This is a note of some well known results. The argument here may be new, and may be complete.
Proposition 1. Let . Then for every closed, invariant hyperbolic set .
See Theorem 15 of Bochi–Viana’s paper. Note that Proposition 1 also applies to Anosov case, in the sense that implies that and is Anosov.
Proof. Suppose for some hyperbolic set. Then the stable and unstable foliations/laminations are absolutely continuous. Hopf argument shows that is (essentially) saturated by stable and unstable manifolds. Being a closed subset, is in fact saturated by stable and unstable manifolds, and hence open. So .
Proposition 2. There exists a residual subset , such that for every , for every closed, invariant hyperbolic set .
Proof. Let be an open subset such that , . Given , let be the set of maps with . It follows from Proposition 1 that is dense. We only need to show the openness. Pick an . Since , there exists such that . So there exists such that . In particular, for every . The genericity follows by the countable intersection of the open dense subsets .
The dissipative version has been obtained in Alves–Pinheiro’s paper
Proposition 3. Let . Then for every closed, transitive hyperbolic set . In particular, implies that and is Anosov.
See Theorem 4.11 in R. Bowen’s book when is a basic set.