Dynamique symboliquedes systemes 2D et des arbres infinis
Soutenance de these,encadree par Marie-Pierre Beal et Mathieu Sablik
Nathalie Aubrun
LIGM, Universite Paris-Est
22 juin 2011
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 1 / 24
Outline
1 Symbolic dynamicsWhat are symbolic dynamics ?SubshiftsClasses of subshifts
2 Motivation1D subshifts vs. 2D subshiftsTwo orthogonal approaches
3 A study of 2D-shiftsThe projective subactionHochman’s resultImprovement
4 Tree-shiftsTree-shift: exampleTree automata and tree-shiftsOn AFT tree-shiftsConjugacy of tree-SFT
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 1 / 24
Symbolic dynamics
Outline
1 Symbolic dynamicsWhat are symbolic dynamics ?SubshiftsClasses of subshifts
2 Motivation1D subshifts vs. 2D subshiftsTwo orthogonal approaches
3 A study of 2D-shiftsThe projective subactionHochman’s resultImprovement
4 Tree-shiftsTree-shift: exampleTree automata and tree-shiftsOn AFT tree-shiftsConjugacy of tree-SFT
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 1 / 24
Symbolic dynamics What are symbolic dynamics ?
Discrete dynamical systems
(X ,F ) is a discrete dynamical system if:X is a topological space, called the phase spaceF is a continuous map ∶ X → X
x●
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 2 / 24
Symbolic dynamics What are symbolic dynamics ?
Discrete dynamical systems
(X ,F ) is a discrete dynamical system if:X is a topological space, called the phase spaceF is a continuous map ∶ X → X
x●
F(x)●
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 2 / 24
Symbolic dynamics What are symbolic dynamics ?
Discrete dynamical systems
(X ,F ) is a discrete dynamical system if:X is a topological space, called the phase spaceF is a continuous map ∶ X → X
x●
F(x)●
F 2(x)●
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 2 / 24
Symbolic dynamics What are symbolic dynamics ?
Discrete dynamical systems
(X ,F ) is a discrete dynamical system if:X is a topological space, called the phase spaceF is a continuous map ∶ X → X
x●
F(x)●
F 2(x)●
F 3(x)●
F 4(x)●
F 5(x)●
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 2 / 24
Symbolic dynamics What are symbolic dynamics ?
Coding of the orbits
X = ⋃ni=1 Xi a partition of the phase space X
a color ai associated with each Xi
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 3 / 24
Symbolic dynamics What are symbolic dynamics ?
Coding of the orbits
X = ⋃ni=1 Xi a partition of the phase space X
a color ai associated with each Xi
orbit (F n(x))n∈N coded by a sequence y ∈ {a1, . . . , an}N
●
● ●
●
●
●
. . .
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 3 / 24
Symbolic dynamics Subshifts
Subshifts: topological definition
A finite alphabet, AN (or AZ, AN2or AZ2
) the configurations spacethe configurations space is endowed with the prodiscrete topology ⇒compact spacenatural action of N (or Z, N2 or Z2) by translation: the shift σ
σj(x)i = xi+j for all x ∈ AN
(Topological) Definition: subshift
A subshift is a closed and σ-invariant subset of AN (or AZ, AN2or AZ2
).
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 4 / 24
Symbolic dynamics Subshifts
Subshifts: topological definition
A finite alphabet, AN (or AZ, AN2or AZ2
) the configurations spacethe configurations space is endowed with the prodiscrete topology ⇒compact spacenatural action of N (or Z, N2 or Z2) by translation: the shift σ
σj(x)i = xi+j for all x ∈ AN
(Topological) Definition: subshift
A subshift is a closed and σ-invariant subset of AN (or AZ, AN2or AZ2
).
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 4 / 24
Symbolic dynamics Subshifts
Subshifts: combinatorial definition
A finite alphabetpattern p ∈ AS, where S ⊂ N is finitethe pattern p = appears in the configuration
(Combinatorial) Definition: subshift
Let F be a set of finite patterns. The subshift defined by the set of forbiddenpatterns F is the set
TF = {x ∈ AN,no pattern of F appears in x} .
Proposition
The toplogical and combinatorial definitions coincide.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 5 / 24
Symbolic dynamics Subshifts
Subshifts: combinatorial definition
A finite alphabetpattern p ∈ AS, where S ⊂ N is finitethe pattern p = appears in the configuration
(Combinatorial) Definition: subshift
Let F be a set of finite patterns. The subshift defined by the set of forbiddenpatterns F is the set
TF = {x ∈ AN,no pattern of F appears in x} .
Proposition
The toplogical and combinatorial definitions coincide.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 5 / 24
Symbolic dynamics Subshifts
Subshifts: combinatorial definition
A finite alphabetpattern p ∈ AS, where S ⊂ N is finitethe pattern p = appears in the configuration
(Combinatorial) Definition: subshift
Let F be a set of finite patterns. The subshift defined by the set of forbiddenpatterns F is the set
TF = {x ∈ AN,no pattern of F appears in x} .
Proposition
The toplogical and combinatorial definitions coincide.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 5 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Subshifts of finite type (SFT)
Sets of configurations that avoid a finite set of forbidden patterns:alphabet { , } on None forbidden pattern
{x ∈ { , }N,∃i ∈ N ∪ {+∞}, (xj = ⇔ j ≤ i)}
Definition: subshift of finite type (SFT)
A subshift of finite type (SFT) is a subshift that can be defined by a finite set offorbidden patterns.
simplest class with respect to the combinatorial definition2D-SFT ≡ Wang tilings
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 6 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Subshifts of finite type (SFT)
Sets of configurations that avoid a finite set of forbidden patterns:alphabet { , } on None forbidden pattern
{x ∈ { , }N,∃i ∈ N ∪ {+∞}, (xj = ⇔ j ≤ i)}
Definition: subshift of finite type (SFT)
A subshift of finite type (SFT) is a subshift that can be defined by a finite set offorbidden patterns.
simplest class with respect to the combinatorial definition2D-SFT ≡ Wang tilings
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 6 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Sofic subshifts
Factor map Φ ∶ AZ2 → BZ2given by a local map φ:
x ∈ AZ2Φ(x) ∈ BZ2
Definition: sofic susbhift
A sofic subshift is the factor of an SFT.
Recodings of SFT, using local rules.
In 1D, sofic subshifts are exactly those recognized by finite automata.
In higher dimension, decide wether a subshift is sofic or not is a difficult problem.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 7 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Sofic subshifts
Factor map Φ ∶ AZ2 → BZ2given by a local map φ:
x ∈ AZ2Φ(x) ∈ BZ2
Definition: sofic susbhift
A sofic subshift is the factor of an SFT.
Recodings of SFT, using local rules.
In 1D, sofic subshifts are exactly those recognized by finite automata.
In higher dimension, decide wether a subshift is sofic or not is a difficult problem.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 7 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Sofic subshifts
Factor map Φ ∶ AZ2 → BZ2given by a local map φ:
x ∈ AZ2Φ(x) ∈ BZ2
Definition: sofic susbhift
A sofic subshift is the factor of an SFT.
Recodings of SFT, using local rules.
In 1D, sofic subshifts are exactly those recognized by finite automata.
In higher dimension, decide wether a subshift is sofic or not is a difficult problem.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 7 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Sofic subshifts
Factor map Φ ∶ AZ2 → BZ2given by a local map φ:
x ∈ AZ2Φ(x) ∈ BZ2
Definition: sofic susbhift
A sofic subshift is the factor of an SFT.
Recodings of SFT, using local rules.
In 1D, sofic subshifts are exactly those recognized by finite automata.
In higher dimension, decide wether a subshift is sofic or not is a difficult problem.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 7 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Sofic subshifts
Factor map Φ ∶ AZ2 → BZ2given by a local map φ:
x ∈ AZ2Φ(x) ∈ BZ2
Definition: sofic susbhift
A sofic subshift is the factor of an SFT.
Recodings of SFT, using local rules.
In 1D, sofic subshifts are exactly those recognized by finite automata.
In higher dimension, decide wether a subshift is sofic or not is a difficult problem.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 7 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Sofic subshifts
Factor map Φ ∶ AZ2 → BZ2given by a local map φ:
x ∈ AZ2Φ(x) ∈ BZ2
Definition: sofic susbhift
A sofic subshift is the factor of an SFT.
Recodings of SFT, using local rules.
In 1D, sofic subshifts are exactly those recognized by finite automata.
In higher dimension, decide wether a subshift is sofic or not is a difficult problem.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 7 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Sofic subshifts
Factor map Φ ∶ AZ2 → BZ2given by a local map φ:
x ∈ AZ2Φ(x) ∈ BZ2
Definition: sofic susbhift
A sofic subshift is the factor of an SFT.
Recodings of SFT, using local rules.
In 1D, sofic subshifts are exactly those recognized by finite automata.
In higher dimension, decide wether a subshift is sofic or not is a difficult problem.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 7 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Effective subshifts
Definition: effective susbhiftAn effective subshift is a subshift that can be defined by a recursively enumerableset of forbidden patterns.
reasonnable susbhiftthis class naturally appears as projective subactions of 2D-SFT
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 8 / 24
Symbolic dynamics Classes of subshifts
Classes of subshifts: Effective subshifts
Definition: effective susbhiftAn effective subshift is a subshift that can be defined by a recursively enumerableset of forbidden patterns.
reasonnable susbhiftthis class naturally appears as projective subactions of 2D-SFT
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 8 / 24
Motivation
Outline
1 Symbolic dynamicsWhat are symbolic dynamics ?SubshiftsClasses of subshifts
2 Motivation1D subshifts vs. 2D subshiftsTwo orthogonal approaches
3 A study of 2D-shiftsThe projective subactionHochman’s resultImprovement
4 Tree-shiftsTree-shift: exampleTree automata and tree-shiftsOn AFT tree-shiftsConjugacy of tree-SFT
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 8 / 24
Motivation 1D subshifts vs. 2D subshifts
1D subshifts vs. 2D subshifts
1D-subshifts 2D-subshiftsEmptyness of SFT ✓ ×
Periodicity in SFT∀ SFT,∃ periodic configuration ∃ aperiodic SFT
Decomposition theorem sofic subshifts
Conjugacy of SFTN ∶ ✓Z ∶ ? ×
Recognizers for SFTfinite
local automataRecognizers
for sofic subshifts finite automata textile systems
✓: decidable problem×: undecidable problem?: open problem
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 9 / 24
Motivation Two orthogonal approaches
Two orthogonal approaches
Two attempts to understand 2D-subshifts :
study 2D-SFT through operations acting on themstudy subshifts defined on a structure between dimensions 1 and 2
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 10 / 24
Motivation Two orthogonal approaches
Two orthogonal approaches
Two attempts to understand 2D-subshifts :study 2D-SFT through operations acting on them
study subshifts defined on a structure between dimensions 1 and 2
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 10 / 24
Motivation Two orthogonal approaches
Two orthogonal approaches
Two attempts to understand 2D-subshifts :study 2D-SFT through operations acting on themstudy subshifts defined on a structure between dimensions 1 and 2
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 10 / 24
Motivation Two orthogonal approaches
Operations on 2D-SFT
factor map operation (Fact): SFT ↝ sofic subshifts
operations that preserves SFT: product (P), finite type (FT) and spatialextensionoperation mainly studied: projective subaction (SA)
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 11 / 24
Motivation Two orthogonal approaches
Operations on 2D-SFT
factor map operation (Fact): SFT ↝ sofic subshiftsoperations that preserves SFT: product (P), finite type (FT) and spatialextension
operation mainly studied: projective subaction (SA)
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 11 / 24
Motivation Two orthogonal approaches
Operations on 2D-SFT
factor map operation (Fact): SFT ↝ sofic subshiftsoperations that preserves SFT: product (P), finite type (FT) and spatialextensionoperation mainly studied: projective subaction (SA)
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 11 / 24
Motivation Two orthogonal approaches
Tree-shifts
Structure in-between N and N2 ?
N N2
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 12 / 24
Motivation Two orthogonal approaches
Tree-shifts
Structure in-between N and N2 : free semi-group with two generators M2.
N M2 N2
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 12 / 24
A study of 2D-shifts
Outline
1 Symbolic dynamicsWhat are symbolic dynamics ?SubshiftsClasses of subshifts
2 Motivation1D subshifts vs. 2D subshiftsTwo orthogonal approaches
3 A study of 2D-shiftsThe projective subactionHochman’s resultImprovement
4 Tree-shiftsTree-shift: exampleTree automata and tree-shiftsOn AFT tree-shiftsConjugacy of tree-SFT
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 12 / 24
A study of 2D-shifts The projective subaction
The projective subaction
Idea: study subdynamics of 2D-subshifts.Principle: given a subgroup G of Z2 and a 2D-susbhift T on alphabet A, andconsider the G-subshift defined by
SAG (T) = {y ∈ AG ∶ ∃x ∈ T such that y = xT} .
G = {(x , y) ∈ Z2 ∶ y = x}
Proposition (A.&Sablik)
The class of effective subshifts is stable under projective subaction.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 13 / 24
A study of 2D-shifts The projective subaction
The projective subaction
Idea: study subdynamics of 2D-subshifts.Principle: given a subgroup G of Z2 and a 2D-susbhift T on alphabet A, andconsider the G-subshift defined by
SAG (T) = {y ∈ AG ∶ ∃x ∈ T such that y = xT} .
G = {(x , y) ∈ Z2 ∶ y = x}
Proposition (A.&Sablik)
The class of effective subshifts is stable under projective subaction.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 13 / 24
A study of 2D-shifts The projective subaction
The projective subaction
Idea: study subdynamics of 2D-subshifts.Principle: given a subgroup G of Z2 and a 2D-susbhift T on alphabet A, andconsider the G-subshift defined by
SAG (T) = {y ∈ AG ∶ ∃x ∈ T such that y = xT} .
G = {(x , y) ∈ Z2 ∶ y = x}
Proposition (A.&Sablik)
The class of effective subshifts is stable under projective subaction.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 13 / 24
A study of 2D-shifts The projective subaction
The projective subaction
Idea: study subdynamics of 2D-subshifts.Principle: given a subgroup G of Z2 and a 2D-susbhift T on alphabet A, andconsider the G-subshift defined by
SAG (T) = {y ∈ AG ∶ ∃x ∈ T such that y = xT} .
G = {(x , y) ∈ Z2 ∶ y = x}
Proposition (A.&Sablik)
The class of effective subshifts is stable under projective subaction.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 13 / 24
A study of 2D-shifts The projective subaction
The projective subaction
Idea: study subdynamics of 2D-subshifts.Principle: given a subgroup G of Z2 and a 2D-susbhift T on alphabet A, andconsider the G-subshift defined by
SAG (T) = {y ∈ AG ∶ ∃x ∈ T such that y = xT} .
G = {(x , y) ∈ Z2 ∶ y = x}
Proposition (A.&Sablik)
The class of effective subshifts is stable under projective subaction.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 13 / 24
A study of 2D-shifts The projective subaction
Projective subactions of 2D-SFT
Pavlov and Schraudner (2010):what can be realized as projective subactions of 2D-SFT ?
any sofic Z-subshift of positive entropyany zero-entropy sofic Z-subshift, with some conditions on its periods
there exist classes of Z-subshifts which are not realizable as projectivesubdynamics of any Zd SFT.
⇒ no complete characterization projective subactions of 2D-SFT.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 14 / 24
A study of 2D-shifts The projective subaction
Projective subactions of 2D-SFT
Pavlov and Schraudner (2010):what can be realized as projective subactions of 2D-SFT ?
any sofic Z-subshift of positive entropyany zero-entropy sofic Z-subshift, with some conditions on its periods
there exist classes of Z-subshifts which are not realizable as projectivesubdynamics of any Zd SFT.
⇒ no complete characterization projective subactions of 2D-SFT.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 14 / 24
A study of 2D-shifts The projective subaction
Projective subactions of 2D-SFT
Pavlov and Schraudner (2010):what can be realized as projective subactions of 2D-SFT ?
any sofic Z-subshift of positive entropyany zero-entropy sofic Z-subshift, with some conditions on its periods
there exist classes of Z-subshifts which are not realizable as projectivesubdynamics of any Zd SFT.
⇒ no complete characterization projective subactions of 2D-SFT.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 14 / 24
A study of 2D-shifts Hochman’s result
Hochman’s result
. . . no complete characterization projective subactions of 2D-SFT. . .
Consider factor map operations in addition to projective subactions.
Theorem (Hochman 2010)
Any effective Zd -subshift may be obtained by SA and Fact operations on aZd+2-SFT.
Natural question: is it possible to use one dimension less ?
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 15 / 24
A study of 2D-shifts Hochman’s result
Hochman’s result
. . . no complete characterization projective subactions of 2D-SFT. . .
Consider factor map operations in addition to projective subactions.
Theorem (Hochman 2010)
Any effective Zd -subshift may be obtained by SA and Fact operations on aZd+2-SFT.
Natural question: is it possible to use one dimension less ?
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 15 / 24
A study of 2D-shifts Hochman’s result
Hochman’s result
. . . no complete characterization projective subactions of 2D-SFT. . .
Consider factor map operations in addition to projective subactions.
Theorem (Hochman 2010)
Any effective Zd -subshift may be obtained by SA and Fact operations on aZd+2-SFT.
Natural question: is it possible to use one dimension less ?
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 15 / 24
A study of 2D-shifts Improvement
Improvement of Hochman’s result
Theorem
Any effective Zd -subshift may be obtained by SA and Fact operations on aZd+1-SFT.
Two different proofs:Durand, Romaschenko & Shen 2010, using self-similar tilingsA.& Sablik 2010, adaptation of Robinson’s construction
Many applications:correspondance between an order on subshifts and an order on languagesmultidimensional effective subshifts are sofic (A.& Sablik 2011)construction of a tiles set whose quasi-periodic tilings have a non-recursivelybounded periodicity function (Ballier& Jeandel 2010). . .
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 16 / 24
A study of 2D-shifts Improvement
Improvement of Hochman’s result
Theorem
Any effective Zd -subshift may be obtained by SA and Fact operations on aZd+1-SFT.
Two different proofs:Durand, Romaschenko & Shen 2010, using self-similar tilingsA.& Sablik 2010, adaptation of Robinson’s construction
Many applications:correspondance between an order on subshifts and an order on languagesmultidimensional effective subshifts are sofic (A.& Sablik 2011)construction of a tiles set whose quasi-periodic tilings have a non-recursivelybounded periodicity function (Ballier& Jeandel 2010). . .
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 16 / 24
A study of 2D-shifts Improvement
Improvement of Hochman’s result
Theorem
Any effective Zd -subshift may be obtained by SA and Fact operations on aZd+1-SFT.
Two different proofs:Durand, Romaschenko & Shen 2010, using self-similar tilingsA.& Sablik 2010, adaptation of Robinson’s construction
Many applications:correspondance between an order on subshifts and an order on languagesmultidimensional effective subshifts are sofic (A.& Sablik 2011)construction of a tiles set whose quasi-periodic tilings have a non-recursivelybounded periodicity function (Ballier& Jeandel 2010). . .
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 16 / 24
Tree-shifts
Outline
1 Symbolic dynamicsWhat are symbolic dynamics ?SubshiftsClasses of subshifts
2 Motivation1D subshifts vs. 2D subshiftsTwo orthogonal approaches
3 A study of 2D-shiftsThe projective subactionHochman’s resultImprovement
4 Tree-shiftsTree-shift: exampleTree automata and tree-shiftsOn AFT tree-shiftsConjugacy of tree-SFT
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 16 / 24
Tree-shifts Tree-shift: example
Example of tree-shift
alphabet A = { , }forbidden patterns: paths containing an even number of between two
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 17 / 24
Tree-shifts Tree automata and tree-shifts
Tree automata
alphabet A = { , }tree automaton A with states Q = {q0,q1,q●}transition rules:
∶ q1
q0 q0
∶ q0
q1 q1
∶ q0
q●,q1 q●,q1
∶ q●
q●,q1 q●,q1
Accepted trees:
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 18 / 24
Tree-shifts Tree automata and tree-shifts
Tree automata
alphabet A = { , }tree automaton A with states Q = {q0,q1,q●}transition rules:
∶ q1
q0 q0
∶ q0
q1 q1
∶ q0
q●,q1 q●,q1
∶ q●
q●,q1 q●,q1
Accepted trees:
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 18 / 24
Tree-shifts Tree automata and tree-shifts
Tree automata and tree-shifts
(finite) tree automata, all states are acceptinga tree is accepted iff there exists a computationdeterministic tree automata ≡ non deterministic tree automatathe set of trees accepted by a tree automaton is a subshift
Proposition (A. & Beal 2009)
A tree-shift is a sofic tree-shift iff it is recognized by a tree automaton.A tree-shift is a tree-SFT iff it is recognized by a local tree automaton.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 19 / 24
Tree-shifts Tree automata and tree-shifts
Tree automata and tree-shifts
(finite) tree automata, all states are acceptinga tree is accepted iff there exists a computationdeterministic tree automata ≡ non deterministic tree automatathe set of trees accepted by a tree automaton is a subshift
Proposition (A. & Beal 2009)
A tree-shift is a sofic tree-shift iff it is recognized by a tree automaton.A tree-shift is a tree-SFT iff it is recognized by a local tree automaton.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 19 / 24
Tree-shifts Tree automata and tree-shifts
Main difference with finite automata on words
Synchronizing block: every calculation (there exist at least one) of A on this blockends in the same state.
Proposition
Any finite automaton on words has a synchronizing word.
But...
Proposition (A.& Beal 2009)
There exists a deterministic, minimal and reduced tree automaton which has nosynchronizing block.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 20 / 24
Tree-shifts Tree automata and tree-shifts
Main difference with finite automata on words
Synchronizing block: every calculation (there exist at least one) of A on this blockends in the same state.
Proposition
Any finite automaton on words has a synchronizing word.
But...
Proposition (A.& Beal 2009)
There exists a deterministic, minimal and reduced tree automaton which has nosynchronizing block.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 20 / 24
Tree-shifts Tree automata and tree-shifts
The context automaton
The context automaton of a tree-shift T is the deterministic tree automatonC = (V ,A,∆) where
V is the set of non-empty contexts of finite blocks appearing in Ttransitions are (contT(u), contT(v)), a → contT(a,u, v), with u, v ∈ L(T).
Proposition (A.& Beal 2010)
The context automaton of a sofic tree-shift is synchronized.
Proposition (A.& Beal 2010)
The context automaton of a sofic tree-shift T has a unique minimal,irreducible and synchronized component S, called the Shannon cover of T.The Shannon cover of a sofic tree-shift is computable.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 21 / 24
Tree-shifts Tree automata and tree-shifts
The context automaton
The context automaton of a tree-shift T is the deterministic tree automatonC = (V ,A,∆) where
V is the set of non-empty contexts of finite blocks appearing in Ttransitions are (contT(u), contT(v)), a → contT(a,u, v), with u, v ∈ L(T).
Proposition (A.& Beal 2010)
The context automaton of a sofic tree-shift is synchronized.
Proposition (A.& Beal 2010)
The context automaton of a sofic tree-shift T has a unique minimal,irreducible and synchronized component S, called the Shannon cover of T.The Shannon cover of a sofic tree-shift is computable.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 21 / 24
Tree-shifts Tree automata and tree-shifts
The context automaton
The context automaton of a tree-shift T is the deterministic tree automatonC = (V ,A,∆) where
V is the set of non-empty contexts of finite blocks appearing in Ttransitions are (contT(u), contT(v)), a → contT(a,u, v), with u, v ∈ L(T).
Proposition (A.& Beal 2010)
The context automaton of a sofic tree-shift is synchronized.
Proposition (A.& Beal 2010)
The context automaton of a sofic tree-shift T has a unique minimal,irreducible and synchronized component S, called the Shannon cover of T.The Shannon cover of a sofic tree-shift is computable.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 21 / 24
Tree-shifts On AFT tree-shifts
On AFT tree-shifts
AFT tree-shifts: class in-between tree-SFT and sofic tree-shiftsgeneralization of AFT 1D-susbhifts, used for coding purposes
AFT tree-shifts: factors of tree-SFT, where the factor map satisfies syntacticproperties (right-resolving, left-closing and having a resolving block)theses properties on the factor map are computable
Proposition (A.& Beal 2010)
It is decidable to say wether a sofic tree-shift is AFT or not.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 22 / 24
Tree-shifts On AFT tree-shifts
On AFT tree-shifts
AFT tree-shifts: class in-between tree-SFT and sofic tree-shiftsgeneralization of AFT 1D-susbhifts, used for coding purposesAFT tree-shifts: factors of tree-SFT, where the factor map satisfies syntacticproperties (right-resolving, left-closing and having a resolving block)theses properties on the factor map are computable
Proposition (A.& Beal 2010)
It is decidable to say wether a sofic tree-shift is AFT or not.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 22 / 24
Tree-shifts On AFT tree-shifts
On AFT tree-shifts
AFT tree-shifts: class in-between tree-SFT and sofic tree-shiftsgeneralization of AFT 1D-susbhifts, used for coding purposesAFT tree-shifts: factors of tree-SFT, where the factor map satisfies syntacticproperties (right-resolving, left-closing and having a resolving block)theses properties on the factor map are computable
Proposition (A.& Beal 2010)
It is decidable to say wether a sofic tree-shift is AFT or not.
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 22 / 24
Tree-shifts Conjugacy of tree-SFT
The conjuguacy problem for tree-SFT
two subshifts are conjugate if they are both factor of the otherconjugate subshifts are the same, up to a recoding
Theorem (A.& Beal 2010)
The conjugacy problem is decidable for tree-SFT.
every conjuguacy can be splitted into a sequence of elementary conjuguacieselementary conjugacies ⇒ unique minimal amalgamation of a tree-SFTtwo tree-SFT are conjugate iff they have the same minimal amalgamation
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 23 / 24
Tree-shifts Conjugacy of tree-SFT
The conjuguacy problem for tree-SFT
two subshifts are conjugate if they are both factor of the otherconjugate subshifts are the same, up to a recoding
Theorem (A.& Beal 2010)
The conjugacy problem is decidable for tree-SFT.
every conjuguacy can be splitted into a sequence of elementary conjuguacieselementary conjugacies ⇒ unique minimal amalgamation of a tree-SFTtwo tree-SFT are conjugate iff they have the same minimal amalgamation
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 23 / 24
Tree-shifts Conjugacy of tree-SFT
The conjuguacy problem for tree-SFT
two subshifts are conjugate if they are both factor of the otherconjugate subshifts are the same, up to a recoding
Theorem (A.& Beal 2010)
The conjugacy problem is decidable for tree-SFT.
every conjuguacy can be splitted into a sequence of elementary conjuguacieselementary conjugacies ⇒ unique minimal amalgamation of a tree-SFTtwo tree-SFT are conjugate iff they have the same minimal amalgamation
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 23 / 24
Conclusion
Conclusion
Many properties on 2D-SFT are uncomputable. . .. . . nevertheless it is possible to partially describe them.
Tree-shifts are very similar to N-subshifts. . .. . . how to characterize monoids with same properties as 1D-subshifts ?
Thank you, Спасибо, Kiitos, Merci !
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 24 / 24
Conclusion
Conclusion
Many properties on 2D-SFT are uncomputable. . .. . . nevertheless it is possible to partially describe them.Tree-shifts are very similar to N-subshifts. . .. . . how to characterize monoids with same properties as 1D-subshifts ?
Thank you, Спасибо, Kiitos, Merci !
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 24 / 24
Conclusion
Conclusion
Many properties on 2D-SFT are uncomputable. . .. . . nevertheless it is possible to partially describe them.Tree-shifts are very similar to N-subshifts. . .. . . how to characterize monoids with same properties as 1D-subshifts ?
Thank you, Спасибо, Kiitos, Merci !
Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 24 / 24