For an ergodic hyperbolic measure $\omega$ of a $C^{1+{\alpha}}$diffeomorphism, there is an $\omega$ full-measured set $\tilde\Lambda$ suchthat every nonempty, compact and connected subset $V$ of$\mathbb{M}_{inv}(\tilde\Lambda)$ coincides with the accumulating set of timeaverages of Dirac measures supported at {\it one orbit}, where$\mathbb{M}_{inv}(\tilde\Lambda)$ denotes the space of invariant measuressupported on $\tilde\Lambda$. Such state points corresponding to a fixed $V$are dense in the support $supp(\omega)$. Moreover,$\mathbb{M}_{inv}(\tilde\Lambda)$ can be accumulated by time averages of Diracmeasures supported at {\it one orbit}, and such state points form a residualsubset of $supp(\omega)$. These extend results of Sigmund [9] from uniformlyhyperbolic case to non-uniformly hyperbolic case. As a corollary, irregularpoints form a residual set of $supp(\omega)$.
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