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 $\mathcal{A}(\mathbf{x})$ of an infinite word $\mathbf{x}$ is the set of infinite words $\mathbf{y}$ such that, for each factor $u$ of $\mathbf{y}$, there exists a factor $v$ of $\mathbf{x}$ 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 $\mathcal{A}(\mathbf{x})$ equals the shift orbit closure $\Omega(\mathbf{x})$. 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.