Global dominated splittings and the C1 Newhouse phenomenom

2006

International audience; We prove that given a compact n-dimensional boundaryless manifold M, n >=2, there exists a residual subset R of the space of C1 diffeomorphisms Diff such that given any chain-transitive set K of f in 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 in [BDP]. It follows from the above result that given a C1-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 m…