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

Abstract

We show that, for every compact n-dimensional manifold, n 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 Ma [Ma3]). In particular, we show that any C 1 -robustly transitive diffeomorphism admits a dominated splitting.