Let $M_0, M_1\in\mathbb R^{k\times k}$ be matrices, and define $M_n=M_{n-1}M_{n-2}$. Then $M_n$ is a product of $F_n$ ($n$th Fibonacci number) many copies of $M_0$ or $M_1$.
How do I compute the limit $M_n^{1/F_n}$ as $n\to\infty$?
This is of course easy if $M_0$ and $M_1$ commute, but I'm not sure how to do it in a more general setting.
I'm particularly interested in the spectral norm $\|M_n\|^{1/F_n}$. Can we show, for example, that $\|M_n\|^{1/F_n} \ge \|\frac{M_0+M_1}{2}\|^{1/F_n}$?