Proper holomorphic maps from the unit disk to some unit ball
  We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$. 
John P. D'Angelo, Zhenghui Huo, Ming Xiao
math.CV