Periodicity and local complexity of Delone sets

Abstract

We study complexity and periodicity of Delone sets by applying an algebraic approach to multidimensional symbolic dynamics. In this algebraic approach, $\mathbb{Z}^d$-configurations $c: \mathbb{Z}^d \to \mathcal{A}$ for a finite set $\mathcal{A} \subseteq \mathbb{C}$ and finite $\mathbb{Z}^d$-patterns are regarded as formal power series and Laurent polynomials, respectively. In this paper we study also functions $c: \mathbb{R}^d \to \mathcal{A}$ where $\mathcal{A}$ is as above. These functions are called $\mathbb{R}^d$-configurations. Any Delone set may be regarded as an $\mathbb{R}^d$-configuration by simply presenting it as its indicator function. Conversely, any $\mathbb{R}^d$-configuration whose support (that is, the set of cells for which the configuration gets non-zero values) is a Delone set can be seen as a colored Delone set. We generalize the concept of annihilators and periodizers of $\mathbb{Z}^d$-configurations for $\mathbb{R}^d$-configurations. We show that if an $\mathbb{R}^d$-configuration has a non-trivial annihilator, that is, if a linear combination of some finitely many of its translations is the zero function, then it has an annihilator of a particular form. Moreover, we show that $\mathbb{R}^d$-configurations with integer coefficients that have non-trivial annihilators are sums of finitely many periodic functions $c_1,\ldots,c_m: \mathbb{R}^d \to \mathbb{Z}$. Also, $\mathbb{R}^d$-pattern complexity is studied alongside with the classical patch-complexity of Delone sets. We point out the fact that sufficiently low $\mathbb{R}^d$-pattern complexity of an $\mathbb{R}^d$-configuration implies the existence of non-trivial annihilators. Moreover, it is shown that if a Meyer set has sufficiently slow patch-complexity growth, then it has a non-trivial annihilator. Finally, a condition for forced periodicity of colored Delone sets of finite local complexity is provided.