The specification property for flows from the robust and generic viewpoint

We prove that if X|Λ has the weak specification property robustly, where Λ is an isolated set, then Λ is a hyperbolic topologically mixing set and, as a consequence, if X is a vector field that has the weak specification property robustly on a closed manifold M, then the flow Xt is a topologically mixing Anosov flow. Also we prove that there exists a residual subset R∈X1(M) so that if X∈R and X has the weak specification property, then Xt is an Anosov flow.