&       & 0 \\
                             & {\lambda}& \ddots &       & \\
                             &     & \ddots & \ddots&   \\
                             &     &        & {\lambda}& 1  \\
                          0  &     &        &       & {\lambda}\end{pmatrix}$$ and write $M$ as $$M = \begin{pmatrix}
    J_{{\lambda}_1, k_1} &                &        &  0 \\
                   & J_{{\lambda}_2, k_2} &        & \\
                   &                & \ddots &  \\
        0          &                &        & J_{{\lambda}_r, k_r} \end{pmatrix},
\label{eq:M}$$ where all $\lambda_i$ are distinct and $0 < |\lambda_i| < 1$ for all $i$. Then $M$ will have dimensions $N \times N$ where $N = k_1 + k_2 + \dots + k_r$.

We consider the two affine maps $$T_m(v) = M v -
 \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ \vdots \\ \vdots  \\ 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}
 \ \ \mathrm{and} \ \
T_p(v) = M v +
 \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ \vdots \\ \vdots  \\ 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}.$$ Here there are $k_1-1$ copies of $0$s follows by one $1$, then $k_2-1$ copies of $0$s follows by one $1$, and so on.

Consider the case with $M$ as a single $k \times k$ Jordan block $J_{{\lambda},k}$.

\[lem:jordan-exp\] We have $$\pi(a_0 a_1 a_2 \dots) 