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Sequences, Groups, and Number Theory pp 177–212Cite as

Birkhäuser

Avoiding or Limiting Regularities in Words

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Abstract

It is commonly admitted that the origin of combinatorics on words goes back to the work of Axel Thue in the beginning of the twentieth century, with his results on repetition-free words. Thue showed that one can avoid cubes on infinite binary words and squares on ternary words. Up to now, a large part of the work on the theoretic part of combinatorics on words can be viewed as extensions or variations of Thue’s work, that is, showing the existence (or nonexistence) of infinite words avoiding, or limiting, a repetition-like pattern. The goal of this chapter is to present the state of the art in the domain and also to present general techniques used to prove a positive or a negative result. Given a repetition pattern P and an alphabet, we want to know if an infinite word without P exists. If it exists, we are also interested in the size of the language of words avoiding P, that is, the growth rate of the language. Otherwise, we are interested in the minimum number of factors P that a word must contain. We talk about limitation of usual, fractional, abelian, and k-abelian repetitions and other generalizations such as patterns and formulas. The last sections are dedicated to the presentation of general techniques to prove the existence or the nonexistence of an infinite word with a given property.

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Authors and Affiliations

  1. CNRS & LIRMM, Montpellier, F-34095, Montpellier Cedex 5, France

    Pascal Ochem

  2. LIP, ENS de Lyon, 46 allée d’Italie, F-69007, Lyon, France

    Michaël Rao & Matthieu Rosenfeld

Authors
  1. Pascal Ochem
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  2. Michaël Rao
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  3. Matthieu Rosenfeld
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Correspondence to Pascal Ochem .

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Editors and Affiliations

  1. IRIF, Université Paris Diderot, Paris, France

    Valérie Berthé

  2. Department of Mathematics, University of Liège, Liège, Belgium

    Michel Rigo

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Ochem, P., Rao, M., Rosenfeld, M. (2018). Avoiding or Limiting Regularities in Words. In: Berthé, V., Rigo, M. (eds) Sequences, Groups, and Number Theory. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69152-7_5

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