  We consider both hyperbolic sets and partially hyperbolic sets attracting a
set of points with positive volume in a Riemannian manifold. We obtain several
results on the topological structure of such sets for diffeomorphisms whose
differentiability is bigger than one. We show in particular that there are no
partially hyperbolic horseshoes with positive volume for such diffeomorphisms.
We also give a description of the limit set of almost every point belonging to
a hyperbolic set or a partially hyperbolic set with positive volume.
