A C^1 -Generic dichotomy for diffeomorphisms

Bonatti, C.; Diaz, L. J.; Pujals, E. R.

A C^1 -Generic dichotomy for diffeomorphisms

Authors

Bonatti, C.

Diaz, L. J.

Pujals, E. R.

Description

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 Mane [Ma3}). In particular, we show that any C^1 -robustly transitive diffeomorphism admits a dominated splitting.
64 pages, published version