Proper maps of ball complements & differences and rational sphere maps

In this paper, we consider proper holomorphic maps of ball complements and differences in complex euclidean spaces of dimension at least two. Such maps are always rational, which naturally leads to a related problem of classifying rational maps taking concentric spheres to concentric spheres, what we call [Formula: see text]-fold sphere maps; a proper map of the difference of concentric balls is a two-fold sphere map. We prove that proper maps of ball complements are in one to one correspondence with polynomial proper maps of balls taking infinity to infinity. We show that rational [Formula: see text]-fold sphere maps of degree less than [Formula: see text] (or polynomial maps of degree [Formula: see text] or less) must take all concentric spheres to concentric spheres and we provide a complete classification of them. We prove that these degree bounds are sharp.