Global dynamic of abelian groups of affine maps on Cn

We show that for any abelian group G of affine maps on Cn, there exists a G-invariant affine subspace E of Cn such that E contains all elements of Cn on which G acts identically, the closure of every orbit in E is a minimal set of G/E and if E≠Cn, there are G-invariant affine subspaces H1,…,Hs (1⩽s⩽n−dim(E)) of Cn of dimension n−1 such that the closure of every orbit in U=Cn\⋃k=1sHk is a minimal set of G/U. Moreover, if G has a dense orbit, all orbits in U are dense in Cn.