Crystallographic number systems

Abstract

We introduce the notion of crystallographic number systems, generalizing matrix number systems. Let Γ be a group of isometries of ${\mathbb{R}^d,g}$ an expanding affine mapping of ${\mathbb{R}^d}$ with ${g\circ\Gamma\circ g^{-1}\subset\Gamma}$ and ${\mathcal{D}\subset\Gamma}$ . We say that ${(\Gamma,g,\mathcal{D})}$ is a Γ-number system if every isometry ${\gamma\in \Gamma}$ has a unique expansion

$$\gamma=g^n\delta_n g^{-n}\,g^{n-1}\delta_{n-1} g^{-(n-1)}\dots g\delta_{1} g^{-1}\,\delta_0,$$

for some ${n\in \mathbb{N}}$ and ${\delta_0,\ldots,\delta_n\in \mathcal{D}}$ . A tile can be attached to a Γ-number system. We show fundamental topological properties of this tile: they admit the fixed point of g as interior point and tesselate the space by the whole group Γ. Moreover, we give several examples, among them a class of p2-number systems, where p2 is the crystallographic group generated by the π-rotation and two independent translations.