Abelian returns in Sturmian words

TL;DR

This paper explores abelian variants of return words in Sturmian words, providing new characterizations and structural insights that connect abelian returns to the fundamental properties of these words.

Contribution

It introduces two abelian return concepts, characterizes Sturmian words via these, and analyzes the structure and periodicity connections of abelian returns.

Findings

A recurrent infinite word is Sturmian iff each factor has 2 or 3 abelian returns.

Characterization of factors with exactly two abelian returns in Sturmian words.

Connections established between abelian returns and periodicity in words.

Abstract

Return words constitute a powerful tool for studying symbolic dynamical systems. They may be regarded as a discrete analogue of the first return map in dynamical systems. In this paper we investigate two abelian variants of the notion of return word, each of them gives rise to a new characterization of Sturmian words. We prove that a recurrent infinite word is Sturmian if and only if each of its factors has two or three abelian (or semi-abelian) returns. We study the structure of abelian returns in Sturmian words and give a characterization of those factors having exactly two abelian returns. Finally we discuss connections between abelian returns and periodicity in words.