Abelian Closures of Infinite Binary Words

TL;DR

This paper investigates the structure of abelian closures of infinite binary words, revealing that non-Sturmian aperiodic words have infinitely many minimal subshifts within their abelian closure, unlike Sturmian words.

Contribution

It characterizes the abelian closures of binary words, showing a stark difference between Sturmian and non-Sturmian words in their minimal subshift structure.

Findings

Sturmian words have abelian closures equal to their shift orbit closure.

Non-Sturmian aperiodic binary words have infinitely many minimal subshifts in their abelian closure.

The result highlights a structural distinction in the abelian closure of binary words.

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 (the shift orbit closure 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 binary uniformly recurrent words, Sturmian words are exactly those words for which $\mathcal{A}(\mathbf{x})$ equals the shift orbit closure $\Omega(\mathbf{x})$. In this paper we show that, contrary to larger alphabets, the abelian closure of a uniformly recurrent aperiodic binary word which is not Sturmian contains infinitely many minimal subshifts.