Global dominated splittings and the C1 Newhouse phenomenon

We prove that given a compact n-dimensional boundaryless manifold M, n >= 2, there exists a residual subset R of the space of C-1 diffeomorphisms Diff(1) (M) such that given any chain-transitive set K of f is an element of 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 C-1-generic diffeomorphism f, then either the nonwandering set Omega(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 diffeomorphisms given by Mane (1982).