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Abelian Returns in Sturmian Words

Svetlana Puzynina, Luca Q. Zamboni, Abelian Returns in Sturmian Words. TUCS Technical Reports 1046, TUCS, 2012.

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.

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BibTeX entry:

@TECHREPORT{tPuZa12,
  title = {Abelian Returns in Sturmian Words},
  author = {Puzynina, Svetlana and Zamboni, Luca Q.},
  number = {1046},
  series = {TUCS Technical Reports},
  publisher = {TUCS},
  year = {2012},
  keywords = {Sturmian word, return word, abelian equivalence},
  ISBN = {978-952-12-2740-0},
}

Belongs to TUCS Research Unit(s): FUNDIM, Fundamentals of Computing and Discrete Mathematics

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