Abelian Closures of Infinite 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 A(x) of (the shift orbit closure of) an infinite word x is the set of infinite words y such that, for each factor u of y, there exists a factor v of 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 A(x) equals the shift orbit closure Ω(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.