A C^1 -Generic dichotomy for diffeomorphisms

Details A C^1 -Generic dichotomy for diffeomorphisms A C^1 -Generic dichotomy for diffeomorphisms

2006-10-17

arxiv.org/abs/math/0610527v1

Ann. of Math. (2), vol. 158 (2003), no.2, 355--418

Mathematics

Dynamical Systems

64 pages, published version

Scientific paper

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.

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