On attractors and their basins

Abstract

We prove that the map assigning to a given vector field the Lebesgue measure of the union of the basins of its attractors is lower semicontinuous in a residual subset of vector fields. Moreover, we prove that the Lebesgue measure of the union of the basins of attractors of a generic sectional axiom A vector field is total. For this, we also improve a result of Morales about sectional-hyperbolic sets. We also remark that homoclinic classes are topologically ergodic and that for a generic tame diffeomorphism, the union of the stable manifolds of the hyperbolic periodic orbits is dense in the manifold.