Diffeomorphisms with various $C^1$ stable properties

Let $M$ be a smooth compact manifold and $\Lambda$ be a compact invariant set. In this paper we prove that for every robustly transitive set $\Lambda$, $f|_\Lambda$ satisfies a $C^1-$generic-stable shadowable property (resp., $C^1-$generic-stable transitive specification property or $C^1-$generic-stable barycenter property) if and only if $\Lambda$ is a hyperbolic basic set. In particular, $f|_\Lambda$ satisfies a $C^1-$stable shadowable property (resp., $C^1-$stable transitive specification property or $C^1-$stable barycenter property) if and only if $\Lambda$ is a hyperbolic basic set. Similar results are valid for volume-preserving case.