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For the real case we will consider two subcases. Let $0 < {\lambda}\leq \mu < 1$ and consider $$M = \begin{pmatrix}         {\lambda}& 0  \\ 0 & \mu \end{pmatrix}.$$ This we will call the [*positive real case*]{}. This was the case considered in [@HS]. The second subcase is $$M = \begin{pmatrix}-{\lambda}& 0  \\ 0 & \mu \end{pmatrix},$$ which we will call the [*mixed real case*]{}. In both cases we take $u = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$, which is clearly cyclic.

In the real positive case, $\pi: \{m,p\}^{\mathbb{N}}\to A_{{\lambda}, \mu}$, we have $\pi(a_0 a_1 a_2 \dots) = \left(\sum_{i=0}^\infty a_i {\lambda}^i, \sum_{i=0}^\infty a_i \mu^i\right) \in {\mathbb{R}}^2$, whereas in the real mixed case, $\pi: \{0,1\}^{\mathbb{N}}\to A_{-{\lambda}, \mu}$, we have $$\pi(a_0 a_1 a_2 \dots) = \left( \sum_{i=0}^\infty a_i (-{\lambda})^i, \sum_{i=0}^\infty a_i \mu^i\right).$$

It is easy to see that all other real cases can be reduced to one of these two. For example, there is a symmetry from $(-{\lambda}, \mu)$ to $({\lambda}, -\mu)$. To see this, write $(x,y) = \left(\sum a_i (-{\lambda})^i, \sum a_i \mu^i\right) \in A_{-{\lambda}, \mu}$. Taking $a'_i = (-1)^i a_i \in \{\pm1\}$ we see that $(x,y) = \left(\sum a'_i {\lambda}^i, \sum a'_i (-\mu)\right) \in A_{{\lambda}, -\mu}$.

For the Jordan block case 