Nonexpansive Z^2 subdynamics and Nivat's conjecture

Abstract

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

Cited by in corpus (2)

• Bounded complexity, mean equicontinuity and discrete spectrum
• An Alphabetical Approach to Nivat's Conjecture