  In this paper we study an abelian version of the notion of return word. Our
main result is a new characterization of Sturmian words via abelian returns.
Namely, we prove that a word is Sturmian if and only if each of its factors has
two or three abelian returns. In addition, we describe the structure of abelian
returns in Sturmian words, and discuss connections between abelian returns and
periodicity.
