Abelian returns in Sturmian words
Combinatorics
2012-04-27 v1
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.
Cite
@article{arxiv.1204.5755,
title = {Abelian returns in Sturmian words},
author = {Svetlana Puzynina and Luca Q. Zamboni},
journal= {arXiv preprint arXiv:1204.5755},
year = {2012}
}