Nonexpansive Z(2)-Subdynamics and Nivat's Conjecture
Transactions of the American Mathematical Society
For a finite A, and eta: Z -> A the Morse-Hedlund Theorem states that eta is periodic if and only if there exists n E N such that the block complexity function P-eta(n) satisfies P-eta(n) <= n, and this statement is naturally studied by analyzing the dynamics of a Z-action associated with ai. In dimension two, we analyze the subdynamics of a Z(2)-action associated with eta: Z(2) -> A and show that if there exist n,k is an element of N such that the ii x k rectangular complexity (n, k) satisfies P,7(n k) < nk, then the periodicity of eta is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist n, k E N such that P-eta(n, k) <= nk/2, then eta is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.
Cyr, Van and Kra, Bryna. "Nonexpansive Z(2)-Subdynamics and Nivat's Conjecture." Transactions of the American Mathematical Society 367, no. 9 (2015) : 6487-6537.