On Abelian Subshifts

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 $n\ge 1$.

S. Puzynina—Partially supported by Russian Foundation of Basic Research (grant 18-31-00118).

M. A. Whiteland—Supported by the Yrjö, Ville and Kalle Väisälä Foundation.