Complexity, periodicity, and expansiveness Van Cyr Northwestern - - PowerPoint PPT Presentation

Complexity, periodicity, and expansiveness

Van Cyr

Northwestern University June 4, 2013

University of Crete

Van Cyr Complexity, periodicity, and expansiveness

Introduction

A=finite set, X = AZd . For α, β ∈ X, d(α, β) = 2− min{

x:α( x)=β( x)}.

For S ⊂ Zd and α ∈ AZd , α|S=restriction of α to S. T

v : X → X is the translation by

v: (T

vα)(

x) := α( x + v).

Van Cyr Complexity, periodicity, and expansiveness

Introduction

A=finite set, X = AZd . For α, β ∈ X, d(α, β) = 2− min{

x:α( x)=β( x)}.

For S ⊂ Zd and α ∈ AZd , α|S=restriction of α to S. T

v : X → X is the translation by

v: (T

vα)(

x) := α( x + v). If η ∈ X, the block complexity function Pα : Nd → N is Pη(n1, . . . , nd) := ˛ ˛ ˛{(T

vα)|[1,n1]×···×[1,nd ] :

v ∈ Zd} ˛ ˛ ˛ . Xη := O(η).

Van Cyr Complexity, periodicity, and expansiveness

Introduction

A=finite set, X = AZd . For α, β ∈ X, d(α, β) = 2− min{

x:α( x)=β( x)}.

For S ⊂ Zd and α ∈ AZd , α|S=restriction of α to S. T

v : X → X is the translation by

v: (T

vα)(

x) := α( x + v). If η ∈ X, the block complexity function Pα : Nd → N is Pη(n1, . . . , nd) := ˛ ˛ ˛{(T

vα)|[1,n1]×···×[1,nd ] :

v ∈ Zd} ˛ ˛ ˛ . Xη := O(η). Theorem(Morse-Hedlund, 1940): α ∈ AZ is periodic iff there exists n ∈ N such that Pα(n) ≤ n.

Van Cyr Complexity, periodicity, and expansiveness

Introduction

A=finite set, X = AZd . For α, β ∈ X, d(α, β) = 2− min{

x:α( x)=β( x)}.

For S ⊂ Zd and α ∈ AZd , α|S=restriction of α to S. T

v : X → X is the translation by

v: (T

vα)(

x) := α( x + v). If η ∈ X, the block complexity function Pα : Nd → N is Pη(n1, . . . , nd) := ˛ ˛ ˛{(T

vα)|[1,n1]×···×[1,nd ] :

v ∈ Zd} ˛ ˛ ˛ . Xη := O(η). Theorem(Morse-Hedlund, 1940): α ∈ AZ is periodic iff there exists n ∈ N such that Pα(n) ≤ n. Conjecture(M. Nivat, 1997): If α ∈ AZ2 and there exist n, k ∈ N such that Pη(n, k) ≤ nk, then α is periodic.

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1.

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist)

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k).

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k). Sander-Tijdeman(‘00): For d > 2 and all n, there exists aperiodic η ∈ AZd such that Pη(n, . . . , n) = 2nd−1 + 1.

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k). Sander-Tijdeman(‘00): For d > 2 and all n, there exists aperiodic η ∈ AZd such that Pη(n, . . . , n) = 2nd−1 + 1. Sander-Tijdeman(‘02): If there exists n such that Pη(n, 2) ≤ 2n then η is periodic.

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k). Sander-Tijdeman(‘00): For d > 2 and all n, there exists aperiodic η ∈ AZd such that Pη(n, . . . , n) = 2nd−1 + 1. Sander-Tijdeman(‘02): If there exists n such that Pη(n, 2) ≤ 2n then η is periodic. Epifanio-Koskas-Mignosi(‘03): If there exist n, k ∈ N such that Pη(n, k) ≤

nk 144 , then

η is periodic. (also claimed

nk 100 works) Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k). Sander-Tijdeman(‘00): For d > 2 and all n, there exists aperiodic η ∈ AZd such that Pη(n, . . . , n) = 2nd−1 + 1. Sander-Tijdeman(‘02): If there exists n such that Pη(n, 2) ≤ 2n then η is periodic. Epifanio-Koskas-Mignosi(‘03): If there exist n, k ∈ N such that Pη(n, k) ≤

nk 144 , then

η is periodic. (also claimed

nk 100 works)

Quas-Zamboni(‘04): If there exist n, k ∈ N such that Pη(n, k) ≤ nk

16 , then η is

periodic.

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k). Sander-Tijdeman(‘00): For d > 2 and all n, there exists aperiodic η ∈ AZd such that Pη(n, . . . , n) = 2nd−1 + 1. Sander-Tijdeman(‘02): If there exists n such that Pη(n, 2) ≤ 2n then η is periodic. Epifanio-Koskas-Mignosi(‘03): If there exist n, k ∈ N such that Pη(n, k) ≤

nk 144 , then

η is periodic. (also claimed

nk 100 works)

Quas-Zamboni(‘04): If there exist n, k ∈ N such that Pη(n, k) ≤ nk

16 , then η is

periodic. Durand-Rigo(’11): Nivat-like bound on the recurrent complexity function is equivalent to definability in Presburger arithmetic.

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k). Sander-Tijdeman(‘00): For d > 2 and all n, there exists aperiodic η ∈ AZd such that Pη(n, . . . , n) = 2nd−1 + 1. Sander-Tijdeman(‘02): If there exists n such that Pη(n, 2) ≤ 2n then η is periodic. Epifanio-Koskas-Mignosi(‘03): If there exist n, k ∈ N such that Pη(n, k) ≤

nk 144 , then

η is periodic. (also claimed

nk 100 works)

Quas-Zamboni(‘04): If there exist n, k ∈ N such that Pη(n, k) ≤ nk

16 , then η is

periodic. Durand-Rigo(’11): Nivat-like bound on the recurrent complexity function is equivalent to definability in Presburger arithmetic. (works in dimension d)

Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k). Sander-Tijdeman(‘00): For d > 2 and all n, there exists aperiodic η ∈ AZd such that Pη(n, . . . , n) = 2nd−1 + 1. Sander-Tijdeman(‘02): If there exists n such that Pη(n, 2) ≤ 2n then η is periodic. Epifanio-Koskas-Mignosi(‘03): If there exist n, k ∈ N such that Pη(n, k) ≤

nk 144 , then

η is periodic. (also claimed

nk 100 works)

Quas-Zamboni(‘04): If there exist n, k ∈ N such that Pη(n, k) ≤ nk

16 , then η is

periodic. Durand-Rigo(’11): Nivat-like bound on the recurrent complexity function is equivalent to definability in Presburger arithmetic. (works in dimension d) C.-Kra(‘12): If there exist n, k ∈ N such that Pη(n, k) ≤ nk

2 , then η is periodic. Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results

Cassaigne(‘99): Classified all η whose complexity function is given by Pη(n, k) = nk + 1. (No minimal examples exist) Berthe-Vuillon(‘00): Can code two circle rotations and get complexity Pη(n, k) = nk + n + k. Example of a uniformly recurrent coloring with complexity Pη(n, k) = nk + min(n, k). Sander-Tijdeman(‘00): For d > 2 and all n, there exists aperiodic η ∈ AZd such that Pη(n, . . . , n) = 2nd−1 + 1. Sander-Tijdeman(‘02): If there exists n such that Pη(n, 2) ≤ 2n then η is periodic. Epifanio-Koskas-Mignosi(‘03): If there exist n, k ∈ N such that Pη(n, k) ≤

nk 144 , then

η is periodic. (also claimed

nk 100 works)

Quas-Zamboni(‘04): If there exist n, k ∈ N such that Pη(n, k) ≤ nk

16 , then η is

periodic. Durand-Rigo(’11): Nivat-like bound on the recurrent complexity function is equivalent to definability in Presburger arithmetic. (works in dimension d) C.-Kra(‘12): If there exist n, k ∈ N such that Pη(n, k) ≤ nk

2 , then η is periodic.

C.-Kra(’13): If there exists n ∈ N such that Pη(n, 3) ≤ 3n, then η is periodic.

Van Cyr Complexity, periodicity, and expansiveness

Expansive and Nonexpansive subspaces for Zd-actions

Definition: If X is a compact metric space and T is a continuous Zd-action, then a subspace V ⊆ Rd is expansive for T if ∃R, δ > 0 such that: whenever x, y ∈ X satisfy d(T

vx, T vy) < δ for all d(

v, V ) < R, then x = y.

Van Cyr Complexity, periodicity, and expansiveness

Expansive and Nonexpansive subspaces for Zd-actions

Definition: If X is a compact metric space and T is a continuous Zd-action, then a subspace V ⊆ Rd is expansive for T if ∃R, δ > 0 such that: whenever x, y ∈ X satisfy d(T

vx, T vy) < δ for all d(

v, V ) < R, then x = y. Expansiveness is an open condition in Gk(Rd).

Van Cyr Complexity, periodicity, and expansiveness

Expansive and Nonexpansive subspaces for Zd-actions

Definition: If X is a compact metric space and T is a continuous Zd-action, then a subspace V ⊆ Rd is expansive for T if ∃R, δ > 0 such that: whenever x, y ∈ X satisfy d(T

vx, T vy) < δ for all d(

v, V ) < R, then x = y. Expansiveness is an open condition in Gk(Rd). Nonexpansive subspaces are common:

Van Cyr Complexity, periodicity, and expansiveness

Expansive and Nonexpansive subspaces for Zd-actions

Definition: If X is a compact metric space and T is a continuous Zd-action, then a subspace V ⊆ Rd is expansive for T if ∃R, δ > 0 such that: whenever x, y ∈ X satisfy d(T

vx, T vy) < δ for all d(

v, V ) < R, then x = y. Expansiveness is an open condition in Gk(Rd). Nonexpansive subspaces are common: Theorem (Boyle-Lind, ‘97) If X is an infinite compact metric space and T is a continuous Zd-action, then for all 0 ≤ k < d there is a k-dimensional subspace of Rd that is nonexpansive for T.

Van Cyr Complexity, periodicity, and expansiveness

Expansive and Nonexpansive subspaces for Zd-actions

Definition: If X is a compact metric space and T is a continuous Zd-action, then a subspace V ⊆ Rd is expansive for T if ∃R, δ > 0 such that: whenever x, y ∈ X satisfy d(T

vx, T vy) < δ for all d(

v, V ) < R, then x = y. Expansiveness is an open condition in Gk(Rd). Nonexpansive subspaces are common: Theorem (Boyle-Lind, ‘97) If X is an infinite compact metric space and T is a continuous Zd-action, then for all 0 ≤ k < d there is a k-dimensional subspace of Rd that is nonexpansive for T. Corollary Suppose η ∈ AZ2, T is the action by translation, and Xη := O(η).

Van Cyr Complexity, periodicity, and expansiveness

Expansive and Nonexpansive subspaces for Zd-actions

Definition: If X is a compact metric space and T is a continuous Zd-action, then a subspace V ⊆ Rd is expansive for T if ∃R, δ > 0 such that: whenever x, y ∈ X satisfy d(T

vx, T vy) < δ for all d(

v, V ) < R, then x = y. Expansiveness is an open condition in Gk(Rd). Nonexpansive subspaces are common: Theorem (Boyle-Lind, ‘97) If X is an infinite compact metric space and T is a continuous Zd-action, then for all 0 ≤ k < d there is a k-dimensional subspace of Rd that is nonexpansive for T. Corollary Suppose η ∈ AZ2, T is the action by translation, and Xη := O(η). Every

• ne-dimensional subspace of R2 is expansive for T if and only if Xη is finite.

Van Cyr Complexity, periodicity, and expansiveness

Expansive and Nonexpansive subspaces for Zd-actions

Definition: If X is a compact metric space and T is a continuous Zd-action, then a subspace V ⊆ Rd is expansive for T if ∃R, δ > 0 such that: whenever x, y ∈ X satisfy d(T

vx, T vy) < δ for all d(

v, V ) < R, then x = y. Expansiveness is an open condition in Gk(Rd). Nonexpansive subspaces are common: Theorem (Boyle-Lind, ‘97) If X is an infinite compact metric space and T is a continuous Zd-action, then for all 0 ≤ k < d there is a k-dimensional subspace of Rd that is nonexpansive for T. Corollary Suppose η ∈ AZ2, T is the action by translation, and Xη := O(η). Every

• ne-dimensional subspace of R2 is expansive for T if and only if Xη is finite.

Question(Boyle-Lind, ‘97): In dimension 2, if there is a unique one-dimensional nonexpansive subspace, must there be a vector v ∈ Z2 \ { 0} such that T

v = Id? Van Cyr Complexity, periodicity, and expansiveness

Expansive and Nonexpansive subspaces for Zd-actions

Definition: If X is a compact metric space and T is a continuous Zd-action, then a subspace V ⊆ Rd is expansive for T if ∃R, δ > 0 such that: whenever x, y ∈ X satisfy d(T

vx, T vy) < δ for all d(

v, V ) < R, then x = y. Expansiveness is an open condition in Gk(Rd). Nonexpansive subspaces are common: Theorem (Boyle-Lind, ‘97) If X is an infinite compact metric space and T is a continuous Zd-action, then for all 0 ≤ k < d there is a k-dimensional subspace of Rd that is nonexpansive for T. Corollary Suppose η ∈ AZ2, T is the action by translation, and Xη := O(η). Every

• ne-dimensional subspace of R2 is expansive for T if and only if Xη is finite.

Question(Boyle-Lind, ‘97): In dimension 2, if there is a unique one-dimensional nonexpansive subspace, must there be a vector v ∈ Z2 \ { 0} such that T

v = Id?

Hochman(‘11): No. (Also showed that every closed subset of G1(R2) can be realized as the set of nonexpansive directions for a Z2 action.)

Van Cyr Complexity, periodicity, and expansiveness

Nonexpansiveness and Nivat’s conjecture

Theorem (C.-Kra, ‘12) Suppose η : Z2 → A and there exists n, k ∈ N such that Pη(n, k) ≤ nk. Then η is singly (and not doubly) periodic if and only if there is a unique subspace of R2 which is nonexpansive for (Xη, T).

Van Cyr Complexity, periodicity, and expansiveness

Nonexpansiveness and Nivat’s conjecture

Theorem (C.-Kra, ‘12) Suppose η : Z2 → A and there exists n, k ∈ N such that Pη(n, k) ≤ nk. Then η is singly (and not doubly) periodic if and only if there is a unique subspace of R2 which is nonexpansive for (Xη, T). Moreover this subspace is a rational line through the

• rigin in R2 and every period vector for η is contained in it.

Van Cyr Complexity, periodicity, and expansiveness

Nonexpansiveness and Nivat’s conjecture

Theorem (C.-Kra, ‘12) Suppose η : Z2 → A and there exists n, k ∈ N such that Pη(n, k) ≤ nk. Then η is singly (and not doubly) periodic if and only if there is a unique subspace of R2 which is nonexpansive for (Xη, T). Moreover this subspace is a rational line through the

• rigin in R2 and every period vector for η is contained in it.

Nivat’s conjecture (Modified version): If η ∈ AZ2 and there exist n, k ∈ N such that Pη(n, k) ≤ nk then there is at most one nonexpansive direction for the Z2-action on O(η) by translation.

Van Cyr Complexity, periodicity, and expansiveness

Nonexpansiveness and Nivat’s conjecture

Theorem (C.-Kra, ‘12) Suppose η : Z2 → A and there exists n, k ∈ N such that Pη(n, k) ≤ nk. Then η is singly (and not doubly) periodic if and only if there is a unique subspace of R2 which is nonexpansive for (Xη, T). Moreover this subspace is a rational line through the

• rigin in R2 and every period vector for η is contained in it.

Nivat’s conjecture (Modified version): If η ∈ AZ2 and there exist n, k ∈ N such that Pη(n, k) ≤ nk then there is at most one nonexpansive direction for the Z2-action on O(η) by translation. Theorem (C.-Kra, ‘12) If η : Z2 → A and there exist n, k ∈ N such that Pη(n, k) ≤ nk

2 , then there is at most

• ne nonexpansive direction for (Xη, T).

Van Cyr Complexity, periodicity, and expansiveness

Nonexpansiveness and Nivat’s conjecture

Theorem (C.-Kra, ‘12) Suppose η : Z2 → A and there exists n, k ∈ N such that Pη(n, k) ≤ nk. Then η is singly (and not doubly) periodic if and only if there is a unique subspace of R2 which is nonexpansive for (Xη, T). Moreover this subspace is a rational line through the

• rigin in R2 and every period vector for η is contained in it.

Nivat’s conjecture (Modified version): If η ∈ AZ2 and there exist n, k ∈ N such that Pη(n, k) ≤ nk then there is at most one nonexpansive direction for the Z2-action on O(η) by translation. Theorem (C.-Kra, ‘12) If η : Z2 → A and there exist n, k ∈ N such that Pη(n, k) ≤ nk

2 , then there is at most

• ne nonexpansive direction for (Xη, T).

Theorem (C.-Kra, ‘13) If η : Z2 → A and there exists n ∈ N such that Pη(n, 3) ≤ 3n, then there is at most

• ne nonexpansive direction for (Xη, T).

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Complexity function: For η ∈ AZ2 and S ⊂ Z2 we define Pη(S) := ˛ ˛ ˛ n (T

vη)|S :

v ∈ Z2o˛ ˛ ˛ to be the number of distinct η-colorings of S.

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Complexity function: For η ∈ AZ2 and S ⊂ Z2 we define Pη(S) := ˛ ˛ ˛ n (T

vη)|S :

v ∈ Z2o˛ ˛ ˛ to be the number of distinct η-colorings of S. Discrepancy function: For S ⊂ Z2 finite, convex, and nonempty set Dη(S) := Pη(S) − |S| .

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Complexity function: For η ∈ AZ2 and S ⊂ Z2 we define Pη(S) := ˛ ˛ ˛ n (T

vη)|S :

v ∈ Z2o˛ ˛ ˛ to be the number of distinct η-colorings of S. Discrepancy function: For S ⊂ Z2 finite, convex, and nonempty set Dη(S) := Pη(S) − |S| . Lemma: If x ∈ S ⊂ Z2 then either Dη(S \ {x}) ≤ Dη(S),

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Complexity function: For η ∈ AZ2 and S ⊂ Z2 we define Pη(S) := ˛ ˛ ˛ n (T

vη)|S :

v ∈ Z2o˛ ˛ ˛ to be the number of distinct η-colorings of S. Discrepancy function: For S ⊂ Z2 finite, convex, and nonempty set Dη(S) := Pη(S) − |S| . Lemma: If x ∈ S ⊂ Z2 then either Dη(S \ {x}) ≤ Dη(S), or Dη(S \ {x}) = Dη(S) + 1 and every η-coloring of S \ {x} extends uniquely to an η-coloring of S.

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Complexity function: For η ∈ AZ2 and S ⊂ Z2 we define Pη(S) := ˛ ˛ ˛ n (T

vη)|S :

v ∈ Z2o˛ ˛ ˛ to be the number of distinct η-colorings of S. Discrepancy function: For S ⊂ Z2 finite, convex, and nonempty set Dη(S) := Pη(S) − |S| . Lemma: If x ∈ S ⊂ Z2 then either Dη(S \ {x}) ≤ Dη(S), or Dη(S \ {x}) = Dη(S) + 1 and every η-coloring of S \ {x} extends uniquely to an η-coloring of S. Definition: A finite, convex set S ⊂ R is a generating set for η if, for all vertices x ∈ V (S), every η-coloring of S \ {x} extends uniquely to an η-coloring of S.

Van Cyr Complexity, periodicity, and expansiveness

Lemma If η : Z2 → A and there exist n, k ∈ N such that Pη(n, k) ≤ nk for some n, k ∈ N, then η has a generating set. Proof. R := [1, n] × [1, k] ∩ Z2. We have Dη(R) ≤ 0.

Van Cyr Complexity, periodicity, and expansiveness

Lemma If η : Z2 → A and there exist n, k ∈ N such that Pη(n, k) ≤ nk for some n, k ∈ N, then η has a generating set. Proof. R := [1, n] × [1, k] ∩ Z2. We have Dη(R) ≤ 0. Let S ⊂ R be minimal in the collection

• f all nonempty, convex subsets of R such that Dη(S) ≤ Dη(R).

Van Cyr Complexity, periodicity, and expansiveness

Lemma If η : Z2 → A and there exist n, k ∈ N such that Pη(n, k) ≤ nk for some n, k ∈ N, then η has a generating set. Proof. R := [1, n] × [1, k] ∩ Z2. We have Dη(R) ≤ 0. Let S ⊂ R be minimal in the collection

• f all nonempty, convex subsets of R such that Dη(S) ≤ Dη(R). S has at least two

elements because Dη({x}) = |A| − 1 > 0, so any convex subset of S has larger discrepancy.

Van Cyr Complexity, periodicity, and expansiveness

Lemma If η : Z2 → A and there exist n, k ∈ N such that Pη(n, k) ≤ nk for some n, k ∈ N, then η has a generating set. Proof. R := [1, n] × [1, k] ∩ Z2. We have Dη(R) ≤ 0. Let S ⊂ R be minimal in the collection

• f all nonempty, convex subsets of R such that Dη(S) ≤ Dη(R). S has at least two

elements because Dη({x}) = |A| − 1 > 0, so any convex subset of S has larger

• discrepancy. If w ∈ E(S) and |w| :=

˛ ˛w ∩ Z2˛ ˛ then S \ {w} is convex.

Van Cyr Complexity, periodicity, and expansiveness

Lemma If η : Z2 → A and there exist n, k ∈ N such that Pη(n, k) ≤ nk for some n, k ∈ N, then η has a generating set. Proof. R := [1, n] × [1, k] ∩ Z2. We have Dη(R) ≤ 0. Let S ⊂ R be minimal in the collection

• f all nonempty, convex subsets of R such that Dη(S) ≤ Dη(R). S has at least two

elements because Dη({x}) = |A| − 1 > 0, so any convex subset of S has larger

• discrepancy. If w ∈ E(S) and |w| :=

˛ ˛w ∩ Z2˛ ˛ then S \ {w} is convex. Pη(S \ {w}) − |S \ {w}| ≥ Pη(S) − |S| + 1

Van Cyr Complexity, periodicity, and expansiveness

Lemma If Pη(n, k) ≤ nk for some n, k ∈ N then η has a generating set. Proof. R := [1, n] × [1, k] ∩ Z2. Let S ⊂ R be a minimal, convex set such that Dη(S) = Dη(R). |S| > 1 (since Dη({x}) = |A| − 1 > 0 ≥ Dη(R)). If w ∈ E(S) and |w| := ˛ ˛w ∩ Z2˛ ˛ then S \ {w} is convex. Pη(S \ {w}) − |S| + |w| ≥ Pη(S) − |S| + 1

Van Cyr Complexity, periodicity, and expansiveness

Lemma If Pη(n, k) ≤ nk for some n, k ∈ N then η has a generating set. Proof. R := [1, n] × [1, k] ∩ Z2. Let S ⊂ R be a minimal, convex set such that Dη(S) = Dη(R). |S| > 1 (since Dη({x}) = |A| − 1 > 0 ≥ Dη(R)). If w ∈ E(S) and |w| := ˛ ˛w ∩ Z2˛ ˛ then S \ {w} is convex. Pη(S) − Pη(S \ {w}) ≤ |w| − 1.

Van Cyr Complexity, periodicity, and expansiveness

Lemma If Pη(n, k) ≤ nk for some n, k ∈ N then η has a generating set. Proof. R := [1, n] × [1, k] ∩ Z2. Let S ⊂ R be a minimal, convex set such that Dη(S) = Dη(R). |S| > 1 (since Dη({x}) = |A| − 1 > 0 ≥ Dη(R)). If w ∈ E(S) and |w| := ˛ ˛w ∩ Z2˛ ˛ then S \ {w} is convex. Pη(S) − Pη(S \ {w}) ≤ |w| − 1. ≤ |w| − 1 colorings of S \ {w} that do not uniquely extend to colorings of S

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Lemma If there exist n, k ∈ N such that Pη(n, k) ≤ nk and N ≥ n, K ≥ k, then the Z2-SFT whose allowed colorings of [1, N] × [1, K] are those occurring in η, has entropy zero.

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Lemma If there exist n, k ∈ N such that Pη(n, k) ≤ nk and N ≥ n, K ≥ k, then the Z2-SFT whose allowed colorings of [1, N] × [1, K] are those occurring in η, has entropy zero. (Also true in dimension d.)

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Lemma If there exist n, k ∈ N such that Pη(n, k) ≤ nk and N ≥ n, K ≥ k, then the Z2-SFT whose allowed colorings of [1, N] × [1, K] are those occurring in η, has entropy zero. (Also true in dimension d.) Lemma If Pη(n, k) ≤ nk and S is a generating set for η, then every nonexpansive direction is parallel to an edge of ∂S.

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Lemma If there exist n, k ∈ N such that Pη(n, k) ≤ nk and N ≥ n, K ≥ k, then the Z2-SFT whose allowed colorings of [1, N] × [1, K] are those occurring in η, has entropy zero. (Also true in dimension d.) Lemma If Pη(n, k) ≤ nk and S is a generating set for η, then every nonexpansive direction is parallel to an edge of ∂S. No irrational nonexpansive directions.

Van Cyr Complexity, periodicity, and expansiveness

Restrictions on nonexpansive directions

Lemma If there exist n, k ∈ N such that Pη(n, k) ≤ nk and N ≥ n, K ≥ k, then the Z2-SFT whose allowed colorings of [1, N] × [1, K] are those occurring in η, has entropy zero. (Also true in dimension d.) Lemma If Pη(n, k) ≤ nk and S is a generating set for η, then every nonexpansive direction is parallel to an edge of ∂S. No irrational nonexpansive directions. Restrictions on the period (when the nonexpansive direction is unique).

Van Cyr Complexity, periodicity, and expansiveness

A unique nonexpansive direction

Van Cyr Complexity, periodicity, and expansiveness

A unique nonexpansive direction

Van Cyr Complexity, periodicity, and expansiveness

“Nonexpansiveness produces periodicity”

S =generating set. Suppose w ∈ E(S) and every line parallel to w that has nonempty intersection with S intersects it in at least |w ∩ S| − 1 integer points.

Van Cyr Complexity, periodicity, and expansiveness

“Nonexpansiveness produces periodicity”

S =generating set. Suppose w ∈ E(S) and every line parallel to w that has nonempty intersection with S intersects it in at least |w ∩ S| − 1 integer points.

Van Cyr Complexity, periodicity, and expansiveness

“Nonexpansiveness produces periodicity”

S =generating set. Suppose w ∈ E(S) and every line parallel to w that has nonempty intersection with S intersects it in at least |w ∩ S| − 1 integer points. Every coloring of S \ w that occurs in this strip must extend non-uniquely.

Van Cyr Complexity, periodicity, and expansiveness

“Nonexpansiveness produces periodicity”

S =generating set. Suppose w ∈ E(S) and every line parallel to w that has nonempty intersection with S intersects it in at least |w ∩ S| − 1 integer points. Every coloring of S \ w that occurs in this strip must extend non-uniquely. There are at most |w ∩ S| − 1 such colorings.

Van Cyr Complexity, periodicity, and expansiveness

“Nonexpansiveness produces periodicity”

S =generating set. Suppose w ∈ E(S) and every line parallel to w that has nonempty intersection with S intersects it in at least |w ∩ S| − 1 integer points. Every coloring of S \ w that occurs in this strip must extend non-uniquely. There are at most |w ∩ S| − 1 such colorings. Every (vertical) line that intersects S intersects in at least |w ∩ S| − 1 integer points.

Van Cyr Complexity, periodicity, and expansiveness

“Nonexpansiveness produces periodicity”

S =generating set. Suppose w ∈ E(S) and every line parallel to w that has nonempty intersection with S intersects it in at least |w ∩ S| − 1 integer points. Every coloring of S \ w that occurs in this strip must extend non-uniquely. There are at most |w ∩ S| − 1 such colorings. Every (vertical) line that intersects S intersects in at least |w ∩ S| − 1 integer points. The coloring of the strip is periodic by the (one dimensional) Morse-Hedlund theorem.

Van Cyr Complexity, periodicity, and expansiveness

“Nonexpansiveness produces periodicity”

Van Cyr Complexity, periodicity, and expansiveness

Balanced set

Van Cyr Complexity, periodicity, and expansiveness

Balanced set

Van Cyr Complexity, periodicity, and expansiveness

Idea of the argument

Van Cyr Complexity, periodicity, and expansiveness

Idea of the argument

Van Cyr Complexity, periodicity, and expansiveness

Idea of the argument

Van Cyr Complexity, periodicity, and expansiveness

Idea of the argument

Van Cyr Complexity, periodicity, and expansiveness

Idea of the argument

Van Cyr Complexity, periodicity, and expansiveness