Dynamics of non abelian affine homotheties group of C^n

Abstract: In this paper we study the action of non abelian subgroup G generated by affine homotheties on C^n. We prove that there exist a subgroup H of C{0}, a G-invariant affine subspace E of C^n and b in E such that the closure of any orbit G(z) is equal to H(z-a)+E, z in C^n. In particular, every orbit in E is dense in it. Moreover, if the complementary U=C^n \E is non empty, every orbit of U is minimal in it.