Aperiodic two-dimensional words of small abelian complexity

Research output


In this paper we prove an abelian analog of the famous Nivat's conjecture linking complexity and periodicity for two-dimensional words: We show that if a two-dimensional recurrent word contains at most two abelian factors for each pair (n;m) of integers, then it has a periodicity vector. Moreover, we show that a two-dimensional aperiodic recurrent word must have more than two abelian factors infinitely often. On the other hand, there exist aperiodic recurrent words with abelian complexity bounded by 3, as well as aperiodic words having abelian complexity 1 for some pairs (m;n).

Original languageEnglish
Article numberP4.15
JournalElectronic Journal of Combinatorics
Issue number4
Publication statusPublished - 11 Oct 2019


Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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