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.
Keywords
attractors, basin, sectional axiom A, sectional hyperbolic,
basin of attraction