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

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