Generic sink-or-source characterization of non-volume-hyperbolic homoclinic classes

Let $f\in\operatorname{Diff}^r(M)$, let $P$ be a saddle, and let $\operatorname{Hom}(P,f)$ be its homoclinic class. A compact invariant set is volume-hyperbolic when its extremal dominated bundles have uniform volume contraction and expansion in the appropriate time directions. A homoclinic class is the Hausdorff limit of sinks or sources when a sequence of such periodic orbits converges to it in the Hausdorff topology.

Generic sink-or-source conjecture. For any integer $r\geq 1$, for any diffeomorphism $f$ in a residual subset $\mathcal R\subset\operatorname{Diff}^r(M)$, a homoclinic class $\operatorname{Hom}(P,f)$ is not volume-hyperbolic if and only if it is the Hausdorff limit of a sequence of sinks or sources.

This is presented as a consequence of the preceding generic mechanical non-volume-hyperbolicity conjecture and the paper's theorem on mechanically non-volume-hyperbolic classes. Its resolution is therefore conditional in the source.