{\lambda}_1^{k_1-2}} \sum_{i=0}^\infty a_i {\lambda}_1^i \\
        \vdots \\
        \frac{d}{d {\lambda}_1} \sum_{i=0}^\infty a_i {\lambda}_1^i \\
        \sum_{i=0}^\infty a_i {\lambda}_1^i \\
        \vdots \\
        \frac{1}{(k_r-1)!}\frac{d^{k_r-1}}{d {\lambda}_r^{k_r-1}} \sum_{i=0}^\infty a_i {\lambda}_r^i \\
        \frac{1}{(k_r-2)!}\frac{d^{k_r-2}}{d {\lambda}_r^{k_r-2}} \sum_{i=0}^\infty a_i {\lambda}_r^i \\
        \vdots \\
        \frac{d}{d {\lambda}_r} \sum_{i=0}^\infty a_i {\lambda}_r^i \\
        \sum_{i=0}^\infty a_i {\lambda}_r^i
        \end{pmatrix}
    & =
        \begin{pmatrix}
        \sum_{t=0}^{n-1} u_{t-n} B_t^{(k_1-1)}({\lambda}_1) \\
        \sum_{t=0}^{n-1} u_{t-n} B_t^{(k_1-2)}({\lambda}_1) \\
        \vdots \\
        \sum_{t=0}^{n-1} u_{t-n} B_t^{(1)}({\lambda}_1) \\
        \sum_{t=0}^{n-1} u_{t-n} B_t({\lambda}_1) \\
        \vdots \\
        \sum_{t=0}^{n-1} u_{t-n} B_t^{(k_1-1)}({\lambda}_r) \\
        \sum_{t=0}^{n-1} u_{t-n} B_t^{(k_1-2)}({\lambda}_r) \\
        \vdots \\
        \sum_{t=0}^{n-1} u_{t-n} B_t^{(1)}({\lambda}_r) \\
        \sum_{t=0}^{n-1} u_{t-n} B_t({\lambda}_r)
        \end{pmatrix} \\
    & = B \begin{pmatrix} u_{-n} \\ u_{-n+1} \\ \vdots \\ u_{-1} \end{pmatrix}
     = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_N \end{pmatrix},\end{aligned}$$ which proves the desired result.

\[rmk:nonempty\] It is worth observing that if $M$ is an $N \times N$ matrix with distinct eigenvalues sufficiently close to (but less than) $1$ in absolute value, then the $N$-dimensional attractor $A$ will have non-empty interior. Here “sufficiently close” depends only on $N$. This follows from essentially the same proof as in [@HS Theorem 3.4] using the polynomial $$P(x) = x^{m n + 2} - x^{nm} + b_{m-1} x^{(m-1) n} + b_{m-2} x^{(m-2) n} +  \dots + b_{0}$$ and $n$ even. We can choose the $b_i$ of this 