  It has been conjectured that every stable manifold arising from a holomorphic
automorphism, that acts hyperbolically on a compact invariant set, is
biholomorphic to complex Euclidean space. Such stable manifolds are known to be
biholomorphic to the basin of a uniformly attracting family of holomorphic
maps. It is shown that the basin of a uniformly attracting family of
holomorphic maps is biholomorphic to complex Euclidean space and this resolves
the conjecture on the biholomorphism type of such stable manifolds
affirmatively.
