A C1-generic dichotomy for diffeomorphisms:: Weak forms of hyperbolicity or infinitely many sinks or sources

We show that, for every compact n-dimensional manifold, n greater than or equal to 1, there is a residual subset of Diff(1)(M) of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities: Either it is contained in the closure of an infinite set of sinks or sources (Newhouse phenomenon), or it presents some weak form of hyperbolicity called dominated splitting (this is a generalization of a bidimensional result of Mane [Ma3]). In particular, we show that any C-1-robustly transitive diffeomorphism admits a dominated splitting.