Non-Expansive Matrix Based number Systems

Abstract: Let $M = \left(\begin{matrix} 1 & 1 \ 0 & 1 \end{matrix}\right)$ be a $2 \times 2$ Jordan block with eigenvalue $1$, and let $\mathcal{D} = {\left(\begin{smallmatrix}0 \ 1 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 \ -1 \end{smallmatrix} \right)}$. In this paper, we answer a question of Caldwell, Hare, and Vávra about the minimal length representation of $\left( \begin{smallmatrix} a \ b \end{smallmatrix} \right) = \sum_{i=0}^{k-1} M^i d_i$ with $d_i \in \mathcal{D}$. Further, we extend the work of Caldwell, Hare, and Vávra to consider the case of $n \times n$ Jordan blocks with eigenvalue $-1$.