Pablo González Sequeiros Joint work with F. Alcalde and Á...

6th European Congress of Mathematics Kraków, July 1, 2012

Pablo González Sequeiros Joint work with F. Alcalde and Á. Lozano

Objective

Theorem

The continuous hull of any repetitive and aperiodic planar tiling is affable

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Objective

Theorem

The continuous hull of any repetitive and aperiodic planar tiling is affable

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

T. Giordano, H. Matui, I. Putnam, C. Skau, Orbit equivalence for Cantor minimal Z2-systems. J. Amer. Math. Soc., 2008

Summary

1.  Tilings

2.  Laminations defined by tilings

3.  Affable equivalence relations

4.  Affability Theorem for planar tilings

5.  Robinson Inflation

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Theorem

The continuous hull of any aperiodic and repetitive planar tiling is affable

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Definition

1. Tilings

A planar tiling T is a partition of into tiles, polygons touching face-to-face obtained by translation from a finite set of prototiles.

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Definition

A planar tiling T is a partition of into tiles, polygons touching face-to-face obtained by translation from a finite set of prototiles.

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

Definition

Patch M

A planar tiling T is a partition of into tiles, polygons touching face-to-face obtained by translation from a finite set of prototiles.

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

Definition

A tiling T is aperiodic if it has no translation symmetries:

A set of prototiles P is said to be aperiodic if any tiling obtained from P is aperiodic.

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

A tiling T is repetitive if for any radius r > 0, there exists R = R (r) > 0 such that any ball B(x,R) contains a translated copy M+v of any patch M with diameter

Definition

r

R

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1. Tilings

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Let be the set of tilings T constructed from a finite set of prototiles P

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Let be the set of tilings T constructed from a finite set of prototiles P

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Let be the set of tilings T constructed from a finite set of prototiles P

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Let be the set of tilings T constructed from a finite set of prototiles P

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Gromov-Hausdorff topology

The leaf through T 2 is its orbit

Proposition

If P is finite, then is a compact metrizable space laminated by the orbits

of the natural action of .

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Total transversal Fixing a base point in any prototile of P, we obtain a Delone set DT in any tiling T 2

r-discrete and R-dense

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Total transversal Fixing a base point in any prototile of P, we obtain a Delone set DT in any tiling T 2

r-discrete and R-dense

The subspace is a totally disconnected and compact

total transversal of .

Proposition

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Foliated atlas / Flow box decomposition

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Foliated atlas / Flow box decomposition

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Minimal subsets

Leaves without holonomy

The continuous hull of T is the closure of its orbit

A tiling T 2 is repetitive if and only if its continuous hull is minimal.

Proposition

The orbit of any aperiodic tiling is homeomorphic to .

Lemma

2. Laminations defined by tilings

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

The lamination defines on the total transversal X an étale equivalence relation (EER)

which represents its transverse dynamics.

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Theorem

The continuous hull X of any repetitive and aperiodic planar tiling T is a minimal transversally Cantor lamination without holonomy.

The lamination defines on the total transversal X an étale equivalence relation (EER)

which represents its transverse dynamics.

Definition

Two EER’s R and R’ on X and X’ are orbit equivalent if there exists a homeomorphism such that .

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

2. Laminations defined by tilings

Theorem

The continuous hull X of any repetitive and aperiodic planar tiling T is a minimal transversally Cantor lamination without holonomy.

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Theorem

The continuous hull of any aperiodic and repetitive planar tiling is affable

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

3. Affable equivalence relations

Let R be an EER on a totally disconnected space X

R is compact (CEER) if R\4 is compact subset of X x X

Definition [Giordano-Putnam-Skau, Renault]

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

3. Affable equivalence relations

Let R be an EER on a totally disconnected space X

R is compact (CEER) if R\4 is compact subset of X x X

Definition [Giordano-Putnam-Skau, Renault]

R is approximately finite (AF) if there exists an increasing

sequence of CEERs Rn such that

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

3. Affable equivalence relations

Let R be an EER on a totally disconnected space X

R is compact (CEER) if R\4 is compact subset of X x X

Definition [Giordano-Putnam-Skau, Renault]

R is approximately finite (AF) if there exists an increasing

sequence of CEERs Rn such that

R is said to be affable if it is orbit equivalent to an AF equivalence relation

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

3. Affable equivalence relations

Let R be an EER on a totally disconnected space X

R is compact (CEER) if R\4 is compact subset of X x X

Definition [Giordano-Putnam-Skau, Renault]

R is approximately finite (AF) if there exists an increasing

sequence of CEERs Rn such that

A transversally Cantor lamination (M, F) without holonomy is affable if the equivalence relation R induced on any total transversal T is affable

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Bratteli diagrams

V0

V1

X = Space of infinite paths from V0

…

V2

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

3. Affable equivalence relations

Bratteli diagrams

V0

V1

X = Space of infinite paths from V0

…

V2

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

3. Affable equivalence relations

Theorem [Giordano-Putnam-Skau, Renault]

Any AF equivalence relation is isomorphic to the tail equivalence relation on the infinite path space of a certain Bratteli diagram.

Theorem [Giordano-Putnam-Skau]

Any minimal affable equivalence relation on a compact space is orbit equivalent to a minimal Z-action (Bratteli-Vershik system).

Bratteli diagrams

V0

V1

X = Space of infinite paths from V0

…

V2

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

3. Affable equivalence relations

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Theorem

The continuous hull of any aperiodic and repetitive planar tiling is affable

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1.  Applying the inflation process to obtain an affable equivalence subrelation R1 of R

2.  Defining the boundary of R1 and studying its properties

3.  Applying the Absorption Theorem

Sketch of the proof

For any flow box decomposition B of X there exists another one B’ inflated of B:

Inflation [Bellisard-Benedetti-Gambaudo]

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Compact subrelations Rn

Let a sequence of flow box decompositions obtained by inflation

For any n 2 N and T 2 X, we define:

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Compact subrelations Rn

Let a sequence of flow box decompositions obtained by inflation

For any n 2 N and T 2 X, we define:

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Compact subrelations Rn

For any n 2 N and T 2 X, we define:

Let a sequence of flow box decompositions obtained by inflation

Proposition

The direct limit is a minimal open AF equivalence subrelation of R.

Rn is a CEER

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Compact subrelations Rn

For any n 2 N and T 2 X, we define:

Let a sequence of flow box decompositions obtained by inflation

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1.  Applying the inflation process to obtain an affable equivalence subrelation R1 of R

2.  Defining the boundary of R1 and studying its properties

3.  Applying the Absorption Theorem

Sketch of the proof

Boundary of R1

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

For any n 2 N and T 2 X, we define:

Boundary of R1

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

For any n 2 N and T 2 X, we define:

Boundary of R1

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

For any n 2 N and T 2 X, we define:

Boundary of R1

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

For any n 2 N and T 2 X, we define:

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Boundary of R1

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

1.  Applying the inflation process to obtain an affable equivalence subrelation R1 of R

2.  Defining the boundary of R1 and studying its properties

3.  Applying the Absorption Theorem

Sketch of the proof

Theorem [Giordano-Matui-Putnam-Skau]

Absorption

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

-  Let R be a minimal AF equivalence relation on the Cantor set X

-  Let K be a CEER on a closed subset Y ⊂ X such that

…

Then RÇK is affable

-  Let R be a minimal AF equivalence relation on the Cantor set X

-  Let K be a CEER on a closed subset Y ⊂ X such that

Then RÇK is affable

Theorem [Giordano-Matui-Putnam-Skau]

Absorption

Y is R-thin for any R-invariant measure ,

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

-  Let R be a minimal AF equivalence relation on the Cantor set X

-  Let K be a CEER on a closed subset Y ⊂ X such that

Then RÇK is affable

Theorem [Giordano-Matui-Putnam-Skau]

Absorption

R|Y remains étale and K is transverse to R|Y

Y is R-thin for any R-invariant measure ,

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

-  Let R be a minimal AF equivalence relation on the Cantor set X

-  Let K be a CEER on a closed subset Y ⊂ X such that

Then RÇK is affable

Theorem [Giordano-Matui-Putnam-Skau]

Absorption

R|Y remains étale and K is transverse to R|Y

Y is R-thin for any R-invariant measure ,

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

-  Let R be a minimal AF equivalence relation on the Cantor set X

-  Let K be a CEER on a closed subset Y ⊂ X such that

Then RÇK is affable

Theorem [Giordano-Matui-Putnam-Skau]

Absorption

R|Y remains étale and K is transverse to R|Y

Y is R-thin for any R-invariant measure ,

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

-  R1 minimal AF equivalence relation on X

-  K a CEER on Y =

Then R=R1Ç K affable

Theorem [Giordano-Matui-Putnam-Skau]

Any R-equivalence class separates into a uniformly bounded number of R1-equivalence classes

for any R1-invariant measure

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

Absorption

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

Step 1: Reduction to square tilings

Theorem [Sadun-Williams]

The continuous hull of a any planar tiling is orbit equivalent to the continuous hull of a tiling whose tiles are marked squares

We can assume that P is a finite set of marked squares and all leaves in the continuous hull X are endowed with the max-distance

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

Step 2: Partial decomposition

Let B1 be a finite family of compact flow boxes

where the plaques P1,i are square patterns of side N1 and the axis

is N1-dense in X.

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

Step 3: Inflated decomposition

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

B’1 , flow box decomposition with plaques P’

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

Step 3: Inflated decomposition

B’1 , flow box decomposition with plaques P’

, sequence of flow box decompositions obtained by Robinson inflation

B’’1 , inflated flow box decomposition with plaques P’’

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

Step 3: Inflated decomposition

Step 4: Properties of the boundary

Proposition

The boundary is R1-thin.

The isoperimetric ratio of the plaques tends to cero uniformly:

Lemma

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

Proposition

Any R-equivalence class separates into at most into 4 R1-equivalence classes.

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

Step 4: Properties of the boundary

Step 5: Absorption

Applying inductively the Absorption Theorem, we can exhaust R by a countable family of affable increasing EERs:

R is affable

AFFABILITY OF LAMINATIONS DEFINED BY REPETITIVE PLANAR TILING Pablo González Sequeiros

5. Robinson inflation

LAMINACIÓNS affableS Pablo González Sequeiros

Note: Figures of Robinson tilings have been made from the applet P. Ofella (with C. Woll): Robinson Tiling, from The Wolfram Demonstrations Project. http://demonstrations.wolfram.com/RobinsonTiling/