On abelian closures of infinite non-binary words

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 S2( 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 ) = S2( x ) and which do not. We observe that for Arnoux-Rauzy words, we always have a strict inclusion S2( x ) subset of 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. (c) 2024 Elsevier B.V. All rights reserved.