  Bieberbach constructed in 1933 domains in $ \bf {C}^2$ which were
biholomorphic to $ \bf {C}^2$ but not dense. The existence of such domains was
unexpected. The special domains Bieberbach considered are basins of attraction
of a cubic H\'enon map. This classical method of construction is one of the
first applications of dynamical systems to complex analysis. In this paper, the
boundaries of the real sections of Bieberbach's domains will be calculated
explicitly as the stable manifolds of the saddle points. The real filled Julia
sets and the real Julia sets of Bieberbach's map will also be calculated
explicitly and illustrated with computer generated graphics. Basic differences
between real and the complex dynamics will be shown.
