Hyperbolic Invariant Sets With Positive Measures

Abstract

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 $C{C}^{1+a}$, $a>0$, Anosov diffeomorphism without the usual Hopf argument.