Fractal tiles associated to generalised radix representations and shift radix systems

Paul Surer

University Leoben, Austria

For $\bf r\in \mathbb{R}^{d}$ define the mapping

$$\tau_{\bf r}:\mathbb{Z}^{d} \rightarrow \mathbb{Z}^{d},{\bf x}=(x_{1},\dots,x_{d}) \mapsto (x_{2},\dots,x_{d},-\lfloor {\bf r\cdot x}\rfloor).$$

$\tau_{\bf r}$ is called a shift radix system (SRS) if $\forall {\bf x} \in \mathbb{Z}^{d} \exists n \in \mathbb{N}:\tau_{\bf r}^{n}({\bf x})={\bf0}$. Shift radix systems are a class of dynamical systems, introduced in 2005 by Akiyama et al., and are strongly related to other well known notions of number systems as $\beta$-expansion or canonical number systems. Let

$$\begin{split} \mathcal{D}_{d}:= & \left\{\bf r \in \mathbb{R}... ...t \tau_{\bf r} \mbox{ is an SRS} \right.\right\}. \end{split}$$

Further denote by $R(\mathbf{r})$ the companion matrix with the characteristic polynomial $x^d+r_{d-1}x^{d-1}+\cdots+r_0$. For $\mathbf{r} \in {\rm int }{{\mathcal{D}}_d}$, $\mathbf{x} \in \mathbb{Z}^d$ define

$$T_{\mathbf{r},n}(\mathbf{x})=\left\{\mathbf{x} \in {\mathbb{Z}}^d \left\vert \tau_{\bf r}^n\mathbf{z}=\mathbf{x}\right. \right\}$$

and

$$T_{\mathbf{r}}(\mathbf{x})=\lim_{n \rightarrow \infty} R(\mathbf{r})^n T_{\mathbf{r},n}(\mathbf{x})$$

(using the limit with respect to a Hausdorff metric). Then $T_{\mathbf{r}}(\mathbf{x})$ is called an SRS-tile. We will give basic properties of such tiles and see, how they are related to tiles induced by Pisot numbers and canonical number systems.

Supported by FWF Project Nr. P9610-N13.