is there any characterisation of all the holomorphic or meromorphic functions from the open unit disk to itself?
As an example of what I mean by characterisation, the holomorphic functions on the Riemann sphere are given by rational functions and the meromorphic functions on the torus are given by elliptic functions.
EDIT: as pointed out by Potato, the class of functions I am asking about is probably far too wide. So let me add the following two conditions:
1) the function is surjective
2) the function has a continuous extension to the unit circle