  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.
