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arXiv · 2502.18162

Dimensions and entropies for an expansive homeomorphism

Ercai Chen·Tassilo Küpper·Yunxiang Xie
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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.

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Ercai Chen, Tassilo Küpper, Yunxiang Xie. 2025-02-25. Dimensions and entropies for an expansive homeomorphism. https://arxiv.org/abs/2502.18162

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