Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:

Consider the dynamical system with $x_i \in \mathbb C^2:$

$$ x_{i} = \left(\begin{matrix} z &&-1 \\ 1 && 0 \end{matrix} \right)^ix_0.$$

I would like to understand whether one can obtain a sharp bound on $\Vert x_i \Vert$ for $N \ge i \ge 0$ just in terms of

$\Vert x_0 \Vert$, $\Vert x_N \Vert$ and $z \neq 2.$

**Observations:** It seems that this dynamical system is rather simple in the sense that the matrix $A=\left(\begin{matrix} z &&-1 \\ 1 && 0 \end{matrix} \right)$ is diagonalizable for $z \neq 2.$

More precisely, the eigenvalues are $$\tfrac{1}{2} \left(z \pm \sqrt{z^2-4 }\right)$$ with eigenvectors $$\left(\tfrac{1}{2} \left(z \pm \sqrt{z^2-4 }\right), 1\right).$$

It is also worth noticing that this system is invertible since $\operatorname{det}(A)=1$

so it is believable that once the boudary norms for $x_0$ and $x_N$ are known. Everything else should be fixed as well.

Of course there are trivial bounds like $\Vert x_i \Vert \le \Vert A \Vert^i \Vert x_0 \Vert$ but I am looking for something more refined.

**Motivation:** This is the discrete analogue of $-y''(x)=zy(x),$ see for details and in the continuous world it is almost trivial to bound $\vert y(x_{\text{middle}}) \vert$ in terms of $\vert y(x_0) \vert$ and $\vert y(x_1)\vert$ where $x_0 \le x_{\text{middle}} \le x_1.$