A C1 -Generic dichotomy for diffeomorphisms
C. Bonatti, L. J. Diaz, E. R. Pujals
Abstract
We show that, for every compact n-dimensional manifold, n≥ 1, there is a residual subset of Diff1(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 C1 -robustly transitive diffeomorphism admits a dominated splitting.
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