Recently I face up with a problem, which I realized that have close connection with the following problem. $\{ f_{n} \}_{n=1}^{\infty}$ is analytic map from $C^{n}$ to $C^{n} $\ $U$ where U is open neighborhood of 0, whether f is a normal family.

I know when n=1, this is really Montel Normal family criterion, However I did know whether it is true for high dimension. also I heard that the for any two topological equivalent simple connected domain in$ C^{n}$ $(n\geq 2)$, the probability of holomorphic equivalent for this two domain is 0. I want to know what is the precise statement for this theorem.

Any advice and comments will be appreciated.