Super-stationary set, subword problem and the complexity

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

doi:10.1016/j.disc.2009.02.001

Cited by 7 publications

(6 citation statements)

References 3 publications

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“…Instead of a union of P(ξ), we have a characterization of a super-stationary set as an intersection of P(ξ) satisfying the condition (#). In [9], the relation between the representations of a super-stationary set as a union and as an intersection of the sets P(ξ) is discussed. (There is an error in the proof of Theorem 2 in [9], the corrected version is available at the author's home page: http://www14.plala.or.jp/kamae) Theorem 5.…”

Section: For a Nonempty Closed Set ωmentioning

confidence: 99%

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Uniform sets and super-stationary sets over general alphabets

Kamae¹

2011

Ergod. Th. Dynam. Sys.

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Uniform sets and super-stationary sets over the binary alphabet have been extensively studied. In this paper, they are generalized to general alphabets. We generalize the fact that any uniform set contains a super-stationary set so that any uniform complexity is realized by a super-stationary set. This gives a formula to calculate the uniform complexity functions. We also give characterizations of the class of super-stationary sets in general settings in two somewhat different ways than in the binary case. Super-stationary sets are considered as phenomena which are independent of the time scale, but sensitive only to the direction of time, or dependent just on the order of events in time series. Hence, characterizations of super-stationary sets give insights into what is time, what looks like a history without a description of time duration, or what remains meaningful after we lose the quantitative sense of time.

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Section: For a Nonempty Closed Set ωmentioning

confidence: 99%

“…Theorem 2. (T. Kamae, H. Rao, B. Tan, Y.-M. Xue [9]) The class of superstationary sets over the alphabet {0, 1}other than {0, 1} N coincides with the class of sets ξ∈Ξ P(ξ) with nonempty finite sets Ξ ⊂ {0, 1} + .…”

Section: Introductionmentioning

confidence: 99%

Uniform sets and super-stationary sets over general alphabets

Kamae¹

2011

Ergod. Th. Dynam. Sys.

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“…It gives an extra tool for the study of sequences of low complexity, but, as we show in Section 2 below, it becomes simply (#A) n under the (relatively mild) condition of weak mixing for the associated symbolic dynamical system. To study a wider class of systems than the non-weakly mixing ones, here we present the dual notion of minimal pattern complexity, p * L (n), and for infinite words or symbolic dynamical systems consider together the three functions p * L (n) ≤ p L (n) ≤ p * L (n); the case where these three functions are the same has been extensively studied by Kamae et al [8,12] under the name of uniform (pattern) complexity. During the refereeing process of the present paper, our three complexity function have been further studied by Kamae [9].…”

Section: Preliminariesmentioning

confidence: 99%

Three complexity functions

Ferenczi

Hubert

2011

RAIRO-Theor. Inf. Appl.

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“…The class of super-stationary sets is characterized in [5] in terms of super-subwords. In this case, we say that has a primitive factor [ • ψ], where [ • ψ] is the isomorphic class containing • ψ in the sense that two non-empty closed subsets and of {0, 1} N Partitions by congruent sets and optimal positions 615 are said to be isomorphic if there is an isometric bijection between them.…”

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confidence: 99%

“…The basic terminology and notations follow from [2,5]. For further related results, see [1,3,4,[6][7][8][9].…”

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confidence: 99%

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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Super-stationary set, subword problem and the complexity

Cited by 7 publications

References 3 publications

Uniform sets and super-stationary sets over general alphabets

Uniform sets and super-stationary sets over general alphabets

Three complexity functions

Partitions by congruent sets and optimal positions

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