  We show that for every compact 3-manifold $M$ there exists an open subset of
$\diff ^1(M)$ in which every generic diffeomorphism admits uncountably many
ergodic probability measures which are hyperbolic while their supports are
disjoint and admit a basis of attracting neighborhoods and a basis of repelling
neighborhoods. As a consequence, the points in the support of these measures
have no stable and no unstable manifolds. This contrasts with the higher
regularity case, where Pesin theory gives us the stable and the unstable
manifolds with complementary dimensions at almost every point. We also give
such an example in dimension two, without local genericity.
