Nonexpansive Z² subdynamics and Nivat's conjecture

For a finite alphabet $\A$ and $\eta\colon \Z\to\A$, the Morse-Hedlund Theorem states that $\eta$ is periodic if and only if there exists $n\in\N$ such that the block complexity function $P_\eta(n)$ satisfies $P_\eta(n)\leq n$, and this statement is naturally studied by analyzing the dynamics of a $\Z$-action associated to $\eta$. In dimension two, we analyze the subdynamics of a $\ZZ$-action associated to $\eta\colon\ZZ\to\A$ and show that if there exist $n,k\in\N$ such that the $n\times k$ rectangular complexity $P_{\eta}(n,k)$ satisfies $P_{\eta}(n,k)\leq nk$, 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 $n,k\in\N$ such that $P_{\eta}(n,k)\leq \frac{nk}{2}$, then $\eta$ is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.