Abstract
Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The Abelian subshift \(\mathcal A_{\varvec{x}}\) of an infinite word \(\varvec{x}\) is the set of infinite words \(\varvec{y}\) such that, for each factor u of \(\varvec{y}\), there exists a factor v of \(\varvec{x}\) which is Abelian equivalent to u. The notion of Abelian subshift gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which \(\mathcal A_{\varvec{x}}\) equals the shift orbit closure \(\varOmega _{\varvec{x}}\). On the other hand, the Abelian subshift of the Thue-Morse word contains uncountably many minimal subshifts. In this paper we undertake a general study of Abelian subshifts. In particular, we characterize the Abelian subshifts of recurrent aperiodic balanced words and the Abelian subshifts of ternary words having factor complexity \(n+2\) for all \(n\ge 1\).
Published Version
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