A characterization of substitutive sequences using return words
Discrete Mathematics
A chararcterization of sturmian words by return words
European Journal of Combinatorics
Words avoiding Abelian inclusions
Journal of Automata, Languages and Combinatorics
Generalized balances in Sturmian words
Discrete Applied Mathematics
Abelian Squares are Avoidable on 4 Letters
ICALP '92 Proceedings of the 19th International Colloquium on Automata, Languages and Programming
Burrows--Wheeler transform and Sturmian words
Information Processing Letters
Some properties of the factors of Sturmian sequences
Theoretical Computer Science
Characterisations of balanced words via orderings
Theoretical Computer Science
On a generalization of Abelian equivalence and complexity of infinite words
Journal of Combinatorial Theory Series A
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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.