Low complexity Tilings of the Plane Jarkko Kari Department of - - PowerPoint PPT Presentation

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 crystals: Atoms attach to each other in a limited number of ways = ⇒ periodic arrangement of the atoms

Our goal is to understand fundamental underlying principles that connect local rules to the global regularities

• bserved in the structures.

Our setup: multidimensional symbolic dynamics (=tilings)

Configurations are infinite d-dimensional grids of symbols.

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

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

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

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

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

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

We call c a low complexity configuration if P(c, D) ≤ |D| for some finite D ⊆ Z2.

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

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

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.

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 ∈ AZ and n ∈ N. If c has at most n distinct subwords of length n then c is periodic.

Best known general bound in 2D: Theorem (Cyr, Kra): If P(c, D) ≤ 1

2|D| for some rectangle D

then c is periodic.

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

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

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

In 3D and higher dimensional cases the conjecture is false Non-periodic c D is n × n × n cube P(c, D) = 1 + n2 + . . .

In 3D and higher dimensional cases the conjecture is false Non-periodic c D is n × n × n cube P(c, D) = 1 + n2 + n2

In 3D and higher dimensional cases the conjecture is false Non-periodic c D is n × n × n cube P(c, D) = 1 + n2 + n2 < n3 = |D| for large n.

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

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

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

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

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.

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|.

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

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

D-patterns are viewed as |D|-dimensional vectors.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 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)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 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.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 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.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

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

• 1

1

Further algebraization: represent configuration c as a formal Laurent power series (negative exponents included).

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

c ← →

• (i,j)∈Z2

c(i, j)xiyj

Further algebraization: represent configuration c as a formal Laurent power series (negative exponents included).

1 2

• 1
• 2

1 1 1 2 1

• 2

1 1

• 1

2 2

0x y2

• 2

x y 1

2

• 1

2 1 2

• 1
• 1
• 1
• 1

1

• 1
• 2
• 1

2

• 2
• 2
• 2
• 1
• 2
• 2

2

• 2

1

0x y 0x y 0x y 0x y 0x y 0x y 0x y 0x y 0x y x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1

c ← →

• (i,j)∈Z2

c(i, j)xiyj

Further algebraization: represent configuration c as a formal Laurent power series (negative exponents included).

1 2

• 1
• 2

1 1 1 2 1

• 2

1 1

• 1

2 2

0x y2

• 2

x y 1

2

• 1

2 1 2

• 1
• 1
• 1
• 1

1

• 1
• 2
• 1

2

• 2
• 2
• 2
• 1
• 2
• 2

2

• 2

1

0x y 0x y 0x y 0x y 0x y 0x y 0x y 0x y 0x y x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 + ...+ + + + ...+ ...+ ...+ ...+ +... +... +... +... +... + + + + + + + + + + + + + + + +

c ← →

• (i,j)∈Z2

c(i, j)xiyj

c(X) =

• I∈Z2

cIXI Notations

• X = (x, y)
• For I = (i, j) ∈ Z2 we denote by

XI = xiyj the monomial that represents cell I.

c(X) =

• I∈Z2

cIXI

• Multiplying by monomial XJ gives translation by J ∈ Z2:

XJ · c(X) =

• I∈Z2

cIXI+J

• So c(X) is J-periodic if and only if XJ · c(X) = c(X), i.e.,

(XJ − 1)c(X) = 0

• We say that (Laurent) polynomial f(X) annihilates c(X) if

f(X)c(X) = 0.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

Recall:

• 1

1

P(c, D) ≤ |D| = ⇒ a vector with constant inner product with all D-patterns of c

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

Recall:

• 1

1

P(c, D) ≤ |D| = ⇒ a vector with constant inner product with all D-patterns of c = ⇒ a non-zero polynomial f(X) such that f(X)c(X) is a constant configuration

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

Recall:

• 1

1

P(c, D) ≤ |D| = ⇒ a vector with constant inner product with all D-patterns of c = ⇒ a non-zero polynomial f(X) such that f(X)c(X) is a constant configuration = ⇒ c(X) has a non-zero annihilator f(X)(x − 1)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

Conclusion: Every low complexity configuration c(X) ∈ Z[[X±1]] has a non-zero annihilator f(X) ∈ Z[X±1].

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1

Conclusion: Every low complexity configuration c(X) ∈ Z[[X±1]] has a non-zero annihilator f(X) ∈ Z[X±1]. We can also do all calculations modulo p. Conclusion: Every low complexity configuration c(X) ∈ Fp[[X±1]] has a non-zero annihilator f(X) ∈ Fp[X±1].

The algebraic SFT defined by a fixed polynomial f(X) ∈ Fp[X±1] is the set Xf = {c(X) | f(X)c(X) = 0}

• f all configurations over field Fp that are annihilated by f(X).

Conclusion: Every low complexity configuration belongs to an algebraic SFT.

Example (Ledrappier SFT): Configurations of F2 annihilated by f(x, y) = 1 + x + xy. Space-time diagrams of the XOR-cellular automaton: Two states {0, 1}, local rule adds (modulo 2) the value of the right neighbor to the current state:

Example (Ledrappier SFT): Configurations of F2 annihilated by f(x, y) = 1 + x + xy. Space-time diagrams of the XOR-cellular automaton: Two states {0, 1}, local rule adds (modulo 2) the value of the right neighbor to the current state:

Example (Ledrappier SFT): Configurations of F2 annihilated by f(x, y) = 1 + x + xy. Space-time diagrams of the XOR-cellular automaton: Two states {0, 1}, local rule adds (modulo 2) the value of the right neighbor to the current state:

Example (Ledrappier SFT): Configurations of F2 annihilated by f(x, y) = 1 + x + xy. Space-time diagrams of the XOR-cellular automaton: Two states {0, 1}, local rule adds (modulo 2) the value of the right neighbor to the current state:

Example (Ledrappier SFT): Configurations of F2 annihilated by f(x, y) = 1 + x + xy. Space-time diagrams of the XOR-cellular automaton: Two states {0, 1}, local rule adds (modulo 2) the value of the right neighbor to the current state:

There are many non-periodic configurations in Ledrappier. . .

There are also two-periodic configurations. . .

More generally: Space-time diagrams of any linear cellular automaton form an algebraic SFT, e.g., 3-neighbor XOR: and a sample space-time diagram

From Fp back to Z: Theorem (JK, M.Szabados 2015): If a configuration c(X)

• ver Z has a non-trivial annihilator then it has an annihilator

∆(X) = (1 − XI1)(1 − XI2) . . . (1 − XIm) for some non-zero Ii ∈ Z2.

From Fp back to Z: Theorem (JK, M.Szabados 2015): If a configuration c(X)

• ver Z has a non-trivial annihilator then it has an annihilator

∆(X) = (1 − XI1)(1 − XI2) . . . (1 − XIm) for some non-zero Ii ∈ Z2. If all calculations in c(X)∆(X) are done modulo p we get Corollary: Every low-complexity configuration over Fp has an annihilator of form ∆(X).

We have that any low complexity c(X) in the Ledrappier SFT is annihilated by f(X) = 1 + x + xy (definition of Ledrappier) ∆(X) = (1 − XI1) . . . (1 − XIm) (our theorem)

We have that any low complexity c(X) in the Ledrappier SFT is annihilated by f(X) = 1 + x + xy (definition of Ledrappier) ∆(X) = (1 − XI1) . . . (1 − XIm) (our theorem) Polynomials f(X) and ∆(X) have no common factors:

• All irreducible factors of ∆(X) are line polynomials (all

terms aligned on a single line).

1 1 2

1 + 2x2y + x6y3

We have that any low complexity c(X) in the Ledrappier SFT is annihilated by f(X) = 1 + x + xy (definition of Ledrappier) ∆(X) = (1 − XI1) . . . (1 − XIm) (our theorem) Polynomials f(X) and ∆(X) have no common factors:

• All irreducible factors of ∆(X) are line polynomials (all

terms aligned on a single line).

• f(X) has no line polynomial factors.

Elimination theory: The resultant Resx(f, g)

• f polynomials f(x, y) and g(x, y).
• A polynomial in variable y only: x has been eliminated.
• Generated by f and g so

Resx(f, g) = α(X)f(X) + β(X)g(X) for some polynomials α and β.

• If f and g do not have common factors then Resx(f, g) is

not zero.

The annihilators of a low complexity c in Ledrappier f(X) = 1 + y + xy ∆(X) = (1 − XI1)(1 − XI2) . . . (1 − XIm) have no common factors so the resultants Resx(f, ∆) and Resy(f, ∆) are vertical and horizontal line polynomials generated by f(X) and ∆(X). f and ∆ annihilate c = ⇒ Resx(f, ∆) and Resy(f, ∆) annihilate c Conclusion: c is two-periodic.

Theorem (JK, E.Moutot 2019). Let c be a low-complexity configuration of an algebraic SFT Xf over the finite field Fp. If the defining polynomial f of the SFT

• has no line polynomial factors then c is two-periodic.
• has line polynomial factors only in one direction then c is

periodic in this direction.

Theorem (JK, E.Moutot 2019). Let c be a low-complexity configuration of an algebraic SFT Xf over the finite field Fp. If the defining polynomial f of the SFT

• has no line polynomial factors then c is two-periodic.
• has line polynomial factors only in one direction then c is

periodic in this direction. The theorem applies to the space-time diagrams of any linear cellular automaton over Fp.

Back to Nivat’s conjecture: A low complexity configuration c over Z has a special annihilator ∆(X) = (1 − XI1)(1 − XI2) . . . (1 − XIm)

• If m = 1 then c is periodic.
• Also otherwise, c is a sum of m periodic power series.

• Example. The 3D counter example

is a sum of two periodic configurations. It is annihilated by polynomial (1 − y)(1 − x).

Using the special annihilator ∆(X) = (1 − XI1)(1 − XI2) . . . (1 − XIm) we obtain (proof skipped): Theorem (JK, E.Moutot 2019): If c is uniformly recurrent and P(c, D) ≤ |D| for some rectangle D then c is periodic. In fact: Any convex shape D works in place of a rectangle.

Every non-empty subshift contains a uniformly recurrent configuration = ⇒ No low complexity aperiodic SFT’s: Any subshift that contains a low complexity configuration (w.r.t. a convex shape) also contains a periodic configuration.

Every non-empty subshift contains a uniformly recurrent configuration = ⇒ No low complexity aperiodic SFT’s: Any subshift that contains a low complexity configuration (w.r.t. a convex shape) also contains a periodic configuration. = ⇒ Decidability of the domino problem: Given a set of at most |D| allowed patterns of some convex shape D, there is an algorithm to determine if a configuration exists whose D-patterns are all allowed.

Every non-empty subshift contains a uniformly recurrent configuration = ⇒ No low complexity aperiodic SFT’s: Any subshift that contains a low complexity configuration (w.r.t. a convex shape) also contains a periodic configuration. = ⇒ Decidability of the domino problem: Given a set of at most |D| allowed patterns of some convex shape D, there is an algorithm to determine if a configuration exists whose D-patterns are all allowed. Standard reasoning:

• A semi-algorithm to test if a given SFT is empty.
• A semi-algorithm to test if a given SFT contains a periodic

configuration.

Every non-empty subshift contains a uniformly recurrent configuration = ⇒ No low complexity aperiodic SFT’s: Any subshift that contains a low complexity configuration (w.r.t. a convex shape) also contains a periodic configuration. = ⇒ Decidability of the domino problem: Given a set of at most |D| allowed patterns of some convex shape D, there is an algorithm to determine if a configuration exists whose D-patterns are all allowed. Best possible: The following problem is undecidable for any fixed α > 1. Given a set of at most α|D| allowed patterns of some square shape D, is there a configuration whose D-patterns are all allowed ?

Open questions:

• What can one say for non-convex shapes ? Is there an

algorithm to test emptyness ? Note that there exist low complexity uniformly recurrent configurations that are non-periodic.

• Higher dimensional cases ? Algorithms and aperiodicity
• f low complexity SFTs in higher dimensions ?
• The original Nivat’s conjecture is still open. Potentially

true for any convex shape in place of the rectangle (as suggested by Julien Cassaigne).

Open questions:

• What can one say for non-convex shapes ? Is there an

algorithm to test emptyness ? Note that there exist low complexity uniformly recurrent configurations that are non-periodic.

• Higher dimensional cases ? Algorithms and aperiodicity
• f low complexity SFTs in higher dimensions ?
• The original Nivat’s conjecture is still open. Potentially

true for any convex shape in place of the rectangle (as suggested by Julien Cassaigne).

T h a n k Y

• u

Recall: Configuration c is uniformly recurrent if ∀ finite patterns p that appear in c ∃ number N s.t. there is a copy of p in c inside every N × N square

Recall: Configuration c is uniformly recurrent if ∀ finite patterns p that appear in c ∃ number N s.t. there is a copy of p in c inside every N × N square

p=

Recall: Configuration c is uniformly recurrent if ∀ finite patterns p that appear in c ∃ number N s.t. there is a copy of p in c inside every N × N square

p=

Recall: Configuration c is uniformly recurrent if ∀ finite patterns p that appear in c ∃ number N s.t. there is a copy of p in c inside every N × N square

p=

Recall: Configuration c is uniformly recurrent if ∀ finite patterns p that appear in c ∃ number N s.t. there is a copy of p in c inside every N × N square

p=

Recall: Configuration c is uniformly recurrent if ∀ finite patterns p that appear in c ∃ number N s.t. there is a copy of p in c inside every N × N square

p=

Recall: Configuration c is uniformly recurrent if ∀ finite patterns p that appear in c ∃ number N s.t. there is a copy of p in c inside every N × N square

p=

Recall: Configuration c is uniformly recurrent if ∀ finite patterns p that appear in c ∃ number N s.t. there is a copy of p in c inside every N × N square

p=