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arXiv · 1110.6091

Variational equalities of entropy in nonuniformly hyperbolic systems

Chao Liang·Gang Liao·Wenxiang Sun·Xueting Tian
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Abstract

In this paper we prove that for an ergodic hyperbolic measure $ω$ of a $C^{1+α}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $ω$-full measured set $\widetildeΛ$ such that for every invariant probability $μ\in \mathcal{M}_{inv}(\widetildeΛ,f)$, the metric entropy of $μ$ is equal to the topological entropy of saturated set $G_μ$ consisting of generic points of $μ$: $$h_μ(f)=h_{\top}(f,G_μ).$$ Moreover, for every nonempty, compact and connected subset $K$ of $\mathcal{M}_{inv}(\widetildeΛ,f)$ with the same hyperbolic rate, we compute the topological entropy of saturated set $G_K$ of $K$ by the following equality: $$\inf\{h_μ(f)\mid μ\in K\}=h_{\top}(f,G_K).$$ 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 $\mathcal{M}_{inv}(\widetildeΛ,f)$ contains an open subset of $\mathcal{M}_{erg}(M,f)$.

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Chao Liang, Gang Liao, Wenxiang Sun, Xueting Tian. 2016-09-24. Variational equalities of entropy in nonuniformly hyperbolic systems. https://doi.org/10.1090/tran%2F6780

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