In joint work with Mário Bessa (UBI) and Joana Torres (UMinho), we obtained the following results:

- dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. 

- a volume-preserving and C1-stably ergodic flow can be C1-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

- a Hamiltonian system H ∈ C2(M,R) is globally hyperbolic if any of the following statements hold: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property.

- for a C2-generic Hamiltonian H , the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits, forms a dense subset of M. As a consequence, any robustly transitive regular energy hypersurface of a C2-Hamiltonian is partially hyperbolic.