TY - JOUR

T1 - On abelian closures of infinite non-binary words

AU - Karhumäki, Juhani

AU - Puzynina, Svetlana

AU - Whiteland, Markus A.

PY - 2024/9/1

Y1 - 2024/9/1

N2 - 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). In this paper, we investigate how this property extends to non-binary words. We consider the abelian closures of most natural generalizations of Sturmian words to non-binary alphabets, such as balanced words and minimal complexity words. We characterize the abelian closures of words in these families and show that in both families, there exist both words which satisfy the property A(x)=Ω(x) and which do not. We observe that for Arnoux-Rauzy words, we always have a strict inclusion Ω(x)⊂A(x). We also consider abelian closures of general subshifts and make some initial observations of their abelian closures and pose some related open questions.

AB - 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). In this paper, we investigate how this property extends to non-binary words. We consider the abelian closures of most natural generalizations of Sturmian words to non-binary alphabets, such as balanced words and minimal complexity words. We characterize the abelian closures of words in these families and show that in both families, there exist both words which satisfy the property A(x)=Ω(x) and which do not. We observe that for Arnoux-Rauzy words, we always have a strict inclusion Ω(x)⊂A(x). We also consider abelian closures of general subshifts and make some initial observations of their abelian closures and pose some related open questions.

KW - Abelian equivalence

KW - Minimal subshifts

KW - Sturmian words

KW - Symbolic dynamical systems

UR - https://www.mendeley.com/catalogue/aa16b5e7-3bc7-3663-8d90-4f1210aa3e72/

U2 - 10.1016/j.disc.2024.114069

DO - 10.1016/j.disc.2024.114069

M3 - Article

VL - 347

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 9

M1 - 114069

ER -