Global dynamic of abelian groups of affine maps on Cn

We show that for any abelian group G of affine maps on C-n, there exists a G-invariant affine subspace epsilon of C-n such that epsilon contains all elements of C-n on which G acts identically, the closure of every orbit in epsilon is a minimal set of G/epsilon and if epsilon not equal C-n there are G-invariant affine subspaces H-1, ..., H-s (1 <= s <= n - dim(epsilon)) of C-n of dimension n - 1 such that the closure of every orbit in U = C-n\U-k=1(s) H-k is a minimal set of G/U. Moreover, if G has a dense orbit, all orbits in U are dense in C-n. (C) 2013 Elsevier B.V. All rights reserved.