We prove that for C~1 generic diffeomorphisms, every isolated compact invariant set Λ which satisfies a mild condition on the hyperbolicity of periodic points in Λ (called the L-NUH condition, see definition 1.1) is hyperbolic. In parallel, we prove that for C~1 diffeomorphisms, every compact invariant set which satisfies Katok's periodic closing property and the L-NUH condition on periodic points is hyperbolic, which is a generalized result of Castro et al (2007 Nonlinearity 20 75-85) for C~2 case with a periodic closing property (called periodic shadowing property in Castro et al (2007 Nonlinearity 20 75-85)).
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