The Specification Property for Flows from the Robust and Generic Viewpoint

We prove that if $X{\mathrm{\mid }}_{\mathrm{\Lambda }}$ has the weak specification property robustly, where $\mathrm{\Lambda }$ is an isolated set, then $\mathrm{\Lambda }$ 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 $\text{SR}\in \text{Mundo}$ so that if $X\in \text{SR}$ and $X$ has the weak specification property, then $X_t$ is an Anosov flow.