An attractor Λ for a 3-vector field X is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that C singularhyperbolic attractors, for some α > 0, always have zero volume, thus extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular...

We prove that every 3-manifold possesses a C, volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold possesses a volume-preserving, C flow with discrete closed trajectories and no fixed points (as well as a PL flow with the same geometry), which is needed for the fir...

In this paper, we show the followings: (i) If a volume preserving diffeomorphism f belongs to the C-interior of the set of all volume preserving diffeomorphims having the ergodic shadowing property then it is transitive Anosov. Moreover, (ii) if a C-generic volume-preserving diffeomorphism f has the ergodic shadowing property then it is transitive Anosov. M.S.C. 2010: 37C50, 37D20.