A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources

Abstract

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