Répétition soutenance

Dynamique symbolique
des syst`mes 2D et des arbres inﬁnis
e
Soutenance de th`se,
e
encadr´e par Marie-Pierre Beal et Mathieu Sablik
e ´

Nathalie Aubrun
LIGM, Universit´ Paris-Est
e

22 juin 2011

Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 1 / 24

Outline
1 Symbolic dynamics
What are symbolic dynamics ?
Subshifts
Classes of subshifts
2 Motivation
1D subshifts vs. 2D subshifts
Two orthogonal approaches
3 A study of 2D-shifts
The projective subaction
Hochman’s result
Improvement
4 Tree-shifts
Tree-shift: example
Tree automata and tree-shifts
On AFT tree-shifts
Conjugacy of tree-SFT


Symbolic dynamics

Outline
1 Symbolic dynamics
Subshifts
2 Motivation
Hochman’s result
Improvement
4 Tree-shifts
Tree-shift: example
On AFT tree-shifts


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 space
F is a continuous map X → X

x





F (x)

x





F 2 (x)
F (x)

x





F 2 (x) F 4 (x)
F (x)

F 3 (x)
x F 5 (x)



Coding of the orbits

X= n
i=1 Xi a partition of the phase space X
a color ai associated with each Xi



Coding of the orbits

X= n
i=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

...


Symbolic dynamics Subshifts

Subshifts: topological definition

2 2
A finite alphabet, AN (or AZ , AN or AZ ) the configurations space
the configurations space is endowed with the prodiscrete topology ⇒
compact space
natural action of N (or Z, N2 or Z2 ) by translation: the shift σ

σ j (x)i = xi+j for all x ∈ AN



Subshifts: topological definition

2 2
A finite alphabet, AN (or AZ , AN or AZ ) the configurations space
the configurations space is endowed with the prodiscrete topology ⇒
compact space
natural action of N (or Z, N2 or Z2 ) by translation: the shift σ

σ j (x)i = xi+j for all x ∈ AN

(Topological) Definition: subshift
2 2
A subshift is a closed and σ-invariant subset of AN (or AZ , AN or AZ ).



Subshifts: combinatorial definition

A finite alphabet
pattern p ∈ AS , where S ⊂ N is finite
the pattern p = appears in the configuration




A finite alphabet

(Combinatorial) Definition: subshift
Let F be a set of finite patterns. The subshift defined by the set of forbidden
patterns F is the set

TF = x ∈ AN , no pattern of F appears in x .




A finite alphabet

(Combinatorial) Definition: subshift
Let F be a set of finite patterns. The subshift defined by the set of forbidden
patterns F is the set

TF = x ∈ AN , no pattern of F appears in x .

Proposition
The toplogical and combinatorial definitions coincide.


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 N
one forbidden pattern

x ∈{ , }N , ∃i ∈ N ∪ {+∞}, (xj = ⇔ j ≤ i)



Classes of subshifts: Subshifts of finite type (SFT)

Sets of configurations that avoid a finite set of forbidden patterns:
alphabet { , } on N
one 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 of
forbidden patterns.

simplest class with respect to the combinatorial definition
2D-SFT ≡ Wang tilings



Classes of subshifts: Soﬁc subshifts

Factor map Φ AZ → B Z given by a local map φ:
2 2

Φ(x) ∈ B Z
2
x ∈ AZ
2




2 2

Φ(x) ∈ B Z
2
x ∈ AZ
2

Definition: sofic susbhift
A sofic subshift is the factor of an SFT.



2 2

Φ(x) ∈ B Z
2
x ∈ AZ
2

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.



Classes of subshifts: Effective subshifts

Definition: effective susbhift
An effective subshift is a subshift that can be defined by a recursively enumerable
set of forbidden patterns.



Classes of subshifts: Effective subshifts

Definition: effective susbhift
An effective subshift is a subshift that can be defined by a recursively enumerable
set of forbidden patterns.

reasonnable susbhift
this class naturally appears as projective subactions of 2D-SFT


Motivation

Outline
1 Symbolic dynamics
Subshifts
2 Motivation
Hochman’s result
Improvement
4 Tree-shifts
Tree-shift: example
On AFT tree-shifts


Motivation 1D subshifts vs. 2D subshifts


1D-subshifts 2D-subshifts
Emptyness of SFT ×
∀ SFT,
Periodicity in SFT ∃ aperiodic SFT
∃ periodic configuration
Decomposition theorem sofic subshifts
N
Conjugacy of SFT ×
Z ?
finite
Recognizers for SFT
local automata
Recognizers
finite automata textile systems
for sofic subshifts

: decidable problem
×: undecidable problem
?: open problem


Motivation Two orthogonal approaches


Two attempts to understand 2D-subshifts :




study 2D-SFT through operations acting on them




study 2D-SFT through operations acting on them
study subshifts deﬁned on a structure between dimensions 1 and 2



Operations on 2D-SFT

factor map operation (Fact): SFT soﬁc subshifts




operations that preserves SFT: product (P), ﬁnite type (FT) and spatial
extension




operations that preserves SFT: product (P), ﬁnite type (FT) and spatial
extension
operation mainly studied: projective subaction (SA)



Tree-shifts

Structure in-between N and N2 ?

N N2



Tree-shifts

Structure in-between N and N2 : free semi-group with two generators M2 .

N M2 N2


A study of 2D-shifts

Outline
1 Symbolic dynamics
Subshifts
2 Motivation
Hochman’s result
Improvement
4 Tree-shifts
Tree-shift: example
On AFT tree-shifts


A study of 2D-shifts The projective subaction

Idea: study subdynamics of 2D-subshifts.
Principle: given a subgroup G of Z2 and a 2D-susbhift T on alphabet A, and
consider the G-subshift deﬁned by
SAG (T) = y ∈ AG ∃x ∈ T such that y = xT .

G = {(x, y ) ∈ Z2 y = x}



Idea: study subdynamics of 2D-subshifts.
Principle: given a subgroup G of Z2 and a 2D-susbhift T on alphabet A, and
consider 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.


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 entropy
any zero-entropy soﬁc Z-subshift, with some conditions on its periods




there exist classes of Z-subshifts which are not realizable as projective
subdynamics of any Zd SFT.




there exist classes of Z-subshifts which are not realizable as projective
subdynamics of any Zd SFT.
⇒ no complete characterization projective subactions of 2D-SFT.


A study of 2D-shifts Hochman’s result

Hochman’s result

. . . no complete characterization projective subactions of 2D-SFT. . .



Hochman’s result


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 a
Zd+2 -SFT.



Hochman’s result


Consider factor map operations in addition to projective subactions.

Theorem (Hochman 2010)
Zd+2 -SFT.
Natural question: is it possible to use one dimension less ?


A study of 2D-shifts Improvement

Improvement of Hochman’s result

Theorem
Zd+1 -SFT.




Theorem
Zd+1 -SFT.
Two diﬀerent proofs:
Durand, Romaschenko & Shen 2010, using self-similar tilings
A.& Sablik 2010, adaptation of Robinson’s construction




Theorem
Zd+1 -SFT.
Two different proofs:
Durand, Romaschenko & Shen 2010, using self-similar tilings
A.& Sablik 2010, adaptation of Robinson’s construction
Many applications:
correspondance between an order on subshifts and an order on languages
multidimensional effective subshifts are sofic (A.& Sablik 2011)
construction of a tiles set whose quasi-periodic tilings have a non-recursively
bounded periodicity function (Ballier& Jeandel 2010)
...


Tree-shifts

Outline
1 Symbolic dynamics
Subshifts
2 Motivation
Hochman’s result
Improvement
4 Tree-shifts
Tree-shift: example
On AFT tree-shifts


Tree-shifts Tree-shift: example

Example of tree-shift

alphabet A = { , }
forbidden patterns: paths containing an even number of between two


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 q

q0 q0 q1 q1 q , q1 q , q1 q , q1 q , q1



Tree automata
alphabet A = { , }
tree automaton A with states Q = {q0 , q1 , q }
transition rules:
q1 q0 q0 q

q0 q0 q1 q1 q , q1 q , q1 q , q1 q , q1
Accepted trees:




(ﬁnite) tree automata, all states are accepting
a tree is accepted iﬀ there exists a computation
deterministic tree automata ≡ non deterministic tree automata
the set of trees accepted by a tree automaton is a subshift




(finite) tree automata, all states are accepting
a tree is accepted iff there exists a computation
deterministic tree automata ≡ non deterministic tree automata
the set of trees accepted by a tree automaton is a subshift

Proposition (A. & Bál 2009)
e
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.



Main difference with finite automata on words

Synchronizing block: every calculation (there exist at least one) of A on this block
ends in the same state.
Proposition
Any finite automaton on words has a synchronizing word.



Main difference with finite automata on words

Synchronizing block: every calculation (there exist at least one) of A on this block
ends in the same state.
Proposition
Any finite automaton on words has a synchronizing word.

But...
Proposition (A.& Bál 2009)
e
There exists a deterministic, minimal and reduced tree automaton which has no
synchronizing block.



The context automaton

The context automaton of a tree-shift T is the deterministic tree automaton
C = (V , A, ∆) where
V is the set of non-empty contexts of ﬁnite blocks appearing in T
transitions are (contT (u), contT (v )), a → contT (a, u, v ), with u, v ∈ L(T).





e
The context automaton of a soﬁc tree-shift is synchronized.





e
The context automaton of a sofic tree-shift is synchronized.

e
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.


Tree-shifts On AFT tree-shifts

On AFT tree-shifts

AFT tree-shifts: class in-between tree-SFT and soﬁc tree-shifts
generalization of AFT 1D-susbhifts, used for coding purposes



On AFT tree-shifts

AFT tree-shifts: factors of tree-SFT, where the factor map satisﬁes syntactic
properties (right-resolving, left-closing and having a resolving block)
theses properties on the factor map are computable



On AFT tree-shifts

AFT tree-shifts: factors of tree-SFT, where the factor map satisﬁes syntactic
properties (right-resolving, left-closing and having a resolving block)
theses properties on the factor map are computable

e
It is decidable to say wether a soﬁc tree-shift is AFT or not.


Tree-shifts Conjugacy of tree-SFT

The conjuguacy problem for tree-SFT

two subshifts are conjugate if they are both factor of the other
conjugate subshifts are the same, up to a recoding





Theorem (A.& B´al 2010)
e
The conjugacy problem is decidable for tree-SFT.





Theorem (A.& B´al 2010)
e
The conjugacy problem is decidable for tree-SFT.

every conjuguacy can be splitted into a sequence of elementary conjuguacies
elementary conjugacies ⇒ unique minimal amalgamation of a tree-SFT
two tree-SFT are conjugate iﬀ they have the same minimal amalgamation


Conclusion

Conclusion

Many properties on 2D-SFT are uncomputable. . .
. . . nevertheless it is possible to partially describe them.


Conclusion

Conclusion

Tree-shifts are very similar to N-subshifts. . .
. . . how to characterize monoids with same properties as 1D-subshifts ?


Conclusion

Conclusion

Tree-shifts are very similar to N-subshifts. . .
. . . how to characterize monoids with same properties as 1D-subshifts ?

Thank you, Спасибо, Kiitos, Merci !


Répétition soutenance

Recommended

Recommended

More Related Content

What's hot

What's hot (19)

Viewers also liked

Viewers also liked (20)

Similar to Répétition soutenance

Similar to Répétition soutenance (6)

Répétition soutenance