Variational equalities of entropy in nonuniformly hyperbolic systems

Abstract:

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 \[ \inf \{h_\mu (f)\mid \mu \in K\}=h_{\mathrm {top}}(f,G_K).\] In particular, for every invariant (usually nonergodic) measure $\mu \!\in \! \mathcal {M}_{inv}(\widetilde {\Lambda },f)$, we have \[ h_\mu (f)=h_{\mathrm {top}}(f,G_{\mu }).\] 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.