Diffeomorphisms with various $C^1$ stable properties

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

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