Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence

  • Juhani Karhumäki
  • Aleksi Saarela
  • Luca Q. Zamboni
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8633)


In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k ∈ ℕ1 ∪ { + ∞ } where ℕ1 denotes the set of positive integers. Two finite words u and v in A * are said to be k-abelian equivalent if for all x ∈ A * 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 ~ k on A *, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = + ∞). Given an infinite word w ∈ A ω , we consider the associated complexity function \(\mathcal P^{(k)}_w : \mathbb N_1 \rightarrow \mathbb N_1\) 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.


Factor Complexity Symbolic Dynamic Usual Notion Unbounded Function Binary Word 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Juhani Karhumäki
    • 1
  • Aleksi Saarela
    • 1
  • Luca Q. Zamboni
    • 1
    • 2
  1. 1.Department of Mathematics and Statistics & FUNDIMUniversity of TurkuTurkuFinland
  2. 2.Université Lyon 1, CNRS UMR 5208, Institut Camille JordanUniversité de LyonVilleurbanne CedexFrance

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