Let $A$ be a real $n\times n$ matrix, $A_n = A^n$, and
$$ \bar A_n = \lbrace\alpha A_n, \alpha\in \mathbb{R}\rbrace.$$
I am interested in characterizing the behavior of $\bar A_n$ when $n\rightarrow \infty$. My initial lead is to start with a Jordan decomposition of $A$. But I guess that it is a standard question that should already be studied, so I am looking for references. I have seen several documents addressing the convergence of the sequence $A_n$ to $0$ or $\infty$. Can someone indicate references for the 'projective version' of the problem ?
Thank you.