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Variational equalities of entropy in nonuniformly hyperbolic systems. (English) Zbl 1360.37075

Summary: In this paper we prove that for a nonuniformly hyperbolic system \( (f,\widetilde {\Lambda})\) and for every nonempty, compact and connected subset \( K\) with the same hyperbolic rate in the space \( \mathcal {M}_{inv}(\widetilde {\Lambda},f)\) of invariant measures on \( \widetilde {\Lambda}\), the metric entropy and the topological entropy of basin \( G_K\) are related by the variational equality \[ \displaystyle \inf \{h_\mu (f)\mid \mu \in K\}=h_{\mathrm {top}}(f,G_K). \] In particular, for every invariant (usually nonergodic) measure \( \mu \!\in \! \mathcal {M}_{inv}(\widetilde {\Lambda},f)\), we have \[ \displaystyle h_\mu (f)=h_{\mathrm {top}}(f,G_{\mu}). \] We also verify that \( \mathcal {M}_{inv}(\widetilde {\Lambda},f)\) contains an open domain in the space of ergodic measures for diffeomorphisms with some hyperbolicity. As an application, the historical behavior is shown to occur robustly with a full positive entropy for diffeomorphisms beyond uniform hyperbolicity.

MSC:

37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37B40 Topological entropy
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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