Variational equalities of entropy in nonuniformly hyperbolic systems

TL;DR

This paper establishes a variational principle linking metric entropy and topological entropy for hyperbolic measures in nonuniformly hyperbolic systems, with applications to various classes of diffeomorphisms.

Contribution

It proves a variational equality for entropy of invariant measures and saturated sets in nonuniformly hyperbolic systems, extending classical results to broader dynamical contexts.

Findings

Metric entropy equals topological entropy of generic point sets.

Topological entropy of saturated sets equals the infimum of measure-theoretic entropies.

Results apply to Katok's nonuniform hyperbolic diffeomorphisms, Ma{}'s transitive partially hyperbolic systems, and Bonatti-Viana's non-partially 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 $\widetilde{\Lambda}$ 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 $\mathcal{M}_{erg}(M,f)$.