Idempotent rational map? or rational map whose orbit under iteration is bounded?

Question

I saw that the map $f:\mathbb{P}^1 \rightarrow \mathbb{P}^1$, $f[x_0,x_1]=[x_0+x_1,x_0-x_1]$ satisfies $f^2=$id, the power means iteration.

It's fairly interesting me. So my question is,

1. What rational maps on $\mathbb{P}^n$ satisfying that the $n$-th interation is itself.

2. Moreover, let's denote the set of all iterations of $f$ to $O_f$. What is the condition that $O_f$ is finite?

Answer 1

Any rational map on $\newcommand{\P}{\Bbb P}\P^n(K)$ with finite order must come from a linear map $F:K^{n+1}\to K^{n+1}$, which we can think of as a matrix. It will have order $m$ iff $F^m=\lambda I$ for some $I$ and $m$ is minimal for this to hold. This reduces the problem to a problem in matrix theory.