Let Bn = {z ∆ Cn : |z| < 1} be the unit ball in Cn. The problem
of classifying proper holomorphic mappings between Bn and BN has attracted
considerable attention (see [Fo 1992] [DA 1988] [DA 1993] [W 1979] [H 1999][HJ 2001]
for extensive references) since the work of Poincare [P 1907][T 1962] and Alexander
[A 1977]. Let us denote by Prop(Bn,BN) the collection of proper holomorphic
mappings from Bn to BN. It is known [A 1977] that any map F ∆ Prop(Bn,Bn)
must be biholomorphic and must be equivalent to the identity map. Here we say that
f, g ∆Prop(Bn,BN)
are equivalent if there are automorphisms σ ∆ Aut(Bn) and
τ ∆ Aut(BN)) such that f = τ ∘ g ∘ σ. For general N > n,
the discovery of inner functions
indicates that Prop(Bn,BN)
is too complicated to be classified. Hence we may
focus on Rat(Bn,BN), the collection of all rational proper holomorphic mappings
from Bn to BN).