Automorphisms, cubes and the Robinson tiling · Automorphisms, cubes and the Robinson tiling based...

Automorphisms, cubes and the Robinson tilingbased on a joint work with Wenbo Sun and on a joint work with

Fabien Durand, Alejandro Maass and Samuel Petite

Sebastian Donoso F.

The Hebrew University of Jerusalem

June 2016

Sebastian Donoso Fuentes

Brief “introduction” to topological dynamics


A topological dynamical system is a pair (X ,G) where X is acompact metric space and G is a group of homeomorphisms.In this talk, G is Z or Z2.

When T is a homeomorphism of X , we write (X ,T ) to denote Zdynamical system (X ,{T n ∶ n ∈ Z}).

Let G = Z2 and let S,T ∈ G be two generators. We say that(X ,S,T ) is a system with commuting transformations S and T .

In this talk we assume that (X ,S,T ) is a minimal system,meaning that the G-orbit of any point is dense in X . This isequivalent to say that the only closed invariant sets are X and∅.


A factor map π is a continuous onto map

Y Y

X X

π

g

π

g

π ○ g = g ○ π.

Remark: Even if Y is a subshift, it may be convenient to consider factors which are not

subshifts.


An equivalent way to think of a factor map of (X ,G) is throughrelations.

Let R ⊆ X ×X be a relation:● closed;● invariant (g × g)R =R for all g ∈ G.● of equivalence.

We can define the canonical projection

X → X/R

Conversely, if π∶X → Y is a factor map, Y can be identified withX/Rπ where Rπ = {(x ,x ′) ∈ X ×X ∶ π(x) = π(x ′)}.


In general, if R is only a relation, we can considerσ(R) to be the smallest closed, invariant, equivalence relationcontaining R.


Automorphisms in topological dynamics

An automorphism of (X ,G) is an homeomorphism φ∶X → Xsuch that

φ ○ g = g ○ φ. for all g ∈ G

We write

Aut(X ,G) = {φ automorphism of (X ,G)}.

If G is abelian

G ⊂ Z(Aut(X ,G)).


LemmaLet (X ,G) be a minimal topological dynamical system. ThenAut(X ,G) acts freely on X (elements different from the identityhave no fixed points).

Proof: For φ ∈ Aut(X ,G)

Fix(φ) ∶= {x ∈ X ∶ φx = x}

is closed (φ is continuous) and invariant (φ commutes with G).By minimality Fix(φ) is equal to X or ∅.


Automorphisms through factor maps

A factor map π∶ (X ,G)→ (Y ,G) is compatible with Aut(X ,G) if

π(x) = π(x ′)⇐⇒ π(φ(x)) = π(φ(x ′)) for every φ ∈ Aut(X ,G)

In this case, we can define the projection

π∶Aut(X ,G)→ Aut(Y ,G).

π(φ)π(x) = π(φx).


In terms of relations, X → X/R is compatible with Aut(X ,G) if

(x ,x ′) ∈R if and only if (φx , φx ′) ∈R

for every φ ∈ Aut(X ,G).

Fact: Many important factors in topological dynamics arecompatible!


A useful criterion

DefinitionLet π∶ (X ,G)→ (Y ,G) be a factor map. The factor π is almostone-to-one if there exists y ∈ Y such that ∣π−1(y)∣ = 1.

Proposition

Let π∶ (X ,G)→ (Y ,G) be an almost one-to-one extension withπ compatible with Aut(X ,G). Then

π∶Aut(X ,G)→ Aut(Y ,G)

is an injection.


Proof: It suffices to analyse φ ∈ Ker(π).

Take y ∈ Y with π−1(y) = {x}.

Then φ(x) = x , and since the action is free

φ = id.

Example: If (X , σ) is a Toeplitz subshift, then Aut(X ,G) isabelian.


How to create compatiblefactors/relations?


Some special relations

Product systems appear a lot in ergodic theory.

Let (Y , σ) and (W , τ) be minimal topological dynamicalsystems. The system

(Y ×W , σ × id, id × τ)

is the the product of (Y , σ) and (W , τ).

In terms of the dynamics, we can think that these systems (andtheir factors) have “simple” behavior.


A relation:

Definition (A cube for two commuting transformations)

Let (X ,S,T ) be a minimal system with commutingtransformations. We define the set of cubes QS,T to be the set

{(x ,Snx ,T mx ,SnT mx) ∶ x ∈ X ,n,m ∈ Z} ⊆ X 4.

An element in QS,T is called a dynamical cube.

T mx SnT mx

x Snx


Example

Let (Y ×W , σ × id, id × τ) be a minimal product system.

A typical cube has the form

(y , τmw) (σny , τmw)

(y ,w) (σny ,w)


(y ,w ′) ?

(y ,w) (y ′,w)


(y ,w ′) (y ′,w ′)

(y ,w) (y ′,w)

Conclusion: The last coordinate of a point in QS,T isdetermined by the other ones.


Theorem (D., Sun (2014))

Let (X ,S,T ) be a minimal system with commutingtransformations. The following are equivalent:

1 (X ,S,T ) is a factor of a product system;2 One can deduce the last coordinate of a dynamical cube.


“Cubes induce factors”. We define relations using the cubes:

RS = {(x ,y) ∈ X ×X ∶ (x ,y ,a,a) ∈ QS,T for some a ∈ X}

a a

x y

x and y are RS related


RT = {(x ,y) ∈ X ×X ∶ (x ,b,y ,b) ∈ QS,T for some b ∈ X}

y b

x b

x and y are RT related

Finally we defineRS,T =RS ∩RT .


Example: RS in a product system.

(y ,w ′) (y ′,w ′)

(y ,w) (y ′,w)

Hence (y ,w) = (y ′,w) and

RS = ∆Y×W =RT =RS,T .


Example: RS in a product system.

(y ,w ′) (y ′,w ′)

(y ,w) (y ′,w)

=

Hence (y ,w) = (y ′,w) and

RS = ∆Y×W =RT =RS,T .


TheoremLet (X ,S,T ) be a minimal system with commutingtransformations. Then (X ,S,T ) has a product extension if andonly if RS,T = ∆X .


Example: Morse

Consider the Morse substitution:


One can iterate this substitution in a natural way:

first, second and third iteration of the substitution


Example

Iterating the process we get a point in x ∈ {0,1}Z2. We let X

denote orbit closure of x under translations. We have that(X , σ(1,0), σ(0,1)) is a minimal system with commutingtransformations. (From now S = σ(1,0), T = σ(0,1)).


Example

Proposition

We have that RS,T (X) = ∆X . Subsequently (X ,S,T ) has aproduct extension.

In fact, (unsurprisingly) the Theorem gives that the “cover” isthe product of two one dimensional Thue-Morse sequences.


More generally

Let (X , σX ) and (Y , σY ) be two one dimensional shift spaceswith alphabets AX and AY . Let x = (xi)i∈Z ∈ X , y = (yj)j∈Z ∈ Y .

Consider z = (zi,j)i,j∈Z where zi,j = (xi ,yj) and Z its closure orbitunder the shift transformations.

Let ϕ be a function defined on AX ×AY .


⋱ ⋮ ⋮ ⋮ ⋰

⋯ (xi−1,yj+1) (xi ,yj+1) (xi+1,yj+1) ⋯

⋯ (xi−1,yj) (xi ,yj) (xi+1,yj) ⋯

⋯ (xi−1,yj−1) (xi ,yj−1) (xi+1,yj−1) ⋯

⋰ ⋮ ⋮ ⋮ ⋱


⋱ ⋮ ⋮ ⋮ ⋰

⋯ ϕ(xi−1,yj+1) ϕ(xi ,yj+1) ϕ(xi+1,yj+1) ⋯

⋯ ϕ(xi−1,yj) ϕ(xi ,yj) ϕ(xi+1,yj) ⋯

⋯ ϕ(xi−1,yj−1) ϕ(xi ,yj−1) ϕ(xi+1,yj−1) ⋯

⋰ ⋮ ⋮ ⋮ ⋱


Proposition

Any minimal Z2 subshift with a product extension is obtained inthis way.

A comment for some specialists: This can also be characterized in terms of enveloping

semigroups. A system has a product extension if and only if

E(X , ⟨S,T ⟩) = E(X ,S)E(X ,T) and E(X ,S) and E(X ,T) commute


Example: Robinson tiling

I refer to the work of Galher, Julien and Savinien.Consider the following set of tiles and their rotations andreflections :

Let ARob denote this set of tiles. Then #ARob = 28.


We can continue this process and we get a point x ∈ AZ2

Rob.

Let X denote the orbit closure of this point under the shifttransformations.

The system (X ,S,T ) is a minimal system with commutingtransformations (it is the unique minimal inside the Robinsontiling).


After some computations, the RS,T relation on X is “almosttrivial”.

Let π∶X → X/RS,T be the quotient map. Then π is an almostone-to-one extension and compatible.

“Regular” points (the ones with only one infinite order supertile)are no related to other points.

Therefore we can inject Aut(X ,S,T ) in Aut(X/RS,T ,S,T )


Important property: the fiber with maximal cardinality is unique(up to shift).

It consists of the configurations with four infinite ordersupertiles.


How two four infinite supertiles points are related


Proposition

The group of automorphisms of (X ,S,T ) consists of the shifts.

Proof:Let φ be a automorphism of the Robinson tiling andπ∶X → X/RS,T be the quotient map. Let x with four infiniteorder supertiles. Then there exist n,m ∈ Z with

π(φ)(π(x)) = π(φ(x)) = SnT mπ(x)

Hence π(φ) = SnT m in X/RS,T and (by injectivity)

φ = SnT m in X

.


Merci pour votre attention.


Automorphisms, cubes and the Robinson tiling · Automorphisms, cubes and the Robinson tiling based...

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