We prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point p∈M that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. p∈M that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.

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We analyze the Lyapunov spectrum of the relative Ruelle operator associated with a skew product whose base is an ergodic automorphism and whose fibers are full shifts. We prove that these operators can be approximated in the $C^0$-topology by positive matrices with an associated dominated splitting.Comment: The article now contains a section on decay of correlations of relative transfer operator

We consider a compact 3-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C 1residual (dense G δ ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M . This is the flowsetting counterpart of Newhouse's Theorem 1.3 [17]. Our result follows from two theorems, the first one is the 3-dimensional continuous-time version of a theorem of Xia [21] and says that if Λ is a hyperbolic invariant set for some class C 1 zero divergence vector field X on M , then either X is Anosov, or else Λ has empty interior. The second one is a version, for our 3-dimensional class, of Theorem 2 of Saghin-Xia [20] and says that, if X is not Anosov, then for any open set U ⊆ M there exists Y arbitrarily close to X such that Y t has an elliptical closed orbit through U .

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