Abstract. We prove that a cohomology free flow on a manifold M fibers over a diophantine translation on T β 1 where β1 is the first Betti number of M .

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For a generic conservative diffeomorphism of a closed connected 3-manifold M , the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is non-uniformly hyperbolic and ergodic. This is the 3-dimensional version of the well-known result by 4], stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result inspired and answers in the positive in dimension 3 a conjecture by Avila-Bochi [2].

We obtain a local topological and dynamical description of expansive attractors on surfaces. The main result is that expansive attractors on surfaces are hyperbolic and have local product structure, except possibly at a finite number of periodic points, which can be either sinks, singularities orépines. Some open questions concerning this kind of dynamics are posed.

In 3-dimensional manifolds, we prove that generically in Diff 1 m (M 3 ), the existence of a minimal expanding invariant foliation implies stable Bernoulliness.