Abelian returns in Sturmian words

TL;DR

This paper introduces an abelian perspective on return words in Sturmian words, providing a new characterization based on the number of abelian returns per factor and exploring their structural properties.

Contribution

It offers a novel characterization of Sturmian words through abelian returns, linking combinatorial properties with abelian structures.

Findings

Sturmian words have factors with two or three abelian returns.

A new characterization of Sturmian words based on abelian returns.

Connections between abelian returns and periodicity are discussed.

Abstract

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.