## Proper Holomorphic Mappings Between Balls

This chapter discusses proper mappings between balls; these are of interest in part because of their abundance and in part because the symmetry of the ball allows for interesting mathematics. It examines consideration to rational proper holomorphic mappings between balls. Such mappings give a higher dimensional generalization of Blaschke products. The generalization of ordinary multiplication turns out to be the tensor product on a subspace. The chapter considers invariance properties of mappings under fixed-point-free finite unitary groups. Assuming enough boundary smoothness, such mappings are necessarily rational. In every positive codimension, however, there are proper mappings, continuous on the closed ball, whose components are not rational functions. Since the automorphism group of the ball is so large, superficially different mappings can often be spherically equivalent. There is an alternate approach to the understanding of polynomial proper mappings between balls.