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- Stable weakly shadowable volume-preserving systems are volume-hyperbolicPublication . Bessa, Mário; Lee, Manseob; Vaz, SandraWe prove that any C^1-stable weakly shadowable volume-preserving diffeomorphism defined 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.