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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)

Abstract

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.

Keywords

Factor Complexity Symbolic Dynamic Usual Notion Unbounded Function Binary Word 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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