Ergodic Properties of Invariant Measures for C^{1+\alpha} nonuniformly hyperbolic systems

For an ergodic hyperbolic measure $\omega$ of a $C^{1+\alpha }$ diffeomorphism, there is an $\omega$ full-measured set $\tilde{\mathrm{\Lambda }}$ such that every nonempty, compact and connected subset $V$ of ${\mathbb{M}}_{inv}(\tilde{\mathrm{\Lambda }})$ coincides with the accumulating set of time averages of Dirac measures supported at {\it one orbit}, where ${\mathbb{M}}_{inv}(\tilde{\mathrm{\Lambda }})$ denotes the space of invariant measures supported on $\tilde{\mathrm{\Lambda }}$. Such state points corresponding to a fixed $V$ are dense in the support $supp(\omega )$. Moreover, ${\mathbb{M}}_{inv}(\tilde{\mathrm{\Lambda }})$ can be accumulated by time averages of Dirac measures supported at {\it one orbit}, and such state points form a residual subset of $supp(\omega )$. 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 $supp(\omega )$.