On Abelian Closures of Infinite Non-binary Words

TL;DR

This paper explores the properties of abelian closures of infinite words beyond binary cases, focusing on non-binary words like balanced and minimal complexity words, and investigates their structure and open questions.

Contribution

It extends the study of abelian closures to non-binary words, analyzing their structure and posing new open problems in the field.

Findings

Abelian closures of non-binary words exhibit diverse structures.

Sturmian words are characterized by their abelian closure properties.

Open questions remain about the abelian closures of general 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 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.