Let M be a smooth compact manifold and Λ be a compact invariant set.In this article,we prove that,for every robustly transitive set Λ,f | Λ satisfies a C1-genericstable shadowable property(resp.,C1-generic-stable transitive specification property or C1-generic-stable barycenter property) if and only if Λ is a hyperbolic basic set.In particular,f | Λ satisfies a C1-stable shadowable property(resp.,C1-stable transitive specification property or C1-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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