This analysis reveals connections between dimension and various entropies in expansive homeomorphisms, indicating the relevance of hyperbolic metrics.
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 log b>0>-log 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.
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Chen et al. (2025) studied this question.
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