Nonexpansive $\mathbb {Z}^2$-subdynamics and Nivat’s Conjecture

Abstract:

For a finite alphabet $\mathcal {A}$ and $\eta \colon \mathbb {Z}\to \mathcal {A}$, the Morse-Hedlund Theorem states that $\eta$ is periodic if and only if there exists $n\in \mathbb {N}$ 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 $\mathbb {Z}$-action associated with $\eta$. In dimension two, we analyze the subdynamics of a $\mathbb {Z}^2$-action associated with $\eta \colon \mathbb {Z}^2\to \mathcal {A}$ and show that if there exist $n,k\in \mathbb {N}$ 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 $n,k\in \mathbb {N}$ 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.