Periodicity and local complexity of Delone sets
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908343105224704 |
|---|---|
| author | Herva, Pyry Kari, Jarkko |
| author_facet | Herva, Pyry Kari, Jarkko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20709 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Periodicity and local complexity of Delone sets Herva, Pyry Kari, Jarkko Dynamical Systems Discrete Mathematics Combinatorics 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. |
| title | Periodicity and local complexity of Delone sets |
| topic | Dynamical Systems Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2504.20709 |