Hyperbolic periodic points for chain hyperbolic homoclinic classes

In this paper we establish a closing property and a hyperbolic closing property for thin trapped chain hyperbolic homoclinic classes with one dimensional center in partial hyperbolicity setting. Taking advantage of theses properties, we prove that the growth rate of the number of hyperbolic periodic points is equal to the topological entropy. We also obtain that the hyperbolic periodic measures are dense in the space of invariant measures.Comment: 15 pages, 1 figure

Variational equalities of entropy in nonuniformly hyperbolic systems

In this paper we prove that for an ergodic hyperbolic measure $\omega$ of a $C^{1+\alpha}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $\omega$-full measured set $\widetilde{\Lambda}$ such that for every invariant probability $\mu\in \mathcal{M}_{inv}(\widetilde{\Lambda},f)$, the metric entropy of $\mu$ is equal to the topological entropy of saturated set $G_{\mu}$ consisting of generic points of $\mu$: $$h_\mu(f)=h_{\top}(f,G_{\mu}).$$ Moreover, for every nonempty, compact and connected subset $K$ of $\mathcal{M}_{inv}(\widetilde{\Lambda},f)$ with the same hyperbolic rate, we compute the topological entropy of saturated set $G_K$ of $K$ by the following equality: $$\inf\{h_\mu(f)\mid \mu\in K\}=h_{\top}(f,G_K).$$ In particular these results can be applied (i) to the nonuniformy hyperbolic diffeomorphisms described by Katok, (ii) to the robustly transitive partially hyperbolic diffeomorphisms described by ~Ma{\~{n}}{\'{e}}, (iii) to the robustly transitive non-partially hyperbolic diffeomorphisms described by Bonatti-Viana. In all these cases $\mathcal{M}_{inv}(\widetilde{\Lambda},f)$ contains an open subset of $\mathcal{M}_{erg}(M,f)$.Comment: Transactions of the American Mathematical Society, to appear,see http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2016-06780-X

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 $\tilde\Lambda$ such that every nonempty, compact and connected subset $V$ of $\mathbb{M}_{inv}(\tilde\Lambda)$ coincides with the accumulating set of time averages of Dirac measures supported at {\it one orbit}, where $\mathbb{M}_{inv}(\tilde\Lambda)$ denotes the space of invariant measures supported 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 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 $supp(\omega)$.Comment: 19 page