The rectangle complexity of functions on two-dimensional lattices

Sander, J. W.; Tijdeman, R.

doi:10.1016/s0304-3975(01)00281-x

Cited by 47 publications

(18 citation statements)

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“…In a first step toward the conjecture, Sander and Tijdeman [20] showed that if there is some n such that P η (n, 2) ≤ 2n (or such that P η (2, n) ≤ 2n), then η is periodic. Soon after, Epifanio, Koskas and Mignosi [10] proved a weak version of 6488 VAN CYR AND BRYNA KRA the conjecture showing that if P η (n, k) ≤ nk 144 for some n and k, then η is periodic; Quas and Zamboni [17] improved the constant to 1 16 .…”

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confidence: 99%

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Nonexpansive $\mathbb {Z}^2$-subdynamics and Nivat’s Conjecture

Cyr

Kra

2015

Trans. Amer. Math. Soc.

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For a finite alphabet A and η : Z → A, the Morse-Hedlund Theorem states that η is periodic if and only if there exists n ∈ N such that the block complexity function P η (n) satisfies P η (n) ≤ n, and this statement is naturally studied by analyzing the dynamics of a Z-action associated with η.In dimension two, we analyze the subdynamics of a Z 2 -action associated with η : Z 2 → A and show that if there exist n, k ∈ N such that the n × k rectangular complexity P η (n, k) satisfies P η (n, k) ≤ nk, then the periodicity of η is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist n, k ∈ N such that P η (n, k) ≤ nk 2 , then η is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words. download from IP 157.182.150.22. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Licensed to West Virginia Univ. Prepared on Tue Jul 14 03:26:28 EDT 2015 for download from IP 157.182.150.22. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use NONEXPANSIVE Z 2 -SUBDYNAMICS AND NIVAT'S CONJECTURE 6489 constant δ > 0 such that whenever x, y ∈ X satisfy d(T u x, T u y) < δ for all u with d( u, V ) < r, then x = y. If V = R d , we recover the usual definition of expansiveness. Any subspace that is not expansive is called a nonexpansive subspace. They showed that Z d -dynamical systems with nonexpansive subspaces are common: Theorem 1.3 (Boyle and Lind [4]). Let X be an infinite compact metric space with a continuous Z d -action. For each 0 ≤ k < d, there exists a k-dimensional subspace of R d that is nonexpansive.When restricting to d = 2 and the context of X = X η , a simple corollary is that η is doubly periodic if and only if every subspace of R 2 is expansive. (As throughout the paper, we mean this with respect to the Z 2 -action on X η by translation.) When there exist n, k ∈ N such that P η (n, k) ≤ nk, the connection between expansive subspaces of R 2 and periodicity of η goes deeper. We show:If there exist n, k ∈ N such that P η (n, k) ≤ nk and there is a unique nonexpansive 1-dimensional subspace for the Z 2 -action (by translation) on X η , then η is periodic but not doubly periodic, the unique nonexpansive line L is a rational line through the origin, and every period vector for η is contained in L.Thus Nivat's Conjecture reduces to:then there is at most one nonexpansive 1-dimensional subspace for the Z 2 -action (by translation) on X η . Licensed to West Virginia Univ. Prepared on Tue Jul 14 03:26:28 EDT 2015 for download from IP 157.182.150.22. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 6490 VAN CYR AND BRYNA KRA nonexpansiveness. While expansive implies one-sided expansive and one-sided nonexpansive implies nonexpansive, the converse statements do not necessarily hold. A similar notion was studied in [2] and [3].1.3. Another reformulation of the conjecture. ...

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confidence: 99%

“…Further partial results connected to Nivat's Conjecture and its generalizations are given in [1,6,9,[18][19][20], and we refer the reader to [5,8,12] for additional discussion.…”

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confidence: 99%

Nonexpansive $\mathbb {Z}^2$-subdynamics and Nivat’s Conjecture

Cyr

Kra

2015

Trans. Amer. Math. Soc.

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“…A discussion of limitations to the periodicity principle ends [14]. In [15] the authors prove (PP) for all (2 × n)-blocks in Z 2 .…”

Section: Introductionmentioning

confidence: 99%

“…In [14] and [15] the authors studied the periodicity principle of Morse and Hedlund for more general configurations. First of all it was observed that for 1-dimensional functions f condition (1) can be replaced by |P f (A)| ≤ |A| (2) for some configuration A and again periodicity follows [14,Theorem 1].…”

Section: Introductionmentioning

confidence: 99%

Low complexity functions and convex sets in $\mathbb{Z}^k$

Sander¹,

Tijdeman²

2000

Math Z

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“…Periodicity has been established if the number of n×m patterns is at most αnm for progressively larger and larger constants α < 1: First for α = 1/144 in [5], then for α = 1/16 in [11], and finally for the best known constant α = 1/2 in [3]. It is also known that having at most 2n patterns of size 2 × n and having at most 3n patterns of size 3 × n imply periodicity [12,4]. In [7] we introduced an algebraic approach that leads to the following asymptotic result: A non-periodic coloring of the grid can only have finitely many pairs (n, m) such that the number of distinct n × m patterns is at most nm.…”

Section: Introductionmentioning

confidence: 99%

Nivat's conjecture and pattern complexity in algebraic subshifts

Kari

Moutot

2019

Theoretical Computer Science

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We study Nivat's conjecture on algebraic subshifts and prove that in some of them every low complexity configuration is periodic. This is the case in the Ledrappier subshift (the 3-dot system) and, more generally, in all twodimensional algebraic subshifts over Fp defined by a polynomial without line polynomial factors in more than one direction. We also find an algebraic subshift that is defined by a product of two line polynomials that has this property (the 4-dot system) and another one that does not.

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The rectangle complexity of functions on two-dimensional lattices

Cited by 47 publications

References 4 publications

Nonexpansive $\mathbb {Z}^2$-subdynamics and Nivat’s Conjecture

Nonexpansive $\mathbb {Z}^2$-subdynamics and Nivat’s Conjecture

Low complexity functions and convex sets in $\mathbb{Z}^k$

Nivat's conjecture and pattern complexity in algebraic subshifts

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