The specification property for flows from the robust and generic viewpoint

Abstract

We prove that if $X{|}_{\Lambda }$ 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 $X_t$ is a topologically mixing Anosov flow. Also we prove that there exists a residual subset $\mathcal{R}\in {\mathfrak{X}}^1(M)$ so that if $X\in \mathcal{R}$ and X has the weak specification property, then $X_t$ is an Anosov flow.