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/