Pablo González Sequeiros Joint work with F. Alcalde and Á ...webhome.auburn.edu/~kuperkm/dynamicsKrakow2012/Sequeiros_6… · Pablo González Sequeiros Joint work with F. Alcalde

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


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


Theorem

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


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.


Definition



1. Tilings

Definition

Patch M



1. Tilings


1. Tilings


1. Tilings


1. Tilings


1. Tilings


1. Tilings


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.


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


1. Tilings


2. Laminations defined by tilings

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











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 .



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



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



Foliated atlas / Flow box decomposition



Foliated atlas / Flow box decomposition



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



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

which represents its transverse dynamics.



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 .



Theorem

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


Theorem



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








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


Bratteli diagrams

V0

V1

X = Space of infinite paths from V0

…

V2



Bratteli diagrams

V0

V1


…

V2



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


…

V2





Theorem



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]


Compact subrelations Rn

Let a sequence of flow box decompositions obtained by inflation

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










Proposition

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

Rn is a CEER









Sketch of the proof

Boundary of R1



Boundary of R1



Boundary of R1



Boundary of R1




Boundary of R1





Sketch of the proof

Theorem [Giordano-Matui-Putnam-Skau]

Absorption


-  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





Absorption

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






Absorption

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







Absorption








Absorption




-  R1 minimal AF equivalence relation on X

-  K a CEER on Y =

Then R=R1Ç K affable


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

for any R1-invariant measure


Absorption


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



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.



Step 3: Inflated decomposition



B’1 , flow box decomposition with plaques P’




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’’




Step 4: Properties of the boundary

Proposition

The boundary is R1-thin.

The isoperimetric ratio of the plaques tends to cero uniformly:

Lemma



Proposition

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



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



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/

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Pablo González Sequeiros Joint work with F. Alcalde and Á ...webhome.auburn.edu/~kuperkm/dynamicsKrakow2012/Sequeiros_6… · Pablo González Sequeiros Joint work with F. Alcalde