Nonexpansive Z^2 subdynamics and Nivat's conjecture

For a finite alphabet $\A$ and $η\colon \Z\to\A$, 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 to $η$. In dimension two, we analyze the subdynamics of a $\ZZ$-action associated to $η\colon\ZZ\to\A$ 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 $P_η(n,k)≤ \fracnk2$, then $η$ is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.