Variations of the Morse-Hedlund theorem for k-abelian equivalence

  • Juhani Karhumäki
  • Aleksi Saarela
  • Luca Q. Zamboni


In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k ≥ 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = ∞). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper.


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How to Cite
Karhumäki, J., Saarela, A., & Zamboni, L. Q. (2017). Variations of the Morse-Hedlund theorem for k-abelian equivalence. Acta Cybernetica, 23(1), 175-189.
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