Derived Sequences of Arnoux–Rauzy Sequences

Medková, Kateřina

doi:10.1007/978-3-030-28796-2_20

Cited by 5 publications

(6 citation statements)

References 14 publications

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“…Clearly, the derived word to the empty word ε in u is the word u itself. The simple form of the morphisms ϕ i defined by (2) gives immediately the following claim, which can be also deduced from the results in [8] or [9]. Claim 10.…”

Section: Proposition 8 ([6]mentioning

confidence: 72%

On non-repetitive complexity of Arnoux-Rauzy words

Medková¹,

Pelantová²,

Vandomme³

2020

Preprint

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The non-repetitive complexity nrC u and the initial non-repetitive complexity inrC u are functions which reflect the structure of the infinite word u with respect to the repetitions of factors of a given length. We determine nrC u for the Arnoux-Rauzy words and inrC u for the standard Arnoux-Rauzy words. Our main tools are S-adic representation of Arnoux-Rauzy words and description of return words to their factors. The formulas we obtain are then used to evaluate nrC u and inrC u for the d-bonacci word.

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Section: Proposition 8 ([6]mentioning

confidence: 72%

On non-repetitive complexity of Arnoux-Rauzy words

Medková¹,

Pelantová²,

Vandomme³

2020

Preprint

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“…In this section, we will derive a matrix formula for the critical exponent and the asymptotic critical exponent of AR sequences. For regular AR sequences, a formula based on the S-adic representation is given in [16]. Previously, Justin and Pirillo [22] provided a formula for the asymptotic critical exponent of AR sequences fixed by a morphism.…”

Section: Critical Exponent Of Ar Sequencesmentioning

confidence: 99%

“…For d = 2, this class coincides with the class of binary balanced sequences, hence the repetition threshold is well known. The paper [16] is devoted to a particular subclass of episturmian sequences -regular Arnoux-Rauzy sequences. It is proved there that the dbonacci sequence u d has the minimal critical and asymptotic critical exponent among all regular d-ary Arnoux-Rauzy sequences.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic repetitive threshold of balanced sequences

Dvorakova¹,

Opočenská

Pelantová

2023

Math. Comp.

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The critical exponent E ( u ) E(\mathbf u) of an infinite sequence u \mathbf u over a finite alphabet expresses the maximal repetition of a factor in u \mathbf u . By the famous Dejean’s theorem, E ( u ) ≥ 1 + 1 d − 1 E(\mathbf u) \geq 1+\frac 1{d-1} for every d d -ary sequence u \mathbf u . We define the asymptotic critical exponent E ∗ ( u ) E^*(\mathbf u) as the upper limit of the maximal repetition of factors of length n n . We show that for any d > 1 d>1 there exists a d d -ary sequence u \mathbf u having E ∗ ( u ) E^*(\mathbf u) arbitrarily close to 1 1 . Then we focus on the class of d d -ary balanced sequences. In this class, the values E ∗ ( u ) E^*(\mathbf u) are bounded from below by a threshold strictly bigger than 1. We provide a method which enables us to find a d d -ary balanced sequence with the least asymptotic critical exponent for 2 ≤ d ≤ 10 2\leq d\leq 10 .

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“…To abbreviate the notation of elements of the monoid M A , we put K. Medková in [8] studies derived words of Arnoux-Rauzy words. She considers all Arnoux-Rauzy (not only standard) words, but she describes derived words only to prefixes of infinite words.…”

Section: Example 1: Standard Episturmian Morphismsmentioning

confidence: 99%

“…Proposition 9 (Theorem 24 in [8]). Let L z ∈ M A , z ∈ A * , be a primitive morphism and u be its fixed point.…”

Section: Example 1: Standard Episturmian Morphismsmentioning

confidence: 99%

On Substitutions Closed Under Derivation: Examples

Košík¹,

Starosta²

2019

Lecture Notes in Computer Science

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We study infinite words fixed by a morphism and their derived words. A derived word is a coding of return words to a factor. We exhibit two examples of sets of morphisms which are closed under derivation -any derived word with respect to any factor of the fixed point is again fixed by a morphism from this set. The first example involves standard episturmian morphisms, and the second concerns the period doubling morphism.

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Derived Sequences of Arnoux–Rauzy Sequences

Cited by 5 publications

References 14 publications

On non-repetitive complexity of Arnoux-Rauzy words

On non-repetitive complexity of Arnoux-Rauzy words

Asymptotic repetitive threshold of balanced sequences

On Substitutions Closed Under Derivation: Examples

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