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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