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 slide 1
Symbolic dynamical systems Symbolic dynamical A Z A Z σ : → systems ( x i ) i ∈ Z �→ ( x i +1 ) i ∈ Z Codes Sliding block codes A subset X of A Z is a symbolic dynamical system or subshift if: Coding of finite words Division of subshifts Pseudovarieties X is topologically closed � Wreath product The De Bruijn σ ( X ) ⊆ X � automaton Inverse image σ − 1 ( X ) ⊆ X � Example Classes closed under division ω -semigroups ζ -semigroups slide 2
Symbolic dynamical systems Symbolic dynamical A Z A Z σ : → systems ( x i ) i ∈ Z �→ ( x i +1 ) i ∈ Z Codes Sliding block codes A subset X of A Z is a symbolic dynamical system or subshift if: Coding of finite words Division of subshifts Pseudovarieties X is topologically closed � Wreath product The De Bruijn σ ( X ) ⊆ X � automaton Inverse image σ − 1 ( X ) ⊆ X � Example Classes closed under division ω -semigroups L ( X ) = { u ∈ A + : u = x i x i +1 . . . x i + n for some x ∈ X , i ∈ Z , n ≥ 0 } . ζ -semigroups slide 2
Symbolic dynamical systems Symbolic dynamical A Z A Z σ : → systems ( x i ) i ∈ Z �→ ( x i +1 ) i ∈ Z Codes Sliding block codes A subset X of A Z is a symbolic dynamical system or subshift if: Coding of finite words Division of subshifts Pseudovarieties X is topologically closed � Wreath product The De Bruijn σ ( X ) ⊆ X � automaton Inverse image σ − 1 ( X ) ⊆ X � Example Classes closed under division ω -semigroups L ( X ) = { u ∈ A + : u = x i x i +1 . . . x i + n for some x ∈ X , i ∈ Z , n ≥ 0 } . ζ -semigroups L ( X ) is factorial and prolongable � If L is a factorial prolongable language of A + , then there is a unique � subshift of A Z such that L = L ( X ) : X ⇄ L ( X ) slide 2
Symbolic dynamical systems Symbolic dynamical A Z A Z σ : → systems ( x i ) i ∈ Z �→ ( x i +1 ) i ∈ Z Codes Sliding block codes A subset X of A Z is a symbolic dynamical system or subshift if: Coding of finite words Division of subshifts Pseudovarieties X is topologically closed � Wreath product The De Bruijn σ ( X ) ⊆ X � automaton Inverse image σ − 1 ( X ) ⊆ X � Example Classes closed under division ω -semigroups L ( X ) = { u ∈ A + : u = x i x i +1 . . . x i + n for some x ∈ X , i ∈ Z , n ≥ 0 } . ζ -semigroups L ( X ) is factorial and prolongable � If L is a factorial prolongable language of A + , then there is a unique � subshift of A Z such that L = L ( X ) : X ⇄ L ( X ) A subshift X is called sofic if L ( X ) is rational. slide 2
� � � Codes A code between two subshifts X and Y is a continuous map from X Symbolic dynamical � systems to Y that respects the shift operation: Codes Sliding block codes σ Coding of finite words X X Division of subshifts Pseudovarieties f f Wreath product σ � Y The De Bruijn Y automaton Inverse image Example Classes closed under Conjugation : bijective code � division ω -semigroups Two subshifts are conjugate if there is a conjugation between them � ζ -semigroups slide 3
� � � Codes A code between two subshifts X and Y is a continuous map from X Symbolic dynamical � systems to Y that respects the shift operation: Codes Sliding block codes σ Coding of finite words X X Division of subshifts Pseudovarieties f f Wreath product σ � Y The De Bruijn Y automaton Inverse image Example Classes closed under Conjugation : bijective code � division ω -semigroups Two subshifts are conjugate if there is a conjugation between them � ζ -semigroups Open problem: is conjugation of sofic subshifts decidable? � slide 3
Sliding block codes Let x ∈ A Z . Given a map f : A k → B , we can code x with f : 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
Sliding block codes Let x ∈ A Z . Given a map f : A k → B , we can code x with f : Symbolic dynamical systems Codes choose m and n such that k = m + n + 1 ; � 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
Sliding block codes Let x ∈ A Z . Given a map f : A k → B , we can code x with f : Symbolic dynamical systems Codes choose m and n such that k = m + n + 1 ; � Sliding block codes Coding of finite words make y i = f ( x [ i − m,i + n ] ) . � Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division ω -semigroups ζ -semigroups slide 4
Sliding block codes Let x ∈ A Z . Given a map f : A k → B , we can code x with f : Symbolic dynamical systems Codes choose m and n such that k = m + n + 1 ; � Sliding block codes Coding of finite words make y i = f ( x [ i − m,i + n ] ) . � Division of subshifts Pseudovarieties Wreath product The De Bruijn . . . x i − 4 x i − 3 x i − 2 x i − 1 x i x i +1 x i +2 x i +3 . . . automaton Inverse image f Example � Classes closed under division . . . y i − 2 y i − 1 y i y i +1 y i +2 . . . ω -semigroups ζ -semigroups slide 4
Sliding block codes Let x ∈ A Z . Given a map f : A k → B , we can code x with f : Symbolic dynamical systems Codes choose m and n such that k = m + n + 1 ; � Sliding block codes Coding of finite words make y i = f ( x [ i − m,i + n ] ) . � Division of subshifts Pseudovarieties Wreath product The De Bruijn . . . x i − 4 x i − 3 x i − 2 x i − 1 x i x i +1 x i +2 x i +3 . . . automaton Inverse image f Example � Classes closed under division . . . y i − 2 y i − 1 y i y i +1 y i +2 . . . ω -semigroups ζ -semigroups slide 4
Sliding block codes Let x ∈ A Z . Given a map f : A k → B , we can code x with f : Symbolic dynamical systems Codes choose m and n such that k = m + n + 1 ; � Sliding block codes Coding of finite words make y i = f ( x [ i − m,i + n ] ) . � Division of subshifts Pseudovarieties Wreath product The De Bruijn . . . x i − 4 x i − 3 x i − 2 x i − 1 x i x i +1 x i +2 x i +3 . . . automaton Inverse image f Example � Classes closed under division . . . y i − 2 y i − 1 y i y i +1 y i +2 . . . ω -semigroups ζ -semigroups slide 4
Sliding block codes Let x ∈ A Z . Given a map f : A k → B , we can code x with f : Symbolic dynamical systems Codes choose m and n such that k = m + n + 1 ; � Sliding block codes Coding of finite words make y i = f ( x [ i − m,i + n ] ) . � Division of subshifts Pseudovarieties Wreath product The De Bruijn . . . x i − 4 x i − 3 x i − 2 x i − 1 x i x i +1 x i +2 x i +3 . . . automaton Inverse image f Example � Classes closed under division . . . y i − 2 y i − 1 y i y i +1 y i +2 . . . ω -semigroups ζ -semigroups the map � F : ( x i ) i ∈ Z �→ ( y i ) i ∈ Z is a code and all codes have this form slide 4
Sliding block codes Let x ∈ A Z . Given a map f : A k → B , we can code x with f : Symbolic dynamical systems Codes choose m and n such that k = m + n + 1 ; � Sliding block codes Coding of finite words make y i = f ( x [ i − m,i + n ] ) . � Division of subshifts Pseudovarieties Wreath product The De Bruijn . . . x i − 4 x i − 3 x i − 2 x i − 1 x i x i +1 x i +2 x i +3 . . . automaton Inverse image f Example � Classes closed under division . . . y i − 2 y i − 1 y i y i +1 y i +2 . . . ω -semigroups ζ -semigroups the map � F : ( x i ) i ∈ Z �→ ( y i ) 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 � slide 4
Coding of finite words Let u = u 1 . . . u n ∈ A + , where u i ∈ A . Given a map f : A k → B , we Symbolic dynamical systems can use f to code u , through the following map ¯ f : Codes Sliding block codes if n < k then ¯ f ( u ) = 1 Coding of finite words � Division of subshifts if n ≥ k then Pseudovarieties � Wreath product The De Bruijn ¯ f ( u ) = f ( u 1 . . . u k ) f ( u 2 . . . u k +1 ) . . . f ( u n − k +1 . . . u n ) automaton Inverse image Example Classes closed under division ω -semigroups ζ -semigroups slide 5
Division of subshifts Symbolic dynamical For an alphabet A , let $ be a letter which is not in A . systems Denote by A $ the alphabet A ∪ { $ } . 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
Division of subshifts Symbolic dynamical For an alphabet A , let $ be a letter which is not in A . systems Denote by A $ the alphabet A ∪ { $ } . Codes Sliding block codes A subshift X of A Z is a divisor of a subshift Y of B Z if there is a code Coding of finite words � F : A Z → B Z $ such that X = F − 1 Y . Division of subshifts Pseudovarieties Wreath product The De Bruijn automaton Inverse image Example Classes closed under division ω -semigroups ζ -semigroups slide 6
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