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.

History

As a consequence of the Riemann mapping theorem, there are no Fatou–Bieberbach domains in the case n = 1. Pierre Fatou and Ludwig Bieberbach first explored such domains in higher dimensions in the 1920s, hence the name given to them later. Since the 1980s, Fatou–Bieberbach domains have again become the subject of mathematical research.

References

• Fatou, Pierre: "Sur les fonctions méromorphes de deux variables. Sur certains fonctions uniformes de deux variables." C.R. Paris 175 (1922)
• Bieberbach, Ludwig: "Beispiel zweier ganzer Funktionen zweier komplexer Variablen, welche eine schlichte volumtreue Abbildung des ${\displaystyle {\mathcal {R}}_{4}}$ auf einen Teil seiner selbst vermitteln". Preussische Akademie der Wissenschaften. Sitzungsberichte (1933)
• Rosay, J.-P. and Rudin, W: "Holomorphic maps from ${\displaystyle \mathbb {C} ^{n}}$ to ${\displaystyle \mathbb {C} ^{n}}$". Trans. Amer. Math. Soc. 310 (1988)