Language structure of pattern Sturmian words

Kamae, Teturo; Rao, Hui; Tan, Bo; Xue, Yumei

doi:10.1016/j.disc.2006.03.043

Cited by 11 publications

(7 citation statements)

References 16 publications

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“…Our theorem was essentially proved in [4] for the case n = 1. We sometimes identify x ∈ R n with the vector O x, where O is the origin.…”

Section: Partitions By Congruent Sets and Optimal Positions 625mentioning

confidence: 98%

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Partitions by congruent sets and optimal positions

Xue

Kamae²

2010

Ergod. Th. Dynam. Sys.

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Let X be a metrizable space with a continuous group or semi-group action G. Let D be a non-empty subset of X . Our problem is how to choose a fixed number of sets in {g −1 D; g ∈ G}, say σ −1 D with σ ∈ τ , to maximize the cardinality of the partitionAn infinite subset of G is called an optimal position of the triple (X, G, D) if #P({σ −1 D; σ ∈ τ }) = p * X,G,D (k) holds for any k = 1, 2, . . . and τ ⊂ with #τ = k. In this paper, we discuss examples of the triple (X, G, D) admitting or not admitting an optimal position. Let X = G = R n (n ≥ 1), where the action g ∈ G to x ∈ X is the translation x − g. If D is the ndimensional unit ball, then p * X,G,D (k) = 2 n i=0 k − 1 i holds and the triple (X, G, D) admits an optimal position. In fact, if n ≥ 2 and is an infinite subset of G such that for some δ with 0 < δ < 1, ⊂ {x ∈ R n ; x = δ}, and that any subset of with cardinality n + 1 is not on a hyperplane, then is an optimal position of the triple (X, G, D). We have determined the primitive factor of the uniform sets coming from these optimal positions. We also show that in the above setting with n = 2 and the unit square D in place of the unit disk D, the maximal pattern complexity is unchanged and p * X,G,D (k) = k 2 − k + 2, but there is no optimal position.

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“…Our theorem was essentially proved in [4] for the case n = 1. We sometimes identify x ∈ R n with the vector O x, where O is the origin.…”

Section: Partitions By Congruent Sets and Optimal Positions 625mentioning

confidence: 98%

“…For q ∈ R n , let ω q be as in (4). Then ω q • ψ(i) = ω q (r 0 + w i ) = 1 if and only if r 0 + w i ∈ B n (q, 1).…”

Section: Partitions By Congruent Sets and Optimal Positions 625mentioning

confidence: 99%

Partitions by congruent sets and optimal positions

Xue

Kamae²

2010

Ergod. Th. Dynam. Sys.

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“…The pattern complexity of one-dimensional Toeplitz words have been studied in [10,5,7]; it is shown [10] that the one-dimensional simple Toeplitz words are pattern Sturmian words. In this section we generalize the construction of Toeplitz words to higher dimensions and obtain a class of words which is considerably large.…”

Section: Higher Dimensional Simple Toeplitz Wordsmentioning

confidence: 99%

Maximal pattern complexity of higher dimensional words

Rao

Zhou

et al. 2010

Journal of Combinatorial Theory, Series A

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This paper studies the pattern complexity of n-dimensional words. We show that an n-recurrent but not n-periodic word has pattern complexity at least 2k, which generalizes the result of [T. Kamae, H. Rao, Y.-M. Xue, Maximal pattern complexity of two dimension words, Theoret. Comput. Sci. 359 (1-3) (2006) 15-27] on two-dimensional words. Analytic directions of a word are defined and its topological properties play a crucial role in the proof. Accordingly n-dimensional pattern Sturmian words are defined. Irrational rotation words are proved to be pattern Sturmian. A new class of higher dimensional words, the simple Toeplitz words, are introduced. We show that they are also pattern Sturmian words.

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“…and ξ is a simple Toeplitz sequence. It would certainly be of interest to find additional examples or to establish limitations on existing ones beyond the known ones (compare, e.g., [20]).…”

Section: Introductionmentioning

confidence: 99%

Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals

Damanik

Embree

Gorodetski

2015

Progress in Mathematics

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We consider discrete Schrödinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schrödinger operators are well understood. In particular, it is known that for every Sturmian potential, the associated Schrödinger operator has zero-measure spectrum and purely singular continuous spectral measures. We conjecture that the same statements hold in the more general class of pattern Sturmian potentials. We prove partial results in support of this conjecture. In particular, we confirm the conjecture for all pattern Sturmian potentials that belong to the family of Toeplitz sequences. Date

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Language structure of pattern Sturmian words

Cited by 11 publications

References 16 publications

Partitions by congruent sets and optimal positions

Partitions by congruent sets and optimal positions

Maximal pattern complexity of higher dimensional words

Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals

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