Dimensions and entropies for an expansive homeomorphism

TL;DR

This paper explores the intricate relationships among dimension, entropy, and Lyapunov exponents in expansive homeomorphisms, proving conjectures and establishing variational principles to deepen understanding of hyperbolic dynamical systems.

Contribution

It introduces a hyperbolic metric with distinct Lyapunov exponents and proves the Eckmann-Ruelle Conjecture for expansive systems, linking entropy and dimension in new ways.

Findings

Proved the Eckmann-Ruelle Conjecture for expansive systems.

Established variational principles for various entropy types.

Demonstrated relationships between entropy, dimension, and Lyapunov exponents.

Abstract

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 $\alpha$-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.