Hypercyclicity and k-Transitivity (k>=2) for abelian semigroup of affine maps on C^n

In this paper, we prove that the minimal number of affine maps on C^n, required to form a hypercyclic abelian semigroup on C^n is n+1. We also prove that the action of any abelian group finitely generated by affine maps on C^n, is never k-transitive for k>=2.