Variational equalities of entropy in nonuniformly hyperbolic systems

Abstract

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 $\tilde{\Lambda }$ such that for every invariant probability $\mu \in {\mathcal{M}}_{inv}(\tilde{\Lambda },f)$, 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 ${\mathcal{M}}_{inv}(\tilde{\Lambda },f)$ 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{ñ}{é}, (iii) to the robustly transitive non-partially hyperbolic diffeomorphisms described by Bonatti-Viana. In all these cases ${\mathcal{M}}_{inv}(\tilde{\Lambda },f)$ contains an open subset of ${\mathcal{M}}_{erg}(M,f)$.

Transactions of the American Mathematical Society, to appear,see http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2016-06780-X/

References in corpus (2)

• Genericity of zero Lyapunov exponents
• Bowen measure for derived from Anosov diffeomorphims

Cited by in corpus (6)

• Ergodic Average Of Typical Orbits And Typical Functions
• On the topological entropy of saturated sets for amenable group actions
• Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic
• Distributional chaos in multifractal analysis, recurrence and transitivity
• The Conditional Variational Principle for Maps with the Pseudo-orbit Tracing Property
• Entropy of irregular points that are not uniformly hyperbolic