  The Morse-Hedlund Theorem states that a bi-infinite sequence $\eta$ in a
finite alphabet is periodic if and only if there exists $n\in\N$ such that the
block complexity function $P_\eta(n)$ satisfies $P_\eta(n)\leq n$. In dimension
two, Nivat conjectured 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
$\eta$ is periodic. Sander and Tijdeman showed that this holds for $k\leq2$. We
generalize their result, showing that Nivat's Conjecture holds for $k\leq3$.
The method involves translating the combinatorial problem to a question about
the nonexpansive subspaces of a certain $\ZZ$ dynamical system, and then
analyzing the resulting system.
