For an expansive homeomorphism, we investigate the relationship among dimension, entropy, and Lyapunov exponents. Motivated by Young's formula for surface diffeomorphisms, which links dimension and measure-theoretic entropy with hyperbolic ergodic measures, we construct the hyperbolic metric with two distinct Lyapunov exponents b>0>- a. We then examine the relationships between various types of entropies (entropy, r-neutralized entropy, and α-estimating entropy) and dimensions. We further prove the Eckmann-Ruelle Conjecture for expansive topological dynamical systems with hyperbolic metrics. Additionally, we establish variational principles for these entropy quantities.
Chen et al. (Tue,) studied this question.