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Low complexity Tilings of the Plane Jarkko Kari Department of Mathematics and Statistics University of Turku, Finland We study how local constraints enforce global regularities This is a common phenomenon is sciences. For example, formation of

  1. Low complexity Tilings of the Plane Jarkko Kari Department of Mathematics and Statistics University of Turku, Finland

  2. We study how local constraints enforce global regularities This is a common phenomenon is sciences. For example, formation of crystals: Atoms attach to each other in a limited number of ways = ⇒ periodic arrangement of the atoms

  3. Our goal is to understand fundamental underlying principles that connect local rules to the global regularities observed in the structures. Our setup: multidimensional symbolic dynamics (=tilings)

  4. Configurations are infinite d -dimensional grids of symbols.

  5. For a fixed finite shape D , we observe the D -patterns in the configuration.

  6. For a fixed finite shape D , we observe the D -patterns in the configuration.

  7. For a fixed finite shape D , we observe the D -patterns in the configuration.

  8. For a fixed finite shape D , we observe the D -patterns in the configuration.

  9. For a fixed finite shape D , we observe the D -patterns in the configuration.

  10. A quantity to measure local complexity: the pattern complexity P ( c, D ) = number of D -patterns in c .

  11. We call c a low complexity configuration if P ( c, D ) ≤ | D | for some finite D ⊆ Z 2 .

  12. Configuration c is periodic if it is invariant under a non-zero translation.

  13. Configuration c is two-periodic if it is invariant under non-zero translations in two different directions. This implies periodicity in every rational direction.

  14. Open problem: Nivat’s conjecture Consider d = 2 and rectangular D . Conjecture (Nivat 1997) If c has low complexity with respect to some rectangle then c is periodic.

  15. Open problem: Nivat’s conjecture Consider d = 2 and rectangular D . Conjecture (Nivat 1997) If c has low complexity with respect to some rectangle then c is periodic. This would extend the one-dimensional case d = 1: Morse-Hedlund theorem: Let c ∈ A Z and n ∈ N . If c has at most n distinct subwords of length n then c is periodic.

  16. Best known general bound in 2D: Theorem (Cyr, Kra): If P ( c, D ) ≤ 1 2 | D | for some rectangle D then c is periodic.

  17. In 3D and higher dimensional cases the conjecture is false Non-periodic c

  18. In 3D and higher dimensional cases the conjecture is false Non-periodic c D is n × n × n cube

  19. In 3D and higher dimensional cases the conjecture is false Non-periodic c D is n × n × n cube P ( c, D ) = 1 + . . .

  20. In 3D and higher dimensional cases the conjecture is false Non-periodic c D is n × n × n cube P ( c, D ) = 1 + n 2 + . . .

  21. In 3D and higher dimensional cases the conjecture is false Non-periodic c D is n × n × n cube P ( c, D ) = 1 + n 2 + n 2

  22. In 3D and higher dimensional cases the conjecture is false Non-periodic c D is n × n × n cube P ( c, D ) = 1 + n 2 + n 2 < n 3 = | D | for large n .

  23. Patterns − → Configurations: Given a set P of allowed D -patterns, the collection X P of all configurations whose D -patterns are allowed is the subshift of finite type (SFT) defined by P .

  24. Patterns − → Configurations: Given a set P of allowed D -patterns, the collection X P of all configurations whose D -patterns are allowed is the subshift of finite type (SFT) defined by P .

  25. Patterns − → Configurations: Given a set P of allowed D -patterns, the collection X P of all configurations whose D -patterns are allowed is the subshift of finite type (SFT) defined by P .

  26. Patterns − → Configurations: Given a set P of allowed D -patterns, the collection X P of all configurations whose D -patterns are allowed is the subshift of finite type (SFT) defined by P .

  27. Famous result by R.Berger (1962): • Undecidability of the domino problem : It is undecidable if a given SFT is empty. • There exist aperiodic SFTs : non-empty SFTs that only contain non-periodic configurations.

  28. Famous result by R.Berger (1962): • Undecidability of the domino problem : It is undecidable if a given SFT is empty. • There exist aperiodic SFTs : non-empty SFTs that only contain non-periodic configurations. We are interested in the analogous problems restricted to low complexity SFTs , where the number of allowed D -patterns is at most | D | .

  29. We study configuration c using algebra, so we first replace symbols by integers:

  30. We study configuration c using algebra, so we first replace symbols by integers: 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1

  31. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 D -patterns are viewed as | D | -dimensional vectors.

  32. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 D -patterns are viewed as | D | -dimensional vectors. (1 , 1 , 1 , 0)

  33. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 D -patterns are viewed as | D | -dimensional vectors. (1 , 1 , 1 , 0) (1 , 1 , 0 , 1)

  34. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 D -patterns are viewed as | D | -dimensional vectors. (1 , 1 , 1 , 0) (1 , 1 , 0 , 1) (0 , 0 , 1 , 0)

  35. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 D -patterns are viewed as | D | -dimensional vectors. (1 , 1 , 1 , 0) (1 , 1 , 0 , 1) (0 , 0 , 1 , 0) (0 , 0 , 1 , 1)

  36. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 • If P ( c, D ) < | D | then there is a vector orthogonal to all D -patterns of c . Indeed: the number P ( c, D ) of distinct vectors is less than the dimension | D | of the linear space.

  37. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 • If P ( c, D ) < | D | then there is a vector orthogonal to all D -patterns of c . • Even if P ( c, D ) = | D | there is a vector with constant inner product with all D -patterns of c .

  38. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1  (1 , 1 , 1 , 0)     (1 , 1 , 0 , 1)   ⊥ (1 , − 1 , 0 , 0) (0 , 0 , 1 , 0)      (0 , 0 , 1 , 1) 

  39. 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1  (1 , 1 , 1 , 0)     (1 , 1 , 0 , 1)   ⊥ (1 , − 1 , 0 , 0) (0 , 0 , 1 , 0)    1   (0 , 0 , 1 , 1)  -1 0 0

  40. Further algebraization: represent configuration c as a formal Laurent power series (negative exponents included). 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 1 1 0 1 0 1 0 1 0 1 � c ( i, j ) x i y j c ← → ( i,j ) ∈ Z 2

  41. Further algebraization: represent configuration c as a formal Laurent power series (negative exponents included). 2 -1 1 2 -2 2 0 2 2 0 x y 2 1 x y 0 x y 1 x y 1 x y 1 2 0 1 -2 1 1 1 -1 1 0 x y 1 x y 1 x y 1 x y 0 x y 0 0 1 0 2 0 0 -2 -1 0 1 x y 0 x y 1 x y 1 x y 0 x y -2 -1 -1 -1 -1 1 -1 2 0 -1 1 x y 1 x y 0 x y 1 x y 0 x y -2 1 -2 -1 -2 -2 -2 0 2 -2 1 x y 0 x y 0 x y 1 x y 1 x y � c ( i, j ) x i y j c ← → ( i,j ) ∈ Z 2

  42. Further algebraization: represent configuration c as a formal Laurent power series (negative exponents included). 2 -1 1 2 -2 2 0 2 2 0 x y 2 ...+ + + + + 1 x y 0 x y 1 x y +... 1 x y 1 2 0 1 -2 1 1 1 -1 1 ...+ + + + + 0 x y +... 1 x y 1 x y 1 x y 0 x y 0 0 1 0 2 0 0 -2 -1 0 1 x y 0 x y + + + + 1 x y ...+ 1 x y +... 0 x y -2 -1 -1 -1 -1 1 -1 2 0 -1 + + + ...+ 1 x y 1 x y +... 0 x y 1 x y 0 x y + -2 1 -2 -1 -2 -2 -2 0 2 -2 ...+ 1 x y 0 x y 0 x y + + 1 x y + + 1 x y +... � c ( i, j ) x i y j c ← → ( i,j ) ∈ Z 2

  43. � c I X I c ( X ) = I ∈ Z 2 Notations • X = ( x, y ) • For I = ( i, j ) ∈ Z 2 we denote by X I = x i y j the monomial that represents cell I .

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