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

Sander, J. W.; Tijdeman, R.

doi:10.1007/pl00004798

Cited by 4 publications

(2 citation statements)

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“…Associated to any such symbol sequence is a dynamical system with a Z-action (the closure of its orbit under the shift) and this result can be interpreted as a condition for this dynamical system to be finite. In the context of higher dimensional symbolic dynamical systems, with symbols on the lattice Z n and a corresponding Z n -action, the question has also been raised to what extent growth restrictions on the number of local symbol patterns (in rectangular patches) enforce periodicity, see for example Sander and Tijdeman [27,28,29] and Berthé and Vuillon [2,3] for a discussion of the two dimensional case. Symbolic dynamics analogues of the patch-counting function are called "permutation numbers" or "complexity functions" in this context, cf.…”

Section: Introductionmentioning

confidence: 99%

Local Complexity of Delone Sets and Crystallinity

Lagarias¹,

Pleasants²

2002

Can. math. bull.

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Abstract. This paper characterizes when a Delone set X in R n is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set X, let N X (T) count the number of translation-inequivalent patches of radius T in X and let M X (T) be the minimum radius such that every closed ball of radius M X (T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a "gap in the spectrum" of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to X being an ideal crystal.Explicitly, for N X (T), if R is the covering radius of X then either N X (T) is bounded or N X (T) ≥

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Section: Introductionmentioning

confidence: 99%

Local Complexity of Delone Sets and Crystallinity

Lagarias¹,

Pleasants²

2002

Can. math. bull.

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“…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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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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