Nonexpansive ℤ²-subdynamics and Nivat’s conjecture

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 P_η(n) satisfies P_η(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 P_η(n, k) satisfies P_η(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.