A note on Eckmann-Ruelle's conjecture

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.