

ζ ψζ(F −1Y). Then

L(F −1Y) = {u ∈ A+ | ∃x ∈ A˜

ω, y ∈ Aω : xuy ∈ F −1Y}

= {u ∈ A+ | ∃x ∈ A˜

ω, y ∈ Aω : ψζ(xuy) ∈ ψζ(F −1Y)}

= ψ−1

+ {t ∈ T+ | ∃x ∈ A˜ ω, y ∈ Aω : ψ˜ ω(x) t ψω(y) ∈ ψζ(F −1Y)}

Hence L(F −1Y) is recognized by (Z ◦ Dk−1)+, thus Synt(L(F −1Y)) ≺ (Z ◦ Dk−1)+. Then

Synt(L(F −1Y)) ≺ Z+ ◦ Dk−1 by Remark 13. By Proposition 11 we have Z+ = Synt(L(Y)).

A topological proof

slide 20

Let ΩAV be the projective limit of all elements from V. Endowed with the product topology, it is a compact semigroup. If V contains the pseudovariety N of semigroups with a zero, then A+ is (isomorphic to) a subsemigroup of ΩAV.

A topological proof

slide 20

Let ΩAV be the projective limit of all elements from V. Endowed with the product topology, it is a compact semigroup. If V contains the pseudovariety N of semigroups with a zero, then A+ is (isomorphic to) a subsemigroup of ΩAV. Proposition 16. (Almeida) Let L be a language of A+. Then Synt(L) ∈ V if and only if

L is open in ΩAV.

A topological proof

slide 20

Let ΩAV be the projective limit of all elements from V. Endowed with the product topology, it is a compact semigroup. If V contains the pseudovariety N of semigroups with a zero, then A+ is (isomorphic to) a subsemigroup of ΩAV. Proposition 16. (Almeida) Let L be a language of A+. Then Synt(L) ∈ V if and only if

L is open in ΩAV.

Proposition 17. Let V be a pseudovariety containing Sl and N. Consider a block map

f : Ak → B. Then the map ¯ f : A+ → B∗ has a (unique) continuous extension ¯ f : ΩA(V ∗ Dk−1) → ΩBV ∪ {1}.

A topological proof

slide 20

Let ΩAV be the projective limit of all elements from V. Endowed with the product topology, it is a compact semigroup. If V contains the pseudovariety N of semigroups with a zero, then A+ is (isomorphic to) a subsemigroup of ΩAV. Proposition 16. (Almeida) Let L be a language of A+. Then Synt(L) ∈ V if and only if

L is open in ΩAV.

Proposition 17. Let V be a pseudovariety containing Sl and N. Consider a block map

f : Ak → B. Then the map ¯ f : A+ → B∗ has a (unique) continuous extension ¯ f : ΩA(V ∗ Dk−1) → ΩBV ∪ {1}.

We then recover: Theorem 18. Let V be a pseudovariety containing Sl and N. Consider a code

F : AZ → BZ with window size k. Let Y be a subshift of BZ. Then: Synt(L(Y)) ∈ V = ⇒ Synt(L(F −1Y)) ∈ V ∗ Dk−1.