On substitution invariant Sturmian words: an application of Rauzy fractals

Berthé, Valérie; Ei, Hiromi; Ito, Shunji; Rao, Hui

doi:10.1051/ita:2007026

Cited by 28 publications

(28 citation statements)

References 28 publications

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“…As an almost immediate application of Corollary 5.2 we recover the following result originally proved by Yasutomi in [19] and later reproved by Berthé, Ei, Ito and Rao in [4] and independently by Fagnot in [12]. We say an infinite word is pure morphic if it is a fixed point of some morphism different from the identity.…”

Section: Verification Of Casesupporting

confidence: 64%

Infinite self-shuffling words

Charlier

Kamae

Puzynina

et al. 2014

Journal of Combinatorial Theory, Series A

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In this paper we introduce and study a new property of infinite words: An infinite word x ∈ A N , with values in a finite set A, is said to be k-self-shuffling (k ≥ 2) if x admits factorizations:In other words, there exists a shuffle of k-copies of x which produces x. We are particularly interested in the case k = 2, in which case we say x is self-shuffling. This property of infinite words is shown to be independent of the complexity of the word as measured by the number of distinct factors of each length. Examples exist from bounded to full complexity. It is also an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic word contains a non self-shuffling word in its shift orbit closure. While the property of being self-shuffling is a relatively strong condition, many important words arising in the area of symbolic dynamics are verified to be self-shuffling. They include for instance the Thue-Morse word fixed by the morphism 0 → 01, 1 → 10. As another example we show that all Sturmian words of slope α ∈ R \ Q and intercept 0 < ρ < 1 are self-shuffling (while those of intercept ρ = 0 are not). Our characterization of self-shuffling Sturmian words can be interpreted arithmetically in terms of a dynamical embedding and defines an arithmetic process we call the stepping stone model. One important feature of self-shuffling words stems from its morphic invariance: The morphic image of a self-shuffling word is self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications particularly in the area of substitutive dynamical systems. For example, as a consequence of our characterization of selfshuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic.

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Section: Verification Of Casesupporting

confidence: 64%

Infinite self-shuffling words

Charlier

Kamae

Puzynina

et al. 2014

Journal of Combinatorial Theory, Series A

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“…The equivalence of (2) and (4) follows from [45]. The equivalence of (2) and (3) appears in [26], and relies on earlier results, see references in [26,11]. The equivalence of (1) and (3) is due to [32,14].…”

Section: Introductionmentioning

confidence: 65%

Selfdual substitutions in dimension one

Berthé

Frettlöh²,

Sirvent

2012

European Journal of Combinatorics

Self Cite

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There are several notions of the 'dual' of a word/tile substitution. We show that the most common ones are equivalent for substitutions in dimension one, where we restrict ourselves to the case of two letters/tiles. Furthermore, we obtain necessary and sufficient arithmetic conditions for substitutions being selfdual in this case. Since many connections between the different notions of word/tile substitution are discussed, this paper may also serve as a survey paper on this topic.

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“…It should be note that some authors call a morphism what we call a substitution, and by a substitution refer to what we call a non-erasing substitution [4]. When ϕ is non-erasing, it can also be extended to a map ϕ ω : S ω → S ω defined by ϕ ω (s 1 s 2 · · ·) = ϕ(s 1 )ϕ(s 2 ) · · · .…”

Section: Finite Words Infinite Words and Substitutionsmentioning

confidence: 97%

The stable set of a self-map

Godelle¹

2010

Advances in Applied Mathematics

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The attracting set and the inverse limit set are important objects associated with a self-map on a set. By stable set of the self-map we mean the projection of the inverse limit set. It is included in the attracting set, but is not equal in the general case. Here we determine whether the equality holds in several particular cases, among which are the case of a dense range continuous function on a Hilbert space, and the case of a substitution over right-infinite words.

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On substitution invariant Sturmian words: an application of Rauzy fractals

Cited by 28 publications

References 28 publications

Infinite self-shuffling words

Infinite self-shuffling words

Selfdual substitutions in dimension one

The stable set of a self-map

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