Fatou-Bieberbach area

A Fatou-Bieberbach area is a real sub-area of , which is biholomorphically equivalent to , i.e. H. an open one is called Fatou-Bieberbach domain, if there is a bijective holomorphic function and a holomorphic inverse function . ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \Omega \subset \mathbb {C} ^{n}\;(\Omega \neq \mathbb {C} ^{n})}$ ${\displaystyle f\colon \Omega \rightarrow \mathbb {C} ^{n}}$ ${\displaystyle f^{-1}\colon \mathbb {C} ^{n}\rightarrow \Omega }$

history

As a consequence of Riemann's mapping theorem, there are no Fatou-Bieberbach areas in the case . In higher dimensions, Fatou-Bieberbach areas were first discovered in the 1920s by Pierre Fatou and Ludwig Bieberbach and later named after their discoverers. Since the 1980s, Fatou-Bieberbach areas have been the subject of mathematical research again. ${\displaystyle n=1}$

swell

• Pierre Fatou: Sur les fonctions méromorphes de deux variables , Sur certaines fonctions uniformes de deux variables . Comptes rendus hebdomadaires des séances de l'Académie des sciences, Volume 175 (1922), pp. 862-865, 1030-1033.
• Ludwig Bieberbach: Example of two whole functions of two complex variables, which convey a simple, true-to-volume mapping of the to a part of itself ${\displaystyle {\mathcal {R}}_{4}}$ . Prussian Academy of Sciences. Meeting reports, 1933, pp. 476–479.
• J.-P. Rosay, W. Rudin: Holomorphic maps from to ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$ . Transactions of the American Mathematical Society, Volume 310 (1988), Issue 1, pp. 47-86.