  Hayashi has extended a result of Ma\~n\'e, proving that every diffeomorphism
$f$ which has a $C^1$-neighborhood $\mathcal{U}$, where all periodic points of
any $g\in\mathcal{U}$ are hyperbolic, it is an Axiom A diffeomorphism. Here, we
prove the analogous result in the volume preserving scenario.
