Abstract. We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok's closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization.In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao's style number (Definition 1.3) which are bases for discussing the approximation properties in the main result.
For an ergodic hyperbolic measure ω of a C 1+α diffeomorphism, there is a ω full-measured setΛ such that every nonempty, compact and connected subset V of M inv (Λ) coincides with the accumulating set of time averages of Dirac measures supported at one orbit, where M inv (Λ) denotes the space of invariant measures supported onΛ. Such state points corresponding to a fixed V are dense in the support supp(ω). Moreover M inv (Λ) can be accumulated by time averages of Dirac measures supported at one orbit, and such state points form a residual subset of supp(ω). These extend results of Sigmund [9] from uniformly hyperbolic case to non-uniformly hyperbolic case. As a corollary, irregular points form a residual set of supp(ω).
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