Preprint A380/2005
Global dominated splittings and the $C^1$ Newhouse phenomenon

Sylvain Crovisier | Abdenur, Flavio | Bonatti, Christian

**Keywords: **
dominated splittings | homoclinic classes | aperiodic classes | generic properties | Newhouse phenomenon

We prove that given an $n$-dimensional boundaryless compact manifold $M$, with $n \geq 2$, then there exists a residual subset $\mathcal{R}$ of the space of $C^1$ diffeomorphisms $Diff^1(M)$ such that given any chain-transitive set $K$ of $f \in \mathcal{R}$ 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 dichomoty for homoclinic classes in [BDP].
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 ``\emph{$C^1$ Newhouse phenomenon}''). This result answers a question in [BDP] and generalizes the generic dichotomy for surface diffeomorphisms in [M].