Fatou–Bieberbach domain

In mathematics, a Fatou–Bieberbach domain is a proper subdomain of ${\displaystyle \mathbb {C} ^{n}}$, biholomorphically equivalent to ${\displaystyle \mathbb {C} ^{n}}$. That is, an open set ${\displaystyle \Omega \subsetneq \mathbb {C} ^{n}}$ is called a Fatou–Bieberbach domain if there exists a bijective holomorphic function ${\displaystyle f:\Omega \rightarrow \mathbb {C} ^{n}}$ whose inverse function ${\displaystyle f^{-1}:\mathbb {C} ^{n}\rightarrow \Omega }$ is holomorphic. It is well-known that the inverse ${\displaystyle f^{-1}}$ can not be polynomial.