Mapping B n into B 2 n − 1

In this paper, we are concerned with the classification problem of proper holomorphic maps between balls in complex spaces. Write Bn = {z ∈ Cn : |z| < 1} and Prop(Bn,BN ) for the collection of all proper holomorphic maps from Bn into BN . We recall that f, g ∈ Prop(Bn,BN ) are said to be equivalent if there are elements σ ∈ Aut(Bn) and τ ∈ Aut (BN ) such that f = τ ◦ g ◦ σ . It is a well-known result of Poincaré [Po] and Alexander [Alx] that when N = n > 1, then any f ∈ Prop(Bn,Bn) is equivalent to the identity map. For the case of N > n > 1, due to the discovery of inner functions, it is clear that solving the classification problem in Prop(Bn,BN ) is unrealistic. Therefore, one focuses on the important subclass of mappings, Rat(Bn,BN ), the collection of all rational proper holomorphic mappings from Bn into BN . And here, there are already many non-trivial and interesting questions ([DA1]). A first result along these lines is due to Webster [We], who showed that Rat(Bn,Bn+1) has only one equivalence class for n > 2. This was proven to hold also for Rat(Bn,BN ) by Faran [Fa2] in the larger codimensional case: N ≤ 2n − 2. For the case N ≥ 2n − 1, the collection of all equivalence classes, denoted by R̃(n, N), of Rat(Bn,BN ) carries a real algebraic structure from the work of [Fo1], [BER]. However, the specific description of R̃(n, N) remains to be quite mysterious in general. In [DA2], D’Angelo discovered a continuous family of mutually inequivalent polynomial proper embeddings from Bn into B2n (see Example 3), which in particular indicates that the set R̃(n, N) contains infinitely many elements when N ≥ 2n. On the other hand, in a paper of Faran [Fa1] in the early 80’s, it was shown that R̃(2, 3) has exactly four elements. This then leads to the natural question of