2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74573If 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.Dynamical Systems37C40; 37D20text