Nonexpansive ℤ2-subdynamics and Nivat’s conjecture

Van Cyr, Bryna Kra

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


For a finite alphabet A and η : ℤ → A, the Morse-Hedlund Theorem states that η is periodic if and only if there exists n ∈ ℕ such that the block complexity function Pη(n) satisfies Pη(n) ≤ n, and this statement is naturally studied by analyzing the dynamics of a ℤ-action associated with η. In dimension two, we analyze the subdynamics of a ℤ2-action associated with η : ℤ2 → A and show that if there exist n, k ∈ ℕ such that the n × k rectangular complexity Pη(n, k) satisfies Pη(n, k) ≤ nk, then the periodicity of η is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist n, k ∈ ℕ such that (formula presented) then η is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.

Original languageEnglish (US)
Pages (from-to)6487-6537
Number of pages51
JournalTransactions of the American Mathematical Society
Issue number9
StatePublished - 2015


  • Block complexity
  • Nivat’s Conjecture
  • Nonexpansive subdynamics
  • Periodicity
  • ℤ-subshift

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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