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

We prove that given a compact $n$-dimensional boundaryless manifold $M$, $n \geq 2$, there exists a residual subset $\mathcal{R}$ of the space of $C^1$ diffeomorphisms $\mathrm{Diff}^1(M)$ 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 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 Mañé (1982).