Holomorphic endomorphisms of $\mathbb{C}^n$ and countable subsets

Following the work of Jean-Pierre Rosay and Walter Rudin we will study actions of holomorphic endomorphisms of $\mathbb{C}^n$ on countable subsets. We will prove the generalization of the Mittag-Leffler interpolation: there exists a locally volume-preserving holomorphic mapping having prescribed values on a discrete subset. We will also show that any dense subset can be mapped to another dense subset by an automorphism. The situation is different for discrete subsets. We call the subset tame if it is conjugate to an arithmetic projection by an automorphism. We will give an example of a non-tame subset and prove some interesting theorems.