For every ergodic hyperbolic measure ω of a C~(1+α) diffeomorphism, there is an ω-full-measure set ? (the union of ?_l D supp.(ω|_(∧l)), the support sets of ω on each Pesin block ∧_1, l=1, 2,...) such that every non-empty, compact and connected subset V ? Closure(M_(inv)(?)) coincides with V_f (x), where Minv(?) denotes the space of invariant measures supported on ? and V_f (x) denotes the accumulation set of time averages of Dirac measures supported at one orbit of some point x. For each fixed set V, the points with the above property are dense in the support supp(ω) In particular, points satisfying V_f (x) D Closure(M_(inv)(?)) are dense in supp(ω) Moreover, if supp(ω) is isolated, the points satisfying V_f (x) Closure(M_(inv)(?)) form a residual subset of supp(ω) These extend results of K. Sigmund [On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285-299] (see also M. Denker, C. Grillenberger and K. Sigmund [Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, Berlin, Ch. 21]) from the uniformly hyperbolic case to the non-uniformly hyperbolic case. As a corollary, irregularC points form a residual set of supp(ω)
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