Holomorphic dynamics in several variables

Remarks

The main cases are $X = \mathbb{P}^k$ with $f$ a holomorphic endomorphism of algebraic degree $d \ge 2$, and $X = \C^2$ with $f$ a polynomial automorphism.

Example: $f[z_0 : \cdots : z_k] = [z_0^d : \cdots : z_k^d]$ with $d \ge 2$ is a holomorphic self-map of $\mathbb{P}^k$. It is well defined because the $z_j^d$ vanish together only when all $z_j$ vanish. If one coordinate $z_i$ has strictly larger modulus than all the others, then the ratios $(z_j/z_i)^{d^m}$ tend to $0$ uniformly near $[z]$. So the orbit tends to the coordinate point with $1$ in place $i$, the iterates converge near $[z]$, and $[z]$ lies in the Fatou set.

Example: a Hénon map $(x, y) \mapsto (p(x) - a y, x)$ with $a \ne 0$ and $\deg p \ge 2$ is a holomorphic automorphism of $\C^2$.

Non-example: the map $(z, w) \mapsto (\bar z^2, w)$ of $\C^2$ is smooth but not holomorphic, so its iteration is not part of holomorphic dynamics.

For $k = 1$ the same definitions give one-variable complex dynamics.