Global dominated splittings and the $C^1$ Newhouse phenomenon

Abstract

We prove that given a compact n-dimensional boundaryless manifold M, n ≥ 2, there exists a residual subset R of the space of 1 difieomorphisms Diff 1 (M) such that given any chain-transitive set K of f ∈ R, then either K admits a dominated splitting or else K is contained in the closure of an infinite number of periodic sinks/sources. This result generalizes the generic dichotomy for- homoclinic classes given by Bonatti, Diaz, and Pujals (2003). It follows from the above result that given a 1 -generic diffeomorphism f, then either the nonwandering set Ω(f) may be decomposed into a finite number of pairwise disjoint compact sets each of which admits a dominated splitting, or else f exhibits infinitely many periodic sinks/sources (the C 1 Newhouse phenomenon). This result answers a question of Bonatti, Diaz, and Pujals and generalizes the generic dichotomy for surface difieomorphisms given by Mane (1982).