A note on Eckmann-Ruelle's conjecture

TL;DR

This paper explores a specific class of nonuniformly hyperbolic sets and establishes a relationship between entropy, Lyapunov exponents, and escape rates, contributing to the understanding of Eckmann-Ruelle's conjecture.

Contribution

It introduces a new class of hyperbolic sets where the supremum of entropy minus Lyapunov exponents equals the escape rate, advancing the theoretical framework of dynamical systems.

Findings

Established the equality between entropy minus Lyapunov exponents and escape rate for the new class of sets.

Extended the understanding of Eckmann-Ruelle's conjecture in the context of nonuniform hyperbolicity.

Provided a mathematical framework linking hyperbolic dynamics with escape phenomena.

Abstract

We introduce a class of $C^{1+\alpha}$ isolated nonuniformly hyperbolic sets $\Lambda$ for which $\sup_{\mu \in {\mathcal M}_f}\{h(\mu) - \chi^+(\mu)\}$ equals the rate of escape from $\Lambda$, where $\chi^+(\mu)$ is the average of the sum of positive Lyapunov exponents counted with their multiplicity.