Abelian Properties of Words

Puzynina, Svetlana

doi:10.1007/978-3-030-28796-2_2

Cited by 5 publications

(5 citation statements)

References 45 publications

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“…An interesting open question is to characterize words for which A(x) = Ω(x). Among uniformly recurrent binary words, this property gives a characterization of Sturmian words, but the characterization does not extend to usual generalizations of Sturmian words over non-binary alphabets: neither for balanced words, nor for words of minimal complexity, nor for Arnoux-Rauzy words [10,13,24].…”

Section: Additional Remarks On the Structure Of The Abelian Closures ...mentioning

confidence: 99%

“…Two finite words u and v are called abelian equivalent if, for each letter a of the underlying alphabet Σ, the words u and v contain equally many occurrences of a. The notion has been studied in the relation of abelian complexity of infinite words [2,18,27,28], abelian repetitions and avoidance [3,4,14,23,26], other topics [9,21,22,25]; see also [24] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…In this note we consider the so-called abelian closures of infinite binary words. This notion is a fairly recent one, and has thus far been considered only in the works [10,13,24], where the terms "abelianization" and "abelian subshift" were used. The notion is motivated by a notion in discrete symbolic dynamics, namely, the shift orbit closure of a word.…”

Section: Introductionmentioning

confidence: 99%

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Abelian Closures of Infinite Binary Words

Puzynina¹,

Whiteland²

2020

Preprint

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Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The abelian closure A(x) of (the shift orbit closure of) an infinite word x is the set of infinite words y such that, for each factor u of y, there exists a factor v of x which is abelian equivalent to u. The notion of an abelian closure gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which A(x) equals the shift orbit closure Ω(x). In this paper we show that, contrary to larger alphabets, the abelian closure of a uniformly recurrent aperiodic binary word which is not Sturmian contains infinitely many minimal subshifts.

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Section: Additional Remarks On the Structure Of The Abelian Closures ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Abelian Closures of Infinite Binary Words

Puzynina¹,

Whiteland²

2020

Preprint

Self Cite

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“…Two words u, v ∈ A * are abelian equivalent, denoted by u ∼ ab v, if |u| a = |v| a for every a ∈ A where |u| a denotes the number of occurrences of the letter a in u. The notion has been studied in the relation of abelian complexity of infinite words [4,14,15,23], abelian repetitions and avoidance [3,10,18], and other topics [6,17,21]; see also [20] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On the 2-binomial complexity of the generalized Thue-Morse words

Lü¹,

Chen²,

Zhou³

et al. 2021

Preprint

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In this paper, we study the 2-binomial complexity b tm,2 (n) of the generalized Thue-Morse words t m for every integer m ≥ 3. We obtain the exact value of b tm,2 (n) for every integer n ≥ m 2 . As a consequence, b tm,2 (n) is ultimately periodic with period m 2 . This result partially answers a question of M. Lejeune, J. Leroy and M. Rigo [Computing the k-binomial complexity of the Thue-Morse word, J. Comb. Theory Ser. A, 176 (2020) 105284].

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“…Of course, there is research on abelian equivalence beyond avoidance. We refer the reader to the recent survey [29]. We remark that the study of paths of graphs whose vertex colorings do not constitute an abelian 2-power has been initiated in [19,38].…”

Section: Introductionmentioning

confidence: 99%

Avoiding abelian powers cyclically

Peltomäki,

Whiteland

2020

Preprint

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We study a new notion of cyclic avoidance of abelian powers. A finite word w avoids abelian N-powers cyclically if for each abelian N-power of period m occurring in the infinite word w ω , we have m ≥ |w|. Let A(k) be the least integer N such that for all n there exists a word of length n over a k-letter alphabet that avoids abelian N-powers cyclically. Let A ∞ (k) be the least integer N such that there exist arbitrarily long words over a k-letter alphabet that avoid abelian N-powers cyclically.We prove that 5 ≤ A(2) ≤ 8, 3 ≤ A(3) ≤ 4, 2 ≤ A(4) ≤ 3, and A(k) = 2 for k ≥ 5. Moreover, we show that A ∞ (2) = 4, A ∞ (3) = 3, and A ∞ (4) = 2.

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Abelian Properties of Words

Cited by 5 publications

References 45 publications

Abelian Closures of Infinite Binary Words

Abelian Closures of Infinite Binary Words

On the 2-binomial complexity of the generalized Thue-Morse words

Avoiding abelian powers cyclically

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