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In this short paper we prove some results concerning volume-preserving
Anosov diffeomorphisms on compact manifolds. The first theorem is that
if a $C^{1 + \alpha}$, $\alpha >0$, volume-preserving diffeomorphism
on a compact connected manifold has a hyperbolic invariant set with
positive volume, then the map is Anosov. The same result had been
obtained by Bochi and Viana [2]. This result is not
necessarily true for $C^1$ maps. The proof uses a Pugh-Shub type of
dynamically defined measure density points, which are different from
the standard Lebesgue density points. We then give a direct
proof of the ergodicity of $C^{1+\alpha}$ volume preserving Anosov
diffeomorphisms, without using the usual Hopf arguments or the
Birkhoff ergodic theorem. The method we introduced also has
interesting applications to partially hyperbolic and non-uniformly
hyperbolic systems.