Title: Proper maps of ball complements & differences and rational sphere maps

Abstract: We consider proper holomorphic maps of complements and differences of balls in complex euclidean space of dimension at least $2$. We prove that every proper map of ball complements gives a polynomial proper map of balls, and conversely, every polynomial proper map of balls whose norm goes to infinity at infinity is a proper map of ball complements. The case of ball differences is more complex and naturally leads to what we call rational $m$-fold sphere maps, that is, rational maps taking $m$ zero-centric spheres to $m$ zero-centric spheres. A proper map of the difference of zero-centric balls is automatically a rational $2$-fold sphere map. We show that a polynomial $m$-fold sphere map of degree $m$ or less is an $\infty$-fold sphere map, that is, a map that takes infinitely many (and hence every) zero-centric spheres to zero-centric spheres. Similarly, every rational $m$-fold sphere map of degree less than $m$ is an $\infty$-fold sphere map. We then show that $\infty$-fold sphere maps are up to a unitary transformation direct sums of a finite number of homogeneous sphere maps. We construct rational $m$-fold degree-$m$ sphere maps that do not take any other zero-centric sphere to a zero-centric sphere. In particular, every first-degree rational proper map of the difference of zero-centric balls is a unitary composed with an affine linear embedding; however, there exist nonpolynomial second-degree rational maps of a difference of balls.