Download A Topological Study of Tilings

Introduction
Besicovitch topology
Cantor topology
Topological games
A Topological Study of Tilings
Grégory Lafitte1
Michaël Weiss2
1 Laboratoire d’Informatique Fondamentale de Marseille (LIF), CNRS –
Aix-Marseille Université, 39, rue Joliot-Curie, F-13453 Marseille Cedex 13, France
2 Centre
Universitaire d’Informatique, Université de Genève, Battelle bâtiment A, 7
route de Drize, 1227 Carouge, Switzerland
Theory and Applications of Models of Computation, 2008
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Domino Problem
Does a given tile set tile the plane?
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Wang Tilings
A Wang tile is a unit size square with colored edges.
A tile set "τ " is a finite set of Wang tiles.
To tile consists in placing the tiles in the grid Z2 , respecting
the color matching. A tiling P is associated to a tiling
function fP : Z2 → τ .
Domino Problem
Does a given tile set tile the plane?
Berger’s proof
The problem is undecidable. Tilings are
equivalent to Turing machine.
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Different kinds of tile sets
A tile set can be:
Finite
Periodic
Aperiodic
Non-recursive
Complete. . .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Different kinds of tile sets
A tile set can be:
Finite
Periodic
Aperiodic
Non-recursive
Complete. . .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Different kinds of tile sets
A tile set can be:
Finite
Periodic
Aperiodic
Non-recursive
Complete. . .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Different kinds of tile sets
A tile set can be:
Finite
Periodic
Aperiodic
Non-recursive
Complete. . .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Different kinds of tile sets
A tile set can be:
Finite
Periodic
Aperiodic
Non-recursive
Complete. . .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Different kinds of tile sets
A tile set can be:
Finite
Periodic
Aperiodic
Non-recursive
Complete. . .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Different kinds of tile sets
A tile set can be:
Finite
Periodic
Aperiodic
Non-recursive
Complete. . .
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Motivation
Two commonly-used topologies on cellular automata
Proximity between tilings and cellular automata
Adapt these topologies to tilings
Understand the behavior of the space of tilings
Study computation from a geometrical point of view
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Motivation
Two commonly-used topologies on cellular automata
Proximity between tilings and cellular automata
Adapt these topologies to tilings
Understand the behavior of the space of tilings
Study computation from a geometrical point of view
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Motivation
Two commonly-used topologies on cellular automata
Proximity between tilings and cellular automata
Adapt these topologies to tilings
Understand the behavior of the space of tilings
Study computation from a geometrical point of view
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Motivation
Two commonly-used topologies on cellular automata
Proximity between tilings and cellular automata
Adapt these topologies to tilings
Understand the behavior of the space of tilings
Study computation from a geometrical point of view
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
Motivation
Two commonly-used topologies on cellular automata
Proximity between tilings and cellular automata
Adapt these topologies to tilings
Understand the behavior of the space of tilings
Study computation from a geometrical point of view
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
What we have, what we expect
Results
How does the tiling space behave inside the mapping
space
What both topologies can say on tilings
Topological games to study meagerness
Expectations
Games to solve problems as:
Which kinds of structures are more likely to appear?
What is most common ? Periodic or aperiodic?
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
What we have, what we expect
Results
How does the tiling space behave inside the mapping
space
What both topologies can say on tilings
Topological games to study meagerness
Expectations
Games to solve problems as:
Which kinds of structures are more likely to appear?
What is most common ? Periodic or aperiodic?
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
What we have, what we expect
Results
How does the tiling space behave inside the mapping
space
What both topologies can say on tilings
Topological games to study meagerness
Expectations
Games to solve problems as:
Which kinds of structures are more likely to appear?
What is most common ? Periodic or aperiodic?
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
What we have, what we expect
Results
How does the tiling space behave inside the mapping
space
What both topologies can say on tilings
Topological games to study meagerness
Expectations
Games to solve problems as:
Which kinds of structures are more likely to appear?
What is most common ? Periodic or aperiodic?
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
What we have, what we expect
Results
How does the tiling space behave inside the mapping
space
What both topologies can say on tilings
Topological games to study meagerness
Expectations
Games to solve problems as:
Which kinds of structures are more likely to appear?
What is most common ? Periodic or aperiodic?
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Wang Tilings
Motivation
What we have, what we expect
Results
How does the tiling space behave inside the mapping
space
What both topologies can say on tilings
Topological games to study meagerness
Expectations
Games to solve problems as:
Which kinds of structures are more likely to appear?
What is most common ? Periodic or aperiodic?
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Traditional topology on cellular automata
Limsup of the different bits of two configurations
How to compare tile sets?
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Traditional topology on cellular automata
Limsup of the different bits of two configurations
How to compare tile sets?
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Traditional topology on cellular automata
Limsup of the different bits of two configurations
How to compare tile sets?
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
S=0
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Traditional topology on cellular automata
Limsup of the different bits of two configurations
How to compare tile sets?
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
S=0,0
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Traditional topology on cellular automata
Limsup of the different bits of two configurations
How to compare tile sets?
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
S=0,0,1/5
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Traditional topology on cellular automata
Limsup of the different bits of two configurations
How to compare tile sets?
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
S=0,0,1/5,2/7,2/9,...
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Traditional topology on cellular automata
Limsup of the different bits of two configurations
How to compare tile sets?
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
S=0,0,1/5,2/7,2/9,...
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
S=0
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
S=0, 2/9
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
S=0, 2/9, 3/25
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
S=0, 2/9, 3/25, 10/49
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
The metric
S=0, 2/9, 3/25, 10/49, 15/81...
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Properties
2
MB : {{1, . . . , s}Z }s>1 with the Besicovitch metric
TB : Tilings with the Besicovitch metric
Motivation
Understand the behavior of TB in MB
Property
The classes of equivalent tilings in TB are still uncountably
many
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Properties
2
MB : {{1, . . . , s}Z }s>1 with the Besicovitch metric
TB : Tilings with the Besicovitch metric
Motivation
Understand the behavior of TB in MB
Property
The classes of equivalent tilings in TB are still uncountably
many
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Properties
2
MB : {{1, . . . , s}Z }s>1 with the Besicovitch metric
TB : Tilings with the Besicovitch metric
Motivation
Understand the behavior of TB in MB
Property
The classes of equivalent tilings in TB are still uncountably
many
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Properties
2
MB : {{1, . . . , s}Z }s>1 with the Besicovitch metric
TB : Tilings with the Besicovitch metric
Motivation
Understand the behavior of TB in MB
Property
The classes of equivalent tilings in TB are still uncountably
many
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Results
Metric spaces
There is a countable infinite set A of Wang tilings such that
any two tilings of A are almost at distance 1
There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spaces
TB is meager in MB
TB is a Baire space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Results
Metric spaces
There is a countable infinite set A of Wang tilings such that
any two tilings of A are almost at distance 1
There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spaces
TB is meager in MB
TB is a Baire space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Results
Metric spaces
There is a countable infinite set A of Wang tilings such that
any two tilings of A are almost at distance 1
There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spaces
TB is meager in MB
TB is a Baire space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Results
Metric spaces
There is a countable infinite set A of Wang tilings such that
any two tilings of A are almost at distance 1
There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spaces
TB is meager in MB
TB is a Baire space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Results
Metric spaces
There is a countable infinite set A of Wang tilings such that
any two tilings of A are almost at distance 1
There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spaces
TB is meager in MB
TB is a Baire space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Introduction
Another traditional cellular automata metric
Consider the biggest common sequence of two
configurations
For tilings, the biggest common pattern
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Introduction
Another traditional cellular automata metric
Consider the biggest common sequence of two
configurations
For tilings, the biggest common pattern
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Introduction
Another traditional cellular automata metric
Consider the biggest common sequence of two
configurations
For tilings, the biggest common pattern
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Introduction
Another traditional cellular automata metric
Consider the biggest common sequence of two
configurations
For tilings, the biggest common pattern
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Introduction
Another traditional cellular automata metric
Consider the biggest common sequence of two
configurations
For tilings, the biggest common pattern
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
d=2
-n
n
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Introduction
Another traditional cellular automata metric
Consider the biggest common sequence of two
configurations
For tilings, the biggest common pattern
0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 1 0 1
d=2
-n
n
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
i=2
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
i=3
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
i=9
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Definition
i=9
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Properties
2
MC : {{1, . . . , s}Z }s>1 with the Cantor metric
TC : Tilings with the Cantor metric
Motivation
Have a better understanding of the local structure
Property
TC is a 0-dimensional space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Properties
2
MC : {{1, . . . , s}Z }s>1 with the Cantor metric
TC : Tilings with the Cantor metric
Motivation
Have a better understanding of the local structure
Property
TC is a 0-dimensional space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Properties
2
MC : {{1, . . . , s}Z }s>1 with the Cantor metric
TC : Tilings with the Cantor metric
Motivation
Have a better understanding of the local structure
Property
TC is a 0-dimensional space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Properties
2
MC : {{1, . . . , s}Z }s>1 with the Cantor metric
TC : Tilings with the Cantor metric
Motivation
Have a better understanding of the local structure
Property
TC is a 0-dimensional space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Results
Metric spaces
There exists an open ball of MC that does not contain any
Wang tiling
There exists a countable set of Wang tilings dense in TC
Topological spaces
TC is a Baire space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Results
Metric spaces
There exists an open ball of MC that does not contain any
Wang tiling
There exists a countable set of Wang tilings dense in TC
Topological spaces
TC is a Baire space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Definition
The topological space
Results
Metric spaces
There exists an open ball of MC that does not contain any
Wang tiling
There exists a countable set of Wang tilings dense in TC
Topological spaces
TC is a Baire space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Motivation
Banach-Mazur
Games with perfect information with two players
Makes possible the study of meagerness of subsets of a
topological space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Motivation
Banach-Mazur
Games with perfect information with two players
Makes possible the study of meagerness of subsets of a
topological space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Motivation
Banach-Mazur
Games with perfect information with two players
Makes possible the study of meagerness of subsets of a
topological space
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Banach-Mazur Games
A topological space X and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 ⊂ U1 , and so on. . .
T
If Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
C
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Banach-Mazur Games
A topological space X and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 ⊂ U1 , and so on. . .
T
If Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
C
U
Lafitte, Weiss
1
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Banach-Mazur Games
A topological space X and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 ⊂ U1 , and so on. . .
T
If Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
C
U
Lafitte, Weiss
2
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Banach-Mazur Games
A topological space X and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 ⊂ U1 , and so on. . .
T
If Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
Lim U
C
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Banach-Mazur Games
A topological space X and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 ⊂ U1 , and so on. . .
T
If Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
Lim U
C
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Outline
1
Introduction
Wang Tilings
Motivation
2
Besicovitch topology
Definition
The topological space
3
Cantor topology
Definition
The topological space
4
Topological games
Introduction
Games on tilings
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of X
Player I chooses a pattern m
Player II chooses a pattern m0 such that m ∈ m0 , and so
on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TC is meager in MC
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of X
Player I chooses a pattern m
Player II chooses a pattern m0 such that m ∈ m0 , and so
on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TC is meager in MC
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of X
Player I chooses a pattern m
Player II chooses a pattern m0 such that m ∈ m0 , and so
on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TC is meager in MC
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of X
Player I chooses a pattern m
Player II chooses a pattern m0 such that m ∈ m0 , and so
on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TC is meager in MC
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of X
Player I chooses a pattern m
Player II chooses a pattern m0 such that m ∈ m0 , and so
on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TC is meager in MC
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of X
Player I chooses a pattern m
Player II chooses a pattern m0 such that m ∈ m0 , and so
on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TC is meager in MC
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 such that U2 ⊂ U1 , and
so on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TB is meager in MB
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 such that U2 ⊂ U1 , and
so on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TB is meager in MB
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 such that U2 ⊂ U1 , and
so on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TB is meager in MB
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 such that U2 ⊂ U1 , and
so on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TB is meager in MB
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 such that U2 ⊂ U1 , and
so on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TB is meager in MB
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of X
Player I chooses an open set U1 of X
Player II chooses an open set U2 such that U2 ⊂ U1 , and
so on. . .
Player II wins if the final tiling is in C
X \ C is meager if Player II has a winning strategy
Theorem
TB is meager in MB
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Further works
Games seems a promising tools for tilings
Study the meagerness of subsets of tilings
Obtain strong results on the probability of a certain
structure to appear
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Further works
Games seems a promising tools for tilings
Study the meagerness of subsets of tilings
Obtain strong results on the probability of a certain
structure to appear
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Further works
Games seems a promising tools for tilings
Study the meagerness of subsets of tilings
Obtain strong results on the probability of a certain
structure to appear
Lafitte, Weiss
A Topological Study of Tilings
Introduction
Besicovitch topology
Cantor topology
Topological games
Introduction
Games on tilings
Shen’s question
Tilings with errors
Finite hole
Aperiodic tile set stability
Consider any aperiodic tile set, can we always remove a
finite border of the hole to complete the tiling?
Lafitte, Weiss
A Topological Study of Tilings