Stable weakly shadowable volume-preserving systems are volume-hyperbolic

We prove that any C ¹-stable weakly shadowable volume-preserving diffeomorphism de-fined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.