176 research outputs found

### 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

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