On Abelian Subshifts

Karhumäki, Juhani; Puzynina, Svetlana; Whiteland, Markus A.

doi:10.1007/978-3-319-98654-8_37

Cited by 3 publications

(12 citation statements)

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“…In this paper, we study how the characterization of Sturmian words as aperiodic uniformly recurrent words with A(x) = Ω(x) extends to non-binary alphabets. We then study the abelian closures of certain generalizations of Sturmian words, and some preliminary results have been reported at DLT 2018 conference [15]. Besides that, we discuss abelian closures of subshifts in general.…”

Section: Introductionmentioning

confidence: 99%

On Abelian Closures of Infinite Non-binary Words

Karhumäki,

Puzynina,

Whiteland

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 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 uniformly recurrent binary words, periodic and aperiodic Sturmian words are exactly those words for which A(x) equals the shift orbit closure Ω(x). Furthermore, for an aperiodic binary word that is not Sturmian, its abelian closure contains infinitely many minimial subshifts. In this paper we consider the abelian closures of well-known families of non-binary words, such as balanced words and minimal complexity words. We also consider abelian closures of general subshifts and make some initial observations of their abelian closures and pose some related open questions.

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Section: Introductionmentioning

confidence: 99%

On Abelian Closures of Infinite Non-binary Words

Karhumäki,

Puzynina,

Whiteland

2020

Preprint

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“…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%

“…For some words and families of words, for example, Sturmian words, the equality holds: Ω(x) = A(x). Moreover, the property Ω(x) = A(x) characterizes Sturmian words among uniformly recurrent binary words [13]. On the other hand, it is easy to see that the abelian closure of the Thue-Morse word TM, defined as the fixed point (starting with 0) of the morphism 0 → 01, 1 → 10, is {ε, 0, 1} • {01, 10} N (see, e.g., [13] for a proof.)…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the property Ω(x) = A(x) characterizes Sturmian words among uniformly recurrent binary words [13]. On the other hand, it is easy to see that the abelian closure of the Thue-Morse word TM, defined as the fixed point (starting with 0) of the morphism 0 → 01, 1 → 10, is {ε, 0, 1} • {01, 10} N (see, e.g., [13] for a proof.) So, contrary to Sturmian words, the abelian closure of the Thue-Morse is huge compared to Ω TM : essentially, it is a morphic image of the full binary shift.…”

Section: Introductionmentioning

confidence: 99%

“…In many cases we are able to prove that the abelian closure actually contains uncountably many minimal subshifts. We remark that in the non-binary case, there exist words with finitely many (and more than one) minimal subshifts; for example, some balanced aperiodic words are like that (announced in [13], see also Example 2.5).…”

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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Abelian Properties of Words

Puzynina

2019

Lecture Notes in Computer Science

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On Abelian Subshifts

Cited by 3 publications

References 9 publications

On Abelian Closures of Infinite Non-binary Words

On Abelian Closures of Infinite Non-binary Words

Abelian Closures of Infinite Binary Words

Abelian Properties of Words

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