2012/04/25 by Svetlana Puzynina, Puzynina, Svetlana, Luca Q. Zamboni +1
Mathematics · #68R15 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:68R15
paper · pdf · doi:10.48550/arxiv.1204.5755
arxiv created 2012/04/25 · arxiv updated 2012/04/27
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.