  If a $C^{1 + a}$, $a >0$, volume-preserving diffeomorphism on a compact
manifold has a hyperbolic invariant set with positive volume, then the map is
Anosov. We also give a direct proof of ergodicity of volume-preserving
$CC^{1+a}$, $a>0$, Anosov diffeomorphism without the usual Hopf argument.
