Fatou–Bieberbach domain

Example

Let $n \ge 2$. A Fatou–Bieberbach domain is an open set $\Omega \subseteq \C^n$ with $\Omega \ne \C^n$ that is biholomorphic to $\C^n$.

A concrete one comes from an automorphism of $\C^2$. Let $F(z, w) = (w/2, \, z/2 + w^2)$. It is a polynomial map with polynomial inverse $(u, v) \mapsto (2v - 8u^2, \, 2u)$, so it is a holomorphic automorphism of $\C^2$. The origin is a fixed point, and the Jacobian matrix of $F$ there has eigenvalues $1/2$ and $-1/2$. So the origin is an attracting fixed point. Let $\Omega$ be its basin of attraction, the set of points $q$ with $F^m(q) \to 0$ as $m \to \infty$. It is a known theorem that the basin of attraction of an attracting fixed point of a holomorphic automorphism of $\C^n$ is biholomorphic to $\C^n$. The point $(3/8, 3/4)$ is a second fixed point of $F$, so it is not in $\Omega$. Hence $\Omega$ is a proper open subset of $\C^2$ biholomorphic to $\C^2$.