Fatou-Bieberbach domain

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In mathematics, a Fatou-Bieberbach domain comprises a proper subdomain of $\mathbb{C}^n$ which is biholomorphically equivalent to $\mathbb{C}^n$ ; i.e. one calls an open $\Omega \subset \mathbb{C}^n \; (\Omega \neq \mathbb{C}^n)$ a Fatou-Bieberbach domain if there exists a bijective holomorphic function $f:\Omega \rightarrow \mathbb{C}^n$ and a holomorphic inverse function $f^{-1}:\mathbb{C}^n \rightarrow \Omega$ .

[edit] History

As a consequence of the Riemann mapping theorem, there are no Fatou-Bieberbach domains in the case of $n = 1$ . 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.

[edit] References

Fatou, Pierre: "Sur les fonctions méromorphs 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 $\mathcal{R}_4$ auf einen Teil seiner selbst vermitteln". Preussische Akademie der Wissenschaften. Sitzungsberichte (1933)
Rosay, J.-P. and Rudin, W: "Holomorphic maps from $\mathbb{C}^n$ to $\mathbb{C}^n$ ". Trans. A.M.S. 310 (1988)