Global dynamic of abelian groups of affine maps on

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